Shock Waves (Graduate Studies in Mathematics) 1470465671, 9781470465674
This book presents the fundamentals of the shock wave theory. The first part of the book, Chapters 1 through 5, covers t
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Table of contents :
Preface
Chapter 1. Introduction
Chapter 2. Preliminaries
1. Method of Characteristics
2. Development of Singularities
3. Weak Solutions, Rankine-Hugoniot Condition
4. Expansion Waves
5. Non-uniqueness, Entropy Condition
6. Notes
7. Exercises
Chapter 3. Scalar Convex Conservation Laws
1. Riemann Problem
2. Hopf Equation
3. Wave Interactions, Constructing Solutions
4. Well-Posedness Theory
5. Generalized Characteristics, Nonlinear Regularization
6. 𝑁-Waves, Inviscid Dissipation
7. Entropy Pairs
8. Generalized Entropy Functional
9. Notes
10. Exercises
Chapter 4. Burgers Equation
1. Heat Equation
2. Hopf-Cole Transformation
3. Inviscid Limit
4. Nonlinear Waves
5. Linearized Hopf-Cole Transformation
6. Green’s Functions
7. Nonlinearity
8. Metastable States
9. Notes
10. Exercises
Chapter 5. General Scalar Conservation Laws
1. Viscous Shock Profiles
2. Riemann Problem
3. 𝐿₁ Stability
4. Scattering Wave Patterns
5. Entropy Pairs
6. Multi-Dimensional Laws
7. Notes
8. Exercises
Chapter 6. Systems of Hyperbolic Conservation Laws: General Theory
1. Hyperbolicity
2. Entropy and Symmetry
3. Symmetry and Energy Estimate
4. Local Existence of Smooth Solutions
5. Euler Equations in Gas Dynamics
6. Shock Waves
7. Notes
8. Exercises
Chapter 7. Riemann Problem
1. Linear System
2. Simple Waves
3. Hugoniot Curves
4. Riemann Problem I
5. Examples I
6. Riemann Problem II
7. Examples II
8. Notes
9. Exercises
Chapter 8. Wave Interactions
1. Interaction of Infinitesimal Waves
2. A 2×2 System and Coordinates of Riemann Invariants
3. A 3×3 System
4. General Analysis
5. Notes
6. Exercises
Chapter 9. Well-Posedness Theory
1. Glimm Scheme
2. Nonlinear Functional
3. Wave Tracing
4. Existence Theory
5. Stability Theory
6. Generalized Characteristics and Expansion of Rarefaction Waves
7. Large-Time Behavior
8. Regularity
9. Decay and 𝑁-Waves
10. Some Basics of Numerical Computations
11. Notes
12. Exercises
Chapter 10. Viscosity
1. Nonlinear Waves for Scalar Laws
2. Wave Interaction for Systems
3. Physical Models
4. The 𝑝-System
5. General Dissipative Systems
6. Notes
7. Exercises
Chapter 11. Relaxation
1. A Simple Relaxation Model
2. Examples
3. Gas In Thermal Non-equilibrium
4. The Boltzmann Equation in Kinetic Theory
5. Notes
6. Exercises
Chapter 12. Nonlinear Resonance
1. Moving Source
2. Sub-shocks
3. Non-strict Hyperbolicity
4. Vacuum
5. Boundary
6. Kinetic Boundary Layers and Fluid-like Waves
7. Shock Profiles for Difference Schemes
8. Notes
9. Exercises
Chapter 13. Multi-Dimensional Gas Flows
1. Linear Waves
2. Discontinuity Waves
3. Potential Flows
4. Self-Similar Flows and the Ellipticity Principle
5. Characteristics and Simple Waves
6. Hodograph Transformation
7. The Shock Polar
8. Prandtl Paradox
9. Notes
10. Exercises
Chapter 14. Concluding Remarks
1. Development of Singularities
2. Local and Global Behavior for Gas Flows with Shock Waves
3. Nonlinear Waves for Viscous Conservation Laws
4. Well-Posedness Theory for Weak Solutions
5. Kinetic Theory and Fluid Dynamics
6. Multiple Effects
Bibliography
Index
Citation preview
GRADUATE STUDIES I N M AT H E M AT I C S
215
Shock Waves Tai-Ping Liu
10.1090/gsm/215
Shock Waves
GRADUATE STUDIES I N M AT H E M AT I C S
215
Shock Waves Tai-Ping Liu
EDITORIAL COMMITTEE Marco Gualtieri Bjorn Poonen Gigliola Staffilani (Chair) Jeff A. Viaclovsky Rachel Ward 2020 Mathematics Subject Classification. Primary 35L65, 35L67, 35L40, 76L05, 76N10.
For additional information and updates on this book, visit www.ams.org/bookpages/gsm-215
Library of Congress Cataloging-in-Publication Data Names: Liu, Tai-Ping, 1945- author. Title: Shock waves / Tai-Ping Liu. Description: Providence, Rhode Island : American Mathematical Society, [2021] | Series: Graduate studies in mathematics, 1065-7339 ; 215 | Includes bibliographical references and index. Identifiers: LCCN 2021009618 | ISBN 9781470466251 (paperback) | 9781470465674 (hardcover) | 9781470466244 (ebook) Subjects: LCSH: Shock waves–Mathematical models. | AMS: Partial differential equations – Hyperbolic equations and systems – Hyperbolic conservation laws. | Partial differential equations – Hyperbolic equations and systems – Shocks and singularities for hyperbolic equations. | Partial differential equations – Hyperbolic equations and systems – First-order hyperbolic systems. | Fluid mechanics – Shock waves and blast waves in fluid mechanics. | Fluid mechanics – Compressible fluids and gas dynamics, general – Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics. Classification: LCC QA930 .L58 2021 | DDC 515/.3535–dc23 LC record available at https://lccn.loc.gov/2021009618
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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1
26 25 24 23 22 21
To the memory of my parents My Mother for her kindness and love of mathematical analysis My Father for his patience and joy of philosophical contemplation
Contents
Preface
xiii
Chapter 1. Introduction
1
Chapter 2. Preliminaries
5
§1. Method of Characteristics
5
§2. Development of Singularities
7
§3. Weak Solutions, Rankine-Hugoniot Condition
9
§4. Expansion Waves
13
§5. Non-uniqueness, Entropy Condition
16
§6. Notes
18
§7. Exercises
19
Chapter 3. Scalar Convex Conservation Laws
21
§1. Riemann Problem
22
§2. Hopf Equation
24
§3. Wave Interactions, Constructing Solutions
26
§4. Well-Posedness Theory
32
§5. Generalized Characteristics, Nonlinear Regularization
40
§6. N -Waves, Inviscid Dissipation
50
§7. Entropy Pairs
57
§8. Generalized Entropy Functional
59
§9. Notes
68
§10. Exercises
69 vii
viii
Chapter 4. Burgers Equation
Contents
71
§1. Heat Equation
71
§2. Hopf-Cole Transformation
78
§3. Inviscid Limit
79
§4. Nonlinear Waves
81
§5. Linearized Hopf-Cole Transformation
84
§6. Green’s Functions
85
§7. Nonlinearity
92
§8. Metastable States
98
§9. Notes
100
§10. Exercises
100
Chapter 5. General Scalar Conservation Laws
103
§1. Viscous Shock Profiles
103
§2. Riemann Problem
106
§3. L1 Stability
109
§4. Scattering Wave Patterns
111
§5. Entropy Pairs
113
§6. Multi-Dimensional Laws
117
§7. Notes
120
§8. Exercises
120
Chapter 6. Systems of Hyperbolic Conservation Laws: General Theory123 §1. Hyperbolicity
124
§2. Entropy and Symmetry
126
§3. Symmetry and Energy Estimate
128
§4. Local Existence of Smooth Solutions
132
§5. Euler Equations in Gas Dynamics
136
§6. Shock Waves
141
§7. Notes
144
§8. Exercises
145
Chapter 7. Riemann Problem
147
§1. Linear System
148
§2. Simple Waves
149
§3. Hugoniot Curves
152
§4. Riemann Problem I
158
Contents
ix
§5. Examples I
164
§6. Riemann Problem II
170
§7. Examples II
175
§8. Notes
178
§9. Exercises
178
Chapter 8. Wave Interactions
181
§1. Interaction of Infinitesimal Waves
181
§2. A 2 × 2 System and Coordinates of Riemann Invariants
183
§3. A 3 × 3 System
191
§4. General Analysis
194
§5. Notes
201
§6. Exercises
202
Chapter 9. Well-Posedness Theory
205
§1. Glimm Scheme
207
§2. Nonlinear Functional
211
§3. Wave Tracing
218
§4. Existence Theory
230
§5. Stability Theory
233
§6. Generalized Characteristics and Expansion of Rarefaction Waves
243
§7. Large-Time Behavior
249
§8. Regularity
254
§9. Decay and N -Waves
260
§10. Some Basics of Numerical Computations
272
§11. Notes
273
§12. Exercises
276
Chapter 10. Viscosity
279
§1. Nonlinear Waves for Scalar Laws
280
§2. Wave Interaction for Systems
292
§3. Physical Models
301
§4. The p-System
302
§5. General Dissipative Systems
313
§6. Notes
314
§7. Exercises
315
x
Contents
Chapter 11. Relaxation
317
§1. A Simple Relaxation Model
318
§2. Examples
326
§3. Gas In Thermal Non-equilibrium
330
§4. The Boltzmann Equation in Kinetic Theory
334
§5. Notes
346
§6. Exercises
347
Chapter 12. Nonlinear Resonance
349
§1. Moving Source
350
§2. Sub-shocks
357
§3. Non-strict Hyperbolicity
361
§4. Vacuum
368
§5. Boundary
373
§6. Kinetic Boundary Layers and Fluid-like Waves
379
§7. Shock Profiles for Difference Schemes
380
§8. Notes
384
§9. Exercises
386
Chapter 13. Multi-Dimensional Gas Flows
387
§1. Linear Waves
388
§2. Discontinuity Waves
390
§3. Potential Flows
393
§4. Self-Similar Flows and the Ellipticity Principle
395
§5. Characteristics and Simple Waves
399
§6. Hodograph Transformation
401
§7. The Shock Polar
403
§8. Prandtl Paradox
407
§9. Notes
413
§10. Exercises
413
Chapter 14. Concluding Remarks
415
§1. Development of Singularities
415
§2. Local and Global Behavior for Gas Flows with Shock Waves
416
§3. Nonlinear Waves for Viscous Conservation Laws
418
§4. Well-Posedness Theory for Weak Solutions
421
§5. Kinetic Theory and Fluid Dynamics
423
Contents
§6. Multiple Effects
xi
425
Bibliography
427
Index
435
Preface
This book presents the fundamentals of shock wave theory. Shock waves are present in many natural situations as a consequence of nonlinear constitutive relations. Mathematical analysis of shock waves is based mostly on conservation laws. Consider the case of gas dynamics. There is the conservation of mass. The conservation of momentum follows from Newtonian physics. During the nineteenth century, the conservation of energy and the second law of thermodynamics were formulated. This was the scientific backdrop when Stokes [119] and Riemann [112] did their pioneering work on shock waves in the mid-nineteenth century. More conservation laws were subsequently formulated in the study of electro-magnetism, nonlinear elasticity, high-temperature gas dynamics, and other physical phenomena. There has been important, continuing progress in the development of shock wave theory since the time of Stokes and Riemann. Mathematical study of shock waves requires thinking beyond the standard theory for partial differential equations. Shock wave theory is one of basic mathematical theories that had impacts on other fields in the mathematical sciences. The study of shock waves has helped to initiate new mathematical theory. The theory can involve sophisticated mathematical techniques. Around the mid-twentieth century, it was recognized that the basic notions of shock wave theory can be understood with relative ease when scalar conservation laws are considered. The first part of the book, Chapter 2 through Chapter 5, covers the basic elements of shock wave theory by analyzing scalar conservation laws. Shock wave theory originates from consideration of natural phenomena, and so efforts are made to carry out exact mathematical analysis of the solution behavior by starting from intuitive
xiii
xiv
Preface
geometric considerations. Solutions for hyperbolic conservation laws exhibit several striking kinds of solution behavior. In many physical situations viscous effects are important. To gain a basic understanding of these effects, in Chapter 4 we consider the Burgers equation, which is the simplest viscous conservation law. The main focus of the analysis is also on the explicit solution behavior. This first part of the book requires only the prerequisite of multi-variable calculus, and is suitable for an undergraduate course. For the study of most natural phenomena, it is necessary to consider systems of conservation laws. The study of shock waves after the midnineteenth century focused mainly on gas dynamics. The key nonlinearity of the Euler equations in gas dynamics causing shocks to occur was identified, and elementary waves were constructed. Incorporation of the second law of thermodynamics through the admissibility conditions (entropy conditions) was carried out for the Euler equations in gas dynamics. A general mathematical theory for systems of conservation laws was subsequently developed, applicable to systems beyond the equations in gas dynamics. The second part of the book presents this general theory for systems of hyperbolic conservation laws in Chapter 6 through Chapter 9. Chapter 6 discusses the general theory of hyperbolic partial differential equations. Basic notions of hyperbolicity and finite speed of propagation of waves are explained. The equivalency of symmetric hyperbolicity and the existence of convex entropy is established. Progress toward shock wave theory based on the local existence theory of smooth solutions for general systems is also illustrated. The shock wave theory for systems of hyperbolic conservation laws in one spatial dimension constitutes an important chapter in nonlinear partial differential equations and in nonlinear analysis. The next three chapters deal with this theory. Chapter 7 studies the fundamental Riemann problem, motivating the search for admissibility conditions and the construction of elementary waves. Chapter 8 studies the rich subject of nonlinear interaction of elementary waves. These topics are extensions of those for the scalar laws studied in the first part of the book. However, the basic states are now vectors and it is essential to keep in mind both the state space and the physical space. The analysis of wave interaction suggests various interaction measures to capture the nonlinear effects under different situations. With the basic understanding of wave behavior thus obtained, the well-posedness theory for the general initial value problem is then presented in Chapter 9. The existence theory, established via the Glimm scheme [54], and the subsequent analysis of continuous dependence on initial data and the solution behavior require explicit, constructive solution algorithms. This leads
Preface
xv
to a most significant well-posedness theory for weak solutions of quasilinear evolutionary partial differential equations. Shock waves occur in various continuum media [53]. In a gas flow shock waves result from the compression of acoustic waves. There is a bow shock in front of a supersonic moving object. As the Earth moves in the solar wind, its magnetic field causes a large magnetohydrodynamics shock. An earthquake starts with longitudinal pressure waves, which form shock waves and in turn induce surface shear waves. In shallow water, surface shock waves are produced near the shore, leading to tsunamis. Due to the drastic occurring across them, shock waves constitute an essential signature of many natural phenomena and reflect the basic properties of the media carrying them. Because of their unique features, shock waves have been used as tools in the physical and medical sciences. With their steep gradients, shock waves are easily observed and often used to test material properties. Shock waves are used to compress steel by a definite ratio to test its strain response under great stress. For a high-temperature gas with several modes of internal energies, a shock tube is used to measure various physical parameters [122, 127]. Shock wave lithotripsy is a standard medical practice used to break up kidney stones; see [23]. Industrial diamond can be produced by the compression from an imploding shock sphere. The third and final part of the book, Chapter 10 to Chapter 14, goes back to the original motivation of shock wave theory by focusing on specific physical models. There has been very substantial progress in the general theory for hyperbolic conservation laws since 1950, as presented, for instance, in the first two parts of the book. The general theory provides useful concepts and techniques for exploring specific physical effects. This third part of the book can serve as a reference for readers interested in exploring new frontiers of shock wave theory. Besides exact analysis, formal calculations and intuitive arguments are presented. Potentially interesting research directions are also suggested in these chapters. Chapter 10 discusses the effect of dissipation, such as viscosity and heat conductivity in gas dynamics. The simplest model is the Burgers equation studied in Chapter 4. A main concern is the coupling of nonlinear flux and the dissipation. Dissipation represents a form of non-local constitutive response, in the sense that the flux depends not only on the basic dependent variables, but also on their differentials. The study of viscous conservation laws enables the study of nonlinear waves for dissipative systems such as the Navier-Stokes equations in gas dynamics and the Boltzmann equation in kinetic theory. A more pronounced non-local response, the mechanism of relaxation, is presented in Chapter 11. The relaxation phenomenon is
xvi
Preface
ubiquitous in science as a consequence of memory and other long-term effects. Chapter 11 also includes an introduction to the Boltzmann equation in kinetic theory. Chapter 12 studies several topics of physical interest in the presence of resonance. The nonlinear nature of shock waves gives rise to rich nonlinear resonance phenomena. Basic elements of multi-dimensional Euler equations in gas dynamics are presented in Chapter 13. A new perspective on the difficult subject of multi-dimensional gas flows with shocks is provided. The final chapter, Chapter 14, concludes with some of the author’s personal perspectives, including suggestions of open research directions. Historical perspectives are indicated in the Notes toward the end of each chapter. This is to help readers gain further understanding of the material covered in the chapter. There is no attempt to provide a complete list of references. Rather, the references either are directly related to the topics covered in the book or provide further reading for possible future research in shock wave theory. The presentation of the book is self-contained. For the most part, multi-variable calculus is the only prerequisite. The book covers major topics of modern shock wave theory and is a suitable reference book for researchers in the mathematical sciences. The book can be used as an introductory text for advanced undergraduate and graduate students in mathematics, engineering, and the physical sciences. Each chapter ends with Exercises for students. Basic Notation A ≡ B means that the quantity A is defined to be B. Without ambiguity, f (x) ≡ 0 also means that the function f (x) is zero for all values of x. The limiting values of a function u(x) from the left and from the right are written as u(x − 0) ≡ lim u(x − y), u(x + 0) ≡ lim u(x + y). y→0,y>0
y→0,y>0
Rm denotes m-dimensional real space. Bold letters u, x etc. denote vectors in Rm for some m > 1. The inner product of two vectors u = (u1 , u2 , . . . , um ) and v = (v1 , v2 , . . . , vm ) in Rm is given by (u, v) ≡ u1 v1 + u2 v2 + · · · + um vm . Plain letters x, y, f, g, . . . denote scalar quantities, in R1 . ∇x ≡ (∂x1 , ∂x2 , . . . , ∂xm ) denotes the partial derivative with respect to the variables x = (x1 , x2 , . . . , xm ) ∈ Rm . The divergence of the function f (x) is denoted by m m ∂f (x) = fxj (x). ∇x · f (x) ≡ ∂xj j=1
j=1
Preface
xvii
M, B etc. denote matrixes, with I denoting the identity matrix. Shock waves are highlighted in figures by thicker lines. The author would like to thank the referees and members of the Editorial Committee for their helpful suggestions. Careful reading of the manuscript and critical comments from a referee are much appreciated.
10.1090/gsm/215/01
Chapter 1
Introduction
The most basic equations of shock wave theory are conservative laws of partial differential equations. The basic formulation procedure is as follows. Let x ∈ Rm be the spatial variables, t ∈ R the time variable, and u = u(x, t) ∈ Rn the physical quantities, per unit volume, under consideration. Consider any fixed region Ω in Rm with outer normal n(x) ∈ Rm to its boundary ∂Ω at x, as shown in figure 1.01.
n(x) x Ω ∂Ω
Figure 1.01. Spatial domain Ω.
The amount of the physical quantities in Ω at time t is u(x, t) dx. Ω
Let F(x, t) ∈ Rm × Rn be the flux matrix so that F(x, t) · n is the amount of the physical quantities flowing through the hyperplane normal to n per 1
2
1. Introduction
unit time and per unit area dS(x). Thus the amount, per unit time, of the physical quantities flowing out of the region Ω through its boundary ∂Ω is F(x, t) · n(x) dS(x). ∂Ω
Suppose that there is a source g(x, t) ∈ Rn , representing the production of the physical quantities per unit time and per unit volume. The production, per unit time, of the physical quantities in the region Ω due to the source g(x, t) is g(x, t) dx. Ω
The rate of change with respect to time of the amount of the physical quantities in Ω in Rm equals the flux into Ω through its boundary ∂Ω plus the production due to the source in the region Ω: d u(x, t) dx = − F(x, t) · n(x) dS(x) + g(x, t) dx. dt Ω ∂Ω Ω Assuming that the functions are smooth, by the divergence theorem, F(x, t) · n(x) dS(x) = ∇x · F(x, t) dx. ∂Ω
Ω
Thus we have d u(x, t) dx + ∇x · F(x, t) dx = g(x, t) dx. dt Ω Ω Ω As Ω is an arbitrary fixed domain in Rm , we have the balance laws ut (x, t) + ∇x · F(x, t) = g(x, t). In many physical situations it is appropriate to assume that the flux matrix F(x, t) depends smoothly on the local density u(x, t) of the physical quantities under consideration, and that the source g(x, t) depends smoothly on the local density u(x, t) and the independent variables (x, t): F(x, t) = F(u(x, t)),
g(x, t) = g(u(x, t), x, t).
In such a situation, we have the integral balance laws (0.1) d u(x, t) dx = −F(u(x, t)) · n(x) dS(x) + g(u(x, t), x, t) dx dt Ω ∂Ω Ω and, for a smooth function u(x, t), (0.2)
ut + ∇x · F(u) = g(u, x, t), balance laws.
In the absence of the external source, g ≡ 0, we have (0.3)
ut + ∇x · F(u) = 0, hyperbolic conservation laws.
1. Introduction
3
An important example of hyperbolic conservation laws is the system of Euler equations in gas dynamics, where the physical quantities u under consideration are the mass, momentum and energy of the gas per unit volume: ⎛
⎞ ⎛ ⎞ ρ mass density u ≡ ⎝ ρv ⎠ = ⎝momentum density⎠ . ρE energy density The mass per unit volume is the density ρ. The momentum is the mass times the gas velocity v = (v1 , . . . , vm )t ∈ Rm , and so the momentum per unit volume is ρv. E is the total energy per unit mass. The total energy per unit volume is therefore ρE. The total energy is the sum of the kinetic energy and the internal energy: ρE = ρ
|v 2 | + ρe, 2
with e the internal energy per unit mass. In considering the flux F, first notice that there is always the convection of a given quantity A by the gas velocity v across the boundary ∂Ω of a given domain Ω: Av · n, flux of A due to convection. We now consider the flux for mass, momentum and energy. The mass flux consists of only the convection ρv · n. Thus the above procedure gives ρt + ∇x · (ρv) = 0, conservation of mass. Consider the flux of the ith momentum ρvi , i = 1, . . . , m. The flux due to convection is ρvi v · n. Besides this, there is the momentum flux due to the force acting on the boundary ∂Ω. The Euler equations in gas dynamics are based on the hypothesis that the only force is the pressure force; no shear or other forces are present. The pressure force is isotropic and normal to the boundary ∂Ω. Thus the pressure force on ∂Ω is pn, where the pressure p is a scalar quantity. The ith momentum flux due to pressure is therefore pni . The above procedure yields (ρvi )t + ∇x · (ρvi v) + pxi = 0, i = 1, . . . , m, conservation of momentum. Similarly, the pressure force does work and contributes to the energy flux: (ρE)t + ∇x · (ρvE + pv) = 0, conservation of energy.
4
1. Introduction
We thus have the Euler equations in gas dynamics:
m ⎛ ⎞ ⎛ ⎞ (ρv ) ρ
m j=1 j xj ⎜ ρv1 ⎟ ⎜ j=1 (ρv1 vj )xj + px1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ ⎟ .. (0.4) ⎜ . ⎟ +⎜ ⎟ = 0. . ⎜ ⎟ ⎜ m ⎟ ⎝ρvm ⎠ ⎝ j=1 (ρvm vj )xj + pxm ⎠
m ρE t j=1 (ρEvj + pvj )xj This is often written in the short form ⎛ ⎞ ⎛ ⎞ ρv ρ ⎝ ρv ⎠ + ∇x · ⎝ρv ⊗ v + pI⎠ = 0. (0.5) ρEv + pv ρE t In (0.5) the pressure is a given function of the density and the internal energy: p = p(ρ, e), constitutive relation. The first column vector in (0.5) is the density function u, and the second column vector is the flux function F(u), which is a function of u through the constitutive relation. There are several other basic physical models in the form of hyperbolic conservation laws. In many physical situations, it may be more appropriate to assume that the flux matrix depends not only the local value u(x, t), but also on its gradient ∇x u. There is then an extra dissipation term added to the conservation laws: (0.6) ut + ∇x · F(u) = ∇x · B(u, ε)∇x g(u) , viscous conservation laws. An important example of viscous conservation laws is the system of NavierStokes equations in gas dynamics with the dissipation parameters ε = (μ, κ), where μ represents the viscosity coefficients and κ is the heat conductivity coefficient. It may be appropriate to assume that the flux matrix depends on the past history u(x, τ ), τ ≤ t, of the state u as for elasticity with memory. It may be necessary to assume that the state u is given in terms of the mean values in some microscopic variables ξ, as moments of some density function f (x, ξ, t): P (ξ)f (x, ξ, t) dξ,
u(x, t) = Rn
as in the kinetic theory. We start our analysis with hyperbolic conservation laws, as they are the most basic physical models.
10.1090/gsm/215/02
Chapter 2
Preliminaries
To illustrate some of the basic elements of shock wave theory, we consider in this chapter the scalar conservation law in one spatial dimension: (0.1)
ut + f (u)x = 0,
u ∈ R, f (u) ∈ R, x ∈ R.
1. Method of Characteristics Assuming that the solution u(x, t) is smooth, (0.1) is equivalent to the transport equation (1.1)
ut + λ(u)ux = 0,
λ(u) ≡ f (u).
The classical method of characters for solving the transport equation reduces the partial differential equation (1.1) to ordinary differential equations in certain characteristic directions as follows. Consider the characteristic curves x = x(t) with characteristic speed λ(u) = f (u): d x(t) = λ(u(x(t), t)). dt By the chain rule, (1.1) and (1.2) imply that the solution is constant along any characteristic curve: d u(x(t), t) = ut + x (t)ux (x(t), t) = ut + λ(u)ux (x(t), t) = 0. (1.3) dt In other words, u(x, t) is freely transported along each characteristic curve: (1.2)
u(x(t), t) = u(x(0), 0). Since the characteristic speed λ(u) is a function of u, it is also constant along the characteristic curve: λ(u(x(t), t)) = f (u(x(t), t)) = f (u(x(0), 0)) = λ(u(x(0), 0)). 5
6
2. Preliminaries
This implies that the solutions of the differential equation (1.2) are straight lines: x(t) = λ(u(x(0), 0))t + x(0) = λ(u(x(t), t))t + x(0). Suppose that we want to solve the initial value problem with given initial values u0 (x):
ut + f (u)x = 0, (1.4) u(x, 0) = u0 (x), −∞ < x < ∞. Through a given location (x, t), draw the characteristic line given by dx/dt = λ(u(x, t)), which reaches the initial time at (x−λ(u(x, t))t, 0); see figure 2.01. We thus obtain a relation for the solution u(x, t) in terms of the initial values u0 at t = 0: (1.5)
u(x, t) = u(x − λ(u(x, t))t, 0) = u0 (x − λ(u(x, t))t).
This relation for the solution u(x, t) is implicit, as the right-hand side of (1.5) involves also the solution itself.
t (x, t) dx dt
= λ(u) u = constant (x − λ(u)t, 0)
x
Figure 2.01. Method of characteristics.
The scalar balance law, with u ∈ R, in several space dimensions, x ∈ Rm with m > 1, is ut + ∇x · f (u) = g(u, x, t),
f (u) ∈ Rm
and has the characteristic formulation
d dt x(t) = f (u(x(t), t)), (1.6) d dt u(x(t), t) = g(u, x(t), t).
2. Development of Singularities
7
2. Development of Singularities The characteristic speed λ = λ(u) varies with the state u. Depending on the solution u(x, t), the characteristic lines can diverge, λx > 0, or compress, λx < 0, as time t increases; see figure 2.02. When they compress, the solution first develops a singularity of the type |ux | → ∞, and then the characteristic lines begin to intersect and the solution becomes multi-valued, as seen in figure 2.02. From scientific considerations, it is natural to expect the solution to be a single-valued function u = u(x, t) of the independent variables (x, t). As we will see, there is a way to make the solution a single-valued function by allowing the solution to contain discontinuous shock waves.
t Expansion dx dt
Multi-valued
= λ(u) Compression
x Figure 2.02. Compression and expansion waves.
The balance law (2.1)
ut + f (u)x = g(u, x, t)
is called quasilinear, as the nonlinear term f (u)x occurs at the highest order, here the first order, of differentiation. For quasilinear equations, the characteristic speed λ(u) = f (u) depends on the solution u, and this can cause the shock wave to emerge at a later time from smooth initial values. In contrast, the equation (2.2)
ut + a(x, t)ux = g(u, x, t),
with a(x, t) a given function, is called semilinear in that the nonlinearity in u occurs at a lower order, here in the zeroth-order term g(u, x, t), of
8
2. Preliminaries
differentiation. In this case, the characteristic curves d x(t) = a(x, t) dt do not depend on the solution u and therefore do not intersect. Instead of a shock forming, i.e. |ux | → ∞, the solution itself may blow up, |u| → ∞. Along a characteristic curve, the transport equation (2.2) becomes the ordinary differential equation d u = g(u, x(t), t), dt whose solution may blow up as a consequence of the nonlinearity in u of the source g(u, x, t). Example (Traffic flows). Physical models, such as the Euler equations in gas dynamics, (0.5), are systems, where u ∈ Rn with n > 1. Scalar conservation laws, where u ∈ R, are derived from the systems as approximations. As an illustration, consider a simple model for traffic flows. Suppose that the spatial scale is taken such that it is appropriate to view the flow of cars as a continuum. Let ρ(x, t) be the density of cars, the number of cars per unit length of highway, and v(x, t) the average velocity. Then the flux f (x, t), the number of cars passing a check point per unit time, is due to convection: f (x, t) = ρ(x, t)v(x, t), flux function. Assuming that no cars leave or enter the road at the location x, we have the conservation law (2.3)
ρt + fx = ρt + (ρv)x = 0.
The density ρ and the velocity v are in general not directly related, and so we need an additional equation, besides the above continuity equation, to govern the two dependent variables (ρ, v). In the situation where the traffic flow is stationary and homogeneous in space, ρ(x, t) constant, it is observed that as the density ρ increases, the flux f first increases, and then the traffic begins to slow and the flux f = ρv eventually decreases to zero at some gridlock density ρ1 . The equilibrium flux is a concave function f = F (ρ) with its maximum at the saturation density ρ0 , as shown in figure 2.03, such that (2.4)
F (ρ) < 0,
F (0) = F (ρ1 ) = 0,
F (ρ0 ) = 0.
In the situation where the traffic flow varies moderately in space and in time, the flux f (x, t) can be accurately approximated to depend on the local equilibrium density ρ(x, t) for stationary and space-homogeneous traffic flows: f (x, t) = F (ρ(x, t)).
3. Weak Solutions, Rankine-Hugoniot Condition
9
F (ρ)
ρ0 ρ1
ρ
Figure 2.03. Traffic flow.
In this approximation, (2.3) becomes a self-contained scalar conservation law for the density ρ: (2.5)
ρt + F (ρ)x = 0.
In general, the characteristic speed λ(ρ) = F (ρ) is different from the car speed v = F (ρ)/ρ. From the above analysis, it is the decrease in the characteristic speed, F (ρ)x < 0, that causes the eventual appearance of shock waves. For the concave flux, F (ρ) < 0, we have λ(ρ)x = F (ρ)ρx < 0 when there are more cars ahead, ρx > 0. For the simple concave flux F (ρ) = Aρ(ρ1 − ρ), we have f = ρv = Aρ(ρ1 − ρ) and so v = A(ρ1 − ρ), and when the car speed is slower ahead, vx = −Aρx < 0, shock waves eventually appear.
3. Weak Solutions, Rankine-Hugoniot Condition The analysis of the previous section gives rise to the question of how to interpret the conservation laws for non-smooth solutions. There are two ways to deal with this basic issue. The first is to go back to the integral version, (0.1) of Chapter 1, of the conservation laws (0.1) in this chapter. For one spatial dimension, x ∈ R, the domain Ω is an interval (x1 , x2 ) and the integral version is formulated as a definition. Definition 3.1. A locally integrable function u(x, t) is a weak solution of ut + f (u)x = 0 if for any x1 < x2 , d x2 u(x, t) dx = f (u(x1 , t)) − f (u(x2 , t)). (3.1) dt x1
10
2. Preliminaries
The second version of weak solutions is in the sense of the theory of distributions. Definition 3.2. A locally integrable function u(x, t), t > 0, is a weak solution of ut + f (u)x = 0 with initial data u(x, 0) = u0 (x) if ∞ ∞
(3.2) 0
−∞
uφt + f (u)φx (x, t) dx dt +
∞ −∞
u0 (x)φ(x, 0) dx = 0
for any smooth function φ(x, t) with bounded support in (x, t). Note that for smooth solutions (3.2) is a consequence of the conservation law (0.1) through the integration by parts. Both integral versions (3.1) and (3.2) make sense for non-smooth functions, and the weak solutions for the differential equation (0.1) are thus defined. It turns out that these two versions are equivalent. In both integral versions, it is easy to see that if a solution is smooth, then it satisfies the conservation law (0.1) in the usual sense of calculus. However, when the solution is not smooth and contains a jump discontinuity, the notion of weak solutions imposes a condition across the jump. Theorem 3.3. Suppose that u(x, t) is a piecewise smooth weak solution of the conservation law ut + f (u)x = 0. Then a jump discontinuity (u− , u+ ) in u(x, t) along x = x(t), as shown in figure 2.04, where (3.3)
u− = u− (t) ≡ u(x(t) − 0, t),
u+ = u+ (t) ≡ u(x(t) + 0, t),
satisfies the relation
(3.4) σ(u+ − u− ) = f (u+ ) − f (u− ),
σ = σ(u− , u+ ) ≡ x (t), Rankine-Hugoniot condition.
Proof. We use the first version, Definition 3.1, and take two constants x1 and x2 around x(t), such that x1 < x(t) < x2 . Away from the curve of discontinuity x = x(t), the weak solution around x = x(t) is classical and so we may use the conservation law ut = −f (u)x in the usual calculus sense to
3. Weak Solutions, Rankine-Hugoniot Condition
11
deduce that d x2 d x(t) d x2 (3.5) u(x, t) dx = u(x, t) dx + u(x, t) dx dt x1 dt x1 dt x(t) x2 x(t) ut (x, t) dx + x (t)u(x(t) − 0, t) + ut (x, t) dx − x (t)u(x(t) + 0, t) = x1
x(t)
x(t)
= x1
x(t)
=
x1
ut (x, t) dx + x (t)u− (t) +
−f (u)x (x, t) dx+x (t)u− (t)+
x2
ut (x, t) dx − x (t)u+ (t)
x(t) x2
−f (u)x (x, t) dx−x (t)u+ (t)
x(t)
= f (u(x1 , t)) − f (u− (t)) + x (t)u− (t) − f (u(x2 , t)) + f (u+ (t)) − x (t)u+ (t). Comparing (3.5) with (3.1), we conclude that x (t)(u+ − u− ) = f (u+ ) − f (u− ). Thus the speed, denoted by σ = σ(u− , u+ ) ≡ x (t), of the discontinuity (u− , u+ ) satisfies the Rankine-Hugoniot condition (3.4). Geometrically, the Rankine-Hugoniot condition says that the speed σ = of the discontinuity (u− , u+ ) is the slope of the graph of the flux function f (u); see figure 2.04. Physically, the Rankine-Hugoniot condition (3.4) ensures that the integral conservation law (3.1) holds across a discontinuity in the solution. x (t)
f (u)
t shock wave
x1
u−(t) u+ (t)
x = x(t)
σ x2
u−
u+ x
u
Figure 2.04. Rankine-Hugoniot condition.
Remark 3.4. The Rankine-Hugoniot condition (3.4) can be interpreted directly in the sense of distributions. For this we bring up the basic notion of the delta funciton δ(x) in the theory of distributions: It is not a function in the classical sense and can be defined by the key property of being the derivative of a discontinuity function:
0 for x < 0, (3.6) H(x) ≡ δ(x) ≡ H (x). 1 for x > 0,
12
2. Preliminaries
Thus any function h(x) that is smooth except for a jump at x = x0 ,
hl (x) for x < x0 , (3.7) h(x) = hr (x0 + 0) − hl (x0 − 0) = α, hr (x) for x > x0 , has derivative (3.8)
h (x) = αδ(x − x0 ) +
hl (x) for x < x0 , hr (x) for x > x0 .
The theory of distributions is based on the validity of integration by parts in classical calculus. With this, the above property (3.6) yields, for any function φ(x) that is smooth and zero outside a bounded interval, ∞ ∞ φ(x)δ(x − x0 ) dx = φ(x)H (x − x0 ) dx −∞ −∞ ∞ ∞ φ (x)H(x − x0 ) dx = − φ (x) dx = φ(x0 ). ≡− −∞
x0
Thus we have another key property of the delta function: ∞ φ(x)δ(x − x0 ) dx = φ(x0 ). (3.9) −∞
Clearly, from (3.6), δ(x) is zero for x = 0. Thus (3.9) implies another defining property: ∞ (3.10) δ(x) dx = 1. δ(x) = 0 for x = 0, −∞
Now suppose we have a function u(x, t), that satisfies the conservation law ut +f (u)x = 0 in the usual calculus sense except for a jump discontinuity along x = x(t). The limiting states of u(x, t) are denoted by (u− , u+ ) = (u− (t), u+ (t)), where u− (t) ≡ u(x(t) − 0, t) and u+ (t) ≡ u(x(t) + 0, t). The flux f (u) has a jump of magnitude f (u + ) − f (u− ) and so, in the sense of distributions, f (u)x = f (u+ ) − f (u− ) δ(x − x(t)) at x = x(t). Similarly, at x = x(t), ux = (u+ − u− )δ(x − x(t)). Around a fixed time t = t0 , the jump discontinuity travels approximately along the line x = x(t0 ) + x (t0 )(t − t0 ):
u− for ξ < 0, φ(ξ) = u+ for ξ > 0; u(x, t) = ˜ φ(ξ),
ξ = x − x(t0 ) + x (t0 )(t − t0 ).
At (x, t) = (x(t0 ), t0 ), we have ˜ −x (t0 )φ (ξ) = −x (t0 )ux = −x (t0 )(u+ − u− )δ(x − x(t0 )). ut = Thus we have, for the general time t for which the jump persists, (3.11) ut + f (u)x = −x (t)(u+ − u− ) + f (u+ ) − f (u− ) δ(x − x(t)).
4. Expansion Waves
13
Thus the Rankine-Hugoniot condition makes the delta functions cancel so that the conservation law holds in the sense of distributions. Definition 3.2 is the formal distributional definition involving all the test functions φ(x, t). This intuitive reasoning provides a useful interpretation of weak solutions in terms of delta functions.
4. Expansion Waves While characteristic lines can compress to result in shock waves, they can also diverge to form expansion waves, another basic wave type. Take the case where f (u) = 0 for u between two given states. Suppose that the initial data u(x, 0) takes values between these two states and that λ(u(x, 0)) = f (u(x, 0)) is increasing in x. Then expansion waves form globally in time. This occurs in the following two situations, illustrated in figure 2.05. In the first situation, u(x, 0) is an increasing function and the flux function f (u) is convex over an interval (u1 , u2 ): (4.1) f (u) > 0 for u1 < u < u2 , u1 ≤ u(x, 0) ≤ u2 , and
∂ ∂ u(x, 0) ≥ 0 or, equivalently, f (u(x, 0)) ≥ 0. ∂x ∂x
In the second situation, u(x, 0) is a decreasing function and f (u) is concave ¯1 ): over an interval (¯ u2 , u u) < 0 for u ¯1 > u ¯>u ¯2 , u ¯2 ≤ u(x, 0) ≤ u ¯1 (4.2) f (¯ and
∂ ∂ u(x, 0) ≤ 0 or, equivalently, f (u(x, 0)) ≥ 0. ∂x ∂x
In both situations, the characteristic lines are diverging and the solutions can be constructed by the method of characteristics to exist globally in positive time; see figure 2.05. Remark 4.1. In figure 2.04 and figure 2.05, and in figure 2.08 and figure 2.09 to follow, each contains a graph of the flux function f (u) in the state space u and another diagram showing the propagation of waves in the physical space (x, t). It is necessary for the study of quasilinear hyperbolic equations to consider the state space, to indicate the values taken by a solution u(x, t), and the physical space, to picture the propagation of the solution in (x, t) space. For the study of systems, more sophisticated, simultaneous consideration of the state space and the physical space is needed. Expansion waves can also be formed from discontinuous initial data.
14
2. Preliminaries
f (u)
u¯2 u¯
u¯1
u1 u u2 t
u1
u
t
u
u2
u¯1 x
u¯
u¯2 x
Figure 2.05. Expansion waves.
Proposition 4.2. Suppose that f (u) = 0 for all u between ul and ur and that f (ul ) < f (ur ). Then the solution of the initial value problem (4.3)
ut + f (u)x = 0, ul , x < 0, u(x, 0) = ur x > 0,
is the centered rarefaction wave (ul , ur ), defined in terms of the inverse function [f ]−1 of f (u): ⎧ ⎪ ⎨ul for x < f (ul )t, (4.4) u(x, t) = [f ]−1 ( xt ) for f (ul )t < x < f (ur )t, ⎪ ⎩ ur for x > f (ur )t. Proof. There are two possibilities, (ul , ur )=(u1 , u2 ) as in (4.1) and (ul , ur ) ¯2 ) as in (4.2), and we are trying to solve = (¯ u1 , u
u1 , x < 0, (4.5) u(x, 0) = u2 , x > 0; (4.6)
u ¯1 , x < 0, u ¯(x, 0) = u ¯2 , x > 0.
4. Expansion Waves
15
t x t x t
= λ(u1 ) u1
t = λ(u) x t
x t
= λ(u2 )
x t
u2
= λ(¯ u) x t
= λ(¯ u1 ) u¯1
x
= λ(¯ u2 )
u¯2
x
Figure 2.06. Centered rarefaction wave.
Since the two characteristic values from the initial states diverge, f (u1 ) < u1 ) < f (¯ u2 ), the method of characteristics yields f (u2 ) and f (¯
u1 for x < f (¯ u1 )t, u1 for x < f (u1 )t, u ¯ (x, t) = (4.7) u(x, t) = u2 )t. u2 for x > f (u2 )t; u2 for x > f (¯ For the first case, the characteristics from the initial time do not reach the wake region f (u1 )t < x < f (u2 )t, and the solution u(x, t) needs to be constructed there. This is done as follows: As the two characteristics dx/dt = f (u1 ) and dx/dt = f (u2 ) diverge and don’t compress to form a shock, one may expect the solution to be continuous in the wake region so that the method of characteristics applies. In the wake region, the characteristic lines start at the origin, x = x(t) = αt. Here α = x/t is the characteristic speed. The characteristic analysis in Section 1 yields x (4.8) λ(u(x, t)) = f (u(x, t)) = for f (u1 )t < x < f (u2 )t. t The same construction applies to the second case, u ¯2 < u < u ¯1 . We thus obtain the centered rarefaction wave solution shown in figure 2.06, ⎧ ⎪ ⎨u1 for x < f (u1 )t, u(x, t) = [f ]−1 ( xt ) for f (u1 )t < x < f (u2 )t, ⎪ ⎩ u for x > f (u2 )t; ⎧ 2 (4.9) ⎪u u1 )t, ¯1 for x < f (¯ ⎨ u ¯(x, t) = [f ]−1 ( xt ) for f (¯ u1 )t < x < f (¯ u2 )t, ⎪ ⎩ u2 )t. u ¯2 for x > f (¯ Here [f ]−1 is the inverse function of f : f (u) = w if and only if [f ]−1 (w) = u. Note that the inverse function [f ]−1 exists because, in both cases, f (u) = 0 for all states u under consideration. It is easy to check that these are continuous solutions to the conservation law ut + f (u)x = 0. The expression
16
2. Preliminaries
(4.4) is the unified expression for the two expressions in (4.9) for the centered rarefaction wave (ul , ur ). Example. Consider the conservation law ut +(u3 /3)x = 0. The flux function f (u) = u3 /3 is convex for u > 0 and concave for u < 0. Thus there are two ¯2 < u ¯1 < 0. In the possibilities for the rarefaction wave, 0 < u1 < u2 and u 2 wave region the characteristic speed f (u) = u = x/t, and x x −1 x −1 x for u > 0; u = [f ] ( ) = − for u < 0. u = [f ] ( ) = t t t t Take the case of (¯ u1 , u ¯2 ) = (−2, −3). We have f (¯ u1 ) = 4 and f (¯ u2 ) = 9. The initial value problem
u3 −2 for x < 0, = 0, u(x, 0) = ut + x 3 −3 for x > 0 ⎧ ⎪ for x < 4t, ⎨−2 x u(x, t) = − t for 4t < x < 9t, ⎪ ⎩ −3 for x > 9t.
has solution
5. Non-uniqueness, Entropy Condition The discontinuity in (4.3) is resolved by the centered rarefaction wave (4.4). But it can also be resolved by a weak solution, the centered rarefaction shock, as shown in the left diagram of figure 2.07:
ul for x < st, f (ur ) − f (ul ) . (5.1) u(x, t) = s= ur − ul ur for x > st, t
t dx dt
dx dt
= λ(ul )
dx dt
=s dx dt
= λ(ur )
=s
u− dx dt
= λ(u− )
u+ dx dt
x
= λ(u+ )
x
Figure 2.07. Rarefaction shock and compression shock.
Thus the initial value problem (4.3) has more than one solution. This non-uniqueness is contrary to common expectations for the theory of evolutionary equations. One expects the initial value problem to be well-posed in that it has exactly one solution and the solution depends continuously
5. Non-uniqueness, Entropy Condition
17
on the initial data. It turns out that the present non-uniqueness is due to the missing information intrinsic in the modeling of the physical situation by hyperbolic conservation laws and is related to the concept of entropy in thermodynamics. We will dwell more upon this key issue later on. For now we simply state that the situation can be remedied by imposing the entropy condition so that rarefaction shocks are disallowed. The term entropy is used here because of its direct relation to the concept of entropy in the second law of thermodynamics, a central topic that will be explained in several contexts later. Definition 5.1. Suppose that the flux function f (u) satisfies f (u) = 0 for all states u between u− and u+ . A discontinuity (u− , u+ ) in a weak solution of the conservation law ut + f (u)x = 0 is admissible if it satisfies the admissibility condition f (u− ) > s > f (u+ ), entropy condition for convex laws.
(5.2)
Here s is the speed of the discontinuity given by the Rankine-Hugoniot condition: f (u+ ) − f (u− ) . s = σ(u− , u+ ) = u+ − u− An admissible discontinuity is called a shock wave. Consider a convex law, that is, the flux function f (u) has no inflection point and is either convex or concave. Then the entropy condition (5.2) is equivalent to
u− > u+ when f (u) > 0, figure 2.08, (5.3) u− < u+ when f (u) < 0, figure 2.09. f (u)
t dx dt
s
=s
u− u− u +
u+
dx dt
= f (u− )
u
dx dt
= f (u+ )
x
Figure 2.08. Shock wave for f (u) > 0.
In our previous analysis, shock waves are formed through compression. The entropy condition (5.2) says that the shock wave (u− , u+ ) is compressive in that the characteristic lines dx/dt = u± on either side of the shock wave impinge on the shock wave in the forward time direction; see right
18
2. Preliminaries
t
f (u) dx dt
u−
s
=s u+
u−
u+ dx dt
= f (u− )
dx dt
= f (u+ )
x
u Figure 2.09. Shock wave for f (u) < 0.
diagrams of figure 2.08 and figure 2.09. As information propagates along the characteristics, the compressibility of shock waves means that the information is received, and not created, at the shocks. As a consequence, the entropy condition makes the solution operator stable, a desirable mathematical property. Enforcement of the entropy condition in Definition 5.1 is sufficient for a mathematical theory for convex conservation laws. This is the topic of Chapter 3. Example. Consider the flux function f (u) = u4 /12+u2 /2, which is convex, f (u) = u2 + 1 > 0. Therefore (u− , u+ ) satisfies the entropy condition if u− > u+ . Take the shock wave (3, −1), which has speed 4
2
4
(−1) 3 + (−1) − ( 12 + f (−1) − f (3) 2 = 12 s = σ(3, −1) = −1 − 3 −1 − 3 the shock wave (3, −1) through the origin is
3, for x < 83 t, u(x, t) = −1, for x > 83 t.
32 2 )
8 = ; 3
6. Notes Shock wave theory was initiated by Stokes [119] and Riemann [112] in their study of the gas dynamics equations. That smooth initial data can give rise to singularities was first pointed out by Stokes. Riemann solved the initial value problem in the case where the initial data consists of two constant states, which we now call the Riemann problem. The book by Courant and Friedrichs [32] reports on subsequent studies by researchers in gas dynamics. This classical book contains many special solutions for the Euler equations of gas dynamics. Most of the questions raised in the book still await answers from future research. Around the time of publication of the book in 1948 and of the paper [62] by Hopf in 1950, it was then recognized that it would be instructive to build up shock wave theory from simpler models, the scalar
7. Exercises
19
conservation laws. This and the next three chapters illustrate the theory for scalar conservation laws. General reference books on shock waves include the following: Besides the Courant-Friedrichs book [32], classical references for shock wave theory include Bers [6] on stationary gas flows, Whitham [126] on nonlinear waves, and Smoller [116] on analysis of elementary waves. The book by Dafermos [35] offers a comprehensive account of the mathematical development of the theory of hyperbolic conservation laws. On the physical side, there are Vicenti-Kruger [122] and Zeldovich-Raizer [127] on high-temperature gas dynamics, and Liepmann-Roshko [80] on gas dynamics.
7. Exercises 1. Derive the characteristic equation for v ≡ ux from the conservation law ut + f (u)x = 0. Show by the method of characteristics that |v| tends to ∂ λ(u)(x, 0) < 0 for some infinity at some later time if at initial time t = 0, ∂x x. This is the analytical way of showing that a global smooth solution for (0.1) does not exist. The geometric way is presented in Section 2. 2. Consider the initial value problem
2 ut + u2 + u x = 0, u(x, 0) = sin x, x ∈ R. Find the time T > 0 and locations where the solution u(x, t) first becomes singular. 3. Consider the initial value problem for the semilinear equation
ut + ux = u2 , u(x, 0) = sin x, x ∈ R. Find the time T > 0 and locations where the solution u(x, t) first blows up. 4. We have seen that the first version, Definition 3.1, of weak solutions yields the Rankine-Hugoniot condition (3.4). Use the second version, Definition 3.2, to derive (3.4). 5. Show that a piecewise smooth function u(x, t) is a weak solution of the conservation law ut + f (u)x = 0 if and only if it satisfies the RankineHugoniot condition (3.4) across the discontinuities and satisfies the conservation law in the usual calculus sense away from the discontinuities. Check this for both Definition 3.1 and Definition 3.2. 6. Generalize the notion of weak solutions in Definition 3.2 to the balance law of the form g(u)t +f (u)x = h(x, t, u). Derive the Rankine-Hugoniot condition across a jump discontinuity (u− , u+ ) with speed σ in a weak solution of
20
2. Preliminaries
g(u)t + f (u)x = h(x, t, u). (Hint: The Rankine-Hugoniot condition does not depend on the lower-order term h(x, t, u).) 7. The conservation law ut + (u2 /2)x = 0 implies another conservation law (u2 /2)t + (u3 /3)x = 0 if u(x, t) is smooth. Find a weak solution of the first equation which is not a weak solution of the second equation. (Hint: Consider a shock wave and compare their Rankine-Hugoniot conditions.) 8. Show that conservation law with damping ut + f (u)x = −αu, α > 0, has a smooth global solution if the initial function u(x, 0) has sufficiently small gradient. (Hint: Use the method of characteristics.)
10.1090/gsm/215/03
Chapter 3
Scalar Convex Conservation Laws
We begin the presentation of the rich theory for scalar conservation laws. The main purpose is to study wave phenomena. The scalar theory is interesting in itself and also serves as a prelude to the theory for systems to be presented later. This chapter considers scalar convex hyperbolic conservation laws in one spatial dimension:
(0.1)
ut + f (u)x = 0,
f (u) = 0,
u, x ∈ R.
The significance of the convexity condition, f (u) = 0, is that the characteristic speed λ(u) = f (u) is a monotone function of the state u, λ (u) = f (u) = 0. We will see later in Chapter 7 that the acoustic mode of gas dynamics and pressure waves in nonlinear elasticity have an analogous property. In Section 2 of Chapter 2, we saw that if f (u0 ) = 0 for some state u0 , then the characteristic lines dx/dt = f (u) for u around u0 can compress to form shock waves. In this chapter we consider the strong nonlinearity property f (u) = 0 for all states u under consideration. The entropy condition and the solution of the Riemann problem, two basic elements of the shock wave theory, can be stated in simple way under this strong nonlinearity property. There are striking solution behaviors, unique to the shock wave theory, that we will explore. 21
22
3. Scalar Convex Conservation Laws
1. Riemann Problem There is a particular initial value problem, the Riemann problem, which allows one to study the basic elements of the shock wave theory with relative ease. We consider for the moment the Riemann problem for a general system of hyperbolic conservation laws in one spatial dimension: (1.1)
n ut + f (u) x = 0, u ∈ R , ul , x < 0, Riemann data. u(x, 0) = ur , x > 0,
The initial data consist of two constant states ul and ur . We call the initial value problem (1.1) the Riemann problem (ul , ur ). Proposition 1.1. The solution to the Riemann problem is self-similar with self-similarity variable ξ = xt ; that is, there exists a function φ(ξ) such that x (1.2) u(x, t) = φ(ξ), ξ ≡ . t Proof. Note that the system ut + f (u)x = 0 as well as the Riemann data are invariant, i.e. unchanged, under dilation of the independent variables (x, t) → α(x, t) for any positive constant α. Precisely, introduce a new function v(x, t) ≡ u(αx, αt), so that v t = αut , v x = αux . Then v t + f (v)x = α ut + f (u)x = 0,
ul for αx < 0, or equivalently for x < 0, v(x, 0) = ur for αx > 0, or equivalently for x > 0. Therefore, the new function v(x, t) satisfies the same conservation laws and has the same initial data as the original function u(x, t). Assuming that the Riemann problem has a unique solution, we have v(x, t) = u(x, t), or (1.3)
u(x, t) = u(αx, αt) for any positive constant α. 1 t
and we have x u(x, t) = u( , 1) ≡ φ(ξ), t This yields the expression (1.2).
For any fixed t, take α =
ξ≡
x . t
The number of independent variables is reduced from two to one, from (x, t) to ξ, and so the Riemann problem is much easier to solve than a general initial value problem. This is one of the main reasons that we will also start with the Riemann problem when we consider general systems later
1. Riemann Problem
23
in Chapter 7. Another reason for focusing on the Riemann problem is that the basic issues concerning shock waves need to be addressed immediately. We now come back to the scalar laws u ∈ R with convex flux f (u) = 0. The entropy condition has been proposed in Definition 5.1 of Chapter 2. Shock waves and rarefaction waves constructed in Section 4 and Section 5 of Chapter 2 are sufficient for solving the Riemann problem for the convex flux f (u) = 0. We summarize the results obtained there as a proposition. Proposition 1.2. Consider the Riemann problem (ul , ur ) for the scalar conservation law: ⎧ ⎪ ⎨ut + f (u)x = 0, x, u ∈ R, f (u) = 0, (1.4) ul , x < 0, ⎪ ⎩u(x, 0) = ur , x > 0. The Riemann solution satisfying the entropy condition (5.2), Definition 5.1 of Chapter 2 is as follows (figure 3.01): When f (ul ) > f (ur ),
ul for x < st, (1.5) u(x, t) = centered shock wave; (ul ) ur for x > st, s = f (uurr)−f , −ul when f (ul ) < f (ur ), ⎧ ⎪ ⎨ ul for x < f (ul )t, (1.6) u(x, t) = [f ]−1 ( xt ) for f (ul )t < x < f (ur )t, ⎪ ⎩ ur for x > f (ur )t, centered rarefaction wave. t
t dx dt
ul
=s
x t
ur
= f (ul )
x t
ul x t
ul
ur f (u ) > f (ur ). l Shock wave:
x
= f (ur )
ur = f (u(x, t))
x ur Rarefaction wave : f (ul ) < f (ur ). ul
Figure 3.01. Riemann problem for convex flux.
Under the entropy condition, the above is the only self-similar solution to the Riemann problem. This is seen as follows: If the solution is continuous, then, by the self-similarity property, the characteristic lines originate at the origin, and so their slope f (u)(x, t) = x/t. This implies that the solution is
24
3. Scalar Convex Conservation Laws
a centered rarefaction wave. If the solution contains more than one wave, ¯) to the left followed by a rarefaction wave (¯ u, ur ) to say a shock wave (ul , u the right, then, by geometric consideration of the wave location, the shock ¯) must be less than or equal to the speed f (¯ u) of the left edge of speed σ(ul , u the rarefaction wave (¯ u, ur ). However, from the entropy condition, (5.2) in ¯) < f (¯ u). Thus such a combination of two waves Chapter 2, we have σ(ul , u is not possible. A similar argument using the entropy condition shows that the Riemann solution can be either a single shock or a single rarefaction wave, as given in Proposition 1.2. Besides this geometric consideration, it will be shown later in Section 4 that analytically there is a unique piecewise smooth solution satisfying the entropy condition to the general initial value problem of a conservation law.
2. Hopf Equation The simplest convex conservation law is the inviscid Burgers equation, the 2 so-called Hopf equation, with f (u) = u2 : (2.1)
ut + (
u2 )x = 0, inviscid Burgers equation, or Hopf equation. 2
The simplicity comes from the fact that the characteristic speed equals the state value λ(u) = f (u) = u and so [f ]−1 (u) = u. The speed s of the shock (u− , u+ ) is the arithmetic mean of the end states: (2.2)
s=
(u+ )2 2
2
− (u−2 ) u+ + u− . = u+ − u− 2
Thus the elementary waves for the Hopf equation take a simple form and the solution u(x, t) of the Riemann problem (ul , ur ) is, when ul > ur ,
ul , x < st, ur + ul (2.3) u(x, t) = s= , Hopf centered shock wave; 2 ur , x > st, and when ul < ur , ⎧ ⎪ ⎨ul , x < ul t, (2.4) u(x, t) = xt , ul t < x < ur t, ⎪ ⎩ ur , x > ur t,
Hopf centered rarefaction wave.
The Hopf equation also provides a simple way of calculating the so-called entropy inequality. Multiply the Hopf equation by u to obtain (2.5)
(
u3 u2 )t + ( )x = 0. 2 3
2. Hopf Equation
25
For reasons involving the second law of thermodynamics that will be explained later when we systematically study general systems, we call u2 u3 , ) an (entropy, entropy flux) pair 2 3 for the Hopf equation. We have noted that the new conservation law (2.5) is not equivalent to the Hopf equation in the weak sense; see Exercise 5 of Chapter 2. The reason is that the Rankine-Hugoniot condition (2.2) for the Hopf equation does not imply the Rankine-Hugoniot conditon for (2.5). In fact, across a shock wave (u− , u+ ) with u+ < u− , for the Hopf equation with speed s = (u+ + u− )/2 there is an inequality for (2.5): (η(u), q(u)) = (
(u+ )2 (u+ )2 (u− )3 (u− )3 − + − −s η(u+ )−η(u− ) +q(u+ )−q(u− ) = −s 2 2 3 3 2 2 3 3 (u− ) (u+ ) (u− ) 1 u+ + u− (u+ ) − + − = − |u+ − u− |3 < 0. =− 2 2 2 3 3 12 In other words, the Rankine-Hugoniot condition for (2.5) does not hold and becomes an inequality 1 (2.6) −s η(u+ ) − η(u− ) + q(u+ ) − q(u− ) = − |u+ − u− |3 < 0. 12 Suppose the solution is piecewise smooth and has shock waves (uj− (t), uj+ (t)) along x = xj (t), j = 1, 2, . . . . Then in the sense of distributions (see (3.11) of Chapter 2), (2.7) η(u)t + q(u)x = −xj (t) η(uj+ (t)) − η(u− (t)) + q(uj+ (t)) − q(uj− (t)) δ(x − xj (t)). j
From (2.6) and (2.7), we obtain the entropy inequality (2.8)
(
1 j u2 u3 )t + ( )x = − |u+ (t) − uj− (t)|3 δ(x − xj (t)) ≤ 0. 2 3 12 j
This quantifies, in terms of the increase in entropy, the degree of irreversibility of the process with shock waves. Suppose that the solution u(x, t) tends to the same constant state u ¯ sufficiently fast as |x| → ∞. Then we may integrate the above inequality to yield t2 ∞ (u(x, t2 ) − u ¯ )2 1 j dx + |u (t) − uj− (t)|3 dt (2.9) 2 12 + −∞ t1 j ∞ (u(x, t1 ) − u ¯ )2 dx, for t1 < t2 . = 2 −∞
26
3. Scalar Convex Conservation Laws
In particular, d dt
(2.10)
∞
(2.11) 0
j
∞ −∞
(u(x, t) − u ¯ )2 dx ≤ 0, 2
1 j |u (t) − uj− (t)|3 dt ≤ 12 +
∞ −∞
t > 0; (u(x, 0) − u ¯ )2 dx. 2
These are the entropy estimates. The mathematical entropy u2 /2 is really the negative physical entropy. Thus (2.10) implies that the entropy increases in time as implied by the second law of thermodynamics. The estimate (2.11) implies that the shock waves in a solution with the same limiting value at x = ±∞ decay time-asymptotically. Any convex conservation law ut + f (u)x = 0, f (u) = 0, is equivalent to the Hopf equation for its characteristic speed λ: λ2 )x = 0, λ ≡ f (u). 2 This is easily seen by multiplying the convex conservation law by f (u) and applying the chain rule. However, these two equations are not equivalent in the weak sense as the Rankine-Hugoniot condition for the conservation law is different in general from that for the Hopf equation: (2.12)
λt + (
f (u+ ) − f (u− ) f (u+ ) + f (u− ) f (u+ ) − f (u− ) λ+ + λ− = − − u+ − u− 2 u+ − u− 2 1 = − f (u0 )(u+ − u− )2 6 for some state u0 between u− and u+ . Since the error is of second order, we see that for solutions with weak shock waves, |u+ − u− | small, the Hopf equation is an accurate approximation of the convex conservation law.
(2.13)
We will see later in Chapter 7 that the Hopf equation governs an important class of solutions, the simple waves, for general systems of conservation laws.
3. Wave Interactions, Constructing Solutions Consider the initial value problem
ut + f (u)x = 0, f (u) = 0, (3.1) u(x, 0) = u0 (x). The initial function u0 (x) is an arbitrarily given function. As we have seen, shock waves in general can form from smooth initial functions. A simple compressive wave, as in figure 2.02 of Chapter 2, can give rise to infinitely
3. Wave Interactions, Constructing Solutions
27
many shock waves. The solution can contain complex wave patterns. Nevertheless, it is possible to construct accurate piecewise smooth approximate solutions. The evolution of these piecewise smooth solutions can be analyzed and visualized through a series of interactions of elementary waves. In preparation for such a construction, we start with the analysis of interactions of elementary waves. For definiteness, consider the case f (u) > 0 so that there are two types of elementary waves, a shock wave (u− , u+ ) with u− > u+ and a rarefaction wave (ul , ur ) with ul < ur . 3.1. Interaction of Elementary Waves Example 1. With f (u) > 0, a shock wave (u− , u+ ) satisfies u− > u+ and f (u− ) > σ(u− , u+ ) > f (u+ ) according to the entropy condition, Definition 5.1 of Chapter 2. When two shock waves (u1 , u2 ) and (u2 , u3 ) are next to each other, they must interact because, by the entropy condition, the speed σ(u1 , u2 ) of the left shock is greater than the speed σ(u2 , u3 ) of the right shock, as shown in figure 3.01: σ(u2 , u3 ) < f (u2 ) < σ(u1 , u2 ). The compressibility of the two shocks, f (u1 ) > f (u2 ) and f (u2 ) > f (u3 ), implies the compressibility, f (u1 ) > f (u3 ), of the combined wave (u1 , u3 ). Thus two shock waves (u1 , u2 ) and (u2 , u3 ) simply combine to form the single shock (u1 , u3 ) after they meet, as shown in figure 3.02. t
f (u)
u1 u3
u1
u2 u
u2
u3
x
Figure 3.02. Combining of shocks.
This is clearly a simplifying process. Two shocks combine to form one shock. Moreover, the solution process is irreversible. The solution process cannot be reversed, as knowing that the solution is a single shock (u1 , u3 ) at a later time does not tell us what the solution was at an earlier time. At an earlier time, the solution could consist of two shock waves (u1 , u2 ) and (u2 , u3 ) as in the situation considered here, or it could consist of the single shock (u1 , u3 ).
28
3. Scalar Convex Conservation Laws
Example 2. Consider two rarefaction waves (u1 , u2 ) and (u2 , u3 ) next to each other. They do not approach each other; the right edge of (u1 , u2 ) and the left edge of (u2 , u3 ) both propagate with the characteristic speed f (u2 ), and so they do not interact, as shown in figure 3.03. f (u)
t
u3 u1
u1
u2
u3
u2 u
x
Figure 3.03. Co-existence of rarefaction waves.
Example 3. Consider the interesting case of the interaction of a shock wave and a rarefaction wave. For definiteness, consider the case of a shock wave (u1 , u2 ) to the left and a rarefaction wave (u2 , u3 ) to the right. The entropy condition for the shock (u1 , u2 ) implies that the shock speed is larger than the characteristic speed of the left state of the rarefaction wave, σ(u1 , u2 ) > f (u2 ). Therefore the two waves meet and interact in finite time. The waves are of opposite sign, u2 − u1 < 0 and u3 − u2 > 0. As x increases, the state jumps down across a shock, u1 > u2 , and increases across a rarefaction wave, u2 < u3 . Thus the interaction induces cancellation. For simplicity, take the center of the rarefaction wave (u2 , u3 ) to be (0, 0), so that in the wave region f (u(x, t)) = x/t by (1.6). The interaction process is governed by the Rankine-Hugoniot condition. After the meeting time of the two waves, the shock location x = x(t) and the right state u(x(t), t) = u(x(t) + 0, t) are calculated by solving the two equations f (u(x(t), t)) − f (u1 ) , Rankine-Hugoniot condition; u(x(t), t) − u1 x(t) , centered rarefaction wave. f (u(x(t), t)) = t Consider first the case where the shock is stronger, u2 < u3 < u1 , as in the left graph of figure 3.04. By the mean value theorem, the difference of the speed of the shock (u1 , u(x(t), t)) and the rarefaction characteristic speed f (u(x(t), t)) is x (t) =
σ(u1 , u(x(t), t)) − f (u(x(t), t)) =
f (u(x(t), t)) − f (u1 ) − f (u(x(t), t)) u(x(t), t) − u1 1 u) u1 − u(x(t), t) = f (¯ 2
3. Wave Interactions, Constructing Solutions
29
for some state u ¯ between u(x(t), t) and u1 . Since u1 −u(x(t), t) > u1 −u3 and the flux is convex, f (u) > 0, the difference σ(u1 , u(x(t), t))−f (u(x(t), t)) is positive and strictly bounded away from zero. Consequently, the interaction rate is positive and the rarefaction wave is cancelled in finite time to result in a single shock (u1 , u3 ); see right diagram of figure 3.04. t
f (u)
u1 u2
x = x(t)
u3
u1
u3
u2
u
x
Figure 3.04. Shock-rarefaction interaction I: cancellation of rarefaction wave.
The second case of the shock being weaker than the rarefaction wave, u2 < u1 < u3 , is more interesting; see left graph of figure 3.05. As the shock (u1 , u(x(t)+, t)) becomes weaker due to the cancellation of waves, its speed is well approximated by the arithmetic mean of the two states, (2.13), f (u(x(t)+, t)) + f (u1 ) + O(1)|u(x(t)+, t) − u1 |2 . 2 Consequently, this differential equation is accurately approximated by 1 x(t) d x(t) − f (u1 )t + f (u1 ) , or (x(t) − f (u1 )t) = . x (t) = 2 t dt 2t Solving this yields x (t) =
x(t) = f (u1 )t + O(1)(t + 1) 2 , 1
and so the strength of the shock is 1 x(t) x(t) − f (u1 )t − f (u1 ) = = O(1)(t + 1)− 2 . f (u(x(t)+, t)) − f (u1 ) = t t Thus the interaction of a stronger rarefaction wave with a weaker shock wave takes infinite time to complete, and the shock wave decays at the rate of t−1/2 ; see right diagram of figure 3.05. Example 4. As a final illustration of the interaction of elementary waves, we consider the situation of a shock (u2 , u3 ) interacting with a rarefaction wave (u1 , u2 ) to its left and another rarefaction wave (u3 , u4 ) to its right, as shown in figure 3.06. Consider the interesting case of the shock being weaker than either of the rarefaction waves. After some time, the shock will interact with both of the rarefaction waves, and the states u± (t) of the evolving shock (u− (t), u+ (t)) need to be computed. Suppose the left
30
3. Scalar Convex Conservation Laws
t
f (u)
x = x(t) u3 u2
u1
u1
u1
u3 u2
u
x
Figure 3.05. Shock-rarefaction interaction II: eventual cancellation of shock wave.
rarefaction wave (u1 , u2 ) originates from x = xl and the right rarefaction wave (u3 , u4 ) originates from x = xr . Let the location of the shock at time t be x = x(t). After finite time the shock interacts with both rarefaction waves and we have, by following the characteristic lines on either side of the shock, xl − xr . x(t) = xl + f (u− (t))t = xr + f (u+ (t))t, or f (u− (t)) − f (u+ (t)) = t Therefore, it follows from the convexity f (u) = 0 that, for some positive bounded function O(1), xr − xl . |u+ (t) − u− (t)| = O(1)|f (u+ (t)) − f (u− (t))| = O(1) t Thus a shock canceled by rarefaction waves from both sides decays at the rate of t−1 . This is faster than the rate of t−1/2 for the interaction of a shock with a stronger rarefaction wave from one side as studied in Example 3. The above examples complete the analysis of the interaction of elementary waves, the centered shock and rarefaction waves. t
u+ (t)
u−(t) u1
xl
u2 u3
u4
xr
x
Figure 3.06. Shock-rarefaction interaction III: cancellation of shock wave from both sides.
3. Wave Interactions, Constructing Solutions
31
3.2. Piecewise Smooth Solutions The study of the interaction of elementary waves in the preceding subsection allows us to construct solutions for the initial value problem with more general initial data than Riemann data. Theorem 3.1. Consider the initial value problem
ut + f (u)x = 0, f (u) = 0, u(x, 0) = u0 (x), −∞ < x < ∞. Suppose that the initial function u0 (x) is a step function with a finite number of jumps. Then the solution u(x, t) exists globally in time and is piecewise smooth. Moreover, the number of shock waves in u(·, t) is no more than the number of jumps of u0 (x) and is non-increasing in time. Proof. The discontinuities at t = 0 represent translated Riemann problems. Proposition 1.2 provides explicit solutions of these Riemann problems. A shock or a centered rarefaction wave issues from each point of discontinuity of u0 (x). These waves interact at later times. These interactions can be resolved explicitly as shown in the examples in Subsection 3.1. Thus the solution u(x, t) is explicitly constructed to be piecewise smooth. The number of initial shock waves is less than the number of jumps of the initial function u(x, 0). When two shock waves meet, there is the simplifying mechanism of combining, as illustrated in Example 1 of Subsection 3.1 and figure 3.02, and so the number of shock waves can only decrease in time. We will show later in Theorem 4.5 that, by refining the approximation of general initial data with step functions, the piecewise smooth solutions corresponding to step function initial data tend to the exact solution in the limit. The accuracy of the approximation is uniform in time. For this reason, in studying the solution behavior in the later part of this chapter, we may consider only piecewise smooth solutions. Note that general piecewise smooth initial data may give rise to compression waves, which can result in infinitely many shock waves, and the solution may fail to be piecewise smooth at later times. However, in the above construction, the compression wave is approximated by a series of shock waves, and the number of shock waves in an approximate solution does not increase in time. Therefore each approximate solution is piecewise smooth globally in time. We will see that the local variation of the solution is uniformly bounded for any later time t > 0 even if the initial values contain large oscillations; see Theorem 5.3. In other words, the solution with general initial data in fact becomes almost piecewise smooth immediately.
32
3. Scalar Convex Conservation Laws
4. Well-Posedness Theory As with the theory for general evolutionary partial differential equations, there is the fundamental question of well-posedness of the initial value problem for the conservation law
ut + f (u)x = 0, u(x, 0) = u0 (x). The theory involves showing that the solution to the initial value problem exists, is unique, and depends on its initial data continuously. The existence has been established for initial data being a step function, in Theorem 3.1 of Section 3. In the first subsection we will establish the continuous dependence on initial data in the L1 (x) topology, also for the class of piecewise smooth solutions. In the second subsection the well-posedness theory for general initial data is established by showing that the class of piecewise smooth solutions is dense in the class of general weak solutions. Since the L1 (x) norm is used, we will consider, in the remainder of this chapter, functions in the following space: (4.1) f (x) ∈ F if and only if f (x) is bounded measurable and 0 ∞ |f (x)−ul | dx+ |f (x)−ur | dx < ∞ for some constants ul and ur . −∞
0
4.1. L1 (x) Contraction For the initial value problem to be well-posed, the solution u(x, t) needs to exist, be unique, and depend continuously on the initial function u0 (x). To have continuous dependence, the solution needs to change a little bit when the initial function is changed a little bit. The notion of changing a little bit depends on the way of measuring the distance between functions. One needs to find the appropriate metric for measuring the distance between solutions for the given equations. In the theory of partial differential equations, there are several such metrics, not all of which have direct physical meaning. We use the L1 (x) distance between two solutions u1 (x, t) and u2 (x, t): ∞ |u1 (x, t) − u2 (x, t)| dx. u1 (·, t) − u2 (·, t) L1 (x) ≡ −∞
It turns out that the solution operator is L1 (x) contractive: ∞ ∞ |u2 (x, t) − u1 (x, t)| dx ≤ |u2 (x, t0 ) − u1 (x, t0 )| dx (4.2) −∞
−∞
for 0 ≤ t0 < t.
4. Well-Posedness Theory
33
This holds for general scalar conservation laws; see Section 5 of Chapter 5 for a detailed explanation. For convex laws there is a stronger version; see Theorem 4.1 below. In this subsection, we will establish the L1 (x) contraction property (4.2) for the case where both solutions are piecewise smooth. Suppose that the two solutions cross at x = xj (t), j = 0, ±1, ±2, . . . : (4.3) u1 (x, t) ≥ u2 (x, t) for xj (t) < x < xj+1 (t), j even; u2 (x, t) ≥ u1 (x, t) for xj (t) < x < xj+1 (t), j odd. By the entropy condition for a convex flux, f (u) > 0, a shock (u− , u+ ) jumps down, u− > u+ . Thus at a crossing x = xj (t) for j even, the general situation is that u1 (x, t) is continuous at xj (t) and u2 (x, t) either is also continuous to intersect u1 (x, t) there or jumps down to cross u1 (x, t) there. An analogous situation holds at a crossing x = xj (t) for j odd. Let qj± be the jumps between the solutions at the crossings, as shown in figure 3.07: (4.4) qj± ≡ ± u1 (xj (t), t) − u2 (xj (t) ± 0, t) ≥ 0, for j even; qj± ≡ ± u2 (xj (t), t) − u1 (xj (t) ± 0, t) ≥ 0, for j odd. Note that qj± = 0 if both solutions are continuous at the crossing x = xj (t). Theorem 4.1. Suppose that ui (x, t), i = 1, 2, are two piecewise solutions of convex conservation law ut + f (u)x = 0 and there exist constants u± and M > 0 such that u1 (x, 0) = u2 (x, 0) = u± for ±x > M . Then there exist bounded positive functions Oj (1) such that d ∞ |u2 (x, t) − u1 (x, t)| dx = − Oj (1)qj− qj+ ≤ 0, (4.5) dt −∞ j
where the summation is over the crossings of the two solutions, (4.3) and (4.4). Proof. Consider the case where the two solutions cross at their continuity points, say u1 (xj (t)) = u2 (xj (t)) and u1 (xj+1 (t)) = u2 (xj+1 (t)) for j even, and u1 (x, t) > u2 (x, t) for xj (t) < x < xj+1 (t). We are dealing with weak solutions and so u1 (x, t) and u2 (x, t) may contain shock waves in the interval xj (t) < x < xj+1 (t). Thus we apply the integral conservation law, (3.1) of Chapter 2, for weak solutions. The conservation law (3.1) in Definition 3.1 of Chapter 2 is for a fixed interval of integration. For a varying interval of
34
3. Scalar Convex Conservation Laws
integration here, d xj+1 (t) |u1 (x, t) − u2 (x, t)| dx dt xj (t) d xj+1 (t) d xj+1 (t) u1 (x, t) dx − u2 (x, t) dx = dt xj (t) dt xj (t) x=xj+1 (t) + f (u1 (xj (t), t)) − f (u1 (xj+1 (t), t)) = x (t)u1 (x, t) x=xj (t)
x=xj+1 (t) + f (u2 (xj (t), t)) − f (u2 (xj+1 (t), t)) , − x (t)u2 (x, t)
x=xj (t)
which is zero as we have assumed that u1 (x, t) = u2 (x, t) at x = xj (t) and at x = xj+1 (t).
u(x, t)
u2
q−
u− u q+
u1
x2m (t)
u+ x2 m+1(t)
x
Figure 3.07. L1 contraction.
The situation is different when one of solutions has a shock wave at the crossing. Suppose that u1 (x, t) jumps down at x2m+1 (t) with states u± ≡ u1 (x2m+1 (t) ± 0, t) and speed s = x2m+1 (t), and that u2 (x, t) is continuous there, u ≡ u2 (x2m+1 (t), t), as shown in figure 3.07. Applying the integral conservation law as above, the new input to d ∞ |u2 (x, t) − u1 (x, t)| dx dt −∞ due to the shock wave at the crossing x = x2m+1 (t) has two parts. The input from the integration to the left of x = x2m+1 (t) is su1 (x2m+1 (t) − 0, t) − f (u1 (x2m+1 (t) − 0, t)) − su2 (x2m+1 (t) − 0, t) + f (u2 (x2m+1 (t) − 0, t)) = su− − f (u− ) − su + f (u);
4. Well-Posedness Theory
35
the input from the integration to the right of x = x2m+1 (t) is, noting that u2 > u1 to the right of x = x2m+1 (t), − −su1 (x2m+1 (t) + 0, t) − f (u1 (x2m+1 (t) + 0, t)) + su2 (x2m+1 (t) + 0, t) + f (u2 (x2m+1 (t) + 0, t)) = su+ − f (u+ ) − su + f (u); and the total new input is (4.6) su− − f (u− ) − su + f (u) + su+ − f (u+ ) − su + f (u) = 2s(u− − u) − 2(f (u− ) − f (u)), d where s = dt x2m+1 (t) is the speed of the shock (u− , u+ ) and the RankineHugoniot condition s(u+ − u− ) = f (u+ ) − f (u− ) has been used. By the Rankine-Hugoniot condition, the new input is 2s(u− −u)−2(f (u− )−f (u)) = 2s(u+ − u) − 2(f (u+ ) − f (u)) where u+ ≤ u ≤ u− . We have
s(u − u+ ) − (f (u) − f (u+ )) = (u − u+ )(g(u− ) − g(u)), g(u) ≡
f (u) − f (u+ ) . u − u+
By the mean value theorem, g(u− )−g(u) = g (¯ u)(u− −u) for some u ¯ between ˆ between u ¯ and u and u− , and, by the mean value theorem again, for some u u+ , f (¯ f (ˆ u)(¯ u − u+ ) − f (¯ u) + f (u+ ) u) . u) = = − g (¯ (¯ u − u+ )2 2 Thus we conclude that the new input at the crossing x = x2m+1 (t) is u)(u − u+ )(u− − u) = −Oj (1)qj+ qj− . 2s(u− − u) − 2(f (u− ) − f (u)) = −f (ˆ This completes the proof of the theorem.
We will see in the next subsection that the L1 (x) contraction property holds not only for piecewise smooth solutions, but also for general solutions; see Theorem 4.3. 4.2. Approximation by Piecewise Smooth Solutions The L1 (x) contraction property (4.5) of the solution operator has been shown only for piecewise smooth solutions. We now show that it also holds for any general solution with bounded initial data. This follows from the following theorem. Theorem 4.2. Suppose that u0 (x) is in the function space F defined by (4.1), for some constant states ul and ur . Then the initial value problem for the conservation law ut + f (u)x = 0, f (u) = 0, with the initial values
36
3. Scalar Convex Conservation Laws
u(x, 0) = u0 (x) has a global-in-time solution u(x, t), which can be approximated by piecewise smooth solutions un (x, t), n = 1, 2, . . . , in the L1 (x) norm: ∞ (4.7) |u(x, t) − un (x, t)| dx ≤ 2−n , n = 1, 2, . . . , t ≥ 0. −∞
Moreover, each approximate solution un (x, t) equals ul (or ur ) for x < −Mn − O(1)t (or x > Mn + O(1)t) for some constant Mn > 0, and the number of shock waves in un (x, t) is finite and non-increasing in time. Proof. Since u(x, 0) = u0 (x) is in the function space F defined by (4.1), there exists Mn > 0 such that ∞ −Mn |u(x, 0) − ul | dx + |u(x, 0) − ur | dx ≤ 2−(n+1) , n = 1, 2, . . . . −∞
Mn
Approximate u(x, 0) for |x| < Mn , by step functions un (x, 0) in L1 (x): Mn |u(x, 0) − un (x, 0)| dx ≤ 2−(n+1) , n = 1, 2, . . . . −Mn
Extend the approximate solution un (x, 0) to x ∈ R: un (x, 0) ≡ ul for x < −Mn ;
un (x, 0) ≡ ur for x > Mn .
From the above two estimates, ∞ |u0 (x) − un (x, 0)| dx < 2−n , n = 1, 2, . . . . −∞
By Theorem 3.1, the initial function un (x, 0) yields a piecewise smooth solution un (x, t), t ≥ 0, with the number of shock waves finite and non-increasing in time. The L1 (x) contraction property (4.5) applies to the piecewise smooth solutions and we have, for m, n = 1, 2, . . . and t ≥ 0, ∞ ∞ n m |u (x, t) − u (x, t)| dx ≤ |un (x, 0) − um (x, 0)| dx −∞ −∞ ∞ ∞ ≤ |un (x, 0) − u0 (x)| dx + |u0 (x) − um (x, 0)| dx ≤ 2−n + 2−m . −∞
−∞
un (x, t),
n = 1, 2, . . . , form a Cauchy seThus the approximate solutions quence in L1 (x). As n → ∞, the sequence converges to a function u(x, t) with the property (4.7). Each approximate solution un (x, t) is constructed as a weak solution: ∞ ∞ ∞ n n un (x, 0)φ(x, 0) dx = 0 u φt + f (u )φx (x, t) dx dt + 0
−∞
0
4. Well-Posedness Theory
37
for any test function φ(x, t). Consequently, the limiting function is also a weak solution:
∞ u0 (x)φ(x, 0) dx uφt + f (u)φx (x, t) dx dt + −∞ 0 ∞ ∞ (u − un )φt + (f (u) − f (un ))φx (x, t) dx dt = 0 −∞ ∞ (u0 (x) − un (x, 0))φ(x, 0) dx ≤ O(1)N T 2−n → 0 as n → ∞. +
∞ ∞ 0
0
Here T > 0 is chosen so that the test function φ(x, t) = 0 for t > T , and N is chosen so that |φ(x, t)|, |φt (x, t)|, |φx (x, t)| < N. Thus the limiting function u(x, t) is a weak solution. This completes the proof of the theorem. Theorem 4.3. Suppose that u(x, 0) and v(x, 0) are two bounded functions in the space F defined by (4.1) for some constants ul and ur . Then there exist weak solutions u(x, t) and v(x, t), t ≥ 0, of the convex conservation law ut + f (u)x = 0 such that
∞
(4.8) −∞
|u(x, t2 ) − v(x, t2 )| dx ≤
∞ −∞
|u(x, t1 ) − v(x, t1 )| dx, 0 ≤ t1 ≤ t2 .
Moreover, if u(x, t) and v(x, t) are two weak solutions which can be approximated by piecewise smooth solutions in L1 (x) uniformly in t, then (4.8) holds. Proof. According to Theorem 4.2, there exist piecewise smooth solutions un (x, t) and v n (x, t), n = 1, 2, . . . , which approach in L1 (x), respectively, weak solutions u(x, t) and v(x, t) as n → ∞:
∞ −∞
|u(x, t) − u (x, t)| dx ≤ 2 n
−n
∞
, −∞
|v(x, t) − v n (x, t)| dx ≤ 2−n , t ≥ 0, n = 1, 2, . . . .
Theorem 4.1 is applicable to the piecewise smooth solutions un (x, t) and v n (x, t):
∞ −∞
|u (x, t2 ) − v (x, t2 )| dx ≤ n
n
∞ −∞
|un (x, t1 ) − v n (x, t1 )| dx, n = 1, 2, . . . .
38
3. Scalar Convex Conservation Laws
Consequently, the L1 (x) contraction property holds also for the exact solutions: ∞ ∞ |u(x, t2 ) − v(x, t2 )| dx ≤ |u(x, t2 ) − un (x, t2 )| dx −∞ −∞ ∞ ∞ |v n (x, t2 ) − v(x, t2 )| dx + |un (x, t2 ) − v n (x, t2 )| dx + −∞ −∞ ∞ |un (x, t1 ) − v n (x, t1 )| dx ≤ 2 · 2−n + −∞ ∞ ∞ −n n |u(x, t1 ) − u (x, t1 )| dx + |v(x, t1 ) − v n (x, t1 )| dx ≤2·2 + −∞ −∞ ∞ + |u(x, t1 ) − v(x, t1 )| dx −∞ ∞ ∞ −n |u(x, t1 ) − v(x, t1 )| dx → |u(x, t1 ) − v(x, t1 )| dx, ≤4·2 + −∞
−∞
upon letting n → ∞. This completes the proof of the first part of the theorem. The second part is proved similarly. A simple and immediate consequence of the proof of the above theorem is the following corollary. Corollary 4.4. The L1 (x) contraction property (4.8) holds for any two solutions u(x, t) and v(x, t) which can be approximated by piecewise smooth solutions in L1 (x) unformly in time t. Theorem 4.3 and Corollary 4.4 provide an existence and continuous dependence theory for the general initial value problem. They give uniqueness theory for the solutions as limits of piecewise smooth solutions. Existence theory obtained in the zero dissipation limit would yield such a well-posedness theory, as viscous solutions are smooth; see e.g. Section 6 of Chapter 5. We have the following general remark. Remark 4.5. The procedure in the proofs of Theorem 4.2 and Theorem 4.3 offers a concrete algorithm for reducing the analysis for general solutions to that for piecewise smooth solutions. In the study of qualitative and quantitative properties of general solutions in the subsequent sections of this chapter, we will therefore focus only on piecewise smooth solutions. 4.3. Lp (x) Spaces The L1 (x) norm is the natural one for conservation laws for two reasons. The first reason is physical. The conservation law ut + f (u)x = 0 governs
4. Well-Posedness Theory
39
the evolution of the density u(x, t) of some physical quantity and x2 u(x, t) dx = mass over an interval (x1 , x2 ). x1
We have the integral conservation law, (3.1) of Chapter 2, d x2 u(x, t) dx = f (u(x1 , t)) − f (u(x2 , t)). dt x1 The goal is to study the distribution of the density u(x, t). The L1 (x) distance measures the total difference of the distribution of the mass between the two solutions. For the well-posedness theory, the L1 (x) distance is therefore a natural choice. The second reason is analytical. The solution operator is not well-posed for other commonly used norms, such as the Lp (x) norms for p > 1, defined as follows: ∞ g − h Lp (x) ≡
−∞
|g(x) − h(x)|p dx
1/p
.
A small distance in Lp (x) between two initial functions u1 (x, 0) and u2 (x, 0) can induce a proportionally large distance between the solutions u1 (x, t) and u2 (x, t) at a short time t later, 0 < t 1. We illustrate this for the L2 (x) norm by comparing ∞ ∞ 2 2 u2 (x, t) − u1 (x, t) dx with u2 (x, 0) − u1 (x, 0) dx. −∞
−∞
Consider the example of the compression wave (f (ul ), f (ur )) = (λ0 , −λ0 ) for a positive constant λ0 : ⎧ ⎪ ⎨λ0 for x < −λ0 δ, f (u1 )(x, 0) = − xδ for − λ0 δ ≤ x ≤ λ0 δ, ⎪ ⎩ −λ0 for x > λ0 δ, so that the characteristics focus at x = 0 at time δ and the solution is a shock wave ([f ]−1 (λ0 ), [f ]−1 (−λ0 )) for t > δ. Let the second solution be a translation of the first solution, u2 (x, t) = u1 (x + ε, t). We now compute their difference at time t = 0 and t = δ. At t = 0, their difference is of the order of ε times the x-derivative of the compression wave. As the derivative is of the order of 1/δ, we have λ0 δ ∞ 2 ε2 1 2 dx = O(1)λ0 . u2 (x, 0) − u1 (x, 0) dx = O(1) ε· δ δ −∞ −λ0 δ At time t = δ, u1 consists of the shock (ul , ur ) located at x = 0 and u2 consists of the same shock located at x = ε, and so ∞ 2 u2 (x, δ) − u1 (x, δ) dx = ε(ur − ul )2 = O(1)ε(λ0 )2 . −∞
40
3. Scalar Convex Conservation Laws
Thus the ratio of the L2 (x) norm at time t = δ to that at time t = 0 is ε2 λ 1 12 0 2 2 O(1) λ0 / ε(λ0 ) = O(1)ε , δ δ which can be arbitrarily large when ε δ 1/2 . The same conclusion holds for any Lp (x) norm with p > 1. The L2 (x) norm corresponds to the entropy estimates (2.8)–(2.11); see also Section 7 of this chapter. The above example shows that the difference of entropy between two solutions can increase drastically. There is no natural physical reason for considering the entropy distance between the solutions. On the other hand, for linear equations, the entropy estimates for a single solution carry over to the difference of two solutions by the linear superposition principle; see, for instance, Section 4.1 of Chapter 4 on the heat equation. Thus it is the nonlinearity of the flux that causes the initial value problem to be not well-posed in the spaces Lp (x), p > 1.
5. Generalized Characteristics, Nonlinear Regularization As pointed out in Remark 4.5, for the analysis of properties of a general solution in this and the following sections, we need only focus on piecewise smooth solutions. We will start with the notion of generalized characteristics, which is an effective tool for studying the solution behavior. We then study the regularization effect of the nonlinear flux and the time-asymptotic stability of shock and rarefaction waves. 5.1. Generalized Characteristics The method of characteristics, introduced in Section 1 of Chapter 2, works for smooth solutions of conservation laws. With shock waves, there is the notion of generalized characteristics, which is useful for studying quantitative properties of weak solutions. A generalized characteristic curve follows the characteristic line through a given initial location (x, t) until it hits a shock wave, and then it follows the shock curve. In fact, as shock waves are compressive, when a characteristic line hits a shock, there is no characteristic line to follow and only the shock curve to follow in the forward time direction, as shown in figure 3.08. On the other hand, a characteristic line does not hit a shock wave as it moves backward in time and therefore always can reach the initial time t = 0. Through a location (x, t) on a shock wave, there are two characteristic lines to follow backward in time, either the characteristic line on the right or that on the left side of the shock. These backward characteristics are the characteristic lines; see figure 3.08. Following Remark 4.5, we will consider only piecewise smooth solutions, for which the notion of generalized characteristics can be described easily, as
5. Generalized Characteristics, Nonlinear Regularization
41
t
generalized characteristic backward characteristic
x Figure 3.08. Generalized characteristics and backward characteristics.
done above. It takes some analytical consideration to define the notion for general weak solutions. We will take up this topic again when we consider systems in Chapter 9. 5.2. Functions of Bounded Variation, Regularization of Solution Operator We now use the idea of generalized characteristics to study the regularity of solutions. In Theorem 5.3, although the initial data u(x, 0) is assumed to be bounded, |u| < M , it is allowed to be very oscillatory. The theorem shows that the nonlinearity of the flux, f (u) = 0, has a strong regularizing effect in damping out the oscillation of the initial data u(x, 0) and making the solution of locally bounded variation. The notion of variation of a solution is of significance for conservation laws, as we are interested in wave propagation in the solution. A solution of bounded variation contains a finite number of shock waves with finite strength, and the rest is almost piecewise smooth. This is the scenario we have in mind for actual physical situations. Definition 5.1. The variation Var(h) = Var(h; a, b) of a scalar function h(x) over an interval [a, b] is the upper limit Var(h; a, b) ≡ lim sup
n
|h(xi ) − h(xi−1 |,
i=1
where the sup is taken over all partitions P ≡ (x0 , x1 , . . . , xn−1 , xn ) with a = x0 < x1 < · · · < xn = b, of [a, b]. The function h(x), −∞ < x < ∞, is of locally bounded variation if its variation over any bounded interval is
42
3. Scalar Convex Conservation Laws
finite; it is of bounded total variation if its variation over (−∞, ∞) is finite. A vector function h(x) = (h1 (x), . . . , hm (x)) is of bounded variation if each of its components hj (x), j = 1, . . . , m, is of bounded variation. The following is the characterizing property of functions with bounded variation. Proposition 5.2. A scalar function is of bounded variation over an interval [a, b] if and only if it can be written as the difference of two monotonically non-decreasing bounded functions h(x) = h+ (x) − h− (x), h± (x)
non-decreasing functions.
The variation Vari (h) (or Vard (h)) of h+ (x) (or h− (x)) is called the increasing (or decreasing) variation of h(x). Var(h) = Vari (h) + Vard (h) is the variation of h(x). Proof. Given x such that a≤x≤b and a partition P ≡(x0 , x1 , . . . , xn−1 , xn ), with a = x0 < x1 < · · · < xn = x, of [a, x], set n n + − − (x) ≡ ) − h(x ) , h (x) ≡ − h(x h(xj ) − h(xj−1 ) , h+ j j−1 P P j=1
where
j=1
h(xj ) − h(xj−1 ) for h(xj ) − h(xj−1 ) > 0, 0 for h(xj ) − h(xj−1 ) ≤ 0;
− h(xj ) − h(xj−1 ) for h(xj ) − h(xj−1 ) < 0, h(xj ) − h(xj−1 ) ≡ 0 for h(xj ) − h(xj−1 ) ≥ 0.
+ h(xj ) − h(xj−1 ) ≡
The proposition follows by setting h+ (x) ≡ sup h+ P (x), P
h− (x) ≡ sup h− P (x), P
A continuous function over a finite interval can be of infinite variation; a simple example is π h(x) = x cos , 0 < x ≤ 1, h(0) = 0. x Take x0 = 0, x1 = 1/n, x2 = 1/(n − 1), . . . , xn = 1, so that |h(xi ) − h(xi−1 | = 1/(n − i + 1) + 1/(n − i + 2) and n 1 1 1 + → ∞ as n → ∞ |h(xi ) − h(xi−1 | = 1 + 2 + · · · + 2 2 n−1 n i=1
because of the divergence of the harmonic series. Thus the function h(x) is highly oscillatory around x = 0 and is of infinite variation.
5. Generalized Characteristics, Nonlinear Regularization
43
The next theorem shows that the solution operator for convex conservation laws damps the oscillation of the initial function immediately, and the solution becomes of local bounded variation for any positive time. Even for more oscillatory initial functions such as u(x, 0) = x cos(π/x2 ), the convex flux forces shock waves to occur immediately in the solution u(x, t), t > 0, and the shock waves cause cancellations and the damping of oscillations. Theorem 5.3. Suppose that the initial data is bounded, |u(x, 0)| ≤ M , and measurable. Then the solution u(x, t) to the convex scalar conservation law ut + f (u)x = 0, f (u) = 0, is of locally bounded variation for any positive time t > 0. Moreover, the total variation of u(x, t), t > 0, is uniformly bounded over an interval of length proportional to time. Precisely, we have, for any a, (5.1)
Var(u(·, t); a, a + t) ≤
2 + A1 M , A2
where A1 ≡ max{f (u), −M ≤ u ≤ M },
A2 ≡ min{f (u), −M ≤ u ≤ M }.
Proof. We only consider piecewise smooth solutions. Extension to general bounded solutions can be done by following the general principle of Remark 4.5. The constants A1 and A2 measure the strength of nonlinearity of the flux function f (u). We have from the mean value theorem that the variation of the solution u(·, t) is of the same order as that for the characteristic speed λ(u(·, t)): (5.2)
Var(λ(u)) ≤ A1 Var(u),
A2 Var(u) ≤ Var(λ(u)).
For each fixed location (x, t), draw the backward characteristic to reach the initial time at (x − f (u(x, t))t, 0) so that u(x, t) = u0 (x − f (u(x, t))t). In particular, the solution is also bounded, |u(x, t)| = |u0 (x−f (u(x, t))t)| ≤ M for t ≥ 0. For fixed positive t, divide a given interval a ≤ x ≤ b so that the characteristic speed λ(x) = λ(u(x, t)) = f (u(x, t)) is non-decreasing (or non-increasing) over (xj , xj+1 ) for j odd (or j even), j = 0, . . . , n, where a = x0 < x1 < · · · < xn+1 = b. Through (xj + 0, t) (or (xj+1 − 0, t)), for j odd, draw the characteristic line χj (or χj+1 ) backward in time with speed λj = λ(xj + 0, t) (or λj+1 = λ(xj+1 − 0, t)). The distance between consecutive characteristics is Dj (t) = Dj (0) + (λj+1 − λj )t and so Dj (t) ≥ (λj+1 − λj )t. b−a≥ j odd
j odd
44
3. Scalar Convex Conservation Laws
The increasing variation Vari (λ) of λ(x), x ∈ (a, b), is λj+1 − λj . Vari (λ) = j odd
From the previous two relations we conclude that b−a . Vari (λ) ≤ t The total variation Var(λ) of λ(x, t) = f (u(x, t)) over the interval x ∈ (a, b) is Var(λ) = Vari (λ) + Vard (λ).
(5.3)
On the other hand, λ(x, t) = λ+ (x, t) − λ− (x, t) and Vari (λ) = λ+ (b, t) − λ+ (a, t), Vard (λ) = − λ− (b, t) − λ− (a, t) . From these we have Vari (λ) − Vard (λ) = λ(b, t) − λ(a, t). Therefore
b−a − λ(b, t) − λ(a, t) . (5.4) Var(λ) = 2Vari (λ) − λ(b, t) − λ(a, t) ≤ 2 t Since the solution is uniformly bounded, |u(x, t)| < M , we have |λ(b, t) − λ(a, t)| ≤ 2A1 M . We thus conclude from the above estimate that b−a + 2A1 M. Var(λ) ≤ 2 t This and the estimate (5.2) yield, for b = a + t, b−a A2 Var(u) ≤ (2 + 2A1 M ) = 2 + 2A1 M. t This completes the proof of the theorem. Corollary 5.4. Suppose that the initial function u(x, 0) = u0 (x) is bounded and periodic, u(x, 0) = u(x + X, 0) for x ∈ R, with period X. Then the variation Var(t) = Var(u(·, t)) of the solution over one period decays at the rate of 1/t: X (5.5) Var(t) = O(1) , t > 0. t Proof. Following the proof of Theorem 5.3 above, we take the interval (a, b) = (0, X). It is clear that the solution is periodic and so λ(a) = λ(b). Thus from (5.4), 2X b−a = . Var(λ) ≤ 2 t t This proves the corollary. Note here that the decay rate is 1/t but the coefficient is small for a small period X. A smaller period implies stronger wave cancellation and therefore yields faster decay.
5. Generalized Characteristics, Nonlinear Regularization
45
5.3. Time-Asymptotic Stability of Shock and Rarefaction Waves An important simplifying mechanism of the solution operator due to the nonlinearity of the flux function is that the time-asymptotic behavior of a general solution u(x, t) is determined only by the far states u± = u(±∞, 0) of its initial data u(x, 0). For simplicity of the analysis we assume that the initial data is bounded and assumes the constant states u± outside a bounded interval −M < x < M : ut + f (u)x⎧= 0, f (u) > 0, ⎪ ⎨u− for x < −M, u(x, 0) = u0 (x) for |x| ≤ M, ⎪ ⎩ u+ for x > M, |u0 (x)| ≤ N.
(5.6)
The convexity of the flux function has strong implications for the timeasymptotic stability of shock waves. A compactly supported perturbation of a shock wave converges to the shock wave in finite time. Theorem 5.5. When u− > u+ , the solution of the initial value problem (5.6) approaches the shifted centered shock (u− , u+ ) in finite time,
u− for x < x0 + st, u(x, t) = u+ for x > x0 + st, t > T, and the shock formation time T and the shock shift x0 satisfy, for some β > 0, M , |u+ − u− |β ∞ 0 1 (u0 (x) − u− ) dx + (u0 (x) − u+ ) dx . x0 = u− − u+ −∞ 0
(5.7) T = O(1)
Proof. As before we only consider piecewise smooth solutions, with the extension to general bounded solutions achieved by following the general principle of Remark 4.5. Thus we may draw the generalized characteristics and backward characteristics. Draw generalized characteristics C1 :x = x1 (t) through (−M, 0) and C2 :x = x2 (t) through (M, 0), and let D(t) ≡ x2 (t) − x1 (t) be the distance between them; see figure 3.09. The generalized characteristic x = x1 (t) may be a characteristic line with speed f (u− ) or may move along a shock (u− , u1 (t)). In the former case we set u1 (t) = u− so that the shock speed degenerates to f (u− ). Similarly, the generalized characteristic x = x2 (t) moves along a shock (u2 (t), u+ ). Through the generalized characteristics (x1 (t)+0, t) and (x2 (t)−0, t) at time t we draw the backward
46
3. Scalar Convex Conservation Laws
characteristics, as in figure 3.08, B1 ≡ {(y, s) : y = x1 (t) − f (u1 (t))(t − τ ), 0 < τ < t}, B2 ≡ {(y, τ ) : y = x2 (t) − f (u2 (t))(t − τ ), 0 < τ < t}. The characteristics B1 and B2 meet the initial time t = 0 in the interval (−M, M ), and so x2 (t) − f (u2 (t))t − x1 (t) − f (u1 (t))t ≤ 2M. Thus, for some function function O(M ) bounded by 2M, the distance D(t) satisfies (5.8) D(t) = x2 (t) − x1 (t) = O(M ) + f (u2 (t)) − f (u1 (t)) t. On the other hand, from the Rankine-Hugoniot condition we have D (t) =
f (u+ ) − f (u2 (t)) f (u− ) − f (u1 (t)) − . u+ − u2 (t) u− − u1 (t)
By the entropy condition, u1 (t) ≤ u− and u2 (t) ≥ u+ , and so f (u+ ) − f (u− ) < D (t) < f (u2 (t)) − f (u1 (t)).
t u+
u− C1 : x = x1(t) u1(t)
B1
−M
C2 : x = x2 (t) D (t)
u2(t)
B2
M
x
Figure 3.09. Convergence to shock wave.
Note that the solution u(x, t) is bounded, |u(x, t)| < N , and so, by the convexity f (u) > 0 and the hypothesis u− > u+ , there exists β independent of t, with 0 < β < 1, such that D (t) ≤ (1 − β) f (u+ ) − f (u− ) + β f (u2 (t)) − f (u1 (t)) . From this and (5.8), we obtain a differential inequality for D(t): D(t) − O(M ) + (1 − β) f (u+ ) − f (u− ) . (5.9) D (t) ≤ β t
5. Generalized Characteristics, Nonlinear Regularization
47
Multipy (5.9) by the integrating factor t−β to obtain d −β O(M ) (t D(t)) ≤ t−β −β + (1 − β) f (u+ ) − f (u− ) . dt t This is integrated from t = 1 to yield t −βO(M )s−β−1 + (1 − β)s−β f (u+ ) − f (u− ) ds t−β D(t) ≤ D(1) + 1 = O(1) + f (u+ ) − f (u− ) t1−β . This yields an estimate of D(t):
D(t) ≤ O(1)tβ + f (u+ ) − f (u− ) t.
As β < 1 and f (u+ ) − f (u− ) < 0, we have D(t) = 0 in finite time t = T, with T satisfying (5.7), and the solution consists of a single shock for t > T :
u− for x < x0 + st, u(x, t) = u+ for x > x0 + st, t > T, for some phase shift x0 . The phase shift x0 can be determined through the conservation law ∞ d st (u(x, t) − u− ) dx + (u(x, t) − u+ ) dx = 0, dt −∞ st which follows from the integral conservation law (3.1) of Chapter 2 for weak solutions. Evaluating the conserved quantity at the initial time t = 0 and after the time t = T , we have ∞ 0 (u0 (x) − u− ) dx + (u0 (x) − u+ ) dx = x0 (u− − u+ ), −∞
0
which yields the formula for the phase shift x0 in (5.7). This completes the proof of the theorem. We next consider the case u− < u+ . The solution will be shown to converge to the centered rarefaction wave, (1.6), ⎧ ⎪ ⎨f (u− ), x < f (u− )t, centered rarefaction wave. f (uR )(x, t) = xt , f (u− )t < x < f (u+ )t, ⎪ ⎩ f (u+ ), x > f (u+ )t, Unlike the case of the shock wave, Theorem 5.5, the convergence of the solution to the rarefaction wave takes infinite time. Theorem 5.6. When u− < u+ , the solution of the initial value problem (5.6) approaches the centered rarefaction wave uR in the sense that there exist two curves Ci : {(x, t), x = xi (t)}, i = 1, 2, enclosing the wave region
48
3. Scalar Convex Conservation Laws
of the rarefaction wave uR , x1 (t) ≤ f (u− )t ≤ f (u+ )t ≤ x2 (t), with the following properties: (1) The convergence is at a linear rate in the wave region: 1 |u(x, t) − uR (x, t)| = O(1) for f (u− )t < x < f (u+ )t. t+1 (2) The distance between Ci , i = 1, 2, and the edges of the rarefaction wave uR is of the order t1/2 , f (u− )t − x1 (t) + x2 (t) − f (u+ )t = O(1)(t + 1) 2 , 1
and the convergence rate is of order 1/2 outside the wave region of uR : |u(x, t) − u− | = O(1)(t + 1)− 2 for x1 (t) < x < f (u− )t, 1 |u(x, t) − u+ | = O(1)(t + 1)− 2 for f (u+ )t < x < x2 (t). 1
(3) Outside the region between C1 and C2 , the solution equals the end states:
u− for x < x1 (t), u(x, t) = u+ for x > x2 (t). Consequently, u(x, t) − uR (x, t) Lp (x) = O(1)(t + 1)
− p−1 2p
, p ≥ 1.
Proof. As in the proof of Theorem 5.5 above, we need to consider only piecewise smooth solutions. Draw the generalized characteristics C1 : x = x1 (t) and C2 : x = x2 (t), and let u1 (t) ≡ u(x1 (t) + 0, t), and u2 (t) ≡ u(x1 (t) − 0, t) be the states next to the generalized characteristics; see figure 3.10. When C1 contains no shock, u− = u1 (t), it propagates with speed f (u− ). When C1 contains a shock (u− , u1 (t)), u− > u1 (t), its speed is less than f (u− ) by the entropy condition. Thus the speed of C1 is less than or equal to the speed f (u− ) of the left edge x = f (u− )t of the rarefaction wave uR . C1 starts at (−M, 0) at time t = 0. Consequently, C1 is to the left of the left edge x = f (u− )t of the rarefaction wave uR . A similar situation holds for C2 , and we have x1 (t) ≤ f (u− )t < f (u+ )t ≤ x2 (t). Choose any location (x, t) between C1 and C2 and draw the backward characteristic line with speed f (u(x, t)) through (x, t). The characteristic line meets a point (¯ x, 0) at the initial time, as shown in figure 3.10, where x=x ¯ + f (u(x, t))t. As |¯ x| < M , we deduce from the form of the rarefaction wave uR (x, t) that ¯ x x 1 (5.10) f (u(x, t)) = − = f (uR )(x, t) + O(M ) , x1 (t) < x < x2 (t). t t t This estimate implies that the distance between the solution u(x, t) and the centered rarefaction wave uR (x, t) decays at the rate of 1/t for (x, t) in the
5. Generalized Characteristics, Nonlinear Regularization
49
region x1 (t) < x < x2 (t). In particular, this yields the first statement of the theorem. With the estimate (5.10), the first estimate in (2) implies the second estimate in (2). Thus it remains to show that the distance between C1 (or C2 ) and x− (t) ≡ f (u− )t (or x+ (t) ≡ f (u+ )t) is of the order of (t + 1)1/2 . That is, we need to show that (5.11) E(t) ≡ D(t) − f (u+ ) − f (u− ) t is of order (t + 1)1/2 . By the backward characteristic analysis, as in the stability analysis for shock waves above, the distance D(t) ≡ x2 (t) − x1 (t) between the generalized characteristics C1 and C2 satisfies (5.12) D(t) = f (u2 (t)) − f (u1 (t)) t + O(M ), with O(M ) ≤ 2M.
t x = x1(t)
u1 (t)
D(t)
u2 (t)
(x, t)
x = x2(t)
u+
u− C1
C2 −M
(¯ x, 0)
M
x
Figure 3.10. Convergence to rarefaction wave.
By the Rankine-Hugoniot condition, E (t) =
f (u+ ) − f (u2 (t)) f (u1 (t)) − f (u− ) − − f (u+ ) − f (u− ) . u+ − u2 (t) u1 (t) − u−
From the entropy condition and our hypothesis u− < u+ of expansion, we have u1 (t) ≤ u− < u+ ≤ u2 (t), and so 0 ≤ E (t) < f (u2 ) − f (u1 ) − f (u+ ) − f (u− ) . By the convexity of the flux f (u) > 0, we have from (5.11) and (5.12) that E(t) + O(M ) E (t) ≤ α f (u2 ) − f (u+ ) + f (u− ) − f (u1 ) = α t for some constant α, 0 < α < 1. This can be solved to yield the growth rate of tα for E(t) and the decay rate tα−1 of the strength of the shocks (u− , u1 (t)) and (u2 (t), u+ ) on the generalized characteristics C1 and C2 . As
50
3. Scalar Convex Conservation Laws
the shocks decay, the constant α becomes close to 1/2, and the shock speeds are approximated by f (u+ ) − f (u2 (t)) f (u+ ) + f (u2 (t)) = + O(1)(u2 (t) − u+ )2 , u+ − u2 (t) 2 f (u1 (t)) + f (u− ) f (u1 (t)) − f (u− ) + O(1)(u− − u1 (t))2 . = u1 (t) − u− 2 Thus we have E(t) + O(M )t−1+β , 2t for β close to zero. This is integrated from some positive t0 to yield E (t) =
1
E(t) = O(1)(t + 1) 2 . This completes the proof of the pointwise estimate part of the theorem. Note that the shocks on the edges x = x1 (t) and x = x2 (t) decay at the rate of 1 |u− − u1 (t)| + |u2 (t) − u+ | = O(1)(t + 1)− 2 , which is slower than the rate of t−1 of convergence in the wave region. The Lp (x) decay follows immediately from the pointwise estimate. This completes the proof of the theorem.
6. N -Waves, Inviscid Dissipation For solutions of ut + f (u)x = 0 which tend to zero at x = ±∞, there is the conservation of total mass: ∞ ∞ d ∞ u(x, t) dx = 0, or u(x, t) dx = u(x, 0) dx, for any t ≥ 0. dt −∞ −∞ −∞ This follows from the integral conservation law, Definition 3.1 of Chapter 2. For linear equations there are infinitely many conserved quantities; for instance, the entire initial profile u(x, 0) = u0 (x) is preserved: ut + aux = 0,
u(x, t) = u0 (x − at).
As we will see in the next chapter, Chapter 4, for dissipative equations there is only one conservation law, the total mass. A striking fact in shock wave theory is that the solution to the convex conservation law has exactly two time invariants, one besides the total mass. This is the theory of N -waves. We start with the explicit construction of N -waves for the Hopf equation with initial data being a multiple of the delta function:
2 ut + ( u2 )x = 0, (6.1) u(x, 0) = cδ(x).
6. N -Waves, Inviscid Dissipation
51
In the terminology of linear partial differential equations, the solution may be called the fundamental solution for the Hopf equation. As the conservation law is nonlinear, there is ambiguity in talking about generalized functions such as the delta function. The theory of distributions is intrinsically a linear notion. For instance, nonlinear operations such as squaring the delta function are not defined within the theory, though one may resort to nonstandard analysis. For now, we only need the two defining properties of the delta function δ(x), as described in Remark 3.4 of Chapter 2: ∞ δ(x) dx = 1. (6.2) δ(x) = 0 for x = 0, −∞
We will interpret the solution u(x, t) of (6.1) as being a weak solution of ut + (u2 /2)x = 0 for positive time t > 0, and it tends to cδ(x) in the above sense as time tends to zero: ∞ u(x, t) dx = c for t > 0. (6.3) lim u(x, t) = 0 for x = 0, and t→0+
−∞
Here we have used the usual notation t → 0+ to mean that t → 0 and t > 0. It turns out that there are infinitely many solutions to (6.1) and (6.3). They are the centered N -waves u(x, t) = Np,q (x, t), depending on two parameters p ≤ 0 ≤ q, an example of which is shown figure 3.11: ⎧ √ ⎪ − −2pt, ⎨0, x < √ √ (6.4) Np,q (x, t) = xt , − −2pt < x < 2qt, ⎪ √ ⎩ 0, x > 2qt. The N -wave Np,q (x, t) consists of a centered rarefaction wave sandwiched √ √ by the shock waves at x = − −2pt on the left and at x = 2qt on the right; see figure 3.11. We claim that this is the solution of (6.1) and (6.3) if p + q = c. The locations of the shock waves need to satisfy the RankineHugoniot condition. We now check this. The left shock wave has speed √ p −2pt d − −2pt = √ . =− dt 4t 2 −2pt √ The speed of the left shock (0, − −2pt/t) is, according to the RankineHugoniot condition for the Hopf equation, the arithmetic mean of its end states: √ √ −2pt −2pt 1 )=− , s = (0 − 2 2t 4t which coincides with the expression above. The entropy condition is clearly √ satisfied as − −2pt/t < 0. The Rankine-Hugoniot condition and the entropy condition for the right shock can be similarly checked. It remains to
52
3. Scalar Convex Conservation Laws
Figure 3.11. The N -wave N−1/2,1 .
check the total mass:
∞ −∞
Np,q (x, t) dx =
√ 2qt √ − −2pt
√ √ x ( 2qt)2 ( −2pt)2 dx = − = p + q = c, t 2t 2t t > 0.
We conclude that the initial value problem (6.1) has more than one solution. In fact, it has a one-parameter family of solutions Np,q (x, t) where the parameter q is any positive number satisfying q ≥ c and the other parameter p = c − q. Remark 6.1. An N -wave spreads out at the rate of t1/2 and its height decays at the rate of t−1/2 , as can be seen in (6.4), and so an N -wave is dissipative. It is an invisicid dissipation, i.e. dissipation resulting from the nonlinearity of the flux and not from the viscous effects. The standard dissipation comes from the viscosity. Take the simplest equation with viscosity,
6. N -Waves, Inviscid Dissipation
53
the heat equation ut = uxx , and consider the initial data cδ(x):
ut = uxx , u(x, 0) = cδ(x). The solution is c times the heat kernel, given in Proposition 1.1 of Chapter 4, x2 c e− 4t . u(x, t) = √ 4πt The √ essential support of the heat kernel is {x : x2 /t = O(1)}, or |x| = 4. Np,q (x, t) has support of O(1) t; cf. figure 4.01 in Section 4.1 of Chapter √ √ √ width 2qt +√ −2pt, also of the order of t. The maximum height of the at one of the shocks. The heat kernel also decays N -wave is 1/ t, attained √ at the rate of 1/ t. We know that the conservation law ut + f (u)x = 0 is hyperbolic and has finite speed of propagation. Nevertheless, due to the unboundedness of the delta function initial data, the speed of propagation for the N -wave is infinite at t = 0, similar to the heat kernel. There is, however, a major difference: a general solution of the heat equation has only one time invariant, its total heat, while the N -wave Np,q has two time invariants p and q. The Burgers equation ut + uux = uxx also has one time invariant. The N -waves represent the metastable, but not the time-asymptotic, states for the Burgers equation; see Section 8 of Chapter 4. We now consider the general initial values around the state u = 0: ⎧ ⎪ ⎨ut + f (u)x = 0, f (u) > 0, (6.5) |u(x, 0)| ≤ C for |x| < M, ⎪ ⎩ u(x, 0) = 0 for |x| > M. Here C and M are some fixed positive constants. For simplicity, we make the transformation x → x − f (0)t, f (0)u2 → v 2 , so that we may assume, without loss of generality, (6.6)
f (0) = f (0) = 0,
f (0) = 1.
Lemma 6.2. There are two time invariants for the solution of (6.5): (6.7) p (t) = q (t) = 0,
p(t) ≡ min x
x
u(y, t) dy, −∞
These two time invariants p = p(t) and q = q(t) satisfy ∞ u(x, t) dx = p + q. (6.8) −∞
∞
q(t) ≡ max x
u(y, t) dy. x
54
3. Scalar Convex Conservation Laws
Proof. As in the proof of Theorem 5.5, we apply the principle of Remark 4.5 and so it suffices to consider piecewise smooth solutions. We note here that both the definition of the invariants p and q, (6.7), and Theorems 4.2 and 4.3, which form the basis of the general principle of Remark 4.5, are in the L1 (x) sense. This allows for direct application of the principle. The identities (6.7) are proved by a series of arguments. First, we claim that if the minimum location x = x0 (t) defining p(t) occurs at a continuity point x = x0 (t) of the solution u(x, t), then u(x0 (t), t) = 0. This is clear from the minimum property defining p(t). For instance, if u(x0 (t), t) > 0, then the integral from −∞ to x0 (t) − ε, for ε positive and small, would be smaller than p(t), contradicting the minimal property of p(t). Similarly, if u(x0 (t), t) is negative, then the integral from −∞ to x0 (t) + ε would be smaller than p(t). Thus u(x0 (t), t) = 0 if u(x, t) is continuous at x = x0 (t). Suppose the minimum occurs at a shock wave (u(x0 (t) − 0, t), u(x0 (t) + 0, t)). The above argument yields u(x0 (t) − 0, t) < 0 and u(x0 (t) + 0, t) > 0. This contradicts the entropy condition, u(x0 (t)−0, t) > u(x0 (t)+0, t). Thus we have shown that u(x, t) is continuous at x = x0 (t) and that u(x0 (t), t) = 0. Draw the characteristic line dx/dt = f (u(x0 (t), t)) through (x0 (t), t). The characteristic can be drawn both forward and backward locally in time because u(x, t) is continuous at x = x0 (t). From above and (6.6), dx/dt = f (u(x0 (t), t)) = f (0) = 0, and so 0 = u(x0 (t), s) = u(x0 (t), t) for |s − t| small. By the integral conservation law, x0 (t) d u(y, s) dy = f (u(−∞, s)) − f (u(x0 (t), s)) = f (0) − f (0) = 0. ds −∞ This and the minimum defining property yield x0 (t) x0 (t) p(s) ≤ u(y, s) dy = u(y, t) dy = p(t) for |s − t| small. −∞
This implies that
−∞
p (t)
= 0.
A similar argument shows that the solution u(x, t) is continuous at x = x1 (t), u(x1 (t), t) = 0, and q (t) = 0, where x = x1 (t) is the maximum location defining q(t). Note that, by the minimum and maximum properties, after x0 (t) has been defined through p(t), we may take x1 (t) = x0 (t). In fact, it is clear from the analysis that one can take the same characteristic line x(t) = x1 (t) = x0 (t) starting from a location x(0) = x0 with initial value u(x0 , 0) = 0; see figure 3.12. The identity (6.8) holds and this completes the proof of the lemma. We now use the notion of generalized characteristic curves to study the formation of the N -wave from the general initial value problem (6.5).
6. N -Waves, Inviscid Dissipation
55
Theorem 6.3. The solution of (6.5) tends to Np,q (x, t) time-asymptotically, with x ∞ (6.9) p = min u(y, 0) dy, q = max u(y, 0) dy. x
x
−∞
x
More precisely, there exist curves x = xl (t) and x = xr (t) satisfying (6.10) |− −2pt − xl (t)| + | 2qt − xr (t)| = O(1). The solution is zero outside the interval (xl (t), xr (t)). Inside the interval (xl (t), xr (t)),
√ √ O(1)t−1 for − −2pt < x < 2qt, (6.11) u(x, t) = Np,q (x, t) + 1 O(1)t− 2 otherwise. In particular,
∞
(6.12) −∞
|u(x, t) − Np,q (x, t)| dx = O(1)t− 2 , 1
and so u(x, t) has exactly two time invariants p and q in L1 (x).
t x =x0(t) = x1(t) x = x l (t) u= 0
u l (t)
D(t ) u r (t) Cr
Cl
−M
(x0 ,0)
M
x = x r (t)
u= 0
x
Figure 3.12. Convergence to N -wave.
Proof. In the study of convergence to rarefaction waves in the proof of Theorem 5.6 in Section 5, we first showed the decay of shock waves on the generalized characteristics, and then approximated the speed of a weak shock with the arithmetic mean of the characteristic values. This way, the essential arguments can be reduced to that for the Hopf equation. We will therefore consider, for simplicity, only the Hopf equation, f (u) = u2 /2. Through (−M, 0) (or (M, 0)) draw the generalized characteristics x = xl (t) (or x = xr (t)), as shown in figure 3.12. Since the initial function u0 (x) is
56
3. Scalar Convex Conservation Laws
zero outside (−M, M ), as in (6.5), the solution is zero outside (xl (t), xr (t)). This establishes the first statement of the theorem. Set ul (t) ≡ u(xl (t) + 0, t). By the Rankine-Hugoniot condition (2.2) for the Hopf equation, ul (t) + 0 d ul (t) xl (t) = = . dt 2 2 When it is a characteristic line we take ul (t) = 0; the speed is λ(t) = ul (t) = 0 and the above formula holds also. Similarly, we have ur (t) d xr (t) = , ur (t) ≡ u(xr (t) − 0, t). dt 2 Let D(t) = xr (t) − xl (t) be the distance between the two generalized characteristics. We have from the above d d ur (t) − ul (t) . D (t) = xr (t) − xl (t) = dt dt 2 We now draw the characteristic line Cl backward in time from (xl (t) + 0, t) with speed λ(ul (t)) = ul (t), and similarly draw the characteristic line Cr from (xr (t) − 0, t) with speed λ(ur (t) = ur (t). The characteristic line Cl (or Cr ) meets the initial time at the location x = xl (t) − ul (t)t (or x = xr (t) − ur (t)t). As these locations lie somewhere between x = −M and x = M , we have 0 ≤ xr (t) − ur (t)t − xl (t) − ul (t)t ≤ 2M, or xr (t) − xl (t) = ur (t) − ul (t) t + O1 for some bounded function O1 with 0 < O1 < 2M. From the above estimates we have D(t) = xr (t) − xl (t) = ur (t) − ul (t) t + O1 = 2D (t)t + O1 , which can be solved to yield (6.13)
1
D(t) = O(1)(t + 1) 2 .
Thus the spreading of the solution is of the order of t1/2 , same as the centered N -waves Np,q (x, t) studied before. We next study the decay of the solution. Draw a characteristic line backward in time through a location (x, t), xl (t) ≤ x ≤ xr (t), between the generalized characteristics Cl and C2 . It reaches the initial line again between x = −M and x = M , as in figure 3.09, and so −M ≤ x − u(x, t)t ≤ M, or x (6.14) u(x, t) = + O(1)t−1 , t ≥ 1. t From (6.13) and (6.14), we conclude that the solution decays at the rate of t−1/2 : 1
(6.15)
1 O(1)t 2 + O(1)t−1 = O(1)t− 2 , t ≥ 1. u(x, t) = t
7. Entropy Pairs
57
With the estimates (6.13)–(6.15), the final step in connecting the solution u(x, t) to the N -wave Np,q (x, t) is to use Lemma 6.2 for the two time invariants p = p(t) and q = q(t) given in (6.9). Recall that in the proof of Lemma 6.2, the invariant p is the integral of u(x, t) from −∞ to x0 (t) and that u(x0 (t), t) = 0. Draw the backward characteristic through (x0 (t), t) to meet t = 0 between x = −M and x = M . The speed of the backward characteristic is u(x0 (t), t) = 0. Thus we have −M ≤ x0 (t) = x1 (0) = x0 ≤ M. To sum up, we have
x0 (t)
u(x, t) dx =
p= xl (t)
=
x0 (t)
x + O(1) t−1 dx
xl (t) O(1) xl (t)
(xl (t))2 −1 t + O(1)xl (t)t−1 , t ≥ 1, x + O(1) t−1 dx = 2
and so
xl (t) = − −2pt + O(1).
Similarly, xr (t) =
2qt + O(1).
From (6.14), the solution deviates from the center rarefaction wave x/t by O(1)t−1 , t ≥ 1, between the two shocks at x = xl (t) and x = xr (t). Thus the solution is close to the N -wave there and (6.12) holds. As an immediate consequence, the solution u(x, t) has exactly two time invariants p and q. This completes the proof of the theorem.
7. Entropy Pairs We have studied the entropy inequality, (2.9), (2.10), and (2.11), in Section 2 on the Hopf equation. We now consider the entropy pairs for the general convex conservation law. The analysis for the general theory of entropy pairs for systems and the physical motivation from the second law of thermodynamics will be presented later in Chapter 6. Definition 7.1. A pair of functions (η(u), q(u)) of the state variable u is called an entropy pair for the conservation law ut + f (u)x = 0 if (7.1)
η (u) > 0, convexity,
(7.2)
η (u)f (u) = q (u), compatibility condition,
so that the entropy equality η(u)t + q(u)x = 0 holds for smooth solutions u(x, t) of the conservation law.
58
3. Scalar Convex Conservation Laws
Similar to the Hopf equation, (2.6) and (2.8), we have, for a piecewise smooth solution u(x, t) with shock waves (uj− , uj+ ) located at x = xj (t) at time t, (7.3) η(u)t + q(u)x = −s(η(uj+ (t)) − η(uj− (t))) + q(uj+ (t)) − q(uj− (t)) δ(x − xj (t)). j
The estimate (2.6) for the Hopf equation is generalized in the following proposition. Proposition 7.2. For an admissible shock (u− , u+ ) of a convex conservation law ut + f (u)x = 0, f (u) = 0, we have (7.4)
−s(η(u+ ) − η(u− )) + q(u+ ) − q(u− ) = −O(1)|u+ − u− |3
for some positive bounded function O(1) = O(u+ , u− ). Proof. Let F (u) ≡ −s(η(u) − η(u0 )) + q(u) − q(u0 ),
s=
f (u) − f (u0 ) , u − u0
be the measure of entropy dissipation for a shock (u0 , u). From the compatibility condition, it satisfies F (u) = (s − f (u))
η(u) − η(u0 ) − sη (u) + q (u) u − u0 η(u) − η(u0 ) − η (u) . = (s − f (u)) u − u0
For definiteness, assume f (u) > 0; then the entropy condition yields u < u0 , s > f (u), and that s − f (u) is of the order of −(u − u0 ). The convexity 0) − η (u) is positive of the entropy function η (u) > 0 yields that η(u)−η(u u−u0 and of the order of −(u − u0 ). Therefore F (u) is of the order of (u − u0 )2 . From this and the fact that F (u0 ) = 0 we conclude that F (u) is of the order of −|u − u0 |3 . This proves the proposition. With Proposition 7.2, we have an entropy estimate similar to (2.9) for the Hopf equation: For t2 > t1 and some positive bounded function Oj = O(uj− (t), uj+ (t)), ∞ t2 η(u(x, t2 )) dx + Oj |uj+ (t) − uj− (t)|3 dt (7.5) −∞
j
t1
∞
= −∞
η(u(x, t1 )) dx for 0 ≤ t1 < t2 .
8. Generalized Entropy Functional
59
Remark 7.3. As we have seen for the N -waves, the decay rates for the shock waves are (t + 1)−1 in general, except for the two boundary shock waves, which decay at the rate of (t + 1)−1/2 . This is consistent with the entropy estimate (7.5) in that the time integral of the third power of the shock wave strength is finite: ∞ |uj+ (t) − uj− (t)|3 < ∞. 0
j
Note that the global entropy estimate makes use of the nonlinearity of the flux function, f (u) = 0; the same convexity property yields a more definite decay property, through the above pointwise analysis of the N -waves. We will see later that the entropy inequality η(u)t + q(u)x ≤ 0 holds in general, not just for convex laws. Remark 7.4. In studying partial differential equations, energy estimates constitute an important tool that is commonly used. There can be several energy estimates in the study of a given problem involving partial differential equations. The entropy estimate can be viewed as an energy estimate. However, the entropy estimate is a very particular energy estimate for the following reasons. The first is that the entropy estimate applies to weak solutions, while other energy estimates usually require the solutions to be smooth. For instance, to study the smooth solutions for systems of hyperbolic conservation laws, one needs to estimate the L2 (x) norm of the solutions and their differentials in order to handle the nonlinear terms, see Section 4 of Chapter 6. The L2 (x) norm of the differentials of the solution is obtained by applying the energy estimate for the differentials of the partial differential equations, and this works only for smooth solutions. Only the entropy estimate works for weak solutions. Another reason for considering the entropy estimate is that only special systems are endowed with entropy pairs. For the scalar laws considered in this chapter, any convex function η(u) qualifies as an entropy function. On the other hand, as we will see in Chapter 6 on general systems of hyperbolic conservation laws, only special systems are endowed with entropy pairs. Several basic physical systems are endowed with entropy pairs. For these physical systems, it is natural to start with the entropy estimate.
8. Generalized Entropy Functional We have so far considered two distinct stability analyses; one is the wellposedness in L1 (x) norm in Section 4, and the other is the entropy estimate just discussed in Section 7. The entropy estimate is used to study the dissipation mechanism of the solution operator and is effective only when shock waves are present in the solution. As the entropy function η(u) is
60
3. Scalar Convex Conservation Laws
convex, it is an estimate in the L2 (x) norm. However, we have seen in Subsection 4.3 that the L2 (x) norm is not suitable for the well-posedness theory. Thus the entropy estimate has limited usage for the well-posedness theory. We now present a generalized entropy functional, the so called LiuYang functional, which is useful for the later study, in Section 5 of Chapter 9, of L1 (x) stability for physical systems such as the Euler equations in gas dynamics. The generalized entropy functional E[u(·, t), u ¯(·, t)] is defined for two solutions u(x, t) and u ¯(x, t). It depends on both the variation and the L1 distance of the solutions. Unlike the entropy estimate, the generalized entropy functional is effective in reflecting the convexity of the flux even for smooth solutions. Moreover, it is stable upon perturbation, a property useful for the study of the well-posedness theory for the system. Our discussion is divided into two parts; the first subsection is about the continuous version of the functional. For later use of the functional, it is convenient to consider the discrete version of the functional. Thus we consider the discrete version in the second subsection. 8.1. Continuous Version Following the general principle of Remark 4.5, we will consider the generalized entropy functional for piecewise smooth solutions. Consider first simple examples to illustrate the dissipation measure for the functional. Example 1. Consider a smooth solution u(x, t), which is positive over a finite interval and zero outside the interval. Fix a value u ¯ in the range of u(x, t), u ¯ = u(¯ x(t), t). We first observe that the area under the graph of u(x, t) to the right of x = x ¯(t) is decreasing in time. This is intuitively clear from geometric inspection, since the nonlinearity f (u) > 0 implies that a higher value of u propagates at a faster speed f (u); see figure 3.13. This can be verified by direct computation as follows. First, the characteristic u) and so from the conservation law ut + f (u)x = 0 we speed is x ¯ (t) = f (¯ have ∞ d¯ x(t) d ∞ u(z, t) dz = − ut (z, t) dz (8.1) u ¯+ dt x¯(t) dt x ¯(t) u)¯ u + f (¯ u) − f (0) = −f (¯ v )(¯ u)2 /2 = ˜ −(¯ u)2 < 0, = −f (¯ where v¯ is some value between 0 and u ¯. Remark 8.1. Note here that the L1 (x) distance between u(x, t) and the zero solution is constant in time, as the zero solution and u(x, t) do not cross; see Theorem 4.1 of Section 4. The entropy inequality is an equality, as there is no shock wave in the solution; see (7.3) of Section 7. Thus these
8. Generalized Entropy Functional
61
u(x)
u(x,t1 )
u(x, t2 )
x(t ¯ 1) x¯ (t2 ) u¯ x
Figure 3.13. Generalized entropy functional of Example 1.
two primary stability mechanisms are not effective in reflecting the convexity of the flux in this case. On the other hand, the new integral decreases in time, (8.1). This represents a strong stability property, which is useful for the well-posedness theory for systems that we will study later in Chapter 9. Example 2. The generalized entropy functional between a non-negative solution u(x, t) and the zero solution is defined as the product of the above area times the increment of u ¯ as u ¯ varies over the range of the solution: ∞ ∞ |ux (x, t)| u(z, t) dz dx, u(x, t) ≥ 0. (8.2) E[u(·, t), 0] ≡ −∞
x
The functional involves both the variation and the L1 (x) norm of the solution. By applying the calculation in (8.1) to each interval where ux stays with one sign, straightforward calculations using the conservation law ut + f (u)x = 0 give (8.3)
d E[u(·, t), 0] = dt
∞
|ux | f (u) − f (0) − uf (u) (x, t) dx −∞ ∞ |ux |u2 (x, t) dx ≤ 0 = −O(1) −∞
for some positive bounded function O(1). Example 3. The formulation (8.2) can be generalized straightforwardly to two weak solutions when one is greater than the other, u(x, t) ≥ u ¯(x, t) for
62
3. Scalar Convex Conservation Laws
all x ∈ R:
(8.4) E[u(·, t), u ¯(·, t)] ≡
|ux (x, t)|
∞ −∞
∞
+ −∞
(u(z, t) − u ¯(z, t)) dz dx
∞
x
|¯ ux (x, t)|
−∞
(¯ u(z, t) − u(z, t)) dz dx.
x
Note that we have included two terms in the functional so that a symmetric role is played by the two solutions. This leads to the following definition for the general situation. Again, for the purpose of dealing with general weak solutions u(x, t) and u ¯(x, t), it is sufficient to work with piecewise smooth solutions, as stated in Remark 4.5. Definition 8.2. Consider two weak solutions u(x, t) and u ¯(x, t) for the convex conservation law ut + f (u)x = 0, f (u) > 0. Suppose that two solutions ¯(x, t) for intersect along x = xj (t), j = 0 ± 1, ±2, . . . , with u(x, t) ≥ u ¯(x, t) for xj (t) < x < xj+1 (t), j xj (t) < x < xj+1 (t), j even, and u(x, t) ≤ u odd; see figure 3.16. The generalized entropy, the Liu-Yang functional, is defined as (El+ + El− ), (8.5) E[u(·, t), u ¯(˙,t)] ≡ l
El+ ≡
xl+1 (t)
|(¯ u)x (x, t)|
xl (t)
El− ≡
El+ ≡ El− ≡
x
xl+1 (t)
|(¯ u)x (x, t)|
xl (t) xl+1 (t)
|ux (x, t)|
x
xl+1 (t) x
xl+1 (t)
|ux (x, t)|
xl (t)
Il ≡
u ¯(z, t) − u(z, t) + Il dz dx, l odd;
xl (t)
xl (t)
u ¯(z, t) − u(z, t) dz + Il dx, l odd;
xl+1 (t)
u(z, t) − u ¯(z, t) dz + Il dx, l even;
u(z, t) − u ¯(z, t) dz + Il dx, l even;
x xl (t)
xj+1 (t)
u ¯(x, t) − u(x, t) dx
j>l,j odd xj (t)
+
xj+1 (t)
u(x, t) − u ¯(x, t) dx, l odd;
jl,j even xj (t)
+
u(x, t) − u ¯(x, t) dx
xj+1 (t)
j u(x, t) around
66
3. Scalar Convex Conservation Laws
x = x(t). Take two nearby locations x = x− (t) < x(t) < x+ (t) and denote, as in figure 3.15, (8.12)
q− ≡ (¯ u(x− (t), t), u− ),
q+ ≡ (¯ u(x+ (t), t), u+ ),
x± (t) = λ(q± ).
Then there is a local conservation of the L1 (x) distance between the two solutions: d x+ (t) |¯ u(x, t) − u(x, t)| dx dt x− (t) d x+ (t) d x(t) d x+ (t) = (¯ u(x, t) − u(x, t)) dx = q− dx + q+ dx dt x− (t) dt x− (t) dt x(t) = −λ(q− )q− + λ(α)u− − λ(α)u+ + λ(q+ )q+ = −λ(q− )q− − λ(α)α + λ(q+ )q+ = 0 by (8.11) and (8.12), as the wave q+ is the combination of α and q− . Thus we have the integral conservation law d x+ (t) |u2 (x, t) − u1 (x, t)| dx = 0. (8.13) dt x− (t) This is the reason for calling (8.11) the local conservation law.
u(x)
u¯(x, t) q− q+
x− (t) |α| x(t)
x+ (t)
u(x, t) x
Figure 3.15. Local conservation law.
For the discrete version of Definition 8.2, we approximate the solutions with step functions consisting of shock waves and weak rarefaction shock waves. Consider a solution u(x, t) (or u ¯(x, t)) consisting of waves αi located ¯ at x = i(t) (or βi located at i(t)), i = 1, 2, . . . ; see figure 3.16. The functional
8. Generalized Entropy Functional
(8.5) becomes E(t) ≡
|αi |
max{0, (u − u ¯)(x, t)} dx
i(t)
i
(8.14)
∞
67
i(x)
+ +
i
−∞
|βi |
max{0, (¯ u − u)(x, t)} dx
∞
¯i(t)
max{0, (¯ u − u)(x, t)} dx
¯i(x)
+ −∞
max{0, (u − u ¯)(x, t)} dx .
For the computation of dE/dt, the estimate (8.9) can be straightforwardly generalized to the case where one solution is greater than the other; see Example 3 of the previous subsection. To generalize to the present case for two general solutions u(x, t) and u ¯(x, t), as illustrated in figure 3.16, there is the additional consideration that now the two solutions can cross each other. When they cross by a shock, we may use the discrete conservation law and the argument leading to the L1 contraction estimate (4.5). We therefore have the discrete version of the dissipation of the generalized entropy functional in the following theorem.
u(x)
u(x, t) q + (αj ) q − (αj )
q − (αi ) |αj |
u¯(x, t)
|αi | q + (αi )
x
Figure 3.16. Generalized entropy functional, discrete version.
Theorem 8.5. There exists a positive constant C such that the generalized entropy functional E = E(u, u ¯), (8.14), between two solutions u(x, t) and u ¯(x, t) satisfies dE ≤ −C |αi ||q− (αi )||q+ (αi )| + |βi ||q− (βi )||q+ (βi )| . (8.15) dt i
i
68
3. Scalar Convex Conservation Laws
The estimate (8.15) is rewritten as a sum of the third-order generalized entropy dissipation e(α) for each wave α, dE (8.16) ≤ −C e(α), e(α) ≡ |α||q− (α)||q+ (α)|, dt α where the summation is over all waves α in the solutions u(x, t) and u ¯(x, t). Theorem 8.3 for the continuous version can be derived from Theorem 8.5 by a direct limiting process. Clearly, for some positive constants C1 and C2 , u)] u(·, t) − u ¯(·, t) L1 (x) ≤ E[u(·, t), u ¯(·, t)] (8.17) C1 [TV(u) + TV(¯ u)] u(·, t) − u ¯(·, t) L1 (x) , ≤ C2 [TV(u) + TV(¯ where TV(u) (or TV(¯ u)) denotes the total variation of the solution u(·, t) (or u ¯(·, t)). When we study the well-posedness theory for systems in Chapter 9, the fact that the functional involves the total variation of the solutions is a desirable property.
9. Notes The essence of shock wave theory is that solutions of conservation laws exhibit striking behavior. This chapter focuses mainly on the solution behavior. It starts with intuitive geometric arguments and then follows up with exact analysis. A prime example is the study of N -waves by K. O. Friedrichs [48], which was motivated by consideration of the stationary supersonic gas flow around a supersonic moving airfoil; see also [40]. There is a shock wave in the upstream region due to compression of the gas by the airfoil. This is followed by an expansion wave as the airfoil becomes thinner. As the two gas streams meet behind the airfoil, there is another shock due also to the compression. By taking the distance orthogonal to the airfoil as the time variable, the N -wave theory says that the strength of the shock waves hitting the ground is inversely proportional to the square root of the distance of the airplane from the ground. Consideration of N -waves is a key element in the study of the Burgers equation by Hopf [62]. The N -wave theory will be generalized to general systems in Section 6 of Chapter 9. The idea of generalized characteristics is due to James Glimm; see the monograph by Glimm and Lax [55] for its first usage in systems of hyperbolic conservation laws and Lax [76] for exposition of the shock wave theory. The entropy condition for convex conservation laws, Definition 5.1 of Chapter 2, is intuitive and can be viewed as an impressionistic version of the shock wave theory for the Euler equations in gas dynamics; see Courant-Friedrichs [32]. The related statement for a gas dynamics shock is that the upstream
10. Exercises
69
(or downstream) flow is supersonic (or subsonic) relative to the shock; see (5.15) in Chapter 7. The general notion of entropy pairs in Section 7 was proposed in Lax [74]. In later chapters this notion will be directly related to the second law of thermodynamics when we consider gas dynamics and kinetic theory. In Section 4.3, it is shown that Lp (x) for p > 1, and in particular L2 (x), are not suitable metrics for general stability analysis. However, the relative entropy methodology, which is directly related to the L2 (x) topology, is effective for the so-called weak-strong uniqueness analysis; see Chapter V of [35]. It is also useful for the stability analysis in certain local-in-time and small-perturbation cases, e.g. [78]. There are several ways to show the L1 (x) contraction property, Theorem 4.3. There are various properties such as the maximum principle that are specific to scalar laws and make this a routine task. We have adopted the approach of going through Theorem 4.1 for piecewise smooth solutions. The proof of the theorem, due to Keyfitz [70], is intuitively clear and useful in its own right; for instance, we use it in considering the generalized entropy functional. The generalized entropy functional of Section 8 is due to LiuYang [93].
10. Exercises 1. Show that the combination of two shock waves, as in figure 3.02, is a weak solution according to both Definition 3.1 and Definition 3.2 in Chapter 2. 2. Find two simple initial functions u(x, 0) for the conservation law ut + f (u)x = 0, one continuous and the other a step function, which yield the same solution u(x, t) after some positive time t ≥ T . This strong irreversibility property of the solution operator shows that, in general, it is not possible to identify the initial values u(x, 0) by knowing the solution u(x, t) at some later time t > 0. 3. Find the solution u(x, t) at (x, t) = (1, 2) for the conservation law ut + (u2 + 3u + 2)x = 0 with the initial values
−3 for x < 0, u(x, 0) = 4 for x > 0. 4. Find the solution u(x, t) at (x, t) = (1, 2) for the conservation law ut + (u2 + 3u + 2)x = 0 with the initial values
4 for x < 0, u(x, 0) = −3 for x > 0.
70
3. Scalar Convex Conservation Laws
5. Solve the initial value problem ut +
u2 2
x
= 0,
6. Solve the initial value problem ut +
u4 12
+u
x
= 0,
⎧ ⎪ ⎨1, x < 0, u(x, 0) = 3, 0 < x < 3, ⎪ ⎩ −1, x > 3. ⎧ ⎪ ⎨0, x < −1, u(x, 0) = 3, −1 < x < 1, ⎪ ⎩ 0, x > 1.
7. Show that a bounded solution u(x, t) for the convex conservation law, f (u) > 0, satisfies x2 − x1 (10.1) u(x2 , t) − u(x1 , t) ≤ M , M ≡ max{f (u)}, u t for t > 0 and x2 > x1 . This is called the Oleinik condition. With the entropy condition, it can be viewed as the Oleinik estimate. (Hint: Note that for the convex flux, the characteristics backward in time reach the initial time t = 0. Then estimate the difference of λ(u(x, t)).) 8. Construct the N -wave Np,q (x, t) for general convex conservation laws ut + f (u)x = 0, f (u) > 0, with initial data u(x, 0) = cδ(x); cf. (6.1)–(6.4). For simplicity, consider the case where the flux function f (u) is normalized to satisfy f (0) = f (0) = 0. 9. Consider the conservation law ut + u4 /4 + u2 /2 x = 0. Let the entropy function be η(u) = u2 + u. Find the corresponding entropy flux q(u) and compute the entropy dissipation measure −s(η(u+ )−η(u− ))+q(u+ )−q(u− ) for any given shock (u− , u+ ). Check that the estimate (7.4) holds. 10. Compute the time derivative of the generalized entropy functional E[u(·, t), 0], defined in Section 8, for
u2 + u cos x, |x| < π2 , ut + = 0, u(x, 0) = x 2 0, |x| > π2 before a shock emerges. 11. Compute the time derivative of the generalized entropy functional E[Np,q (·, t), 0] between the N -wave, (6.4), and the zero function. 12. Lemma 6.2 has been proved for piecewise smooth solutions. Prove it for bounded solutions with compact support. (Hint: See the proof of Theorem 4.3.)
10.1090/gsm/215/04
Chapter 4
Burgers Equation
Hyperbolic conservation laws ut + ∇x · f (u) = 0
(0.1)
are the simplest type of systems modeling shock waves. The most basic physical systems taking into account additional dissipative effects are the viscous conservation laws: (0.2) ut + ∇x · f (u) = ∇x · B(u, μ)∇x g(u) . The viscosity matrix B(u, μ) vanishes as the dissipation parameters μ tend to zero, B(u, 0) = 0. In the zero dissipation limit, μ → 0+, the viscous conservation laws (0.2) reduce to the hyperbolic conservation laws (0.1). The rich phenomena for systems resulting from the combined effects of the nonlinearity of the flux f (u) and the viscous term B(u, μ)∇x g(u) will be illustrated later in Chapter 10. In this chapter we consider this effect for the simplest scalar viscous conservation law u2 )x = κuxx , Burgers equation. 2 The Burgers equation can be solved explicitly using the Hopf-Cole transformation. The transformation provides a quantitative understanding of the combined effects of the nonlinear flux and the viscosity. We will start with the heat equation, the simple linear equation with a viscous term.
(0.3)
ut + (
1. Heat Equation Consider the heat equation (1.1)
ut = κuxx . 71
72
4. Burgers Equation
The heat equation corresponds to the integral conservation law, (3.1) of Chapter 2, which in this case is d x2 (1.2) u(x, t) dx = κux (x2 , t) − κux (x1 , t). dt x1 Normalize the heat capacity to be 1 so that u(x, t) is the temperature of the material as well as the heat per unit length. The Fourier law (1.2) says that the heat flux is proportional to −ux , the differential of the temperature, with proportionality constant κ, the heat conductivity coefficient. According to the second law of thermodynamics, heat flows from hotter to colder spots so that κ is positive. The fundamental solution of the heat equation, the heat kernel H(x, t), is the solution of the heat equation with initial data being the delta function,
Ht = κHxx , (1.3) H(x, 0) = δ(x). Proposition 1.1. The heat kernel is (1.4)
H(x, t) = √
x2 1 e− 4κt . 4πκt
Proof. A simple way to find the heat kernel is to use the scaling argument: For given positive constants α, β, and γ, set K(x, t) ≡ αH(βx, γt). Clearly K(x, t) solves the heat equation if β 2 = γ. To check the initial value, we first note from the basic property of the delta function, given in Remark 3.4 of Chapter 2 and (6.2) in Chapter 3, that for any positive constant C, δ(Cx) = 0 for x = 0 and that, after making the change of variable x → y = Cx, ∞ ∞ 1 1 δ(y) dy = , δ(Cx) dx = C C −∞ −∞ and so δ(Cx) =
1 δ(x). C
Thus 1 K(x, 0) = αH(βx, 0) = αδ(βx) = α δ(x). β Therefore K(x, 0) = δ(x) if α = β. We conclude that, with the choice of β 2 = γ, and α = β, both functions K(x, t) and H(x, t) satisfy the same equation and have the same initial value. Assuming there is uniqueness theory for the initial value problem, which there is, we conclude that K(x, t) = H(x, t), or, for any positive α, H(x, t) = αH(αx, α2 t),
1. Heat Equation
73
√ For a fixed √ t > 0, by setting α = 1/ t, one induces a self-similarity variable ξ = x/ t: 1 x 1 x H(x, t) = √ H( √ , 1) ≡ √ φ(ξ), ξ ≡ √ . t t t t Thus the scaling analysis reduces two independent variables (x, t) to one independent variable ξ. Plug the above into the heat equation to yield the ODE for the one-variable function φ(ξ): ∞ 1 1 φ(ξ) dξ = 1. − φ − ξφ = κφ , 2 2 −∞ The above integral constraint comes from the initial delta function. This is solved to yield ξ2 1 e− 4κ . φ(ξ) = √ 4πκ This establishes the formula (1.4) for the heat kernel. Remark 1.2. The heat equation and the conservation laws have several distinct features. 1. For the hyperbolic conservation law ut + f (u)x = 0, the flux f (u) is a function of the local value u; but for the heat equation, the flux −κux is non-local in that it requires knowing the function u(x, t) around x in order to compute ux . In the general theory of partial differential equations, ut + f (u)x = 0 is hyperbolic, while ut = κuxx is parabolic. √ 2. The self-similar variable ξ = x/ t for the heat equation ut = κuxx is a parabolic scaling, while for the hyperbolic conservation law ut + f (u)x = 0 the self-similar variable is a hyperbolic scaling √ ξ = x/t; see (1.2) in Section 1 of Chapter 3. The parabolic scaling x = c t yields 1 1 dx = t− 2 , large for small time and sub-linear for large time. dt 2 In particular, as noted √ below, the essential support of the heat kernel grows in time at the rate t. For small time, it has large propagation speed and so the heat operator has a smoothing effect, as can be seen also from the expression for the heat kernel. 3. Note that at initial time t = 0, the heat is concentrated at x = 0, and yet for any positive time t > 0, H(x, t) is positive for all x ∈ R. In other words, there is infinite speed of propagation. This can also be understood from the large speed of propagation, of the order of t−1/2 for small time. On the other hand, for hyperbolic conservation laws, the propagation speed is the characteristic speed and there is finite speed of propagation. 4. Although there is infinite speed of propagation, one observes √ that the heat kernel is very small: exponentially small, H(x, t) < e−C / 4πκt, for x2 /(4κt) > C. Thus the essential support of the heat kernel is {x : x2 /t =
74
4. Burgers Equation
O(1)κ}, which expands at the rate of t1/2 . The heat kernel decays at the rate of t−1/2 within that essential support; see figure 4.01. This is similar to the situation for the N -waves, though for entirely different reasons; see Remark 6.1 in Chapter 3. Both processes are irreversible.
2
Figure 4.01. Heat diffusion
1 − xt √ e t
and its essential support x2 < 5t.
The significance of the heat kernel is that it contains the basic information of the heat equation and can be used to obtain explicit formulas for more general problems. Consider the general initial value problem for the heat equation
(1.5)
ut = κuxx , u(x, 0) = f (x),
where the initial heat distribution is an arbitrarily given function f (x). Proposition 1.3. The solution to the initial value problem (1.5) for the heat equation is given as the convolution of the heat kernel H(x, t) with the initial heat distribution f (x): (1.6)
∞
u(x, t) = −∞
H(x − y, t)f (y) dy.
1. Heat Equation
75
Proof. The formula (1.6) is derived using the heat kernel and by the principle of linear superposition. Decompose f (x) into a series of localized functions (1.7) f (x) =
∞
fj (x),
j=−∞
f (x) for jΔx ≤ x < (j + 1)Δx, fj (x) = 0 otherwise.
For small Δx, each fj (x) is localized, fj (x) = 0 for x not in (jΔx, (j + 1)Δx], and has total heat
∞ −∞
fj (x) = ˜ f (jΔx)Δx.
This is close to the localized property of the delta function, (3.10) in Chapter 2, and so fj (x) can be approximated by a multiple of δ(x − jΔx): ˜ f (jΔx)Δx δ(x − jΔx). fj (x) = Thus from (1.7), the initial temperature distribution f (x) is approximated by a sum of delta functions: (1.8)
f (x) = ˜
∞
f (jΔx)Δx δ(x − jΔx).
j=−∞
Physically, this is approximating the initial temperature distribution by a sum of point heat sources. The heat equation is linear and the principle of linear superposition applies to give
u(x, ˜ j uj (x, t),
t) = (uj )t = κ(uj )xx , uj (x, 0) = f (jΔx)Δx δ(x − jΔx). A translated delta function δ(x − jΔx) yields the translated heat kernel H(x − jΔx, t) as a solution and so uj (x, t) = f (jΔx)Δx H(x − jΔx, t). Consequently, the solution is approximated by a sum of heat kernels: H(x − jΔx)f (jΔx)Δx. u(x, t) = ˜ j
This is the Riemann sum approximation of the convolution of the heat kernel with the initial distribution. By taking finer and finer approximations of the initial distribution f (x) as a sum of delta functions, we obtain, in the limit
76
4. Burgers Equation
of Δx → 0, the solution formula (1.6) for the general initial value problem (1.5). Next we consider the heat equation with a source:
ut = κuxx + g(x, t), (1.9) u(x, t) = 0. Proposition 1.4. The solution to the heat equation with a source, (1.9), is given as the convolution, in x and in t, of the heat kernel H(x, t) with the the source g(x, t): t ∞ H(x − y, t − s)g(y, s) dy ds. (1.10) u(x, t) = 0
−∞
Proof. Write the source as g(x, t) =
∞
gk (x, t),
k=0
gk (x, t) =
g(x, t) for −∞ < x < ∞, kΔt ≤ t < (k + 1)Δt, 0 otherwise.
This way we approximate the source by delta functions as well: g(x, t) = ˜
∞
g(x, t − kΔt)Δt δ(t − kΔt).
k=0
We now fix a target time T and seek the solution u(x, T ), −∞ < x < ∞, of (1.9). Write T = KΔt with K large and Δt small. The solution u(x, T ) is approximated by the sum of solutions uk (x, T ) affected by the heat source at earlier times kΔt, k = 0, 1, 2, . . . , K − 1: u(x, T ) = ˜
K−1
uk (x, T ),
k=0
where uk solves
(uk )t = κ(uk )xx + g(x, t − kΔt)Δt δ(t − kΔt), uk (x, 0) = 0. Each source g(x, t − kΔt)δ(t − kΔt)Δt represents an impetus of heat source, which is transformed instantly into a temperature distribution g(x, kΔt)Δt at time t = kΔt. In other words, uk solves
(uk )t = κ(uk )xx , uk (x, kΔt) = g(x, kΔt)Δt.
1. Heat Equation
77
By the analysis of the initial value problem (1.6), this initial value problem with initial heat distribution given at time kΔt has the solution representation ∞ uk (x, t) = H(x − y, t − kΔt)g(x, kΔt)Δt dy, −∞ < x < ∞, t > kΔt. −∞
The total effect of the sources on the solution at time T is therefore u(x, T ) = ˜
K−1
uk (x, T ) =
k=0
K−1 ∞ k=0
−∞
H(x − y, T − kΔt)g(x, kΔt)Δt dy
T
∞
= ˜ 0
−∞
H(x − y, T − s)g(y, s) dy ds.
This yields the solution formula (1.10).
Corollary 1.5. The solution of the heat equation with general initial distribution and a source,
ut = κuxx + g(x, t), (1.11) u(x, t) = f (x), is
t
∞
∞
H(x−y, t)f (y) dy+
(1.12) u(x, t) = −∞
H(x−y, t−s)g(y, s) dy ds. 0
−∞
By the principle of linear superposition, the solution of (1.11) is the sum of the solutions given by (1.6) and (1.10) and so we have the above corollary. This is Duhamel’s principle for the heat equation, which expresses the solution of the heat equation with a source in terms of the convolution of the initial distribution f (x) and the source g(x, t) with the heat kernel H(x, t). Remark 1.6. The formula (1.12) can also be derived from functionalanalytic considerations, e.g. the proof of Proposition 6.4. The derivation here is close to the original idea of George Green in starting from physical considerations. Assume that the initial function u(x, 0) = f (x) is bounded and measurable. It can be shown easily that (1.12) solves the heat equation and, as t → 0+, approaches the initial distribution f (x) for almost all x ∈ R, in fact for all Lebesgue points x of the function f (x). One obtains the standard energy estimate for the heat equation ut = κuxx by integrating the heat equation times u: ∞ ∞ ∞ u2 2 dx = (uux )x dx − ux dx. t −∞ 2 −∞ −∞
78
4. Burgers Equation
For simplicity, assume that u(x, t) → 0 as |x| → ∞; then ∞ d ∞ u(x, t)2 (1.13) dx = − ux (x, t))2 dx ≤ 0. dt −∞ 2 −∞ We know that u(x, t) is proportional to the heat of the material. The negative of the entropy is a convex function, and in the linear situation here, u2 /2 is proportional to the negative of the entropy and so (1.13) does not mean the decay of the energy but rather is an entropy estimate. The inequality (1.13) says that the entropy increases as the heat dissipates.
2. Hopf-Cole Transformation The Burgers equation ut + uux = κuxx contains the simplest convex flux term. It plays a crucial role in the study of the interplay of the nonlinear flux and the dissipation because it can be solved explicitly by the Hopf-Cole transformation. The transformation is done through the following steps. We note first that, like the heat equation, the Burgers equation is parabolic and its solutions are smooth. Thus, unlike with the hyperbolic conservation laws, calculus manipulations, such as the chain rule, are applicable to the Burgers equation. The first step of the Hopf-Cole procedure is to form the Hamilton-Jacobi equation for the anti-derivative U of u: (2.1)
1 Ut + (Ux )2 = κUxx , 2
Ux ≡ u.
Then introduce the Hopf-Cole relation (2.2)
U (x, t) = −2κ log φ(x, t) .
Direct calculations using (2.1) and (2.2) yield the heat equation for the new function φ(x, t): φt = κφxx .
(2.3)
Proposition 2.1. The solution u(x, t) to the general initial value problem for the Burgers equation,
ut + uux = κuxx , (2.4) u(x, 0) = u0 (x), is ∞ (2.5)
u(x, t) =
x−y − −∞ t e
∞
−∞ e
(x−y)2 1 − 2κ 4κt
(x−y)2 1 − 4κt − 2κ
y 0−
y 0−
u0 (z) dz
u0 (z) dz
dy
dy
.
3. Inviscid Limit
79
Proof. The heat equation (2.3) with initial value φ(x, 0) is given as the convolution of the initial data with the heat kernel H(x, t), (1.6):
∞ φ(x, t) = −∞ H(x − y, t)φ(y, 0) dy, H(x, t) ≡
2
x √ 1 e− 4κt . 4πκt
In later analysis we may have initial data containing a delta function and so we make it definite by setting x x U (x,t) 1 u(y, t) dy, φ(x, t) = e− 2κ = e− 2κ 0− u(y,t) dy . (2.6) U (x, t) = 0−
Thus the above formula for φ(x, t) becomes ∞ z (x−y)2 1 1 √ e− 4κt e− 2κ 0− u(z,0) dz dy, φ(x, t) = 4πκt −∞ and (2.2) yields the solution formula for U (x, t): ∞ (x−y)2 1 − − 1 y u(z,0) dz √ dy . e 4κt 2κ 0− U (x, t) = −2κ log 4πκt −∞ The Hopf-Cole transformation therefore yields an explicit formula (2.5) for the solution u = Ux to the general initial value problem for the Burgers equation.
3. Inviscid Limit The explicit formula (2.5) for the solution of the Burgers equation can be used to solve the hyperbolic conservation law, the inviscid Burgers equation, or the so-called Hopf equation. In general we have the idea of obtaining solutions to hyperbolic conservation laws through the zero dissipation limit of solutions to viscous conservation laws, stated below for general systems; see Chapter 10. Definition 3.1. A solution u(x, t) to the hyperbolic conservation law ut + ∇x · F(u) = 0 is said to be an inviscid limit solution if it is obtained as the zero dissipation limit u(x, t) = lim u(x, t; ε), ε→0+
where u(x, t; ε) are solutions to the corresponding viscous conservation laws ut + ∇x · F(u) = ∇x · B(u, ε)∇x g(u) . Proposition 3.2. The initial value problem
2 ut + ( u2 )x = 0, (3.1) u(x, 0) = u0 (x), −∞ < x < ∞,
80
4. Burgers Equation
has the inviscid limit solution obtained from the zero dissipation limit of solutions to the Burgers equation. The inviscid limit solution is characterized by (3.2)
u(x, t) = u0 (ξ),
where (x − y)2 + F (x, y, t) = 2t
(3.3)
min F (x, y, t) ≡ F (x, ξ, t),
x = ξ + u0 (ξ)t,
y
y
u0 (z) dz. 0−
Proof. The formula (2.5) for the solution u = u(x, t; κ) to the initial value problem for the Burgers equation ut + (
u2 )x = κuxx , 2
can be rewritten as (3.4)
u(x, 0) = u0 (x)
∞
u(x, t) = u(x, t; κ) =
1 x−y − 2κ F (x,y,t) dy −∞ t e , ∞ − 1 F (x,y,t) 2κ dy −∞ e
where F (x, y, t) is given in (3.3). The zero dissipation limit κ → 0+ is a singular limit, as is clear from the above expression. As κ → 0+, the integrals in the expression for u(x, t; κ) are dominated by the minimum value of F (x, y, t). This is the method of steepest descent, which yields x−ξ , κ→0+ t where y = ξ is the location at which the function F (x, y, t) assumes its minimum value: min F (x, y, t) ≡ F (x, ξ, t). lim u(x, t, κ) =
y
We have
ξ−x + u(ξ, 0) = 0. Fy (x, y, t)y=ξ = t Thus the inviscid limit is characterized by (3.2).
Recall that the Hopf equation can be solved by the method of characteristics described in Section 1 of Chapter 2, d d u(x(t), t) = 0, x(t) = u(x(t), t). dt dt With the solution formula (3.2), the limiting function u(x, t) ≡ lim u(x, t, κ) κ→0+
does have the property that the initial data u(ξ, 0) propagates along the characteristic line x = ξ + u(ξ, 0)t to reach the target (x, t). However, the original method of characteristics works only when the solution contains no
4. Nonlinear Waves
81
shock wave. When the solution contains shock waves, there may be more than one initial location reaching the given target (x, t) through the characteristic lines. The solution formula (3.2) says that to determine which initial information reaches the given location (x, t) requires global consideration of the minimum of F (x, y, t). When the minimum is attained by more than one value, it represents a discontinuity, a shock wave in the inviscid solution. In general, a smooth initial function can give rise to an infinite, even dense, set of shock waves. This solution formula for the Hopf equation provides a concrete example of the notion of weak solutions. Various solution behaviors, such as convergence to N -waves, can be studied using this formula.
4. Nonlinear Waves We now use the solution formula (2.5) obtained by the Hopf-Cole transformation to construct the viscous versions of the shock waves, rarefaction waves and N -waves for the hyperbolic conservation laws studied in Chapter 3. From these constructions, one understands the interesting coupled effects of the nonlinear flux and the dissipation. Proposition 4.1. The Burgers kernel bD , the solution to (bD )t + bD (bD )x = κ(bD )xx , bD (x, 0) = Aδ(x),
(4.1) is given by the formula (4.2)
bD (x, t; κ; A) = √
√ A √κ (e 2κ t ∞
π+
√x 4κt
x2
− 1)e− 4κt A
(e 2κ − 1)e−y2 dy
, Burgers kernel.
Proof. We go through the steps (2.6) in the Hopf-Cole transformation:
A x A, x ≥ 0, e− 2κ , x ≥ 0, Aδ(y) dy = φD (x, 0) = UD (x, 0) = 0, x < 0, 1, x < 0. 0− Thus from (2.3) we have (4.3) φD (x, t) =
0
= −∞
0 −∞
∞
H(x − y, t) dy +
1 √ e 4πκt
(x−y)2 − 4κt
0 ∞
dy +
H(x − y, t)e− 2k dy A
(x−y)2 1 e− 4κt dy 4πκt A 1 − e− 2κ ∞ −y2 √ + e dy. π √x
e− 2k √ A
0
= e− 2κ A
4κt
82
4. Burgers Equation
We have from the Hopf-Cole relation (2.2) that (4.4) UD = −2κ log φD √ √ A A − 2κ 2κ = (−2κ) log e − log π + log π + (e − 1)
∞
e−y dy 2
.
√x 4κt
Finally, from bD = (UD )x , one obtains the explicit expression (4.2) for the Burgers kernel bD .
bD
A>0 x A 0, the convection is to the right, with higher values of u having larger convection. Using the same procedure for calculating the Burgers kernel bD , we can easily calculate Burgers rarefaction and shock waves, as in the following two propositions. Without loss of generality, take these waves to be symmetric with respect to x by choosing a positive constant λ0 > 0 and considering ±λ0 as the initial values. Proposition 4.2. Consider the formation of the Burgers shock profile
λ0 for x < 0, (4.5) (uS )t + uS (uS )x = κ(uS )xx , uS (x, 0) = −λ0 for x > 0.
4. Nonlinear Waves
83
It has the explicit expression (4.6) uS (x, t) = −λ0
√ 0 t ) − e− Erfc( −x−λ 4κt
λ0 x κ
√ 0t ) Erfc( x−λ 4κt
√ 0 t ) + e− Erfc( −x−λ 4κt
λ0 x κ
√ 0t ) Erfc( x−λ 4κt
,
Burgers shock formation, where Erfc is the error function 1 Erfc(z) ≡ √ π
∞
e−y dy. 2
z
The Burgers shock profile bS (ξ) = bS (ξ; λ0 ) = bS (x−st; λ0 ; κ), ξ = x−st, is a traveling wave solution, here with speed s = 0 and strength 2λ0 : (4.7)
bS (x) = lim uS (x, t) = −λ0 tanh( t→∞
λ0 x ). 2κ
The time it takes for the Burgers shock wave bS (x) to form from Riemann initial data (λ0 , −λ0 ) is the thickness of the initial layer. From (4.6), the thickness of the √ initial layer is the time it takes for the error functions Erfc((±x √ − λ0 t)/ 4κt) to be close to 1 for a fixed location x. This is so if λ0 t/ 4κt is larger than a given constant. Therefore the thickness T0 of initial layer is given by λ0 T0 κ (4.8) √ = O(1), or T0 = O(1) , Burgers initial layer thickness. (λ0 )2 4κT0 The thickness of the initial layer T0 is large if the strength of the viscosity κ is large, which is understandable as the dissipation resulting from viscosity delays the formation of the shock profile. T0 is small when the shock strength λ0 is large. This is because for larger λ0 , the effect of compressibility is also larger and thereby accelerates the formation of the shock profile. Proposition 4.3. Consider the Burgers rarefaction wave
−λ0 for x < 0, (4.9) (bR )t + bR (bR )x = κ(bR )xx , bR (x, 0) = λ0 for x > 0. It has the explicit expression e
λ0 x 2κ
√ 0 t ) − e− Erfc( −x+λ 4κt
λ0 x 2κ
√ 0t ) Erfc( x+λ 4κt
e
λ0 x 2κ
√ 0 t ) + e− Erfc( −x+λ 4κt
λ0 x 2κ
√ 0t ) Erfc( x+λ 4κt
(4.10) bR (x, t) = λ0
,
Burgers rarefaction wave. The expression (4.10) contains rich wave structure. We only point out that in the region well within the inviscid rarefaction wave, |x| < λ0 t, and
84
4. Burgers Equation
after the initial layer with length O(1)κ/(λ0 )2 , (4.8), the Burgers rarefaction wave is close to the inviscid wave x/t: 1 1 x + (4.11) bR (x, t) − = O(1) t |x − λ0 t| |x + λ0 t| √ √ κ . for x ∈ (−λ0 t + M 4κt, λ0 t − M 4κt), t > O(1) (λ0 )2 Thus, within the inviscid rarefaction wave region, the hyperbolic expansion dominates the dissipation effect. On the other hand, near the boundary x = ±λ0 t of the inviscid rarefaction wave, the rate of expansion degenerates from (bR )x = O(1)t−1 to the usual dissipation rate of t−1/2 . Thus dissipation dominates around the edges x = ±λ0 t of the inviscid wave region as well as within the initial layer, 0 < t < O(1) (λκ0 )2 . The aforementioned information concerning the viscous waves is provided by the explicit formulas (4.2), (4.6), and (4.10) obtained by the HopfCole transformation.
5. Linearized Hopf-Cole Transformation The Hopf-Cole transformation has a linearized version. Consider a solution ¯ (x, t) and u ¯(x, t) of the Burgers equation with its associated functions U ¯ t) as given in (2.6), φ(x,
2 uxx u ¯t + ( u¯2 )x = κ¯ (5.1) ¯ t) . ¯x = u ¯ (x, t) = −2κ log φ(x, U ¯, U The Hopf-Cole transformation can be applied to solve the Burgers equation linearized around the given Burgers solution u ¯(x, t). Proposition 5.1. Let u ¯(x, t) be a solution of the Burgers equation with as¯ t), (5.1). Consider the Burgers equation linearized ¯ (x, t) and φ(x, sociated U around u ¯: uv)x = κvxx . vt + (¯
(5.2)
The initial value problem for (5.2) has the solution formula
(5.3)
∂ v(x, t) = ∂x
∞
−∞
√ 1 e− 4πκt
(x−y)2 4κt
¯ 0) φ(y, ¯ t) φ(x,
y 0
v(z, 0)dz dy
The initial value problem for the anti-derivative of (5.2), (5.4)
¯wx = κwxx , wt + u
.
6. Green’s Functions
85
has the solution formula ∞ (5.5)
− √1 −∞ 4πκt e
w(x, t) =
(x−y)2 4κt
¯ 0)w(y, 0) dy φ(y, .
¯ t) φ(x,
Proof. This is done by following the steps (2.1), (2.2), and (2.3) for the Hopf-Cole transformation. The Hopf-Cole relation (2.2) for the solution u=u ¯ + v is ¯ + V = −2κ log[φ¯ + ζ], Vx = v. U This is for the Burgers equation. We are interested in the linear equation ¯ ¯ , φ): (5.2). For this we linearize the above Hopf-Cole relation around (U ¯ + V = −2κ log φ¯ − 2κ ζ + O(1)ζ 2 . U φ¯ ¯ = −2κ log φ¯ from Ignore the nonlinear term O(1)ζ 2 and use the relation U (5.1) to arrive at ζ (5.6) V = −2κ ¯ , v = Vx , linearized Hopf-Cole relation. φ Straightforward calculations using (5.6) show that, for the solution v = Vx of the linear equation (5.2), the function ζ satisfies the heat equation ζt = κζxx .
(5.7)
Similar to the original Hopf-Cole transformation, (5.6) and (5.7) are solved to yield the solution of the initial value problem for the linearized Burgers equation (5.2): v(x, t) = Vx (x, t), ∞
(5.8)
V (x, t) =
√ 1 e− 4πκt
−∞
1 ζ(x, t) = − 2κ
∞
−∞
(x−y)2 4κt
√1 e 4πκt
¯ φ(y,0)
y 0
¯ φ(x,t) (x−y)2 − 4κt
v(z,0) dz dy
¯ 0) φ(y,
y 0
,
v(z, 0) dz dy.
This yields the formula (5.3). For (5.4) we have ζ w ≡ −2κ ¯ , ζt = κζxx , φ which is solved to obtain the solution expression (5.5).
6. Green’s Functions We use the linearized Hopf-Cole transformation procedure, Propostion 5.1, to construct the Burgers Green’s functions for the study of wave propagation over particular Burgers solutions. Consider a small perturbation v(x, t) of a particular Burgers solution u ¯(x, t): ¯ ⇒ vt + (¯ uv)x + vvx = vxx . ut + uux = uxx , u = v + u
86
4. Burgers Equation
For a small perturbation the approximate equation is the linearized Burgers equation vt + (¯ uv)x = vxx . Thus the linearized equation governs the propagation of infinitesmal disturbances in the physical space (x, t) of the particular Burgers solution u ¯(x, t). The Green’s function G(x, t; y, s) is the value of the disturbance at (x, t) due to a point source at (y, s) for s < t. For conservation laws, it is natural to consider the anti-derivative of the linear equation ¯wx = κwxx . wt + u Definition 6.1. The Green’s function s < t, −∞ < x, y < ∞,
G = G(x, t; y, s) = G(x, t; y, s; u ¯),
for the Burgers equation linearized around a given Burgers solution u ¯(x, t) is the solution of
¯Gx = κGxx , Gt + u (6.1) G(x, s; y, s) = δ(x − y). Proposition 6.2. The Green’s function G = G(x, t; y, s) = G(x, t; y, s; u ¯), the solution of (6.1), has the expression, (6.2)
¯ s) (x−y)2 φ(y, 1 − e 4κ(t−s) ¯ . G(x, t; y, s) = φ(x, t) 4πκ(t − s)
Proof. The Green’s function is computed using the formula (5.5) from the linearized Hopf-Cole transformation: ∞ G(x, t; y, s) =
−∞
(x−z)2
− 1 ¯ s)δ(z e 4κ(t−s) φ(z, 4πκ(t−s)
√
− y) dz
¯ t) φ(x,
.
This yields the formula (6.2).
In order to calculate the total amount of waves reaching a fixed location (x, t) from a disturbance at (y, s), where s < t and −∞ < y < ∞, we need to consider the backward Green’s function. Definition 6.3. The backward Burgers Green’s function K = K(x, t; y, s) = K(x, t; y, s; u ¯), is the solution of (6.3)
t < s, −∞ < x, y < ∞,
uK)x + κKxx = 0, Kt + (¯ K(x, s; y, s) = δ(x − y).
6. Green’s Functions
87
Proposition 6.4. The solution v(x, t) of
vt + u ¯vx = κvxx + g(x, t), (6.4) v(x, 0) = f (x) is given by Duhamel’s principle: t t K(y, 0; x, t)f (y) dy + (6.5) v(x, t) = 0
∞
K(y, s; x, t)g(y, s) dy ds. −∞
0
Proof. Multiply (6.4) by K(x, t; x0 , t0 ) and integrate to obtain t0 ∞ K(x, t; x0 , t0 ) vt + u ¯vx − κvxx dx dt −∞
0
t0
∞
= 0
−∞
K(x, t; x0 , t0 )g(x, t) dx dt.
Integration by parts using (6.3) yields t0 ∞ K(x, t; x0 , t0 ) vt + u ¯vx − vxx dx dt 0
t0
−∞ ∞
= 0
−∞
− Kt + (¯ uK)x + κKxx v dx dt +
∞
−∞
= v(x0 , t0 ) −
t=t0 K(x, t; x0 , t0 )v(x, t) dx t=0
∞
−∞
K(x, 0; x0 , t0 )f (x) dx.
The proposition follows from the above two identities.
Proposition 6.5. (6.6)
K(x, t; y, s) = G(y, s; x, t).
Proof. For fixed (x0 , t0 ) and (x1 , t1 ), with t1 > t0 ≥ 0, we have from equation (6.3) for K that t1 ∞ G(x, t; x0 , t0 ) ∂t + ∂x u ¯ + κ∂x2 K(x, t; x1 , t1 ) dx dt = 0. t0
−∞
Note that G(x, t0 ; x0 , t0 ) = δ(x − x0 ) and K(x, t1 ; x1 , t1 ) = δ(x − x1 ), and so we have from equation (6.3) for G and integration by parts that t1 ∞ G(x, t; x0 , t0 ) ∂t + ∂x u ¯ + κ∂x2 K(x, t; x1 , t1 ) dx dt 0= t0
=−
−∞ t1
t0
∞ −∞
K(x, x1 ; t, t1 ) ∂t + u ¯∂x − κ∂x2 G(x, t; x0 , t0 ) dx dt t=t1 + G(x, t; x0 , t0 )K(x, t; x1 , t1 ) t=t0
= G(x1 , t1 ; x0 , t0 ) − K(x0 , t0 ; x1 , t1 ).
88
4. Burgers Equation
This completes the proof of the proposition.
With the above proposition, the solution formula (6.5) for (6.4)becomes
t
∞
∞
G(x, t; y, 0)f (y) dy +
(6.7) v(x, t) = −∞
G(x, t; y, s)g(y, s) dy ds. −∞
0
Remark 6.6. The formula (6.7) is a generalization of formula (1.12) for the heat equation. Note that we arrive at the formulas in different ways. Formula (1.12) for the heat equation was derived based on the approximation of the initial function and the source function by sums of delta functions. Here we use the duality argument in Proposition 6.6 to derive formula (6.7). One may apply either method to a given problem. It is interesting to think about the difference in meaning of the two approaches. The heat equation is a partial differential equation with constant coefficients, and so the solution can be viewed as the propagation of waves over a constant state; in contrast, in the general set-up of (6.4), the propagation is over a given non-constant background u ¯ = u ¯(x, t). As a consequence, unlike the Green’s function for the heat equation, which has the simple form H(x, t), the Green’s function for (6.4) is of the more general form G(x, t; y, s). The two situations can be compared if we use the notation H(x, t; y, s) ≡ H(x − y, t − s). For variable-coefficient equations, the propagation of information depends on both the target (x, t) and the location (y, s) of the source, and not just on their difference (x − y, t − s). We now use the formula (6.2) to calculate the Green’s function for specific base functions u ¯(x, t). Proposition 6.7. The Green’s function GD (x, t; y, s) for the Burgers kernel u ¯(x, t) = bD (x, t; A), which solves (6.8)
(GD )t + (bD GD )x = κ(GD )xx , GD (x, s; y, s) = δ(x − y),
has the formula e− 2κ + A
1
e (6.9) GD (x, t; y, s) = 4πκ(t − s)
(x−y)2
− 4κ(t−s)
e
A − 2κ
A
− 1−e √ 2κ π A
+
− 1−e √ 2κ π
∞
e−z dz
∞
e−z 2
√y 4κs √x 4κt
2
. dz
6. Green’s Functions
89
Proof. We have from (4.2) and (4.3) that √ A √κ (e 2κ t ∞
bD (x, t; κ; A) = √ π+
√x 4κt
x2
− 1)e− 4κt A
(e 2κ − 1)e−y2 dy
, 1 − e− 2κ √ π A
φD (x, t) = e− 2κ + A
∞
e−z dz. 2
√x 4κt
¯ t) = The formula (6.9) follows from (6.2) and the above formula for φ(x, φD (x, t). We can compute the Green’s function for the Burgers shock and Burgers rarefaction wave by the same method as above. The shock profile u ¯(x, t) = bS (x), (4.7), is a function of x only, and so its Green’s function has a simpler form, GS (x, t; y, s) = GS (x, t − s; y). Proposition 6.8. The Green’s function for the shock profile, (GS )t + bS (GS )x = κ(GS )xx , GS (x, 0) = δ(x − y),
(6.10) has the form
λ0 y
(6.11)
λ0 y
− 2κ −(λ0 )2 t (x−y)2 e 2κ + e 1 4κ e . GS (x, t; y) = √ e− 4κt λ0 x λ0 x 4πκt e− 2κ + e− 2κ
The Green’s function can be rewritten as a weighted combination of the heat kernel with speeds ±λ0 : (6.12) GS (x, t; y) =
1 + e−
λ0 |y| κ λ0 |x|
1 + e− κ ⎧ ⎪ H(x − y + λ0 t, t) for x > 0, y > 0; ⎪ ⎪ ⎪ ⎨e− λ0κ|x| H(x − y + λ t, t) for x < 0, y > 0; 0 · ⎪ H(x − y − λ0 t, t) for x < 0, y < 0; ⎪ ⎪ ⎪ ⎩ − λ0κ|x| H(x − y − λ0 t, t) for x > 0, y < 0. e
When both the source x0 and the target x are positive, ˜ H(x − y + λ0 t, t), x > 0, y > 0, GS (x, t; y) = and so the propagation of the source is basically according to the heat kernel with speed −λ0 for the Hopf shock. Similarly, when both the source x0 and the target x are negative, ˜ H(x − y − λ0 t, t), x < 0, y < 0, GS (x, t; x0 ) =
90
4. Burgers Equation
The above forms are consistent with propagation of information around the inviscid shock along the characteristic lines with speed −λ0 for x > 0 and λ0 for x < 0,
−λ0 for x < 0, hS (x, t) = λ0 for x > 0, and so when the source and target are of the same sign, the propagation is governed by the dissipative version of the transport equation ut ± λ0 ux = 0. On the other hand, when the source y and the target x are of different signs, the inviscid speed at the target is different from the speed at the source, and there is an exponentially decaying term in x: GS (x, t; y) = ˜ e−
λ0 |x| κ
H(x − y ± λ0 t) for ± y > 0, xy < 0.
Proposition 6.9. The Green’s function GR (x, t; y, s) for wave propagation over the rarefaction wave u ¯(x, t) = bR (x, t) in (4.10), which solves (6.13)
(GR )t + bR (GR )x = κ(GR )xx , GR (x, s; y, s) = δ(x − y),
has the formula (6.14) GR (x, t; y, s) =
(x−y)2
1 4κ(t − s) ·
e
2
− 4κ(t−s) − (λ0 ) (t−s) 4κ
e
e−
λ0 y 2κ
√ 0s ) + e Erfc( −y+λ 4κs
λ0 y 2κ
√ 0s ) Erfc( y+λ 4κs
e−
λ0 x 2κ
√ 0t ) + e Erfc( −x+λ 4κt
λ0 x 2κ
√ 0t ) Erfc( x+λ 4κt
.
The rarefaction wave bR (x, t) evolves in time and in space, and so the Green’s function is of the general form GR (x, t; y, s). As we have seen, bR (x, t) incorporates both the linear hyperbolic expansion rate and the sublinear dissipation rate; see the discussion after Proposition 4.3. It is therefore interesting to understand the Green’s function GR (x, t; y, s), which describes the propagation of disturbances over bR (x, t). The above exact form gives rise to several scales for the behavior of the propagation. Take the case where both the target (x, t) and the location (y, s) of the source are well inside the inviscid rarefaction wave and after the initial layer time O(1)κ/(λ0 )2 in (4.8): √ √ x ∈ (−λ0 t + M 4κt, λ0 t − M 4κt), √ √ y ∈ (−λ0 s + M 4κs, λ0 s − M 4κt0 ),
t > s > O(1)
κ . (λ0 )2
6. Green’s Functions
91
By direct calculations using the exact form (6.14), we have the following quantitative estimate of the Green’s function GR (x, t; y, s): √
GR (x, t; y, s) = ˜
κs λ0 s+y √ κt λ0 t+x
+ +
√ κs λ0 s−y √ κt λ0 t−x
−
e
t(y−sx/t)2 4κs(t−s)
4κs(t − s)
We now analyze this expression. First, the propagation of waves is around the zero line of the exponential, t t(y − sx/t)2 = 0, or x = y, 4κs(t − s) s that is, along the characteristic line through the source at (y, s) of the rarefaction wave according to the inviscid Hopf solution hR (x, t) = x/t for |x/t| < λ0 . The essential support of the information is in the region given by t t t(y − sx/t)2 = O(1), or |x − y| = O(1) κ(t − s) . 4κs(t − s) s s Thus, for short- and intermediate-time propagation, 0 < t − s < t/2, the width of the support is t O(1) κ(t − s) = O(1) κ(t − s). s For short and intermediate times, that is, for source times close to the target time, the effect of the inviscid expansion of the rarefaction wave is secondary and the width O(1) κ(t − s) is of the same order as the linear dissipation. On the other hand, for large-time propagation, t/2 < t − s < t, t κ t, O(1) κ(t − s) = O(1) s s a width linear in time t, which has the same rate of expansion as the inviscid rarefaction wave. Notice that the strength of the source, √
κs λ0 s+y √ κt λ0 t+x
+ +
√ κs λ0 s−y √ κt λ0 t−x
,
also varies significantly. This has to be so as the L1 (x) norm of the Green’s function is kept at the constant 1. As the source and target approach the boundary of the inviscid rarefaction wave, λ0 t − x λ0 s − y λ0 t + x λ0 s + y √
1, √
1,
1, √
1, √ κs κs κt κt the effect of diffusion becomes as important as the inviscid nonlinear expansion. The study of rarefaction waves is interesting even for scalar equations,
92
4. Burgers Equation
and richer wave phenomena occur for systems. These rich wave phenomena are studied here by resorting to the explicit expression obtained by the Hopf-Cole transformation. The above Green’s functions GS (x, t; y, s) and GR (x, t; y, s) are for the anti-derivative variables. From these we can easily construct the Green’s functions gS (x, t; y, s) and gR (x, t; y, s) for the original variables by gy = −Gx : (6.15) (gS )t + (bS gS )x = κ(gS )xx ,
gS (x, s; y, s) = δ(x − y), ∞ (GS )x (x, t; z, s) dz; gS (x, t; y, s) = y
(6.16) (gR )t + (bR gR )x = κ(gR )xx ,
gR (x, s; y, s) = δ(x − y), ∞ (GR )x (x, t; z, s) dz. gR (x, t; y, s) = y
7. Nonlinearity The quadratic nonlinearity uux in the Burgers equation is critical. This can be seen by comparing the Burgers equation with the heat equation for the case of dissipation around the zero state. Being a dissipative equation, the Burgers solution with bounded support is expected to have similar decay rates as the heat kernel: u(x, t) = O(1)t− 2 , ut (x, t) = O(1)t− 2 , ux (x, t) = O(1)t−1 , 3 uxx (x, t) = O(1)t− 2 as t → ∞. 1
3
This is so for the Burgers kernel bD (x, t) in (4.2). From this, we see that the nonlinear term uux has the same dissipation rate as the other terms in the Burgers equation: uux (x, t) = O(1)t 2 t−1 = O(1)t− 2 = ˜ ut (x, t) = ˜ uxx (x, t) as t → ∞. 1
3
In other words, the dissipation phenomenon of Burgers solutions cannot be governed by the heat equation. Instead, one needs to consider the Burgers kernel bD (x, t) = bD (x, t; A) from (4.1) and (4.2). The heat kernel and the Burgers kernel distribute the mass differently; see figure 4.01 and figure 4.02. We compare bD (x, t; A) with AH(x, t) as they carry the same mass: ∞ bD (x, t; A) − AH(x, t) dx = 0, t ≥ 0. −∞
The Burgers kernel is non-symmetric with respect to the x-axis, while the heat kernel is symmetric with respect to the x-axis. It is easy to show by
7. Nonlinearity
93
√ transforming to the self-similar variable ξ ≡ x/ 4κt that the heat kernel and Burgers kernel do not approach each other time-asymptotically: ∞ (7.1) |bD (x, t; A) − AH(x, t)| dx −∞
∞
= −∞
√
√ A √κ (e 2κ t ∞
π+
∞
= −∞
√
√x 4κt
x2
− 1)e− 4κt A
(e 2κ − 1)e−y dy 2
− A√
x2 1 e− 4κt dx 4πκt
2κ(e 2κ − 1) 1 −ξ2 √ dξ = O(1). − A e ∞ A 2 π π + ξ (e 2κ − 1)e−y dy A
In contrast to the Burgers equation, for the nonlinear equation ut + (
up )x = κuxx , p > 2, p
the nonlinear term up−1 ux decays at a higher rate than the other terms in the equation. In this case, the dissipation phenomenon can be accurately governed by the heat equation, ignoring the nonlinear term. We now perform a quantitative analysis of the difference between the heat equation and the Burgers equation for a compactly supported perturbation: ∞ u0 (x) dx. (7.2) u0 (x) = 0 for |x| > M, A ≡ −∞
Consider first the heat equation. Proposition 7.1. The solution to the initial value problem of the heat equation
ut = κuxx , (7.3) u(x, 0) = u0 (x) tends, time-asymptotically, to a multiple of the heat kernel. More precisely, for any constant D > 4κ, ∞ x2 x2 A − 4κt −1 − D(t+1) (7.4) u(x, t) = √ e + O(1)t e , A≡ u0 (x) dx; 4πκt −∞ ∞ 1 |u(x, t) − AH(x, t)| dx = O(1)t− 2 as t → ∞. (7.5) −∞
Proof. Set v0 (x) ≡ u0 (x) − Aδ(x) so that x v0 (y) dy satisfies V0 (x) = 0 for |x| > M. V0 (x) ≡ −∞
94
4. Burgers Equation
From the solution formula (1.6) for the heat equation, ∞ ∞ u(x, t) − AH(x, t) = H(x − y, t)v0 (y) dy = H(x − y, t)(V0 )y (y) dy. −∞
−∞
By the explicit expression of the heat kernel H(x −y, t), integration by parts yields, for any D > 4κ, ∞ ∞ H(x − y, t)(V0 )y (y) dy = − Hy (x − y, t)V0 (y) dy (7.6) −∞
−∞
M
=
O(1)t−1
−M
2 x − y − (x−y)2 − x e 4κt dy = O(1)t−1 e D(t+1) . t
Here we have noted that (x−y)2 x − y − (x−y)2 √ e 4κt = O(1)e− Dt t for any constant D > 4κ. This establishes the estimate (7.4). As a direct consequence of (7.4), the difference between the solution and the multiple of the heat kernel in L1 (x) decays at the rate of t−1/2 , (7.5). We know that u(x, t) and AH(x, t) do not decay in L1 (x) because ∞ ∞ u(x, t) dx = AH(x, t) dx = A, t ≥ 0. −∞
−∞
The proposition says that their difference decays in L1 (x). By the estimate (7.1), we see that the solution to the Burgers equation does not converge to the heat kernel. Instead, it converges to the Burgers kernel. We will use the Green’s function approach to obtain strong pointwise behavior of the solution. The Green’s function is for linear equations, and so we focus on weakly nonlinear phenomena and assume that the initial data u(x, 0) is small. Proposition 7.2. Consider the initial value problem of the Burgers equation
2 ut + u2 x = κuxx , (7.7) u(x, 0) = u0 (x), with the initial data satisfying (7.2) and |u0 (x)| ≤ ε, −∞ < x < ∞. Then, provided that the constant ε is sufficiently small, the solution tends time-asymptotically to the Burgers kernel with the same total mass. More precisely, for any constant D > 4κ, (7.8)
u(x, t) − bD (x, t + 1; A) = O(1)ε(t + 1)−1 e
2
x − D(t+1)
.
7. Nonlinearity
95
In particular, the difference between the solution and the Burgers kernel decays in L1 (x) at the rate of t−1/2 : ∞ (7.9) |u(x, t) − bD (x, t + 1; A)| dx = O(1)t−1/2 as t → ∞. −∞
Moreover, the solution u(x, t) is infinitely differentiable for t > 0: (7.10)
∂m u(x, t) − bD (x, t + 1; A) m ∂x = O(1)ε(t + 1)−1 t− 2 e m
2
x − D(t+1)
, m = 0, 1, 2, . . . .
Proof. Set v ≡ u − bD (x, t + 1; A) and use the fact that bD is a Burgers solution to obtain the equation for v: v2 = κvxx . vt + bD v + 2 x Choose A to be the total mass of u(x, 0) so that the anti-derivative x v(y, t) dy V (x, t) ≡ −∞
satisfies v2 2 and its initial value V (x, 0) decays exponentially as |x| → ∞. Analyze this using the formula (6.7) in terms of the Green’s function GD : t ∞ ∞ v2 GD (x, t; y, 0)V (y, 0) dy + GD (x, t; y, s) (y, s) dy ds. V (x, t) = 2 −∞ 0 −∞ Vt + bD Vx = κVxx −
Differentiating this, we obtain ∞ ∂x GD (x, t; y, 0)V (y, 0) dy (7.11) v(x, t) = −∞
t
∞
+ 0
−∞
∂x GD (x, t; y, s)
v2 (y, s) dy ds. 2
From (6.9), GD is of the same order as the heat kernel: (x−y)2 1 − e 4κ(t−s) , GD (x, t; y, s) = O(1) √ t−s (x−y)2 x − y 1 − e 4κ(t−s) ∂x GD (x, t; y, s) = O(1) √ t−s t−s (x−y)2 1 − D(t−s) e , 0 ≤ s ≤ t, = O(1) t−s
96
4. Burgers Equation
for any constant D > 4κ. This implies, by the same reasoning as in (7.6), that the first integral in (7.11) decays as ∞ 2 − x ∂x GD (x, t; y, 0)V (y, 0) dy = O(1)ε(t + 1)−1 e D(t+1) (7.12) −∞
for any fixed D > 4. This suggests the following ansatz: v(x, t) = O(1)ε(t + 1)−1 e
2
x − D(t+1)
.
With this ansatz, the second integral in (7.11), the nonlinear term, becomes t
∞
∂x GD (x, t; y, s)
(7.13)
−∞ t ∞
0
O(1)ε2
= −∞
0
t 2
= 0
t
2 (x−y)2 1 − D(t−s) − y e (s + 1)−2 e 2D(s+1) dy ds t−s
∞
y2 1 − (x−y)2 − O(1)ε2 e D(t−s) (s + 1)−2 e 2Dκ(s+1) t −∞
∞
O(1)ε2
+ t 2
v2 (y, s) dy ds 2
−∞
(x−y)2 y2 1 − D(t−s) − e (t + 1)−2 e 2Dκ(t+1) t−s
= O(1)ε2 (t + 1)−1 e
2
x − D(t+1)
.
This has an extra small factor ε over the above ansatz for v(x, t). Therefore we have the nonlinear closure, as can be seen from the following iteration process. Since one expects the leading term to be ∞ ∂x GD (x, t; y, 0)V (y, 0) dy, (7.14) v0 ≡ −∞
we write v¯(x, t) ≡ v(x, t) − v0 (x, t) and (7.11) becomes t ∞ (v0 + v¯)2 (y, s) dy ds, ∂x GD (x, t; y, s) v¯(x, t) = 2 0 −∞ which leads to the iterations (7.15) v¯0 (x, t) ≡ 0, t ∞ (v0 + v¯n )2 (y, s) dy ds, n = 0, 1, 2, . . . . ∂x GD (x, t; y, s) v¯n+1 (x, t) ≡ 2 0 −∞ From the estimate (7.14), v0 = O(1)ε(t + 1)−1 e
2
x − D(t+1)
.
7. Nonlinearity
97
We make the induction hypothesis (7.16) v¯j = O(1)ε2 (t + 1)−1 e
2
x − D(t+1)
, j = 1, 2, . . . , n;
(¯ vj − v¯j−1 )(x, t) = O(1)ε2j (t + 1)−1 e
2
x − D(t+1)
, j = 1, 2, . . . , n.
Then from (7.15), (¯ vn+1 − v¯n )(x, t) t ∞ 2v0 + v¯n + v¯n−1 (¯ vn − v¯n−1 )(y, s) dy ds. ∂x GD (x, t; y, s) = 2 0 −∞ From this, the calculation (7.13) yields that the estimate (7.16) holds for j = n + 1. Thus, for ε small, the sequence vn (x, t) ≡ v0 (x, t) + v¯n (x, t), n = 1, 2, . . . , converges to the exact solution v(x, t) = u(x, t) − bD (x, t + 1; A) satisfying (7.8). The estimate (7.9) follows directly from (7.8). To show that the solution u(x, t) is infinitely differentiable for t > 0, we first notice that v0 (x, t), t > 0, is infinitely differentiable, i.e. one can take derivatives of any order of the equation (7.14). This is so because for t > 0, the Green’s function GD (x, t; y, 0) is infinitely differentiable and integrable. The differentiability of v¯n (x, t) is shown by induction on n. Suppose that v¯n (x, t) is infinitely differentiable for t > 0; then we may differentiate the relation (7.15) and distribute the differentiation as follows: ∂m v¯n+1 (x, t) ∂xm t m = (−1)
(7.17)
t 2
+ 0
t 2
∞ −∞
∞ −∞
∂x GD (x, t; y, s)
∂ m (v0 + v¯n )2 (y, s) dy ds ∂y m 2
∂ m+1 (v0 + v¯n )2 (y, s) dy ds, m = 1, 2, . . . . G (x, t; y, s) D ∂xm+1 2
The rationale behind the distribution of the differentiations is to note that v¯n (y, s) is assumed under the induction hypothesis to be infinitely differentiable for s > 0 and in particular for s > t/2, and that the Green’s function GD (x, t; y, s) is infinitely differentiable for t − s > 0. We have noted that ∂x GD (x, t; y, s) is integrable for t − s ≥ 0. The estimate (7.10) is proved by an induction process similar to the above with ansatz provided by the left-hand side of (7.10). This completes the proof of the proposition. A solution to the Burgers equation approaches the Burgers kernel with the same total mass, as shown in Proposition 7.2, while a solution to the heat equation approaches a multiple of the heat kernel with the same total heat, as shown in Proposition 7.6. From (7.1), the heat kernel and Burgers kernel do not approach each other. Thus the critical nonlinearity in the Burgers
98
4. Burgers Equation
equation implies that the heat equation does not govern the time-asymptotic behavior of solutions of the Burgers equation. The infinite differentiability of the solution in the proof of Proposition 7.2 uses the Green’s function approach. Solutions to the initial value problem for parabolic equations in general are infinitely differentiable for any positive time. For the scalar equations considered here, there are other methods, such as the maximum principle, that can be used for the study of solution behavior.
8. Metastable States For the Burgers equation, a solution of finite mass A tends to the self-similar Burgers kernel bD (x, t; A) time-asymptotically, as stated in Proposition 7.6. However, the solution goes through a metastable state for a time period, which depends on the initial data and the strength of the viscosity κ, before it approaches its time-asymptotic state bD (x, t; A). To analyze this, we consider a solution with bounded support and total mass A: (8.1) ut + uux = κuxx ,
u(x, 0) = 0 for |x| > M, ∞ ∞ u(x, 0) dx = u(x, t) dx = A, t ≥ 0. −∞
−∞
Consider the general scaling v(x, t) ≡ αu(βx, γt). For γ = αβ the equation for the function v(x, t) is the Burgers equation with a different viscosity coefficient: κα vt + vvx = vxx . β Then set β = αA so that the total mass for v(x, t) is normalized to 1. In summary, ∞ κ 2 v(x, t) dx = 1, t ≥ 0. v(x, t) ≡ αu(αAx, α At), vt + vvx = vxx , A −∞ As a final scaling, let αA = M so that v(x, 0) is supported within (−1, 1): (8.2) vt + vvx =
κ vxx , A
v(x, t) ≡
M2 M u(M x, t), A A
v(x, 0) = 0 for |x| > 1,
∞
−∞
v(x, t) dx = 1, t ≥ 0.
When the effective Reynolds number R = A/κ is large, that is, when the nonlinearity, measured by A, is large compared to the viscosity κ, v(x, t)
8. Metastable States
99
is accurately approximated by the solution of the inviscid Burgers equation ht + hhx = 0. With the normalization ∞ h(x, 0) = 0 for |x| > 1, h(x, t) dx = 1, t ≥ 0, −∞
we learned from Section 6 of Chapter 3 that h(x, t) tends to the N -wave within time of order 1. On the other hand, the viscosity κ/A has a strong effect on the function v(x, t) after time t = O(1)A/κ. In summary, the solution v(x, t) of (8.2) is close to the N -wave for the period of O(1) < t < O(1)A/κ. Translating this into the original solution using (8.1), we conclude that u(x, t) is close to the N -wave for the time period (8.3)
O(1)
M2 M2 < t < O(1) , duration of metastable N -wave. A κ
Note that for a large effective Reynolds number R = A/κ 1, we have
M 2 /κ and the metastable N -wave appears in the solution as a good approximation for a long period of time.
M 2 /A
The N -waves represent accurate approximation for a long time period, but are not time-asymptotic states of the viscous solutions. Thus, for the Burgers solutions, the N -waves represent metastable states. We therefore conclude that the two limits, the time-asymptotic and zero dissipation limits, do not commute: lim lim u(x, t; κ) = lim lim u(x, t; κ).
κ→0 t→∞
t→∞ κ→0
The first limit is limκ→0 bD (x, t; A), with one time invariant, the total mass A; the second limit is the N -wave, with two time invariants, p and q. We know from the theory for N -waves that p + q = A. We have seen that the Burgers equation has a strong nonliinearity that is critical. Consider a weaker nonlinearity: ut + (
u3 )x = κuxx , 3
u3 )x = 0. 3 It turns out that the solution operator of this inviscid equation has only one time invariant. This can also be understood by studying
3 ut + ( u3 )x = 0, (8.4) u(x, 0) = Aδ(x), ut + (
which, unlike the Burgers case with two time invariants, has only one time invariant and a unique solution; see Exercise 4.
100
4. Burgers Equation
9. Notes The original motivation of Burgers [19, 20] for studying the Burgers equation was to analyze the interplay of the nonlinearity of the flux u2 /2 and the dissipation κuxx in the generation of turbulence for incompressible flows. The Burgers equation was proposed earlier by Bateman [3] based on analytical considerations. Cole [31] derived the Burgers equation from the Navier-Stokes equations in gas dynamics for the “excess velocity” near a sonic state. As it turns out, the inviscid Burgers equation (Hopf equation) and Burgers equation can be derived naturally from the Euler and Navier-Stokes equations in gas dynamics, with the function u(x, t) being the characteristic speed of the acoustic waves; see Chapter 7 and Chapter 10. The important Hopf-Cole transformation was discovered independently by Hopf [62] and Cole [31]. It was noted later that in fact the transformation was proposed in the 1906 book by Forsythe, [45]. After learning about Friedrichs’ N -waves from Burgers, Hopf began to apply the transformation to the study of N -waves for the inviscid equation. In this book, the transformation is called the Hopf-Cole transformation because Hopf’s paper [62] initiated the study of scalar hyperbolic conservation laws by considering the inviscid limit and N -waves. That the Hopf-Cole transformation can transform a strongly nonlinear equation into a linear equation is striking. This has motivated research in areas beyond fluid dynamics; see, for example, [108] for its application in soliton theory and [67] in the study of growth patterns.
10. Exercises 1. Solve the initial value problem
ut = 3uxx + sin x, 2 u(x, 0) = e−x . 2. Derive the heat kernel for multi-dimensional heat equation
ut = κ m i=1 uxi xi , u(x, 0) = δ(x), x ∈ Rm . 3. Generalize Duhamel’s principle (1.12) to the multi-dimensional heat equation with a source:
m ut = κ m i=1 uxi xi + g(x, t), x = (x1 , . . . , xm ) ∈ R , u(x, 0) = f (x).
10. Exercises
101
4. Consider the initial value problem for the heat equation ut = κuxx with bounded initial values u(x, 0) = u0 (x). Show that the solution u(x, t), for t > 0, is infinitely differentiable. 5. Find the Burgers kernel by the self-similarity argument, as was done in finding the heat kernel. (Hint: bD (x, t) = κt ψ( √xκt , A κ ).) 6. Find the solution of the porous media equation with a point source:
ut = (uux )x , u(x, 0) = Aδ(x), where A is a positive constant. 7. Solve the initial value problem (8.4) by showing that the solution consists of a shock wave canceling with a centered rarefaction wave. (Consider the case A > 0 separately from the case A < 0.) 8. In Proposition 7.2, it is shown that a solution with compact initial data to the Burgers equation converges to the Burgers kernel bD . Show that for ut + (u3 /3)x = uxx , the time-asymptotic states are multiples of the heat kernel. 9. Approximate the source g(x, t) by delta functions, similar to the procedure in Section 1, to derive the solution formula (6.7) for the problem (6.4). 10. Solve the initial value problem for the Burgers equation ut + uux =
κuxx , 1 for |x| < 1, u(x, 0) = 0 for |x| > 1, by going through the Hopf-Cole process (2.1), (2.2), and (2.3).
10.1090/gsm/215/05
Chapter 5
General Scalar Conservation Laws
We now consider the general scalar conservation law ut + f (u)x = 0, where f (u) may change signs. This general case has physical correlates. For instance, the stress-strain response for a shear motion in an elastic material is an odd function and therefore cannot be convex. We will see that in this general situation the notion of entropy condition and the resolution of the Riemann problem require new thinking and intricate analysis.
1. Viscous Shock Profiles The first task is to find the entropy condition for this general case. For convex laws, the entropy condition requires a shock to be compressive; see Section 5 of Chapter 2. It is intuitive that a shock should receive and not give information and therefore should be compressive. However, it turns out that this simple geometric consideration is not sufficient to identify the appropriate entropy condition for the general case. For that we turn to the basic idea that the hyperbolic conservation law is not complete in itself and should be supplemented by additional physical considerations. Thus we regard the hyperbolic conservation law ut +f (u)x = 0 as the zero dissipation (vanishing viscosity) limit of the viscous conservation law (uκ )t + f (uκ )x = κ(uκ )xx ,
κ → 0+.
As we have seen in Chapter 4, the viscosity term κ(uκ )xx smooths out the solutions (e.g. Proposition 7.7 in Section 7 of Chapter 4), and so solutions of viscous conservation laws do not contain discontinuous shock waves and 103
104
5. General Scalar Conservation Laws
are classical solutions. We can use the zero dissipation limit to identify the entropy condition for shock waves. Theorem 1.1. A shock wave (u− , u+ ) with speed s for the hyperbolic conservation law ut + f (u)x = 0 can be obtained from the zero dissipation limit if (1.1)
s=
f (u) − f (u− ) f (u+ ) − f (u− ) < for all u between u− and u+ . u+ − u− u − u−
Proof. For a small viscosity coefficient κ, locally in space and time an inviscid shock corresponds to a viscous shock profile, a traveling wave, for the viscous conservation law. A traveling wave is a permanent wave form uκ (x, t) = ψ(x − st) with a fixed speed s. To identify the entropy condition for a shock wave (u− , u+ ) of the inviscid conservation law, we consider such a traveling wave connecting the end states, ψ(±∞) = u± , for the viscous conservation law and study its zero dissipation limit as κ → 0+. It turns out that it is natural to use the variable (x − st)/κ: ⎧ κ κ κ ⎪ ⎨(u )t + f (u )x = κ(u )xx , (1.2) uκ (x, t) = φ(ξ), ξ ≡ x−st κ , ⎪ ⎩ φ(±∞) = u± . The scaling ξ = (x − st)/κ has the effect that when a solution of the form uκ (x, t) = φ(ξ) is plugged into the viscous conservation law, one obtains an ODE for the function φ that is independent of the viscosity coefficient κ,
−sφ (ξ) + f (φ(ξ)) = φ (ξ), (1.3) φ(±∞) = u± , which is integrated to yield
−sφ(ξ) + f (φ(ξ)) + A = φ (ξ), (1.4) φ(±∞) = u± , for some integration constant A. From the scaling ξ = (x − st)/κ, the profile has width κ and approaches the inviscid shock in the zero dissipation limit:
u− for x < st, x − st κ as κ → 0+. → (1.5) u (x, t) = φ κ u+ for x > st, The ODE (1.4) is of the general form φ (ξ) = g(φ(ξ)),
lim φ(ξ) = u− ,
ξ→−∞
lim φ(ξ) = u+ .
ξ→∞
1. Viscous Shock Profiles
105
For this to have a solution, it is necessary and sufficient that (1.6)
(1) g(u− ) = g(u+ ) = 0; (2) g(u) > 0 for all u between u− and u+ when u− < u+ ; (3) g(u) < 0 for all u between u− and u+ when u− > u+ .
In the present case, g(u) ≡ −su + f (u) + A. Condition (1) of (1.6) yields −su− + f (u− ) + A = −su+ + f (u+ ) + A, i.e. (u− , u+ ) satisfies the same Rankine-Hugoniot condition as an inviscid shock for the hyperbolic conservation law, (3.4) in Chapter 2: s=
f (u+ ) − f (u− ) . u+ − u−
Conditons (2) and (3) of (1.6) are summarized as (1.1) in the theorem. In summary, a shock wave (u− , u+ ) for the hyperbolic law can be obtained from the zero dissipation limit if it satisfies the condition (1.1). This completes the proof of the theorem. If we consider also the combination of several discontinuities with the same speed s, each satisfying (1.1), we can form a jump discontinuity with ¯) and a relaxed inequality. For instance, in figure 5.01, both profiles (u− , u ¯) ∪ (¯ u, u+ ) = (¯ u, u+ ) satisfy the condition (1.1). Their combination (u− , u (u− , u+ ) satisfies the relaxed condition, with the line through (u− , f (u− )) and (u+ , f (u+ )) lying above the graph of f (u), but touching the graph at (¯ u, f (¯ u)). This leads to the following Oleinik entropy condition for general scalar conservation laws. Definition 1.2. A shock wave (u− , u+ ) with speed s for the conservation law ut +f (u)x = 0 is admissible if it satisfies the Rankine-Hugoniot condition and the following Oleinik entropy condition: f (u) − f (u− ) f (u+ ) − f (u− ) ≤ for all u between u− and u+ . (1.7) s = u+ − u− u − u− Geometrically, (u− , u+ ) satisfies the Oleinik entropy condition if, on the graph of the flux function f (u), the line connecting (u− , f (u− ), and (u+ , f (u+ )) lies above (or below) the graph when u− > u+ (or u− < u+ ), as shown in figure 5.01. With the notation for the shock speed s = σ(u− , u+ ) used before, the Oleinik entropy condition is written as (1.8) σ(u− , u+ ) ≤ σ(u− , u) for all u between u− and u+ , s = σ(u− , u+ ),
σ(u− , u) ≡
f (u) − f (u− ) . u − u−
It is easy to see that the Oleinik entropy condition implies the compressibility property f (u− ) ≥ s ≥ f (u+ ).
106
5. General Scalar Conservation Laws
f (u) u− u¯
u+
u+
u u− − u u+
u u Figure 5.01. Oleinik entropy condition.
When the flux is convex, f (u) = 0, this reduces to the strong compressibility property f (u− ) > s > f (u+ ), the entropy condition (E) in Definition 5.1 of Chapter 2. Remark 1.3. The Oleinik entropy condition is a consequence of the vanishing viscosity limit. As we have seen in the derivation of the heat equation in Section 1 of Chapter 4, the positivity of the viscosity coefficient κ originates from the second law of thermodynamics in that heat flows from hotter places to colder places. Therefore the Oleinik entropy condition (1.7) is a consequence of the second law of thermodynamics. The same holds for the integral entropy condition (5.5) to be discussed later. This approach will be applied later for the study of systems of hyperbolic conservation laws. The entropy conditions are also termed admissibility conditions, as the conditions may not be directly related to the second law of thermodynamics for some physical models.
2. Riemann Problem We now use the rarefaction waves constructed before, (4.4) of Chapter 2, and shock waves satisfying the Oleinik entropy condition (1.7) in Definition 1.8 to form the solution of the Riemann problem ⎧ ⎪ ⎨ut + f (u)x = 0, x, u ∈ R, (2.1) ul , x < 0, ⎪ ⎩u(x, 0) = ur , x > 0. For the convex flux case, a Riemann solution is either a shock wave or a rarefaction wave, (1.5) or (1.6) of Chapter 3. For a general flux, a Riemann
2. Riemann Problem
107
solution may contain both shock and rarefaction waves. The geometric property of the Oleinik condiiton, figure 5.01, induces the following geometric construction of the Riemann solution. There are two cases to consider. For the case ul > ur , one forms the upper convex envelope of the graph of the flux function f (u). When the envelope coincides with the graph, e.g. (u2 , u1 ) in the left diagram of figure 5.02, we have f (u) < 0. For the part of the envelope consisting of lines detached from the graph, e.g. (ur , u2 ) and (u1 , ul ) in the left diagram of figure 5.02, the corresponding shocks satisfy the Oleinik entropy condition. The solution to the Riemann problem consists of these rarefaction and shock waves, e.g. shock wave (ul , u1 ), rarefaction wave (u1 , u2 ), and another shock wave (u2 , ur ) in the right diagram of figure 5.02. Note that when the envelope leaves the graph of f (u) between the end states ul and ur , the lines are tangent to the graph there, e.g. σ(ul , u1 ) = f (u1 ),
σ(u2 , ur ) = f (u2 )
in the left diagram of figure 5.02. This implies that the shock and rarefaction waves in the solution of the Riemann problem (ul , ur ) are tangent to each other and form a fan-like wave pattern; see the right diagram of figure 5.02. t
f(u) u1
u2 u1 ur
ul u
u2
ul
ur
ul
ur
x
Figure 5.02. Riemann problem: ul > ur .
For the case ul < ur , one forms the lower convex envelope, e.g. left diagram of figure 5.03. When the envelope coincides with the graph, e.g. (ul , u1 ) and (u2 , ur ) in the left diagram of figure 5.03, f (u) > 0 over the intervals and we have rarefaction waves. The parts of the envelope leaving the graph and becoming lines represent shock waves satisfying the Oleinik entropy condition, e.g. (u1 , u2 ) in figure 5.03. We also obtain a fan-like wave pattern, e.g. a combination of the rarefaction wave (ul , u1 ), shock wave (u1 , u2 ), and rarefaction wave (u2 , ur ) in the right diagram of figure 5.03. Example. Consider the Riemann problem
3 2 for x < 0, u = 0, u(x, 0) = ut + 3 x −2 for x > 0.
108
5. General Scalar Conservation Laws
f (u)
t u1 u2
ul
ur
ul
ur u2
u1
x
ur
ul
u
Figure 5.03. Riemann problem: ul < ur .
Since ul = 2 > ur = −2, we draw the upper envelope of the graph of f (u), as in the left diagram of figure 5.04. The line from u = 2 touches the graph of f (u) tangentially at u ¯: 23 3
3
− u¯3 = f (¯ u) = u ¯2 . 2−u ¯ This is solved to give u ¯ = −1. The Riemann problem is solved by the shock (2, −1) with speed u ¯2 = 1 followed by the rarefaction wave (−1, −2), shown √ in the right diagram of figure 5.04. Since f (u) = u2 , [f ]−1 (w) = w, and so within the wave region the solution is x x −1 x , 1 < < 2. [f ] ( ) = t t t
t
u3 3
−2 −1
x t=
2
1
u(x,t) =
−1
u
2 2
x= t
−2 −2
x t
2
x
Figure 5.04. Riemann problem: an example.
For a general flux, the solution of the Riemann problem is a composite wave pattern, a fan-like wave pattern, which may contain both shock and rarefaction waves. As a consequence, there are richer wave interaction phenomena for general flux as than for the convex flux studied in Section 3 of Chapter 3. Take the case of interaction of two shock waves. For convex
3. L1 Stability
109
conservation laws, when two shock waves (u1 , u2 ) and (u2 , u3 ) meet, they combine to form a single shock (u1 , u3 ); see figure 3.01 of Chapter 3. This holds also for general scalar conservation laws when two shock waves are of the same sign, (u2 − u1 )(u3 − u2 ) > 0. It is easy to see geometrically that if both (u1 , u2 ) and (u2 , u3 ) satisfy the Oleinik entropy conditon, then the combined wave (u1 , u3 ) also satisfies the Oleinik entropy condition. However, two shock waves can now be of opposite signs, (u2 − u1 )(u3 − u2 ) < 0, and their interaction can result in a wave pattern containing both shock and rarefaction waves; see figure 5.05. f (u)
t u4
u2 u3
u5
u1 u
u1 u2
u3
x
Figure 5.05. Wave interaction.
3. L1 Stability An analysis of a strong version of the L1 (x) contraction of the solution operator for a convex flux has been done in Section 4 of Chapter 3. We now show for a general flux the L1 (x) contraction property of the solution operator as a direct consequence of the Oleinik entropy condition (1.7). Theorem 3.1. Let u1 (x, t) and u2 (x, t) be two admissible solutions of the scalar conservation law ut + f (u)x = 0 and assume that R |u1 (x, 0) − u2 (x, 0)| dx < ∞. Then (3.1) ∞
−∞
|u1 (x, t2 ) − u2 (x, t2 )| dx ≤
∞
−∞
|u1 (x, t1 ) − u2 (x, t1 )| dx for t1 < t2 .
Proof. The proof follows that of Theorem 4.1 of Chapter 3. The L1 (x) distance is preserved except for the input from the two solutions crossing with a shock wave. Consider the situation shown in figure 5.05, which is the same as figure 3.06 of Chapter 3. We have from (4.6) in the proof of Theorem 4.1 of Chapter 3 that the input from the crossing corresponding to figure 5.06 is 2s(u− − u) − (f (u− ) − f (u)). Here u− > u > u+ and s is the speed of the shock (u− , u+ ). Thus from the Oleinik entropy condition
110
5. General Scalar Conservation Laws
u(x, t)
u2
q−
u− u q+
u1
x2m (t)
u+ x2 m+1(t)
x
Figure 5.06. L1 (x) contraction.
(1.8) and the Rankine-Hugoniot condition for the shock wave (u− , u+ ), the input is f (u) − f (u− ) = 2(u− − u) σ(u− , u+ ) − σ(u− , u) ≤ 0. 2(u− − u) s − u − u− This completes the proof of the theorem.
In the proof of Theorem 3.1, the two solutions are assumed to be piecewise smooth. As pointed out in Remark 4.5 of Chapter 3, for convex laws, a general solution can be approximated by piecewise smooth solutions in the L1 (x) norm. This is done by approximating the initial function by step functions. Then the number of shock waves does not increase in an approximate solution. This is because the interaction of two shock waves leads the shocks to combine and therefore the number of shocks decreases. However, for the general flux, the example given in figure 5.05 shows that the number of waves does not necessarily decrease after interaction. Therefore the reasoning for convex laws does not directly apply to a general flux. The number of waves can increase after interaction. Nevertheless, it can be shown that the number of shock waves in an approximate solution is bounded for all time if the number of inflection points of the flux function f (u) is finite. One sees in figure 5.05 that the increase in number of waves due to interaction is caused by the inflection points of the flux function f (u) and that there is also a finite amount of wave cancellation due to the interaction. This means that for the number of waves to increase indefinitely, the initial total wave strength needs be unbounded. In conclusion, each approximate solution is piecewise smooth and the above proof of Theorem 3.1 is justified. In fact, Theorem 3.1 holds for any flux function. An alternative approach
4. Scattering Wave Patterns
111
is to consider viscous conservation laws and then take the zero dissipation limit; see (6.9). The above proof of Theorem 3.1 offers a clear geometric understanding of the L1 (x) contraction property.
4. Scattering Wave Patterns Solving the Riemann problem requires the construction of the elementary waves, shock and rarefaction waves, and the identification of the entropy condition. There is another basic reason for studying the Riemann problem, namely that the Riemann solutions represent the only scattering, noninteracting wave patterns. This fact results from the simplifying mechanism of the solution operator due to the nonlinearity of the flux. As a consequence, the Riemann solutions represent, locally in time as well as time-asymptotically, the behavior of general solutions. For a convex flux, a solution u(x, t) with limiting states at x = ±∞ tends to either a shock, Theorem 5.5, or a rarefaction wave, Theorem 5.6, of Chapter 3. This is true also for the general situation where the flux f (u) is not linear in the sense that f (u) has isolated zeros. For convex laws, the convergence to a shock can be in finite time and the convergence to a rarefaction wave is at the rate of t−1/2 . However, the rich wave interaction phenomena for the conservation laws with general flux prevent a simple characterization of the process of convergence toward the time-asymptotic states. In contrast to the quantitative analysis in Section 5 and Section 6 of Chapter 3, we adopt a qualitative approach as follows. Suppose that a solution u(x, t) has limiting states u(−∞, t) = ul and u(∞, t) = ur and consists of elementary waves (uj , uj+1 ), j = 0, ±1, ±2, . . . , with (uj , uj+1 ) lying to the left of (uk , uk+1 ) for j < k. There is a natural way to measure the distance between the general wave pattern W = {(uj , uj+1 ), j = 0, ±1, ±2, . . . } and the wave pattern of the solution to the Riemann problem (ul , ur ). The measure is the degree of wave interaction of W defined in following steps. 1. For an elementary wave γ = (u− , u+ ), let λ− (γ) (or λ+ (γ)) be the left (or right) speed of propagation of γ. For instance, in figure 5.02, the solution α ≡ (ul , ur ) has shock wave (ul , u1 ) next to the state ul and shock wave (u2 , ur ) next to the state ur , and we set λ± to be the shock speeds: λ− (α) = σ(ul , u1 ),
λ+ (α) = σ(u2 , ur ).
Another example is figure 5.03, where the solution has rarefaction waves on both ends and we set λ± (α) to be the characteristic speeds: λ− (α) = λ(ul ) = f (ul ),
λ+ (α) = λ(ur ) = f (ur ).
112
5. General Scalar Conservation Laws
2. Given two waves α = (uj , uj+1 ) and β = (uk , uk+1 ), with α with lying to the left of β, the “angle” θ(α, β) between them is defined as (4.1)
θ(α, β) = λ− (β) − λ+ (α).
This difference is of the same order as the angle between the right side of α and the left side of β, and so we conveniently call it the angle between the two waves. Note that when θ(α, β) < 0, the two waves are approaching and will interact in the future. The quantity θ(α, β) measures the degree of the approaching between the waves α and β. For a convex flux, f (u) = 0, two shock waves α and β next to each other interact and θ(α, β) is of the order of −(|α| + |β|). A shock and a rarefaction wave interact with the angle between them proportional to the strength of the shock. For two rarefaction waves, the angle between them is zero and they don’t interact. For a general flux, a shock and a rarefaction wave next to each other may not interact; the same holds for two shock waves. In fact, the solution to the Riemann problem consists of a non-interacting wave pattern and may contain both shock and rarefaction waves. In other words, the notion of degree of interaction should be defined in terms of the angle between waves, and not in terms of wave types. The following definition is natural in view of the above two considerations. Definition 4.1. Consider a wave pattern W = {(uj , uj+1 ), j = 0, ±1, ±2, . . . } consisting of elementary waves, with αj = (uj , uj+1 ) lying to the left of αk = (uk , uk+1 ) for j < k. The degree of interaction of the wave pattern W is defined as (4.2) Q(W) ≡ {|αk αl ||θ(αj , αk )|, k, l = 0, ±1, ±2, θ(αj , αk ) < 0}, where θ(αj , αk ) is the angle between αj and αk , defined in (4.1). It is obvious that when the interaction measure Q(W) = 0, the wave pattern is the one given by the solution of the Riemann problem (ul , ur ). This means that the waves in W are the waves in the Riemann solution, with their ordering from left to right unaltered. It can be shown that if Q(W) is small, then W is a small perturbation of the waves in the Riemann solution. For a given time t, the wave pattern W in u(x, t) in general has a positive interaction measure, Q(t) = Q(W) > 0, and there will be wave interactions at later times. Each interaction reduces the interaction measure, d Q(t) ≤ 0. dt
5. Entropy Pairs
113
It can be shown that time-asymptotically, the wave pattern becomes noninteracting, Q(t) → 0 as t → ∞, and the solution u(x, t) tends to the Riemann wave pattern. Such an approach is qualitative and does not yield the rate of convergence of a solution to the corresponding Riemann solution. In general, it is not simple to obtain the rates of convergence.
5. Entropy Pairs The notion of an entropy pair (η(u), q(u)) for a conservation law ut +f (u)x = 0 has been studied for convex laws; see Chapter 3, Definition 7.1. The pair has the property that η(u) is convex, η (u) > 0, and the compatibility condition q (u) = η (u)f (u) is satisfied so that for smooth solutions of the conservation law we have the entropy equality η(u)t + q(u)x = 0. For convex laws, there is a third-order estimate for the the entropy inequality η(u)t + q(u)x ≤ 0, given in (7.3) and (7.4) of Chapter 3. The estimate is obtained directly from the compressibility of shock waves and the convexity of the flux. For general laws, there is also the entropy inequality, but no third-order dissipation. To show the entropy inequality, one may use the Oleinik entropy condition (1.7) directly. We apply the zero dissipation limit approach, in the same spirit as the derivation of the Oleinik entropy condition in Section 1. Consider the viscous conservation law (uκ )t + f (uκ )x = κ(uκ )xx .
(5.1)
The solutions of the viscous conservation laws are smooth; see e.g. Proposition 7.2 of Section 7 in Chapter 4. Thus we may apply the chain rule to obtain from (5.1) the equation for the entropy pair: η(uκ )t + q(uκ )x = κη (uκ )(uκ )xx .
(5.2)
We now analyze the zero dissipation limit κ → 0+ of the equation for the entropy pair. We will obtain an entropy inequality in the limit. To see this, we first perform the energy estimate by integrating the viscous conservation law (5.1) times uκ to obtain the energy identity: ∞ d ∞ (uκ )2 (x, t) dx + κ ((uκ )x )2 (x, t) dx = 0, and so dt −∞ 2 −∞
∞
(5.3) −∞
(uκ )2 (x, t) dx − 2
∞ −∞
(uκ )2 (x, 0) dx 2 t = −κ 0
∞ −∞
((uκ )x )2 (x, s) dx ds.
114
5. General Scalar Conservation Laws
Here we have assumed for simplicity that the solution uκ (x, t) tends to zero as |x| → ∞. We have seen that as κ → 0+ , shock waves emerge and |uκx | becomes unbounded around the resulting shocks. We have seen in Section 1 that the width of a single shock profile uκ (x, t) = φ((x − st)/κ) is of the order of κ and its derivative (uκ )x = φ /κ is of the order of 1/κ. Therefore, for a shock profile, 2 κ φ (ξ) 2 φ (ξ) 2 1 ) dx = ( ) κ dξ = O(1) . (u )x (x, t) dx = ( κ κ κ Consequently, when the solution contains shocks, ∞ κ ((uκ )x )2 (x, t) dx ds = O(1), −∞
and the right-hand side integral in (5.3) does not go to zero as κ → 0. We next integrate the equation (5.2) for η(u): d dt
∞
κ
∞
η(u )(x, t) dx + κ −∞
∞
−∞
η (uκ )((uκ )x )2 (x, t) dx = 0, or t
κ
∞
η(u )(x, t) dx + κ −∞
0
−∞
η (uκ )((uκ )x )2 (x, s) dx ds ∞ η(uκ )(x, 0) dx. = t∞
−∞
From the above energy analysis, as κ → 0+ , κ 0 −∞ η (u)(ux )2 (x, s) dx ds remains positive when shocks are present. Thus in the zero dissipation limit, κ → 0+ , we have the inequality ∞ ∞ η(u)(x, t) dx ≤ η(u)(x, 0) dx. (5.4) −∞
−∞
The inequality is strict when there are shock waves in the solution before time t. This leads to the following alternative condition, besides the Oleinik entropy condition (1.7), for the admissibility of solutions. Definition 5.1. A weak solution of the conservation law ut + f (u)x = 0 is admissible with respect to the entropy pair (η(u), q(u)) if η(u)t + q(u)x ≤ 0
(5.5)
in the sense of distributions. In other words, for any non-negative smooth function φ(x, t) ≥ 0 with bounded support in t > 0, ∞ ∞ η(u)φt + q(u)φx dx dt ≥ 0. (5.6) 0
−∞
5. Entropy Pairs
115
Theorem 5.2. Suppose that u(x, t) is a weak solution of the conservation law ut + f (u)x = 0 obtained from the zero dissipation limit of solutions of the viscous conservation law: u(x, t) = lim uκ (x, t), κ→0+
(uκ )t + f (uκ )x = κ(uκ )xx .
Then for any entropy pair (η(u), q(u)) with η (u) ≥ 0, η(u)t + q(u)x ≤ 0 in the sense of distributions. Proof. Integrate the identity (5.2) for the entropy pair times a positive test function φ(x, t): ∞ ∞ κ κη (uκ )(uκ )xx φ dx. η(u )t + q(uκ )x φ dx = −∞
−∞
Using integration by parts, this yields ∞ κ η(u )φt + q(uκ )φx dx − −∞ ∞ κ κ 2 η (u )((u )x ) φ dx − κ = −κ −∞
∞ −∞
η (uκ )(uκ )x φx dx.
The first term on the right-hand side is non-positive as η ≥ 0. It remains to show that the second term tends to zero as κ → 0+. Apply the CauchySchwarz inequality to the second term: ∞ ∞ 1 ∞ 1 1 2 2 κ κ κ 2 κ 2 κ κ η (u )(u ) φ dx ≤ |(u ) | dx |η (u )φ | dx . κ x x x x 2 −∞ −∞ −∞ ∞ The uniform boundedness of κ −∞ |(uκ )x |2 dx has been shown by the energy estimate (5.3). Therefore ∞ ∞ 1 2 κ κ κ 2 κ η (u )(u )x φx dx = O(1) κ |η (u )φx | dx , −∞
−∞
which tends to zero as κ → 0+ because the function φ is a smooth test function and the solutions uκ are bounded uniformly in κ. This completes the proof of the theorem. The entropy inequality η(u)t + q(u)x ≤ 0, (5.5), holds if across each shock wave (u− , u+ ) with speed s, (5.7) −s η(u+ ) − η(u− ) + q(u+ ) − q(u− ) ≤ 0. The following proposition shows that the two notions of admissibility, Definition 1.8 and Definition 5.1, are equivalent. Theorem 5.3. A shock wave (u− , u+ ) satisfies the Oleinik entropy condition (1.7) if and only if it satisfies (5.7) for all entropy pairs (η(u), q(u)).
116
5. General Scalar Conservation Laws
Proof. Since the entropy inequality is a consequence of the zero viscosity limit, the entropy inequality (5.7) should be implied by the Oleinik entropy condition (1.7). This can be shown directly. Suppose that the shock (u− , u+ ) satisfies the strict Oleinik entropy condition (1.1) so that it corresponds to a viscous shock profile φ(ξ) satisfying (1.4): −sφ (ξ) + f (φ(ξ))φ (ξ) = φ (ξ), Set
φ(±∞) = u± .
F (ξ) ≡ −s η(φ(ξ) − η(u− ) + q(φ(ξ)) − q(u− ).
We have F (−∞) = 0 and, from the compatibility condition q = η f , F (ξ) = −sη (φ(ξ))φ (ξ) + q (φ(ξ))φ (ξ) = η (φ(ξ)) −sφ (ξ) + f (φ(ξ))φ (ξ) = η (φ(ξ))φ (ξ). Integrate this and use the convexity φ > 0 to obtain F (∞) − F (−∞) = F (∞) = −s η(u+ ) − η(u− ) + q(u+ ) − q(u− ) ∞ ∞ ∞ F (ξ) dξ = η (φ(ξ))φ (ξ) dξ = − η (φ(ξ))(φ (ξ))2 < 0. = −∞
−∞
−∞
This proves (5.7) when (u− , u+ ) satisfies the strict Oleinik entropy condition. When (u− , u+ ) is a combination of shocks satisfying the strict Oleinik entropy condition, we apply the above analysis to each shock with the same speed and sum the resulting entropy inequalities to show that (5.7) holds in general. Suppose next that a shock (u− , u+ ) satisfies the entropy inequality (5.7) for all entropy pairs (η(u), q(u)). We need to show that (u− , u+ ) satisfies the Oleinik entropy condition (1.7). For this we consider the singular entropy pair with parameter k: η(u) = |u − k|,
q(u) = sign(u − k)(f (u) − f (k)).
In this case, η (u) = δ(u − k) and is localized and non-negative in the sense of distributions. Consider a shock (u− , u+ ) with u− < u+ . Choose the parameter k to be between u− and u+ ; then the entropy dissipation is − s η(u+ ) − η(u− ) + q(u+ ) − q(u− ) = −s (u+ − k) − (k − u− ) + f (u+ ) − f (k) − f (k) − f (u− ) = 2 s(k − u− ) − (f (k) − f (u− )) ≤ 0, where the Rankine-Hugoniot condition has been used. This yields the Oleinik entropy condition (1.7) by varying the constant k between u− and u+ . The same holds for the case of u− > u+ . Such singular entropy pairs can be approximated by smooth entropy paris, and so the Oleinik entropy
6. Multi-Dimensional Laws
117
condition holds if the entropy inequality (5.7) holds for all smooth entropy pairs. This completes the proof of the theorem.
6. Multi-Dimensional Laws There is a well-posedness theory for scalar conservation laws in several space dimensions, ut +
(6.1)
m
fi (u)xi = 0.
i=1
We illustrate this using the vanishing viscosity method and the entropy estimates as follows. Consider the entropy pairs (η(u), q(u)) = η(u), (q1 (u), . . . , qm (u)) satisfying η (u) ≥ 0,
qi (u) = η (u)fi (u), i = 1, 2, . . . , m.
There are many choices of entropy pairs; for any chosen entropy η(u) with u η (w)fi (w) dw. Consider η (u) > 0, the entropy flux is given by qi (u) = the associated viscous conservation laws κ
(6.2)
(u )t +
m
fi (uκ )xi = κΔuκ .
i=1
Here Δuκ =
m i=1
∂2 uκ ∂xi ∂xi
Rm .
is the Laplacian in For brevity of notation, we write u = uκ . The same analysis as in the proof of Theorem 5.2 for one spatial dimension applies to the multi-dimensional case. We start with the equation for the entropy pairs: m ∂t η(u) + ∂xi qi (u) = κΔη(u) − κη (u)|∇u|2 . i=1
Take a non-negative smooth function φ(x, t) with bounded support. Multiply the above identity by φ(x, t) and integrate, noting that η (u) > 0, to obtain the entropy inequality ∞
0
Rm
∂t φη(u) +
m i=1
∂xi φqi (u) dx dt +
φ(x, 0)η(u(x, 0) dx Rm
∞
≥κ
Δφη(u) dx dt. 0
Rm
118
5. General Scalar Conservation Laws
By the same analysis as in the proof of Theorem 5.2, the last integral tends to zero as κ → 0 by the energy estimate, and we obtain the entropy inequality for the zero dissipation limit: ∞ m ∂xi φqi (u) dx dt (6.3) ∂t φη(u) + 0
Rm
i=1
+ Rm
φ(x, 0)η(u(x, 0)) dx ≤ 0as κ → 0+.
In the sense of distributions this means that η(u)t + ∇x · q(u) ≤ 0.
(6.4)
For scalar laws, any convex function η(u) qualifies as an entropy function. The entropy inequality (6.3) holds also for the limit of entropy pairs. Thus η(u) can be non-smooth. Because of the abundance of entropy pairs, it turns out that if a function u(x, t) satisfies the entropy inequality for all entropy pairs, then it is a weak solution. Thus we have a third definition of weak solutions, besides the ones in Section 3 of Chapter 2. It should be emphasized that this one works only for scalar laws. Definition 6.1. A bounded measurable function is an admissible weak solution of (6.1) if (6.3) holds for all entropy pairs (η(u), q(u)) and all nonnegative test functions φ(u). We now study the continuous dependence of the solutions on their initial data for hyperbolic conservation laws (6.1) by the zero dissipation limit process. Solutions to the parabolic equation (6.2) are smooth, allowing us to use calculus. Consider two solutions u(x, t) = uκ (x, t) and v(x, t) = v κ (x, t) of (6.2). Let (η(u), q(u)) be an entropy pair. Straightforward calculations yield (6.5) η(u − v)t +
m
η (u − v) fi (u) − fi (v)
i=1 m
−
xi
η (u − v) fi (u) − fi (v) (u − v)xi
i=1
= κΔη(u − v) − κη (u − v)|∇u − ∇v|2 . Integrate this to obtain the inequality m t η(u − v)(x, t) dx − (6.6) Rm
i=1
0
Rm
dx η (u − v) fi (u) − fi (v) xi
≤
Rm
η(u − v)(x, 0) dx.
6. Multi-Dimensional Laws
119
The entropy function η(u) is smooth and convex, η (u) > 0. We take a series of entropy functions that approach the particular one:
0 for u ≤ 0, 0 for u ≤ 0, ˜ (u) = (6.7) η˜(u) = q u for u > 0, f (u) − f (0) for u > 0. ˜ ), (6.6) becomes, for (u − With the choice of the entropy pair (η, q) = (˜ η, q v)(x, t) → 0 as |x| → ∞, (u − v)+ (x, t) dx ≤ (u − v)+ (x, 0) dx, (6.8) Rm
Rm
where (u − v)+ (x, t) is the positive part of the function:
0 when (u − v)(x, t) < 0, (u − v)+ (x, t) = (u − v)(x, t) when (u − v)(x, t) > 0. Changing u − v to v − u in the above analysis, we arrive at the L1 (x) continuous dependence property: |u(x, t) − v(x, t)| dx ≤ |u(x, 0) − v(x, 0)| dx, t > 0. (6.9) Rm
Rm
As the dissipation coefficient κ → 0+ , the L1 continuous dependence property (6.9) holds also for solutions of the hyperbolic conservation laws (6.1). ¯ (u, k), The particular choice of entropy pair η¯ = η¯(u, k) and q¯ (u) = q where η¯(u) ≡ |u − k|,
(6.10)
¯ (u) ≡ sign(u − k)(f (u) − f (k)), q
with the free parameter k ∈ R, is sufficient for the definition of admissible solution in Definition 6.1. Thus an alternative definition, the Kruzkov definition, is the following: Definition 6.2. A bounded measurable function u(x, t) is an admissible weak solution for the initial value problem ut +
(6.11)
m
fi (u)xi = 0,
u(x, 0) = u0 (x),
i=1
if for any non-negative smooth function φ(x, t) of compact support in x ∈ Rm , t > 0, and for any constant k ∈ R, ∞ |u(x, t) − k|φt + sign u(x, t) − k (6.12) 0
Rm
m fi (u(x, t)) − fi (k) φxi dx dt ≥ 0; · i=1
lim u(x, t) = u0 (x, t) for almost all x ∈ Rm .
t→0+
120
5. General Scalar Conservation Laws
By computations similar to those for (6.6)–(6.9), one obtains the following theorem through the vanishing viscosity method. Theorem 6.3. There exists a solution operator for (6.1) satisfying the L1 (x) contraction property; that is, for any two solutions u(x, t) and v(x, t), |u(x, t) − v(x, t)| dx ≤ |u(x, 0) − v(x, 0)| dx, t > 0. (6.13) Rm
Rm
7. Notes The theory of convex scalar conservation laws, presented in Chapter 3, drew inspiration from the theory for the Hopf and Burgers equations [62]. The theory for general scalar conservation laws represents another major step in the development of the shock wave theory. Substantial progress has been made by the Russian school, Oleinik [110] for a single space variable and Kruzkov [71] for several space variables. The zero dissipation limit approach is an effective way of identifying the Oleinik entropy condition for inviscid shock waves of scalar conservation laws, as described in Section 1, for the study of the entropy dissipation measure in Section 5, and also for the well-posedness in Section 6. The shock wave theory for scalar conservation laws as presented in Chapter 3 and this chapter is of significance in its own right. It also serves as a prelude to the study of systems of hyperbolic conservation laws, the focus of Chapter 7 to Chapter 9. For the study of scalar laws in multiple dimensions, the usual focus is on the Hamilton-Jacobi equations. An important corollary of the shock wave theory is generalization of the effective use of the maximum principle for scalar conservation laws, e.g. (6.5)–(6.9), to the Hamilton-Jacobi equations, e.g. [33, 72]. The approach in Section 4 is qualitative and does not yield the rate of convergence of a solution to the corresponding Riemann solution. The convergence to the Riemann solution for general solutions of hyperbolic and viscous conservation laws with rates is an interesting problem.
8. Exercises 1. Solve the Riemann problem (ul , ur ) for ut + (u3 )x = 0 for varying initial states ul and ur . 2. Find examples for which the Oleinik estimate for a conservation law with convex flux, Exercise 4 of Chapter 2, does not hold for a non-convex flux f (u).
8. Exercises
121
3. Show that, for a convex flux f (u) > 0, (5.7) is equivalent to the entropy condition u− > u+ , Definition 5.1 in Chapter 2. 4. In Section 6, by drawing a connection with the viscous shock profile, the Oleinik entropy condition (1.7) is shown to imply the entropy inequality (5.7) for any convex entropy function η. Show this directly without using the viscous profile. 5. Follow the argument in Remark 4.5 of Chapter 3 to show that the entropy inequality η(u)t + q(u)x ≤ 0 holds for the exact solution u(x, t) obtained as a limit of piecewise continuous solutions constructed in Section 3 of Chapter 3. 6. Consider the conservation law ut + um /m x = 0 and the entropy function η(u) = u2 . Here m is an integer greater than 2. Find the corresponding entropy flux q(u) and compute the entropy dissipation measure −s(η(u+ ) − η(u− )) + q(u+ ) − q(u− ) for a shock (u− , u+ ) with u− > 0 and u+ = 0. Compare the result with the estimate (7.4) of Chapter 3 for convex laws. 7. Consider the one-dimensional Hamilton-Jacobi equation ht + f (hx ) = 0. Show that u ≡ hx satisfies a hyperbolic conservation law. In Theorem 3.1 in Section 3 of Chapter 3, piecewise smooth solutions are constructed. Describe the corresponding solutions for the Hamilton-Jacobi equation and show that they are Lipschitz continuous and piecewise differentiable. Note that a Hamilton-Jacobi equation of general form ht + f (∇x u) = 0 can be converted to a hyperbolic conservation law only when the space dimension is one.
10.1090/gsm/215/06
Chapter 6
Systems of Hyperbolic Conservation Laws: General Theory
Studying physical phenomena usually requires the study of the coupling effects of more than one physical quantity. These effects are modeled by systems of equations. Consider the following system of conservation laws for n dependent variables u = (u1 , u2 , . . . , un )T ∈ Rn in m space dimensions x = (x1 , x2 , . . . , xm ) ∈ Rm :
(0.1)
ut + ∇x · F(u) = 0, or ut +
m F j (u) x = 0, j
j=1
with the flux matrix functions (0.2) F(u) = (F 1 (u), F 2 (u), . . . , F m (u)) ∈ Rn×m , F j (u) = (fj1 (u), . . . , fjn (u))T ∈ Rn , j = 1, 2, . . . , m. There are several basic models of this type in continuum physics, e.g. gas dynamics, mechanics, water waves, and magnetohydrodynamics. The Euler equations in gas dynamics, (0.5) of Chapter 1, have n = d+2 dependent variables when the spatial dimension is m = d. There is one equation for mass conservation, d equations for momentum conservation, and one equation for energy conservation. For more complex physical situations, the number of dependent variables increases. For instance, in nonlinear elasticity, there are more momentum equations to take into account momentum flux besides 123
124
6. Systems of Hyperbolic Conservation Laws: General Theory
the pressure and convection. In magnetohydrodynamics, there are magnetic fields, besides the gas velocity, to take into account. It is often more natural to consider systems of the form g(u)t + ∇x · F(u) = 0,
(0.3)
where the conserved vector g(u) is a function of other chosen dependent variables u with basic physical interpretations. For instance, the Euler equations in gas dynamics are of the form (0.1) for u ≡ (ρ, ρv, ρE)T , the density per unit volume of the conserved quantities, namely the mass, momentum, and energy. If one chooses the more basic variables u ≡ (ρ, v, e)T , the density, velocity, and internal energy, then the Euler equations are of the form (0.3). In this chapter we outline the basic theory for the general systems.
1. Hyperbolicity We are interested in systems which are hyperbolic. The classification of partial differential equations is in the infinitesimal sense. To be hyperbolic, a system is required to carry infinitesimal waves in every spatial direction ξ ∈ Rm , |ξ| = 1, with wave speed s(ξ) and wave shape φ(ξ ·x−st) depending on the local value of a chosen state u0 . This informal definition of hyperbolicity has the following precise meaning. Consider a small perturbation εv of a given constant state u0 : u = u0 + εv. A system is hyperbolic if in the limit ε → 0, the normalized perturbation v possesses waves. Plug this into (0.1) and apply the Taylor expansion to obtain m ∇u F j (u0 )v xj = O(1)ε2 . ε vt + j=1
The infinitesimal waves are solutions of the system ignoring the higherorder term O(1)ε2 . Thus the infinitesimal waves are governed by the system linearized around a given constant state u0 , m Aj v xj = 0, Aj = Aj (u0 ) ≡ ∇u F j (u0 ). (1.1) vt + j=1
For a system of the general form (0.3), the linearized system becomes (1.2) A0 v t +
m
Aj v xj = 0,
A0 ≡ ∇u g(u0 ), Aj = Aj (u0 ) ≡ ∇u F j (u0 ).
j=1
For system (1.2) the matrix A0 is assumed to be positive definite. System (0.3) is hyperbolic if for any given state u0 , the corresponding linearized system (1.2) carries waves in any given wave direction ξ. In other
1. Hyperbolicity
125
words, for any unit vector ξ ∈ Rm , there exists a function φ(η) of one variable such that the plane wave in figure 6.01, (1.3)
v(x, t) = φ(η),
η ≡ ξ · x − st,
is a solution of (1.2) for some scalar, real speed s = s(ξ). In particular, the wave shape φ and wave speed s depend only on the wave direction ξ and the given state u0 .
at t + Δ t at t ξ sΔt
v= v0
x space
v =v0
Figure 6.01. Plane wave.
Plug the form (1.3) into the system (1.2) to get (1.4)
m ξj Aj φ = 0. −sA0 + j=1
This leads to the following formal definition. Definition 1.1. System (0.3) is hyperbolic if for any given state u0 and for each unit vector ξ ∈ Sm−1 in Rm , the eigenvalue problem for the associated linear system (1.2), (1.5)
−λA0 + A r = 0,
A = A(ξ) ≡
m
ξj Aj ,
j=1
has real eigenvalues λ = λ(ξ) = λ(ξ, u0 ). The eigenvalue s ≡ λ(ξ) is the wave speed and the eigenvector r = r(ξ) ≡ φ is the corresponding wave shape. As the system (1.2) is the linearization of (0.3) around the given state u0 , both the speed λ(ξ, u0 ) and the wave shape r(ξ, u0 ) depend only on the wave direction ξ and the given base state u0 .
126
6. Systems of Hyperbolic Conservation Laws: General Theory
2. Entropy and Symmetry We now consider the basic notion of entropy pairs for a general system (0.1). This notion has been studied for scalar laws; see Section 7 of Chapter 3 and Section 6 of Chapter 5. For systems with physical contexts, the notion of entropy originates from the second law of thermodynamics. We will see this for the Euler equations in gas dynamics in the next section. Here we present a mathematically basic relationship between the existence of entropy and the symmetrization of the conservation laws. The definition of entropy pairs for general systems generalizes directly that for scalar laws, Definition 7.1 of Chapter 3. Definition 2.1. A pair (η(u), q(u)) is called an entropy pair, with scalar entropy function η(u) and vector entropy flux function q(u) ∈ Rm , for the system of conservation laws ut + ∇x · F(u) = 0 if 1. η(u) is a convex function, that is, η (u) is positive definite; 2. η(u)t + ∇x · q(u) = 0 for smooth solutions u of the conservation laws or, equivalently, the following compatibility condition is satisfied: η (u)F (u) = q (u).
(2.1)
For scalar laws, u ∈ R, any convex function η(u) can be an entropy function. The compatibility condition η (u)f (u) = q (u) is satisfied by choosing the entropy flux function as u η (v)f (v) dv. q(u) ≡ For a system of two conservation laws in one spatial dimension, T 2 T 2 u
= (u, v) ∈ R , f (u) = (f (u, v), g(u, v)) ∈ R , ut + f (u, v)x = 0, vt + g(u, v)x = 0,
the entropy flux q(u, v) is a scalar function for the one space dimension. The compatibility conditions (2.1) for an entropy pair (η(u, v), q(u, v)) are fu fv ηu ηv = qu qv , gu gv or, in component form, ηu fu + ηv gu = qu ,
ηu fv + ηv gv = qv .
Thus the existence of the entropy flux function q(u, v) requires the entropy function η(u, v) to satisfy the compatibility condition (ηu fu + ηv gu )v = quv = qvu = (ηu fv + ηv gv )u .
2. Entropy and Symmetry
127
This is a second-order linear equation for the entropy function η as a function of the independent variables (u, v): (2.2)
fv ηuu + (gv − fu )ηuv − gu ηvv = 0.
The hyperbolicity of the two conservation laws means that fv fu − λ det =0 gu gv − λ has real eigenvalue λ. Thus (fu + gv )2 − 4(fu gv − fv gu ) > 0, or (fu − gv )2 − 4fv gu > 0, which is also the hyperbolicity condition for equation (2.2). Thus the entropy function η(u, v) satisfies the linear hyperbolic equation (2.2). A linear hyperbolic equation can be solved, at least locally, by the method of characteristics. This yields many entropy functions η(u, v). The convexity condition η > 0 can also be attained locally. For specific 2 × 2 systems, the entropy function may exist globally. In general, for a system ut + ∇x · F(u) = 0 of n equations for u = (u1 , u2 , . . . , un )T ∈ Rn , in m space dimensions x = (x1 , x2 , . . . , xm ) ∈ Rm , the entropy flux q(u) = (q 1 (u), . . . , q m (u)) ∈ Rm . The compatibility condition η (u)F (u) = q (u) imposes mn(n − 1)/2 conditions on the entropy function η(u). For scalar laws, n = 1, there is no condition. For 2 × 2 laws, n = 2 and m = 1, there is one condition for the scalar function η(u). As we have seen above, for these two situations, entropy functions always exist. In general, there will be more than one condition and the entropy function in general does not exist, as there would be more than one constraint on the scalar function η(u). In short, the compatibility condition (2.1) is over-determined for general systems and the existence of the entropy pair is expected only for special systems. Physically, the existence of the entropy function is viewed as a constitutive hypothesis for the system. In fact, for many physical systems, there exists essentially only one entropy function, that originating from the second law of thermodynamics. Existence of the entropy function has a fundamental mathematical implication: A system endowed with entropy pairs can be written in symmetric hyperbolic form so that the local well-posedness theory of smooth solutions for symmetric systems applies. It is completely hyperbolic and has energy estimates. We now verify this statement. Recall from (0.2) that the flux matrix functions are of the form F(u) = (F 1 (u), F 2 (u), . . . , F m (u)),
F i (u) = (fi1 (u), . . . , fin (u))T .
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6. Systems of Hyperbolic Conservation Laws: General Theory
We let (2.3)
A0 (u) ≡ η (u),
Ai (u) ≡ η (u)F i (u), i = 1, 2, . . . , m.
˜, The Legendre transformation u → u convex function η(u) is
η(u) → η˜(˜ u) corresponding to a
˜ ≡ η (u), η˜(˜ u) ≡ u. u
(2.4)
From this, the transformation is one-to-one because the convexity of η(u) means that η (u) is positive definite: −1 −1 ˜ ∂u ∂u = η (u) = η˜ (˜ = η˜ (˜ u) , u) = η (u) . (2.5) ˜ ∂u ∂u Theorem 2.2. For a given convex entropy function η(u), a corresponding entropy flux q(u) exists for the system of conservation laws ut +∇x ·F(u) = 0 ˜, if and only if the system of conservation laws in the new variable u m ˜ 0 = η˜ (˜ ˜ i = F (˜ ˜ i (˜ ˜ 0 (˜ u)˜ ut + u)˜ uxi = 0, A u), A u))˜ η (˜ u), A (2.6) A i η (˜ i=1
˜ i , i = 0, 1, . . . , m, and the is a symmetric system with symmetric matrices A ˜ a positive definite matrix A0 . Proof. That the system (2.6) is a consequence of the conservation laws ut + ∇x · F(u) = 0 for smooth solutions follows immediately from (2.4) and (2.5). Suppose that the entropy pair exists. Differentiate the compatibility condition η (u)F i (u) = qi (u), (2.1), to get qi (u) = η (u)F i (u) + η (u)F i (u).
Both qi (u) and fij (u), j = 1, . . . , n, are symmetric. From this and the fact that F i (u) ≡ (fi1 (u), . . . , fin (u))T , we see that both qi (u) and η (u)F i (u) are symmetric. Thus the above identity yields that η (u)F i (u) is symmet˜ i, ˜ i , i = 0, 1, . . . , m, are symmetric. Conversely, if A ric. This shows that A for each i = 0, 1, . . . , m, is symmetric, then by a similar argument, there exists qi (u) such that η (u)F i (u) = qi (u) holds. This completes the proof of the theorem.
3. Symmetry and Energy Estimate Theorem 2.2 relates the physically natural notion of entropy to the notion of symmetric systems. For such a system, there is an energy estimate and the local-in-time existence theory of smooth solutions. Consider systems of the general form as in (2.6), (3.1)
A0 (u)ut +
m i=1
Ai (u)uxi = 0.
3. Symmetry and Energy Estimate
129
Definition 3.1. System (3.1) is symmetric if the matrices Ak (u), k = 0, 1, . . . , m, are symmetric and A0 (u) is positive definite for each state u. It is completely hyperbolic if it is hyperbolic and the eigenvalue problem (1.4) has a complete set of eigenvectors. It is strictly hyperbolic if the eigenvalues are real and distinct. It is clear that a symmetric system is completely hyperbolic and that strict hyperbolicity implies complete hyperbolicity. Strict hyperbolicity is a notion appropriate only for one spatial dimension, m = 1. Consider the eigenvalue problem (1.5) for a symmetric system, with A0 positive definite and A = A(ξ) symmetric and with an eigenpair (λj , r j ). The inner product satisfies (Arj , r j ) = (λj A0 r j , r j ) = λj (A0 r j , r j ), and by the symmetry of A, ¯ j (r j , A0 r j ). (Ar j , rj ) = (r j , Ar j ) = (r j , λj A0 r j ) = λ ¯ j is the complex conjugate of λj . By the positive definiteness of A0 Here λ we conclude that the eigenvalues λj , j = 1, 2, . . . , n, are real. A similar argument shows that (Ar j , r k ) = λj (A0 r j , r k ) = λk (A0 rj , rk ), j, k = 1, 2, . . . , n, and so (A0 r j , r k ) = 0 for λj = λk . For an eigenvalue λ of multiplicity greater than one, we may use the GramSchmidt process to choose a complete set of eigenvectors r j , j = 1, 2, . . . , n. These eigenvectors are properly normalized so that (3.2) Ar j = λj A0 r j , j = 1, 2, . . . , n, (A0 r j , r k ) = δjk for j, k = 1, 2, . . . , n. To illustrate the importance of the notion of symmetry, we start with the analysis of the basic finite speed of propagation property for symmetric hyperbolic systems. Definition 3.2. Consider a system of evolutionary partial differential equations for independent variables (x, t). Take times t0 and t1 , with t0 < t1 , and two regions Γ0 and Γ1 in x space. The region Γ0 is called a domain of dependence at time t0 for solutions at time t1 and x ∈ Γ1 if any two solutions u1 and u2 satisfying u1 (x, t0 ) = u2 (x, t0 ) for x ∈ Γ0 have u1 (x, t1 ) = u2 (x, t1 ) for x ∈ Γ1 . ¯ 1 is called a domain of influence at time t1 for solutions at The region Γ ¯ 0 if any two solutions u1 and u2 satisfying u1 (x, t1 ) = time t0 and x ∈ Γ ¯ ¯ 0. u2 (x, t1 ) for x ∈ Γ1 have u1 (x, t0 ) = u2 (x, t0 ) for x ∈ Γ
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6. Systems of Hyperbolic Conservation Laws: General Theory
For simplicity of analysis, we consider linear systems with constant coefficients: (3.3) A0 ut +
m
Aj uxj = 0,
j=1
A0 positive definite, Ai , i = 0, 1, . . . , m, constant matrices, with eigenvalues λj (ξ), j = 1, . . . , n, as in (1.5). Proposition 3.3. Consider the system (3.3) and set (3.4)
M ≡ max{|λj (ξ)| : ξ ∈ Sm−1 , j = 1, . . . , n}.
For any fixed x0 and t0 < t1 , let (see figure 6.02) (3.5) Γ0 ≡ {(x, t0 ) : |x| < N + M (t1 − t0 )},
Γ1 ≡ {(x, t1 ) : |x| < N }.
Then Γ0 is a domain of dependence at time t0 for solutions at time t1 and x ∈ Γ1 . Let ¯ 0 ≡ {(x, t0 ) : |x| < N }, (3.6) Γ
¯ 1 ≡ {(x, t1 ) : |x| < N + M (t1 − t0 )}. Γ
¯ 1 is a domain of influence at time t1 for solutions at time t0 and Then Γ ¯ x ∈ Γ0 . In particular, a symmetric hyperbolic system has the finite speed of propagation property. Proof. The proof is by a local energy estimate. An energy identity is obtained by taking the inner product of the system (3.3) with the vector u: 1 1 (u, A0 u)t + (u, Aj u)xj = 0. 2 2 m
(3.7)
j=1
Here the symmetry of the matrices has been crucially used for (uxj , Aj u) = (Aj u, uxj ) = (u, Aj uxj ), and so (u, Aj u)xj = (uxj , Aj u) + (u, Aj uxj ) = 2(u, Aj uxj ). Consider the cone region around a given location x0 connecting Γ0 and Γ1 , as shown in figure 6.02, Ω ≡ {(x, t) : t0 < t < t1 , |x − x0 | < N + M (t1 − t)}. The boundary ∂Ω of the region Ω is the union of Γ0 at time t0 , Γ1 at time t1 , both defined in (3.5), and the lateral boundary Γl , defined by Γl ≡ {(x, t) : |x − x0 | = N + M (t1 − t), t0 < t < t1 },
3. Symmetry and Energy Estimate
131
t
Γ1 Γl
x2 Ω x1
Γ0
Figure 6.02. Domain of dependence.
with outer unit normals Γ0 : n0 = (0, −1),
Γ1 : n1 = (0, 1), Γl : n l = √
x − x0 1 (ξ, M ), ξ ≡ . 2 |x − x0 | M +1
In figure 6.02, the space dimension is taken to be two and the position x0 is taken to be the origin. Integrate the energy identity (3.7) over the region Ω:
1 1 (u, A0 u)t + (u, Aj u)xj 2 2 j=1 1 1 (u, A0 u)(x, t0 ) dx + (u, A0 u)(x, t1 ) dx =− 2 2 Γ0 Γ1 m 1 √ + (u, Ak u)ξj (x, t) dS(x, t). M (u, A0 u) + 2 Γl 2 M + 1 j=1 m
(3.8) 0 = Ω
Note that A0 is positive definite and
that M is greater than any eigenvalues λj (ξ), j = 1, . . . , m, and so M A0 + Aj ξj is positive definite: Γl
m 1 0 √ (u, Ak u)ξj (x, t) dS(x, t) M (u, A u) + 2 2 M +1 j=1 m 1 √ u, (M A0 + Ak ξj )u (x, t) dS(x, t) ≥ 0. = 2 Γl 2 M + 1 j=1
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Thus we have from above the energy estimate 1 1 0 (3.9) (u, A u)(x, t1 ) dx ≤ (u, A0 u)(x, t0 ) dx. 2 2 Γ1 Γ0 Consider two solutions u1 and u2 and apply the energy estimate (3.9) to their difference u = u1 −u2 . The estimate (3.9) implies that if two solutions are equal inside the ball Γ0 at initial time t0 , then they are also equal in the ball Γ1 at the later time t1 . Thus Γ0 is a domain of dependence of solutions at time t0 for the solutions over the ball Γ1 at time t = t1 . ¯ with boundary Similarly, we can apply the energy estimate for a region Ω ¯ 0 at time t0 and Γ ¯ 1 at time t1 , Γ ¯ ≡ {(x, t) : t0 < t < t1 , |x − x0 | < N + M (t − t0 )}, Ω to obtain
¯1 Γ
1 (u, A0 u)(x, t1 ) dx ≥ 2
¯0 Γ
1 (u, A0 u)(x, t0 ) dx. 2
This yields the statement on the domain of influence in the proposition.
4. Local Existence of Smooth Solutions We now generalize the energy method to establish the local existence of smooth solutions to the initial value problem for the nonlinear system of conservation laws ut + ∇x · F(u) = 0. Since there is the finite speed of propagation property for hyperbolic systems, the local-in-time existence theory concerns only the local-in-space property of the initial function u(x, 0). Thus, for simplicity, we may take u(x, 0) to be a perturbation of a constant state u0 , and by a translation we may take u0 to be the zero state. For the classical solutions under consideration, we have the entropy equality η(u)t + ∇x · q(u) = 0. In order for a function u(x, t) to be a classical solution, the fluxes F(u(x, t)) need to be differentiable. For this to be so, the solution u(x, t) is required to be bounded on some higher-order differential L2 (x), x ∈ Rm , norms: (4.1) (f (x), g(x))s ≡
s |α|=0
f (x) s ≡
s
Rm
∂xα f (x) · ∂xα g(x) dx,
(∂xα f , ∂xα f ), s = 0, 1, . . . ,
f (x) 2L2 (x) = f (x) 0 ,
|α|=0
α = (α1 , . . . , αm ), |α| ≡
m i=1
αi , ∂xα ≡ ∂xα11 . . . ∂xαmm .
4. Local Existence of Smooth Solutions
133
The Sobolev embedding theorem says that the function is bounded if the L2 (x) norm of its differentials of sufficient high order is bounded. We have, for some positive constant Cs , m (4.2) |f (x)|∞ + |∂x f (x)|∞ ≤ Cs f (x) s for s > + 1. 2 We will apply the energy method to establish the boundedness of u(x, t) s for some s > m/2 + 1 and for t small. Assume that the system of conservation laws ut + ∇x · F(u) = 0 is endowed with an entropy pair (η(u), q(u)), as in Definition 2.1. In the following local existence theorem, we apply Theorem 2.2, but for simplicity of presentation we take η (u) = I so that A0 (u) = I and Aj (u) ≡ ∇u F j (u), j = 1, . . . , m, are symmetric. Theorem 4.1. Consider the initial value problem for the symmetric system
j ut + m j=1 A (u)uxj = 0, (4.3) u(x, 0) = u0 (x). Suppose that the initial data u0 (x) is continuously differentiable and that ∂x u0 (x) l ≤ β for some positive constant β and for some l > m/2. Then there exists a positive time T of the order of 1/β such that (4.3) has a unique classical solution u(x, t) for 0 ≤ t < T. Proof. The first energy estimate is obtained by integrating the entropy equation η(u)t + ∇x · q(u) = 0 to get the entropy estimate η(u(x, t) dx = η(u0 (x) dx. Rm
Rm
By a simple transformation η(u) → η(u) − η(0) − η (0)u,
q → q − η (0)F(u),
we may assume that the original η(u) is equivalent to |u|2 due to the convexity of η. Thus the above entropy estimate yields (4.4)
u(x, t) L2 (x) = ˜ u0 (x) L2 (x) .
The entropy estimate is a fully nonlinear one, and it applies to the original conservation laws. For the estimation of higher differentials, consider first the initial value problem for the system linearized around a given function v:
j ut + m j=1 A (v)uxj = 0, (4.5) u(x, 0) = u0 (x).
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Apply ∂xα to (4.5) to obtain (4.6) (∂xα u)t +
m
Aj (v)(∂xα u)xj
j=1 m j j α α A = (v)∂x uxj − ∂x A (v)uxj . j=1
The energy method integrates the above system times (∂xα u)T for 1 ≤ |α| ≤ l + 1: t (∂xα u)T (∂xα u)t dx dt (4.7) 0
Rm
t + 0
Rm
t = Rm
0
m
(∂xα u)T
(∂xα u)T
Aj (v)(∂xα u)xj dx dt
j=1 m
Aj (v)∂xα uxj − ∂xα Aj (v)uxj dx dt.
j=1
The breakup of terms in (4.6) and in (4.7) has two purposes. The first is to use the symmetry of Aj for the second term on the left-hand side of (4.7): t m (∂xα u)T Aj (v)(∂xα u)xj dx dt (4.8) 0
Rm
=
1 2
j=1 m t j=1
0
Rm
(∂xα u)T Aj (v) x ∂xα u dx dt j
2 = O(1)t sup |∇x v(x, s)|∞ ∇x u(x, s) l , 1 ≤ |α| ≤ l + 1. 0≤s≤t
The second reason is that the right-hand side of (4.7) contains only differentials of u and v up to order |α|, and a Moser-type inequality yields m j A (v)∂xα uxj − ∂xα Aj (v)uxj (4.9) j=1 L2 (x) = O(1) |∇x v(x, s)|∞ ∇x u(x, s) l + |∇x u(x, s)|∞ ∇x v(x, s) l . The first term in (4.7) gives t t d α T α (∂x u) (∂x u)t dx dt = (∂ α u)T (∂xα u) dx dt (4.10) 2dt 0 Rm x 0 Rm t 1 (∂xα u)T (∂xα u) dx . = 2 Rm 0
4. Local Existence of Smooth Solutions
135
From (4.7)–(4.10), α T α (4.11) (∂x u) (∂x u)(x, t) dx = (∂xα u)T (∂xα u)(x, 0) dx m m R R +O(1)t sup |∇x v(x, s)|∞ ∇x u(x, s) l 0≤s≤t
+ |∇x u(x, s)|∞ ∇x v(x, s) l , 1 ≤ |α| ≤ l + 1.
From (4.11) we have the energy estimate for the linear system (4.5) (4.12) ∇x u(x, t) l = O(1) ∇x u0 (x) l + O(1)t sup |∇x v(x, s)|∞ ∇x u(x, s) l + |∇x u(x, s)|∞ ∇x v(x, s) l . 0≤s≤t
By the Sobolev embedding theorem, (4.2), (4.12) implies (4.13) ∇x u(x, t) l = O(1) ∇x u0 (x) l + O(1)t sup ∇x v(x, s) l ∇x u(x, s) l + ∇x u(x, s) l ∇x v(x, s) l 0≤s≤t
for l >
m + 1. 2
Consider the space (4.14)
Lβ ≡ {w : sup ∇x w(x, s) l ≤ β}. 0≤s≤t
The energy estimate (4.13) yields (4.15)
sup ∇x u(x, s) l ≤ O(1)tβ for v ∈ Lβ .
0≤s≤t
Thus the mapping v → u is a mapping from Lβ into Lβ for t = C/β, with C sufficiently small. This establishes the boundedness in the higher-order norms l > m/2 + 1 for the time period t ≤ C/β. One can some continuity in time t of the solution. From (4.11) the gain α function Rm (∂x u)T (∂xα u)(x, t) dx is continuous in t. The function ∂xα u(·, t) is weakly continuous in L2 (x). By the continuity of v t , it is easy to see that this implies the strong continuity of ∂x u(·, t) in f (x) l norm. The final step is to show that the mapping is contractive in the lowerorder norm. Let ui be the solution of (4.5) with v replaced by v i , i = 1, 2, and with both v 1 and v 2 in the space Lβ . Then u ≡ u2 −u1 and v ≡ v 2 −v 1 satisfy
m j j j ut + m j=1 A (v 1 )uxj = j=1 A (v 1 ) − A (v + v 1 ) (u2 )xj , (4.16) u(x, 0) = 0.
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6. Systems of Hyperbolic Conservation Laws: General Theory
Similar to above, integrate (4.16) times uT to obtain (4.17) u(·, t) 2L2 (x)
= O(1)tβ sup 0≤s≤t
u(·, s) 2L2 (x) + v(·, s) L2 (x) u2 (·, s) L2 (x) .
This implies that the mapping is a contraction in L2 (x) norm for t = C/β, with C sufficiently small: 1 sup v(·, s) L2 (x) . (4.18) sup u(·, s) L2 (x) < 2 0≤s≤t 0≤s≤t Note that, while the boundedness is established for higher differentials, (4.15), the contraction is for lower differentials, (4.18). We conclude that the mapping v → u for (4.5) has a fixed point, which solves (4.3). Notice that the energy estimate (4.12) derived from the previous estimates actually yields an a priori estimate for the solution of (4.3): α T α (∂x u) (∂x u)(x, t) dx = (∂xα u)T (∂xα u)(x, 0) dx (4.19) m m R R t + O(1) ∇x u(x, s)|∞ ∇x u(x, s) l ds. 0
This can be viewed as a linear relation for ∇x u(x, t) l with coefficient O(1) ∇x u(x, t)|∞ . Suppose that the coefficient is bounded for t ≤ T ; then by Gronwall’s inequality, ∇x u(x, t) l is bounded for t ≤ T and a classical solution exists. In other words, if the classical solution fails to exist as t → T , then ∇x u(x, t) ∞ → ∞ as t → T . Thus we have the following corollary. Corollary 4.2. Suppose that the classical solution exists for time t < T and fails to exist as t → T . Then a singularity develops for the gradient of the solution: ∇x u(x, t) ∞ → ∞ as t → T . From this corollary, the first differential ∇x u blows up for some x at time T . Consequently, some shock waves would emerge in the solution u around the location x right after time T . So far, the local existence theory for smooth solutions yields no information for the construction of the solution with shocks after time T .
5. Euler Equations in Gas Dynamics We now check the basic property of hyperbolicity and consider the entropy pairs for the Euler equations in gas dynamics, (0.5) in Chapter 1: ⎛ ⎞ ⎛ ⎞ ρv ρ ⎝ ρv ⎠ + ∇x · ⎝ρv ⊗ v + pI⎠ = 0. ρEv + pv ρE t
5. Euler Equations in Gas Dynamics
137
Take a spatial direction ξ ∈ Rm , |ξ| = 1, and consider the projection of the Euler equations in this direction: ⎛ ⎞ ⎛ ⎞ m m ρv ρ ⎝ ρv ⎠ + ⎝ ρv 2 + p ⎠ = 0, v ≡ vj ξj , x ≡ xj ξ j . j=1 j=1 ρEv + pv x ρE t The Euler equations are rotation invariant and so the above system is of the same form as the one-dimensional Euler equations. To show that the multidimensional Euler equations are hyperbolic, we need to check the eigenvalues of the gradient of the flux function for the one-dimensional Euler equations. For simplicity in calculation of the eigenvalues, we use the Euler equations in Lagrangian coordinates: ⎧ ⎪ ⎨τt − vx = 0, (5.1) vt + px = 0, ⎪ ⎩ (e + 12 v 2 )t + (pv)x = 0. We use this occasion to describe the relationship between the Eulerian coordinates and the Lagrangian coordinates for the Euler equations. Let (y, t) be the independent variables for the Eulerian coordinates: ⎛ ⎞ ⎛ ⎞ ρv ρ ⎝ ρv ⎠ + ⎝ ρv 2 + p ⎠ = 0. (5.2) ρEv + pv y ρE t Let (x, t) be the new independent variables, i.e. the variables for the Lagrangian coordinates, (5.1). The relation between the two sets of independent variables is denoted by y = y(x, t). The main feature of the Lagrangian coordinates is that the gas flow paths are along a fixed spatial location, x = constant, ∂y(x, t) = v. ∂t
(5.3)
The mass between two flow paths is constant in time: for x1 and x2 fixed, (5.4)
d dt
y(x2 ,t)
y(x2 ,t)
ρ(y, t) dy = y(x1 ,t)
=−
y(x1 ,t) y(x2 ,t)
y=y(x ,t) ρt (y, t) dy + ρ(y, t)yt (x, t)y=y(x21 ,t)
(ρv)y (y, t) dy + (ρv)(y(x2 ), t) − (ρv)(y(x1 ), t) = 0,
y(x1 ,t)
where, in the second equality above, the conservation of mass equation ρt + (ρv)y = 0 for the Eulerian coordinates and the identity (5.3) have been used. For definiteness, let the scaling of the Lagrangian location x be such that
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6. Systems of Hyperbolic Conservation Laws: General Theory
the mass grows at the same rate as x: y(x,t) ρ(y, t) = y(0,t)
y(x,0)
ρ(y, 0) = x.
y(0,0)
From the above two identities, the relation between the two sets of independent variables is ∂y 1 ∂y = v, = τ, τ ≡ (specific volume). (5.5) ∂t ∂x ρ The continuity equation in (5.1) becomes the compatibility relation ∂2y ∂2y ∂v ∂τ = = = . ∂t ∂t∂x ∂x∂t ∂x The other equations in (5.1) follow from the relations (5.5) and (5.2). Write (5.1) in the general form (0.1): ⎞ ⎛ ⎞ ⎛ −v τ ⎠ ⎝ ⎝ p ⎠, v , f (u) = ut + f (u)x = 0, u = 1 2 pv ⎞ e +⎛ 2v (5.6) 0 −1 0 ⎝ −vpe pe ⎠ . f (u) = pτ 2 vpτ p − v pe vpe where the constitutive relation is assumed to be p = p(τ, e). In the above calculations, u1 = τ, u2 = v, u3 = e + v 2 /2, and so p = p(τ, e) = p(u1 , u3 − (u2 )2 /2). Thus ∂ ∂ ∂ p = pτ , p = −vpe , p = pe . ∂u1 ∂u2 ∂u3 Here τ, v, p and e are, respectively, the specific volume, velocity, pressure and internal energy of the gas. The other state variables include the gas temperature θ and the entropy s. The basic thermodynamics hypothesis states that among the state variables, such as τ, p, e, θ and s, only two of them are independent. The constitutive relations prescribe the functional dependence of the other thermodynamics variables in terms of the specific volume τ and a chosen thermodynamics variable. The constitutive relations are expressed in the following three forms, with the specific volume τ as the first chosen variable and either the internal energy e, entropy s, or temperature θ as the second chosen variable: (5.7) p = p(τ, e) = p¯(τ, s) = p˜(τ, θ),
s = s(τ, e), ¯ s), θ = θ(τ, e) = θ(τ,
The thermodynamics relation (5.8)
de = θ ds − p dτ
e = e¯(τ, s) = e˜(τ, θ).
5. Euler Equations in Gas Dynamics
139
yields 1 p se = , sτ = , p¯s = θpe , p¯τ = pτ − ppe , θ θ where the chain rule is applied to the relation s = s(τ, e) = s(τ, e¯(τ, s)) and p(τ, e¯(τ, s)) = p¯(τ, s). For smooth solutions of the Euler equations (5.1), this implies p 1 p 1 (5.10) s(τ, e)t = sτ τt + se et = τt + et = vx + [−vvt − (pv)x ] = 0. θ θ θ θ Thus we may consider the Euler system (5.1) with the energy equation replaced by the entropy equation st = 0: ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎛ ⎞ τ −v τ 0 −1 0 τ ⎝v ⎠ + ⎝p¯(τ, s)⎠ = 0, or ⎝v ⎠ + ⎝p¯τ 0 p¯s ⎠ ⎝v ⎠ = 0. 0 0 0 s x 0 s t s t x (5.9)
e¯τ = −p, e¯s = θ,
Note that this system is obtained from (5.1) by a nonlinear transformation and so the two systems are equivalent only for smooth solutions. The two flux matrices are therefore similar and so the eigenvalues of the original flux matrix f (u) are the same as those for the new flux matrix, whose eigenvalues are easily computed to be √ √ pτ , λ2 (u) = 0, λ3 (u) = −¯ pτ . (5.11) λ1 (u) = − −¯ Thus the Euler equations are hyperbolic if p¯τ (τ, s) < 0. With this, the right eigenvectors rj (u) of the flux matrix f (u) are easily computed to be constant multiples of the following: (5.12) ⎞ ⎞ ⎛ ⎛ ⎞ ⎛ 1 pe 1 ⎠ , r 2 (u) = ⎝ 0 ⎠ , r 3 (u) = ⎝ ⎠. λ1 λ3 r 1 (u) = ⎝ −p − λ1 v −p − λ3 v pτ In the following theorem, the admissibility of weak solutions is defined as a direct generalization of the definition for scalar laws in Section 7 of Chapter 3 and Section 5 of Chapter 5. The condition is η(u)t + ∇x · q(u) ≤ 0; see Definition 6.4 in Section 6. Theorem 5.1. Under the constitutive hypotheses (5.13)
p˜τ (τ, θ) < 0,
θe (τ, e) > 0,
(η(u), q(u)) ≡ (−s, 0) forms an entropy pair for the Euler equations (5.1). Thus, for admissible weak solutions of (5.1), we have the entropy inequality (−s)t ≤ 0 in agreement with the second law of thermodynamics. Proof. We already know that st = 0 for smooth solutions. For the pair (−s, 0) to be an entropy pair, it remains to show that the negative of the
140
6. Systems of Hyperbolic Conservation Laws: General Theory
physical entropy −s as a function of the conserved variables u = (τ, v, e + v 2 /2)T , −s = −s(τ, e) = −s(u1 , u3 − (u2 )2 /2), is convex: ⎛ ⎞ −sτ τ vsτ e −sτ e A0 ≡ (−s) (u) = ⎝ vsτ e se − v 2 see vsee ⎠ is positive definite. −sτ e vsee −see To show this, we now derive identities from the various functional relations. We have det A0 = se (see sτ τ − (sτ e )2 ). From the thermodynamics relation de = θ ds − p dτ , we have se = 1/θ, sτ = p/θ, θe θτ θpτ − pθτ see = − 2 , sτ e = − 2 , sτ τ = − , θ θ θ2 and so θτ 1 θe pτ θ2 θτ 1 − 2 ( − p 2 ) − τ2 = 4 −pτ θe + (pθe − θτ ) . det A0 = θ θ θ θ θ θ θ Again from the thermodynamics relation, we have 1 p seτ = ( )τ = sτ e = ( )e , and so −θτ = θpe − pθe , θ θ whence the above is simplified to 1 det A0 = 4 −pτ θe + pe θτ . θ Next we use the functional relations p˜(τ, θ) = p(τ, e˜(τ, θ)) and e˜(τ, θ(τ, e)) = e to obtain θτ p˜τ = pτ + pe e˜τ , e˜τ + e˜θ θτ = 0, e˜θ θe = 1, and so p˜τ = pτ + pe (− ), θe whence the above is further simplified to 1 det A0 = − 4 θe p˜τ > 0, θ because θe > 0 and p˜τ < 0 by the hypothesis (5.13). It remains to show the positivity of the sub-matrices: θe θe se − v 2 see vsee = −se see = 3 , det(−see ) = −see = 2 , det vsee −see θ θ which are positive by the second hypothesis in (5.13). This completes the proof of the theorem. We have seen from the general theory in Theorem 2.2 that the Euler system is symmetric hyperbolic because it possesses an entropy pair pτ (τ, s) is (−s, 0). In particular, p¯τ (τ, s) < 0 so that the sound speed −¯ real. This can also be seen directly from the hypotheses (5.13), p˜τ (τ, θ) < 0 ¯ s)), we and θe (τ, e) > 0, as follows: From the relation p¯(τ, s) = p˜(τ, θ(τ,
6. Shock Waves
141
have p¯τ = p˜τ + p˜θ θ¯τ . From the thermodynamics relations e¯s = θ and e¯τ = p ps , and so p¯τ = p˜τ − p˜θ p˜s . From p¯(τ, s˜(τ, θ)) = p˜(τ, θ) we we have θ¯τ = −¯ pθ )2 /˜ sθ . Finally from s˜(τ, θ(τ, e)) = s(τ, e) have p¯s s˜θ = p˜θ and so p¯τ = p˜τ − (˜ pθ )2 θθe , which is we have s˜θ θe = se = 1/θ and conclude that p¯τ = p˜τ − (˜ negative from the hypotheses (5.13). Example (Polyatomic gases). Ideal gases satisfy Boyle’s law pτ = N Rθ, or p˜(τ, θ) = N Rθ/τ . Polyatomic gases are ideal gases satisfying p¯(τ, s) = AeCs τ −γ for γ > 1. The Euler equations (5.1) are hyperbolic as p¯τ (τ, s) < 0 clearly holds. We now carry out the calculations to check that the constitutive hypotheses (5.13) hold for the polyatomic gases. Consider p = p(τ, e) = p(τ, e¯(τ, s)) = p¯(τ, s). This yields p¯s = pe e¯s . From the thermodynamics relation, e¯s = θ and so pe = p¯s /θ. For polyatomic gases, p¯s > 0. θ The other hypothesis p˜τ < 0 in (5.13) follows from Boyle’s law. p¯s = ACeCs τ −γ > 0, and so pe =
6. Shock Waves The definition of weak solutions for the system (0.1) directly generalizes the two versions of weak solutions for the scalar equations, Definition 3.1 and Definition 3.2 of Chapter 2. Definition 6.1. A bounded measurable function u is a weak solution for the system of conservation laws ut +∇x ·F(u) = 0 if for any bounded domain Ω in Rm , d u(x, t) dx = − F(u(x, t)) · n dS(x), (6.1) dt Ω ∂Ω where n is the outer normal to the boundary ∂Ω of the domain Ω, as in figure 1.01 of Chapter 1. Definition 6.2. A bounded measurable function u is a weak solution for the system of conservation laws ut + ∇x · F(u) = 0 if φ(x, 0)u(x, 0) dx = 0 φt u + ∇x φ · F(u) dx dt + (6.2) R
Rm
Rm
for any smooth function φ(x, t) that is zero outside a bounded set in {(x, t) : x ∈ Rm , t ≥ 0}. Consider a weak solution u(x, t) containing a jump discontinuity along hypersurface ψ(x, t) = 0 with propagation direction ξ and speed σ, as shown
142
6. Systems of Hyperbolic Conservation Laws: General Theory
in figure 6.03: ∂ ψ(x, t), ∂t where we have normalized the defining function ψ so that |ξ| = |∇x ψ(x, t)| = 1. We define the states, according to the set direction ξ, on either side of the discontinuity as: (6.3)
(6.4)
ξ ≡ ∇x ψ(x, t), |ξ| = 1,
σ≡
u± ≡ u(x± , t) ≡ lim u(x + αξ, t). α→0±
Either version of the solutions yields (6.5) σ(u+ − u− ) = ξ · F(u+ ) − F(u− ) , Rankine-Hugoniot condition.
u+
ξ
ψ(x, t) = 0 u− Figure 6.03. Shock surface.
Remark 6.3. A shock with normal ξ for the multi-dimensional system of conservation laws ut + ∇x · F(u) = 0 is the same as a shock for the onedimensional system that is the projection of the multi-dimensional system ut + ∇x · F(u) = 0 in the direction ξ: (6.6)
ut + f (u)x = 0, f (u) ≡ ξ · F(u),
x≡
m
j=1 xj ξj .
We will generalize the entropy conditions for scalar laws Definition 5.1 of Chapter 2 and Definition 1.8 of Chapter 5, to general systems in one spatial dimension in the next chapter; see Definition 4.3 and Definition 6.1 of Chapter 7. These entropy conditions therefore have direct generalizations for multi-dimensional systems. The following notion of integral entropy condition is a direct extension of the theory for scalar laws, Definition 5.1 in Chapter 5.
6. Shock Waves
143
Definition 6.4. A weak solution u of the system of conservation laws (0.1) satisfies the integral entropy condition with respect to the entropy pair (η(u), q(u)) if (6.7)
η(u)t + ∇x · q(u) ≤ 0
in the sense of distributions. Just as with the scalar laws, the above integral entropy condition is equivalent to the dissipation inequality across each discontinuity in a weak solution: (6.8) −σ η(u+ ) − η(u− ) + ξ · q(u+ ) − q(u− ) ≤ 0. Note that the inequality satisfies the frame indifference property; when ξ changes its direction, then u− and u+ are switched, σ changes sign, and the entropy inequality remains the same. The integral entropy condition is a consequence of the zero dissipation limit, κ → 0+, of solutions for the viscous conservation laws ut + ∇x · F(u) = κΔx u. Just as for the scalar laws, this is shown by multiplying the viscous conservation laws by η (u) to obtain η(u)t + ∇x · q(u) = κ∇x · η (u)∇x u − κ∇x ut η (u)∇x u. In the zero dissipation limit κ → 0+, the first term on the right-hand side tends to zero in the weak sense, while the second term, due to the convexity η > 0, yields a non-negative value. For the scalar laws, the viscosity matrix has few choices besides multiples of the identity matrix; but there are many more choices for systems. Physical systems usually take the form ut + ∇x · F(u) = ∇x · B(u, ε)∇x g(u) , x ∈ Rm , u ∈ Rn , where B(u, ε) is called the physical viscosity matrix. The corresponding equation for the entropy pairs becomes η(u)t +∇x · q(u) = κ∇x · η (u)B(u)∇x g(u) −κ∇x ut η (u)B(u)g (u)∇x u. The entropy inequality results from the constitutive hypothesis, which says that η (u)B(u)g (u) is non-negative definite; see Section 5 of Chapter 10. A general solution u(x, t) of (0.3) is a function of the independent variables (x, t) and therefore depends on m+1 parameters. There are particular solutions, the simple waves, which depend on fewer than m + 1 parameters. For instance, solutions of the form (6.9)
u(x, t) ≡ φ(η(x, t))
144
6. Systems of Hyperbolic Conservation Laws: General Theory
for some scalar function η(x, t) depend only on one parameter. Plugging this form into the conservation laws (0.3), we have (6.10)
ηt A0 +
m
ηxj Aj φ = 0,
j=1
A0 = A0 (u) ≡ ∇u g(u), Aj = Aj (u) ≡ ∇u F j (u). This is the nonlinear version of the plane wave (1.4) in figure 6.01, with ξ≡
∇x η , |∇x η|
σ≡−
ηt . |∇x η|
This formulation can be used for constructing particular solutions for the nonlinear system (0.3). We will see in the next three chapters that for one spatial dimension, m = 1, the simple waves and shock waves are sufficient for the study of general solutions. The first step is to construct the solution to the Riemann problem in the class of shock and simple waves, which is done in the next chapter.
7. Notes The important observation that the existence of entropy pairs implies symmetry of the system was made by Godunov [57]; see also [49] and [50]. The general mathematical notion of entropy pairs as formulated in Section 2 is due to Lax [74]. The energy method for linearized systems and its generalization to the entropy method for nonlinear systems briefly presented in Section 3 and Section 4 is by now a standard procedure. The local theory is the only general theory for systems of more than one spatial dimension. Substantial progress has been made from the local existence theory toward the shock wave theory. Corollary 4.2 yields no information on the way a shock would develop from a smooth solution. Intricate geometric considerations are called for in understanding the onset of singularity development. Such intricate geometric analysis was initiated in [29]. There have been many advances in this fruitful direction; see references in [30]. Sufficient conditions were found for the development of the singularity. For one spatial dimension, the geometry is simpler, and necessary and sufficient conditions can be found for the development of the singularity; see Lax [75], John [66], and Liu (1997) in the references of [92]. The extension to initial-boundary value problems allows for the study of local existence of wave patterns involving shock waves; see, for instance, Majda [103] for curved shocks and Chen [28] for Mach configurations. For multi-dimensional systems, the energy estimate is based on L2 (x) norms, as in Theorem 4.1, whereas even for scalar laws, the solution operator for weak solutions is stable only in L1 (x); see Subsection 4.3 of
8. Exercises
145
Chapter 3. There is an essential difficulty in finding the appropriate function spaces for solutions of systems with more than one space dimension. In the following three chapters, we will present a successful shock wave theory for systems in one spatial dimension. It is worth noting that the theory does not start with the analysis of classical solutions. Instead, it starts with the analysis of the Riemann problem. There are several reasons for that; chiefly among them is that the Riemann solutions represent the time-asymptotic, as well as local in space and time, wave patterns. As we have seen for scalar laws, approximating the initial data, even smooth ones, by Riemann solutions allows for intuitive analysis of the piecewise continuous solutions; see Remark 4.5 of Chapter 3. This idea extends to systems in one spatial dimension, though the extension is highly non-trivial and not at all straightforward. The analysis in the following three chapters applies only to one spatial dimension. Straightforward generalization of the Riemann solutions to several spatial dimensions is not feasible due to tremendous analytical difficulties. Moreover, the Riemann problem for more than one spatial dimension does not play a central role in illuminating the general wave interactions. Instead, the identification of time-asymptotic and local wave patterns is always a central issue, and it would be desirable to understand this in a more general sense. There is a need to refocus the study of shock waves away from the classical notion of well-posedness and to directly address the physical aspects of the shock wave theory. The time-asymptotic wave patterns with shocks under the effects of the boundary constitute an important issue for study. How to formulate the problems mathematically and to come up with appropriate analytical techniques in order to reach physically meaningful conclusions remains largely an exciting challenge, though some important progress has been made. We will come back to discuss this and other issues in Chapter 13 on multi-dimensional gas flows and Chapter 14 with concluding remarks.
8. Exercises 1. Write the wave equation utt = c2 uxx as a system of first-order partial differential equations and check its hyperbolicity. 2. Write the two-dimensional wave equation utt = c2 uxx + uyy as a hyperbolic system of first-order partial differential equations. 3. In the study of the domain of dependence, the maximum propagation speed M in (3.4) is used to yield the symmetric cone Ω in figure 6.02. Explain how to obtain a non-symmetric cone to more accurately describe the domain of dependence.
146
6. Systems of Hyperbolic Conservation Laws: General Theory
4. Extend the energy method from linear to nonlinear systems for the study of the finite speed of propagation. For this, consider smooth solutions. 5. Compute the eigenvalues for (5.2). Find the domain of dependence on the initial data for the region |x| < 10, t = 11, for the linear Euler equations linearized around the state (ρ0 , v 0 , p0 ) = (1, (1, 2, 3)T , 1). 6. Show that for the isentropic Euler equations τt − vx = 0, vt + p(τ )x = 0, one can take the energy function
τ v2 − p(z) dz η(τ, v) ≡ 2 as an entropy function, and find the corresponding entropy flux q(τ, v). Thus the physical energy can serve as the mathematical entropy. 7. Consider the two-dimensional shallow water wave equations ht + (hu)x + (hv)y = 0, (hu)t + (huv)y = 0, (hv)t + (huv)x = 0. Here h is the depth and (u, v) is the velocity of the water. Find the condition for the system to be strictly hyperbolic. 8. Consider the scalar law ut + f (u)x = 0. Suppose that the initial function u(x, 0) is C ∞ and bounded. Show by the energy method that the solution u(x, t) exists and is C ∞ so long as |ux (x, t)| remains bounded. 9. Consider the Euler equations in the Eulerian coordinates, (5.2), ⎛ ⎞ ⎛ ⎞ ρv ρ ⎝ ρv ⎠ + ⎝ ρv 2 + p ⎠ = 0. ρEv + pv y ρE t Compute the eigenvalues of the flux matrix and find the condition for the system to be hyperbolic. Check this with the condition p¯τ < 0 for the Lagrangian coordinates. 10. Consider the isentropic Euler equations for polyatomic gases, ρt + (ρv)x = 0, (ρv)t + (ρv 2 + ργ )x = 0, 1 < γ ≤ 53 . Show that the system is not completely hyperbolic at vacuum ρ = 0.
10.1090/gsm/215/07
Chapter 7
Riemann Problem
We have seen in Chapter 3 and Chapter 5 that for scalar laws the analysis of the Riemann problem leads to the notion of admissibility condition and the construction of elementary waves. Solutions of the Riemann problem can also be used to construct solutions for general initial value problems and to study the solution behaviors; see Section 3 and Section 4 of Chapter 3. This and the next two chapters generalize this approach to systems. This chapter studies the Riemann problem for systems of hyperbolic conservation laws, u ∈ Rn with n > 1,
(0.1)
⎧ ⎪ x = 0, ⎨ut + f (u) ul for x < 0, ⎪ ⎩u(x, 0) = ur for x > 0.
In the next two chapters the Riemann solutions will be used to construct solutions to the initial value problem with general initial data. The Riemann problem is easier to solve than the general initial value problem because the solution of the Riemann problem is self-similar, u(x, t) = φ(x/t); see Proposition 1.1 in Section 1 of Chapter 3. The system is assumed to be strictly hyperbolic, with the eigenvectors normalized: (0.2) ut + f (u)x = 0, f (u)r i (u) = λi (u)ri (u),
li (u)f (u) = λi (u)li (u),
li (u)r j (u) = δij , i, j = 1, 2, . . . , n, λ1 (u) < λ2 (u) < · · · < λn (u). 147
148
7. Riemann Problem
1. Linear System Consider first the system ut + f (u)x = 0 linearized around a constant state u0 : (1.1)
ut + A0 ux = 0, A0 ≡ f (u0 ).
The flux matrix A0 now has constant eigenvalues λ01 ≡ λ1 (u0 ) < λ02 ≡ λ2 (u0 ) < · · · < λ0n ≡ λn (u0 ) with constant eigenvectors r0j ≡ r j (u0 ) and l0j ≡ lj (u0 ), j = 1, 2, . . . , n, normalized as in (0.2). Decompose the system (1.1) into linear transport equations in the coordinates of the right eigenvectors r 0j , j = 1, . . . , n: (uj )t + λ0j (uj )x = 0, j = 1, . . . , n, uj ≡ l0j · u, u =
n
uj r 0j .
j=1
Decompose the vector ur −ul also in the coordinates of the right eigenvectors r 0j , j = 1, . . . , n, ur − ul ≡
n
aj r 0j , aj = l0j · (ur − ul ).
j=1
The Riemann problem for the scalar linear transport equations j (uj )t + λ0j (u
)x = 0, 0 for x < 0, uj (x, 0) = aj for x > 0
can be solved easily by the method of characteristics to yield
0 for x − λ0j t < 0, j u (x, t) = aj for x − λ0j t > 0. The solution u(x, t) of the Riemann problem (ul , ur ) for (1.1) is expressed in terms of uj : ⎧ 0 ⎪ n ⎨ul for x < λ1 t, uj r 0j = uj for λ0j t < x < λ0j+1 t, j = 1, . . . , n − 1, u(x, t) = ul + ⎪ ⎩ j=1 ur for x > λ0n t, uj ≡ ul +
j
aj r 0j , j = 1, . . . , n − 1.
k=1
The solution consists of waves with speed λj and wave shape aj r 0j , j = 1, . . . , n. From this linear theory and the study of scalar laws in Chapter 3 and Chapter 5, it is natural to envision that a solution of the nonlinear system (0.2) can be decomposed into elementary waves, such as the shock and
2. Simple Waves
149
rarefaction waves, corresponding to each characteristic family. These waves need to be constructed, and the entropy condition needs to be identified. The construction of the elementary waves for systems mirrors that for scalar laws. In studying nonlinear waves for scalar laws, we have seen that there is a need to consider both the graph of the flux function f (u) and the wave propagation in (x, t) space; see figures 2.05, 2.08, and 2.09 of Chapter 2 and figures 5.02 and 5.03 of Chapter 5. For the construction of Riemann solutions for systems, it is also necessary to consider both the graph in u space and the wave in (x, t) space. For systems, the geometry is richer, as the dependent variable u ∈ Rn , n > 1, is now a vector. One needs to visualize the values a wave takes in u space, the state space, as well as the propagation of the wave in (x, t) space, the physical space. In the following description of elementary waves, we will present figures for both the state space and the physical space.
2. Simple Waves We start with a direct generalization of the theory for scalar laws with the construction of simple waves. A simple wave takes values along a characteristic curve.
t
R i(u0) u1
ri(u)
expansion
dx dt
= λ i (u)
u u0 u2
u space
u0
u1
u2 compression x
Figure 7.01. The ith characteristic curve. Simple expansion and compression i-waves.
Definition 2.1. An ith characteristic curve Ri (u0 ) is the integral curve, in the state space, of the vector field r i (u) through the given state u0 , as shown in the left diagram of figure 7.01. Parametrize a ith characteristic curve Ri (u0 ) by a non-singular parameter τ . For definiteness, scale the eigenvector ri (u) so that (2.1)
d u = r i (u), u ∈ Ri (u0 ). dτ
150
7. Riemann Problem
An ith simple wave u(x, t) is a solution of the system of conservation laws which takes values along a given characteristic curve Ri (u0 ): (2.2)
u(x, t) ∈ Ri (u0 ), ut + f (u)x = 0, ith simple wave.
Proposition 2.2. Suppose that the initial function is on Ri (u0 ) and smooth, (2.3)
u(x, 0) ∈ Ri (u0 ), τ0 (x) ≡ τ (u(x, 0)), −∞ < x < ∞.
Then an ith simple wave u(x, t) can be constructed, locally in time, by the following procedure: (1) Determine τ (x, t) by (2.4)
τ (x, t) = τ0 (x0 ) for x = λi (u(x0 , 0))t + x0 , −∞ < x0 < ∞.
(2) Set u(x, t) uniquely to satisfy (2.5)
τ (u(x, t)) ≡ τ (x, t), u(x, t) ∈ Ri (u0 ), −∞ < x < ∞.
Proof. Since the solution u(x, t) is on Ri (u0 ), we have from (2.1) and the chain rule that ∂τ du ∂τ ∂ u(x, t) = = r i (u(x, t)) ∂x ∂x dτ ∂x and similarly ∂τ ∂ u(x, t) = r i (u(x, t)). ∂t ∂t Thus for u(x, t) to be a solution of the system of conservation laws ∂τ ∂τ + λi r i (u(x, t)) = 0, ∂t ∂x we require τ (u(x, t)) to satisfy ut + f (u)x =
∂ ∂ τ (u) + λi (u) τ (u) = 0 ∂t ∂x as u moves along Ri (u0 ). We have thus an explicit way to construct a class of simple waves by the following procedure. Choose any smooth bounded function τ0 (x), −∞ < x < ∞, and set the initial data u(x, 0) by requiring (2.3). Then evolve the parameter τ (x, t) by solving (2.6) using the method of characteristics to yield (2.4). With τ (x, t) thus computed, the solution u(x, t) is uniquely determined by (2.5).
(2.6)
The procedure (2.3)–(2.5) yields at least a local-in-time solution to the conservation laws ut + f (u)x = 0; see right diagram of figure 7.01. A simple ith wave expands if its speed λi (u)(x, t) of propagation increases in x, as in the case of (u0 , u1 ) in the right diagram of figure 7.01. This means that as a function of the state variable u, the ith characteristic
2. Simple Waves
151
speed λi (u) increases as u moves along Ri (u0 ) from u0 to u1 in the left diagram of figure 7.01. In other words, λi (u) increases in the r i (u) direction: ∇λi (u) · r i (u) > 0. In this case, the wave (u0 , u1 ) is an expansion wave. On the other hand, (u1 , u2 ) forms a compressive wave because λi (u) decreases as u moves from u1 to u2 . Recall that the convexity condition f (u) = 0 for the scalar laws is important because the characteristic speed λ(u) = f (u) is monotone as a function of the state variable u. We see that the monotonicity of the characteristic speed λi (u) in the direction of the characteristic vector r i (u) plays a similar role. In both situations, compression and expansion waves result from the monotonicity property. This convexity condition is called the genuine nonlinearity condition for systems: Definition 2.3. An i-characteristic field is called genuinely nonlinear at the state u0 if the ith characteristic value λi is strictly monotone in its characteristic direction r i at u0 : ∇λi (u0 ) · r i (u0 ) = 0. Around such a state u0 we can choose τ (u) = λi (u) as the parameter for the characteristic curve Ri (u0 ), and equation (2.6) becomes the inviscid Burgers equation, the Hopf equation: (2.7)
∂ ∂ λi (u) + λi (u) λi (u) = 0. ∂t ∂x
From the analysis for the inviscid Burgers equation, or Hopf equation, we can construct local-in-time solutions, the i-simple waves for u ∈ Ri (u0 ), shown in figure 7.01, where the eigenvector is oriented so that ∇λi (u0 ) · ri (u0 ) > 0:
(2.8)
expansion wave (u0 , u1 ): λi (u) increases as u moves along Ri (u0 ) from u0 to u1 ; compression wave (u0 , u2 ): λi (u) decreases as u moves along Ri (u0 ) from u0 to u2 .
Proposition 2.4. Suppose that the ith characteristic field is genuinely nonlinear near the state u0 . Then for a state u1 on Ri (u0 ) that is close to u0 and has λi (u0 ) < λi (u1 ), the Riemann problem for the system
u0 for x > 0, u(x, 0) = u1 for x < 0
152
7. Riemann Problem
has the centered rarefaction wave solution
u0 for x < λi (u0 )t, (2.9) u(x, t) = u1 for x > λi (u1 )t, x λi (u)(x, t) = for λi (u0 )t < x < λi (u1 )t, t u(x, t) ∈ Ri (u0 ) between u0 and u1 . Proof. Consider the Riemann problem for the inviscid Burgers equation corresponding to the expansion wave: λt + λλx =
0, λ0 for x < 0, λ(x, 0) = λ1 for x > 0, λ0 ≡ λi (u0 ) < λ1 ≡ λi (u1 ). The solution has the simple form of the centered rarefaction wave, (2.4) of Chapter 3: ⎧ ⎪ ⎨λ0 for x < λ0 t, λ(x, t) = xt for λ0 t < x < λ1 t, ⎪ ⎩ λ1 for x > λ1 t. This and the above considerations, (2.3)–(2.5) and (2.7), yield the expression for the centered rarefaction wave (2.9) for the system.
3. Hugoniot Curves As with the scalar laws, smooth initial data for a system of conservation laws in general give rise to shock waves at later times, and one needs to consider weak solutions with shocks. A jump discontinuity (u− , u+ ) with speed s in a weak solution satisfies (6.5) in Chapter 6, (3.1)
s(u+ − u− ) = f (u+ ) − f (u− ), Rankine-Hugoniot condition.
Remark 3.1. The Rankine-Hugoniot condition for a system is a vector relation. For scalar laws, for any two given states u− and u+ , the RankineHugoniot condition can be satisfied by choosing the speed s to be (f (u+ ) − f (u− ))/(u+ − u− ). On the other hand, for two given end states u− and u+ for a system, in general there does not exist a scalar s which satisfies the Rankine-Hugoniot condition. For a given state u− , the other state u+ cannot be chosen arbitrarily; the two vectors u+ −u− and f (u+ )−f (u− ) are required to be parallel to each other in order for the scalar s to exist. This nonlinear requirement has been studied globally only for specific systems such as the Euler equations for polyatomic gases. For general systems it can
3. Hugoniot Curves
153
be studied locally, for u− close to u+ , by the implicit function theorem; see Theorem 3.3. Definition 3.2. For a given state u0 , the Hugoniot set H(u0 ) consists of all states u with the property that the two vectors u − u0 and f (u) − f (u0 ) are parallel: (3.2) H(u0 ) ≡ {u : σ(u − u0 ) = f (u) − f (u0 ) for some scalar σ = σ(u0 , u)}, Hugoniot set. Example (p-system). For some systems, the Hugoniot set can be expressed explicitly. Consider the p-system, which models the isentropic gas dynamics in Lagrangian coordinates, as in (5.1) of Chapter 6, and also the elasticity:
τt − vx = 0, (3.3) vt + p(τ )x = 0. In the general notation of (0.2), 1 u τ u= = , 2 u v
f (u) =
−v p(τ )
=
−u2 . p(u1 )
By straightforward computations, the flux matrix f (u) and its eigenvalues λj (u) and eigenvectors r j (u), j = 1, 2, are (3.4) 0 −1 , λ1 (u) = − −p (τ ), λ2 (u) = −p (τ ), f (u) = p (τ ) 0 1 1 , r2 = d r1 = c −p (τ ) − −p (τ ) for arbitrary nonzero constants c and d. The system is strictly hyperbolic if p (τ ) < 0. In gas dynamics, p is the pressure and τ = 1/ρ the specific volume, and the condition p (τ ) < 0 says that the pressure increases as the density increases, a reasonable physical hypothesis except at the vacuum state ρ = 0. By direct calculations, p (τ ) ∇λ1 (u) = 2√−p (τ ) 0 , p (τ ) 1 √ = c √p (τ) . ∇λ1 (u) · r 1 (u) = 2 −p (τ ) 0 c 2 −p (τ ) −p (τ ) Similar computations work for λ2 and we have (3.5)
p (τ ) , ∇λ1 (u) · r 1 (u) = c 2 −p (τ )
p (τ ) ∇λ2 (u) · r 2 (u) = −d . 2 −p (τ )
Thus the system is genuinely nonlinear in both of its characteristic fields if p (τ ) = 0. For polyatomic gases, (3.6)
p(τ ) = Aτ −γ ,
γ > 1, A > 0.
154
7. Riemann Problem
γ = 5/3 for monatomic gases, γ = 7/4 for diatomic gases, and γ decreases to 1 as the number of atoms in the gas molecule increases. We have −p (τ ) = Aγτ −γ−1 , sound speed, (3.7) p (τ ) = Aγ(γ + 1)τ −γ−2 > 0, genuine nonlinearity. For the general constitutive law p = p(τ ), the characteristic curves Ri , i = 1, 2, and the integral curves of r i (u), i = 1, 2, are given by the ordinary differential equations dv dv = −p (τ ), R2 (u) : = − −p (τ ). (3.8) R1 (u) : dτ dτ The Rankine-Hugoniot condition is v+ − v− τ+ − τ− = (3.9) s . v+ − v− −(p(τ+ ) − p(τ− )) The speed s can be eliminated to yield the direct relation between the states: (3.10)
(v+ − v− )2 = −(τ+ − τ− )(p(τ+ ) − p(τ− )).
There are two branches of the Hugoniot set: ⎫ ⎧ ⎨ ⎬ p(τ ) − p(τ0 ) τ τ (τ − τ0 ) ; (3.11) H 1 0 = : v = v0 + − v0 ⎭ ⎩ v τ − τ0 ⎫ ⎧ ⎨ ⎬ p(τ ) − p(τ ) τ τ 0 (τ − τ0 ) . : v = v0 − − H2 0 = v0 ⎭ ⎩ v τ − τ0 Thus, for a p-system, the Hugoniot set consists of two globally defined curves. We now continue with the discussion for general conservation laws. In general, the Rankine-Hugoniot condition (3.1), a set of algebraic relations, does not yield an explicit expression between the end states. Nevertheless, from the implicit function theorem there is the following general result. Theorem 3.3. In a small neighborhood around a given state u0 , the Hugoniot set H(u0 ) consists of n Hugoniot curves H j (u0 ), j = 1, 2, . . . , n, with the following properties: (i) H j (u0 ) is tangent to the characteristic curve Rj (u0 ) at u = u0 and the shock speed σ(u, u0 ) tends to λj (u0 ) as u approaches u0 along H j (u0 ). (ii) The characteristic curve Rj (u0 ) and the Hugoniot curve H j (u0 ) have a second-order tangency at u = u0 . For a given state u on H j (u0 ), ¯ | = O(1)|u − u0 |3 ; ¯ on Rj (u0 ) such that |u − u there exists another state u see figure 7.02.
3. Hugoniot Curves
155
(iii) The shock speed is approximated by the arithmetic mean of the characteristic speeds of its end states: (3.12)
σ(u, u0 ) =
λj (u) + λj (u0 ) + O(1)|u − u0 |2 , u ∈ H j (u0 ). 2
Ri(u0) Hi(u0) ε
3
ε u0
u space
Hi(u0) Ri(u0)
Figure 7.02. Hugoniot curve H i (u0 ) and characteristic curve Ri (u0 ).
Proof. Write the difference of the flux f (u) in terms of the matrix average of its gradient f (u): 1 d f (u0 + τ (u − u0 )) dτ = f¯ (u)(u − u0 ); f (u) − f (u0 ) = dτ 0 1 f (u0 + τ (u − u0 )) dτ. f¯ (u) ≡ 0
A state u is on H(u0 ), (3.2), if and only if (3.13) f¯ (u) − σI (u − u0 ) = 0.
For u close to u0 , f¯ (u) is close to f (u0 ), which has distinct eigenvalues. ¯ j (u), j = 1, 2, . . . , n, and therefore Thus f¯ (u) also has distinct eigenvalues λ a complete set of eigenvectors r¯j (u), j = 1, 2, . . . , n: ¯ i (u)¯ r i (u) = λ ri (u), f¯ (u)¯
¯li (u)f¯ (u) = λ ¯ i (u)li (u), ¯li (u) · r ¯ j (u) = δij , i, j = 1, 2, . . . , n.
156
7. Riemann Problem
From (3.13), u ∈ H(u0 ) if and only if u − u0 is parallel to r¯j (u) and ¯ j (u) for some j, 1 ≤ j ≤ n. By the normalization ¯li (u) · r¯j (u) = σ = λ δij , i, j = 1, 2, . . . , n, this is equivalent to ⎞ ⎛ ¯l1 (u) ⎜ .. ⎟ ⎜ . ⎟ ⎟ ⎜ ⎜¯lj−1 (u)⎟ ⎟ · (u − u0 ) = 0 for some j, 1 ≤ j ≤ n. ⎜ u ∈ H(u0 ) if and only if ⎜¯ ⎟ ⎜lj+1 (u)⎟ ⎜ .. ⎟ ⎝ . ⎠ ¯ln (u) The rank of the matrix above is n − 1 and so, by the implicit function theorem, this system of n − 1 equations for the n-vector u can be solved to yield a one-dimensional curve H j (u0 ). As u approaches the base state ¯ j (u) tends to λj (u0 ). This u0 along H j (u0 ), r¯j (u) tends to r j (u0 ) and λ proves the first statement of the theorem. To compare the tangency of Rj (u) and H j (u), we parametrize both curves by their length and choose r j (u) to be a unit vector. Denote by A˙ the derivative of A along the curves with respect to their arc length. Along Rj (u), u˙ = r j (u) by the choice of the parameter. Differentiate twice the jump condition σ(u − u0 ) = f (u) − f (u0 ) along H j (u): ¨. ¨ =σ ¨ (u − u0 ) + 2σ˙ u˙ + σ u f u˙ u˙ + f u Evaluate this at u = u0 , noting that σ = λj (u0 ) there, to get u = 2σr ˙ j (u0 ) + λj (u0 )¨ u. f (u0 )r j (u0 )r j (u0 ) + f (u0 )¨ Similarly, differentiate the characteristic relation f r j = λj r j along Rj (u) and evaluate it at u = u0 to obtain f (u0 )r j (u0 )r j (u0 ) + f (u0 )r˙ j = λ˙ j r j (u0 ) + λj (u0 )r˙ j . Comparing these we have, at u = u0 , u − r˙ j ) = (2σ˙ − λ˙ j )r j . (f − λj I)(¨ The right-hand side is a multiple of r j , while the left-hand side does not contain the component of r j due to the factor f − λj I. Thus both sides are zero: u − r˙ j ) = (f − λj I)¨ u = (f − λj I)r˙ j = 0, 2σ˙ − λ˙ j = 0. (3.14) (f − λj I)(¨ The third statement of the theorem follows from the second identity above. ¨ − r˙ j is parallel to r j . As the parameter is From the first identity we see that u ¨ − r˙ j is perpendicular the arc length, both u˙ and r j are unit vectors and so u ¨ − r˙ j = 0 and the second statement of the theorem is proved. to r j . Thus u This completes the proof of the theorem.
3. Hugoniot Curves
157
We have seen for convex scalar laws that the entropy dissipation across a shock is of third order, (7.4) in Chapter 3. This holds also for systems. Proposition 3.4. Let (η(u), q(u)) be an entropy pair for the system of conservation laws ut + f (u)x = 0, or equivalently, suppose that η (u) is positive definite and the compatibility condition q = η f holds. Then the entropy dissipation across a shock is of third order; that is, for u ∈ H i (u0 ), 1 ≤ i ≤ n, such that |u − u0 | is small, (3.15)
−σ[η − η0 ] + q − q0 = O(1)|u − u0 |3 ,
where σ ≡ σ(u0 , u), η = η(u), η0 = η(u0 ), etc. Moreover, if the jth characteristic field is genuinely nonlinear at u0 , as in Definition 2.3, then the entropy flux is exactly of third order, that is, O(1) is a positive (or negative) bounded function when λj (u) < λj (u0 ) (or λj (u) > λj (u0 )). Proof. As in the proof of Theorem 3.3, we parametrize the Hugoniot curve H i (u0 ) by the arc length and normalize the vector r i to a unit vector. Denote by α and β the Rankine-Hugoniot expression and the entropy jump: α ≡ −σ(u − u0 ) + f − f 0 ,
β ≡ −σ(η − η0 ) + q − q0 ,
where we have used the abbreviations α = α(u), f = f (u), η0 = η(u0 ), etc. Clearly α = 0 and β = 0 at u = u0 . The first and second derivatives of β along the Hugoniot curve are β˙ = −σ(η ˙ − η0 ) − σ η˙ + q, ˙ β¨ = −¨ σ (η − η0 ) − 2σ˙ η˙ − σ η¨ + q¨. From the proof of Theorem 3.3, at u = u0 we have σ = λi , η˙ = η u˙ = η r i , and q˙ = q u˙ = q r i , and so β˙ = −λi η r i + q r i , β¨ = −2ση ˙ r i − λi η¨ + q¨ at u = u0 . Thus, by the compatibility condition η f = q , β˙ = −λi η r i + q r i = −λi η r i + η f r i = η (−λi + f )ri = 0 at u = u0 . ¨ and showed that 2σ˙ = λ˙ i and In the proof of Theorem 3.3 we computed α ¨ = r˙ i at u = u0 . Thus u β¨ = −2ση ˙ r i − λi η¨ + q¨ = −λ˙ i η r i − λi {η ˙r i } + {η f˙ r i } = −λ˙ i η r i − λi {η ˙r i } + {η λ˙i r i } = 0 at u = u0 . This proves the first part of the proposition. To prove the second part of the proposition, we...need to further differentiate the compatibility relation ... to compute α and β and use (3.14) to show that ... (3.16) β = 2η λ˙ i r i r i at u = u0 . We omit the details of the straightforward computations. By the convexity assumption, η r i r i is positive. For the case λj (u) < λj (u0 ) (or λj (u) >
158
7. Riemann Problem
λj (u0 )), we have by ... the genuine...nonlinearity condition (4.1) that λ˙ i < 0 ˙ (or λi > 0), and so β < 0 (or β > 0). The completes the proof of the proposition.
4. Riemann Problem I This section studies the Riemann problem under the assumption that each characteristic field is either genuinely nonlinear or linearly degenerate in the following sense. Definition 4.1. An i-characteristic field is called genuinely nonlinear if the ith characteristic value λi is strictly monotone in its characteristic direction r i , ∇λi (u) · r i (u) = 0 for all states u under consideration. For definiteness, we rescale the eigenvector r i for a genuinely nonlinear field so that (4.1)
∇λi (u) · r i (u) = 1, genuine nonlinearity.
An i-characteristic field is called linearly degenerate if the ith characteristic value λi is constant in its characteristic direction r i : ∇λi (u) · r i (u) = 0, linear degeneracy, for all states u under consideration. Note that the normalization, li (u)r j (u) = δij , in (0.2) leaves one degree of freedom and the normalization (4.1) determines the direction and length of r i and li . For a genuinely nonlinear field we may choose the parameter τ for the i-characteristic curve Ri (u0 ) to be λi and partition Ri (u0 ) as shown in figure 7.03: − (4.2) Ri (u0 ) = R+ i (u0 ) ∪ Ri (u0 ), − λi (u) > λi (u0 ) for u ∈ R+ i (u0 ), λi (u) ≤ λi (u0 ) for u ∈ Ri (u0 ).
Proposition 4.2. Suppose that the ith characteristic field is genuinely nonlinear. Then a centered rarefaction wave (u0 , u1 ) can be formed for state u1 on R+ i (u0 ). Proof. This is a direct consequence of Proposition 2.4. The centered rar efaction wave (u0 , u1 ) has been constructed and is of the form (2.9). Suppose that the i-characteristic field is genuinely nonlinear. Then by (3.12) in Theorem 3.3, in a small neighborhood of u0 we can decompose the Hugoniot curve into two components with the following property, as shown
4. Riemann Problem I
159
in figure 7.03: − (4.3) H i (u0 ) − {u0 } = H + i (u0 ) ∪ H i (u0 ),
λi (u) > σ(u0 , u) > λi (u0 ) for u ∈ H + i (u0 ), λi (u) < σ(u0 , u) < λi (u0 ) for u ∈ H − i (u0 ). Thus for a genuinely nonlinear field, a discontinuity (u− , u+ ) is either + compressive, u+ ∈ H − i (u− ), or expansive, u+ ∈ H i (u− ). As with scalar convex conservation laws, the entropy condition chooses the compressive ones. Definition 4.3. Suppose that the i-characteristic field is genuinely nonlinear. An i-shock wave (u− , u+ ), with u+ ∈ H i (u− ), is admissible if it satisfies ⎧ ⎪ ⎨λi (u+ ) < σ(u− , u+ ) < λi (u− ), (4.4) Lax entropy condition. λj (u− ) < σ(u− , u+ ) for j < i, ⎪ ⎩ σ(u− , u+ ) < λj (u+ ) for i < j, Proposition 4.4. Suppose that u+ ∈ H − i (u− ), with u+ close to u+ . Then the shock wave (u− , u+ ) given by
u− for x < σ(u− , u+ )t, (4.5) u(x, t) = i-shock wave u+ for x > σ(u− , u+ )t, satisfies the Lax entropy condition, Definition 4.3. Proof. The first condition in (4.4) is the main compressibility condition of the Lax entropy condition. When |u+ − u− | is sufficiently small, we have from (3.12) that the shock speed is close to the arithmetic mean of the characteristic speeds: λi (u− ) + λi (u+ ) + O(1)|u+ − u− |2 . 2 Thus the compressibility condition holds for half of the Hugoniot curve when |u+ − u− | is small: σ(u− , u+ ) =
λi (u+ ) < σ(u− , u+ ) < λi (u− ) for u+ ∈ H − i (u− ). The other conditions are needed for consideration of the stability of the shock. When the two states u− and u+ are close to each other, σ(u− , u+ ) is close to λi (u− ) and λi (u+ ) by property (i) of Theorem 3.3. Thus the remaining conditions hold by the hypothesis of strict hyperbolicity. This completes the proof of the proposition. The following proposition is a direct consequence of Proposition 2.4 and Proposition 4.4.
160
7. Riemann Problem
Proposition 4.5. Consider a genuinely nonlinear i-characteristic field and set, as in figure 7.03, − (4.6) W i (u0 ) ≡ R+ i (u0 ) ∪ H i (u0 ),
wave curve for genuinely nonlinear i-field. Then for u1 ∈ W i (u0 ), (u0 , u1 ) forms either an ith rarefaction wave, (2.9), or an ith shock wave, (4.5); see figure 7.04.
Ri+(u0) Hi+(u0)
u1
u0 u2 u space
Hi−(u0) Ri−(u0)
− Figure 7.03. Wave curve W i (u0 ) ≡ R+ i (u0 ) ∪ H i (u0 ) for a genuinely nonlinear i-field.
t
t x t = σ(u0, u2)
dx dt
u0 = λi(u0)
x t
x t
u0
u2 dx dt
= λi(u0)
u1 x t
= λi(u2)
= λi(u1)
= λi(u)
x
x
Figure 7.04. Shock and rarefaction waves for a genuinely nonlinear i-field.
We have the following proposition relating the Lax entropy condition and the dissipation of entropy as stated in Proposition 3.4.
4. Riemann Problem I
161
Theorem 4.6. For a genuinely nonlinear field, a weak shock (u− , u+ ) satisfies the Lax entropy condition if and only if the entropy dissipation condition −σ(η(u+ ) − η(u− )) + q(u+ ) − q(u− ) < 0 holds. Moreover, the dissipation is exactly of third order in the shock strength: For some positive and bounded function O(1), (4.7)
−σ(η(u+ ) − η(u− )) + q(u+ ) − q(u− ) = −O(1)|u+ − u− |3 .
Proof. The proposition is a direct consequence of Proposition 3.4. Let λ˙ i be the derivative of λi along the Hugoniot curve H − i (u0 ), then by the genuine ˙ nonlinearity condition,...(4.1), λi = ∇λi · (−r i ) = −1 at u0 . Therefore the estimate (3.16) gives β = 2η λ˙ i r i r i = −2η r i ri , which is bounded and negative. This completes the proof of the proposition. We now go to the other extreme of linear degeneracy, corresponding to the case of linear flux in the scalar case. In turns out that the Hugoniot and characteristic curves are the same. Proposition 4.7. For linearly degenerate i-field, with ∇λi ·r i ≡ 0, Ri (u0 ) = H i (u0 ) and σ(u0 , u) = λi (u0 ) = λi (u) for any state u ∈ Ri (u0 ). Proof. For any state u on Ri (u0 ), we have from the assumption of linear degeneracy that λi (u) = λi (u0 ), and so differentiation along Ri (u0 ) of the Rankine-Hugoniot expression yields d −λi (u)(u − u0 ) + f (u) − f (u0 ) = −λi (u)u˙ + f (u)u˙ dτ = −λi (u) + f (u) r i (u) = 0. This shows that the Rankine-Hugoniot condition holds for u ∈ Ri (u0 ) and therefore H i (u0 ) = Ri (u0 ). This proves the proposition. As an immediate consequence of the above proposition, we have the following result. Proposition 4.8. Consider a linearly degenerate i-characteristic field and define the wave curve as in the left diagram of figure 7.05: (4.8) W i (u0 ) ≡ Ri (u0 ) = H i (u0 ), wave curve for linearly degenerate i-field. Then for u1 ∈ W i (u0 ), we can construct the contact discontinuity as shown in the right diagram of figure 7.05:
u0 for x < λi (u0 )t = λi (u1 )t, contact discontinuity. (4.9) u(x, t) = u1 for x > λi (u0 )t = λi (u1 )t,
162
7. Riemann Problem
t x t
u1
= λi(u0) = λi(u1) = σ(u0, u1)
Ri(u0) = Hi(u0) = Wi(u0)
u0
u0
u1
u space
x Figure 7.05. Wave curve W i (u0 ) and contact discontinuity for a linearly degenerate i-field.
Proposition 4.5 and Proposition 4.8 allow us to solve the Riemann problem. Theorem 4.9. Suppose that each characteristic field is either genuinely nonlinear or linearly degenerate and that the two states ul and ur are sufficiently close to each other. Then the Riemann problem (0.1) has a unique solution in the class of centered shock waves, centered rarefaction waves, and contact discontinuities.
u1
u0
u i −1
Wi (u i − 1 )
ui
un− 1
W1 (u 0 ) Wn(u n−1 )
u space
un Figure 7.06. Solution to the Riemann problem for systems, I: state space.
4. Riemann Problem I
163
Proof. Parametrize the wave curves W i (u) in (4.6) and (4.8) by nonsingular parameters τi , i = 1, 2, . . . , n, so that du/dτ = r i (u) at u = u0 , as in Theorem 3.3 and Proposition 4.7. For a vector v = (v1 , v2 , . . . , vn ) ∈ Rn near the origin, we assign states ui , i = 0, 1, . . . , n, with u0 = ul , ui ∈ W i (ui−1 ), τi (ui ) − τi (ui−1 ) = vi , i = 1, 2, . . . , n, and denote by u(v) the end state un , i.e. u(v) ≡ un ; see figure 7.06.
t u i−1 u1 u0
un− 1 ui un x
Figure 7.07. Solution to the Riemann problem for systems, II: physical space.
The goal is to show that there is a unique v such that the end state u(v) equals the given right state ur of the Riemann data. This would show that the mapping v → u(v) is one-to-one and onto from a small neighborhood of the origin to a small neighborhood of the left state ul . By the inverse function theorem, it is sufficient to check that the map linearized at the origin v = 0 is one-to-one and onto. To compute the linearized map, vary vi and keep vj = 0 for all j = i; the state u(v) then moves along the wave curve W i (ul ). We just noted that the tangent at vi = 0 is r i (ul ). In other words, u(0, . . . , vi , . . . , 0) − u(0) = r i (ul ). lim vi →0 vi Therefore the linearized map is ⎞ ⎛ r 1 (ul ) ⎜ r 2 (ul ) ⎟ ∂u(v) ⎟ ⎜ = ⎜ .. ⎟ . ∂v v=0 ⎝ . ⎠ r n (ul )
164
7. Riemann Problem
As the vectors r i (ul ), i = 1, 2, . . . , n, are independent, the linearized map is of full rank and the inverse function theorem applies. The solution consists of i-waves (ui−1 , ui ), i = 1, 2, . . . , n, as in figure 7.07. This completes the proof of the theorem.
5. Examples I Example 1 (Isentropic Euler equations). For the full Euler equations, we know that entropy is constant along the particle path in a smooth flow; see (5.10) of Chapter 6. Thus if the gas has constant entropy at some instance, this will remain so at later times until shock waves emerge; and for weak shock waves the entropy variation is of third order, as stated in Proposition 3.4, and therefore small. For this reason, one also considers the isentropic Euler equations, which provide a good approximation of the full Euler equations: (5.1)
ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p)x = 0, p = p(ρ).
The characteristic values λ1 = v −
p (ρ),
λ2 = v +
p (ρ)
are genuinely nonlinear if p¯ (τ ) = 0,
1 p¯(τ ) ≡ p( ). τ
We now study the Riemann problem for polyatomic gases, taking the constant A = 1 in (3.6) and (3.7) by rescaling 5 p(ρ) = ργ , 1 ≤ γ ≤ , 3
c=
p (ρ) =
√ γ−1 γρ 2 .
Solve the differential equations (3.8) and we have the characteristic curves Rj (u0 ): √ γ−1 2 γ − γ−1 ρ 2 − (ρ0 )− 2 } for γ > 1, γ−1 √ R1 (u0 ) = {u : v − v0 = − γ(log ρ − log ρ0 )} for γ = 1; √ γ−1 2 γ − γ−1 ρ 2 − (ρ0 )− 2 } for γ > 1, R2 (u0 ) = {u : v − v0 = − γ−1 √ R2 (u0 ) = {u : v − v0 = γ(log ρ − log ρ0 )} for γ = 1.
(5.2) R1 (u0 ) = {u : v − v0 =
5. Examples I
165
The Hugoniot curves H j (u0 ), j = 1, 2, in (3.11) are (5.3) H 1 (u) = {u : (v − v0 ) = −
p(ρ) − p(ρ0 ) (ρ − ρ0 )}; ρρ0 (ρ − ρ0 )
H 2 (u) = {u : (v − v0 ) =
p(ρ) − p(ρ0 ) (ρ − ρ0 )}. ρρ0 (ρ − ρ0 )
We will exhibit in the next chapter the four possibilities for the Riemann solution; see figure 8.06 of Chapter 8. Note that the characteristic curves Rj (u0 ) can reach the vacuum states ρ = 0 if γ > 1. If two states u1 and u2 are far apart and v1 < v2 , then the Riemann problem (u1 , u2 ) is solved with the vacuum state; see figure 7.08. In other words, if two states of gas are pulling apart with a sufficiently large velocity difference, then a vacuum region is created. This is not so when γ = 1, as in this case the characteristic curves do not reach the vacuum states; see (5.2).
ρ
u1
t u2
u1
ρ=0
v
u2 x
Figure 7.08. Vacuum creation through rarefactions.
Example 2 (Full Euler equations in gas dynamics). We have considered the full Euler equations in gas dynamics in Lagrangian coordinates; see Section 5 of Chapter 6. Here we consider the Euler equations in Eulerian coordinates: (5.4)
ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p)x = 0, (ρE)t + (ρEv + pv)x = 0.
From the thermodynamic relation de = T ds−p dτ, τ = 1/ρ, it is easy to see that the entropy is constant along the flow, according to (5.10) of Chapter 6: (5.5)
st + vsx = 0.
From (5.5) and the first equation in (5.4), we have the conservation law (5.6)
(ρs)t + (ρvs)x = 0.
166
7. Riemann Problem
Thus η(u) ≡ ρs and q(u) ≡ ρvs form an entropy pair if ρs is a convex function of the conserved quantity u = (ρ, ρv, ρE). For polyatomic gases, p = eCs ργ ⇒ s = A(ln p + γ ln ρ) = A ln e + B ln ρ for some positive constants A and B. Thus the entropy, as a function of the conserved quantities, is ρE (ρv)2 − + Bρ ln ρ, (ρs)(ρ, ρv, ρE) = Aρ ln ρ 2ρ2 which can be checked to be a convex function. We next compute the characteristic values and directions. For this, we rewrite the Euler equations in terms of (ρ, v, s) and regard the pressure as a function of (ρ, s): (5.7)
p = p(ρ, s).
The Euler equations become ρt + vρx + ρvx = 0, p vt + vvx + ρρ ρx + pρs sx = 0, st + vsx = 0,
(5.8) or, in matrix form,
(5.9)
ut + Aux = 0, ⎛ ⎞ ⎛ v ρ pρ ⎝ ⎠ ⎝ u= v , A= ρ s 0
ρ v 0
0
ps ⎠ . ρ
v
From the flux matrix A, direct calculations yield
(5.10)
⎞
⎛
⎞ ρ √ √ λ1 = v − pρ = v − c, r 1 = C1 ⎝− pρ ⎠ , 0 ⎛ ⎞ ps λ2 = v, r 2 = C2 ⎝ 0 ⎠ , −pρ ⎞ ⎛ ρ √ √ λ3 = v + pρ = v + c, r 1 = C3 ⎝ pρ ⎠ , 0
with arbitrary constants Cj , j = 1, 2, 3. To ensure the hyperbolicity, we require ∂ p(ρ, s) > 0, (5.11) c2 = ∂ρ which is the same as the previous condition (5.11) in Chapter 6, p¯τ (τ, s) < 0, and p¯(τ, s) = p(1/τ, s) = p(ρ, s). The second speed λ2 = v is the speed of the
5. Examples I
167
gas flow. In the frame moving with the gas flow, the other two characteristics λj , j = 1, 3, represent the speed of acoustic waves: (5.12)
c≡
√
pρ =
∂ p(ρ, s), sound speed. ∂ρ
Recall that the characteristic curves Rj are the integral curves of the vectors r j , j = 1, 2, 3. We will also treat the pressure as a function of the specific volume τ and the entropy s: p = p¯(τ, s) = p(ρ, s). Proposition 5.1. The characteristic family (5.10) for the Euler equations (5.8) has the following properties: (1) The second characteristic family is linearly degenerate, ∇λ2 · r2 ≡ 0, and along its characteristic curves R2 (u0 ), the velocity v and the pressure p are constant, ∇v · r 2 = ∇pr 2 = 0, and the entropy s is monotone. (2) The first and third characteristic families are genuinely nonlinear, ∇λj · r j = 0 for j = 1, 3, if (5.13)
∂2 p¯(τ, s) = 0. ∂τ 2
(3) Along the first and third characteristic directions Rj , j = 1, 3, the entropy s is constant and the density ρ and the velocity v are monotone. (4) Along the Hugoniot curves H j (u0 ), j = 1, 3, the variation of the entropy s is of third order: s(u) − s(u0 ) = O(1)|u − u0 |3 for u ∈ H j (u0 ), j = 1, 3. Moreover, when (5.13) holds, the entropy increases at third order as the gas flows across a shock, that is, (5.14)
s(u) − s(u0 ) = −O1 (1)|u − u0 |3 + for u ∈ H − 1 (u0 ), u ∈ H 3 (u0 ),
for some positive bounded function O1 (1). Proof. With respect to the variables (ρ, v, s), we have ∇v = (0, 1, 0), ∇p = (pρ , 0, ps ), and ∇s = (0, 0, 1), and the statement (1) follows from this and √ √ the second identity in (5.10). We have ∇λ1 = (−( pρ )ρ , 1, −( pρ )s ) and so from the first identity in (5.10), 1 1 1 √ ∇λ1 · r 1 = − ρ √ pρρ − pρ = − √ 3 p¯τ τ . 2 pρ 2 pρ ρ This proves (2). The proof of (3) is similar to that of (1). We only consider the second statement in (4) by assuming that (5.13) holds. By Theorem 4.6
168
7. Riemann Problem
for general systems, for the shock with speed σ, −σ(ρ+ s+ − ρ− s− ) + ρ+ s+ v+ − ρ− s− v− = −O1 (1)|u − u0 |3 < 0. This and the conservation of mass, m ≡ −σρ+ +ρ+ v+ = −σρ− +ρ− v− , yields m(s+ − s− ) = −O1 (1)|u − u0 |3 < 0. For a 1-shock, m < 0 and (4) is proved. Note also that the gas flows across a 1-shock from left to right and across a 3-shock from right to left because v > λ1 = v − c and v < λ3 = v + c. Thus (5.14) says that entropy increases as the gas flows across a shock wave. We will define the notion of Riemann invariant in Definition 2.1 in Section 2 of Chapter 8. From (2) of Proposition 5.1, the pressure p and the velocity v are second Riemann invariants. From (4) of Proposition 5.1, the entropy s is an ith Riemann invariant for i = 1, 3. The Lax entropy condition (4.4) applies when the acoustic modes are p(τ, s) = 0. Consider a 1-shock (u− , u+ ). genuinely nonlinear, (∂ 2 /∂τ 2 )¯ Since λ1 = v − c according to (5.10), the Lax entropy condition says that the shock speed σ satisfies v− − c− > σ > v+ − c+ . The gas flows across a 1-shock from the upstream state u− to the downstream state u+ . Thus the Lax entropy condition implies that (5.15) |v− − σ| > c− , upstream flow supersonic relative to shock; |v+ − σ| < c+ , downstream flow subsonic relative to shock. For the 3-shock, u+ is the upstream state and |v+ − σ| > c+ . The same conclusion holds. To solve the Riemann problem for the full Euler equations, it is convenient to consider the projections onto the (v, p) plane of the Hugoniot curves H i and the characteristic curves Ri . From (1) of Proposition 5.1, the curve H 2 (u0 ) = R2 (u0 ) represents a point (v, p) = (v0 , p0 ) in the (v, p) plane. At least for states u near the initial state u0 , the projections onto the (v, p) plane of the curves H i (u0 ) and Ri (u0 ), i = 1, 3, have the property that v increases as p decreases (or increases) for i = 1 (or i = 3). The Riemann problem (ul , ur ) is solved by drawing the wave curve W 1 (ul ) through ul and drawing the analogous curve through the right state ur : ˆ 3 (ur ) ≡ H + (ur ) ∪ R− (ur ). W 3
3
ˆ 3 (ur ) meet at (vm , pm ). That In the (v, p) plane, the curves W 1 (ul ) and W ˆ 3 (ur ) such that is, there exists a state um on W 1 (ul ) and a state un on W (vm , pm ) = (vn , pn ). The two states um and un are connected by a 2-wave, a contact discontinuity. The Riemann solution consists of the 1-wave (ul , um ), 2-wave (um , un ), and 3-wave (un , ur ). The above solution algorithm yields a unique solution to the Riemann problem for the full Euler equations when the two states ul and ur are
5. Examples I
169
close to each other, as follows from Theorem 4.9. Actually, the Riemann problem for the full Euler equations can be solved globally for any two given initial states. To show this, it sufficient to verify that the wave curves ˆ 3 (ur ) exist globally and that the velocity v and the pressure W 1 (ul ) and W p are monotone along them. The monotonicity property is obvious for the R(u0 )i , i = 1, 3, curves; see property (3) of Proposition 5.1. We now show that the velocity and pressure are globally monotone along the Hugoniot curves. Proposition 5.2. Suppose that (5.16)
pτ (τ, e) < 0,
pe (τ, e) > 0,
p = p(τ, e),
τ = 1/ρ.
Then along the wave curves W i (ul ), i = 1, 3, the pressure p and the velocity v are strictly monotone. Proof. By eliminating the shock speed σ from the Rankine-Hugoniot condition σ(ρ − ρ0 ) = ρv − ρ0 v0 , σ(ρv − ρ0 v0 ) = ρv 2 + p − (ρ0 v02 + p0 ), σ(ρE − ρ0 E0 ) = ρEv + pv − (ρ0 E0 v0 + p0 v0 ), we obtain the relations S1 ≡ (τ − τ0 )(p − p0 ) + (v − v0 )2 = 0, S2 ≡ (τ − τ0 )(p + p0 ) + 2(e − e0 ) = 0. Take the gradient of these with respect to the variable v ≡ (v, p, e): 1 pe ∇v S1 = 2(v − v0 ), τ − τ0 + (p − p0 ), (p − p0 ) , pτ pτ 1 pe ∇v S2 = 0, τ − τ0 + (p + p0 ), 2 − (p + p0 ) . pτ pτ The tangent to the Hugoniot curve is the cross product of the above gradients, pe 2 ∇v S1 × ∇v S2 = 2(τ − τ0 ) 1 − p0 + (p − p0 ), pτ pτ pe 1 (p + p0 ) − 2 , 2(v − v0 ) τ − τ0 + (p + p0 ) . 2(v − v0 ) pτ pτ For H + 3 (u0 ), we have v − v0 < 0, p − p0 > 0, and τ − τ0 < 0, and hypothesis (5.16) implies that pe 2 2(τ − τ0 ) 1 − p0 + (p − p0 ) < 0, pτ pτ
2(v − v0 )
pe (p + p0 ) − 2 > 0. pτ
170
7. Riemann Problem
This shows that moving along H + 3 (u0 ), the velocity v decreases and the pressure p increases. Similarly, moving along H − 1 (u0 ), where v − v0 > 0, the velocity v increases and the pressure p decreases. Note that for ideal gases, and polyatomic gases in particular, hypothesis (5.16) holds: 1 1 1 p(τ, e) = Ae , pτ = −Ae 2 < 0, pe = A > 0. τ τ τ Note also that the hypothesis (5.16) implies, by the thermodynamics relation, that p¯τ = pτ − ppe < 0, p¯ = p¯(τ, s), and so the Euler system is hyperbolic.
6. Riemann Problem II This section studies the Riemann problem for the general situation, without the assumption of each characteristic field being either genuinely nonlinear or linearly degenerate. The first step is to formulate the entropy condition. Definition 6.1. An ith discontinuity wave (u− , u+ ), with u+ ∈ H i (u− ), is admissible if the following entropy condition is satisfied: (6.1) σ(u− , u+ ) ≤ σ(u− , u) for all states u on the Hugoniot curve H i (u− ) between u− and u+ . This is the so-called Liu entropy condition. It is clear that this entropy condition reduces to the Oleinik entropy condition, (1.7) of Chapter 5, for scalar laws. It is equivalent to the Lax entropy condition, (4.4), when the i-characteristic field is genuinely nonlinear. Proposition 6.2. Consider a small neighborhood of the Hugoniot curve H i (u0 ). Moving along the curve, u ∈ H i (u0 ), starting from the initial state u0 , the speed σ(u0 , u) is increasing (or decreasing) if and only if λi (u) > ˙ 0 , u) = 0, σ(u0 , u) = λi (u), σ(u0 , u) (or λi (u) < σ(u0 , u)). When σ(u ¨ (u0 , u) > 0 (or and if σ(u0 , u) has a local minimum (or maximum), i.e. σ σ ¨ (u0 , u) < 0) there, then the i-characteristic field is genuinely nonlinear there, i.e. λ˙ i (u) > 0 (or λ˙ i (u) < 0). Proof. Note that the theorem is obvious when one considers the situation of scalar laws; see e.g. the geometric construction of the Riemann problem, figure 5.02 and figure 5.03 in Chapter 5. For systems, differentiate the Rankine-Hugoniot expression along the Hugoniot curve H i (u0 ): d ˙ −σ(u − u0 ) + f (u) − f (u0 ) = −σ(u ˙ − u0 ) + (f (u) − σ)u. 0= dτ
6. Riemann Problem II
Write u˙ ≡
171
n j=1
αj r j ,
u − u0 ≡
n
βj r j .
j=1
Here r j = r j (u). By Theorem 3.3, for a state u sufficiently close to the initial state u0 , the ri (u) component of the two vectors u − u0 and u˙ have the same sign, αi > 0 and βi > 0. Other coefficients αj and βj , j = i, are smaller. Here we have chosen, for definiteness, the differentiation to be in the r i direction. Plug these into the first identity and compare the r i (u) component to get αi (λi − σ) = σβ ˙ i, and so λi − σ and σ˙ are of the same sign. This proves the first statement of the proposition. Differentiate the above equation again and evaluate it at a state at which λi − σ = σ˙ = 0: ˙ =σ ¨ βi + σ˙ β˙ i , and so αi λ˙ i = σ ¨ βi . α˙ i (λi − σ) + αi (λ˙ i − σ) ¨ are of the same sign and the second statement of the propoThus λ˙ i and σ sition is proved. Corollary 6.3. Suppose that (u− , u+ ) satisfies the Liu entropy condition (6.1). Then (6.2)
λi (u− ) ≥ σ(u+ , u− ) ≥ λi (u+ ).
Proof. As the state u approaches the initial state u− , the speed σ(u− , u) approaches the characteristic speed λi (u− ), by Theorem 3.3, and so the first inequality follows directly from the entropy condition (6.1) upon letting u tend to u− . The entropy condition says that the speed σ reaches a minimum at u = u+ as the state u moves from the initial state u− toward the end state u+ . Thus in particular it is decreasing, dσ/dτ ≤ 0, at the end state u+ , and so the second inequality follows from Proposition 6.2. The corollary says that an admissible discontinuity according to the Liu entropy condition (6.1) is consistent with the Lax entropy condition (4.4). When the i-characteristic field is genuinely nonlinear, we know that a discontinuity either satisfies the Lax entropy condition or is expansive. Thus the Liu entropy condition reduces to the Lax entropy condition for genuinely nonlinear fields. We now construct the wave curve W i (u0 ) through a given state u0 for later use in solving the Riemann problem. For a genuinely nonlinear field, the wave curve is composed of Hugoniot and characteristic curves, (4.6), and for a linearly degenerate field all these curves are the same, (4.8). We now construct the wave curves for the general case, which gives rise to new wave types.
172
7. Riemann Problem
Algorithm for the construction of wave curves W i (u0 ) We will describe the algorithm for constructing the wave curve W i (u0 ) through an initial state u0 in that direction of r i (u0 ). Take a non-singular parameter τ in the direction with τ (u0 ) = 0. For each positive and small τr , we want to identify the state ur on the wave curve W i (u0 ) with τ (ur ) = τr . The state ur has the property that it is connected to ul on the left by a composite wave pattern consisting of ith shock and rarefaction waves. The construction is interesting already for scalar laws. The construction using upper and lower envelopes, illustrated in figure 5.02 and figure 5.03 of Chapter 5, for scalar laws does not work for systems because the Hugoniot curve H i (u0 ) depends on the initial state u0 in a global way. The upper and lower envelopes construction needs to be recast slightly for the present construction. Thus we now first visualize the present construction through figures for the corresponding construction of the Riemann problem (ul , ur ), ur < ul , for scalar conservation laws ut + f (u)x = 0 with the flux function f (u) having finite inflection points. Take the example in figure 5.02 of Chapter 5. The construction is through the upper envelope as described there. We redo this by proceeding from the state ul . The state u1 is the state satisfying the property that σ(ul , u), ur ≤ u ≤ ul , reaches a minimum at u1 ; see the dashed lines in figure 7.09. The state u1 has the property that σ(u1 , u) ≥ λ(u1 ) = f (u1 ) for ur < u < u1 . This is used as the defining property for the state u2 , shown by the dotted lines in figure 7.09: u2 =
min {u : σ(u, u ¯) ≥ λ(u) for all u ¯ such that ur ≤ u ¯ ≤ u.}
ur ≤u≤u1
The reason for the above recasting of the upper envelope construction for scalar laws in terms of extremal properties is that it can be generalized to general systems. For systems, the Hugoniot curve H i (u0 ) depends on its intial state u0 in an algebraic way through the Rankine-Hugoniot relation, while the characteristic curve Ri (u0 ) is the integral curve of the eigenvector r i (u). Unlike with scalar laws, for a given initial state u0 and wave strength |ur − u0 | = τr , the end state ur ∈ W i (u0 ) is not known a priori and needs to be constructed. As the state u moves along the Hugoniot curve H i (u0 ), the speed σ(u0 , u) reaches a minimum at u = u1 for 0 ≤ τ (u) ≤ τr . There are two cases to consider; the first case is that τ (u1 ) = τr . In this case we set ur = u1 ∈ W i (ul ). The corresponding solution is a single shock (ul , ur ) satisfying the Liu entropy condition; see figure 7.10.
6. Riemann Problem II
173
f(u)
u2 u
ur
u1 ul
u ¯
u
t u1
u2
ul
ur
ul
ur
x
Figure 7.09. Riemann problem for scalar laws.
We next consider the second case, 0 ≤ τ (u1 ) < τr . The discontinuity (u0 , u1 ), which is vacuous when τ (u1 ) = 0, satisfies the Liu entropy condition. Moreover, dσ/dτ = 0 at u = u1 , and σ has a minimum there. By Proposition 6.2, λi (u1 ) = σ(u0 , u1 ) and λ˙ i (u) > 0. Thus the shock (u0 , u1 ) can be continued by a rarefaction wave (u1 , u) for u ∈ Ri (u1 ) and close to u1 . These two waves have the same speed λ(u1 ) = σ(u0 , u1 ) at their intersection; see figure 7.11. The defining property of the state u1 also implies that there is no state u ∈ H i (u1 ) satisfying τ (u1 ) < τ (u) < τr , with the property that σ(u1 , u) = λi (u1 ). This is so because otherwise we would
174
7. Riemann Problem
t
σ(ul , u) u ∈ Hi(ul )
σ(ul , u)
λi(ul )
x t
ul
= σ(ul , ur ) ur
σ(ul , ur ) τ
τr
x
Figure 7.10. Riemann problem for systems, I: single i-shock case.
have σ(ul , u) = σ(u0 , u1 ) ≡ σ = λi (u1 ), which leads to σ(u1 − u0 ) = f (u1 ) − f (u0 ),
σ(u − u1 ) = f (u) − f (u1 ) ⇒ σ(u − u0 ) = f (u) − f (u0 ),
and so u ∈ H i (u0 ) and σ(u0 , u) = σ(u0 , u1 ), contradicting the minimality property defining the state u1 . This shows that for any state u ∈ H i (u1 ) such that τ (u1 ) < τ (u) < τr , we have σ(u1 , u) < λi (u1 ). We increase the strength of the rarefaction wave (u1 , u), u ∈ Ri (u1 ), until we reach the state u2 with the largest value of τ under the condition that for any state ¯ ∈ Ri (u1 ) between u1 and u2 , the speed σ(¯ u u, u) is less than λ(¯ u) for all u) with τ (¯ u) < τ (u) < τr . The defining property of the state u2 u ∈ H i (¯ implies that λi (u), u ∈ Ri (u1 ), is monotonically increasing for u between u1 and u2 . Thus, indeed, (u1 , u2 ) forms a rarefaction wave. t
σ(ul ,u) u ∈R i(u1)
u1 u∈Hi(ul)
ul
λi(ul) σ(ul,u1) λi(u r)
τr
ur
τ
x
Figure 7.11. Riemann problem fors system, II: a shock and a rarefaction waves.
From here we again have two cases: either τ (u2 ) = τr or τ (u2 ) < τr . In the first case, we set ur ≡ u2 ∈ W i (u0 ) and the solution consists of the shock wave (u0 , u1 ) followed by the rarefaction wave (u1 , ur ), as shown in figure 7.11. When τ (u2 ) < τr , by the defining property of u2 , there exists a state u3 ∈ H i (u2 ) with the property that σ(u3 , u) < λi (u) for all states u ∈ H i (u3 ) such that τ (u3 ) < τ (u) < τr . There are again two cases. When τ (u3 ) = τr we set ur = u3 , and the solution is the shock (u0 , u1 ) followed by
7. Examples II
175
the rarefaction wave (u1 , u2 ) and then the shock (u2 , u3 ). When τ (u3 ) < τr , we repeat the above procedure, with u3 replacing u1 . This way, the solution is a wave pattern of alternating shock and rarefaction waves. This completes the construction of the wave curves W i (u0 ) for i = 1, 2, . . . , n. With this construction, Theorem 4.9 can be directly extended to the general case. Theorem 6.4. Suppose that each characteristic field is either linearly degenerate or such that the characteristic speed λi (u) has isolated inflection points along each characteristic curve Ri (u0 ) and the two states ul and ur are sufficiently close to each other. Then the Riemann problem (0.1) has a unique solution in the class of i-centered waves, i = 1, 2, . . . , n. Each icentered wave is either a composition of shock and rarefaction waves or a contact discontinuity.
7. Examples II Example 1 (Nonlinear elasticity). Consider a model for an elastic material where u is the position relative to the rest state and the elastic stress p = p(τ ) is a function of the shear strain τ ≡ ux . The velocity is v = ut . For linear elasticity the shear stress is p(τ ) = −μτ , in the opposite direction of the shear strain with μ > 0 the shear modulus. For general nonlinear elasticity the shear stress is an odd function of the shear strain and we have
τt − vx = 0, p(τ ) an odd function. (7.1) vt + p(τ )x = 0, The system is of the same form as the p-system (3.3). Here the first equation is the compatibility condition uxt = utx = 0 and the second equation is Newton’s second law of motion. The shear stress p(τ ) is an odd function for obvious physical reasons. In the case of compression of the elastic material, the stress is not an odd function of the strain and the situation is similar to that of the isentropic Euler equations in Lagrangian coordinates, (2.1). System (7.1) is hyperbolic if p (τ ) < 0, that is, the stress is in the opposite direction to the strain. For the system to be genuinely nonlinear, one needs p (τ ) = 0, which is impossible for an odd function. Thus the system is not genuinely nonlinear and for a nonlinear response, p (τ ) not a constant, it is not linearly degenerate either. The Rankine-Hugoniot condition is −s(τ+ − τ− ) = −v+ − v− , −s(v+ − v− ) = p(τ+ ) − p(τ− ). Thus the shock speed s = σ(u− , u+ ) can be expressed as a function of τ± : s2 = −
p(τ+ ) − p(τ− ) . τ+ − τ−
176
7. Riemann Problem
It is clear from this that the Liu entropy condition for (u− , u+ ) can be expressed in a similar way to the Oleinik entropy condition for scalar laws: For a 1-shock (u− , u+ ), p(τ+ ) − p(τ− ) p(τ ) − p(τ− ) ≤ for all τ between τ− and τ+ ; τ+ − τ− τ − τ− for a 2-shock (u− , u+ ), p(τ ) − p(τ− ) p(τ+ ) − p(τ− ) ≥ for all τ between τ− and τ+ . τ+ − τ− τ − τ− Example 2 (Magnetohydrodynamics). Next we consider the inviscid magnetohydrodynamics equations in Lagrangian coordinates: ⎧ ⎪ τt − v ⎪ 1x = 0, ⎪ ⎪ 2 ⎪ 1 2 ⎪ + p + + b = 0, b v ⎪ 1t 3 ⎪ 2μ0 2 ⎪ x ⎪ ⎪ ⎪ ⎪ v2t − μ10 b∗1 b2 = 0, ⎪ ⎪ x ⎪ ⎪ ⎨v − 1 b∗ b 3t μ0 1 3 x = 0, (7.2) ⎪ ⎪ (τ b2 )t (b∗1 v2 )x = 0, ⎪ ⎪ ⎪ ⎪ (τ b3 )t − (b∗1 v3 )x = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e + 12 v12 + v22 + v32 + 2μτ 0 b22 + b23 ⎪ ⎪ t ⎪ ⎪ ⎪ ⎩+ p + 2μ1 (b22 + b23 ) v1 − μ1 b∗1 b2 v2 + b3 v3 = 0, 0 0 x
(b∗1 , b2 , b3 )t ,
)t ,
p, b = e, and θ represent, where τ = 1/ρ, v = (v1 , v2 , v3 respectively, the specific volume, velocity, pressure, magnetic induction, internal energy, and temperature. In one space dimension, the first component of the magnetic field is a constant, denoted by b∗1 . The thermodynamics relation and the above system also yield that the entropy is constant along particle paths, st = 0. Set the constitutive relation p = p¯(τ, s). Let v 2 + v22 + v32 τ (b22 + b23 ) E =e+ 1 + 2 2μ0 be the total energy. The above system is written as
(7.3)
ut + Aux = 0, ⎛ ⎛ ⎞ 0 τ ⎜ d ⎜ v1 ⎟ ⎜ ⎜ ⎟ b1 b2 ⎜ ⎜ v2 ⎟ ⎜ μ0 τ ⎜ ⎟ ⎜ ⎟ ⎜ u = ⎜ v3 ⎟ , A = ⎜ bμ1 bτ3 ⎜ 0 ⎜τ b2 ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝τ b3 ⎠ ⎝ 0 E 0
−1 0 0 0 0 0 P
0 0 0 0 −b∗1 0
−b1 b2 μ0
0 0 0 0 0 −b∗1
−b1 b3 μ0
0 0 d2 b2 d2 b2 −b∗1 0 μ0 τ 0 0 0 0
−b∗1 μ0 τ
0 0 0
0
⎞
p¯s ⎟ θ ⎟
0⎟ ⎟ ⎟ 0 ⎟, ⎟ 0⎟ ⎟ 0⎠ 0
7. Examples II
177
where P ≡ p+(b22 +b23 )/(2μ0 ), d1 ≡ p¯τ +
p¯ ps 1 2 3 p¯s 1 1 1 p¯s ( − ). + (b2 +b3 )( − ), d2 ≡ θ μ0 2θ τ μ0 τ θ
By straightforward calculations, the characteristic speeds, the eigenvalues of the flux matrix A, are −λf , −λa , −λs , 0, λs , λa , λf . Here ±λf are called the fast wave speeds, ±λa the Alf´ven wave speeds, ±λs the slow wave speeds, and 0 the speed of the gas flow in Lagrangian coordinates. These speeds are given as b∗2 1 , μ0 τ 1 b∗2 + b22 + b23 (λs )2 = − p¯τ + p¯τ + 1 2 μ0 τ
(7.4) λ0 = 0, (λa )2 ≡
−
1 2
b∗2 + b22 + b23 2 b2 + b23 ), − 4¯ pτ ( 2 p¯τ + 1 μ0 τ μ0 τ
(λf )2 = − p¯τ + +
1 2
1 b∗2 + b22 + b23 p¯τ + 1 2 μ0 τ p¯τ +
2 2 b∗2 b2 + b23 1 + b2 + b3 2 ). − 4¯ pτ ( 2 μ0 τ μ0 τ
Like with the Euler equations in gas dynamics, the hypothesis p¯τ < 0 makes all the eigenvalues real and so the magnetohydrodynamics system (7.2) is hyperbolic. There are essential differences from the Euler equations in that the eigenvalues are not necessarily distinct and so the system (7.2) is not strictly hyperbolic. When the component of the magnetic induction in the ven wave has the same speed as the gas wave direction is zero, b∗1 = 0, the Al´ flow, λa = λ0 . This does not lead to a nonlinear effect, as both characteristics are linearly degenerate. When the component of the magnetic induction normal to the wave direction is zero, b2 = b3 = 0, there are two situations, depending on the relation between the Al´ ven speed λa and the speed λg ≡ √ −¯ pτ of the acoustic wave without the magnetic induction:
b∗2 −¯ pτ = λ2g when μ10 τ = λ2a > λ2g = −¯ pτ , 2 λs = b∗2 ∗2 b 1 = λ2a when μ10 τ = λ2a < λ2g = −¯ pτ ;
μb0∗2τ (7.5) b∗2 2 2 2 1 1 = λa when μ0 τ = λa > λg = −¯ pτ , λ2f = μ0 τ ∗2 b pτ . −¯ pτ = λ2g when μ10 τ = λ2a < λ2g = −¯ Thus, when b2 = b3 = 0, λf = max{λa , λg },
λs = min{λa , λg }.
178
7. Riemann Problem
This allows for the possibility of combining fast and Al´ ven waves, as well as slow and Al´ ven waves, into the so-called intermediate waves of magnetohydrodynamics, which gives rise to the interesting issues of the suitable entropy condition and the stability of the associated viscous wave. We will discuss these questions in Chapter 12 on the resonance phenomena.
8. Notes The Riemann problem was introduced in the classical paper on isentropic Euler equations by Riemann [112], which also helped to initiate shock wave theory. Riemann’s theory was complemented and corrected by various authors, and then generalized to the full Euler equations; see CourantFriedrichs [32]. The generalization of the theory for Euler equations in gas dynamics to general hyperbolic conservation laws was done by Peter Lax [73]. The seminal paper [73] introduced the basic notions of genuine nonlinearity, linear degeneracy, and the Lax entropy condition, and solved the Riemann problem. Liu [83] introduced the Liu entropy condition and solved the Riemann problem for general hyperbolic conservation laws. The condition for the p-system was found by Wendroff (1972), and another equivalent condition, the entropy rate admissibility criterion, was proposed by Dafermos (1973); see references in [35]. For the study of gas dynamics, see Chang-Hsiao [22] and references therein.
9. Exercises 1. Consider the isentropic Euler equations (5.1) with constitutive relation p(ρ) = ρ3 /3: ρt + (ρv) x = 0, (ρv)t + ρv 2 +
ρ3 3
x
= 0.
Solve the Riemann problem with Riemann data
(2, −1) for x < 0, (ρ, v)(x, 0) = (2, 1) for x > 0. 2. The isentropic Euler equations for monatomic gases are of the form ρt + (ρv)x = 0, 5 (ρv)t + (ρv 2 + a2 ρ 3 x = 0. Here a is a positive constant. Consider the Riemann problem
(1, −V ) for x < 0, (ρ, v)(x, 0) = (1, V ) for x > 0, for some constant V . Show that
9. Exercises
179
(i) for the compression case, V < 0, the Riemann solution consists of shock waves; (ii) for the expansion case, V > 0, the Riemann solution consists of rarefaction waves. Find the value V0 such that for V > V0 , the solution contains the vacuum state. 3. Consider the system ut + u(u2 + v 2 ) x = 0, vt + v(u2 + v 2 ) x = 0. Show that R1 (u0 ) = H 1 (u0 ) is the circle u2 + v 2 = constant through the state u0 , and that R1 (u0 ) = H 1 (u0 ) is the line through the origin and the state u0 . 4. Consider the full Euler equations (5.4). Show that in a neighborhood of the initial state u0 , the pressure p and the velocity v are both monotone along the characteristic curves Rj (u0 ), j = 1, 3. 5. Consider the shear model (7.1) with stress-strain relation p(τ ) = −τ 3 /3−τ :
τt − vx = 0, 3 vt + − τ3 − τ = 0. x
Construct the wave curve W 1 (u0 ) through the state u0 = (τ0 , v0 ) = (3, −1). 6. Consider hyperbolic conservation laws of the form g(u)t + f (u)x = 0. Define the notion of genuine nonlinearity and the Lax entropy condition for a genuinely nonlinear characteristic field. 7. Consider hyperbolic conservation laws of the form g(u)t + f (u)x = 0. Define the Liu entropy condition for a general characteristic field and show that the condition reduces to the Lax entropy condition when the characteristic field is genuinely nonlinear. 8. Consider a system of hyperbolic conservation laws in several dimensions, (0.1) of Chapter 6, m F j (u) x = 0. ut + ∇x · F(u) = 0, or ut + j
j=1
Generalize the notion of genuine nonlinearity for a characteristic field to the system and formulate the Lax entropy condition. Formulate the Liu entropy condition for a general characteristic field. (Hint: Consider the one-dimensional projection of the system, (6.6) in Section 6.6 of Chapter 6.) 9. Consider the isothermal constitutive relation p = σ 2 ρ. For special relativity, the sound speed σ is less than the speed c of light, σ < c. The isentropic
180
7. Riemann Problem
Euler equations become (see [117]) σ 2 + c2 v 2 v 2 2 (9.1) ρ + 1 + ρ(σ + c ) = 0, c2 c2 − v 2 c2 − v 2 x t v v2 2 2 2 + ρ (σ + c ) + σ = 0. ρ(σ 2 + c2 ) 2 c − v2 t c2 − v 2 x Show that in the region |v| < c, the system is hyperbolic and genuinely nonlinear. (The multi-dimensional version of (9.1), (9.2)
ρc2 + p c2 − v 2
ρc2 + p p + vi = 0, c2 t c2 − v 2 xi 3
−
i=1
ρc2 + p c2 − v
vj 2
t
+
3 2 ρc + p i=1
c2 − v
v j v i + pδij 2
xi
= 0, j = 1, 2, 3,
is endowed with an entropy pair and so the system is symmetrizable; see [105].)
10.1090/gsm/215/08
Chapter 8
Wave Interactions
In this chapter we study the nonlinear interactions between the elementary waves constructed in Chapter 7 for solving the Riemann problem for the general system of conservation laws ut + f (u)x = 0. The interaction of elementary waves for convex scalar laws is rather simple; see Section 3 of Chapter 3. Even for genuinely nonlinear systems the wave interaction phenomena are richer. For instance, two shock waves do not simply combine; the combining also gives rise to waves of other characteristic families. Interaction of two elementary waves can produce infinitely many shock waves at later times. The complexity of the general situation prompts the consideration of the time-asymptotic analysis of interactions between elementary waves. The well-posedness theory presented in Chapter 9 is based on the analysis of wave interactions.
1. Interaction of Infinitesimal Waves As we have seen in Theorem 3.3 of Chapter 7, elementary waves take values close to the characteristic directions r i (u), i = 1, 2, . . . , n. Infinitesimal waves therefore take values in the characteristic directions. An understanding of the interactions of infinitesimal waves will help us in the study of interactions of nonlinear waves later. For the study of infinitesimal waves, consider the decomposition of the coupling measure f (u) of the flux function f (u) in the characteristic directions: (1.1)
j (u) ≡ lj (u)f (u)(rk (u), rl (u)), j, k, l = 1, 2, . . . , n. Ckl
181
182
8. Wave Interactions
Differentiate the characteristic relation f (u)r k (u) = λk (u)rk (u) in the r l (u) direction to obtain f (u)(r k (u), rl (u)) + f (u)(rk (u)(rl (u)) = (λk (u)(rl (u))rk (u) + λk (u)rk (u)(r l (u)). From the normalization li (u)r j (u) = δij , the above yields lj f (r k , r l ) + λj lj r k (r l ) = λk (rl )δjk + λk lj r k (rl ). j j = Ckl (u), Here and in what follows, for simplicity, we write r i = r i (u), Ckl etc. We have from (1.1) and the above identity that
λk (rl ), j = k, j = (λk − λj )lj r k (r l ) + λk (r l )δjk = (1.2) Ckl (λk − λj )lj r k (rl ), j = k.
Thus Ciii = 0 for a genuinely nonlinear ith characteristic field, and Ciii = 0 j with j = k or for a linearly degenerate i-field. The other coefficients Ckl j = l, measure the degree of coupling between waves pertaining to distinct characteristic families. We now illustrate this with a general smooth solution u(x, t) around a constant state u0 . Write the solution in the coordinates of eigenvectors: n uj (x, t)r0j , r 0j ≡ r j (u0 ). u(x, t) = u0 + j=1
Plug this into the hyperbolic conservation laws and apply Taylor expansion to obtain n n j 0 Ckl (uk ul )x r 0j + O(1)|u − u0 |3 x = 0. ujt + λj ujx + ut + f (u)x = j=1
k,l=1
Ignoring the third-order terms, we obtain the coupled quadratic system n j0 0 (1.3) ujt + λj ujx + Ckl (uk ul )x = 0, j = 1, 2, . . . , n. k,l=1 j0 j = Ckl (u0 ). With the linear transHere, similar to the above, we write Ckl 0 port part ujt +λj ujx , information propagates mainly along the characteristic j0 0 0 direction dx dt = λj . From (1.2), the coefficients Cjj = λj (r j ) of the diagoj0 ((uj )2 )x in (1.3) are nonzero for a genuine nonlinear nal nonlinear terms Cjl j-field and zero for a linearly degenerate j-field. They measure the compresj0 , kl = jj, sion and expansion of the j-characteristics. The coefficients Ckl measure the production of waves in the jth characteristic direction resulting from the interaction of waves in the kth and lth characteristic directions. j0 = 0 for some kl = jj, we expect the interaction of a k-wave and When Ckl an l-wave to produce a j-wave.
2. A 2 × 2 System and Coordinates of Riemann Invariants
183
2. A 2 × 2 System and Coordinates of Riemann Invariants To obtain definite quantitative information on the interaction of nonlinear waves with finite strength, one needs to consider particular systems. Consider the p-system
τt − vx = 0, (2.1) vt + p(τ )x = 0. The system is strictly hyperbolic if p (τ ) < 0 and genuinely nonlinear if p (τ ) > 0; see (3.4) and (3.5) of Section 3 in Chapter 7. The characteristic curves are given by the ordinary differential equations, (3.8) in Chapter 7, R1 (u) :
dv = −p (τ ), dτ
R2 (u) :
dv = − −p (τ ). dτ
The relative positions of the characteristic curves Rj (u) and the Hugoniot curves H j (u) are important for the study of wave interactions. Write the first Hugoniot curve, (3.11) of Chapter 7, as H 1 (u0 ) : v = v0 +
√ B(τ − τ0 ),
B = B(τ ) ≡ −
p(τ ) − p(τ0 ) . τ − τ0
By Taylor expansion, B = −p (τ0 ) − p (τ0 )(τ − τ0 )/2 − p (τ0 )(τ − τ0 )2 /6 + O(1)(τ − τ0 )3 and so B(τ0 ) = −p (τ0 ),
B (τ0 ) = −
p (τ0 ) , 2
B (τ0 ) = −
p (τ0 ) . 3
This yields, along H 1 (u0 ) at the initial state u0 , (2.2)
dv √ = B(τ0 ) = −p (τ0 ), dτ
d2 v B −p (τ0 ) √ , (τ = ) = 0 dτ 2 B 2 −p (τ0 )
d3 v 3 (p (τ0 ))2 3 B 1 p (τ0 ) √ ) − ( = (τ ) = − . 0 dτ 3 2 B 2 −p (τ0 ) 16 (−p (τ0 ))3/2 dv = The characteristic curve R1 (u0 ) is defined by the differential equation dτ −p (τ ) and so, along R1 (u0 ) at the initial state u0 , dv = −p (τ ) = −p (τ0 ), (2.3) dτ τ =τ0 2 −p (τ0 ) d v , = ( −p (τ )) = dτ 2 2 −p (τ0 ) τ =τ0 1 (p (τ0 ))2 −p (τ ) 1 p (τ0 ) d3 v ) − = ( = − . dτ 3 2 −p (τ0 ) 4 (−p (τ0 ))3/2 2 −p (τ ) τ =τ0
184
8. Wave Interactions
The first two derivatives in (2.2) and (2.3) are the same, a general fact by Theorem 3.3 in Chapter 7. But the third differentials have a difference of
−
1 p (τ0 ) 3 (p (τ0 ))2 1 p (τ0 ) 1 (p (τ0 ))2 − − − − 2 −p (τ0 ) 16 (−p (τ0 ))3/2 2 −p (τ0 ) 4 (−p (τ0 ))3/2 =
1 (p (τ0 ))2 > 0. 16 (−p (τ0 ))3/2
This yields the relative position of these two curves near ul as depicted in figure 8.01. R 1+(ul ) H +(ul ) R 2−(un) ul um un H1−(ul ) H1−(um) R1−(ul )
τ
ur R 2+(un)
v
Figure 8.01. Wave interaction I: state space.
t
un ul
ur um
Figure 8.02. Wave interaction I: physical space.
x
2. A 2 × 2 System and Coordinates of Riemann Invariants
185
Consider two weak 1-shock waves (ul , um ) and (um , ur ), with um ∈ and ur ∈ H − 1 (um ). The relative position of R1 and H 1 implies + that there is a state un ∈ H − 1 (ul ) with the property that ur ∈ R2 (un ). This means that the interaction of the two 1-shocks (ul , um ) and (um , ur ) results in a 1-shock wave (ul , un ) followed by a 2-rarefaction wave (un , ur ); see figure 8.01 and figure 8.02. H− 1 (ul )
Let α be the strength of the left shock (ul , um ) and β the strength of the right shock (um , ur ). Then the strength of the combined shock (ul , un ) is close to α + β and the resulting rarefaction wave (un , ur ) has strength of third order, according to Theorem 3.3 in Chapter 7: ˜ + β)3 − α3 − β 3 = 3αβ(α + β), or |un − ur | =(α (2.4)
|un − ur | = O(1)|ul − um ||um − ur |(|ul − um | + |um − ur |).
In other words, the interaction of two shock waves of the same characteristic family follows the principle of linear superposition with a third-order error. The above example shows that interaction of two waves of the same family can produce a wave of another family. In fact, the interaction of a two elementary waves can give rise to infinitely many shock waves as the next example shows. Consider the interaction of a 1-rarefaction wave (ul , um ), with um ∈ R+ 1 (ul ), and a stronger 1-shock wave (um , ur ), with (u ). As the rarefaction wave spreads out, the interaction is not ur ∈ H − m 1 instantaneous and not a simple process. Instead of following the interaction process exactly, and to gain a basic understanding of the interaction, we pretend that the 1-rarefaction wave (ul , um ) is a rarefaction shock and let it interact with the stronger 1-shock wave (um , ur ). Setting aside the fact that the rarefaction shock does not satisfy the Rankine-Hugoniot condition and is not admissible, the interaction is now instantaneous. By the same geometric considerations as above, we see that the resulting wave pattern is ¯ ) followed by a 2-shock (¯ u, ur ), as shown in figure 8.03. a 1-shock (ul , u t
R1+(ul ) H2−(¯ u) ur H1−(um) H1−(ul )
u¯
um u¯
ul ul
τ
v
ur um
Figure 8.03. Interaction of shock and rarefaction shock.
x
186
8. Wave Interactions
To gain an understanding of the actual interaction process, we approximate the rarefaction wave (ul , um ) by a sum of weak rarefaction shocks: (2.5) (ul , um ) =(u ˜ l , u1 ) ∪ (u1 , u2 ) ∪ · · · ∪ (uk−1 , um ); λi (uj−1 ) + λi (uj ) , uj ∈ R1 (ul ), j = 1, . . . , k − 1. 2 We will assess the accuracy of the approximation of rarefaction waves by rarefaction shocks toward the end of this example in Proposition 2.2. σ(uj−1 , uj ) ≡
We now go into more detail of the interaction of a 1-rarefaction wave (ul , um ) and a stronger 1-shock wave (um , ur ) using the approximation (2.5). The interaction of the rarefaction shock (uk , um ) with the shock (um , ur ) results in a 1-shock (uk , ua ) and a weaker 2-shock (ua , ur ); see figure 8.04. The 1-shock (uk , ua ) will interact with the rarefaction shock (uk−1 , uk ) to yield a 1-shock and another 2-shock. The interactions of the 1shock with other rarefaction shocks will result in a series of 2-shocks. Similar to the first example, figure 8.03, the interaction of these 2-shocks in turn yields a series of 1-rarefaction waves. Thus there will be more interactions between the original 1-shock and the newly created 1-rarefaction waves. Eventually, the interactions will result in infinitely many 2-shocks and 1rarefaction waves.
t
ua uk
ul
um
ur x
Figure 8.04. Interaction of shock and rarefaction wave.
The interaction process just described will continue indefinitely. The 2-shocks will combine and the newly produced waves will be of ever smaller ¯ ) and a strength. Time-asymptotically, there will be a 1-shock wave (ul , u
2. A 2 × 2 System and Coordinates of Riemann Invariants
187
2-shock wave (¯ u, ur ), as depicted in figure 8.03 under the simplified analysis there. If one ignores the complex interaction process and focuses on the time-asymptotic state, then the interaction of (ul , um ) and (um , ur ) time-asymptotically results in a non-interacting wave pattern consisting of ¯ ) and a 2-shock wave (¯ a 1-shock wave (ul , u u, ur ), which is the same as the combination of elementary waves in the solution of the Riemann problem (ul , ur ). In other words, figure 8.03 accurately describes the situation in the time-asymptotic sense. This time-asymptotic analysis makes sense for two reasons: First, if two sets of Riemann solutions are close to each other in space, then the interaction of the elementary waves contained therein is completed soon in time, and the time-asymptotic analysis is an accurate approximation of the actual complex interaction process. The second reason is that this analysis predicts the time-asymptotic behavior of a general solution. As the system of conservation laws ut + f (u)x = 0 is invariant under the dilation (x, t) → (αx, αt) for any positive constant α, the above two reasons are basically the same. We will carry out the quantitative analysis of wave interactions in this time-asymptotic sense in Section 4. In the above examples, the focus was on the interaction of two waves of the same characteristic family. Because of the third-order contact of the Hugoniot curve H i (u0 ) and the characteristic curve Ri (u0 ) at the initial state u0 , as described in Theorem 3.3 in Chapter 7, the interaction follows the principle of linear superposition plus a third-order error. For the interaction of waves from distinct families, the error is in general of second order, as predicted in the study of infinitesimal waves, (1.3). We will carry out the quantitative analysis of such interactions for the full Euler equations in the next section. For 2 × 2 systems we have the coordinates of Riemann invariants, and the error is always of third order when the strength of waves is measured in the coordinates of Riemann invariants. Definition 2.1. A scalar function z(u) is called a j-Riemann invariant if z(u) is constant along any jth characteristic curve Rj (u0 ) or, equivalently, (2.6)
∇z(u) · r j (u) = 0, j-Riemann invariant z(u).
Consider a 2 × 2 system ut + f (u)x = 0, u = (u1 , u2 )T ∈ R2 . By the normalization li (u)rj (u) = δij , (0.2) in Chapter 7, the gradient ∇z(u) of a 1-Riemann invariant is orthogonal to r 1 (u), as in (2.6), and is therefore parallel to l2 (u): ∇z(u) = α(u)l2 (u) for a scalar function α(u).
188
8. Wave Interactions
Write l2 (u) = (a(u), b(u)) and parametrize a curve z(u) = constant by u2 = u2 (u1 ). Then 0=
dz(u1 , u2 (u1 )) du2 (u1 ) z u1 a(u1 , u2 ) du2 (u1 ) . = z ⇒ = − = − 1 + z u2 u du1 du1 du1 z u2 b(u1 , u2 )
This is the differential equation for the construction of the function z(u1 , u2 ). The 2-Riemann invariant w(u1 , u2 ) is constructed similarly. We may also write uj = uj (z, w) for j = 1, 2. For smooth solutions, the 2 × 2 system is diagonalized using the Riemann invariants: zt + λ2 zx = α(u)l2 (u)ut + λ2 α(u)l2 (u)ux = α(u)l2 (u) ut + f (u)x = 0. A similar calculation holds for the 2-Riemann invariant w and we have
wt + λ1 wx = 0, (2.7) equations for Riemann invariants. zt + λ2 zx = 0, In the coordinates (w, z) of the Riemann invariants, the characteristic curves are horizontal and vertical lines. For the p-system with p (τ ) < 0 and p (τ ) > 0, we have analyzed the local relative position of the rarefaction wave curves and Hugoniot curves; see figure 8.01. It can be shown that this relative position is global and in the Riemann invariant coordinates the Hugoniot curves are convex, as depicted in figure 8.05.
R2+(u0)
u0 H1−(u0)
R1+(u0) z w
H2−(u0) Figure 8.05. Wave curves for p-system in Riemann invariant coordinates.
There are four cases for the solution of the Riemann problem, illustrated in figure 8.06. The Hugoniot curves deviate from the axes at third order by
2. A 2 × 2 System and Coordinates of Riemann Invariants
189
Theorem 3.3 of Chapter 7. Consequently, when the strength of the waves is measured by the Riemann invariants, the interaction follows the principle of linear superposition with a third-order error. t
ur
um
ul
ul
um
ur x
t
ul
um
ul
um
ur
ur
x t
ur
um
ul ul
um
ur x
t
um
ur
ul
ul
um
ur x
Figure 8.06. Riemann problem for p-systems.
190
8. Wave Interactions
In the case of interaction of two rarefaction waves, the interaction does not alter the wave strength; see figure 8.07.
t ur
um ul
un
un
ul
um
ur
x
Figure 8.07. Interaction of rarefaction waves.
Before we leave this example, we analyze the accuracy of the approximation of a rarefaction wave by rarefaction shocks as in (2.5). Consider a general system of hyperbolic conservation laws and an i-rarefaction wave (u− , u+ ). Approximate the rarefaction wave by a sum of weak rarefaction shocks: (2.8) ˜ 0 , u1 ) ∪ (u1 , u2 ) ∪ · · · ∪ (uk−1 , uk ), u0 = u− , uk = u+ ; (u− , u+ ) =(u σ(uj−1 , uj ) ≡
λi (uj−1 ) + λi (uj ) , uj ∈ Ri (u− ), j = 1, . . . , k. 2
The following proposition quantifies the accuracy of the approximation. Proposition 2.2. Suppose that each rarefaction shock is weak in the partition (2.8): (2.9)
|uj − uj−1 | ≤ ε, j = 1, . . . , k, k = O(1)
|u+ − u− | , ε
for ε small. Then the Rankine-Hugoniot condition and the entropy inequality are violated with a third-order error: (2.10) |−σ(uj−1 , uj )(uj − uj−1 ) + f (uj ) − f (uj−1 )| + | − σ(uj−1 , uj ) η(uj ) − η(uj−1 ) + q(uj ) − q(uj−1 )| = O(1)ε3 , j = 1, . . . , k,
3. A 3 × 3 System
(2.11)
k
191
|−σ(uj−1 , uj )(uj − uj−1 ) + f (uj ) − f (uj−1 )|
j=1
+
k
|−σ(uj−1 , uj ) η(uj ) − η(uj−1 ) + q(uj ) − q(uj−1 )|
j=1
= O(1)|u+ − u− |ε2 . Proof. This proposition is a consequence of Theorem 3.3 and Proposition 3.4 in Chapter 7. Consider a rarefaction shock (uj−1 , uj ). By Theorem 3.3 ¯ j on H i (uj−1 ) such that |¯ uj − uj | = O(1)ε3 . in Chapter 7, there exists u ¯ j ), i.e. Since the Rankine-Hugoniot condition holds for (uj−1 , u ¯ j )(¯ uj − uj−1 ) + f (¯ uj ) − f (uj−1 ) = 0, −σ(uj−1 , u we have from |¯ uj − uj | = O(1)ε3 and the definition of the speed σ(uj−1 , uj ) in (2.8) that − σ(uj−1 , uj )(uj − uj−1 ) + f (uj ) − f (uj−1 ) ¯ j ) − σ(uj−1 , uj ))(uj − uj−1 ) =(σ(uj−1 , u ¯ j )(uj − uj−1 ) + f (uj ) − f (¯ uj ) + σ(uj−1 , u ¯ j ) − σ(uj−1 , uj ))(uj − uj−1 ) = O(1)ε3 + (σ(uj−1 , u λi (uj−1 ) + λi (¯ uj ) ¯j) − (uj − uj−1 ). = O(1)ε3 + σ(uj−1 , u 2 From the last statement of Theorem 3.3 in Chapter 7, the shock speed is close to the arithmetic mean of the characteristic speeds of the end states: ¯j) = σ(uj−1 , u
λi (uj−1 ) + λi (¯ uj ) + O(1)|¯ uj − uj−1 |2 . 2
Thus we obtain the estimate for the first term in (2.10): −σ(uj−1 , uj )(uj − uj−1 ) + f (uj ) − f (uj−1 ) = O(1)ε3 . The second term in (2.10) is estimated by Proposition 3.4 in Chapter 7: ¯ j ) η(¯ −σ(uj−1 , u uj ) − η(uj−1 ) + q(¯ uj ) − q(uj−1 ) = O(1)|¯ uj − uj−1 |3 . The estimate (2.11) follows from (2.10) and the fact that k = O(1)|u+ − u− |/ε. This completes the proof of the proposition.
3. A 3 × 3 System For a general n × n system, where n > 2, in general there do not exist coordinates of Riemann invariants and the error in linear superposition is of
192
8. Wave Interactions
second order. We illustrate this for the full Euler equations, (5.4) in Chapter 7: ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p)x = 0, (ρE)t + (ρEv + pv)x = 0. First we note from Proposition 5.1 of Chapter 7 that the pressure p and the velocity v are 2-Riemann invariants and that the 2-characteristic field is linearly degenerate. Thus the 2-wave curves are expressed as W 2 (u0 ) = R2 (u0 ) = H 2 (u0 ) = {u : v(u) = v(u0 ), p(u) = p(u0 )}. As with the study of the Riemann problem in Section 4 of Chapter 7, for the analysis of wave interactions it is natural to consider the projection of the wave curves onto the (v, p) plane. A 2-wave takes values at a point in the (v, p) plane. The i-Hugoniot curves, i = 1, 3, are H− 1 (u)
= {u : (v − v0 ) = −
p − p0 (ρ − ρ0 ), p > p0 }; ρρ0 (ρ − ρ0 ) p − p0 + (ρ − ρ0 ), p > p0 }. H 3 (u) = {u : (v − v0 ) = ρρ0 (ρ − ρ0 )
For definiteness, consider the case of polyatomic gases, p(ρ, s) = ργ es with 1 < γ ≤ 5/3, and the interaction of a 3-shock (ul , um ) with a 2wave (um , ur ). The result of the interaction is the solution of the Riemann problem (ul , ur ) consisting of a 1-wave (ul , uk ), a 2-wave (uk , un ), and a 3-shock (un , ur ). In the state space, instead of starting with the left state, + we start with the right states and have ul ∈ H + 3 (um ) and un ∈ H 3 (ur ). Recall that on the (v, p) plane, the two states um and ur are at the same + point (vm , pm ) = (vr , pr ) ≡ (v0 , p0 ). However, H + 3 (um ) and H 3 (ur ) are different curves: (3.1) 1/2 along H 3 (um ), v − v0 = − (p0 −1 esm )1/γ − (p−1 es1 )1/γ (p − p0 ) 1/2 v − v0 = − (p0 −1 esr )1/γ − (p−1 es2 )1/γ (p − p0 ) along H 3 (ur ). The entropies s1 and s2 vary with the shock strength. The entropy is of third-order variation along the Hugoniot curves, by Proposition 5.1 in Chapter 7: s1 = sm + O(1)|u − um |3 ,
s2 = sr + O(1)|u − ur |3 .
As we will consider the interaction of weak waves and effects of second order, we take s1 = sm and s2 = sr for simplicity of presentation. Thus (3.1) has
3. A 3 × 3 System
193
the accurate approximation 1/2 (3.2) v − v0 = ˜ −esm /(2γ) p0 −1/γ − p−1/γ (p − p0 ) along H + 3 (um ), 1/2 ˜ −esr /(2γ) p0 −1/γ − p−1/γ (p − p0 ) along H + v − v0 = 3 (ur ). The 1-wave (ul , uk ) resulting from the interaction can be either a rarefaction wave or a shock wave, depending on the sign of the entropy jump across the 2-wave (um , ur ): Case 1 (sm < sr ). From the above Rankine-Hugoniot relation (3.2), for the same given p, the value of v on the Hugoniot curve H + 3 (um ) is larger than (u ). Consequently, in the (v, p) plane, the that on the Hugoniot curve H + r 3 + + curve H 3 (um ) lies to the right of the curve H 3 (ur ); see left diagram of figure 8.08. Thus the characteristic curve R+ 1 (ul ) meets the Hugoniot curve (u ) at a point in the (v, p) plane. This point represents the two states H+ r 3 + + uk ∈ R1 (ul ) and un ∈ H 3 (ur ) with vk = vn and pk = pn . In this case, the 1-wave (ul , uk ) is a rarefaction wave, as depicted in figure 8.08. t
p H3+(um)
H3+(ur)
un sm
sr
H1−(ul) un uk
uk un sm > sr ul
um ur
um
ur
v
x
Figure 8.09. Interaction of shock and contact discontinuity, Case 2.
Case 2 (sm > sr ). In this case, the Hugoniot curve H + 3 (um ) is to the left (u ) in the (v, p) plane and the 1-wave (ul , uk ) of the Hugoniot curve H + r 3 produced by their interaction is a shock wave, as depicted in figure 8.09.
194
8. Wave Interactions
Note that in both cases, the Hugoniot curves have an angle between them of the order of the entropy difference sm − sr in the (v, p) plane. Thus the strength of the newly produced 1-wave (ul , uk ) is of second order in the incoming waves: (3.3)
˜ |ul − um ||sm − sr | = ˜ |ul − um ||um − ur |. |ul − uk | =
The second-order error in linear superposition represents the usual situation of interaction of waves from distinct families.
4. General Analysis As the examples in Section 2 and Section 3 indicate, it is possible to perform a definite quantitative analysis of the time-asymptotic wave pattern resulting from the interaction of two Riemann solutions without examining the complex interaction process. The time-asymptotic wave pattern resulting from the interaction of the solutions of the Riemann problems (ul , um ) and (um , ur ) is the solution of the Riemann problem (ul , ur ). The goal is to obtain quantitative estimates for the three sets of elementary waves in these Riemann solutions. In the first subsection we consider the case where each characteristic field is either genuinely nonlinear or linearly degenerate, so that the Lax’s construction of the Riemann problem applies; see Section 4 of Chapter 7. In the second subsection we consider the general case corresponding to Section 6 of Chapter 7. 4.1. Genuinely Nonlinear and Linearly Degenerate Characteristic Fields In this subsection we assume that each characteristic field is either genuinely nonlinear or linearly degenerate, so that the elementary wave are shock waves, rarefaction waves, and contact discontinuities, as described in Section 4 of Chapter 7. First we define some notation. Suppose that the Riemann problem (ul , ur ) consists of i-waves (ui−1 , ui ), i = 1, 2, . . . , n, where u0 = ul and un = ur . We denote the strength of (ui−1 , ui ) by αi = αi (ul , ur ). These strengths are signed; for instance, in the case of a genuinely nonlinear i-field, we may set αi = λi (ui ) − λi (ui−1 ) so that an i-shock has negative strength and an i-rarefaction wave has positive strength. For a contact discontinuity, the characteristic value λi is constant, and we may choose, for instance in the case of the Euler equations in gas dynamics, the entropy jump αi = s(ui ) − s(ui−1 ) as its strength. All Euclidean strengths are equivalent to each other, and we may make particular choices for the convenience of a given purpose. Theorem 4.1. Suppose that each characteristic field is either genuinely nonlinear or linearly degenerate and that the states ul , um , and ur are close
4. General Analysis
195
γ1
γi
ul α1 αi ul
t
γn ur
αn
β1 βi um
βn ur
x Figure 8.10. Wave interaction III: interaction of two sets of Riemann solutions.
to each other. Let αi , βi , and γi , i = 1, 2, . . . , n, be the strengths of the i-waves in the solutions of the Riemann problems (ul , um ), (um , ur ), and (ul , ur ), respectively; see figure 8.10. Then (4.1)
γi = αi + βi + O(1)D(ul , um , ur ), i = 1, 2, . . . , n,
where D(ul , um , ur ) = Dd + nj=1 Dsj is a measure of nonlinear interactions given by (4.2) ⎧ ⎪ |αj βk |, interaction measure for waves Dd ≡ ⎪ ⎪ ⎪ ⎪ j>k ⎪ ⎪ ⎪ ⎪ from distinct characteristic families; ⎪ ⎨ ⎧ ⎪ ⎨|αj βj |(|αj | + |βj |) ⎪ j ⎪ D ≡ if one or both of αj and βj are shock waves, ⎪ ⎪ ⎪ s ⎩ ⎪ ⎪ ⎪ ⎪ 0, otherwise ⎪ ⎪ ⎩ interaction measure for waves of the same characteristic family. Remark 4.2. In the definition of Dd , the requirement j > k implies that the wave αj to the left has greater speed than the wave βk to the right, and so the two waves head toward each other and should interact. For the definition of Dsj , if both αj and βj are rarefaction waves or contact discontinuities, they are linearly combined and so do not contribute to the interaction measure Dsj . These are the main considerations for the definition of these measures of wave interactions in (4.2).
196
8. Wave Interactions
Proof. Consider first the situation where the solution of (ul , um ) contains only one wave, a j-shock or j-rarefaction wave αi = 0 for all i = j and that the solution of (ul , um ) consists only of waves heading toward the j-wave, i.e. βi = 0 for i ≥ j, as illustrated in figure 8.11.
γj
γ1
αj
γn
β1 βj−1
t x Figure 8.11. Wave interaction IV: interaction of partial set of Riemann solutions.
In this case, the result is clearly trivial if either the strength αj or the total strength of the set of βi -waves is zero: γi = αi + βi , i = 1, 2, . . . , if either αj = 0 or
j−1
|βi | = 0.
i=1
The wave curves are of second-order smoothness, according to Theorem 3.3 of Chapter 7, and so is the Riemann solver. We thus conclude that the deviation from the above linear superposition is the product of these two strengths: (4.3)
γi = αi + βi + O(1)|αj |
Notice that in this case Dd = |αj | is proved in this case.
j−1 i=1
j−1
|βi |.
i=1
|βi | and Ds = 0, and so the theorem
Next we consider the interaction of two j-waves; that is, we assume that αi = βi = 0 for i = j. In the case where the j-characteristic field is linearly degenerate, these waves are contact discontinuities and so their interaction is simply a linear superposition; therefore the theorem holds trivially with Dd = Ds = 0. When the j-characteristic field is genuinely nonlinear and
4. General Analysis
197
both waves are rarefaction waves, the interaction is simply the combination of the two waves and the theorem again holds trivially with Dd = Ds = 0. It remains to consider the case where at least one of the two j-waves is a shock wave. We consider only the case where both j-waves are shock waves; see figure 8.01 and figure 8.02 for a 2 × 2 case. Thus we have um ∈ Hj− (ul ) and ur ∈ Hj− (um ). As the Hugoniot curve H j (u0 ) and the characteristic ¯ ∈ Hj− (ul ), curve Rj (u0 ) are of second-order contact, we can find a state u as shown in figure 8.12, with the property that γ¯j = αj + βj , |¯ u − ur | = O(1)(|αj + βj |3 − |α|3 − |βj |3 ) = O(1)Dsj , Dsj ≡ |αj |βj |(|αj | + |βj ).
Rj (ul ) Hj+(ul )
ul um Hj−(ul ) Hj−(um)
u¯ ur
u space
Figure 8.12. Interaction of two shock waves: state space.
Thus we may view the solution of the Riemann problem (ul , ur ) as being ¯ ) with strength γ¯j = αj + βj under the perturbation (¯ u, ur ) a j-shock (ul , u j of order O(1)Ds . By the continuous dependence of the Riemann solution on its initial states, we conclude that γi = αi + βi + O(1)Dsj , i = 1, 2, . . . , n. Note that here Dd = 0 and Ds = Dsj , and so the theorem is proved in this case of the interaction of two waves of the same characteristic family. Finally, the analysis of the general situation is reduced to the above two cases through a finite iteration process as follows. We allow the n-wave αn to interact with βi , i = 1, 2, . . . , n − 1, and call the resulting waves
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8. Wave Interactions
γi1 , i = 1, 2, . . . , n. This is the same situation as the first kind considered above, with the measure of interaction Ddn , part of the sum in Dd :
n−1 βi , i = 1, 2, . . . , n − 1, γi1 = O(1)Ddn + Ddn ≡ |αn | |βj |. αn , i = n, i=1 Next we allow the new set of waves γi1 , i = 1, 2, . . . , n, to interact with the original n-wave βn . This is done in two steps; first we consider the interaction of the two n-waves γn1 and βn . This is the situation of the second kind considered above. If at least one of γn1 and βn is a shock, then the resulting j-waves, j = 1, 2, . . . , n − 1, are of the order O(1)|γn1 ||βn |(γn1 | + |βn |) = O(1)|αn ||βn |(|αn | + |βn |) + O(1)Ddn |βn | = O(1)(Dsn + Ddn ),
Dsn ≡ |αn ||βn |(|αn | + |βn |).
The resulting n-wave is the linear superposition plus the same error of O(1)(Dsn + Ddn ). The next step is to allow αn−1 to interact with the above resultant waves. By the same analysis as above, this results in an additional error of the order of n−2 |βj |. Dsn−1 + Ddn−1 , Ddn−1 ≡ |αn−1 | j=1
This is continued by allowing αn−2 to interact with the resulting wave
n j pattern and so on. The accumulated interaction measure is j=1 Ds +
n j j=1 Dd = Ds + Dd . The process leads to the estimate of the interaction of two sets of Riemann solutions. This completes the proof of the theorem. The above analysis yields immediately the following proposition on the states of the waves. Proposition 4.3. The estimate (4.1) holds also in the vector sense: (4.4) γ i = αi + β i + O(1)D(ul , um , ur ), γ i ≡ ui − ui−1 ,
αi ≡ v i − v i−1 ,
β i ≡ wi − wi−1 .
Here ui , v i , and wi , i = 0, 1, . . . , n, are the states defining the elementary waves in the solutions of the Riemann problems (ul , ur ), (ul , um ), and (um , ur ), respectively. 4.2. General Characteristic Fields We next consider the general situation where each characteristic field is not necessarily genuinely nonlinear or linearly degenerate. The Riemann problem is studied in Section 6 of Chapter 7 using the Liu entropy condition. We use the same notation as there, with incoming waves αi and β i and
4. General Analysis
199
outgoing waves γ i , i = 1, . . . , n, as in figure 8.11. The measure for the interaction of waves from distinct characteristic families is set to be the same as in (4.2): |αj βk |, (4.5) Dd (ul , um , ur ) ≡ j>k
interaction measure for waves from distinct characteristic families. The measure Dsj = Ds (ul , um , ur ) for the interaction of waves of the same family needs to be changed for the following obvious reason: Even for scalar laws, a Riemann solution can contain both shock and rarefaction waves; see figure 5.02 and figure 5.03 of Chapter 5. As a Riemann solution represents a non-interacting wave pattern, its measure Ds should be zero, even though the wave pattern can contain both shock and rarefaction waves or, in the case of figure 5.03, two non-interacting shock waves. Recall the measure of interaction D(ul , um , ur ) between the Riemann solutions α = (ul , um ) and β = (um , ur ) for a scalar law in Section 4 of Chapter 5: (4.6)
D(ul , um , ur ) = |α||β||θ(α, β)| = |um − ul ||ur − um |θ.
Here θ = θ(α, β) is the difference between the speed of propagation of the right side of α and that of the left side of β. In figure 8.13, for simplicity, θ is depicted as the angle between two solutions. The difference in speeds and the angle are of the same order.
θ ul
ur um
Figure 8.13. Angle θ = θ((ul , um ), (um , ul )) between two Riemann solutions for scalar laws.
For the system, the two i-waves αi and β i in figure 8.10 are not next to each other. Thus the effective angle between them cannot be defined
200
8. Wave Interactions
immediately. We begin by elaborating on the scalar laws situation. For a given wave α = (ul , ur ) of a scalar law, set (4.7) θ− = θ− (α) ≡ λ(ul ) − σ− ,
θ+ = θ+ (α) ≡ σ+ − λ(ur ),
σ− (α) ≡ speed of the wave next to the state ul , σ+ (α) ≡ speed of the wave next to the state ur , [θ](α) ≡ σ+ (α) − σ− (α),
θ(α) = f (ur ) − f (ul ) = λ(ur ) − λ(ul ).
Thus θ− = 0 if the left edge of α is a rarefaction wave, and θ− = λ(ul )−σ− if the left edge of α is a shock with speed σ− . Similarly, we have either θ+ = 0 or θ+ = σ+ − λ(ur ). The quantity [θ] = [θ](α) measures the span of the wave α and is zero if α is a shock. The quantity θ(α) measures the degree of compressibility of the wave α; the wave α is regarded as compressive if θ(α) is negative, even if its span [θ] is positive. These notions are illustrated in figure 8.14, corresponding to figure 5.02 and figure 5.03 of Chapter 5. In (4.6) θ(α, β) = σ+ (α) − σ− (β). The above notions for scalar laws have direct generalizations to systems. Suppose that the Riemann problem (ul , um ) is solved by i-waves αi = (v i−1 , v i ), i = 1, 2, . . . , n where v 0 = ul and v n = um . We define similar angles to the ones in (4.7) for scalar laws: (4.8) θ− (αi ) ≡ λi (v i−1 ) − σ− (αi ),
θ+ (αi ) ≡ σ+ (αi ) − λi (v i ),
σ− (αi ) ≡ speed of the wave next to the state v i−1 , σ+ (αi ) ≡ speed of the wave next to the state v i , [θ](αi ) ≡ σ+ (αi ) − σ− (αi ),
θ(αi ) ≡ λi (v i ) − λ(v i−1 ).
And, similar to (4.6), we define a measure of interaction between the i-waves αi and β i : (4.9) Dsi (ul , um , ur ) ≡ θ(αi , β i )|αi ||β i |, θ(αi , β i ) ≡ θ+ (αi ) + θ− (β i ), (effective) angle between αi and β i . Note that as the two waves αi and β i are not next to each other, the angle between them is not a good measure of the degree of their approaching each other because of the other waves αi+1 , . . . , αn , β 1 , . . . , β i−1 in between. Instead, the above effective angle θ(αi , β i ) is the true measure of the degree of their approaching each other. In the analysis of wave interaction, after the waves in between have interacted with the two waves, as measured by Dd (ul , um , ur ), when the two waves are in direct contact, the angle between them is θ(αi , β i ) plus O(1)Dd (ul , um , ur ). With this new definition of Dsi (ul , um , ur ), i = 1, 2, . . . , n, Theorem 4.1 and Proposition 4.3 hold for general systems, without the genuine nonlinearity or linear degeneracy hypothesis. We skip the proof of the following theorem.
5. Notes
201
t u2
u1
dx = σ+ dt
dx = σ− dt
ur
ul dx = λ(u ) l dt
[ θ] θ−
dx = λ(u r) dt
θ+
x
ur
ul t
u1 u 2 dx = λ(u r) dt
dx = λ(ul) dt
ul θ− = 0
ul
[θ]
ur θ+ = 0
ur
x
Figure 8.14. Angles θ± between the characteristics and the wave edge, and the wave span [θ].
Theorem 4.4. Suppose that the states ul , um , and ur are close to each other. Let αi , βi , and γi , i = 1, 2, . . . , n, be the strengths of the i-waves in the solutions of the Riemann problems (ul , um ), (um , ur ), and (ul , ur ), respectively; see figure 8.10. Then γi = αi + βi + O(1)D(ul , um , ur ), i = 1, 2, . . . , n,
where D(ul , um , ur ) = Dd + nj=1 Dsj is the measure of nonlinear interaction given by (4.5), (4.8), and (4.9). (4.10)
5. Notes The study of interactions of elementary waves is a basic element of shock wave theory. The analysis of the general situation, Theorem 4.4, was done by James Glimm [54] and is a core part of the well-posedness theory to be
202
8. Wave Interactions
covered in the next chapter. Its generalization, Theorem 4.4, is covered in [85]. For one space dimension, there are finitely many types of canonical wave interactions. Due to the greater degrees of freedom, wave interactions for multi-dimensional gas flows are usually studied in the presence of physical boundaries in order to limit and stabilize the possible wave patterns; see, for instance, the classical Courant-Friedrichs book [32].
6. Exercises 1. Construct the Riemann invariants for the isentropic Euler equations with the polyatomic law p(ρ) = ργ , γ ≥ 1. 2. The Hugoniot curve H i (u0 ) and the characteristic curve Ri (u0 ) coincide when the i-characteristic field is linearly degenerate. Some special systems have this property without necessarily being linearly degenerate. Show that the 2 × 2 system (6.1)
ut + (uφ(u, v))x = 0, vt + (vφ(u, v))x = 0
has this property. (Hint: Consider the lines through the origin and the curves φ(u, v) = constant.) 3. For the full Euler equations, show that for γ ≤ 5/3 (or γ > 5/3), the interaction of two 1-shocks produces a 3-rarefaction wave (or a 3-shock wave) for sufficiently weak waves. 4. According to the definition of Riemann invariants, Definition 2.1, show for the Euler equations (5.4) that v and p are 2-Riemann invariants and s is a 1-Riemann invariant. 5. Show by the method of characteristics that the following system of conservation laws with damping possesses smooth global solutions if the initial value is smooth and slow varying:
ut − vx = 0, vt + p(v)x = −v, p (v) < 0. 6. Show that the interaction of two 1-shocks for the p-system with p (τ ) < 0 and p (τ ) > 0 results in a 1-shock and a 2-rarefaction wave for waves of arbitrary strength. 7. Consider the isothermal Euler equations (6.2)
τt − vx = 0, 2 vt + aτ x = 0.
Check that r ≡ v + a log τ and s ≡ v − a log τ are Riemann invariants. Show that in the coordinates of these Riemann invariants, the Hugoniot curves are congruent to each other. In other words, the shape of the Hugoniot curves
6. Exercises
203
H i (u0 ), i = 1, 2, does not depend on the initial state u0 . Consequently, using these Riemann invariants for the measurement of wave strength, the interaction of waves does not increase their total strength [109]. 8. Consider a 2 × 2 system with Riemann invariants r and s and initial data taking values on the curve r = constant. Find a condition on the initial data that guarantees the existence of a global smooth solution. 9. Consider the system of hyperbolic conservation laws with damping, ut + f (u)x = −αu, with damping constant α > 0. Show by the energy method that a smooth solution exists globally in time if the initial data is smooth and sufficiently small. 10. Show that without using the Riemann invariants as coordinates for measuring wave strength, the interaction of waves for 2 × 2 systems may not follow the linear superposition principle with third-order error. We have seen in Section 3 that the error, measured in any coordinates, is of second order for general 3 × 3 systems.
10.1090/gsm/215/09
Chapter 9
Well-Posedness Theory
Consider the general initial value problem for systems of hyperbolic conservation laws, where u ∈ Rn with n > 1:
ut + f (u)x = 0, −∞ < x < ∞, t > 0, (0.1) u(x, 0) = u0 (x), −∞ < x < ∞. A system of hyperbolic conservation laws is quasilinear. With the presence of shock waves in its solutions, the system is highly nonlinear. It is a challenge to construct the solutions and to study their wave behavior. The analysis of weak solutions requires function spaces that have strong geometric structure. The space of functions of bounded total variation, Definition 5.1 in Chapter 3, is such a space from the point of view of geometric measure theory. There is the basic form of Helly’s Theorem: Theorem 0.1 (Helly’s Theorem). Consider a sequence of functions uj (x) = (u1j , . . . , unj )(x), j = 1, 2, . . . , which are uniformly bounded and of bounded variation, Var(ukj ) ≤ TV for k = 1, . . . , n, and j = 1, 2, . . . . Then there ex¯ j (x), j = 1, 2, . . . , which converges to a limiting function ists a subsequence u u(x) = (u1 , . . . , un )(x) in L1 (x) for any bounded interval. The convergence is pointwise and the limiting function is of bounded variation, Var(uk ) ≤ TV for k = 1, . . . , n. One way to prove the theorem is to use the fact that each component is a scalar function of bounded variation and can be written as the difference of two monotone functions; see Proposition 5.2 in Chapter 3. The strong convergence property of Helly’s Theorem implies that the sequence f (¯ uj ), j = 1, 2, . . . , also converges pointwise to f (u) for any given differentiable function f (·). This is useful for the analysis of weak solutions of 205
206
9. Well-Posedness Theory
quasilinear hyperbolic conservation laws. From the usual functional-analytic point of view, the bounded variation space is quite restrictive. On the other hand, a function of bounded variation is close to being piecewise continuous. We have seen in Chapter 3 that for scalar laws, it is fruitful to think of piecewise continuous functions when analyzing the behavior of solutions. This is possible because of the regularizing effect due to the nonlinearity of the flux function. Moreover, when one thinks of shock waves physically, one is intuitively thinking of piecewise continuous functions. As we will see, this is essentially the case also for systems. The main task is to show that the solution is in the space of functions of bounded variation if its initial data is in this restrictive space, and that the solution can be viewed, in some sense, as being piecewise continuous. The elementary waves in the Riemann solutions were constructed in Chapter 7 and are now used for the Glimm scheme of random choice, discussed in Section 1. Unlike general numerical schemes, the Glimm scheme preserves the sharpness of shock waves in its approximate solutions. In Section 2, the wave interaction analysis in Chapter 8 is used to control the nonlinear interactions of waves in the approximate solutions through the Glimm functional. This yields the existence theory, detailed in Section 3 and Section 4, by using a wave tracing mechanism to analyze the evolution in time of the interactions of elementary waves. The construction procedure for the solutions makes it possible to study the solution behavior. Section 5 studies the basic continuous dependence of the solution on its initial data. The generalized entropy functional for scalar laws, introduced in Section 7 of Chapter 3, is generalized to systems for this purpose. The first five sections, Section 1 to Section 5, present a well-posedness theory for systems, a very satisfying achievement in nonlinear analysis. The solution behavior is considered in the second part of this The generalization of the scalar theory in Chapter 3 and Chapter rich theory for systems is illustrated in Section 6 to Section 9. some basic elements of general numerical schemes for conservation presented in Section 10.
chapter. 5 to the Finally, laws are
The existence and uniqueness theory and the solution behavior are two integral parts of the well-posedness theory. A well-posedness theory for weak solutions of hyperbolic conservation laws as developed here is for the Glimm solution algorithm. It is not known whether the solutions obtained by other compactness methods, such as the theory of compensated compactness, depend on their initial data continuously in L1 (x), because the usual compactness argument does not offer sufficient structure for the evolution in time of the waves. The Glimm solution algorithm provides sufficient control
1. Glimm Scheme
207
over the time evolution of the solutions. This makes it possible to study the continuous dependence of the solutions on their initial data in the first part of this chapter as well as the solution behavior in the second part. PART I: Existence and Continuous Dependence of Solutions on Initial Values
1. Glimm Scheme The solution to the initial value problem is constructed using the Glimm scheme. The design of the Glimm scheme for constructing approximate solutions of the initial value problem for (0.1) is based on the following three principles: The first is to recognize that the Riemann problem is an important part of the one-spatial-dimension shock wave theory in that it allows for the consideration of the entropy condition and the construction of elementary waves. The construction of Riemann solutions takes full account of the nonlinearity of the system. The Glimm scheme uses the Riemann solutions as building blocks. The second principle is that both locally and asymptotically in time, a general solution tends to certain Riemann solutions. As time increases, complex wave interactions are expected to subside and so the solution would tend, time-asymptotically, to a non-interacting wave pattern, a Riemann solution. For local behavior, except for countable points of interaction in the (x, t) plane, the solution is either continuous at (x, t) or has a shock going through (x, t). Around one of the countable interaction points (x0 , t0 ), the solution u(x, t) is dominated by a Riemann solution for (x, t) close to (x0 , t0 ), after the complex wave interaction, i.e. t > t0 . The precise analysis of the asymptotic and local behavior will be done later in Section 7 and Section 8. With this scenario in mind, the Glimm approximation by Riemann solutions, done at every time and every spatial step as in figure 9.02, would be accurate uniformly in grid size and in time. We now describe the Glimm scheme. Take grid sizes Δx in space and Δt in time and consider the grid points (jΔx, kΔt), j = 0, ±1, ±2, . . . , k = 0, 1, 2, . . . . At time t = 0, approximate the initial function u(x, 0) by step functions ¯ (x, 0) = u ¯ 0j for (j − 1)Δx < x < jΔx, j = 0, ±1, ±2, . . . . u ¯ 0j are constant states and so there are jumps at the grid points x = Here u jΔx, j = 0, ±1, ±2, . . . . Each jump constitutes a pair of Riemann data and ¯ (x, t) is resolved, as done in Chapter 7, to yield an approximate solution u consisting of elementary waves in the series of Riemann solutions. They issue from (jΔx, 0), j = 0, ±1, ±2, . . . . These elementary waves do not interact for t < Δt when the grid sizes Δx and Δt are assumed to satisfy the basic
208
9. Well-Posedness Theory
Courant-Friedrichs-Lewy (C-F-L) condition (1.1)
Δx ≥ {2|λi (u)|, i = 1, 2, . . . , n} Δt
¯ (x, t) for all states u under consideration. With the C-F-L condition, u is an exact solution of the system of conservation laws for 0 < t < Δt. At later times, these waves will interact to produce a complex wave pattern. Time-asymptotically, the nonlinear interactions will produce a noninteracting wave pattern, the wave pattern in a Riemann solution. The scheme takes as an approximation the Riemann solution immediately at the next time step. As pointed out above, this has the virtue that the Glimm approximation is accurate uniformly in time. The third principle for the construction of the Glimm approximate solutions is to ensure that there is no numerical dissipation in the sense that shock waves are not smoothed out in the approximation procedure. This implies that elementary waves maintain their identity. This allows one to use the mechanism of wave tracing to analyze wave propagation in the approximate solutions. This zero numerical dissipation is achieved by assigning the state between the spatial grid points at the next time step kΔt to be the local value, the value at some location between the grid points of the approximate solution computed from the previous time step kΔt − 0: ¯ (x, kΔt) = u ¯ ((j + ak )Δx, kΔt − 0), (1.2) u jΔx < x < (j + 1)Δx, k = 1, 2, . . . . The usual finite difference schemes assign the states at the next time step by certain types of averaging of the values in the previous time step. Instead, the Glimm scheme takes the states at the next time step t = kΔt to be the local values at the locations x = (j + ak )Δx, j = 0, ±1, ±2, . . . , from the previous time step t = kΔt − 0. The locations x = (j + ak )Δx, j = 0, ±1, ±2, . . . , are determined by the value ak , a random choice of number in the interval (0, 1). Example. We illustrate the Glimm scheme for the propagation of a single shock wave (u− , u+ ) with positive speed s > 0:
u− for x < st, u(x, t) = u+ for x > st. ¯ (x, t) is the same as the exact solution u(x, t) The approximate solution u for 0 < t < Δt. Thus
u− for x < sΔt, ¯ (x, Δt − 0) = u u− for x > sΔt.
1. Glimm Scheme
209
The values at the next time level t = Δt is determined by a parameter a1 with 0 < a1 < 1: ¯ (x, Δt) = u ¯ ((a1 + j)Δx, Δt − 0) for jΔx < x < (j + 1)Δx. u In the present case of the propagation of one shock (u− , u+ ) of positive speed s, ⎧ ⎪ u− for x < 0, ⎪ ⎪ ⎪ ⎨u for 0 < x < Δx, if s > a Δx, − 1 ¯ (x, Δt) = u ⎪u+ for 0 < x < Δx, if s < a1 Δx, ⎪ ⎪ ⎪ ⎩u for x > Δx. + Thus when sΔt > a1 Δx, the shock is accelerated and moved to the right by Δx; when sΔt < a1 Δx, it is decelerated and kept at the location x = 0; see the upper diagrams of figure 9.01. t
t Δx
Δx
Δt
Δt
a1Δx
a1Δ x
x
x
t
t Δx
Δx
Δt
Δt a1Δx
a1Δx
x
x
Figure 9.01. Glimm approximation for a single shock.
Similarly, if the shock has negative speed, s < 0, then when sΔt > (a1 − 1)Δx, the shock is kept at x = 0; when sΔt < (a1 − 1)Δx, it is moved to x = −Δx; see the lower diagrams in figure 9.01: ⎧ ⎪ u− for x < −Δx, ⎪ ⎪ ⎪ ⎨u for − Δx < x < 0, when |s| = −s > (1 − a ) Δx , + 1 Δt ¯ (x, Δt) = u ⎪u− for − Δx < x < 0, when |s| = −s < (1 − a1 ) Δx ⎪ Δt , ⎪ ⎪ ⎩u for x > 0. + Thus at the first time step, although the sharpness of the shock is preserved, its location is allowed to have randomness, depending on the shock speed s and the chosen constant a1 . The situation for the next time level 2Δt
210
9. Well-Posedness Theory
depends on another constant a2 and so on. For a shock with positive speed, its location in the approximate solution at time step nΔt is N Δt:
u+ for x > N Δx, ¯ (x, nΔt) = u u− for x < N Δx, where N = N (s; a1 , a2 , . . . , an ) ≡ number of {k : 1 ≤ k ≤ n, ak < s
Δt }. Δx
At a fixed time t = nΔt, the shock location for the exact solution is x = st = snΔt. The shock location for the approximate solution is x = N Δx. In order for the shock to have accurate location in the limit as n → ∞ and Δt → 0, it is required that snΔt → N Δx, or
Δt N →s . n Δx
This needs to hold for all shocks and so the sequence a needs to be equidistributed in (0, 1). Definition 1.1. A sequence a = {a1 , a2 , . . . } is equi-distributed in (0, 1) if e(n; a) ≡ sup
N (α, n; a)
− α → 0 as n → ∞,
n N (α, n; a) ≡ number of {k : ak < α, 1 ≤ k ≤ n.}
0 λj for i > j. This is the idea behind the following definition of the wave interaction potential functional Qd (J) for waves associated with distinct characteristic fields. We are considering the situation where each characteristic field is either genuinely nonlinear or linearly degenerate. Thus two i-waves α and β have the potential to interact in the future if at least one of them is a shock wave; see Qis (J) below. Thus the potential wave interaction functionals for waves crossing J are defined as (2.3) Q(J) ≡ Qs (J) + Qd (J) = Qis (J)
≡
Qd (J) ≡
n
Qis (J) + Qd (J),
i=1
{|α||β|(|α| + |β|) : α, β i-waves crossing J,
at least one of them a shock wave}, {|α||β| : α an i-wave and β a j-wave crossing J, n ≥ i > j ≥ 1 and α to the left of β}.
The Glimm nonlinear functional is defined as (2.4) F (J) ≡ L(J) + A Qs (J) + Qd (J) . Here the constant A will be chosen to be sufficiently large. The following theorem on the wave interaction and cancellation measures is of fundamental importance. Theorem 2.1. Suppose that the initial data u(x, 0) has small total variation TV. Then the Glimm functional, defined in (2.3) and (2.4), is a nonincreasing function of time. More precisely, for space-like curves Jj , j = 1, 2, with J2 toward larger time than J1 , (2.5)
F (J2 ) ≤ F (J1 ) − C(Ω) − D(Ω),
where Ω is the region between J1 and J2 , and C(Ω) and D(Ω) are the cancellation and interaction measures in the region Ω, defined in (2.1). Proof. This is proved by induction. It suffices to consider the case where J1 and J2 sandwich only one grid point (jΔx, kΔt); see figure 9.04. The induction hypothesis implies that F (J1 ) ≤ F (O), where O is the space-like curve around t = 0. We have F (O) = TV + A(TV)2 + A(TV)3 ≤ 2TV from the smallness assumption on the initial total variation TV. In particular, L(J1 ) ≤ 2TV. With the same notation as above, let (ul , um ) and (um , ur ) be the Riemann problems before interaction and (ul , ur ) the one after interaction at the grid point (jΔx, kΔt), and let the strengths of the elementary waves be αi , βi , and γi , i = 1, 2, . . . , n, respectively. We have D j,k = D(ul , um , ur ). Note
2. Nonlinear Functional
215
that waves crossing J1 and J2 are the same except for the local waves αi and β i , i = 1, 2, . . . , n, crossing J1 , and γ i , i = 1, 2, . . . , n, crossing J2 . There are the following considerations: (1) Waves on the intersection of J1 and J2 are not involved in the above interaction around the grid point (jΔx, kΔt), and so their strengths do not change. Thus they do not contribute to the change in the linear functional L(J). Moreover, the interactions among themselves also do not contribute to the change in the interaction functional Qs (J) + Qd (J). (2) Due to the interaction of local waves, the linear functional L(J) can increase, by an amount of O(1)D j,k , and decrease, by the amount of C j,k , according to (4.10) of Chapter 8: n
(|γi | − |αi | − |βi |) = O(1)D j,k − C j,k .
i=1
Thus L(J2 ) − L(J1 ) = −C j,k + O(1)D j,k . (3) The interactions involving local waves αi , β i , and γ i , i = 1, 2, . . . , n, induce a change in the interaction functional Q(J) in two ways. First, there is the contribution to Qs (J1 ) + Qd (J1 ) of the amount D j,k , due to the potential interaction of waves α and β on J1 . On the other hand, the waves γ i , i = 1, 2, . . . , n, on J2 are in a Riemann solution and are non-interacting according to the definition in (2.3), and therefore they do not contribute to Q(J2 ). Thus −D j,k = contribution to Q(J2 ) − Q(J1 ) due to interaction of local waves αi , β i , and γ i , i = 1, 2, . . . , n. (4) The interactions of the local waves αi , β i , and γ i , i = 1, 2, . . . , n, with waves δ crossing both J1 and J2 also contribute to Q(J2 ) − Q(J1 ). For J2 it is the interaction of γ i , i = 1, 2, . . . , n, with δ, while for J1 it is the interaction of αi and β i , i = 1, 2, . . . , n, with δ. As |γi | can be larger than |αi | + |βi | by an amount of O(1)D j.k , the contribution to D(J2 ) − D(J1 ) is an amount of O(1)D j,k |δ|. Adding up all the waves δ, we obtain O(1)D j,k L(J1 ) = O(1)D j,k TV = contribution to Q(J2 ) − Q(J1 ) due to interaction of waves on J1 ∩J2 with αi , β i , and γ i , i = 1, 2, . . . , n. Summing the terms in (3) and (4) above, we have Q(J2 ) − Q(J1 ) = −D j,k + O(1)D j,k TV,
216
9. Well-Posedness Theory
and, combining this with the consideration in (2), we obtain F (J2 ) − F (J1 ) = −C j,k + O(1)D j,k + A −D j,k + O(1)D j,k TV = −C j,k + D j,k O(1) + A(−1 + O(1)TV) . Thus by choosing the constant A sufficiently large, and for small TV, F (J2 ) − F (J1 ) ≤ −C j,k − D j,k . The proof of the theorem is complete upon summing the estimates over all grid points in the region Ω. An immediate corollary of Theorem 2.1 is the uniform boundedness and time-asymptotic vanishing of the local wave interaction measure. Corollary 2.2. Let (2.6) D(t) ≡ {D j,k : j = 0, ±1, ±2, . . . , kΔt ≥ t}, C(t) ≡ {C j,k : j = 0, ±1, ±2, . . . , kΔt ≥ t}; then (2.7) D(0) = O(1)TV2 ,
d D(t) ≤ 0, dt
lim D(t) = 0,
t→∞
d C(t) ≤ 0, lim C(t) = 0. t→∞ dt Remark 2.3. The above theorem and its corollary hold when the wave interaction potential for waves of the same family is replaced by a stronger quadratic measure. Precisely, we have the following: The estimate (4.10) ¯ sj = of Chapter 8 holds when Dsj = |αj βj |(|αj | + |βj |) is replaced by D functional (2.3), Qis (J) ≡ |αj βj |. Accordingly, in the corresponding Glimm
i ¯ (J) ≡ |α||β|(|α|+|β|) may be replaced by Q |α||β| so that Theorem 2.1 s and Corollary 2.2 hold. In the next two sections we will use this functional for the study of convergence of approximate solutions to the exact solution. The original functional is effective for the study of wave decoupling and will be used for a more precise, quantitative study of N -waves in the subsequent section. C(0) = O(1)TV,
We now turn to the general situation where the characteristics are not necessarily genuinely nonlinear or linearly degenerate. The definition of the interaction measure Dd for waves from distinct characteristic families in (4.2) still applies. The interaction measure Dsj for waves of the same i-characteristic family in (4.2) needs to be changed. For local interactions, there is the measure (4.2) of Chapter 8: Dsi = Dis (ul , um , ur ) = |αi ||βi ||θ(αi , β i )|.
2. Nonlinear Functional
217
The angle θ(αi , β i ) between the waves αi and β i is given in (4.8) and (4.9) of Chapter 8. This notion of angle generalizes naturally to two i-waves far apart on a space-like curve J. This is done as follows, starting first by recalling several notions defined earlier. Consider an i-wave α = (u1 , u2 ). The propagation speed of the wave next to the state u1 is denoted by σ− (α), and the propagation speed next to the state u2 is denoted by σ+ (α). The left compression angle θ− and right compression angle θ+ are defined as θ− (α) = λi (u1 ) − σ− (α),
θ+ (α) = σ+ (α) − λi (u2 ).
The expansion angle of the wave is defined as θ(α) = λi (u2 ) − λi (u1 ). Suppose that the i-waves crossing a space-like curve J are αk , k = 0, ±1, ±2, . . . , with αk1 to the left of αk2 for k1 < k2 . The angle between αk1 and αk2 is (2.8) θ(αk1 , αk2 ) ≡ −θ+ (αk1 ) − θ− (αk2 ) + θ(αk ). k1 σΔt. j(k)Δx
Write φ(x, kΔt) = φ0 + O(1)(k2 − k1 )Δt with φ0 ≡ φ(j(k1 )Δx, k1 Δt) the value of the test function at the initial position of the shock; then the above becomes Ek (Δx, φ, a) = (u− − u+ )(Δx − σΔt)φ0 + O(1)(u+ − u− )(k2 − k1 )(Δt)2 for ak Δx < σΔt; Ek (Δx, φ, a) = (u+ − u− )σΔt φ0 + O(1)(u+ − u− )(k2 − k1 )(Δt)2 for ak Δx > σΔt. This yields the crucial cancellation of the errors when the sequence a is equi-distributed, as in Definition 1.1: Error(Ωk1 ,k2 ) = −
k 2 −1
Ek (Δx, φ, a)
k=k1
2 = O(1)(u+ − u− ) e(k2 − k1 ; a)(k2 − k1 )Δt + (k2 − k1 )Δt . Thus the magnitude of the error is |Error(Ωk1 ,k2 )| = O(1)|u+ − u− |(k2 − k1 )Δt (k2 − k1 )Δt + e(k2 − k1 ; a) . Without taking into account the cancellation of truncation errors between the time steps, each term Ek (Δx, φ, a), k1 ≤ k < k2 , is of the order of O(1)|u+ − u− |Δt and the total error Error(Ωk1 ,k2 ) is O(1)|u+ − u− |(k2 − k1 )Δt. Note that the above estimate contains an extra factor that is the sum of the time duration (k2 − k1 )Δt and the degree of equi-distributedness e(k2 − k1 ; a) of the random sequence of k2 − k1 terms. This extra factor is due to the cancellation of truncation errors at different time levels.
3. Wave Tracing
227
¯ (x, t). Let αi = We now consider a general approximate solution u (ui− (k), ui+ (k)), i = 1, 2, . . . , be the waves with speeds σi (k) and loca¯ (x, t) at t = kΔt − 0. These waves are classified into types tions ji (k) in u I, II and III as in Proposition 3.1. The random-choice scheme, (1.3) and (1.4), for the scenario in figure 9.01, gives
ji (k) for ak Δx > σi (k)Δt > 0, (3.11) ji (k + 1) = ji (k) + 1 for ak Δx < σi (k)Δt > 0;
ji (k) for Δx − ak Δx > −σi (k)Δt > 0, ji (k + 1) = ji (k) − 1 for Δx − ak Δx < −σi (k)Δt. Therefore the truncation errors are (3.12) Ek (Δx, φ, a) {(ui+ − ui− ) =
ji (k)Δx+σi (k)Δt
φ(x, kΔt) dx : ak Δx > σi (k)Δt > 0} ji (k)Δx
i
{(ui− − ui+ ) + + {(ui− −ui+ )
j
j(k)Δx
φ(x, kΔt) dx : Δx−ak Δx > −σi (k)Δt > 0}
j(k)kΔx+σi (k)Δt
j
+
φ(x, kΔt) dx : ak Δx < σi (k)Δt > 0} ji (k)Δx+σi (k)Δt
j
(ji (k)+1)Δx
{(ui+ −ui− )
j(k)Δx+σi (k)Δt
φ(x, kΔt) dx : Δx −ak Δx < −σi (k)Δt}.
(j(k)−1)Δx
With the above expression, we see that assessment of the truncation error can be done by evaluating the truncation error caused by each sub-wave α = αi = (ui− (k), ui+ (k)) and then summing the errors for all the subwaves. The main error is that due to waves of type I. The analysis of a wave α of type I is similar to that for a single shock. It is somewhat more complicated because of the variation Δαi of its wave shape and the variation of its speed, Δσ(α) ≡ max{σ(k), k1 ≤ k < k2 , } − min{σ(k), k1 ≤ k < k2 }. These variations induce an extra truncation error beyond that for the singleshock case. Clearly, the variation Δα induces an error of order |Δα|(k2 − k1 )Δt. The variation Δσ(α) induces extra error because the cancellation due to the equi-distributedness of the random sequence is now not as precise. When the random point ak is outside an interval of length Δσ(α)Δt/Δx, the wave α moves in a deterministic way according to (3.11). On the other hand, when ak lies within the interval, the wave can move to the right at one time step and move to the left at another time step. This somewhat complicates
228
9. Well-Posedness Theory
the cancellation analysis. The number of ak , k1 ≤ k < k2 , that lie in the interval of length |Δσ(α)|Δt/Δx is |Δσ(α)|Δt/Δx plus e(k2 − k1 ; a), and so the extra error is bounded by O(1)|α| |Δσ(α)| + e(k2 − k1 ; a) (k2 − k1 )Δt. This is added up, for all type I waves α, to yield an extra truncation error of the order of |α||Δσ(α)| + TVe(k2 − k1 ; a) , O(1)(k2 − k1 )Δt α∈I
which, by the estimate (3.6), is O(1)(k2 − k1 )Δt D(Ωk1 ,k2 ) + TVe(k2 − k1 ; a) . Finally, the error due to type II and type III waves is (k2 − k1 )Δt times the total strength of these waves. From the estimates (3.7) and (3.8) in Proposition 3.1, the error due to type II and type III waves is therefore O(1) C(Ωk1 ,k2 ) + D(Ωk1 ,k2 ) (k2 − k1 )Δt. This completes the proof of the proposition.
¯ (x, t) and a wave pattern We now compare the approximate solution u of linear superposition of nonlinear waves over a mesoscopic time period k1 Δt ≤ t < k2 Δt. Definition 3.4. Consider the wave partition of an approximate solution ¯ (x, t) for the region Ωk1 ,k2 . Let αji , for i = 1, . . . , n and j = 0, ±1, ±2, . . . , u ¯ (x, t) at t = k1 Δt. The linear superposition be the partitioned waves in u ˜ (x, t) of u ¯ (x, t) over the time interval k1 Δt ≤ t < k2 Δt is approximation u defined as the linear superposition of αji , i = 1, . . . , n and j = 0, ±1, ±2, . . . , ¯ at propagating along lines and with the same locations and speeds as in u time k1 Δt. The next proposition gives an estimate for the L1 (x) distance between ¯ (x, t) and u ˜ (x, t) at the end time of Ωk1 ,k2 . u ¯ (x, t) Proposition 3.5. The difference between the approximate solution u ˜ (x, t) satisfies and its linear superposition approximaton u ∞ ˜ (x, k1 Δt)| dx = 0; |¯ u(x, k1 Δt) − u (3.13) −∞ ∞ ˜ (x, k2 Δt)| dx |¯ u(x, k2 Δt) − u −∞ = O(1) D(Ωk1 ,k2 ) + C(Ωk1 ,k2 ) + TV · e(k2 − k1 ; a) (k2 − k1 )Δt. Here e(k2 − k1 ; a) is the measure of the equi-distributedness of the random sequence a in Definition 1.1.
3. Wave Tracing
229
Proof. The first half of the proposition is trivial because, by construction, ˜ (x, t) are the partitioned ¯ (x, t) = u ˜ (x, t) at time t = k1 Δt. The waves in u u ¯ (x, t) at time t = k1 Δt. These sub-waves propagate with conwaves in u ˜ (x, t) according to the linear superposition construction stant speeds in u ¯ (x, k1 Δt) are during time t ∈ (k1 Δt, k2 Δt). All three types of waves in u ˜ (x, t) during the time period k1 Δt < t < k2 Δt. During the time kept in u ¯ (x, t) and are not acperiod, new type II and type III waves appear in u ˜ (x, t). These type II and type III waves have total strength counted for in u O(1) D(Ωk1 ,k2 ) + C(Ωk1 ,k2 ) , according to (3.7) and (3.8). Each wave propagates over an interval in x space of order O(1)(k2 Δt − k1 Δt) because of the finite speed of propagation. Therefore type II and type III waves lead to ¯ (x, k2 Δt) and u ˜ (x, k2 Δt) of the order of an L1 (x) distance between u Error1 ≡ O(1) D(Ωk1 ,k2 ) + C(Ωk1 ,k2 ) (k2 Δt − k1 Δt). ¯ (x, k2 Δt) and u ˜ (x, k2 Δt) due to To calculate the L1 (x) difference between u type I waves, there are two considerations. The first involves the change in strength |Δα| and the change in speed |Δσ(α)| of a type I wave α in the ¯ (x, t) as time evolves. Waves in u ˜ (x, t) propagate approximate solution u with constant speed. The L1 (x) distance caused by this follows from the estimate (3.6) and is of the order O(1)D(Ωk1 ,k2 )(k2 Δt − k1 Δt), which is dominated by Error1 above. The second consideration is that, while the ˜ (x, t) propagate along straight lines, the waves in u ¯ (x, t) jump waves in u at the grid time t = kΔt, k1 ≤ k ≤ k2 . We need to assess the difference in location of each type I sub-wave α at time t = k2 Δt between the two ˜ (x, t) has speed σ(α(k1 )) and propagates a functions. The wave α in u distance of σ(α(k1 ))·(k1 Δt−k1 Δt). To calculate the propagation distance of ¯ (x, t), we first use its initial speed σ(α(k1 )) α in the approximate solution u to calculate the jumps at the grid times according to the Glimm scheme, (1.4). The situation is the same as the propagation of a single shock wave shown in figure 9.01, and the distance of propagation of α is σ(α(k1 )) · (k1 Δt − k1 Δt) + O(1)e(k2 − k1 ; a)(k2 − k1 )Δt. Thus the difference of the two propagation distances is O(1)e(k2 − k1 ; a)(k2 − k1 )Δt, which yields an L1 (x) ¯ (x, k2 Δt) and u ˜ (x, k2 Δt) of the order of |α| · O(1)e(k2 − distance between u k1 ; a)(k2 − k1 )Δt. Summing all type I waves, the total contribution to the L1 (x) distance is of the order of Error2 ≡ TV · O(1)e(k2 − k1 ; a)(k2 − k1 )Δt. The variation Δσ(α) of the speed of α induces uncertainty in the location of the wave. When the random number ak , k1 ≤ k ≤ k2 , lies within this range of length Δσ(α), the jump can be to the right or to the left. This uncertainty yields an error in the propagation distance of the order of e(k2 − k1 , a)Δσ(α)(k2 − k1 )Δt. The corresponding error in the L1 (x) distance is |α| · O(1)e(k2 − k1 , a)Δσ(α)(k2 − k1 )Δt. Summing all the waves gives a total
230
9. Well-Posedness Theory
error of O(1)e(k2 − k1 , a)D(Ωk1 ,k2 )(k2 − k1 )Δt according to (3.6). This is the same as Error2 . ¯ (x, k2 Δt) and u ˜ (x, k2 Δt) is In conclusion, the L1 (x) distance between u the sum Error1 + Error2 . This completes the proof of the proposition.
4. Existence Theory We now use Theorem 2.1, Proposition 3.3, and Proposition 3.5 to study the convergence and consistency of the Glimm scheme. Theorem 4.1. Suppose that the initial data u(x, 0) is of small total variation TV and that the random sequence a in (1.2) is equi-distributed in the sense of Definition 1.1. Then there is a sequence of Glimm approximate solutions which converges to a weak solution of the system of hyperbolic conservation laws ut + f (u)x = 0 as the mesh sizes Δx and Δt go to zero. Moreover, the solution is Lipschitz continuous in L1 (x), that is, ∞ |u(x, t2 ) − u(x, t1 )| dx = O(1)TV|t2 − t1 |, t1 , t2 ≥ 0. (4.1) −∞
Proof. The proof consists of two parts. The first part is to show that the ¯ converge to a limiting function u for a sequence approximate solutions u of mesh sizes going to zero. The second part is to show that the limiting function u is a weak solution of the initial value problem. Convergence The convergence through a sequence of mesh sizes Δxj → 0 follows from the uniform boundedness of the total variation of the approximate ¯ (x, t) = u ¯ (x, t; Δx) = u ¯ (x, t; Δx, a) by Theorem 2.1. It is proved solutions u as follows. For a fixed time t ≥ 0, by Theorem 0.1, Helly’s Theorem, there ¯ (x, t) converges to a exists a sequence of step sizes Δxi → 0 such that u limiting function u(x, t) pointwise in x ∈ R. Consider rational numbers tk ≥ 0, k = 1, 2, . . . . Apply Helly’s Theorem to find a sequence Δx1i , i = ¯ (x, t1 ; Δx1i ) converges to a limiting function u(x, t1 ). 1, 2, . . . , such that u Repeat the process to find a subsequence, labeled Δx2i , i = 1, 2, . . . , of ¯ (x, t2 ; Δx2i ) converges to a limiting function Δx1i , i = 1, 2, . . . , such that u 2 u(x, t2 ). By choice, Δxi , i = 1, 2, . . . , is a subsequence of Δx1i , i = 1, 2, . . . , ¯ (x, t; Δx2i ) converges to a limiting function u(x, t) for and consequently u both t = t1 and t = t2 . Continue the process to find sequences Δxki , i, k = ¯ (x, t; Δxki ) converges to a limiting function u(x, t) for 1, 2, . . . , such that u t = t1 , t2 , . . . , tk . Consider the sequence Δxii , i = 1, 2, . . . . The tail of this sequence Δxii , i = N, N + 1, . . . , is a subsequence of Δxki , i, k = 1, 2, . . . , for N > i. This is the famous Cantor’s diagonal process method. It follows from
4. Existence Theory
231
¯ (x, t; Δxii ) converges pointwise in x to a limiting function the above that u u(x, t) as i → ∞ for all rational times t = t1 , t2 , . . . . For the convergence at irrational times, we need the following L1 (x) Lipschitz continuity, (4.1), first on the level of the approximate solutions: ∞ ¯ (x, t1 )| dx = O(1)TV(t2 − t1 ), t2 > t1 ≥ 0. |¯ u(x, t2 ) − u (4.2) −∞
This follows from the finite speed of propagation of waves in the approximate ¯ (x, t1 ) is bounded by the total strength ¯ (x, T2 ) − u solutions. The difference u of waves crossing the vertical line x = constant between times t1 and t2 . This is bounded by the total strength of waves in the domain of dependence (x − C(t2 − t1 ), x + C(t2 − t1 )) at time t1 on the vertical line: ¯ (x, t1 )| = O(1)Var(¯ u(·, t1 ); x − C(t2 − t1 ), x + C(t2 − t1 )). |¯ u(x, t2 ) − u Here C is any constant greater than Δx/Δt. Integrate the above with respect to x over R and change the order of integrations to obtain d¯ u (y, t ) 1 ¯ (x, t1 )| dx ≤ |¯ u(x, t2 ) − u dx dy dy −∞ −∞ x−C(t2 −t1 ) x+C(t2 −t1 ) ∞ ∞ d¯ d¯ u(x, t1 ) u(x, t1 ) dy = 2C(t2 − t1 ) = dx dx dx dx, −∞ x−C(t2 −t1 ) −∞
∞
∞
x+C(t2 −t1 )
which is O(1)TV(t2 − t1 ) by Theorem 2.1. With (4.2), the limiting function u(x, t) over rational times can be completed to all times by continuity in L1 (x). Consistency It remains to show that the Glimm scheme is consistent in that the limiting function u(x, t) is a weak solution of the system of conservation ¯ (x, t) is an exact solution between the time laws. An approximate solution u grid levels, and so we have that for any test function φ(x, t), the truncation error is concentrated at the grid times kΔt, k = 0, 1, . . . :
¯ φ dx = − u
¯ φt + f (¯ u)φx + u
(4.3) t≥0
t=0
Ek (Δx, φ, a) ≡
∞ −∞
∞
Ek (Δx, φ, a),
k=0
¯ (x, kΔt) − u ¯ (kΔt − 0)φ(x, kΔt) dx. u
As the test function φ(x, t) is zero outside a bounded set, say φ(x, t) = 0 for |x| ≥ M and t ≥ M for some positive constant M , the above sum has finite
232
9. Well-Posedness Theory
non-zero terms: ¯ φt + f (¯ u)φx + u t≥0
¯ φ dx = − u t=0
K
Ek (Δx, φ, a),
k=0
K=
M . Δt
Write K = AB and divide the truncation error sum into A groups, each with B terms, using the notation of Proposition 3.1 and Proposition 3.3: K
Ek (Δx, φ, a) =
A
Error(Ω(j−1)B,jB ),
j=1
k=0
Error(Ω(j−1)B,jB ) ≡
jB−1
Ek (Δx, φ, a),
k=(j−1)B
Ω(j−1)B,jB ≡ {(x, t) : −∞ < x < ∞, (j − 1)BΔt ≤ t < jBΔt}. By Proposition 3.3, Error(Ω(j−1)B,jB ) = O(1)BΔt TV BΔt + e(B; a) + C(Ω(j−1)B,jB ) + D(Ω(j−1)B,jB ) . From this and (4.3), the total truncation error of the approximate solution ¯ (x, t) is u ¯ φt + f (¯ ¯ φ dx u)φx + u u (4.4) t≥0
=
A
t=0
O(1)BΔt TV BΔt + e(B; a) + C(Ω(j−1)B,jB ) + D(Ω(j−1)B,jB )
j=1
= O(1)ABΔt TV BΔt + e(B; a) + O(1)BΔt C(Ω0,AB ) + D(Ω0,AB ) M C(Ω0,AB ) + D(Ω0,AB ) = O(1)M TV BΔt + e(B; a) + O(1) A M M + e(B; a) + O(1) TV . = O(1)M TV A A Here we have noted that ABΔt = M is the size of the support of the test function φ(x, t) and that the cancellation and interaction measures are additive, A C(Ω(j−1)B,jB ) + D(Ω(j−1)B,jB ) = C(Ω(0,AB ) + D(Ω(0,AB ), j=1
and by Corollary 2.2, C(Ω(0,AB ) + D(Ω(0,AB ) = O(1)TV.
5. Stability Theory
233
Finally, as the mesh size Δt → 0, we may choose A, B → ∞ so that M/A → 0 and e(B; a) → 0. We thus conclude from (4.4) that ¯ φt + f (¯ ¯ φ dx → 0 as Δt → 0. u u u)φx + t≥0
t=0
This completes the proof of the consistency of the Glimm scheme for an equi-distributed sequence a. We thus conclude that the Glimm approximate solutions approach a limiting function u(x, t), which is a weak solution when the sequence a is equi-distributed. This completes the proof of the theorem.
5. Stability Theory A fundamentally important question is whether there is a well-posedness theory for weak solutions of the gas dynamics equations (5.4) with the additional consideration of the second law of thermodynamics in the form of the entropy condition, as discussed in Chapter 7. We know that this is so for scalar conservation laws; the solution operator is L1 (x) contractive, as shown in Section 4 of Chapter 3 and Section 5 of Chapter 5. The scalar theory is established through certain constructive solution operators; the one presented in Chapter 3 and Chapter 5 entails approximation by piecewise smooth solutions or through the zero dissipation limit. The L1 (x) contraction property yields the key property that a solution depends continuously on its initial data. We now show that this is true also for general systems of conservation laws through the Glimm solution algorithm. Consider the system of conservation laws ut + f (u)x = 0, u = (u1 , u2 , . . . , un )T ∈ Rn . The generalized entropy functional approach applies to gas dynamics equations. It applies to general systems when each characteristic field is either genuinely nonlinear or linear degenerate; see Section 3 of Chapter 7. For simplicity, we will carry out the analysis for the case where each characteristic field is genuinely nonlinear; generalization to the case where some characteristic fields are linearly degenerate is straightforward. The goal is to show that solutions depend continuously on their initial values in the L1 (x) norm. That is, we aim to show that for any two solutions u(x, t) and v(x, t) obtained by the Glimm solution algorithm, ∞ ∞ u(x, 0) − v(x, 0) dx, t > 0. u(x, t) − v(x, t) dx ≤ C (5.1) −∞
−∞
234
9. Well-Posedness Theory
The idea is to construct a functional H(t) with the property that it is nonincreasing in time and is equivalent to the L1 (x) distance of the two solutions: (5.2)
d H(t) ≤ 0, H(t) = H[u(·, t), v(·, t)], dt ∞ u(x, t) − v(x, t) dx ≤ H(t) ≤ C2 C1 −∞
∞ −∞
u(x, t) − v(x, t) dx.
The construction and analysis of the functional H(t) is done in steps. 5.1. Mesoscale The functional H(t) can be defined for two general weak solutions. It should be emphasized that only by considering solutions constructed by the Glimm scheme will there be sufficient structure for the solutions to allow analysis of the time evolution of the functional. Thus in the present approach, the functional H(t) is first constructed for the corresponding approximate solu¯ (x, t) and v ¯ (x, t). Following the general approach outlined in Remark tions u 3.2, the exact solution on the macroscopic scale (x, t) is reached from the approximate solution on the microscopic scale (kΔt, jΔx) by going through the analysis on the mesoscopic scale. Thus we first consider the evolution of the functional H(t) over a region of mesoscopic scale, Ωk1 ,k2 ≡ {(x, t) : −∞ < x < ∞, k1 Δt ≤ t < k2 Δt}, by applying Proposition 3.1 and Proposition 3.5. The waves in Ωk1 ,k2 for ¯ (x, t) and v ¯ (x, t) are partitioned for tracing. The function u ¯ (x, k1 Δt) both u ¯ (x, k1 Δt) consists of consists of waves αj = (uj− , uj+ ), j = 1, 2, . . . , and v waves β j = (v j− , v j+ ), j = 1, 2, . . . . For simplicity of later analysis, the approximate solutions are regarded as step functions for each given time. This discretization is achieved when any given rarefaction wave is approximated by a series of weak rarefaction shocks, each of strength less than a small number ε. An i-rarefaction wave ¯ (x, k1 Δt) is partitioned by taking αj = (uj− , uj+ ), uj+ ∈ R+ i (uj− ), in u (u ) between u and uj+ so that |um − um−1 | ≤ ε for u1 , . . . , ul in R+ j− j− i m = 1, . . . , l. Each sub-wave α ≡ (um , um−1 ) is regarded as a rarefaction shock with speed σ(α) ≡ (λi (um ) + λi (um−1 ))/2; see (2.8) of Chapter 8. By Proposition 2.2 in Chapter 8, each rarefaction shock satisfies the RankineHugoniot condition up to an error of order ε3 and violates the entropy inequality also up to an error of ε3 . With each rarefaction wave approximated by a series of rarefaction shocks, the original wave partitions are accordingly refined. We then con˜ (x, t) and v ˜ (x, t) for sider the linear superposition of nonlinear waves, u
5. Stability Theory
235
¯ (x, t) and v ¯ (x, t), respectively, and apply Proposition 3.5: u ∞ ˜ (x, t1 Δt)| dx = 0, |¯ u(x, t1 Δt) − u (5.3) −∞ ∞ ˜ (x, t1 Δt)| dx = 0, |¯ v (x, t1 Δt) − v −∞ ∞ ˜ (x, t2 Δt)| dx = O(1) D(¯ |¯ u(x, t2 Δt) − u u, Ωk1 ,k2 ) + C(¯ u, Ωk1 ,k2 ) −∞ + TV · ε + e(k2 − k1 ; a) (k2 − k1 )Δt; ∞ ˜ (x, t2 Δt)| dx = O(1) D(¯ |¯ v (x, t2 Δt) − v v , Ωk1 ,k2 ) + C(¯ v , Ωk1 ,k2 ) −∞ + TV · ε + e(k2 − k1 ; a) (k2 − k1 )Δt. Note that there is the extra error O(1)TV · ε(k2 − k1 )Δt due to the approximation of the rarefaction waves by weak rarefaction shocks. We also use D(¯ u, Ωk1 ,k2 ) to denote the wave interaction measure for the approximate ¯ in the region Ωk1 ,k2 , etc. solution u With the above set-up, the focus is on defining the functional H(t) for ˜ (x, t) and v ˜ (x, t) and for the the linear superposition of nonlinear waves u mesoscale time period. 5.2. Distance Functions The distance between two solutions needs to be defined to be consistent with ˜ and v ˜ so that the the approximation of the solutions by step functions u Liu-Yang functional for scalar laws, Theorem 8.5 in Section 8 of Chapter 3, can be generalized to systems. ˜ (or v ˜ ) located at x = Let αj (or β j ), j = 1, 2, . . . , be waves in u ˆ ˜ (x, t) to j(t) (or x = j(t)) at time k1 Δt ≤ t < k2 Δt. The distance from u ˜ (x, t) is measured by solving the Riemann problem (˜ ˜ (x, t)) using v u(x, t), v the Hugoniot curves: (5.4)
˜ (x, t), un = v ˜ (x, t), uj ∈ H j (uj−1 ), j = 1, 2, . . . , n. u0 = u
˜ (x, t) to u ˜ (x, t) is defined by solving the Symmetrically, the distance from v ˜ (x, t)) instead: Riemann problem (˜ v (x, t), u (5.5)
˜ (x, t), v n = u ˜ (x, t), v j ∈ H j (v j−1 ), j = 1, 2, . . . , n. v0 = v
+ Note that, instead of using H − j and Rj curves as done for solving the Riemann problem in (4.6) of Chapter 7, here H j , j = 1, 2, . . . , n are used. This is consistent with the approximation of rarefaction waves by rarefaction shocks. By a linear change of variables u, if necessary, we can assume that the i-component ui of u is strictly monotone along the characteristic curve
236
9. Well-Posedness Theory
Ri and the Hugoniot curve H i . This allows us to use the ith scalar conservation law (ui )t + f i (u)x = 0 in the system to measure the ith component ˜ (x, t) to v ˜ (x, t) and set the wave speeds as follows: distance q i from u (5.6)
q i = q i (x, t) ≡ uii − uii−1 , λ(q i ) ≡ σ(ui−1 , ui ), i = 1, 2, . . . , n.
˜ (x, t) to u ˜ (x, t) and its speed Similarly, the ith component distance qˆi from v are defined as (5.7)
i , λ(ˆ q i ) ≡ σ(v i−1 , v i ), i = 1, 2, . . . , n. qˆi = qˆi (x, t) ≡ vii − vi−1
Analogously, the ith scalar conservation law (ui )t + f i (u)x = 0 is used to measure an i-wave: (5.8) |α| ≡ |αi |, αi ≡ ui+ − ui− , λ(α) ≡ σ(u− , u+ ) for an i-wave α = (u− , u+ ), i = 1, . . . , n. With the definition of the above notions based on the scalar conservation law (ui )t + f i (u)x = 0, the basic local conservation law for scalar equations, (8.11) of Chapter 3, can be generalized to systems. For a wave α = (u− , u+ ) j ˜ (x ± 0, t), denote by q± ˜ located at (x, t) where u± = u (α) = q j (x ± in u 0, t), j = 1, . . . , n, the distances around α. j ˜ satisfy (α) around an i-wave α in u Lemma 5.1. The distances q±
i i (5.9) q+ (α) = q− (α) + αi + O(1) e(α) + qd (α) , j j (α) = q− (α) + O(1) e(α) + qd (α) , j = 1 . . . , n, j = i, q+ and there are the local conservation laws i i i i (α))q+ (α) = λ(q− (α))q− (α) + O(1) e(α) + qd (α) , (5.10) λ(α)αi + λ(q+ j i i (α)||q+ (α)|, qd (α) ≡ |α| |q± (α)|. e(α) ≡ |α||q− j =i
Proof. The distances are constructed by solving Riemann problems using the Hugoniot curves only, (5.4), (5.5), (5.6), and (5.7). We may view j (α), j = 1, 2, . . . , n, as waves resulting from the interaction of α and q− j (α), j = 1, 2, . . . , n. Note that being an i-wave, α has the set of waves q+ zero strength except for the ith strength αi . Thus we may follow the same analysis as in Section 4 of Chapter 8 for the interaction of two Riemann solutions, even though the waves here are only compression or expansion shocks, not rarefaction waves. This establishes the first two estimates in j (α), j = 1, 2, . . . , n, as waves resulting (5.9). Alternatively, we may view q+ j (α), j = 1, 2, . . . , n. from the interaction of the two sets of waves α and q−
5. Stability Theory
237
We therefore see that in the above definition of qd (α) we can use either j j qd (α) ≡ |α| |q− (α)| or qd (α) ≡ |α| |q+ (α)|. j =i
j =i
qj ,
j = 1, 2, . . . , n, are measured This proves (5.9). Recall that the distances j j using the scalar conservation laws (u )t + (f (u))x = 0. Therefore the local conservation laws (5.10) for systems are proved by generalizing the local conservation laws for scalar equations, (8.11) in Chapter 3, and using the two estimates in (5.9). 5.3. The Functional ˜ (·, t)]. The first There are three parts to the functional H(t) = H[˜ u(·, t), v part L(t) is equivalent to the L1 (x) distance between the two solutions: ∞ ∞ ˜ (x, t)| dx ≤ L(t) ≤ C2 ˜ (x, t)| dx; |˜ u(x, t) − v |˜ u(x, t) − v (5.11) C1 −∞
L(t) ≡
n
Li (t), Li (t) ≡
i=1
−∞
∞ −∞
|q i (x, t)| + |ˆ q i (x, t)| dx.
The second part of H(t) registers the effect on the L1 (x) distance due to crossing waves of distinct characteristic families and is defined as ∞ ∞ n i { |αj | |q i (y, t)| dy, αj an l-wave} (5.12) Qd (t) ≡ −∞
l=i+1
i−1 { +
+
l=1 n
−∞
{
l=i+1
i−1 { + l=1
∞
|αj |
∞ −∞
∞
−∞
j(t)
j(t)
|β j |
|β j |
|q i (y, t)| dy, αj an l-wave}
−∞ ∞
ˆ j(t) ˆ j(t) −∞
|ˆ q i (y, t)| dy, β j an l-wave}
|ˆ q i (y, t)| dy, β j an l-wave}; Qd (t) ≡
n
Qid (t).
i=1
In the above, we recall that ˜ (x, t) consists of waves αj , j = 12, . . . , located at x = j(t), and (5.13) u ˜ (x, t) consists of waves β j , j = 1, 2, . . . , located at x = ˆj(t). v The third functional, the generalized entropy functional (Liu-Yang functional), extends that for the scalar laws, (8.14) in Section 8 of Chapter 3, to
238
9. Well-Posedness Theory
systems: Ei (t) ≡
{|αj |
∞
| min{0, q i (x, t)}| dx
j(t)
(5.14)
j(x)
+ +
{|βj |
∞
ˆ j(t)
−∞
max{0, q i (x, t)} dx , αj an i-wave}
| min{0, qˆi (x, t)}| dx ˆ j(x)
+ −∞
max{0, qˆi (x, t)} dx , β j an i-wave}; E(t) =
n
Ei (t).
i=1
The desired functional H(t) is the linear combination of the above: H(t) ≡ L(t) + K1 Qd (t) + K2 E(t)
(5.15)
for sufficiently large K1 and K2 . 5.4. Coupling Analysis The above functionals are designed to monitor the effects of the coupling of waves on the L1 (x) distance between the solutions. We now assess these effects. For scalar laws, the dissipation of the generalized entropy functional E[u, v] has been quantified in Theorem 8.3 and Theorem 8.5 of Chapter 3. It will be shown later—see (5.22)—that the dissipation measure for the generalized entropy functional E(t) for systems as defined in (5.14) is given by (5.16) e(t) =
n
ei (t), ei (t) =
{ei (αj ), αj i-waves},
i=1 i i (α)||q+ (α)|, α an i-wave. ei (α) = e(α) ≡ |α||q−
The following measure qd will turn out to be the appropriate measure of the decoupling mechanism—see (5.21)—of the functional Qd defined in (5.12): (5.17) qd (t) =
n
qdi (t), qdi (t) =
{qdi (αj ), αj an i-waves},
i=1
qdi (α) = qd (α) ≡ |α|
j =i
j |q± (α)|, α i-wave.
5. Stability Theory
239
The above notions are from the perspective of u(x, t) to v(x, t). The corresponding notions from v(x, t) to u(x, t) are (5.18) eˆ(t) =
n
eˆi (t), eˆi (t) =
i i {|β j ||ˆ q− (β j )||ˆ q+ (β j )|, β j i-waves},
i=1
qˆd (t) =
n
qˆdi (t), qˆdi (t) =
{|βj |
i=1
j |q± (β)|, β j i-waves}.
j =i
In the following, we will carry out the analysis from the perspective of u(x, t) to v(x, t); the analysis for the reverse situation is the same. We now start the analysis for the time evolution of the functionals ˜ (x, t) and v ˜ (x, t) Qid (t), and E i (t), defined for linear superpositions u of shocks and rarefaction shocks for the region Ωk1 ,k2 . Due the approximation by linear superposition and the approximation of rarefaction waves by weak rarefaction shocks, we will see that there is a common error term of the form
Li (t),
u, Ωk1 ,k2 ) + C(¯ u, Ωk1 ,k2 ) (5.19) LSk1 ,k2 ≡ D(¯ + D(¯ v , Ωk1 ,k2 ) + C(¯ v , Ωk1 ,k2 ) + TV · ε. Lemma 5.2. d i L (t) = O(1) e(t) + qd (t) + O(1)LSk1 ,k2 , k1 Δt ≤ t < k2 Δt. (5.20) dt Proof. From the definition (5.11) of Li (t), we have d i i i L (t) = λ(α)(|q− (α| − |q+ (α)|). dt α Trivially, we have
i i λ(q− (α))|q− (α)| =
α
i i λ(q+ (α))|q+ (α)|.
α
Thus we may rewrite the above as d i L (t) = Ai (α), dt α i i i i i i (α)| − |q+ (α)|) − λ(q+ (α))|q+ (α)| + λ(q− (α))|q− (α)|. Ai (α) ≡ λ(α)(|q− i (α) and q i (α) are of the same sign; the other We consider the case where q+ − case is aided by the L1 (x) contraction analysis, Theorem 4.1 of Chapter 3. i (α) ≥ 0, Thus when ±q+ i i i i i i (α) − q+ (α)) − λ(q+ (α))q+ (α) + λ(q− (α))q− (α). ±Ai (α) = λ(α)(q−
240
9. Well-Posedness Theory
When α is a j-wave, j = i, then from (5.9), i i q− (α) = q+ (α) + O(1) e(α) + qd (α) , and Ai (α) = O(1) e(α) + qd (α) . When α is an i-wave, estimate (i) follows from the local conservation laws in (5.9). This proves the estimate (5.20). Lemma 5.3. d i ¯ i (t) + O(1)TV e(t) + qd (t) + O(1)LSk ,k Qd (t) ≤ −Cq (5.21) d 1 2 dt ¯ for some positive constant C. Proof. Consider the time derivative of a term in Qid (t): ∞ d i ∞ j i d |α | |q (t)| dx = |α | |q j (t)| dx, j < i. dt dt x(α) x(α) ∞ ∞ j The term x(α) |q (t)| dx is similar to a part, x¯(α) |q j (t)| dx, of Lj (t). Here j (α)) according to the x=x ¯(α) denotes the propagation of the j-wave (0, q+ jth conservation law. We have, from the analysis in establishing (5.20), that ∞ i d |q j (t)| dx = O(1)|αi | e(α) + qd (α) . |α | dt x¯(α)
From j < i and the strict hyperbolicity, the difference of their speeds is positive: d d x(α) − x ¯(α) ≥ [λ], dt dt where [λ] ≡ min{|λi (u1 ) − λj (u2 )|, i = j, for all u1 and u2 under consideration} is positive and bounded away from zero. Thus we have ∞ x¯(α) i d j i d |q (t)| dx = |α | |q j (t)| dx + O(1)|α| e(α) + qd (α) |α | dt x(α) dt x(α) j (α) + |αi | e(α) + qd (α) . ≤ −[λ]|αi |q+ The proof of the lemma is complete by this and the definition of qdi (t) in (5.17). The constant C¯ in (5.21) can be taken to be [λ]. Lemma 5.4. d i E (t) = −ei (t) + O(1)TV e(t) + qd (t) + O(1)LSk1 ,k2 . (5.22) dt Proof. The estimate follows from the analysis in the above two lemmas and the estimate for scalar laws, Theorem 8.5 of Chapter 3, using the local conservation laws (5.10).
5. Stability Theory
241
Lemma 5.5. (5.23) H(k2 Δt) − H(k2 Δt − 0) = O(1) LSk1 ,k2 + TV(e(k2 − k1 , a) + ε) (k2 − k1 )Δt. Proof. The functional H(k2 Δt − 0) is defined by the linear superposition of ¯ ˜ and v ˜ at time t = k2 Δt − 0, while H(k2 Δt) is defined by u wave patterns u ¯ at time t = k2 Δt. The L1 (x) distance between u ¯ and u ˜ (or between and v ¯ and v ˜ ) at time k2 Δt is estimated by Proposition 3.5: v ∞ ˜ (x, k2 Δt)| dx |¯ u(x, k2 Δt) − u (5.24) −∞ u, Ωk1 ,k2 ) + TV · e(k2 − k1 ; a) (k2 − k1 )Δt, = O(1) D(¯ u, Ωk1 ,k2 ) + C(¯ ∞ ˜ (x, k2 Δt)| dx |¯ v (x, k2 Δt) − v −∞ v , Ωk1 ,k2 ) + TV · e(k2 − k1 ; a) (k2 − k1 )Δt. = O(1) D(¯ v , Ωk1 ,k2 ) + C(¯ As the functional H(t) is equivalent to the L1 (x) distance, the estimate (5.23) follows from (5.24). Note that, because of the further approximation ˜ and v ˜ , an extra error of the rarefaction wave with rarefaction shocks in u term O(1)TV · ε(k2 − k1 )Δt is included in (5.23). This completes the proof of the lemma. 5.5. Stability With the coupling analysis in the above lemmas, we are ready to state and prove the main theorem on stability. Theorem 5.6. Suppose that both u(x, t) and v(x, t) are solutions with small total variation as constructed by the Glimm scheme and that their L1 (x) distance at time t = 0 is small. Then as the approximate solutions tend to the exact solution as Δt → 0, in the limit the above functional H(t) for approximate solutions, (5.11)–(5.15), satisfies d H(t) ≤ 0 for t > 0. dt Consequently, the solution operator is stable in L1 (x) norm; that is, for some positive constant C independent of time, ∞ ∞ |u(x, t2 )−v(x, t2 )| dx ≤ C |u(x, t1 )−v(x, t1 )| dx, 0 ≤ t1 < t2 < ∞. −∞
−∞
Proof. The proof is based on Lemma 5.2, Lemma 5.3, Lemma 5.4, and Lemma 5.5, and follows the general procedure of the proof of the consistency
242
9. Well-Posedness Theory
part of Theorem 4.1. Thus the time interval (t1 , t2 ) is divided into AB time steps, t2 − t1 = ABΔt, t1 = k0 Δt, t2 = (k0 + AB)Δt. For each time zone Ωk1 ,k2 with k2 = k1 + B and k1 = k0 , . . . , k0 (A − 1), we apply the above lemmas to obtain d ¯ d (t) + O(1)TV(e(t) + qd (t)) H(t) ≤ O(1) qd (t) + e(t) + K1 −Cq dt + K2 −e(t) + O(1)TV(e(t) + qd (t)) + O(1)LSk1 ,k2 = O(1) − K1 C¯ + (K1 + K2 )O(1)TV qd (t) + O(1) + (K1 + K2 )O(1)TV − K2 e(t) + O(1)LSk1 ,k2 . This yields, for small TV, by choosing the constants K1 and K2 to be larger than O(1), d H(t) ≤ O(1)LSk1 ,k2 , k1 Δt ≤ t < k2 Δt. dt Integrate this over k1 Δt ≤ t < k2 Δt and use Lemma 5.5 to obtain H(k2 Δt) ≤ H(k1 Δt) + O(1) LSk1 ,k2 + TV e(k2 − k1 , a) + ε (k2 − k1 )Δt. When the above is summed over l = 0, . . . , A−1, the terms O(1)LSk1 ,k2 (k2 − k1 )Δt add up to u, Ωk0 ,k0 +AB ) O(1) D(¯ u, Ωk0 ,k0 +AB ) + C(¯ v , Ωk0 ,k0 +AB ) AΔt + TV · ε(t1 − t2 ) + D(¯ v , Ωk0 ,k0 +AB ) + C(¯ t2 − t1 + TV · ε(t1 − t2 ) → 0 as B → ∞ and letting ε → 0. B − k , a) + ε (k2 − k1 )Δt, adds up to O(1)TV The other term, O(1)TV e(k 2 1 e(k2 − k1 , a) + ε (t2 − t1 ), which goes to zero since k2 − k1 → ∞ in the limit. Here the approximation by rarefaction shocks is taken to be finer, ε → 0 as Δt → 0. Therefore H(t2 ) ≤ H(t1 ) in the limit. This completes the proof of the theorem. = O(1)(TV)2
Remark 5.7. The convergence of the Glimm approximation to the exact solution of the system of conservation laws, as usual, is through a subsequence of the grid sizes (Δx, Δt). For any sequence (Δx)l , (Δt)l → 0, there is a subsequence for which the corresponding approximate solutions tend to an exact solution. By Theorem 5.6, the exact solution is unique within this construction. It is clear from these considerations that the approximate solutions tend to the exact solution as the grid sizes Δx, Δt → 0. This finishes the first part of the well-posedness theory. This part is the classical study of the existence and continuous dependence on initial data
6. Generalized Characteristics and Expansion of Rarefaction Waves
243
of solutions. For the consideration of weak solutions here, this depends on the explicit construction of Glimm approximate solutions and analysis of the wave evolution in time. The definite understanding of the solution algorithm naturally leads to the second part of the study of solution behavior, which is therefore also viewed as part of the well-posedness theory. PART II: Solution Behavior
6. Generalized Characteristics and Expansion of Rarefaction Waves The proof of convergence of Glimm approximate solutions in Section 4 can be made more definite. From the basic estimate (2.10), the interaction measure D(Ω) and cancellation measure C(Ω) are bounded uniformly in the mesh ¯ ¯ sizes. We rename them D(Ω) and C(Ω) as they refer to the approximate ¯ (x, t). They converge, for a sequence of mesh sizes going to zero, solution u in weak* topology to limiting bounded measures D(Ω) and C(Ω) for the exact solution u(x, t). Here Ω is any region in the (x, t) space. Let E be a set corresponding to point measures in D(Ω) and C(Ω). Since the measures D(Ω) and C(Ω) are bounded, the set E is at most countable. We will see in Section 8 that a point in E represents interaction points. ¯ (x, t) is of uniform bounded variation in x An approximate solution u for each fixed time t. From Helly’s Theorem, Theorem 0.1, and the well¯ (x, t) converges to the limiting solution posedness theorem, Theorem 5.6, u u(x, t) almost everywhere as the mesh size Δt → 0. We can also show that for a fixed time t, the convergence is pointwise except at countably many points. This is analyzed by writing a scalar function of bounded variation ¯2 (x) as the difference of two monotone increasing functions, u ¯(x) = u ¯1 (x) − u and u(x) = u1 (x) − u2 (x). When both u1 and u2 are continuous at x, the convergence at x is local in the sense that for any given ε > 0, there exists δ > 0 such that |¯ u(y, t) − u(y, t)| < ε for |y − x| < δ and Δt < δ. On the other hand, when u(·, t) is continuous at x, there is a possibility that there is a jump at x for both u1 and u2 and these jumps cancel out in the limit Δt → 0. In that case, the convergence is not local in the above sense. For the Glimm approximate solutions, this would correspond to a point cancellation measure and so the point (x, t) ∈ E. We conclude that ¯ (·, t) to u(·, t) is local in the sense that for any given the convergence of u ε > 0, there exists δ = δ(x) > 0 such that (6.1)
|¯ u(y, t) − u(y, t)| < ε for |y − x| < δ and Δt < δ,
244
9. Well-Posedness Theory
for any x ∈ R which is a continuity point of u(·, t) and such that (x, t) is not in E. We will show later that if x is a continuity point of u(·, t), then (x, t) does not belong to E; see part (2) of Theorem 8.1 on the regularity of the solution in Section 8. Thus (6.1) holds for all continuity points of u(·, t). In ¯ (·, t), the ith compression (or expansion) waves the approximate solution u ¯ i− (or dX ¯ i+ ) according to the variation in λi . Thus form the measure dX ¯ i+ ) is a non-positive (or non-negative) measure. By the local ¯ i− (or dX dX convergence property (6.1), these measures converge to the corresponding ones dXi± for the limiting exact solution u(·, t). The notion of generalized characteristics is an effective tool for studying the solution behavior for scalar laws. A forward-in-time generalized characteristic curve propagates with characteristic speed until it hits a shock wave, and then it propagates with the shock speed. The backward-in-time characteristics for scalar laws are straight lines. These curves are considered for piecewise continuous solutions in Section 5 of Chapter 3. In Remark 4.5 in Section 4 of Chapter 3, it is pointed out that, in analyzing the behavior of general solutions, it is sufficient to consider piecewise continuous solutions. We may follow this general approach for systems of conservation laws; however, there are essentially new issues that will arise. For a system of conservation laws, there is the wave partition for the ¯ (x, t) in a given region Ωk1 ,k2 , discussed in Section approximate solution u 3, over the time period (t1 , t2 ) where t1 = k1 Δt and t2 = k2 Δt. Over the time period, a backward-in-time ith characteristic is defined for any ith rarefaction wave of type I and propagates with characteristic speed λi , and the forward-in-time generalized characteristics are defined the same way as for scalar laws, i.e. such a curve propagates with characteristic speed and, upon hitting a type I shock, follows the shock. An essential difference from the scalar laws is that the speed of these curves changes due to the crossing of waves pertaining to jth characteristic fields, j = i. Although the approxi¯ (x, t) is piecewise continuous between the time grid levels, mate solution u there are jumps at the time grid levels. In particular, the characteristic and generalized characteristic curves for the approximate solutions jump at grid times. Through analysis on the mesoscopic scale, it is seen that one can work with continuous curves instead.
Lemma 6.1. Consider the Glimm approximate solutions with initial data of small total variation TV. For a given characteristic or generalized characteristic curve C between t1 and t2 , with t2 − t1 = N 2 Δt, denote by C˜ the continuous curve with the same speed as C between the grid times. Then the ˜ distance dist(C, C)(t) between C and C˜ at time t, t1 < t < t2 , satisfies, as
6. Generalized Characteristics and Expansion of Rarefaction Waves
245
Δt → 0, TV ˜ (6.2) dist(C, C)(t) = O(1) e(N ; a) + (t − t1 ) N if C˜ has same position as C at time t1 ; TV ˜ (t2 − t) dist(C, C)(t) = O(1) e(N ; a) + N if C˜ has same position as C at time t2 . In particular, the curves C and C˜ approach each other at the rate (Δt)1/2 + e((Δt)−1/2) as Δt → 0. Proof. We prove only the first part at time t = t2 . We use the analysis for the mesoscale t − t1 ∈ Ik ≡ (kN Δt, (k + 1)N Δt), k = 0, . . . , N − 1. Let Var(k) be the variation of the speed of the curve C˜ in Ik . The variation is mainly due to waves of other characteristics crossing the curve, and so the total variation is N −1 Var(k) = O(1)TV. k=0
During each time interval Ik , the increase in distance is of the amount O(1)Var(k)N Δt due to the variation of speed and of O(1)e(N ; a)N Δt arising from the randomness of the scheme; see Definition 1.1. The distance at time t2 is the sum of these variations over k = 0, . . . , N − 1: ˜ 2) = dist(C, C)(t
N −1
O(1)Var(k)N Δt + O(1)e(N ; a)N Δt
k=0
TV + e(N ; a) (t2 − t1 ). N This establishes the estimate (6.2) for t = t2 . The last statement follows from the estimate (6.2) by noting that since t2 − t1 = N 2 Δt, Δt is of the order of N −1/2 . = O(1)
Proposition 6.2. Except for an amount of O(1)D(Ωk1 ,k2 ) + O(1)C(Ωk1 ,k2 ), each location within the ith rarefaction waves in the limiting measure dXi+ for the exact solution u(·, t2 ) is the starting point of backward-in-time ith rarefaction curves. These curves are Lipschitz continuous, with speed λi . Each curve is the limit of a corresponding backward-in-time ith characteristic ¯ (x, t), uniformly for the duration of the curve for the approximate solution u time period (t1 , t2 ). Proof. The backward-in-time characteristics are defined for type I rarefaction waves. Of the ith rarefaction waves in the time zone Ωk1 ,k2 , those ¯ k ,k ) + not of type I make up a total amount of the order of O(1)D(Ω 1 2
246
9. Well-Posedness Theory
¯ k ,k ). The variation of the speed λi of an ith characteristic curve O(1)C(Ω 1 2 is due to waves of other families crossing the curve, and therefore is of the order of TV. We also know that the convergence of the approximate solutions to the exact solution is local in the sense of (6.1). From these, the proposition follows. The backward-in-time i-characteristics have the natural property of conservation of i-waves. Proposition 6.3. Suppose that C1 to the left and C2 to the right are two backward i-characteristics and Ω is the region between C1 , C2 , time t1 , and time t2 where t1 < t2 . Then the following conservation of waves holds: (6.3) X± (t2 ) = X± (t1 ) − Ci (Ω) + O(1)D(Ω), X+ (t) ≡ amount of i-rarefaction wave in Ω at time t, X− (t) ≡ amount of i-compression wave in Ω at time t. Proof. By its construction, no i-wave crosses an i-characteristic curve. The only variation of i-waves in time is due to cancellation and interaction in Ω. This verifies the proposition. A key to the study of behavior of solutions is the analysis of the rate of expansion of rarefaction waves. Consider first the convex scalar law ut + f (u)x = 0,
f (u) > 0,
λ(u) = f (u).
Consider two generalized characteristics, C1 to the left and C2 to the right, for a piecewise continuous solution u(x, t). Let I(t) be the distance between C1 and C2 at time t. Divide I(t) into subintervals Ik so that λ(u(x, t)) is increasing (or decreasing) for x ∈ Ik , k even (or k odd); see figure 9.07. Draw backward characteristic lines starting at the end points of Ik , k even; then we have that the distance Ik (t) at time t equals the variation of λ(u) over Ik times t − t0 plus a part of I(t0 ). Summing over all even k, we obtain an estimate for the total amount of rarefaction waves, the increasing variation X+ (t) of λ(u(x, t)), between C1 and C2 : (6.4)
X+ (t)(t − t0 ) ≤ I(t), or X+ (t) ≤
I(t) . t − t0
For a system, consider two generalized ith characteristics, C1 to the left and C2 to the right, between time t0 and t with t0 < t. Let I(t) be the interval between them at time t and Ω the region between C1 , C2 , time t0 , and time t. We have the following generalization of the estimate (6.4) for scalar laws to systems.
6. Generalized Characteristics and Expansion of Rarefaction Waves
247
C2
C1 I2i (t) Ω2i − C2i
t
I2j (t)
I(t) s + C2i
I2j (s)
s − O(1)I2j (s)
I(t0) x Figure 9.07. Expansion wave.
Proposition 6.4. The total amount of i-rarefaction waves X+ (t) on the interval I(t) at time t between C1 and C2 satisfies (6.5)
X+ (t) ≤
I(t) ˜ 0 , t) + O(1)D(Ω) + O(1)D(Ω), 1 + O(1)X(t t − t0
˜ 0 , t) is the total amount of j-waves, j = i, crossing I(t0 ) or C1 where X(t and C2 between time t0 and time t. Proof. We proceed as with the above analysis for scalar laws, dividing I(t) into a finite number of subintervals Ik (t) so that the increasing (or decreasing) variation of dXi+ (or dXi− ) for the exact solution u(·, t) for i-waves is concentrated in Ik for k even (or odd). Draw backward-in-time i-characteristics Ck− and Ck+ , which correspond to the limits of type I waves ¯ (·, t) in Ik , k even, and denote by Ωk the in the approximate solution u ¯ k− (t)) be ¯ k+ (t) (or X region between them, as shown in figure 9.07. Let X the total amount of ith rarefaction waves (or ith compression waves) in Ik . As the type I waves cover the i-rarefaction waves in I(t) minus an amount of O(1)D(Ωk1 ,k2 ) by Proposition 6.2, we have, for a small amount ε due to approximation of u by u ¯ and also due to the finiteness of the number of subintervals, ¯ k+ (t) = X+ (t) + O(1)D(Ωk ,k ) + O(1)ε. X (6.6) 1 2 k even
248
9. Well-Posedness Theory
As Δt → 0, we can increase the number of subintervals and let ε → 0. For simplicity of presentation we will ignore the perturbation ε in the following formulations. By Proposition 6.2, there are corresponding backward-in-time ith characteristic curves C¯k± . In the following analysis we will use their ¯ . At corresponding continuous curves C˜k± for the approximate solution u time s, t0 < s < t, the speed of the curve Ck− (or Ck+ ) is the characteristic + speed λi , denoted by λ− k (s) (or λk (s)). By Lemma 6.1, these curves are close to each other for small Δt and so the width Ik (s) of Ωk at time s is of the same order for all three sets of curves, Ck± , C¯k± , and C˜k± . Thus we may choose Δt small enough so that Ik (s) is of the same order for all s and k. We carry out the analysis for the corresponding approximate solution. We have, for the width Ik (s) of Ωk at time s, d − Ik (s) = λ+ k (s) − λk (s), ds
t0 < s < t, k even.
There is no jump at the grid times because we are using the continuous ˆ k (t) the amount of curves C˜i and so the above holds for all s. Denote by X j-waves, j = i, in Ik (t). We have from above that d ¯ k− (s) + X ¯ k+ (s) + O(1)X ˆ k (s). Ik (s) = X ds By the choice of subintervals, in Ik (t) for k even, the amount X− (t) of compression waves is small, and, for the sake of simplicity of presentation, we assume that X− (s) = 0 and so d ¯ k− (s) + O(1)X ˆ k (s). Ik (s) = X ds ˜ k (t) be the increment of X ˜ k (t, t0 ) at time t, i.e. the increment of the Let dX amount of j-waves, j = i, which cross Ck− or Ck+ at time t. In (2.11), the interaction measure after time t is denoted by D(t). For the present set-up, ¯ k (s) be the increment of the interaction measure for the region in Ωk let dD ˆ k (s) is due to either the j-waves crossing C ± during at time s. The amount X k the time interval between s − O(1)Ik (s) and s, or the waves produced by interaction in Ωk during the same time period (s − O(1)Ik (s), s), as depicted in figure 9.07: s ˜ k (τ ) + dD ¯ k (τ )]. ˆ [dX Xk (s) = O(1) s−O(1)Ik (s)
Thus it follows from the above that (6.7)
d ¯ k+ (s) + O(1) Ik (s) = X ds
s s−O(1)Ik (s)
¯ k (τ ) . ˜ k (τ ) + dD dX
7. Large-Time Behavior
249
Integrate (6.7) from time t0 to time t and change the order of integration to get an integral equation for Ik : s s ˜ k (τ ) + dD ¯ k (s) . ¯ Ik (s) = Ik (t0 ) + Ik (τ ) dX Xk+ (τ ) dτ + O(1) t0
t0
By the conservation of i-waves in the region Ωk , Proposition 6.3, we have ¯ k+ (s) ≥ X ¯ k+ (t) + O(1)D(Ω ¯ k ), t0 ≤ s ≤ t. X Thus we have from the above that ¯ k+ (t) + O(1)D(Ω ¯ k) Ik (s) ≥ Ik (t0 ) + (s − t0 ) X s ˜ k (τ ) + dD ¯ k (τ ) . + O(1) Ik (τ ) dX t0
This integral inequality for I(t) is solved to yield ¯ k+ (t) + O(1)D(Ω ¯ k) (6.8) Ik (s) ≥ Ik (t0 ) + (s − t0 ) X ¯ k) . ˜ k (t0 , s) + O(1)D(Ω · 1 + O(1)X ¯ k+ (t) is increasing in s, as X ¯ k+ (t) is positive. From this The term (s − t0 )X we have ¯ k) ˜ k (to , t) + O(1)D(Ω ¯ k+ (t) ≤ Ik (t) 1 + O(1)X (6.9) X t − t0 ¯ k ), k even. + O(1)D(Ω Sum the estimate (6.9) over even k, let Δt → 0, and use (6.6) to finally obtain the estimate (6.5). Note that in the estimate (6.9) the multiplicative factor ˜ k (t0 , t) + O(1)D(Ωk ) is dominated by the corresponding factor 1 + O(1)X ˜ 1 + O(1)X(t0 , t) + O(1)D(Ω) in (6.5) because waves crossing Ck± are either those crossing the boundary of Ω or those produced through interaction in ¯ ¯ k ) in (6.9) sums to O(1)D(Ω) in (6.5). This Ω. The additive term O(1)D(Ω completes the proof of the proposition.
7. Large-Time Behavior The design of the Glimm approximation is based on the belief that the Riemann solutions would dominate the large-time as well as local-in-time behavior of a general solution; see Section 1. This and the following sections confirm this belief for the solutions obtained by the Glimm solution algorithm. This section studies the large-time behavior and the next section considers the local behavior of solutions. For simplicity, we assume that each characteristic field is either genuinely nonlinear or linearly degenerate, as defined in Section 4 of Chapter 7.
250
9. Well-Posedness Theory
¯ (x, t) = u ¯ (x, t; Δx, Δt) tend to an When the approximate solutions u ¯ exact solution u(x, t) as Δx, Δt → 0, the interaction measure D(Ω) and ¯ ¯ , defined in (2.1), cancellation measure C(Ω) for the approximate solution u over a region Ω converge to limiting measure, D(Ω) and C(Ω) for the exact solution u, for sequences Δxk , Δtk → 0 as k → ∞. In the remaining analysis of this chapter, for simplicity of presentation, we will not distinguish between approximate and exact solutions. The analysis in the last two sections gives an indication of how this approach can be rigorously justified. Consider the initial value problem with initial data having small total variation, TV ≡ Var(u0 (x)) 1,
ut + f (u)x = 0, (7.1) u(x, 0) = u0 (x), −∞ < x < ∞. It is clear that the end states of the solution are determined by those of the initial data, u(−∞, t) = u0 (−∞) ≡ ul ,
u(∞, t) = u0 (∞) ≡ ur ,
t > 0.
The Riemann problem (ul , ur ) pertaining to the end states at x = ±∞ is solved by i-waves (ui−1 , ui ), i = 1, . . . , n, with u0 = ul and un = ur . Divide the (x, t) space into regions ˜ 1 t}, Ω ˜ n−1 t}, ˜ n ≡ {(x, t) : x > λ ˜ 1 ≡ {(x, t) : x < λ (7.2) Ω ˜ i−1 t < x < λ ˜ i t}, i = 2, . . . , n − 1, ˜ i ≡ {(x, t) : λ Ω ˜i < ˜ i , i = 1, . . . , n − 1, are constants chosen such that λi (u) < λ where λ λi+1 (u) for all states u under consideration. The elementary waves are classified into three groups: (7.3) (ui−1 , ui ), a shock wave for i ∈ I, (ui−1 , ui ), a rarefaction wave for i ∈ II , (ui−1 , ui ), a contact discontinuity for i ∈ III . Theorem 7.1. The solution u(x, t) of the general initial value problem (7.1) approaches, time-asymptotically, the solution of the Riemann problem (ul , ur ) in the following sense, referring to (7.3): (1) For each i ∈ I, there exists a shock (u− (t), u+ (t)) along a curve ˜ i for t sufficiently large such that (u− (t), u+ (t)) approaches x = xi (t) in Ω ˜ i and for x < xi (t) (or x > xi (t)), (ui−1 , ui ) as t → ∞. Inside the region Ω the solution u(x, t) approaches the state u− (or u+ ) as t → ∞. (2) For each i ∈II , the solution approaches the rarefaction wave (ui−1 , ui ) ˜ i as t → ∞. in the region Ω
7. Large-Time Behavior
251
(3) For each i ∈ III , the solution approaches a linear traveling wave with speed λi (ui−1 ) = λi (ui ) connecting the state ui−1 to the left and the state ˜ i as ui to the right and taking values on Ri (ui−1 ) = Ri (ui ) in the region Ω t → ∞. Proof. The total amount of wave interaction and cancellation after time T goes to zero as T → ∞, by (2.12). Therefore, for any given small positive constant ε, there exists T such that the amount of interaction and cancellation after time t = T is less than ε: D(T ) + C(T ) ≤ ε.
(7.4)
The solution u(x, T ) is of bounded variation in x and so there exists a positive constant M such that the variation of u(x, T ) is mostly contained in the interval (−M, M ): Var{u(x, T ), |x| ≥ M } ≤ ε.
(7.5)
Γ− 1
Γ− i+1
Γ+ i
Γ− i
Γ+ i+1
I(t)
Γ+ 1
Γ− n Ωi
¯i Ω
Ω1
Γ+ n
Ωi+1
Ωn
T0 ¯n Ω
¯0 Ω
t x
(−M, T )
(M, T )
T
Figure 9.08. Forward generalized characteristics.
Through (±M, T ) draw generalized i-characteristics Γ± i , i = 1, . . . , n. ± and Γ , j = k, intersect in finite time, before time T0 = T + The curves Γ± j k O(1)M ; see figure 9.08. Denote by Ωi , i = 1, . . . , n, be the region between + − ¯ ¯ Γ− i and Γi and after time T0 . Let Ω0 (or Ωn ) the region to the left of Γ1 (or + − + ¯ i the region between Γ and Γ , i = 1, . . . , n − 1; to the right of Γn ) and Ω i i+1 see figure 9.08. The j-waves in the region Ωi , for j > i (or j < i), come from the waves in u(x, T ), x < −M (or x > M ), plus those due to interaction and cancellation, and therefore by (7.4) and (7.5) their total strength is of ¯ i , i = 0, 1, . . . , n, have total strength of the order of ε. Similarly, waves in Ω
252
9. Well-Posedness Theory
¯ i is close to a constant state u ¯ i: the order of ε. Thus the solution in Ω ⎧ ¯ ⎪ ⎨ul for (x, t) ∈ Ω0 , ¯ i , i = 1, . . . , n − 1, (7.6) u(x, t) = O(1)ε + u ¯ i for (x, t) ∈ Ω ⎪ ⎩ ¯ n. ur for (x, t) ∈ Ω To show that the solution approaches the Riemann solution, we need to ¯ i = ui for i = 1, . . . , n − 1 and that the waves in show that we may take u Ωi are close to the elementary waves (ui−1 , ui ), i = 1, . . . , n. The Riemann problem has a unique solution in the class of elementary waves, by Theorem 4.9 in Chapter 7. Thus it suffices to show that the solution in Ωi is close to an i-elementary wave. In Ωi , the total amount of j-waves, j = i, is of the order of ε and so it is small and the analysis for scalar laws can be generalized here. Consider first the situation where the ith characteristic field is genuinely nonlinear and we have either case I or case II in the theorem. Consider first ui ) − λi (¯ ui−1 ) < −ε1/3 , where the compressive i-waves dominate the case λi (¯ in Ωi . Fix a time t > T0 , and denote by X+ (t) (or X− (t)) the amount of ith expansion (or compression) waves in Ωi at time t. Draw backward i-characteristics C ± through points at time t on Γ± i to lie inside Ωi . The analysis for the proof of Theorem 5.5 in Chapter 3 on the convergence to shock waves can now be generalized as follows: Let the speed of C ± at time t be λ± ≡ λi (u± ). Then ¯ = X+ (t) + X− (t) + O(1)ε, t > T0 , λ+ − λ− = X+ (t) + X− (t) + O(1)X(t) − ¯ where X(t) is the amount of k-waves, k = i, between Γ− i and Γi at time t. ¯ Here we have noted that X(t) = O(1)(D(T0 ) + C(T0 )) = O(1)ε for t > T0 by (7.4). The speed of the backward characteristics C ± changes due to the k-waves, k = i, crossing, which is also of the amount of O(1)ε. Thus the + width I(t) between Γ− i and Γi at time t is I(t) = (t − T0 ) X+ (t) + X− (t) + O(1)ε + O(1)M = (t − T0 ) λ+ − λ− + O(1)ε + O(1)M, t > T0 .
The rate of change I (t) is given by the Rankine-Hugoniot condition for the the generalized characteristics Γ± i , which, together with the above estimate, leads to an estimate similar to (5.9) in Chapter 3: I (t) ≤ β
I(t) − O(M ) − O(1)ε + (1 − β)(λi (¯ ui ) − λi (¯ ui−1 )) + O(1)ε, t − T0
for some constant β close to 1/2. Similar to the scalar case, this differui ) − ential inequality for I(t) can be integrated to yield that, because λi (¯ 1/3 ui−1 ) < −ε and ε is small, the distance I(t) becomes zero for some λi (¯
7. Large-Time Behavior
253
larger t. Thus the solution in Ωi becomes a single shock wave with strength greater than ε1/3 plus smaller waves of the order of ε after some time t > T0 . In fact, it can be shown that the solution is dominated by a single shock already at time T0 . First notice that all the expansion waves in Ωi are cancelled due to the eventual coalescence of the generalized characteristics. Thus the amount X+ (T0 ) of expansion waves at time T0 is less than C(Ωi ) = O(1)ε. So the compression waves dominate at the initial time T0 . Moreover, if there are two groups of compression waves, each with strength greater than ε1/3 , then their combining would yield an interaction measure greater than ε2/3 ε, contradicting (7.4). Consequently, there can be only a single shock of strength greater than ε1/3 , with the other waves being of smaller total strength, less than ε2/3 . Thus the solution in Ωi is dominated by a single shock for time t > T0 . ui ) − λi (¯ ui−1 ) > −ε1/3 , where either the exNext consider the case λi (¯ pansion i-waves dominate or the amount of compressive waves is small. The ui )−λi (¯ ui−1 )+ two generalized characteristics diverge at a rate of at least λi (¯ O(1)ε due to the possible shocks on them. When they weakly converge, a weak shock of strength less than O(1)ε1/3 is formed and the solution in Ωi is ¯ i−1 = u ¯ i +O(1)ε1/3 . In the general a small perturbation of a constant state, u situation, the two characteristics do not meet. We have from the estimate (6.5) on the expansion of rarefaction waves that the distance between the ui ) − λi (¯ ui−1 ))(t − T0 )(1 + two characteristics at time t > T0 is at least (λi (¯ O(1)ε) + O(1)ε. Through a point (x, t) ∈ Ωi , draw a backward ith characteristic; then, by a similar argument to that for the proof of Theorem 5.6 in Chapter 3, it can be shown that the solution in Ωi tends to a i-rarefaction wave. Notice that unlike the situation considered in Theorem 5.6 in Chapter 3, the initial data considered here, (7.1), may approach its end states ul and ur at a slow rate. Thus no rate is expected for the convergence to i-rarefaction waves for the general situation considered here. Finally, consider the case where the ith characteristic field is linearly degenerate, case (3) in the theorem. As above, the total amount of j-waves in Ωi , for j = i, is of the order of ε. For a linearly degenerate field, the wave (u− , u+ ) is a contact discontinuity and takes values along the Ri curve and propagates with speed λi (u− ) = λi (u+ ); see Proposition 4.7 in Chapter 7. In the case of no j-waves, j = i, in Ωi , there are only i-waves in Ωi with speed λi (¯ ui−1 ) = λi (¯ ui ), and Ri (¯ ui−1 ) = Ri (¯ ui ), according to (7.6). These i-waves therefore form a linear traveling wave in Ωi . In the present situation, there is a perturbation by the amount ε of j-waves, j = i, in Ωi and so Ri (¯ ui−1 ) = Ri (¯ ui ) + O(1)ε. From the above, the solution u(x, t) is close to the solution of the Riemann problem (ul , ur ) in each domain Ωi , i = 1, . . . , n, in the sense stated
254
9. Well-Posedness Theory
in the theorem. Note that the region Ωi , i = 1, . . . , n shown in figure 9.08, ˜ i of (7.1) after some large time. This completes the proof of is included in Ω the theorem. In Theorem 7.1, the rarefaction waves can spread out in any way, depending on the initial data. The linear waves for linearly degenerate fields can take any shape, the same situation as for the linear equation with constant coefficients, ut + cux = 0. In the case where the initial data takes the limiting values ul and ur within a bounded interval, the rates of convergence to the Riemann wave pattern can be obtained. We will study an interesting case of this in Section 9.
8. Regularity The usual visualization of a weak solution with shock waves is as a piecewise continuous function with jump discontinuities, and these discontinuities satisfy the Rankine-Hugoniot condition and the entropy condition. In the example of interaction of a shock wave and a rarefaction wave of the same family, illustrated in figure 8.04 of Chapter 8, infinitely many shock waves result from such a simple wave interaction. Thus the visualization of a piecewise continuous solution needs to be interpreted in a more general sense. This section studies the regularity of weak solutions obtained through the Glimm solution algorithm, and makes definite this more general sense. For simplicity, we assume in the following theorem that each characteristic field is genuinely nonlinear, so that the nonlinear regularization effect can be stated in a simple way. The theorem has a natural generalization to the case where each characteristic field can be neither genuinely nonlinear nor linearly degenerate, as in Section 6 of Chapter 7, when we consider the Riemann problem. Theorem 8.1. Let u(x, t) be a weak solution of the system of hyperbolic conservation laws with small total variation that is obtained by the Glimm solution algorithm. Assume that each characteristic field is genuinely nonlinear. Then the solution has the following properties: (1) There exist countably many Lipschitz curves of shock waves. Except for at most countably many interaction points on a shock curve, across the shock curve the solution has a jump discontinuity, which satisfies the Rankine-Hugoniot condition and the entropy condition. Moreover, for suffi¯ (x, t; Δx, Δx) ciently small grid sizes Δx and Δt, the approximate solution u is dominated by a shock wave in a small neighborhood around the shock curve in the exact solution. (2) There exist countably many points of interaction. On each point of interaction (x0 , t0 ), the solution is approximated by the solution of the
8. Regularity
255
Riemann problem (ul , ur ), with ul ≡ u(x0 − 0, t0 ) and ur ≡ u(x0 + 0, t0 ), for (x, t) near (x0 , t0 ) and t > t0 ; for t < t0 , the solution is close to a combination of compression waves. √ The strength |ul − ur | of the Riemann data is at least of the order of δ, where δ is the strength of the combined interaction and cancellation measure at (x0 , t0 ). (3) The solution u(x, t) is continuous outside the countably many shock curves and the countably many points of interaction. The variation of the approximate solutions around a continuity point for the exact solution is small, uniformly as the mesh sizes tend to zero. The variation around a given continuity point depends only on the strength of the combined interaction and cancellation measure around the point. Proof. The local behavior of the solution u(x, t) around a given location (x, t) is analyzed by the following two considerations: The first consideration concerns the solution u(x, t). For a fixed time t0 , the solution u(x, t0 ) is a function of bounded variation in x. For a given location x0 , the solution u(x, t0 ) is either continuous at x = x0 or there is a jump, u− = u+ , u± ≡ u(x0 ±0, t0 ). The second consideration pertains to the measure D(Ω)+C(Ω). The measure D(Ω) + C(Ω) is bounded and consists of a continuous measure and countably many point measures. The measure D(Ω) + C(Ω), (x0 , t0 ) ∈ Ω, either has a point measure at the given location (x0 , t0 ) or has no point measure there. We extend the analysis of large-time behavior in the preceding section to the present setting. There are the following three situations. (1) Continuity points Suppose that the solution u(x, t0 ) is continuous at x = x0 and that there is no point measure of D(Ω) + C(Ω) at (x0 , t0 ). Thus, given a small constant ε, there exists an interval Γ0 around (x0 , t0 ) with the property that the total variation of the solution u(x, t0 ) over Γ0 is less than ε. Moreover, there exists a neighborhood Ω0 of (x0 , t0 ) such that D(Ω0 ) + C(Ω0 ) < ε. Now we take Γ0 to be an interval centered at (x0 , t0 ) and let Ω0 be a parallelogram with center (x0 , t0 ), Γ0 as its horizontal axis, and sides with slopes ±M , where M is a constant larger than |λi (u)|, i = 1, . . . , n; see figure 9.09. Take a horizontal line Γ1 (or Γ2 ) in Ω0 at an earlier time t < t0 (or a later time t > t0 ), and let Ω1 (or Ω2 ) be the region inside Ω0 and between Γ1 and Γ0 (or between Γ0 and Γ2 ). Denote by TVi the variation of the solution u over the interval Γi , i = 0, 1, 2. We have TV0 < ε. The choice of the slope ±M of the boundary of Ω implies that the domain of dependence of the solution over Γ2 is included in Γ0 . Thus we have from the analysis of wave interactions, e.g. estimate (2.10), that TV2 ≤ TV0 +O(1)D(Ω2 ). In the
256
9. Well-Posedness Theory
Γ2 Ω0
Ω2 (x0 , t0 ) ¯0 Γ
Γ0 Ω1
C−
C+ Γ1
t x Figure 9.09. Domain of influence and dependence.
definition (2.3) of potential wave interaction, we have Q(Γ0 ) = O(1)(TV0 )2 and so D(Ω2 ) ≤ (TV0 )2 . Therefore, for ε small, the variation of the solution over Γ2 is small, TV2 ≤ 2ε. The variation TV0 over Γ0 may be smaller than the variation TV1 over the earlier time interval Γ1 because of the cancellation C(Ω1 ) over the region Ω1 between the two intervals, and so we can only conclude that TV1 ≤ TV0 + C(Ω1 ) + O(1)D(Ω1 ). Since the cancellation and interaction measures over the bigger region Ω0 are both less than ε, we have TV1 ≤ O(1)ε. In conclusion, the variation of the solution over any horizontal interval in Ω is small, of the order of ε. Take any time-like line Γ in Ω0 . Waves crossing Γ are among those crossing Γ0 , with modifications due to cancellations and interactions in the region Ω0 . Thus the variation of the solution over Γ is also small, of the order of ε. In summary, |u(x, t) − u(x0 , t0 )| = O(1)ε, (x, t) ∈ Ω0 . Thus the solution u as a function of (x, t) is continuous at (x0 , t0 ). In fact, we now show that the variation depends mainly on the interaction measure ε by the following consideration: Choose a small sub-interval ¯ 0 | = δ0 |Γ0 | = (δ0 )2 δ0 for a ¯ 0 of Γ0 around (x0 , t0 ), say |Γ0 | = δ0 and |Γ Γ small δ0 . Draw backward ith characteristics C± through the end points of ¯ 0 . The time span of these backward characteristics in Ω0 is of the order Γ ¯ 0 | = (δ0 )2 . The estimate (6.5) of |Γ0 | = δ0 , which is large compared to |Γ
8. Regularity
257
for the expansion waves shows that the amount of expansion waves on Γ0 is ¯ 0 |/|Γ0 | + D(Ω)) = O(1)(δ0 + ε). This implies that if the bounded by O(1)(|Γ solution is continuous at (x0 , t0 ), then the variation of u(·, t0 ) is small over a small interval around x = x0 . (2) Shock curves Γ− j Γ+ j
Ω0
Γ0
Ωj (x0 , t0 ) ¯0 Γ
Γ0 Ω0
t x Figure 9.10. Generalized characteristic of ones family.
Suppose that the solution u(x, t0 ) has a jump at x = x0 , u± ≡ u(x0 ± 0, t0 ), u− = u+ , and that D(Ω) + C(Ω) has no point measure at (x0 , t0 ). Let γ ≡ |u− − u+ | be the size of the jump. Choose a region Ω0 around ¯ 0 be a sub-interval of Γ0 , as in figure 9.08. By the an interval Γ0 , and let Γ hypotheses, these can be chosen to have the following properties: (1) D(Ω0 ) + C(Ω0 ) < ε. ¯ 0 is less than (2) The total variation of the solution u(x, t0 ) over Γ0 − Γ ¯ 0 is of the order of γ. ε, and its variation over Γ ¯ 0 is much (3) The length δ0 of Γ0 is small and the length (δ0 )2 of Γ smaller. ¯ 0 draw two jth generalized Through the end points of the interval Γ ± characteristics Γj and let Ωj be the region inside Ω0 and between Γ− j and + Γj , j = 1, . . . , n, as shown in figure 9.10. The amount of j-waves in the region Ωj varies with time due to the cancellations and interactions. Thus this amount is constant in time, except
258
9. Well-Posedness Theory
for a small variation of the order of ε. We claim that there exists k, 1 ≤ k ≤ n, with the following properties: √ (i) The amount of j-waves inside Ωj , j = k, is less than ε. (ii) The amount of k-waves inside Ωk is γ + O(1)ε. (iii) Inside the region Ωk , the solution is dominated a single k-shock wave of strength γ + O(1)ε. Statement (i) is proved by contradiction; if there exist j1 = j2 such that √ the amount of j-waves inside Ωj is larger than ε for both j = j1 and j = j2 , then these waves would interact in Ω0 and contribute to D(Ω0 ) of an amount √ exceeding ( ε)2 = ε, in violation of the above property (1). Property (ii) follows from (i) and the above property (2). From (i) and (ii), it follows from Theorem 3.3 in Chapter 7 that u+ ∈ Rk (u− ) + O(1)γ 3 + O(1)ε. We use this to verify the property (iii). The analysis is similar to that in the proof of Theorem 7.1 in generalizing the analysis in Chapter 3 for scalar equations. There are two cases to consider; the k-waves in Ωk are either compressive, λk (u+ ) < λk (u− ), or expansive, λk (u+ ) > λk (u− ). In the region Ωk , k-waves dominate. For the compressive case, apply the arguments for Theorem 5.5 in Chapter 3 to show that a k-shock of strength γ + O(1)γ 3 + O(1)ε forms in Ωk . Note that the region Ωk is inside Ω0 at time t0 − Cδ0 for some positive constant C. The width of Ωk at time t0 is (δ0 )2 , the variation of the speed of Γ± k is at most of the order of ε, and therefore the width of Ωk at time t = t0 − Cδ0 is at most M ≡ (δ0 )2 + O(1)δ0 ε. From the second estimate in (5.7) of Theorem 5.5 in Chapter 3, the formation of a relatively strong k-shock in Ωk is before time t1 = t0 − Cδ0 + O(1)
M (δ0 )2 + O(1)δ0 ε = t − Cδ + O(1) , 0 0 1 γβ γ2
where we have noted that the constant β in the estimate (5.7) is 1/2+O(1)γ for a weak shock. For the given jump γ, we choose the region Ω0 sufficiently small so that the width δ0 of Γ0 and the perturbation ε are less than γ. Then the above estimate yields t1 < t0 − Cδ0 /2. Thus the relatively strong shock forms before time t0 − Cδ0 /2. The k-shock therefore dominates the solution around (x0 , t0 ). Its speed changes due to wave crossing and interaction, which contribute small amounts, and so the relatively strong k-shock is Lipschitz continuous in a small neighborhood of (x0 , t0 ). For the approximate solution there is a correction term, such as in Lemma 6.1, and we have 1 1 (δ0 )2 + O(1) δ0 ε + e((Δt)− 2 , a) + Δt 2 Cδ0 , t1 = t0 − Cδ0 + O(1) 1 γ2 and so t1 < t0 − Cδ0 /2 and the shock exists in the approximate solution for a sufficiently small mesh size Δt.
8. Regularity
259
For the expansive case, we have from Theorem 6.5 that the k-waves expand in Ωk and so this is in contradiction to having a discontinuity of strength γ at time t0 . This completes the description of the solution in this case. (3) Points of interaction Suppose that the combined interaction and cancellation measure D(Ω)+ C(Ω) has a point measure of strength γ > 0 at (x0 , t0 ). Let the limiting values of the solution u(x, t0 ) at x = x0 be ul ≡ u(x0 − 0, t0 ) and ur ≡ u(x0 + 0, t0 ). For now we allow the possibility of no jump, ul = ur , though we will show that there is a jump: ul = ur at a non-zero interaction and cancellation measure point.
(8.1)
Choose a small positive constant ε. Around the point (x0 , t0 ) draw an interval Γ (or Γ0 ) of length δ0 (or (δ0 )2 ) such that the variation of the solution u(x, t0 ), x ∈ Γ − Γ0 , is less than ε and its variation over Γ0 is of the order of |ul − ur )| + O(1)ε. Γ− i+1 Γ− i
Γ+ i+1
Γ+ i
Γ− n
Ωi+1
Γ+ n
Ω+ i+1
Ω+ i
Γ − Γ0
Γ − Γ0
(x0, t0) Ω0
Ω0
Ω− i+1
Ωn
Ω− i
t x Figure 9.11. Generalized characteristics of n families.
Draw a parallelogram Ω (or Ω0 ) with axis Γ (or Γ0 ), as in figure 9.11. Through the end points of Γ0 draw generalized characteristics Γ± i , i = − (or Ω ) the regions inside Ω after (or before) the 1, . . . , n, and denote by Ω+ i i generalized characteristics intersect Γ0 ; see figure 9.11. The region between + i+1 , and Ω0 (or Ωn ) is the region inside Ω that Ω+ i and Ωi+1 is denoted by Ω
260
9. Well-Posedness Theory
is to the left (or right) of these generalized characteristics. For grid sizes Δx and Δt sufficiently small, the measure is concentrated inside Ω0 : D(Ω − Ω0 ) + C(Ω − Ω0 ) ≤ ε,
D(Ω0 ) + C(Ω0 ) ≥ γ − ε.
As the mesh sizes tend to zero, the size δ0 of the interval Γ0 also shrinks to zero, so that the interval shrinks to the point (x0 , t0 ). The analysis for case (1), continuity points, applies to the region Ω0 on the left and the region Ωn on the right, and we have u(x, t) = ul + O(1)ε for (x, t) ∈ Ω0 , u(x, t) = ur + O(1)ε for (x, t) ∈ Ωn . Apply the analysis for case (2), shock curves, to the regions Ω+ i to obtain that , is dominated by either an i-rarefaction wave the solution u(x, t), (x, t) ∈ Ω+ i or an i-shock (ui−1 , ui )(x, t), i = 1, . . . , n. As the mesh sizes tend to zero, the regions tend to the point (x0 , t0 ) and these states ui , i = 0, 1, . . . , n, with u0 = ul and un = ur , solve the Riemann problem (ul , ur ). In other words, the solution u(x, t) for (x, t) near (x0 , t0 ) and for t > t0 is dominated by the Riemann solution (ul , ur ) with ul ≡ u(x0 −0, t−0) and ur ≡ u(x0 +0, t−0). The total amount of j-waves in the region Ω− i , j = i, is small, as it is the same as the amount of waves crossing Γ − Γ0 plus the interaction and cancellation measure in Ω−Ω0 . Thus the analysis of waves in Ω− i is similar to that for scalar laws. Moreover, the region Ω− has small width, of the order i − 2 of (δ0 ) , near the top at t = t0 . The time span of the region Ωi is δ0 . Thus the analysis of expansion of rarefaction waves, (6.5), yields that the amount 2 of i-rarefaction waves in Ω− i near the top is of the order of (δ0 ) /δ0 = δ0 . − Thus the waves entering the region Ω0 in Ωi are mostly compressive waves, and so the cancellation measure C(Ω0 ) in Ω0 is small. The measure Dd (Ω) of wave interaction between distinct characteristic families dominates the ˜ Dd (Ω0 ) = ˜ γ. The interaction measure Dd (Ω) interaction measure D(Ω0 ) = comes from the interaction of waves at the top of Ω− i , i = 1, . . . , n, and thus there exists at least one i, 1 ≤ i ≤ n, such that the amount of compressive √ γ. The amount of i-waves at the top of Ω− i is at least of the order of i-waves at time t0 may change due to interaction by an amount of the order √ of γ, and so it is also at least of the order of γ. We have thus shown that there is a jump in the solution u(x, t0 ) at x = x0 , at least of the order of √ √ γ, i.e. |ur − ul | is nonzero and at least of the order of γ. This proves the claim (8.1). This completes the discussion of the regularity of the solution.
9. Decay and N -Waves Section 7 was on the qualitative analysis of large-time behavior of solutions. This section presents a quantitative analysis of the time-asymptotic shape of
9. Decay and N -Waves
261
the solution when the initial function is a compactly supported perturbation of a constant state u0 : (9.1) ut + f (u)x = 0, u(x, 0) = u0 for |x| > M, TV ≡ Var(u(·, 0)) small. Assume that the characteristic fields are genuinely nonlinear. The simple waves for a genuinely nonlinear field satisfy the inviscid Burgers equation, (2.7) in Chapter 7. Time-asymptotically, waves associated with a genuinely nonlinear field would follow the inviscid Burgers equation. For convex scalar laws, a solution with spatial compact support tends to an N -wave with two time invariants p and q; see Section 6 of Chapter 3. The convergence rate is t−1/2 in L1 (x). We now show that there is an N -wave as time-asymptotic state for each characteristic field. For each i-field, there are two time invariants pi and qi . However, with the coupling of distinct characteristic fields, the convergence for the system is at the slower rate of t−1/4 in L1 (x). As in the previous sections, we will work with the Glimm approximation solutions with initial values of small total variation TV. ¯− Γ i Γ− i
Γ− 1
¯+ Γ i
Γ+ i
Γ− n
Γ+ 1
Γ+ n
Ω1 −M (t1 )
M (t1 )
t1
t x
−M (t0 )
M (t0 )
t0
Figure 9.12. Decoupling of waves, I.
First we explain a key element in the quantitative analysis, the thirdorder coupling of waves. From the hyperbolicity, the support of u(x, t) is contained in a finite interval: u(x, t) = u0 for |x| > C1 t + M ≡ M (t), for some positive constant C1 > |λi (u)|, i = 1, . . . , n, for all u under consideration. Given a positive time t0 ≥ M , draw generalized i-characteristics + Γ− i (or Γi ), i = 1, 2, . . . , n, through x = −M (t0 ) (or x = M (t0 )). By the strict hyperbolicity, [λ] ≡ min{|λi (u1 ) − λj (u2 )|, i = j, for all u1 and u2 under consideration}
262
9. Well-Posedness Theory
is positive and bounded away from zero. These generalized characteristic curves from distinct families intersect before time t1 , as shown in figure 9.12, and for t0 ≥ M we have t1 ≤ t0 +
M (t0 ) C1 t 0 + M ≤ t0 + ≤ C2 t 0 [λ] [λ]
for a positive constant C2 . ¯− Γ i
Γ− i+1
¯+ Γ i Di (t)
¯+ Γ 1
Ωi
¯− Γ 1
¯− Γ n
Ωi
¯+ Γ n t2
t −M (t1 )
x
t1 M (t1 )
Figure 9.13. Decoupling of waves, II.
¯ − (or Γ ¯ + ), Repeat this process by drawing generalized i-characteristics Γ i i i = 1, 2, . . . , n, through x = −M (t1 ) (or x = M (t1 )). These generalized characteristics associated with distinct families intersect before time t2 ; see ¯ − and Γ ¯ + , and figure 9.12 and figure 9.13. Let Ωi be the region between Γ i i + − i ¯ and Γ ¯ , as depicted in figure 9.13. Similar let Ω be the region between Γ i i+1 to the estimate for t1 above, the time t2 is of the same order as t0 : M < t0 < t2 ≤ Ct0 , C ≡ (C2 )2 .
(9.2) For t > t2 , set
i (t) ≡ amount of ith shock waves at time t; (9.3) X− i (t) ≡ amount of ith rarefaction waves at time t; X+ i i (t) + X+ (t) = amount of ith waves at time t; X i (t) ≡ X−
X(t) ≡
n
Xi (t) = amount of waves at time t;
i=1
˜ i (t) ≡ amount of i-waves outside Ωi at time t. X
9. Decay and N -Waves
263
Proposition 9.1. The coupling of the waves after time t2 is of third order in terms of the wave strength at time t0 ; that is, for t > t2 , ˜ i (t) = O(1)(X(t0 ))3 , i = 1, . . . , n; X 1 i X± (t) = X i (t) + O(1)(X(t0 ))3 , i = 1, . . . , n − 1; 2
(9.4) D(t) = O(1)(X(t0 ))3 ,
(9.5)
u(x, t) = u0 + O(1)(X(t0 ))3 for (x, t) ∈ Ωi , i = 1, . . . , n − 1.
Proof. Let D(t) = Dd (t) + Ds (t) be the amount of interaction after time t. We know that Dd (t) = O(1)(X(t))2 and Ds (t) = O(1)(X(t))3 from (2.3), and so X(t1 ) = X(t0 ) + O(1)D(t0 ) = X(t0 ) + O(1)(X(t0 ))2 . The i-waves in Ωi , j = i, after time t1 come from the interaction of waves after time t0 and is of the amount O(1)(X(t))2 . This implies that the measure Dd (t1 ) of wave interaction between waves of distinct characteristic fields is of the order of Dd (t1 ) = O(1)X(t1 )(X(t0 ))2 = O(1) X(t0 ) + O(1)(X(t0 ))2 (X(t0 ))2 = O(1)(X(t0 ))3 . This establishes the first estimate in (9.4). Note that this third-order separation of waves is achieved by noting that the wave interaction Ds for the same characteristic family is of third order and that invoking strict hyperbolicity twice minimizes the effect of the wave interaction Qd for distinct characteristic families. Outside of Ωi , the i-waves at time t > t2 are those produced by interaction after time t1 , and so the second estimate in (9.4) follows from the first. There is third-order contact of Hugoniot and characteristic curves, according to Theorem 3.3 of Chapter 7, and so the first and second estimates in (9.1) imply that u(x1 , t) ∈ Ri (u(x0 , t)) + O(1)(X(t0 ))3 for any (x0 , t) ∈ Ωi−1 , (x1 , t) ∈ Ωi , t > t2 . As the end states at x = ±∞ are both equal to the base state u0 , the above implies that the solution is close to the base state between the generalized characteristics of distinct families; in other words, the estimate (9.5) holds. Finally, the third estimate in (9.4) follows from (9.5) and the first estimate in (9.4).
264
9. Well-Posedness Theory
Recall from (7.2) that the (x, t) space is divided into regions ˜ 1 t}, ˜ 1 ≡ {(x, t) : x < λ Ω
˜ n−1 t}, ˜ n ≡ {(x, t) : x > λ Ω ˜ i−1 x < λ ˜ i t}, i = 2, . . . , n − 1, ˜ i ≡ {(x, t) : λ Ω
˜ i , i = 1, . . . , n − 1, chosen such that with the constants λ ˜ i < λi+1 (u) for all states u under consideration. λi (u) < λ Theorem 9.2. Consider the solution of the system of conservation laws with small initial total variation TV constructed by the Glimm scheme. Then, as t → ∞, (9.6)
X(t) = O(1)TV(t + 1)− 2 , 1
D(t) = O(1)(TV)3 (t + 1)− 2 . 3
Proof. From the first estimate in (9.4), D(t2 ) = O(1)(X(t0 ))3 ; and from (9.2), t2 ≤ Ct0 . Thus, assuming that the first estimate in (9.6) holds, we have 1 3 D(t2 ) = O(1)(X(t0 ))3 = O(1) TV(t0 + 1)− 2 t − 1 3 3 2 + 1 2 = O(1)(TV)3 (t2 + 1)− 2 . = O(1) TV C In other words, the first estimate in (9.6) implies the second estimate. With the third-order wave coupling estimate in Proposition 9.1, we can generalize the method for scalar equations, presented in Section 6 of Chapter 3, to obtain the decay of the solutions for systems as follows. By (9.4), for t ≥ t2 , the estimate (6.5) for the expansion waves becomes ¯i ¯ i (t) ≤ D (t) 1 + O(1)(X(t0 ))3 + O(1)(X(t0 ))3 , t > t2 , (9.7) X + t − t2 i ¯ − and Γ ¯ + at time t > t2 , figure 9.13, ¯ D (t) ≡ distance between Γ i i i ¯+ ¯ − and Γ ¯ +. X (t) ≡ amount of i-rarefaction waves between Γ i
i
¯ The distance D(t) is governed by the Rankine-Hugoniot condition. From Theorem 3.3 of Chapter 7, the speed of Cl (or Cr ) is approximated by the arithmetic mean of the characteristic speeds λl (or λr ) and λ0 = λi (0) with perturbations of the order of O(1)(X(t0 ))3 of both sides of the shock. Thus we have λr − λl d ¯i D (t) = O(1)(X(t0 ))3 + + O(1)[(λr − λ0 )2 − (λl − λ0 )2 ] dt 2 1 = ( + O(1)X(t0 ))(λr − λl ) + O(1)(X(t0 ))3 . 2
9. Decay and N -Waves
265
We have from the third estimate in (9.4) that ¯ i (t) + O(1)(X(t0 ))3 . ¯ i (t) = X ¯ i (t) + (X(t0 ))3 = 1 X λr − λl ≤ X − + 2 From (9.7) and the above estimates we obtain the estimate for the distance ¯ i (t) between Γ ¯ − and Γ ¯ +: D i i (9.8)
¯ i (t) 1 D d ¯i D (t) ≤ + O(1)X(t0 ) + O(1)(X(t0 ))3 , dt 2 t − t2 i 1 1 d ¯i ¯ (t) + O(1)(X(t0 ))3 , t > t2 . D (t) ≤ + O(1)X(t0 ) X dt 2 2
One then uses the estimates (9.7) and (9.8) to finish the proof by induction: X i (t) ≤ Hn t− 2 for t < Tn ≡ 2n T0 , n = 0, 1, 2, . . . . 1
(9.9)
To start the induction, set T0 = ε− 5 M, 11
(9.10)
H0 ≡ ε− 10 M 2 , 1
1
ε ≡ 2TV
so that γ0 ≡ H0 (T0 )− 2 = ε and (9.9) holds for n = 0. Assume that (9.9) holds for n ≤ p. We now apply the set-up above with 1
2
t0 = (Hp Tp ) 5 ,
γp ≡ Hp (t0 )− 2 , 1
t2 = Ct0 ,
where C is uniformly bounded and greater than 1; see figure 9.12 and figure 9.13. From the induction hypothesis, X i (t0 ) ≤ γp . ¯ i in (9.8) from t = t0 to t = Tp , Integrate the first differential equation for D 1 making use of the induction hypothesis X i (t) ≤ Hp t− 2 for t < Tp : ¯ i (Tp ) ≤ 1 (1+O(1)γp )Hp ((Tp ) 12 −(Ct0 ) 21 )+O(1)(γp )3 (Tp −Ct0 )+ D ¯ i (Ct0 ). D 2 Next apply the second estimate (9.8) for t ∈ (Tp , Tp+1 ): ¯ i (t) ≤ D
t − Ct0 Tp − Ct0
1 +O(1)γp 2
¯ i (Tp ) + O(1)(γp )3 (t − Ct0 ). D
¯ i (Ct0 ) = O(1)t0 , we have from the above two estimates that Noting that D ¯i
D (t) ≤
1 +O(1)γp
1 1 1 (1 + O(1)γp )Hp ((Tp ) 2 − (Ct0 ) 2 ) 2 3 + O(1)(γp ) (Tp − Ct0 ) + O(1)t0 + O(1)(γp )3 (t − Ct0 ).
t − Ct0 Tp − Ct0
2
266
9. Well-Posedness Theory
With this, we conclude from (9.4) and (9.7) that ¯i
(9.11) X (t) = X (t) + O(1)(X (t0 )) ≤ i
i
3
t − Ct0 Tp − Ct0
O(1)γp
(t − Ct0 )− 2 1
(Tp ) 2 (Ct0 ) 2 t0 · (1 + O(1)γp )Hp + O(1)(γp )3 . 1 + O(1) 1 (Tp − Ct0 ) 2 (Tp − Ct0 ) 2 1
1
The induction is complete if Hp+1 is found to satisfy 1 1 t − Ct O(1)γp (Tp ) 2 (Ct0 ) 2 0 − 21 (t − Ct0 ) (1 + O(1)γp )Hp 1 Tp − Ct0 (Tp − Ct0 ) 2 1 t0 + O(1) + O(1)(γp )3 ≤ Hp+1 t− 2 for t ∈ (Tp , Tp+1 ). 1 (Tp − Ct0 ) 2
This is so if (9.12) Hp+1
1 1 2T − Ct O(1)γp 1 (Tp ) 2 (Ct0 ) 2 Tp 2 p 0 ≥ (1 + O(1)γp )Hp 1 Tp − Ct0 Tp − Ct0 (Tp − Ct0 ) 2 t0 + O(1) + O(1)(γp )3 . 1 2 (Tp − Ct0 )
We assert that this can be satisfied with a choice of Hp+1 having the property (9.13) H0 = ε− 10 M ≤ Hp ≤ Hp+1 1
p
≤ Hp 3O(1)γp (1 + O(1)γp ) + O(1)2− 10 ε 5 ≤ 2H0 . 1
This is the consequence of the following simple estimates. From the definitions T0 ≡ ε− 5 M, H0 = ε− 10 M 2 , Tp ≡ 2p T0 , t0 ≡ (Hp Tp ) 5 , γp = Hp (t0 )− 2 11
we deduce that
1
1
2
1
1 Hp 4 1 γp 5 = 2− 5 ≤ 2− 10 γp−1 Hp−1
where the last inequality comes from the estimate in (9.13) that we have assumed. Thus p p γp ≤ 2− 10 γ0 = ε2− 10 , which is small and decays exponentially in p. With this it is easy to see that (9.13) should hold if the terms not related to γp on the right-hand side of (9.12) can be satisfied for small ε. The basic estimate for this is the ratio 3p 3 2 3 2 2 32 t0 = (Tp )− 5 (Hp ) 5 ≤ (Tp )− 5 (2H0 ) 5 = 2− 5 M − 5 ε 25 , Tp
9. Decay and N -Waves
267
which is small and tends to zero exponentially in p. For instance, it implies that the first ratio on the right-hand side of (9.12) satisfies 2Tp − Ct0 ≤ 3. Tp − Ct0 The other terms on the right-hand side of (9.12) are estimated similarly. This completes the proof of the theorem. With the decay of the total variation of the solution, one can study in more detail the wave distribution and in particular the convergence to N -waves. Theorem 9.3. Suppose that each characteristic field of the system of hyperbolic conservation laws ut + f (u)x = 0 is genuinely nonlinear. Then there exist 2n time invariants pi and qi , i = 1, . . . , n, such that the solution of (9.1) converges to the linear combination of N -waves N i (x, t) = Npi ,qi (x − λi (u0 )t, t)ri (u0 ), i = 1, . . . , n. There is an explicit description of the solution, (9.22), which yields the convergence rate t−1/4 in L1 (x). 2
Proof. For a given time t, set t0 = t 5 and t2 = Ct0 < t, and follow the set¯ − : x = xl (t) up of Proposition 9.1. Consider generalized characteristics Γ i ¯ + : x = xr (t) as in figure 9.13. Consider the lines separating the and Γ i characteristics: λj (u0 ) + λj+1 (u0 ) }, j = 1, . . . , n − 1; 2 Γ0 ≡ {(x, t) : x = (λ1 (u0 ) − 1)t}, Γn ≡ {(x, t) : x = (λn (u0 ) + 1)t}.
Γj ≡ {(x, t) : x = μj t, μj ≡
After a finite time O(1)M , the support of the solution is contained in the region between Γ0 and Γn . In the region between Γi−1 and Γi , μi−1 t < x < μi t, we have from Theorem 9.2 that the amount of i-waves decays at the rate of t−1/2 , and the amount of j-waves, for j = i, decays at the rate of t−3/2 . Moreover, by Theorem 9.2 and (9.5), along Γi−1 and Γi one has u = u0 + O(1)t−3/2 . This implies that, in the weak sense, the solution is governed accurately by the Hopf equation: λt + (
λ2 )x = ν i (x, t), λ ≡ λi (u) − λ0 t, λ0 ≡ λi (u0 ). 2
The measure ν i (x, t) dx comes from two sources, the first is the amount of j-waves for j = i, which is of the order t−3/2 as just noted, and the second is due to the fact that i-shock waves do not exactly satisfy the Hopf equation; there is a third-order error (t−1/2 )3 = t−3/2 as seen in (3.12) of Section 3 in
268
9. Well-Posedness Theory
Chapter 7. Thus we have ∞ μi t t
ν i (x, s) dx ds = O(1)(TV)2 t− 2 . 1
μi−1 t
The Hopf equation has two time invariants, by Theorem 6.3 of Chapter 3, so the above approximate Hopf equation yields two time invariants timeasymptotically: x 1 (λ(x, t) − λ0 ) dx + O(1)(TV)2 t− 2 , (9.14) pi = inf x
μi−1 t
μi t
qi = max x
(λ(x, t) − λ0 ) dx + O(1)(TV)2 t− 2 , λ0 ≡ λi (u0 ). 1
x
Γi−1
Γi
x=λ0 t
x xl x
xr
t
tβ
x0
t2/5
Figure 9.14. Wave distribution.
Draw generalized i-characteristics x = xl (t) (or x = xr (t)) from the point on Γi−1 (or Γi ) at time t2/5 , as in figure 9.14. Draw a backward characteristic through a location (x, t), with xl (t) < x < xr (t), to reach the time t2/5 at x0 . The change in speed of the backward characteristic after time s due to waves crossing it is of the order of O(1)(TV)2 (s + 1)−3/2 , and so its location at time t is t 3 x = x0 + 2 λi (x, t) + O(1)(TV)2 (s + 1)− 2 ds t5
= λi (x, t)(t − t 5 ) + O(1)(TV)2 (t + 1)− 5 = x0 + λi (x, t)t + O(1)t 5 . 2
1
2
9. Decay and N -Waves
269
2
Since x0 = λ0 t + O(1)t 5 , the above implies that the solution is close to the centered i-rarefaction wave for xl (t) < x < xr (t)–(9.4) and (9.5): (9.15) λi (x, t) =
3 x − λ0 t + O(1)t− 5 , t 3 u(x, t) ∈ Ri (u0 ) + O(1)t− 2 for xl (t) < x < xr (t).
Next consider the region left of the generalized characteristic, μi−1 t < x < xl (t), and draw a backward i-characteristic to meet the boundary at time tβ for some β with 2/5 < β < 1; see figure 9.14. The change in the speed of the backward characteristic curve is due to waves crossing the curve and is of the order of (TV)3 (tβ )−3/2 . By (9.4), the curve has speed λ0 + O(1)(TV)3 t−3β/2 and is on Γi−1 at time tβ . Thus 3β x = μi−1 tβ + (t − tβ ) λ0 + O(1)(TV)3 t− 2 , or λ0 t − x = (λ0 − μi−1 )tβ + O(1)(TV)3 t1−
3β 2
.
Since β > 2/5, we have β > 1 − 3β/2. Note also that λ0 − μi−1 is positive and of order 1, and TV is small. Thus the above yields C1 tβ ≤ λ0 t − x ≤ C2 tβ for some positive constant 0 < C1 < C2 . The waves on the interval (μi−1 t, x) at time t are those produced after time tβ . These waves have total strength O(1)(TV)3 t−3β/2 and so u(x, t) = u(μi−1 t, t) + O(1)(TV)3 t−3/2 . We know from (9.5) and (9.6) that u(x, t) = u0 + O(1)(TV)3 t−3/2 on Γi−1 , where x = μi−1 t. Thus we have from the above estimate that u(x, t) = u0 + O(1)(TV)3 t−3β/2 + O(1)(TV)3 t−3/2 = u0 + O(1)(TV)3 t−3β/2 = u0 + O(1)(TV)3 |x − λ0 t|− 2 . 3
A similar estimate holds for the region xr (t) < x < μi t, and we have thus shown that (9.16) u(x, t) = u0 + O(1)(TV)3 |x − λ0 t|− 2
3
for μi−1 t < x < xl (t) and for xr (t) < x < μi t. 9.2, in Since the decay of the solution is at the rate of t−1/2 by Theorem −3/2 can be changed to |(x − λ0 t)| + the estimate (9.16) the factor |x − λ0 t|
270
t1/3
9. Well-Posedness Theory −3/2
. Thus from (9.15) and (9.16),
(9.17) λi (x, t) − λ0 ⎧ 1 − 3 3 ⎪ 2 ⎪ for μi−1 t < x < xl (t), ⎨O(1)(TV) |(x − λ0 t)| + t 3 3 x−λ t − 3 0 5 = + O(1)(TV) t for xl (t) < x < xr (t), t ⎪ ⎪ ⎩O(1)(TV)3 |(x − λ t)| + t 13 − 23 for x (t) < x < μ t. 0
r
i
We use this to relate the locations of the two generalized characteristics x = xl (t) and x = xr (t) to the time invariants pi and qi given in (9.14). From the first and third equations in (9.17), (9.18) μi t xl (t) 1 (λ (x, t) − λ ) dx + (λ (x, t) − λ ) dx = O(1)(TV)3 t− 6 . i 0 i 0 xr (t) μi−1 t Thus the integrations outside the generalized characteristics are decaying in time and do not contribute to the time invariants pi and qi . From the second equation in (9.17) the integral between the generalized characteristics are
λ0 t
(λi (x, t) − λ0 ) dx = −
xl (t) xr (t)
(λi (x, t) − λ0 ) dx =
λ0 t
2 3 1 λ0 t − xl (t) + O(1)(TV)3 t− 5 λ0 t − xl (t) , 2t
2 3 1 xr (t) − λ0 t + O(1)(TV)3 t− 5 xr (t) − λ0 t . 2t
From the decay rate of t−1/2 of the solution given in Theorem 9.2, the second equation in (9.17) implies that 1 xr (t) − λ0 t = O(1)t− 2 , t
1 λ0 t − xl (t) = O(1)t− 2 . t
The above estimates therefore yield
λ0 t
(9.19)
(λi (x, t) − λ0 ) dx = −
xl (t)
xr (t) λ0 t
2 1 1 λ0 t − xl (t) + O(1)(TV)3 t− 10 , 2t
(λi (x, t) − λ0 ) dx =
2 1 1 λ0 t − xr (t) + O(1)(TV)3 t− 10 . 2t
The estimates (9.18) and (9.19) yield x 2 1 1 λ0 t − xl (t) + O(1)(TV)3 t− 10 , λ(x, t) − λ0 dx = − inf x 2t μi−1 t μi t 2 1 1 λ0 t − xr (t) + O(1)(TV)3 t− 10 . λ(x, t) − λ0 dx = max x 2t x
9. Decay and N -Waves
271
This and (9.14) for the time invariants pi and qi give the estimates of the locations of the generalized characteristics: (9.20) λ0 t − xl (t) =
−2pi t + O(1)(TV)3 t− 10 , 1
xr (t) − λ0 t =
1 2qi t + O(1)(TV)3 t− 10 .
Consider the generic case of pi < 0 < qi . From (9.17) and (9.20), we see that two relatively strong shock waves eventually emerge on the generalized characteristics and the shock strengths are
3 −2pi + O(1)(TV)3 t− 5 , t 3 2qi λi (xr (t) − 0, t) − λi (xr (t) + 0, t) = + O(1)(TV)3 t− 5 . t
(9.21) λi (xl (t) − 0, t) − λi (xl (t) + 0, t) =
After the emergence of the two relatively strong shocks, say after time T , we may redo the above analysis as follows: Let Cl and Cr be the generalized characteristics through (μi−1 T, T ) and (μi T, T ), which eventually coincide with the two relatively strong shock curves after some finite time T1 > T . For (x, t), t > T1 , between Cl and Cr draw a backward characteristic to meet time T1 at (x0 , T1 ) between Cl and Cr . The backward characteristic has speed λi (x, t) + O(1)(TV)3 (s + 1)−3/2 , T1 < s < t, and so t λi (x, t) + O(1)(TV)3 (s + 1)−3/2 ds = λi (x, t)t + O(1). x = x0 + T1
This improves the second estimate of (9.17). For (x, t) between Γi−1 and Γi but outside of Cl and Cr , we repeat the process and conclude from the above, (9.17) and (9.20), that
(9.22)
λi (x, ⎧ t) − λ0 − 23 3 i ⎪ ⎪O(1)(TV) |λ0 t − x| for μi−1 t < x − λ0 t < − −2p , ⎪ ⎨ t −2p 2q i i 0t = x−λ + O(1)(TV)3 t−1 for − t t < x − λ0 t < t , ⎪ ⎪ ⎪ ⎩O(1)(TV)3 |x − λ t|− 23 for 2qi < x − λ t < μ t. 0 0 i t
This completes the description of the solution between Γi−1 and Γi . By direct calculations using (9.22), μi t 1 |λi (x, t) − λ0 − N (x − λ0 t, t; pi , qi )(x, t)| dx = O(1)(t + 1)− 4 . (9.23) μi−1 t
This completes the proof of the theorem.
272
9. Well-Posedness Theory
10. Some Basics of Numerical Computations There are several distinct issues concerning the numerical computation of discontinuity waves. Contact discontinuities are linear waves in the sense that they can split, and thus numerical viscosity easily blurs and spreads the waves. Artificial compression is one way to remedy this. Shock waves are compressive and stable; however, there can be subtle behaviors of shockcapturing schemes depending on the relation between the C-F-L number and the shock speed; see Section 7 of Chapter 12. Though the wave location is random, as a numerical scheme the Glimm scheme keeps the discontinuity waves sharp. In general, one needs to track the shocks in order to have sharp discontinuities. Another basic issue is obtaining an accurate shock speed globally in time. The speed of a shock wave satisfies the Rankine-Hugoniot condition obtained by the integral conservation laws. A fundamental principle in designing a numerical scheme is to insist that it be a conservative scheme in the following sense. Let Δx and Δt be the grid lengths. In order for the actual information propagation speeds λi , i = 1, 2, . . . , n, to be smaller than the numerical speed, the grid lengths need to satisfy the C-F-L condition (10.1)
Δx ≥ {|λj (u)|, j = 1, 2, . . . , n} Δt for all u under consideration
¯ are constant between Suppose that the numerical approximate solutions u the grid points: ¯ (x, kΔt) = u ¯ kj for jΔx < x < (j + 1)Δx, (10.2) u k = 0, 1, 2, . . . , j = 0, ±1, ±2, . . . . ¯ kj , j = 0, ±1, ±2, . . . , the numerical scheme deGiven the constant states u ¯ k+1 , j = 0, ±1, ±2, . . . , at the next time level. For termines the states u j k a conservative scheme, this is done by providing a numerical flux f¯ j and forming the approximate conservation laws (10.3)
¯ kj ¯ k+1 −u u j Δt
k k f¯ j+1 − f¯ j + = 0, j = 0, ±1, ±2, . . . . Δx
Conservative schemes have the desired property of capturing the right shock speeds. An important example is the Godunov scheme, which resolves the discontinuity at the grid points and finds the numerical flux accordingly. Thus at x = jΔx, solve the Riemann problem (ul , ur ), where ¯ kj−1 , ur = u ¯ kj , ul = u
11. Notes
273
and let φ(x/t) be its solution. The numerical flux is the exact flux f (u) of the computed Riemann solution in the grid direction: k f¯ j = f (φ(0)).
The Godunov scheme has the desirable property that the entropy condition is enforced by incorporating into the construction the admissible elementary waves of the Riemann solutions. For the same reason, the Godunov scheme is an upwind scheme in that it takes into account the direction of wave propagation. The entropy condition is guaranteed also by numerical dissipation. A basic example of diffusive schemes is the Lax-Friedrichs scheme. In designing a numerical scheme, consideration of its efficiency is through the Taylor expansion. As shock waves are jump discontinuities, Taylor expansion cannot be applied and the efficiency of a scheme for a shock can be first order at best. As we have seen, in a general solution, there can be infinitely many shock waves. Thus the efficiency of a general finite difference scheme is limited. For the Glimm scheme, which averages over general initial data, its efficiency is shown to be half order. On the other hand, it is easy to see that if a scheme is first order for smooth solutions, the efficiency ought to be also first order away from shocks. A general solution, though it may contain infinitely many shocks, is in practice piecewise smooth if very small shocks are ignored according to the regularity theory stated in Section 8. Thus it would be desirable to relate the efficiency of a numerical scheme and the regularity of the solutions. Analysis of numerical schemes in the presence of shock waves is a challenging problem; see, for instance, examples in Section 5 and Section 7 of Chapter 12 on the phenomenon of nonlinear resonance.
11. Notes The classical paper of James Glimm [54] initiated a new era in the study of the well-posedness theory for systems of hyperbolic conservation laws. The idea of using the Riemann solutions as building blocks was proposed by Godunov for computational purposes [56]. Oleinik [110] started the use of the function space of bounded variation for the study of scalar conservation laws. The Glimm scheme presented in Section 1 and the Glimm functional, (2.3), were introduced in Glimm [54]. The Glimm analysis of wave interactions, Theorem 4.1 of Chapter 8, forms the basis of Glimm’s theory. The Glimm functional may be viewed as a natural consequence of this general analysis of wave interactions. The existence theory obtained in [54] is of probabilistic nature; approximate solutions tend to an exact solution for almost all sequences a in the random-choice method (1.2). The deterministic
274
9. Well-Posedness Theory
version, introduced by Liu [84], shows that the limiting function is a weak solution when the sequence a is equi-distributed, Theorem 4.1. This is done by the wave tracing method covered in Section 3. The important paper of Glimm-Lax [55], the sequel of Glimm [54], introduced the notion of generalized characteristics and initiated the study of solution behavior. The notion of wave measures, (2.1), is a natural continuation of the wave analysis in [54] and is used in [55]. The notion of generalized characteristics is defined on the level of Glimm approximate solutions to monitor the evolution of waves. The difficulty of handling the wave evolution in [55] is somewhat lessened by using the wave tracing technique as done here in Section 3 to Section 6. There are two basic approaches to the L1 (x) stability theory presented in Section 5; one is the homotopy approach initiated in Bressan [13], and the other is the functional approach initiated in Liu-Yang [93]. Both approaches require control of the time evolution of the elementary waves. There are two techniques for such control, one is the front tracking method proposed by Dafermos [34], and the other is the wave tracing method of Liu [84] presented in Section 3. For some idea of the front tracking method, see figure 8.04 of Chapter 8, where both shock and rarefaction waves are approximated by discontinuity waves. The front tracking method constructs approximate solutions based on the Glimm algorithm so that either the homotopy or the functional approach can be carried out. The wave tracing method analyzes the Glimm approximate solutions through wave partitioning. The homotopy approach was first realized using the front tracking technique by Bressan-Colombo [14] for 2 × 2 systems; see also references in Bressan [17]. The functional approach was realized using the front tracking technique in Bressan-Liu-Yang [16] and using the wave tracing method in Liu-Yang [94]. The measurement of the distance between solutions using Hugoniot curves, (5.4), is natural for the front tacking method; see [17]. The stability theory as presented in Section 5 is an elaboration of [94] by explicit analysis on the microscopic, mesoscopic, and macroscopic scales. The wave tracing method can be applied to the general situation when the characteristic fields are not genuinely nonlinear or linearly degenerate. The notion of generalized characteristics can be defined on the level of Glimm approximate solutions. Theorem 2.4 and Corollary 2.5 are the basic estimates for the analysis of existence, time-asymptotic behavior, and regularity of solutions for the general situation; see [85]. This is a qualitative, not quantitative, analysis; no convergence rates are obtained. The notion of generalized characteristics has been defined directly for some classes of functions of bounded variation, and without resorting to the
11. Notes
275
Glimm scheme. It is used to study the solution behavior for 2 × 2 systems; see Chapter XII of [35]. The question of uniqueness of solutions is a central part of traditional well-posedness theory. The stability theory in Section 5 yields the uniqueness of solutions within the general framework of the Glimm solution algorithm. The general Glimm solution algorithm uses Riemann solutions as building blocks. Thus this solution operator is also called the Riemann semigroup. For scalar laws, one way to establish the uniqueness is to approximate the solutions by piecewise functions, e.g. Theorem 4.3 and Corollary 4.4 in Section 4 of Chapter 3. Similary, the aforementioned two approaches to the L1 (x) stability theory yield the same solution. Based on the establishment of the Riemann semigroup, e.g. Section 4 and Section 5, there are perturbationtype uniqueness results, e.g. [15]. There are no stability results for solutions obtained by the method of compensated compactness. General compactness arguments do not provide sufficient understanding of the time evolution of the solutions. In the difficult study of well-posedness theory for weak solutions, the main effort should be to establish a semigroup property for certain solution algorithms. In Section 4 of Chapter 14 there is a general discussion of the well-posedness theory for weak solutions of evolutionary partial differential equations. The shock wave theory provides an impetus for the development of nonlinear analysis. When the theory of compensated compactness was introduced, it was applied to hyperbolic conservation laws as a benchmark test of its effectiveness; see e.g. [120], [38], and [37]. In Section 7, the study of the large-time behavior is without decay rates. The analysis with decay rates for systems with genuinely nonlinear or linearly degenerate characteristic fields was done by Liu in 1977; see references in [85]. The decay of solutions in Section 9 is taken from an unpublished note by Glimm-Lax [55]. The study of N -waves was done for 2 × 2 systems by DiPerna in 1974, and for general genuinely nonlinear systems by Liu in 1977 and 1987; see [92] and references therein. For systems such as Euler equations with linearly degenerate fields, the solution converges to a combination of N -waves and linear waves, with convergence rates of t−1/4 and t−1/2 , respectively [92]. The study of regularity of solutions in Section 8 was done by DiPerna in 1976 for 2 × 2 systems in 1976 and for general systems later; see [85] and references therein. The qualitative analysis in Section 7 of the convergence to the Riemann wave pattern can be done for the general situation when the characteristic fields are not genuinely nonlinear or linearly degenerate using the notion of wave interaction measures for the general case; see (4.9) and Theorem 4.4
276
9. Well-Posedness Theory
in Chapter 8. The convergence rates for the general case depend on the degeneracy of the characteristic fields, e.g. Exercise 8. Because of the existence of the coordinates of Riemann invariants, discussed in Section 2 of Chapter 8, the analysis for systems of hyperbolic conservation laws often starts with 2 × 2 systems. The Glimm-Lax paper [55] makes use of the third-order interaction measure to generalize the regularizing property for scalar laws, Theorem 5.3 of Chapter 3 , to 2×2 systems. It is not clear if the regularizing property would hold for general genuinely nonlinear systems. In the remarkable paper by Bianchini-Bressan [9], Glimm’s functional is generalized to conservation laws with artificial viscosity, ut + f (u)x = κuxx . The general well-posedness theory for hyperbolic systems is then obtained through the zero dissipation limit, κ → 0+. It remains challenging to study the problem for systems with physical viscosity, such as the Navier-Stokes equations in gas dynamics. A related problem is the zero mean free path limit for the Boltzmann equation of kinetic theory; see Section 4 of Chapter 11.
12. Exercises 1. Consider the interaction of two shock waves for scalar convex conservation law, as shown in figure 3.02 of Chapter 3. Show that the Glimm scheme converges in this case. 2. Construct the Glimm functional for scalar convex conservation laws. 3. Show by elementary analysis that the Glimm approximation converges for a centered rarefaction wave for a system of hyperbolic conservation laws. 4. Take the sequence an to have the error e(n, a) √ = O(1)(log n)/n, as in the case of an equal to the non-integral part of n 2 or the van der Corput sequence. Find the convergence rate, in the L1 (x) norm, of the Glimm scheme in the case of a single shock wave. 5. Same as Exercise 4, but for the case of a centered rarefaction wave. 6. Same as Exercise 4, but for the case of interaction of two shock waves, as in figure 8.02 of Chapter 8. 7. Consider a system of hyperbolic conservation laws with all its characteristic fields linearly degenerate. Show that the initial value problem (9.1) converges to a linear superposition of linear waves, u(x, t) →
n i=1
φi (x − λ(u0 )t)ri (u0 ) as t → ∞,
12. Exercises
277
for some scalar functions φi (x) satisfying φ(x) = O(1)TV(1+|x|)−3/2 . Show also that the convergence rate is t−1/2 in L1 (x). 8. Consider the scalar conservation law ut + (up /p)x = 0, where p is an integer greater than 2. Construct the solution resulting from the interaction of the shock (1, 0) issuing from (x0 , 0) = (−1, 0) and the rarefaction wave (0, 1) issuing from the origin (x1 , 0) = (0, 0). Show that the solution decays time-asymptotically with decay rate depending on p. 9. Consider the Hopf equation ut + (u2 /2)x = 0 and the N -wave Np,q (x, t), (6.4) of Chapter 3. Write down the explicit form of the generalized entropy functional E[Np,g (·, t), 0], (8.5) of Chapter 3, for the N -wave and the zero solution. Compute its rate of change with respect to time, d E[Np,g (·, t), 0], dt and check with Theorem 8.3 of Chapter 3.
10.1090/gsm/215/10
Chapter 10
Viscosity
This chapter considers conservation laws with dissipative effects, i.e. viscous conservation laws. The study of shock wave theory with dissipative effects is important in its own right from physical considerations. Analytically, nonlinear coupling of the flux and the dissipation gives rise to rich phenomena. The study of shock wave theory for viscous conservation laws is considerably less developed than that for hyperbolic conservation laws. Nevertheless, there has been notable progress. We will examine viscous conservation laws from several points of view. One basic relation of viscous conservation laws to hyperbolic conservation laws was considered in Section 1 of Chapter 5. There we have seen for scalar conservation laws the basic role of viscosity in defining the entropy condition and establishing the energy estimate through the zero dissipation limit. In Chapter 4, we considered the simplest scalar viscous conservation law, the Burgers equation, and addressed some of the basic issues using the Hopf-Cole transformation. We now consider systems of viscous conservation laws. General physical models of conservation laws with dissipation take the following form: (0.1)
ut + ∇x · F(u) = ∇x · B(u, ε)∇x g(u) ,
x ∈ Rm , u ∈ Rn .
The viscosity matrix B(u, ε) depends on the dissipation parameters ε = ε(u). When the dissipation parameters are set to zero, the viscosity matrix vanishes, B(u, 0) = 0, and the system becomes a system of hyperbolic conservation laws (0.2)
ut + ∇x · F(u) = 0,
x ∈ Rm , u ∈ Rn . 279
280
10. Viscosity
The identity matrix B = εI, with ε a positive constant, is called the artificial viscosity: (0.3)
ut + ∇x · F(u) = εuxx ,
x ∈ Rm , u ∈ Rn ,
Physical systems of the form (0.1) are not uniformly parabolic, but hyperbolic-dissipative and quasilinear. A prime example is the NavierStokes equations of gas dynamics. Before considering systems, we will start with the stability of nonlinear waves for scalar conservation laws. The rich wave interaction phenomena for systems are then illustrated.
1. Nonlinear Waves for Scalar Laws We are interested in nonlinear waves for scalar conservation laws. For the study of wave phenomena for scalar laws, it is sufficient to consider artificial viscosity: ut + f (u)x = uxx ,
u ∈ R.
1.1. Perturbation of Constant State To study the simplest nonlinear behavior, consider the convex flux and the perturbation of a constant state. By translation and rescaling, we may assume, without loss of generality, that the constant state is u0 = 0 and (1.1)
ut + f (u)x = uxx ,
f (0) = f (0) = 0, f (0) = 1,
u ∈ R.
In studying the large-time behavior, as t → ∞, we will use the notation O(1)(t + 1)−α+0 to mean O(1)(t + 1)−α+δ for any given positive constant δ, with the bound O(1) depending on δ. In Section 7 of Chapter 4, it is shown that the convergence of a solution of finite mass to the Burgers kernel is at the rate of (t + 1)−1 in L∞ (x) norm and the rate of (t + 1)−1/2 in L1 (x) norm; see (7.8) and (7.9). This is extended to general convex scalar laws with a slightly lower rate. The approach here is to use the heat kernel as the Green’s function, just as the Burgers Green’s function is used in Section 7 of Chapter 4. Proposition 1.1. Consider the initial value problem for (1.1) with small and compactly supported initial data,
O(1)ε for |x| < M, ε 1, (1.2) u(x, 0) = 0 otherwise.
1. Nonlinear Waves for Scalar Laws
281
The solution converges to the Burgers kernel θ(x, t; c) = bD (x, t; 1; c), given in (4.1) and (4.2) of Chapter 3, time-asymptotically: − x2 (1.3) u(x, t) − θ(x, t + 1; c) = O(1)ε (t + 1)−1 + ε(t + 1)−1+0 e D(t+1) , ∞ u(x, 0) dx = O(1)ε, c≡ −∞
for any constant D > 4. In particular, ∞ (1.4) |u(x, t) − θ(x, t + 1; c)| dx = O(1)ε(t + 1)−1/2+0 as t → ∞. −∞
Proof. The perturbation v ≡ u − θ satisfies v2 + O(1)v 3 + O(1)θ3 x . vt = vxx + −vθ − 2 In (1.3), the total mass of the Burgers kernel is set to that of the initial value, and so it is natural to consider the anti-derivative V (x, t) of the perturbation v(x, t): x ∞ v(x, t) dx = 0 for t ≥ 0, V (x, t) ≡ v(y, t) dy, −∞ −∞
O(1)ε for |x| < M, V (x, 0) = 0 otherwise. The equation for v can be viewed either as the heat equation vt = vxx with the source (−vθ − v 2 + O(1)θ3 )x , or more accurately as the linearized Burgers equation vt + (vθ)x = vxx around the Burgers kernel θ with source (−v 2 + O(1)θ3 )x . For our purposes, it is sufficient to use the first version and so by Duhamel’s principle, Corollary 1.5 in Section 1 of Chapter 4, its solution is the convolution with the heat kernel H(x, t): ∞ H(x − y, t)v(y, 0) dy v(x, t) = t + 0
−∞ ∞
v2 + O(1)v 3 + O(1)θ3 y (y, s) dy ds, H(x − y, t − s) −vθ − 2 −∞ x2 1 e− 4κt . H(x, t) = H(x, t; κ) = √ 4πκt
To take advantage of the zero total mass of the perturbation, the contribution from the initial data is expressed in terms of the anti-derivative V (x, t): ∞ ∞ H(x − y, t)v(y, 0) dy = − Hy (x − y, t)V (y, 0) dy. −∞
−∞
282
10. Viscosity
Note that Hy (x − y, t) has a higher time decay rate than H(x − y, t): (1.5)
(x−y)2 1 2(x − y) − (x−y)2 Hy (x − y, t) = √ e 4κt = O(1)t−1 e− Dt 4πκt 4κt
for any constant D > 4κ. Thus ∞ −1 Hy (x − y, t)V (y, 0) dy = O(1)εt −∞
M
e−
(x−y)2 Et
.
−M
M (x−y)2 (x−˜ y )2 By the mean value theorem, −M e− Dt dy = 2M e− Dt for some y˜ with −M < y˜ < M. The above easily yields ∞ x2 H(x − y, t)v(y, 0) dy = O(1)εt−1 e− Dt , t > 1. −∞
For 0 < t < 1, the integral is localized and bounded, and so we conclude that the contribution from the initial value is ∞ 2 − x H(x − y, t)v(y, 0) dy = O(1)ε(t + 1)−1 e D(t+1) , t ≥ 0. −∞
This leads us to make the following a priori assumption on the solution for the purpose of assessing the contribution from the nonlinear term: x2
v(x, t) = O(1)ε(t + 1)−1 e− Dt , possible ansatz for solution. From (4.2) in Chapter 4, θ(x, t + 1; c) = bD (x, t + 1; 1, c) with c = O(1)ε, we have 2 1 − x θ(x, t + 1; c) = O(1)ε e 4κ(+1)t . 4π(t + 1) We now analyze the contribution from the nonlinear terms −vθ − v 2 /2 + O(1)v 3 +O(1)θ3 . From (4.2) of Chapter 4, with θ(x, t+1; c) = bD (x, t+1; 1, c) and c = O(1)ε, we have 2 1 − x e 4κ(+1)t . θ(x, t + 1; c) = O(1)ε 4π(t + 1)
From this and the ansatz for v(x, t), −vθ −
2 v2 − y + O(1)v 3 + O(1)θ3 = O(1)ε2 (s + 1)−3/2 e D(s+1) . 2
The derivative of the heat kernel H(x − y, t − s) has an extra decay rate of t−1/2 , as seen in (1.5). Thus we make a similar assumption on the source: (−vθ −
2 v2 − y + O(1)v 3 + O(1)θ3 )y = O(1)ε2 (s + 1)−2 e D(s+1) , D > 4. 2
1. Nonlinear Waves for Scalar Laws
283
To optimize the estimate, we evaluate the integral by putting the differential in y on the nonlinear source for small time, 0 < s < t/2, and on the heat kernel for large time, t/2 < s < t: t ∞ v2 + O(1)v 3 + O(1)θ3 y (y, s) dy ds H(x − y, t − s) −vθ − 2 0 −∞ t/2 ∞ v2 + O(1)v 3 + O(1)θ3 (y, s) dy ds Hy (x − y, t − s) −vθ − = 2 0 −∞ t ∞ v2 + O(1)v 3 + O(1)θ3 y (y, s) dy ds H(x − y, t − s) −vθ − + 2 t/2 −∞ t/2 ∞ 2 (x−y)2 − − y (t − s)−1 e D(t−s) (s + 1)−3/2 e D(s+1) = O(1)ε2 0
−∞ ∞
t
+ −∞
t/2
(t − s)−1/2 e
(x−y)2
− D(t−s)
(s + 1)−2 e
2
y − D(s+1)
dy ds.
We can check the semigroup property of the heat kernel by direct calculation: ∞ ∞ 2 (x−y)2 t+1 x2 − − y − (y− s+1 x)2 − D(t+1) t+1 e D(t−s) e D(s+1) dy = e D(t−s)(s+1) e dy −∞ −∞ x2 Dπ(t − s)(s + 1) − D(t+1) e . = (t + 1) Thus the integral of the nonlinear terms is estimated as t ∞ H(x − y, t − s)(vθ + v 2 + O(1)θ3 )y (y, s) dy ds 0
−∞
= O(1)ε
2
t/2 0
(t − s)−1/2 (s + 1)−1 (t + 1)−1/2 ds
t
+
2 − x (s + 1)−3/2 (t + 1)−1/2 ds e D(t+1)
t/2
− x2 = O(1)ε2 (t + 1)−1 log(t + 1) + (t + 1)−1 e D(t+1) . Except for the log(t +1) factor, the final expression is much smaller than the x2
ansatz O(1)ε(t+1)−1 e− Dt assumed for sufficiently small ε, and the standard method of nonlinear closure applies. To take care of the log(t + 1) factor, we change the above ansatz for v(x, t) to x2
v(x, t) = O(1)ε(t + 1)−1+δ e− Dt , final ansatz for solution, for any given positive constant δ. By going through the computations as above, it is easily shown that the nonlinear terms yield the same expression as the ansatz, but with an extra factor of ε. Therefore this new ansatz can
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be used as the final ansatz for the closure of the nonlinear term as in the proof of Proposition 7.2, e.g. (7.14) and (7.15), in Section 4 of Chapter 4. The completes the proof of the proposition. 1.2. Viscous Shocks For hyperbolic conservation laws, the compression of shock waves makes the shock stable, and the shift of the shock location due to perturbation can be computed by the conservation law, Theorem 5.5 in Section 5 of Chapter 3. We now consider shock wave stability for viscous conservation laws. Suppose that the perturbation u(x, 0) − φ(x) decays at the rate of |x|α , α > 1, as x → ±∞, so that u(x, 0) − φ(x) is integrable over −∞ < x < ∞. From the conservation law, ∞ ∞ (u(x, t) − φ(x + x0 )) dx = (u(x, 0) − φ(x + x0 )) dx −∞ −∞ ∞ ∞ (u(x, 0) − φ(x)) dx + (φ(x) − φ(x + x0 )) dx, t > 0. = −∞
The integral A(x0 ) ≡ A(0) = 0,
−∞
∞
− φ(x + x0 )) dx satisfies ∞ φ (x + x0 ) dx = −(u+ − u− ), A (x0 ) = − −∞ (φ(x)
−∞
and so A(x0 ) = −x0 (u+ − u− ): ∞ [φ(x + x0 ) − φ(x)] dx = x0 (u+ − u− ). (1.6) −∞
Thus, with the proper choice of the shift x0 , the perturbation has zero total mass: ∞ v(x, t) dx = 0, t > 0, v(x, t) ≡ u(x, t) − φ(x + x0 ), (1.7) −∞ ∞ 1 x0 ≡ (u(x, 0) − φ(x)) dx. u+ − u− −∞ For simplicity, assume that the flux is convex, f (u) > 0. By a change of coordinate x → x − st, we may assume that the shock is stationary, s = 0, and the shock profile φ(x), with φ(±∞) = u± , satisfies (1.8)
u+ < u− ,
d φ(x) < 0, dx
d d f (φ(x)) = f (φ(x)) φ(x) < 0. dx dx
We illustrate first the stability of shock profiles without rates of convergence by a simple energy method, using only the compressibility of the shock profiles.
1. Nonlinear Waves for Scalar Laws
285
Proposition 1.2. The shock profile is time-asymptotically stable upon small perturbation: u(x, t) → φ(x + x0 ) as t → ∞, where the shift of the profile x0 is given in (1.7). Proof. The perturbation v(x, t) carries zero total mass, as stated in (1.7), and so we can define x v(y, t) dy, so that V (±∞, t) = 0. V (x, t) ≡ −∞
The equation for this anti-derivative V (x, t) is (1.9)
Vt + f (Vx + φ) − f (φ) = Vxx , or Vt + f (φ)Vx + O(1)(Vx )2 = Vxx .
To perform the energy estimate, multiply (1.9) by V and integrate: t ∞ ∞ V (x, t)2 V (x, s)2 dx + dx ds −f (φ)x 2 2 −∞ 0 −∞ ∞ t ∞ V (x, 0)2 dx. (O(1)V + 1)(Vx )2 (x, s) dx ds = + 2 0 −∞ −∞ Using the compressibility property −f (φ(x))x > 0 from (1.8), and assuming that the perturbation is small so that O(1)V + 1 > 1/2, the above yields an effective energy estimate: t ∞ ∞ ∞ 2 2 v (x, t) dx + (vx ) (x, s) dx ds = O(1) v 2 (x, 0) dx. −∞
−∞
0
This implies that
∞ ∞
−∞
t
−∞
(vx )2 (x, s) dx ds → 0 as t → ∞.
Continue the energy estimate for the differentiation of (1.9) to obtain, for 0 < t1 < t2 , t2 ∞ ∞ ∞ (vx )2 (x, t2 ) dx + (vxx )2 (x, s) dx ds = O(1) (vx )2 (x, t1 ) dx. −∞
−∞
t1
This and the above yield ∞ −∞
−∞
(vx )2 (x, t) dx → 0 as t → ∞.
From this, we use the standard Sobolev inequality: (v(x, t))2 = 2 (1.10)
x −∞
vvx (y, t) dy ≤
|v(x, t)| ≤
x
v 2 (y, t) dy −∞
×
x −∞
(vx )2 (y, t) dy, or
v(·, t) L2 (x) vx (·, t) L2 (x) , −∞ < x < ∞.
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Here the first inequality is the Cauchy-Schwarz inequality. This implies the desired decay result v(x, t) → 0 as t → ∞. Details are similar to the second half of the analysis of stability of rarefaction waves below, e.g. the proof of Proposition 1.4, and are therefore omitted. For scalar hyperbolic conservation laws, a compact perturbation of the shock (u− , u+ ) converges to the shock in finite time; see Theorem 5.5 of Chapter 3. Similar behavior holds also for the scalar viscous conservation laws. From the structure of Green’s functions for the Burgers equation, covered in Section 6 of Chapter 4, the propagation around a shock profile is mainly along the inviscid characteristics. Due to the exponential tail of the diffusion, a compactly supported perturbation of a shock profile does not vanish in finite time, but decays exponentially fast. We consider more general initial data to illustrate the propagation of the perturbation using the weighted energy method. There is a direct relation between the decay rate of the perturbation and the rate of convergence in time of the solution. We use the same set-up as in the above proposition and consider the perturbation of the shock v(x, t) = u(x, t)−φ(x+x0 ) with zero total mass and anti-derivative V (x, t) satisfying (1.9). Proposition 1.3. Suppose that the perturbation of a stationary shock profile φ(x), with φ(±∞) = u± , decays at an algebraic rate: (1.11)
v(x, 0) = O(1)δ(1 + |x|)−α ,
vx (x, 0) = O(1)δ(1 + |x|)−α−1 ,
for some postive constant α > 1, and a small constant δ. Then u(x, t) − φ(x + x0 ) = O(1)δ(1 + |x| + Λt)−α+0 , t > 0, for any positive constant Λ < min{|f (u− )|, |f (u+ )|}. Proof. We use weighted energy method, integrating (1.9) times a weight function W (x, t): ∞ Wt V2 d ∞ V 2 (x, t) Wx dx + − f (φ) − f (φ)x W dx − W (x, t) dt −∞ 2 W W 2 −∞ ∞ ∞ 2 W 1 + O(1)V Vx ) dx + Wx V Vx dx = 0. + −∞
−∞
The main propagation directions at the far fields |x| = ∞ are the hyperbolic characteristics, f (φ(x)) → f (u± ) as x → ±∞. From the entropy condition for a stationary shock, f (u− ) > 0 > f (u+ ), the propagation is toward the shock. Let Λ be a positive constant representing a speed slower than f (u− ): Λ = af (u− ) for some constant a, 0 < a < 1.
1. Nonlinear Waves for Scalar Laws
287
Choose the weight function corresponding to the slower speed Λ: W (x, t) ≡ (|x| + Λt + X)β , with the positive constant X to be chosen later. From now on we only consider the interesting case where the shock is weak, |u+ − u− | 1. The shock speed, zero here, is close to the arithmetic mean of the characteristic speed, f (u+ ) + f (u− ) = O(1)ε2 , ε ≡ f (u− ) 1. The width of the profile is of the order of ε−1 , and in that interval one has f (φ)x = ˜ ε2 for |x| < O(1)ε−1 . Outside of the transitional interval, f (φ)(x) = f (u− ) + O(1)ε2 = ε + O(1)ε2 for x < −O(1)ε−1 , f (φ)(x) = f (u+ ) + O(1)ε2 = −ε + O(1)ε2 for x > O(1)ε−1 . The constant X defining the weight function W (x, t) is set to ˜ Λ−1 . Cε−1 = f (u− )−1 = We now use the above identities to check the main coefficient in the energy identity: −
Wx βΛ β · sign(x) Wt − f (φ) − f (φ)x = − − f (φ) − f (φ)x W W |x| + Λt + X |x| + Λt + X ⎧ β (u ) + O(1)ε2 − f (φ) ⎪ −Λ + f ⎪ − x ⎨ |x|+Λt+X β 2 ≥ |x|+Λt+X f (u− )(1 − a) + O(1)ε > 0 for |x| > O(1)ε−1 , = ⎪ ⎪ β ⎩ ˜ O(1) Xε + ε2 for |x| < O(1)ε−1 . |x|+Λt+X O(1)ε − f (φ)x =
By choosing X −1 = C −1 ε for some C large, we have O(1)ε/X − ε2 > 0 and the main coefficient is positive. The other terms in the energy identity are easier to estimate and we conclude that V 2 (x, t) d ∞ dx ≤ 0, W (x, t) dt −∞ 2 and so
∞
(|x| + Λt + X) −∞
βV
2 (x, t)
2
dx ≤
∞ −∞
(|x| + X)β
V 2 (x, 0) dx. 2
From the assumption on the initial data, the last integral is bounded for β < 2α−3. With this we apply the weighted the equation ∞energy method for β 2 for v(x, t) to obtain the boundedness of −∞ (|x| + Λt + X) v (x, t) dx for ∞ β < 2α − 1 and the boundedness of −∞ (|x| + Λt + X)β (vx )2 (x, t) dx for
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10. Viscosity
β < 2α + 1. Apply the Sobolev inequality to obtain x ∂ β 2 (|x| + Λt + X) v (x, t) = (|x| + Λt + X)β v 2 (x, t) dx −∞ ∂x ∞ ∞ β(|x| + Λt + X)β−1 v 2 (x, t) dx + 2 (|x| + Λt + X)β vvx (x, t) dx ≤ −∞ −∞ ∞ β(|x| + Λt + X)β−1 v 2 (x, t) dx ≤
−∞
∞
(|x|+Λt+X)
+2
β−1 2
∞
v (x, t) dx·
−∞
−∞
(|x|+Λt+X)β+1(vx )2 (x, t) dx
1 2
.
Take any β < 2α; then the above integrals are bounded and of order δ 2 : (|x| + Λt + X)β v 2 (x, t) = O(1)δ 2 for any β < α. This completes the proof of the proposition.
1.3. Rarefaction Waves Rarefaction waves differ from shock waves in three aspects. Rarefaction waves are expansive, while shock waves are compressive. For shock waves, the shift of the location needs to be exactly computed by the conservation law; for rarefaction waves, there is a continuum of ways of shifting and the location of the wave cannot be exactly defined. The latter has been explained in Section 6 of Chapter 4. Because of this, in the analysis of shock stability just considered, we started with the equation for the anti-derivative V of the perturbation v; but in the analysis below for the rarefaction wave, we will start directly with the equation for the perturbation v. The third difference is that there are exact shock profiles, but there is no explicit expression for exact rarefaction waves for viscous conservation laws. For rarefaction waves, the best that can be done is to construct approximate rarefaction waves that are of sufficient time-asymptotic accuracy. Consider the rarefaction wave (ul , ur ), ul < ur , for the convex conservation law ut + f (u)x = uxx , f (u) > 0. As explained in (2.12) of Chapter 3, the rarefaction waves for the inviscid equation ut + f (u)x = 0 satisfy the inviscid Burgers, or Hopf, equation λt + λλx = 0, where λ ≡ f (u); in contrast, for the viscous equation ut +f (u)x = uxx , the rarefaction waves satisfy the Burgers equation only approximately: λt + λλx = λxx + f (u)(ux )2 . By simple translation, we can assume that the rarefaction wave is symmetric, −f (ul ) = f (ur ) ≡ λ0 . We will view the solution as a perturbation of the
1. Nonlinear Waves for Scalar Laws
289
Burgers rarefaction wave bR (x, t) in (4.10) of Chapter 4, e
λ0 x 2κ
√ 0 t ) − e− Erfc( −x+λ 4κt
λ0 x 2κ
√ 0t ) Erfc( x+λ 4κt
e
λ0 x 2κ
√ 0 t ) + e− Erfc( −x+λ 4κt
λ0 x 2κ
√ 0t ) Erfc( x+λ 4κt
bR (x, t) = λ0
.
By direct calculations, (1.12) ⎧ √ (x+λ0 t)2 ⎪ − k+1 − ⎪ 2 e 5t O(1)t for x < −λ t + t, k = 1, 2, . . . ; ⎪ 0 ⎪ ⎪ k ⎨ ∂ x −(k+1) + (x + λ t)−(k+1) ( ) + O(1) (λ t − x) ∂k 0 0 k t √ √ bR (x, t) = ∂x ⎪ ∂xk t + t < x < λ t − t, k = 0, 1, . . . ; for −λ 0 0 ⎪ ⎪ ⎪ √ (x−λ0 t)2 ⎪ ⎩O(1)t− k+1 − 2 e 5t for x > λ0 t − t, k = 1, 2, . . . . The approximate rarefaction wave uR satisfies the viscous conservation law 2 with a truncation error of the order of (bR )x : 2 f (uR ) (uR )x , uR ≡ (f )−1 (bR ), f (uR ) 2 ⇒ (uR )t + f (uR )x = (uR )xx + O(1) (bR )x .
(1.13) (uR )t + f (uR )x = (uR )xx +
To focus on the time-asymptotic analysis, consider the Burgers rarefaction wave bR (x, t+T ) with large starting time T . We first use the energy method to show the time-asymptotic stability of rarefaction waves without rates of decay of the perturbation. Proposition 1.4. Consider a solution u(x, t) of the viscous conservation law with initial data u(x, 0) being a small perturbation of bR (x, T ). Then v(x, t) ≡ u(x, t) − bR (x, T + t) → 0 as t → ∞. Proof. The equation for the perturbation is 5 2 f (k) (uR ) k v + O(1)v 6 x = vxx + O(1) (bR )x . vt + f (uR )v + k! k=2
Consider the energy identity by integrating the equation times v: t ∞ 1 ∞ v2 2 f (uR )x + O(1)(uR )x v 3 + O(1)v 6 v(x, t) dx + 2 −∞ 2 0 −∞ ∞ t ∞ 2 1 2 2 +(vx ) (x, s) dx ds = v(x, 0) dx+ O(1) (bR )x v(x, s) dx ds. 2 −∞ 0 −∞ By f (uR ) = bR from (1.13), the second term on the left-hand side is nonnegative: t ∞ t ∞ v2 v2 (x, s) dx ds = (x, s) dx ds ≥ 0, f (uR )x (bR )x 2 2 0 −∞ 0 −∞
290
10. Viscosity
because of the expansion of the rarefaction wave, (bR )x > 0, which is the key property of its stability. There is another desirable term in the identity, t ∞ (vx )2 (x, s) dx ds ≥ 0, 0
−∞
which comes from the dissipation of the viscous conservation law. One needs to show that these two desirable terms dominate the other terms in the energy identity to complete the energy estimate. The third term, the integral of O(1)(uR )x v 3 , is dominated by the second term because the energy estimate will show that |v(x, s)| is small for a small perturbation. The fourth term, the integral of O(1)v 6 , is dealt with by the Sobolev inequality (1.10): ∞ 2 ∞ v (vx )2 (v(x, s))4 ≤ (y, s) dy ds × (y, s) dy ds. 4 2 −∞ 2 −∞ From this, we can estimate the fourth term in the energy identity: t 0
∞ −∞
O(1)v (x, s) dx ≤ O(1)
t
6
∞
(vx )2 (x, s) dx ds −∞ ∞ 2 · max v 2 (x, s) dx .
0
0 4, ⎧ (x−λi (t+1))2 ⎪ − D(t+1) −3/4 ⎪ e ⎪O(1)(t + 1) ⎪ ⎪ √ ⎪ ⎪ for x > λ (t + 1) − (λ − λ )( t + 1), ⎪ i i j ⎨ −3/2 (2.16) I(x, t) = O(1)|x − λi (t + 1)| ⎪ √ ⎪ ⎪ for λ (t + 1) < x < λ (t + 1) − (λ − λ )( t + 1), ⎪ j i i j ⎪ ⎪ ⎪ (x−λj (t+1))2 ⎪ ⎩O(1)(t + 1)−3/2 e− D(t+1) for x < λj (t + 1). Here, for definiteness, we have assumed that i > j, λi > λj , and so the i-characteristic x = λi t lies to the right of the j-characteristic x = λj t. Proof. To estimate (2.15) one needs to consider the relative positions of the essential support of the heat kernel H(x − y − λi (t − s), t − s), shown in figure 4.01 of Chapter 4, and the source θj2 (y, s), depicted in figure 10.01. This yields the form of the solution in (2.16).
Figure 10.01. Coupling of viscous waves, together with their essential supports.
The function θj is either the Burgers or the heat kernel; in either case, it satisfies the diffusion equation in the λj direction with a higher-order error: (θj )2 s + λj (θj )2 y = (θj )2 yy + O(1)(s + 1)−3/2 θ. Here we take the liberty of writing Hy = O(1)(s + 1)−1/2 H, when actually ¯ with H ¯ having slightly wider support, e.g. it is Hy = O(1)(s + 1)−1/2 H,
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297
(1.5). This is for simplicity of presentation and should cause no ambiguity. From this we can write (θj )2 )y as its dissipation in the λi direction plus an higher-order error: (λi − λj ) (θj )2 y = (θj )2 )s + λi (θj )2 y − (θj )2 yy + O(1)(s + 1)−3/2 θ. To compute the integral I(x, √ √t) we divide the time period into small times s < t and not-small times t < s < t and put the differentiation operator ∂y in a different place through integration by parts: √ t ∞
I(x, t) = 0
−∞
Hy (x − y − λi (t − s), t − s)(θj )2 (y, s) dy ds
t ∞ 1 H(x − y − λi (t − s), t − s) + λi − λj √t −∞ · (θj )2 s + λi (θj )2 y − (θj )2 yy + O(1)(s + 1)−3/2 θ(y, s) dy ds. Here we have used the above expression for (θj )2 y . The heat kernel propagates in the λi direction: G = G(y, s) ≡ H(x − y − λi (t − s), t − s).
Gs + λi Gy = Gyy ,
Thus the second integral above is simplified through integration by parts, t ∞ 2 2 2 G(y, s) (θ ) + λ ) − (θ ) (θ dy ds j i j j √ s y yy t −∞ ∞ √ √ 2 = (θj ) (x, t) − G(y, t)(θj )2 (y, t) dy, −∞
and the above identity becomes I(x, t) = −
√ t ∞
Gy (y, s)θ2 (y, s) dy ds 0 −∞ ∞ √ √ 1 1 2 (θj ) (x, t) − G(y, t)(θj )2 (y, t) dy + λi − λj λi − λj −∞ t ∞ + O(1) √ G(y, s)(s + 1)−3/2 θj (y, s) dy ds t
−∞
≡ I1 +
1 (θj )2 (x, t) + I2 + I3 . λi − λj
The function θj and the function G and its differential Gy are of the order θ(y, s) =(s ˜ + 1)−1/2 e
−
(y−λj (s+1))2 4(s+1)
,
G =(t ˜ − s)−1/2 e
−
(x−y−λi (t−s))2 4(t−s)
Gy = O(1)(t − s)
−1 −
e
,
(x−y−λi (t−s))2 D(t−s)
,
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10. Viscosity
for any D > 4. Complete the square for the y variable for the two exponentials in the integrand to obtain e
−
(y−λj (s+1))2 D(s+1)
e
−
(x−y−λi (t−s))2 D(t−s)
=e
−
(t+1)[(y−(λi −λj )(t−s)+x(s+1))/(t+1)]2 D(s+1)(t−s)
e
−
[x−λi (t−s)−λj (s+1)]2 D(t+1)
.
Note that, besides the exponential term, there is a factor of (s + 1)−1/2 in θ, and there is an additional factor of (t − s)−1/2 in Hy as compared to H. Thus the spatial integration yields ∞ −α θ (y, s) dy j −∞ H(x − y − λi (t − s), t − s)(s + 1) 1 e = O(1)(s + 1)−α √t+1
∞ −∞
−
[x−λi (t−s)−λj (s+1)]2 D(t+1)
,
Hy (x − y − λi (t − s), t − s)(s + 1)−α θj (y, s) dy = O(1)(s + 1)
−α
1
e (t − s)(t + 1)
−
[x−λi (t−s)−λj (s+1)]2 D(t+1)
.
As (θj )2 = O(1)(s + 1)−1/2 θ, the above expressions yield I1 = O(1)
√
t
(s + 1)
−1/2
(t + 1)
−1/2
(t − s)
0
−1/2 −
e
[x−λi (t−s)−λj (s+1)]2 D(t+1) √
ds,
√
[x−λi (t− t)−λj ( t+1)]2 √ −1/2 −1/2 − D(t+1) (t + 1) e , I2 = O(1)( t + 1) t [x−λi (t−s)−λj (s+1)]2 − D(t+1) ds. I3 = O(1) √ (s + 1)−3/2 (t + 1)−1/2 e
t
√ ˜ t and so We focus on the large-time behavior, t > 1. For I2 , t − t = √ √ x−λ (t+1) −[ √ i +(λi −λj )( t+1)/ D(t+1)]2 D(t+1) I2 = O(1)(t + 1)−3/4 e = O(1)(t + 1)
x−λi (t+1) 2 √ −3/4 −[ D(t+1) +O(1)]
e
= O(1)(t + 1)
−3/4 −
e
(x−λi (t+1))2 D(t+1)
.
For I1 , by a change of variables, √ 2 (√t+1)/√t+1 x−λi (t+1) √ −3/4 −1/2 −[ D(t+1) +(λi −λj )τ / D] τ e I1 = O(1)(t + 1) √ 1/ t+1
= O(1)(t + 1)−3/4 e
x−λ (t+1) −[ √ i +O(1)]2 D(t+1)
= O(1)(t + 1)−3/4 e
For I3 , with a change of variable (λi − λj )(s + 1) x − λi (t + 1) + ξ≡ D(t + 1) D(t + 1)
−
(x−λi (t+1))2 D(t+1)
.
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299
we have
B
I3 = A
%−3/2 2 (t + 1) x − λ i (t + 1)−3/4 ξ − e−ξ dξ, D(t + 1) √ (λi − λj )( t + 1) + x − λi (t + 1) A≡ , D(t + 1) $
x − λj (t + 1) B≡ . D(t + 1)
√ In the case where A > 0 and x > λi (t+1)−(λi −λj )( t+1), i.e. in the√region to the right of the wave region λj (t + 1) < x < λi (t + 1) − (λi − λj ) t + 1, $ I3 = O(1)(t + 1)
−3/4
x − λi (t + 1) A− D(t + 1)
%−3/2 e−A
2
= O(1)(t + 1)−3/4 e−
(x−λi (t+1))2 D(t+1)
.
Similarly, to the left of the wave region x < λj (t + 1), or B < 0, $ I3 = O(1)(t + 1)
−3/4
x − λi (t + 1) B− D(t + 1)
%−3/2 e−B
2
= O(1)(t + 1)−3/2 e−
(x−λj (t+1))2 D(t+1)
.
√ In the wave region λj (t+1) < x < λi (t+1)−(λi −λj )( t+1), or A < 0 < B, $
I3 = O(1)(t + 1)
−3/4
x − λi (t + 1) − D(t + 1)
%−3/2
= O(1)|x − λi (t + 1)|−3/2 .
This completes the proof of the lemma.
We now study the initial value problem (2.8) with the set-up (2.9), (2.10), and (2.11) and define
(2.17)
φij (x, t) ≡
⎧ (x−λi (t+1))2 −3/4 − ⎪ D(t+1) (t + 1) e ⎪ ⎪ ⎪ √ ⎪ ⎪ for x > λi (t + 1) − (λi − λj )( t + 1); ⎪ ⎪ ⎪ ⎪ ⎪ |x − λi (t + 1)|−3/2 ⎪ ⎪ √ ⎪ ⎪ ⎪ for λj (t + 1) < x < λi (t + 1) − (λi − λj )( t + 1); ⎪ ⎪ 2 ⎪ (x−λj (t+1)) ⎪ ⎪ ⎪ (t + 1)−3/2 e− D(t+1) ⎪ ⎪ ⎪ ⎨ for x < λ (t + 1), i, j = 1, 2, . . . , n, j < i; j
(x−λi (t+1)) ⎪ ⎪ (t + 1)−3/4 e− D(t+1) ⎪ ⎪ √ ⎪ ⎪ ⎪ for x < λi (t + 1) + (λj − λi )( t + 1); ⎪ ⎪ ⎪ ⎪|x − λ (t + 1)|−3/2 ⎪ i ⎪ ⎪ √ ⎪ ⎪ ⎪ for λ i (t + 1) + (λj − λi )( t + 1) < x < λj (t + 1); ⎪ ⎪ ⎪ (x−λj (t+1))2 ⎪ ⎪ −3/2 − D(t+1) ⎪ (t + 1) e ⎪ ⎪ ⎩ for x > λj (t + 1), i, j = 1, 2, . . . , n, j > i. 2
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Theorem 2.2. The solution of (2.8) has the following form: (2.18) u(x, t) = u0 +
n
θi (x, t)ri (u0 ) +
i=1
n
vi (x, t)ri (u0 ),
i=1
vi (x, t) = O(1)ε2
φij (x, t),
j =i
where the coupling functions φij (x, t) are given in (2.17) and the diffusion wave θi (x, t) is either the heat or the Burgers kernel, (2.10) or (2.11). In particular, ∞ n 1 |u(x, t) − u0 − θi (x, t)ri (u0 )| dx = O(1)ε2 (t + 1)− 4 . (2.19) −∞
i=1
Proof. The equation (2.14) for vi is viewed as the heat equation with a source: (vi )t + λi (u0 )(vi )x = (vi )xx + S, S≡−
i Cjk (θi θj )x + O(1)|v|2 + O(1)|θ|3 x .
(j,k) =(i,i)
The proof uses Duhamel’s principle: ∞ H(x − y − λj (u0 )t, t)vi (y, 0) dy (2.20) vi (x, t) = −∞
t
∞
+ 0
−∞
H(x − y − λj (u0 )(t − s), t − s)S(y, s) dy ds.
i (θ θ ) , the coupling The known terms in the source S are − (j,k) =(i,i) Cjk i j x terms of waves in the leading term θ. The diffusion waves θi and θj , i = j, propagate in different directions and so their product θi θj is small. Thus i ((θ )2 ) , k = i. Since θ , i = the most important source terms are −Cik i x i 1, 2, . . . , n, are the Burgers and heat kernels, (2.10) and (2.11), the main term in the second integral of (2.20) is of the form of I(x, t) in (2.15). By similar calculations to those in the proof of Lemma 2.1, it can be shown that the other terms in the source are smaller than the ansatz (2.18). This completes the proof of the theorem. Remark 2.3. The above rate of (t + 1)−1/4 in L1 (x) for the convergence of the solution to the diffusion waves is the same as the convergence rate of solutions for the hyperbolic conservation laws to N -waves, as studied in Section 6 of Chapter 9. Note, however, that each Np,q -wave has two time invariants p and q, whereas a diffusion wave θi has only one time invariant ci . We saw this already for the Burgers equation, in Section 5 of Chapter 4.
3. Physical Models
301
Remark 2.4. For systems with physical viscosity, the Green’s function contains singular waves. Nevertheless, for large time, the heat kernel still dominates; see Section 4. Consequently, the time-asymptotic behavior holds for systems with physical viscosity in a form similar to Theorem 2.2. The Green’s function for systems with artificial viscosity used above is the collection of scalar heat kernels H(x − y − λj (u0 )t, t), j = 1, 2, . . . , n. For physical systems of the form ut + f (u)x = B(u, ε)∇x g(u) x , the corresponding linearized system is the same as the linear part of the quadratic system (2.2): ut + f (u0 )ux = B(u0 , ε0 )g (u0 )uxx . The construction of the Green’s function for this linearized system requires non-trivial analysis; see Section 4 of this chapter.
3. Physical Models We now give some examples of systems with physical viscosity. Example 3.1. Isentropic gas flows in Eulerian coordinates are modeled by the system (3.1)
ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p)x = (μvx )x ,
and in Lagrangian coordinates take the form of the p-system (3.2)
τt − vx = 0, vt + p(τ )x = ( μτ vx )x ,
where ρ is the density, τ = 1/ρ the specific volume, v the velocity, p = p(ρ) the pressure, and μ > 0 the viscosity coefficient. The p-system is the simplest system of viscous conservation laws with physical viscosity. System (3.2) can be written in the form (0.1) with 0 τ −v 0 0 g(u) = . , f (u) = , B(u, μ) = (3.3) u = μ , v v p 0 τ Equations (3.2) are also used for modeling viscoelasticity for either longitudinal or shear motion. Example 3.2. The full Navier-Stokes equations for compressible flows in one space dimension are, in Eulerian coordinates, (3.4)
ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p)x = (μvx )x , (ρE)t + (ρEv + pv)x = (μvvx + κθx )x ,
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and, in Lagrangian coordinates, τt − vx = 0, vt + px = ( μτ vx )x , Et + (pv)x = ( κτ θx + μτ vvx )x .
(3.5)
Here the new variables e and θ are, respectively, the internal energy and the temperature of the fluid, and E = e + v 2 /2 the total energy. The pressure p and the temperature θ are given by the constitutive relations p = p(ρ, e) and θ = θ(ρ, e). From (3.5) and the thermodynamics relation de = θ ds − p dτ we have the identity κ κ 2 μ 2 (3.6) st = θx )x + 2 θx + vx , τθ τθ τ which is in agreement with the second law of thermodynamics in that the entropy increases in the distributional sense considered here. Example 3.3. The one-dimensional equations of magnetohydrodynamics in Lagrangian coordinates are
(3.7)
τt − v 1x = 0, v1t + p + 2μ1 0 (b22 + b23 ) x = ( ντ v1x )x , 1 ∗ v2t − μ0 b1 b2 x = ( μτ v2x )x , v3t − μ10 b∗1 b3 x = ( μτ v3x )x , (b2 τ )t − b∗1 v2 x = σμ10 τ b2x x , (b3 τ )t − (b∗1 v3 )x = σμ10 τ b3x x , e + 12 v12 + v22 + v32 + 2μτ 0 b22 + b23 t + p + 2μ1 0 (b22 + b23 ) v1 − μ10 b∗1 b2 v2 + b3 v3 x = ντ v1 v1x + μτ (v2 v2x + v3 v3x ) + κτ θx + σμ12 τ (b2 b2x + b3 b3x ) . 0
x
Besides the dissipation parameters μ and κ in the Navier-Stokes equations, there are the magnetic permeability μ0 and the electrical resistivity 1/σ. The corresponding inviscid system is (7.2) in Chapter 7. As in Example 3.2, all thermodynamic variables are given functions of τ and e. A common feature of these systems is that the continuity equation is nondissipative as dictated by physics. As a consequence, for physical systems of the form (3.2), the viscosity matrix B in (3.3) is semi-positive definite and not positive definite. In other words, (3.2) is hyperbolic-parabolic rather than uniformly parabolic.
4. The p-System To gain understanding of general physical systems, consider the 2×2 system (3.2). We will examine this system from several viewpoints.
4. The p-System
303
4.1. Shock Profiles Consider a shock profile, a traveling wave solution of (3.2), with constant viscosity μ: x − st τ φ φ τ . (4.1) (x, t) = (ξ), (±∞) = ± , ξ = v ψ ψ v± μ Plug this into (3.2) to obtain the differential equations
−sφ − ψ = 0, (4.2) −sψ + p(φ) = ( φ1 ψ ) . Theorem 4.1. Assume that p (τ ) < 0, so that the inviscid system corresponding to the p-system (3.2) is strictly hyperbolic. A shock profile connecting u− and u+ , with u+ ∈ Hj (u− ), exists for the p-system (3.2) if and only if it satisfies the Rankine-Hugoniot condition and the strict Liu entropy condition σ(u− , u+ ) < σ(u− , u) for all states u on Hj (u− ) between u− and u+ . Proof. Notice that the first equation in (4.2), the continuity equation, is of first order and can be integrated to yield the algebraic relation (4.3)
A − sφ − ψ = 0,
A = στ− + v− = sτ+ + v+ .
The second equation integrates to the first-order differential equation 1 B − σψ + p(φ) = ψ , B = sv− − p(τ− ) = sv+ − p(τ+ ). φ That both A and B are constant implies the Rankine-Hugoniot condition −(v+ − v− ) τ+ − τ− . = s v+ − v− p(τ+ ) − p(τ− ) This is necessary for the shock profile to exist. From the Rankine-Hugoniot condition, we have s=± −
p(τ+ ) − p(τ− ) . τ+ − τ−
From the assumption of hyperbolicity of the inviscid system, p (τ ) < 0, one of the shock speeds is positive and the other is negative. For definiteness, consider the positive speed s > 0. Use the algebraic relation (4.3) to eliminate φ from the second equation to obtain a first-order differential equation for ψ: B − σψ + p(φ) =
1 ψ, φ
A − sφ − ψ = 0, ψ − A ψ − A ) . or ψ = g(ψ) ≡ B − sψ + p( s s
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10. Viscosity
For a general pressure function p(τ ), the upstream velocity v+ can be larger or smaller than the downstream velocity v− . For definiteness, consider the case v− < v+ ; then the shock profile exists if and only if g(ψ) > 0 for v− < ψ < v+ , or (4.4)
B − sψ + p(φ) > 0 for v− < ψ < v+ and A − sφ − ψ = 0.
Suppose that the strict Liu entropy condition holds for the shock, (6.1) of Chapter 7: s = σ(u− , u+ ) < σ(u− , u) for all states u on H2 (u− ) between u− and u+ . Note here that σ > 0 corresponds to the Hugoniot curve H2 (u− ). As already noted for scalar laws in Section 1 of Chapter 5, we need to take the strict inequality for the entropy condition here. If (4.4) is violated in that there is a state u = (φ, ψ) such that B − σψ + p(φ) = 0, then it is easy to see that u ∈ H2 (u− ) and σ(u− , u) = σ and the strict entropy condition is violated. Thus B − σψ + p(φ) = 0. It remains to show that B − σψ + p(φ) < 0 and that (4.4) implies the strict entropy condition. The proof is straightforward. This completes the proof of the theorem. For physical systems with more than two equations, in general it is not possible to reduce to a single first-order equation. To illustrate the approach for a general system, consider a shock profile for a general system with artificial viscosity, ut + f (u)x = κuxx ,
u(x, t) = φ(ξ) ∈ Rn ,
ξ=
x − st . κ
Integrating the second-order differential equations satisfied by the profile φ(ξ) connecting u− and u+ yields the first-order equations (4.5)
φ (ξ) = −s(φ(ξ) − u± ) + f (φ(ξ)) − f (u± ),
u+ ∈ H j (u− ).
We know that a shock satisfying the Liu entropy condition is a Lax shock in that λj (u− ) ≥ s ≥ λj (u+ ); see (6.2) in Chapter 7. For simplicity, consider the generic case of λj (u− ) > s > λj (u+ ). Linearizing the system (4.5) about the end states u− and u+ gives, respectively, (4.6) φ (ξ) = (f (u− ) − sI)φ(ξ), φ (ξ) = f (u+ ) − sI φ(ξ). The matrix f (u± ) − sI has eigenvalues λi (u± ) − s, i = 1, 2, . . . , n. For a weak shock, with u− close to u+ , the shock speed is close to λj (u− ) and so by strict hyperbolicity we have s > λi (u± ) for i < j,
s < λi (u± ) for i > j.
4. The p-System
305
In the generic case of λj (u− ) > s > λj (u+ ), f (u− ) − sI has j + 1 negative and n − j − 1 positive eigenvalues; f (u+ ) − sI has j negative and n − j positive eigenvalues . The first linear system in (4.6) has a (j+1)-dimensional unstable space, while the second linear system has a (n − j − 1)-dimensional stable space in Rn . Thus, by the standard theory of dynamical systems, there exists a (j + 1)dimensional sub-manifold U in Rn with the property that trajectories leave u− at ξ = −∞ along U . At ξ = ∞ trajectories in a (n − j)-dimensional sub-manifold S reach u+ . The total dimension of these two manifolds is j + 1 + n − j = n + 1. The dimension of the whole space is n. Suppose that there is no critical point for the differential equations between the two end states. Then the above implies, by center manifold reduction, that there exists a one-dimensional trajectory C flowing from u− to u+ . Suppose, to the contrary, that such a critical point u exists, that is, the right-hand side of (4.5) is zero: −s(u − u± ) + f (u) − f (u± ) = 0. This violates the strict Liu entropy condition. Without going through the detailed analysis, we state the theorem. Theorem 4.2. A shock profile connecting u− and u+ , with u+ ∈ Hj (u− ) and u− close to u+ , exists for the system of viscous conservation laws ut + f (u)x = κuxx if and only if the strict Liu entropy condition is satisfied: σ(u− , u+ ) < σ(u− , u) for all states u on Hj (u− ) between u− and u+ . The shock profile approaches the discontinuous shock (u− , u+ ) for the system of hyperbolic conservation laws ut + f (u)x = 0 as κ → 0 + . 4.2. Propagation of Discontinuities Systems with physical viscosity are quasilinear and hyperbolic-parabolic. Solutions for a uniformly parabolic system with rough initial data smooth out immediately; see the proof of (7.10) in Proposition 7.2 of Chapter 4. For a system with physical viscosity, as it is not uniformly parabolic, there is the propagation of singularities of the initial data into the solution at later times. Consider the p-system (3.2), τt − vx = 0, vt + px = ( μτ vx )x . Suppose that the initial data (τ, v)(x, 0) has a discontinuity at x = 0: limx→0± (τ, v)(x, 0) = (τ± , v± )(0). Suppose that the singularity propagates
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10. Viscosity
along the curve x = x(t). Denote the states of the jump by lim
x→x(t)±0
(τ, v)(x, t) = (τ± , v± )(t), t > 0.
Then the weak formulation of the p-system yields the jump condition x (t)[τ ](t) = −[v](t), x (t)[v](t) = [p](t) − [ μτ vx ](t), where [A](t) = A+ (t) − A− (t), with A± (t) ≡ A(x(t) ± 0, t), denotes the jump of a quantity A at time t. There is the viscosity term for the momentum equation, so no jump is expected for the velocity, [v](t) = 0, and the first equation yields x (t) = 0, which says that the discontinuity propagates along the particle path in the present setting of Lagrangian coordinates. The second equation then yields [p](t) = [ μτ vx ](t). This and the first equation of the p-system imply that &μ ' ∂ τt (t) = [g(τ (t))], [p(τ )](t) = τ ∂t
g(τ ) ≡
τ
μ(α) dα. α
Here we allow the viscosity to depend on τ, μ = μ(τ ). The function p(τ ) is strictly decreasing, and it is clear that g(τ ) is a strictly increasing function. Therefore the above differential equation [g(τ )] (t) = [p(τ )](t) yields (4.7)
[τ ](t) = ˜e
−O(1)
(c0 )2 t μ0
for a base sound speed c0 and viscosity μ0 and some positive, bounded function O(1). Note that the existence of the discontinuity depends on the hyperbolicity, p (τ ) = −c2 < 0, of the inviscid system. Note also that in the zero dissipation limit μ0 → 0+, the discontinuity vanishes. For the full Navier-Stokes equations in gas dynamics, (3.5), similar analysis shows that the discontinuity also propagates along the particle path. The corresponding inviscid system, the Euler equations in gas dynamics, possess contact discontinuities propagating also along the particle path; see Section 4 of Chapter 7. However, these two discontinuities are not directly related. In the inviscid limit, the discontinuity for the Navier-Stokes equations (3.5) vanishes and the contact discontinuity for the Euler equations surfaces in the limit. The discontinuities for the Navier-Stokes equations belong to the initial layer due to the dissipative mechanism of viscosity. The occurrence of the initial layer is due to the resolution of the singularity of the initial data by the dissipation mechanism. This is reflected also in the appearance of delta functions in the Green’s function as we see in the next subsection.
4. The p-System
307
4.3. Green’s Function Consider the p-system (3.2) linearized around the constant specific volume τ0 and zero velocity v0 = 0: τ τ 0 −1 0 0 τ (4.8) + = , or v t v x −c2 0 0 μ v xx 0 0 0 1 , B= . ut + Aux = Buxx , A = 2 0 μ c 0 Here, for simplicity of notation, we write μ for μ/τ0 from (3.2), and c = −p (τ0 ) is the sound speed at τ0 . Also, we use (τ, v) to denote the deviation of (τ, v) from the base state (τ0 , 0). Consider the Green’s function G(x, t; y, s) = G(x − y, t − s): 0 −1 0 0 Gx = Gxx , Gt + −c2 0 0 μ (4.9) G(x, 0) = δ(x)I. The Green’s function is an operator, represented by a matrix function with initial value being the identity matrix I concentrated at the origin x = 0. The characteristic information, (3.4) in Chapter 7, for the hyperbolic system ut + Aux = 0 is 1 1 , r2 = α , (4.10) λ1 = −c, λ2 = c, r 1 = α −c c 1 1 −c 1 , l2 = c 1 , l1 = 2αc 2αc where the coefficients α and 1/(2αc), for α any constant, are chosen for the normalization lj r k = δjk , j, k = 1, 2. Express u in the coordinates of the right eigenvectors r j , j = 1, 2, for the inviscid system, 2 u = r 1 r 2 w, or u = wj r j , j=1
so that the convection matrix A is diagonalized, −c 0 l1 , A r1 r2 = l2 0 c and the system (4.8) becomes −c 0 μ/2 μ/2 wx = w xx . (4.11) wt + 0 c μ/2 μ/2
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We consider the decoupled convected heat equations by ignoring the cross diffusion, the off-diagonal viscosity coefficients of the viscosity matrix: −c 0 μ/2 0 (4.12) (w∗ )t + (w∗ )x = (w∗ )xx . 0 c 0 μ/2 Remark 4.3. For dissipative systems such as the p-system under consideration here, the solutions dissipate and become slowly varying as time becomes large. For parabolic equations such as the heat equation and Burgers equation in Chapter 4, the delta function in the initial data is resolved with a heat kernel type of singularity and the Green’s function becomes smooth immediately. On the other hand, as the analysis of the propagation of initial discontinuities in Section 3 indicates, the delta function in the initial data for the Green’s function should propagate into the solution at later times. Explicit construction of the Green’s function will reflect these facts. An effective method is the Fourier transform: ∞ e−ixη g(x, t) dx. gˆ(η, t) ≡ −∞
The inverse Fourier transform is g(x, t) =
1 2π
∞
eixη gˆ(η, t) dx.
−∞
Approximate this integral by the Riemann sum g(x, t) = ˜
∞
gˆ(jΔη, t)eijΔηx Δη.
j=−∞
Thus the function g(x, t) is approximated by a sum of terms of the form eiηx gˆ(η, t), η = jΔη. The function eiηx = cos(ηx) + i sin(ηx) is slowly varying for low-frequency modes, |η| small, and is highly oscillatory for high-frequency modes, |η| large. Thus, for the analysis of the large-time behavior of the Green’s function G, it ˆ around η = 0. As the Green’s function suffices to consider the behavior of G has as its initial value the delta function, there is an initial layer where the Green’s function contains singularities. This is studied by considering the ˆ around |η| = ∞. The leading terms for the small- and largebehavior of G time behavior are the delta function and the convected heat kernels. We ˆ for large will therefore look for the Fourier transform of these functions in G and small |η|. The Fourier transform of the delta function is ∞ ˆ e−ixη δ(x) dx = e−ixη |x=0 = 1. (4.13) δ(x) = −∞
4. The p-System
309
The Fourier transform of the convected heat kernel can also be calculated straightforwardly: 1 − (x−dt)2 (4.14) (Hd )t + d(Hd )x = ε(Hd )xx , Hd (x, t; ε) = e 4εt , 4πεt ∞ 1 − (x−dt)2 2 ˆ d (η, t; ε) = e 4εt dx = e−idηt−εη t . e−ixη H 4πεt −∞ Take the Fourier transform of (4.9) to get 0 iη ˆ ˆt = G, G c2 iη −μη 2 (4.15) ˆ 0) = I. G(ξ, This differential equation in t is solved to yield ⎛ λ t ⎞ λ+ e − −λ− eλ+ t eλ+ t −eλ− t iη λ+ −λ− λ+ −λ− ˆ ⎠. (4.16) G(η, t; μ, c) = ⎝ λ+ eλ+ t −λ− eλ− t eλ+ t −eλ− t iη λ+ −λ− λ+ −λ− Here μ (4.17) λ± = λ± (η; μ, c) ≡ − η 2 ∓ iη 2 are the eigenvalues of 0 iη . ic2 η −μη 2
1 c2 − μ2 η 2 4
The expansion of the eigenvalues at |η| = ∞ is (4.18) λ+ = −
c4 c2 − 3 η −2 + O(1)η −4 as |η| → ∞, μ μ c4 c2 + 3 η −2 + O(1)η −4 as |η| → ∞. λ− = −μη 2 + μ μ
From this, c2 1 1 0 0 1 − t ˆ + (4.19) G(η, t; μ, c) = e e μ 0 0 1 0 iημ 1 −μη2 t 0 1 2 0 0 e + + O(1)η −2 as |η| → ∞. + e−μη t 0 1 1 0 iημ 2
− cμ t
The expansion of the eigenvalues around η = 0 is 1 μ2 (4.20) λ+ = −iηc − μη 2 + iη 3 + O(1)η 4 as |η| → 0, 2 8c 1 μ2 λ− = iηc − μη 2 − iη 3 + O(1)η 4 as |η| → 0. 2 8c
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This yields 1 1 μ 1 2 0 2 2 2c + iη − 4c ˆ + O(1)η (4.21) G(η, t; μ, c) = e(icη− 2 μη )t μ 1 1 0 2c 2 4c 1 μ 1 − 2c 0 (−icη− 21 μη 2 )t 2 2 4c +e + iη + O(1)η as |η| → 0. μ 1 1 0 − 4c − 2c 2 These asymptotic formulas yield explicit expressions for the leading terms in the Green’s function. Theorem 4.4. The Green’s function G(x, t; μ, c) has the following properties: 1. For 0 < t < 1, (4.22) G(x, t; μ, c) = e
2
− cμ t
x2 1 0 0 1 0 − 4μt +O(1)e−σ|x| e δ(x) +√ 0 1 0 0 4πμt
for some positive constant σ. 2. For t ≥ 1,
1 1 (x−ct)2 1 1 0 − 2c − 2 +√ e 2μt 1 1 − 2c 0 0 2πμt 2 μ 1 1 (x+ct)2 (x−ct)2 1 1 0 − − 2μt 4c 2 2c + ∂ √ e 2μt e +√ x μ 1 1 0 − 4c 2πμt 2πμt 2c 2 μ (x+ct)2 1 (x+ct)2 (x−ct)2 3 0 − 4c e 2μt + O(1)t− 2 e− Dt + e− Dt + ∂x √ μ 0 − 4c 2πμt
(4.23) G(x, t; μ, c) = e
2
− cμ t
δ(x)
+ O(1)e−σ|x|−σt for some positive constant σ and for any constant D > 2μ. By Remark 4.3, the explicit form of the leading terms in (4.22) and (4.23) of the Green’s function are obtained directly from the leading terms in (4.19) and (4.21). The complete proof of the theorem requires inversion of the Fourier transform using complex contour integrations. We don’t go into the details here. We now examine the significance of the leading terms. The delta function δ(x) in G, the first term in (4.22) and (4.23), decays exponentially in time and pertains only to the specific volume. This is consistent with the propagation of the discontinuities in the solution as studied in (4.7). Thus the delta function belongs to the initial layer. For small time, 0 < t < 1, there is a heat kernel, the second term in (4.22), which, together with the first term, resolves the initial singularity δ(x)I. On the other hand, the heat kernel after the initial time, t ≥ 1, is represented by the second and third terms in (4.23), which propagate with acoustic speeds ±c and with viscosity
4. The p-System
311
coefficient μ/2. Thus they correspond to the diagonal system (4.12). The off-diagonal elements relate to the faster-decaying fourth and fifth terms in (4.23). The heat kernels propagating with speed ±c have a direct correspondence with the inviscid system. Thus the inviscid system corresponds to the large-time behavior of the solutions for the viscous system. This is in agreement with the scaling analysis done earlier in Section 8 of Chapter 4. Similarly, directly from the expansions (4.19) and (4.21), there is the following expression for the differential of the Green’s function. Theorem 4.5. The differential ∂x G(x, t; μ, c) of the Green’s function has the following properties: 1. For 0 < t < 1, 1 − cμ2 t 1 0 0 1 (4.24) ∂x G = e δ (x) + e δ(x) 0 0 1 0 μ 1 2 x x2 1 1 0 0 0 1 − 4μt − 4μt + √ e e + O(1)e−σ|x| + ∂x √ 0 1 1 0 μ 4πμt 4πμt 2
− cμ t
for some positive constant σ. 2. For t ≥ 1, for any D > 2μ, 1 1 1 (x−ct)2 − 2c − 2 e 2μt (4.25) G(x, t; μ, c) = ∂x √ 1 1 − 2c 2πμt 2 μ 1 1 1 1 (x+ct)2 (x−ct)2 0 − 2μt − 2μt 2 2c 4c + ∂x √ + ∂x √ e e μ 1 1 0 − 4c 2πμt 2πμt 2c 2 1 (x+ct)2 μ (x+ct)2 (x−ct)2 0 − 4c e 2μt O(1)t−2 e− Dt + e− Dt + ∂x √ μ 0 − 4c 2πμt + O(1)e−σ|x|−σt for some positive constant σ and for any constant D > 2μ. The singularity of δ (x) in (4.24) has the same direction as that of δ(x) for G in (4.22), as expected. Note that the new singularity of δ(x) in (4.24) pertains to the off-diagonal elements. Similarly, singularities in the new directions also appear in (4.23). These singularities are found by going through the expansion on the Fourier transform of the Green’s function. 4.4. Energy Estimate Consider the following linear system resulting from conservation laws with artificial viscosity: ut + Aux = κuxx .
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The basic energy estimate is obtained by integrating the equation times uT : 1 2 d |u| dx + κ|ux |2 dx = 0. dt R 2 R The key hypothesis for the above to hold is symmetry of the flux matrix A. This is also the case for hyperbolic systems; see Section 3 of Chapter 6. For physical models, the viscosity matrix is degenerate and so the specific form of the equations needs to be used for the energy estimate. Consider the linear equations (4.8): τt − vx = 0, vt − τx = μvxx .
(4.26)
Here, for simplicity, we have scaled away the sound speed and assumed c = 1. The first estimate is a standard one obtained by integrating the sum of the first equation in (4.26) times τ plus the second equation times v: τ 2 + v2 − (τ v)x + μ(vx )2 = μ(vvx )x , t 2
(4.27) (4.28) R
τ 2 + v2 (x, t) dx + 2
t 0
R
μ(vx )2 dx dt =
R
τ 2 + v2 (x, 0) dx. 2
The same procedure applied to the differential of the system yields as estimate on the higher differential: (4.29) R
(τx )2 + (vx )2 (x, t) dx + 2
t 0
μ(vxx )2 dx dt R (τx )2 + (vx )2 (x, 0) dx. = 2 R
However, unlike in the parabolic case, the second term in the above energy estimate (4.28) contains only (vx )2 and not the (τx )2 term. To study smooth solutions, the next step is crucial and uses the coupling of the system (4.26) to obtain an estimate for (τx )2 . This is done by multiplying the system by 0 −1 : (τx , vx ) times a skew-symmetric matrix 1 0 (4.30) (τ, v)x
0 −1 1 0
τ 0 −1 0 −1 τ + (τ, v)x v t 1 0 0 −1 v t 0 −1 0 0 τ = (τ, v)x . 1 0 0 μ v xx
5. General Dissipative Systems
313
The first term in the above identity is of divergence form, τt vx − τx vt = (τ vx )x − (τ vt )x , and so the above yields t t (4.31) (τx )2 dx dt = (vx )2 − μτx vxx dx dt 0 0 R R − (τ vx )(x, t) + (τ vx )(x, 0) dx. + R
From (4.28), (4.29), and (4.31) one obtains t (τx )2 + (vx )2 + τ 2 + v 2 (τx )2 + (vx )2 dx dt (x, t) dx + (4.32) 2 0 R R 2 (τx ) + (vx )2 + τ 2 + v 2 = O(1) (x, 0) dx. 2 R This represents the first step toward a general theory for physical systems as outlined in the next section.
5. General Dissipative Systems Consider a system of the following general form: (5.1) ut + ∇x · F(u) = ∇x · B(u)∇x u . The most basic property of the physical system (5.1) is the existence of an entropy pair (η(u), q(u)). Besides the requirement for hyperbolic systems, Definition 2.1 of Chapter 6, there is the dissipation requirement η B ≥ 0 for the viscosity matrix B: There exists an entropy pair (η(u), q(u)) such that η(u) is strictly convex, η > 0, the compatibility condition η F = q holds, and the dissipation requirement η B ≥ 0 is satisfied. This gives the basic entropy estimate t η(u(x, t) dx + η(u(x, 0) dx. ∇x uη B∇x u dx dt = Rm
0
Rm
Rm
The entropy estimate is a wholly nonlinear estimate. The estimates (4.27) and (4.28) constitute a linear version of the entropy estimate for the psystem. Consider the p-system τt − vx = 0, vt + p(τ )x = μτ vx x . Integrate the first equation times −p(τ ) plus the second equation times v to yield an equation of divergence form, v2 τ μ μ − vvx x , p(w) dw + vp(τ ) x = − (vx )2 + 2 τ τ t
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which yields the identity 2 τ t μ v − (vx )2 dx dt p(w) dw (x, t) dx + 2 τ 0 R R 2 τ v − p(w) dw (x, 0) dx. = 2 R This is the entropy estimate for the p-system with the entropy pair τ v2 − p(w) dw, η(u) = vp(τ ). η(u) = 2 The entropy η(u) is the sum of the kinetic energy and the potential energy. Since the p-system is for isentropic flows, there is no conservation of energy. The energy now serves as the mathematical entropy. To have a global-in-time theory, the general system (5.1), though not parabolic, needs to be dissipative. For instance, instead of considering the p-system, consider τt − τx = 0, vt + p(τ )x = μτ vx x . There is no coupling; the first equation is a self-contained transport equation, τ (x, t) = τ (x + t, 0), and not dissipative. The requirement of coupling for general systems of the form (5.1) is expressed as follows: (5.2)
r(u; ξ) is not in the null space of B(u).
Here r(u; ξ) is any right eigenvector of f (u; ξ) and f (u; ξ) ≡ F(u) · ξ is the projection of the flux in any direction ξ ∈ Rm , |ξ| = 1. Finally, the general system (5.1) is required to satisfy the third hypothesis of having block structure as for the p-system. With the above three hypotheses, which are satisfied by basic physical models, the system is dissipative and a global smooth solution near a constant exists, as for the p-system.
6. Notes The analysis carried out in Section 10.4 for the p-system has been extended by varying degrees to more general systems. The construction of shock profiles, described in Section 4.1, for general viscous conservation laws is done using the center manifold theory of differential equations, e.g. [47, 111]. For shocks of moderate strength, a viscous shock profile exists if and only if the entropy condition holds. Shock profiles of large strength can be constructed for some models, such as the Navier-Stokes equations in gas dynamics [52]. The analysis of propagation of discontinuities for physical systems has been carried out for compressible Navier-Stokes equations; see [60]. The analysis for more general systems should start with identification of delta functions
7. Exercises
315
in the Green’s function, as in Section 4.3. The construction of Green’s functions for more general systems was done by Liu-Zeng in 1997; see [100] and references therein. The only existing construction of Green’s functions in several space dimensions is that for the compressible Navier-Stokes equations, following the technique for constructing the Green’s function for the Boltzmann equation; see the reference Liu-Yu (2006) in [97]. It would be interesting to construct Green’s functions for other physical systems, such as the magneto-hydrodynamics equations and viscoelasticity equations. The global theory for smooth and small initial data is established using the energy method; see Section 4.4 for the p-system. Generalization to general systems was done first for the Navier-Stokes equations in gas dynamics [64, 69, 106]. There is a global-in-time theory for general systems, the Kawashima-Shizuta theory with the aforementioned three hypotheses; see Section 5, [115], and [68]. Small, smooth initial data will yield global solutions, which decay time-asymptotically in energy norms. As just mentioned, the Green’s function in one space dimension for such general systems has been constructed. It would be interesting to construct the Green’s function for such a general system in several space dimensions. For studies of large-time behavior of solutions that are not dissipative, see [128] and references therein. Systems with partial dissipation need to be more systematically studied. The study of nonlinear stability of viscous shock waves was done first for the case where the perturbation is of zero total mass, by Goodman [58] and Matsumura-Nishihara [107]. The study of wave propagation over a shock profile was done by Liu in 1985, Liu-Zeng in 1997, and Liu-Zeng in 2015; see [100] and references therein. This long line of analysis starts with the simplest example of nonlinear wave interactions for a system, Theorem 2.2; see [88]. It would be interesting to study the propagation over waves with stronger nonlinear properties, such as a combination of shock and rarefaction waves. The study of scalar waves in Proposition 1.3 and Proposition 1.4 would be the natural starting point. The study of wave propagation in more than one space dimension is related to the aforementioned construction of Green’s functions and is an important topic for future research. It would be desirable to bring the study of one-dimensional viscous waves to the level of the analysis of wave interaction for inviscid waves, as presented in Chapter 8. A basic topic is the interaction of boundary and interior waves, e.g. [10]. We will touch upon this topic in Chapter 12.
7. Exercises 1. To obtain the optimal convergence rates, (7.8) and (7.9), in Section 7 of Chapter 4, the Burgers Green’s function GD (x, t; y, s) is used for Duhamel’s principle for a solution with compact support of the Burgers equation, (7.11).
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10. Viscosity
Apply the same approach to convex scalar conservation laws to obtain a refined estimate, e.g. (1.3) in Section 1: − x2 u(x, t) − θ(x, t + 1; c) = O(1)ε (t + 1)−1 + ε2 (t + 1)−1+0 e D(t+1) . 2. Use the Green’s function for the Burgers shock, (6.11) and (6.12) in Section 6 of Chapter 4, to study the shock stability of the Burgers shock in the pointwise sense, similar to the approach for rarefaction waves in Proposition 1.5. 3. Consider the isentropic Navier-Stokes equations (3.1): ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p(ρ))x = μ(ρ)vx x , with the viscosity coefficient μ = μ(ρ) being a function of the density ρ. Suppose that the constitutive law is p(ρ) = ργ with γ > 1 and the viscosity coefficient is of the form μ(ρ) = ρα α > 0. Construct the shock profiles for the system. 4. Consider the full Navier-Stokes equations in Eulerian coordinates, (3.4). Describe the propagation of a discontinuity starting from x = 0 at the initial time t = 0. 5. Construct the large-time and small-time leading terms of the Green’s function for the linearized full Navier-Stokes equations in Eulerian coordinates, (3.4). Compare the small-time result with the description in problem 4. 6. Find a non-decaying perturbation of a constant state for the full NavierStokes equations (3.5) with zero heat conductivity, κ = 0, and positive viscosity, μ > 0. Show that the coupling hypothesis (5.2) does not hold for (3.5) when κ = 0.
10.1090/gsm/215/11
Chapter 11
Relaxation
Relaxation is a common phenomenon in various natural situations. It occurs in high-temperature gas dynamics, elasticity with memory, kinetic theory, etc. In such situations, the conserved quantities depend also on other variables, sometimes called the internal variables. As the system relaxes and reaches the equilibrium states, the internal variables assume the form of a given function of the conserved quantities. Thus the resulting conservation laws govern the evolution of the equilibrium states. In kinetic theory, for instance, the internal variables form an infinite-dimensional space and the conservation laws are the gas dynamics equations. This chapter presents a general theory for a simple model and uses some physical examples to illustrate the general ideas and methodology. Consider the simple example of the traffic flow model ρt + F (ρ)x = 0, (2.5) of Chapter 2. The model assumes that the traffic flux ρv is given by the stationary flux ρv = F (ρ). This can be made more realistic by assuming that it takes time, of the order of τ , for the flux of a general flow to relax to the stationary flux:
ρt + (ρv)x = 0, (ρv)t + (ρv 2 )x =
F (ρ)−ρv . τ
The equilibrium equation, called the Euler equation, is the scalar conservation law ρt + F (ρ)x = 0 when the relaxation to the equilibrium state F (ρ) − ρv = 0 is immediate as τ → 0+. Relaxation induces dissipation. Viscous conservation laws, called Navier-Stokes equations, can be derived by various expansions. There are the so-called Chapman-Enskog expansion and Hilbert expansion, notions borrowed from the theory of the Boltzmann 317
318
11. Relaxation
equation in kinetic theory. Relaxation phenomena are interesting, particularly in the presence of shock waves. We start with a simple model to illustrate the general theory of hyperbolic conservation laws with relaxation. We then concentrate on specific physical models, the thermal non-equilibrium Euler equations and the Boltzmann equation.
1. A Simple Relaxation Model Consider a 2 × 2 system of hyperbolic conservation laws with relaxation: (1.1)
ut + f (u, v)x = 0, . vt + g(u, v)x = V (u)−v τ
The first equation is a conservation law. The second equation postulates that the second variable, the internal variable v, has a tendency to relax to its equilibrium state v = V (u) within a time period of the order of the relaxation time τ . 1.1. Frozen and Equilibrium Speeds There are two limiting considerations. The first is the immediate wave propagation, which ignores the lower-order term representing the slowerin-time relaxation process, or, equivalently, takes the relaxation time to be large, τ → ∞, to obtain the corresponding 2 × 2 conservation laws, referred to as the frozen system: (1.2)
ut + f (u, v)x = 0, vt + g(u, v)x = 0.
The system is assumed to be strictly hyperbolic: fu fv rj (u, v) = λj (u, v)rj (u, v), j = 1, 2, (1.3) gu gv λ1 (u, v) < λ2 (u, v), frozen speeds. The characteristic speeds λi (u, v), i = 1, 2, are called the frozen characteristic speeds as they represent the instantaneous speeds of wave propagation when the relaxation process is frozen, or ignored. The second limiting consideration is that the relaxation is immediate, τ → 0+, so that the equilibrium state v = V (u) is attained and we obtain the self-contained equilibrium conservation law, the Euler equation: (1.4) ut + F (u)x = 0, Euler equation, F (u) ≡ f (u, V (u)), equilibrium flux; Λ(u) ≡ F (u), equilibrium speed.
1. A Simple Relaxation Model
319
Following the tradition in the kinetic theory for the Boltzmann equation, discussed in Section 4 later, the equation (1.4) is called the Euler equation for the relaxation model (1.1). The equilibrium speed Λ(u) represents the speed of wave propagation after the relaxation process is completed. 1.2. Sub-characteristic Condition By the self-similar transformation x → αx and t → αt, with α a positive constant, the relaxation model (1.1) remains of the same form, but with the relaxation time τ → ατ. Thus as α becomes small, the relaxation time becomes small and the solutions approach the equilibrium values and are therefore governed by the equilibrium conservation law (1.4). For fixed αt, the time t becomes large for small α. Therefore, time-asymptotically, the wave pattern in the solution of the relaxation system (1.1) is the same as in the solution of the equilibrium equation (1.4). For a short time, waves propagate with the frozen speeds λj (u, v), j = 1, 2. In order for the waves to propagate with the equilibrium speed Λ(u) time-asymptotically, the following sub-characteristic condition should hold: (1.5)
λ1 (u, v) < Λ(u) < λ2 (u, v)
for v near its equilibrium value V (u). This natural condition on the propagation of information can be compared with the C-F-L condition for numerical schemes; see (10.1) of Chapter 9. The C-F-L condition requires that the domain of dependence of the numerical solutions include the domain of dependence of the exact solutions. This is needed in order for the numerical solutions to have the possibility of approaching the exact solutions as the mesh sizes tend to zero. 1.3. Chapman-Enskog Expansion The sub-characteristic condition (1.5) is now shown to be directly related to the dissipative property induced by the relaxation. This is analyzed by the so-called Chapman-Enskog expansion, originally introduced for the Boltzmann equation in kinetic theory. The expansion is based on two hypotheses: First, the deviation from the equilibrium is small, |v − V (u)| 1, and the primary propagation direction follows the equilibrium speed, ˜ 0. ∂t + Λ∂x = Second, the solution is slowly varying in that higher differentials are smaller than the lower ones; for instance, |∂(x,t) u| |u|,
|∂(x,t) v| |v|.
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11. Relaxation
This slowly varying hypothesis is appropriate when one is studying dissipation phenomena. With the above two hypotheses, the second equation in (1.1) is simplified to V (u) − v = vt + g(u, v)x = ˜ V (u)t + g(u, V (u))x = ˜ Λ(u)V (u)x + g(u, V (u))x , τ so the second equation in (1.1) is replaced by the approximation v = V (u) − τ − Λ(u)V (u)x + g(u, V (u))x , Chapman-Enskog relation. Plug the Chapman-Enskog relation into the first equation in (1.1) to obtain ut + f u, V (u) − τ −Λ(u)V (u)x + g(u, V (u))x x = 0. By Taylor expansion, and applying the above hypotheses again, this is further approximated by (1.6) ut + F (u)x = μ(u)ux x , μ(u) ≡ τ fv Λ(u)V (u) + gu + gv V (u) . Here the functions fv , gu , and gv are evaluated at equilibrium, e.g. fv = fv (u, V (u)). The equilibrium speed is d f (u, V (u)) = fu + fv V (u). du From this, straightforward calculations show that the above viscosity coefficient μ(u) can be expressed in terms of the characteristic speeds and we have (1.7) ut + F (u)x = μ(u)ux x , Navier-Stokes equation, μ(u) = τ Λ(u) − λ1 (u, V (u)) λ2 (u, V (u)) − Λ(u) . Λ(u) = F (u) =
The sub-characteristic condition (1.5) and the positivity of the dissipation parameter, μ(u) > 0, are therefore equivalent. This is natural, as the hypotheses leading to the viscous conservation law (1.7) are valid for diffuse waves and for large time, and it is the time-asymptotic consideration that leads to the derivation of the sub-characteristic condition (1.5). Drawing an analogy with kinetic theory, we call the viscous conservation law (1.7) the system of Navier-Stokes equations associated with the relaxation model (1.1). 1.4. Hilbert Expansion The Hilbert expansion in kinetic theory can be applied systematically to derive simpler equations when the relaxation time is small, τ 1. The set-up of the Hilbert expansion depends on the physical situation under consideration. To study large-time phenomena, the time variable t is replaced with
1. A Simple Relaxation Model
321
t/τ , and (1.1) becomes Q(u) , or τ 2 ut + τ f (u)x = Q(u); τ u f (u, v) 0 u= , f (u) = , Q(u) = . v g(u, v) V (u) − v
(1.8) τ ut + f (u)x =
The new time of order 1 corresponds to the original time of order τ −1 1 for small relaxation times, τ 1. Consider the case of weakly nonlinear expansion around a constant state u0 and expand the solution of (1.8) as u = u0 + τ u1 + τ 2 u2 + . . . .
(1.9)
Plug the expression (1.9) into the system (1.8) and compare the terms of the same order in τ . The zeroth-order term τ 0 yields (1.10) Q(u0 ) = 0 or u0 = (u0 , V (u0 ))T , an equilibrium constant state. The τ , τ 2 , and τ 3 terms yield the following systems: (1.11)
0 = Q (u0 )u1 ;
(1.12)
1 f (u0 )(u1 )x = Q (u0 )u2 + Q (u0 )u1 u1 ; 2
1 (1.13) (u1 )t + f (u0 )(u2 )x + f (u0 )(u1 u1 )x 2 1 = Q (u0 )u3 + Q (u0 )u1 u2 + Q (u0 )u1 u1 u1 . 6 T In terms of the components uj = (uj , vj ) , j = 1, 2, 3, these are (1.14) (1.15)
(1.16)
fu0 gu0
v1 = V0 u1 ; 0 fv0 u1 = ; gv0 v1 x V0 u2 + 12 V0 (u1 )2 − v2
1 fuu0 fvv0 fu0 fv0 u2 (u1 )2 u1 + + v1 t gu0 gv0 v2 x 2 guu0 gvv0 (v1 )2 x fuv0 0 + (u1 v1 )x = , guv0 V0 u1 u2 − v3
where fu0 ≡ fu (u0 , V (u0 )), etc. From (1.14) and the first equation in (1.15), 0 = fu0 (u1 )x + fv0 (v1 )x = fu0 + V0 fv0 (u1 )x = Λ0 (u1 )x . Thus, to obtain a non-trivial solution, (u1 )x = 0, the equilibrium characteristic speed Λ0 at the base state u0 needs to be zero: (1.17)
Λ0 = Λ(u0 ) = 0.
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11. Relaxation
In kinetic theory, this would be the small Mach number hypothesis for the Hilbert expansion. From the second equation in (1.15), (1.18)
1 v2 = V0 u2 + V0 (u1 )2 − gu0 (u1 )x + gv0 (v1 )x . 2
From (1.14), (1.17), and (1.18), the first equation in (1.16) becomes (u1 )t + fv0
1
V0 (u1 )2 − gu0 + V0 gv0 (u1 )x 2 x 1 fuu0 + 2V0 fuv0 + (V0 )2 fvv0 (u1 )2 = 0. + 2 x
From the expression (1.4) for the equilibrium flux function F (u) = f (u, V (u)), the above becomes 1 F0 (u1 )2 = fv0 gu0 + V0 gv0 (u1 )x . (u1 )t + 2 x x Using the same computations as in deriving the viscosity coefficient μ for the Chapman-Enskog expansion, (1.6) and (1.7), in the present case of Λ0 = 0, (1.17), the above can be written as (1.19) (u1 )t +
1 2
F0 (u1 )2
= μ0 (u1 )xx , x μ0 = fv0 gu0 + V0 gv0 = −λ1 (u0 , V (u0 ))λ2 (u0 , V (u0 )).
This is the Burgers equation resulting from the Hilbert expansion. The viscosity coefficient μ0 = −λ10 λ20 = (λ20 − Λ0 )(Λ0 − λ10 ) is positive by the sub-characteristic condition. Notice that in the above derivation, the first equation, the conservation law, gives the final single conservation law as an approximation of the original system. The role of the second equation, the rate equation, is to provide the expression for vi in terms of uj , j ≤ i, and their differentials, so that the final conservation law becomes a self-contained equation. This is the general procedure for the Hilbert expansion applied to the Boltzmann equation in kinetic theory. Borrowing the notions from kinetic theory, the Hilbert expansion carried out above for the relaxation model is done under the hypotheses of small relaxation time, (1.9), weak nonlinearity, i.e. small perturbation of an equilibrium state, (1.10), small Mach number, (1.17), and large-time asymptotics, (1.8). For the Boltzmann equation, several fluid dynamics equations can be derived for different physical situations. 1.5. Shock Waves A discontinuous shock (u− , u+ ) = ((u− , v− )T , (u+ , v+ )T ) for the relaxation model (1.1) with speed σ satisfies the same Rankine-Hugoniot condition and
1. A Simple Relaxation Model
323
entropy condition as for the hyperbolic conservation laws (1.2): f (u+ , v+ ) − f (u− , v− ) u+ − u− = −σ . v+ − v− g(u+ , v+ ) − g(u− , v− ) Jump discontinuities tend to be smoothed out by the dissipation mechanism induced by the relaxation. In their place, smooth shock profiles, like those for viscous conservation laws, emerge as permanent wave forms. Actually, the dissipation induced by relaxation is not as strong as that for viscous conservation laws, and a strong traveling wave may contain discontinuities as sub-shocks. We will consider this situation in Chapter 11 as part of the study of resonance phenomena. Moreover, as a hyperbolic system, discontinuities of moderate strength in the initial data propagate into the solution at later times and are smoothed out by dissipation only time-asymptotically. For now, we only study the smooth shock profile (u, v)(ξ) of (1.1): u u (x, t) = (ξ), ξ ≡ x−σt τ ; v v u u1 u u2 (ξ) → as ξ → −∞, (ξ) → as ξ → ∞; (1.20) v1 v2 v v f (u, v) 0 u (ξ) + (ξ) = (ξ). −σ g(u, v) V (u) − v v Note that the differential equations in (1.20) do not depend on the relaxation time τ due to the choice of the self-similar variable ξ = (x − σt)/τ . As a consequence, the shock profile tends to a jump discontinuity in the zero relaxation limit, τ → 0+. The profile becomes flat at ξ = ±∞ and so the end states (u1 , v1 ) and (u2 , v2 ) are constant solutions of (1.1). A solution of (1.1) with initial value being a constant state (¯ u, v¯) is governed by the ordinary differential equations ⎧
⎪ ⎨ut = 0, u(x, t) = u ¯, t or (1.21) , vt = V (u)−v τ ⎪ v(x, t) = V (¯ u) + v¯ − V (¯ u) e − τ . ⎩ (u, v)(x, 0) = (¯ u, v¯), Thus the solution (u, v)(x, t) tends to the equilibrium state (¯ u, V (¯ u)) at an exponential rate e−t/τ . In particular, the only constant-state solutions are the equilibrium states, and so the end states (u1 , v1 ) and (u2 , v2 ) of the shock profile satisfy (1.22)
vj = V (uj ), f (uj , vj ) = f (uj , V (uj )) = F (uj ), j = 1, 2.
By (1.22), the first differential equation in (1.20) is integrated from ξ = −∞ to yield (1.23) 0 = −σ(u − u1 ) + f (u, v) − f (u1 , v1 ) = −σ(u − u1 ) + f (u, v) − F (u1 ).
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As x → ∞, (u, v) → (u2 , v2 ) and f (u, v) → F (u2 ) and so the above implies that F (u2 ) − F (u1 ) (1.24) 0 = −σ(u2 − u1 ) + F (u2 ) − F (u1 ), or σ = . u2 − u1 Therefore the end states of the shock profile satisfy the same RankineHugoniot condition as the shock (u1 , u2 ) for the scalar equilibrium conservation law (1.4). In particular, we know from the theory for scalar laws that the speed σ of a shock of moderate strength is close to the equilibrium speed Λ(u) = F (u). The first conservation law in (1.1) is assumed to be strongly coupled to the second relaxation equation: fv (u, v) = 0.
(1.25)
The following proposition shows that, not only do the end states of a shock profile satisfy the Rankine-Hugoniot condition for the equilibrium equation, the existence of the shock profile for the full system is ensured by the Oleinik entropy condition, (1.7) of Chapter 5, for the corresponding inviscid shock for the equilibrium equation. Proposition 1.1. A smooth shock profile, as given in (1.20), of sufficiently small strength exists if and only if the corresponding equilibrium shock (u1 , u2 ) satisfies the strong Oleinik entropy condition for (1.4): (1.26) σ =
F (u) − F (u1 ) F (u2 ) − F (u1 ) < for all u between u1 and u2 . u2 − u1 u − u1
Proof. Rewrite the differential equations in (1.20) as 0 fu fv u (ξ) = (ξ). (1.27) −σI − gu gv V (u) − v v The speed σ of the shock is governed by the Rankine-Hugoniot condition for the equilibrium equation, (1.24). Therefore, for a shock of sufficiently small strength, its speed σ is close to the equilibrium speed Λ and so the matrix on the left-hand side of (1.27) is non-singular by the sub-characteristic condition (1.5). A necessary condition for the existence of the shock profile is for the right-hand side of the differential equations (1.27) to be nonzero, v = V (u), for u between u1 and u2 along the profile. If the Oleinik entropy condition (1.26) is violated, σ=
F (u) − F (u1 ) for some u between u1 and u2 , u − u1
then we have from (1.23) that f (u, v) = F (u) = f (u, V (u)). By the coupling condition (1.25), fv = 0, we conclude that v = V (u) and so (u, v) is a critical state for the differential equations (1.27) and there is no profile connecting
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325
(u1 , v1 ) and (u2 , v2 ). By similar arguments we conclude that the profile exists if the entropy condition (1.26) holds. When the shock strength is not moderate, the differential equations (1.27) can become singular as the shock speed σ may equal the frozen speed λj , j = 1, 2. In this case there will be sub-shocks. We discuss this in Chapter 12 on resonance. 1.6. Generalized Riemann Problem The theory for the equilibrium conservation law (1.4) describes the global behavior of solutions of the relaxation system (1.1) in the following sense. The time-asymptotic behavior of a general solution u(x, t) of the conservation law (1.4) is determined by its initial values ul = u(−∞, 0) and ur = u(∞, 0) at x = ±∞; see Section 7 of Chapter 9. That is, u(x, t) tends to the wave pattern in the solution of the Riemann problem (ul , ur ). This is the case also for the solution (u, v)(x, t) of the relaxation model (1.1) in the following sense. Given the initial values (u, v)(x, 0) for (1.1) with limiting values (ul , vl ) = (u, v)(−∞, 0) and (ur , vr ) = (u, v)(∞, 0), the limiting constant states approach the equilibrium states at the exponential rate e−t/τ in (1.21). Consequently, after an initial layer of duration t = O(1)τ , the end states are very close to their respective equilibrium values (ul , V (ul )) and (ur , V (ur )). For time-asymptotic analysis, it suffices to consider the case where the limiting values are the equilibrium values for all time: (u, v)(−∞, t) = (ul , V (ul )),
(u, v)(∞, t) = (ur , V (ur )).
The following notion of generalized Riemann problem will be useful for identifying the time-asymptotic states. Definition 1.2. Consider two equilibrium states (ul , V (ul )) and (ur , V (vr )). Let u(x, t) = φ(x/t) be the solution of the Riemann problem (ul , ur ) for the equilibrium conservation law (1.4) as constructed in Chapter 5 (for systems in Chapter 7). The wave pattern obtained by replacing a rarefaction wave ψ(x, t) in φ(x/t) by its equilibrium values (ψ, V (ψ)(x, t)) and replacing a shock wave (u− , u+ ) in φ(x/t) by the corresponding shock profile for (1.1) is called the solution of the generalized Riemann problem T T (ul , V (ul )) , (ur , V (vr )) for the relaxation system (1.1). Notice that the solution of the generalized Riemann problem described above is not an exact solution of the relaxation model (1.1) because the corresponding rarefaction waves of the equilibrium values are not solutions and the tails of the corresponding shock profiles would overlap with nearby
326
11. Relaxation
waves. Nevertheless, the wave pattern represents a time-asymptotic solution of the relaxation system by the following considerations. For a given rarefaction wave ψ(x, t) of the equilibrium equation (1.4), the corresponding local equilibrium rarefaction function, defined by (u, v)(x, t) ≡ (ψ, V (ψ))(x, t), satisfies the first equation of (1.1): ut + f (u, v)x = ψt + f (ψ, V (ψ))x = ψt + F (ψ)x = 0. However, (ψ, V (ψ)) does not satisfy the second equation of (1.1) exactly, but with a truncation error E(x, t): vt + g(u, v)x −
V (u) − v V (ψ) − V (ψ) = ψt + g(ψ, V (ψ))x − , or τ τ
V (u) − v + E(x, t), E(x, t) ≡ ψt + g(ψ, V (ψ))x . τ By the linear expansion of the rarefaction wave ψ(x, t), (4.8) of Chapter 2, the trucation error E(x, t) decays at the rate of t−1 and so (ψ, V (ψ) is an accurate time-asymptotic solution of the relaxation system (1.1). There is another truncation error in the solution of the generalized Riemann problem, which is due to the overlapping of shock profiles and local equilibrium rarefaction functions. As the rarefaction waves and the shock waves in the Riemann solution (ul , ur ) for the equilibrium equation (1.4) are non-interacting and a shock profile tends to its end states exponentially fast, the overlap gives rise also to truncation error decaying at the rate of t−1 . In conclusion, the solution to the generalized Riemann problem is a time-asymptotic solution of the full system (1.1). vt + g(u, v)x =
The waves in the solution of the generalized Riemann problem are timeasymptotically non-interacting. A general solution of the relaxation model (1.1) is expected to approach the solution of the corresponding generalized Riemann problem. The generalized Riemann problem offers a simple and definite way to study the time-asymptotic states of general solutions. For instance, the solution would contain discontinuity waves time-asymptotically if the shock profiles in the solution of the generalized Riemann solution contain discontinuous sub-shocks.
2. Examples The above theory can be generalized straightforwardly to the general 2 × 2 system (2.1)
ut + f (u, v)x = 0, vt + g(u, v)x = h(u, v).
2. Examples
327
For the system to be a relaxation model, we need the coupling hypothesis as well as the hypothesis on the source h(u, v): (2.2)
fv = 0,
hv < 0.
Besides the model for traffic flows, we mention here some more examples. The first example concerns river flow. Let h be the height of the water, v the velocity, g the gravitational constant, cf the friction coefficient between the water and the river bottom, and α the inclination angle of the river. The shallow water model is
ht + (hv)x = 0, (2.3) α 2 (hv)t + (hv 2 + g cos 2 h )x = −Cf v + (g sin α)h. Equilibrium is reached when the friction force Cf v and the gravitational force (g sin α)h are equal, Cf v = (g sin α)h, or v = V (h) ≡
g sin α h. cf
The equilibrium conservation law and equilibrium speed Λ(h) are ht + F (h)x = 0, F (h) ≡
g sin α 2 h , cf
Λ(h) = F (h) =
2g sin α h. cf
It is easily shown that the sub-characteristic condition holds if h
0, a < 0, a > 0, and a(s), a (s), a (s) → 0 as s → ∞ so that it is a fading memory. Although this model is not of the form discussed so far, analogous ideas can be applied. The frozen system is the one with no memory:
ut − vx = 0, (2.6) vt − f (u)x = 0, with frozen characteristic speeds λ1 (u, v) = − f (u),
λ2 (u, v) =
f (u).
The equilibrium conservation laws are obtained when the memory is instantaneous, a (s) = −δ(s), and in this case the integral is simply g(s) evaluated at s = 0, σ(x, t) = f (u(x, t)) − g(u(x, t)), so that the system is reduced to the one with no memory and with p(u) = f (u) − g(u):
ut − vx = 0, (2.7) vt − (f (u) − g(u))x = 0. This corresponds to the Euler equations in the set-up of the last section. The equilibrium speeds are Λ1 (u, v) = − f (u) − g (u), Λ2 (u, v) = f (u) − g (u). The sub-characteristic condition is (2.8)
λ1 (u, v) < Λ1 (u, v) < Λ2 (u, v) < λ2 (u, v),
which holds if (2.9)
0 < g (u) < f (u).
To derive the Navier-Stokes equation through the Chapman-Enskog expansion, we approximate the state g(u(x, τ )), τ < t, in (2.5) by its current value g(u(x, t)) plus a non-local value of the form of the differential of g(u(x, t)): g(u(x, τ )) − g(u(x, t)) = ˜ g(u(x, t))t (τ − t) = g (u(x, t))u(x, t)t(τ − t). Plug this into the integral in (2.5) to obtain the viscous system, the NavierStokes approximation by the Chapman-Enskog expansion: ⎧ ⎪ ⎨ut − vx = 0, (2.10) vt − (f (u) − g(u))x = (μg (u)vx )x , ⎪ 0 ⎩ μ ≡ −∞ |a (s)s| ds.
2. Examples
329
The sub-characteristic condition (2.9) implies the hyperbolicity of the Euler equations (2.7) and the positivity of the dissipation coefficient, μg (u) > 0, in (2.10). The system with fading memory, (2.4) and (2.5), can be rewritten as a relaxation model in the special case of a(s) = e−Cs , C a positive constant, as follows: Introduce a new dependent variable t t a (t − τ )g(u(x, τ )) dτ = Ce−C(t−τ ) g(u(x, τ )) dτ. w≡− −∞
−∞
We then have σ = f (u)−w and, by the chain rule applied to the derivative of the integral with respect to t, wt = C(g(u) − w). Thus we have the following equivalent system of three equations: ⎧ ⎪ ⎨ut − vx = 0, (2.11) vt − (f (u) − w)x = 0, ⎪ ⎩ wt = C(g(u) − w). Here τ ≡ C −1 acts as the relaxation time. This is a relaxation system with more than two equations. The third example is a kinetic model with three discrete speeds, taken to be −1, 0, and 1, each with its density function, f− , f0 , and f+ : ⎧ 1 2 ⎪ ⎨(f− )t − (f− )x = −f− f+ + 4 (f0 ) , (2.12) (f0 )t = f− f+ − 14 (f0 )2 , ⎪ ⎩ (f+ )t + (f+ )x = −f− f+ + 14 (f0 )2 , Broadwell model. The interaction of particles with speeds 1 and −1 yields a particle with speed zero, and this accounts for the f− f+ terms on the right-hand sides of the equations. Particles with zero speed lie on the two-dimensional space orthogonal to the x direction. When particles with zero speed interact, equal amounts of particles with speeds 1 and −1 result. This accounts for the (f0 )2 terms on the right-hand sides of the equations. The density ρ and the momentum m are the conserved quantities: (2.13) ρ ≡ f− + 2f0 + f+ , mass density, m ≡ (−1)f− + (0)f0 + (1)f+ = f+ − f− , momentum density. From the system (2.12), there are conservation laws for ρ and m: (2.14)
ρt + mx = 0, conservation of mass, mt + (f+ + f− )x = 0, conservation of momentum.
The equilibrium states satisfy 4f− f+ = (f0 )2 , and by straightforward calculations, f+ + f− = ρ − 2f0 ,
(f0 )2 = 4f− f+ = (f+ + f− )2 − m2 ,
330
11. Relaxation
which can be solved for f+ + f− in terms of the conserved quantities (ρ, m): 1 2 1 f+ + f− = − ρ + 2ρ + 4m2 at equilibrium states. 3 3 The Euler equation for (2.12) is obtained by substituting the above relation into the conservation laws (2.14):
ρt + mx = 0, (2.15) mt + − 13 ρ + 13 2ρ2 + 4m2 x = 0. The characteristic speeds of the Euler equations (2.15) are the equilibrium speeds, while the speeds −1, 0, and 1 of the Broadwell model (2.12) are the frozen speeds.
3. Gas In Thermal Non-equilibrium For monatomic gases, the internal energy consists of only the translational energy e0 . As the pressure and temperature are functions of the translational energy and the density, p = p(ρ, e0 ) and T = T (ρ, e0 ), the Euler equations (0.5) of Chapter 1 are self-contained with e = e0 . The Euler equations (3.1)
ρt + ∇x · (ρv) = 0, (ρv)t + ∇x · (ρv ⊗ v + pI) = 0, (ρE)t + ∇x · (ρvE + pv) = 0
have d +2 equations, where d is the spatial dimension, and the same number of dependent variables (ρ, v, e), with the gas velocity v ∈ Rd . For non-monatomic gases, besides the translational energy there are rotational, vibrational, and other forms of internal energy. For small temperature variations, the other modes of internal energy are approximately in fixed proportions, with the translational energy e = ˜ Ce0 for some constant C. In such a situation p = p(ρ, e0 ) is well approximated by p = (ρ, e/C) and the Euler equations are self-contained. For the physical situation where there can be large temperature variations, as in the case of high-temperature gas dynamics, the various modes of internal energy are not in fixed proportions and p(ρ, e0 ) cannot be accurately approximated by p(ρ, e/C) for a fixed constant C. In such a situation the Euler equations are not self-contained. Consider the simplest case where only one mode of internal energy, say the vibrational energy, is not in equilibrium and the other modes, such as the translational and rotational energies, are in equilibrium and in fixed proportions. Write the total internal energy as e = e0 + q, with e0 being the density of the energy modes in equilibrium and q the density of the non-equilibrium energy mode. The pressure can be expressed as p = p(ρ, e0 ). The total energy per unit volume is E = e + |v|2 /2 = (e0 + q) + |v|2 /2.
3. Gas In Thermal Non-equilibrium
331
There are now d + 3 dependent variables, with the extra one being the nonequilibrium internal energy q. An extra equation for q needs to be added to the usual Euler equations of gas dynamics, (3.1): ⎧ ρt + ∇x · (ρv) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(ρv)t + ∇x · (ρv ⊗ v + pI) = 0, (ρE)t + ∇x · (ρvE + pv) = 0, (3.2) ⎪ ⎪ 0 )−q ⎪ (ρq)t + ∇x · (ρvq) = ρ Q(ρ,e ⎪ κ(ρ,e0 ) , ⎪ ⎪ 2 ⎩ p = p(ρ, e0 ), E = e0 + q + |v2 | . The relaxation time κ(ρ, e0 ) is assumed to be positive and Q(ρ, e0 ) is the equilibrium value for q. From the second energy equation in (3.2) the internal energy q has the tendency to approach its equilibrium value Q(ρ, e0 ) with relaxation time κ(ρ, e0 ). The equilibrium relation q = Q(ρ, e0 ) induces an implicit relation between e0 and e: e = e0 + q = e0 + Q(ρ, e0 ) ≡ E(ρ, e0 ), e0 = E0 (ρ, e), p = p(ρ, e0 ) = p(ρ, E0 (ρ, e)) ≡ P (ρ, e). The Euler equations (3.1) with the constitutive relation p = P (ρ, e) are ⎧ ⎪ ⎨ρt + ∇x · (ρv) = 0, equilibrium Euler equations. (3.3) (ρv)t + ∇x · (ρv ⊗ v + P I) = 0, ⎪ ⎩ (ρE)t + ∇x · (ρvE + P v) = 0, To calculate the characteristic speeds of the above systems, for simplicity we consider the Euler equations in one space dimension and in Lagrangian coordinates: ⎧ ⎪ τt − vx = 0, ⎪ ⎪ ⎪ ⎨v + p = 0, t x non-equilibrium Euler equations; (3.4) 1 2 ⎪ + q + v ) + (pv) = 0, (e 0 t x ⎪ 2 ⎪ ⎪ ⎩q = Q−q , t κ
(3.5)
(3.6)
⎧ ⎪ τt − vx = 0, ⎪ ⎪ ⎪ ⎨v + p = 0, t x ⎪(e0 + q + 12 v 2 )t + (pv)x = 0, ⎪ ⎪ ⎪ ⎩q = 0, t ⎧ ⎪ ⎨τt − vx = 0, vt + Px = 0, ⎪ ⎩ (e + 12 v 2 )t + (P v)x = 0,
frozen Euler equations;
equilibrium Euler equations.
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11. Relaxation
Instead of using the density ρ as a basic dependent variable, in the Lagrangian coordinates, the specific volume τ = 1/ρ is used. Thus the relaxation time is κ(τ, e0 ) and the equilibrium value for q is Q(τ, e0 ). We assume that the Boltzmann statistical theory applies to both internal energies e0 and q. From the theory, there exist an entropy s0 and a temperature θ0 relating to e0 , and also a non-equilibrium entropy s1 and temperature θ1 relating to q. The thermodynamic relations are (3.7) de0 = θ0 ds0 − p dτ for equilibrium internal energy, dq = θ1 ds1 for non-equilibrium internal energy, s = s0 + s1 , de = θ0 ds + (θ1 − θ0 )ds1 − p dτ for total internal energy. On the equilibrium manifold q = Q(τ, e0 ), (3.8) e = e0 + q = e0 + Q(τ, e0 ) ≡ E(τ, e0 ), e0 = E0 (τ, e), p = p(τ, e0 ) = p(τ, E0 (τ, e)) ≡ P (τ, e). It follows from the thermodynamics relations (3.7) that for the frozen system (3.5), (s0 )t = 0, and for the equilibrium system (3.6), st = 0. Thus the calculations in (5.11) of Chapter 6 apply, and we have the following: The characteristic speeds of (3.6), the equilibrium characteristic speeds, are given as (3.9) Λ1 = − −P¯τ (τ, s), Λ2 = 0, Λ3 = −P¯τ (τ, s), P¯ (τ, s) ≡ P (τ, e); the characteristic speeds of (3.4) and (3.5), the frozen characteristic speeds, are pτ (τ, s0 ), λ2 = 0, λ3 = −¯ pτ (τ, s0 ), p¯(τ, s0 ) ≡ p(τ, e0 ). (3.10) λ1 = − −¯ In the above λ2 = 0 has double multiplicity. The two sound speeds (3.11) Ce ≡ −P¯τ (τ, s), equilibrium sound speed, pτ (τ, s0 ), frozen sound speed, cf ≡ −¯ are compared at an equilibrium state q = Q(τ, e0 ). From the implicit relations in (3.8) and the thermodynamics relations (3.7), we have pτ (τ, s0 ) = −pτ − ppe0 , (3.12) (cf )2 = −¯ pQe0 − Qτ pe0 . (Ce )2 = P¯τ (τ, s) = (cf )2 − 1 + Qe0 We assume that (3.13)
pτ < 0, pe0 > 0, Qτ < 0, Qe0 > 0.
In (3.13), the first two inequalities are the same as (5.13) in Chapter 6, and the next two inequalities say that the equilibrium value Q(τ, e0 ) for the
3. Gas In Thermal Non-equilibrium
333
non-equilibrium internal energy q increases as the density ρ and the internal energy e0 in equilibrium increase. With the physical assumptions (3.13), we have (3.14)
0 < Ce < cf , sub-characteristic property.
From the third equation in (3.4), q = Q − κqt . The Chapman-Enskog expansion approximates the system (3.4) according to the principles used in ˜ Section 1. The solution is assumed to be close to the equilibrium q Q: ˜ Q − κQt . q = Q − κqt = The partial derivative ∂t is expressed in terms of ∂x using the equilibrium system (3.6): ˜ Qτ τt + Qe0 (E0 )t = Qτ τt + Qe0 (E0 )τ τt + (E0 )e et Qt = Qτ τt + Qe0 (e0 )t = = Qτ vx + Qe0 (E0 )τ vx + (E0 )e (−vvt − (P v)x ) = Qτ vx + Qe0 (E0 )τ vx + (E0 )e (vPx − (P v)x ) Q − PQ τ e0 vx . = Qτ + Qe0 ((E0 )τ − (E0 )e P ) vx = 1 + Qe0 We have thus obtained the Chapman-Enskog relation Q − PQ τ e0 vx . (3.15) q =Q−κ 1 + Qe0 This is used to approximate the pressure p in (3.4). First, from e = e0 + q and at equilibrium e = E0 + Q(τ, E0 ) we have, from (3.15), Q − PQ τ e0 e0 − E0 = Q(τ, E0 ) − q = κ vx . 1 + Qe0 This yields ˜ P (τ, e) + pe0 (e0 − E0 ), p(τ, e0 ) = p(τ, E0 ) + p(τ, e0 ) − p(τ, E0 ) = Q − PQ τ e0 vx . p = P + κpe0 1 + Qe0 Plugging this into the conservation laws in (3.4), we obtain the Navier-Stokes equations for (3.4): ⎧ ⎪ ⎨τt − vx = 0, (3.16) vt + Px = (μvx )x , ⎪ ⎩ (e + 12 v 2 )t + (P v)x = (μvvx )x . Note that this is the system of Navier-Stokes equations with zero heat conductivity. The positivity of the viscosity coefficient μ, as expected, is related
334
11. Relaxation
to the sub-characteristic property (3.14), P Qe0 − Qτ (3.17) μ = κpe0 = κ (cf )2 − (Ce )2 > 0. 1 + Qe0 Remark 3.1. Both the non-equilibrium system (3.4) and the Navier-Stokes equations with zero heat conductivity (3.16) are not dissipative and possess linear non-dissipative waves; see Exercises. By a similar analysis, one sees that a shock profile of moderate strength for (3.4) exists if the end states satisfy the Rankine-Hugoniot condition and Liu entropy condition (6.1) of Chapter 7 for the equilibrium system (3.6).
4. The Boltzmann Equation in Kinetic Theory In fluid dynamics, the basic dependent variables are macroscopic variables, such as the fluid velocity v, pressure p, density ρ, and other thermodynamic variables. These variables are functions of the spatial variable x and time t, e.g. ρ = ρ(x, t). The kinetic theory starts with the velocity distribution function f(x, t, ξ), with the inclusion of an extra independent variable, the microscopic velocity ξ ∈ R3 . With knowledge of the density distribution function f(x, t, ξ), the macroscopic variables, functions of (x, t), are expressed as the moments: (4.1) ρ(x, t) ≡
f(x, t, ξ) dξ, density, ξf(x, t, ξ) dξ, momentum, m(x, t) ≡
R3
R3
m v = (v1 , v2 , v3 ) ≡ , fluid velocity, ρ |ξ|2 f(x, t, ξ) dξ, total energy, ρE(x, t) ≡ R3 2 |ξ − v|2 f(x, t, ξ) dξ, internal energy, ρE(x, t) ≡ 2 R3 ij (ξi − vi )(ξj − vj )f(x, t, ξ) dξ, p (x, t) ≡ R3 ij
P = (p )1≤i,j≤3 , stress tensor 1 (ξ − v) |v − ξ|2 f(x, t, ξ) dξ, heat flux. q(x, t) ≡ 2 3 R The pressure is formally defined as 1 2 1 |ξ − v(x, t)|2 f(x, t, ξ) dξ = ρe. p = (p11 + p22 + p33 ) = 3 3 R3 3
4. The Boltzmann Equation in Kinetic Theory
335
We now go into some details of the physical meaning of the macroscopic variables just defined. The above definition of the density ρ = ρ(x, t), mass per unit volume, is clear, as the integration in ξ sums over particles with varying microscopic velocity in the volume. The same goes for the definitions of the momentum m(x, t) and the total energy ρE(x, t). Here E is the total energy per unit mass and ρE is the total energy per unit volume. Note from |ξ|2 /2 = |ξ − v|2 /2 − |v|2 /2 + ξ · v and from the above definition of the momentum that
|ξ|2 f(x, t, ξ) dξ R3 2 |v|2 |ξ − v|2 f(x, t, ξ) dξ − f(x, t, ξ) dξ = 2 2 R3 R3 ξf(x, t, ξ) dξ +v· R3 |v|2 |v|2 |ξ − v|2 |ξ − v|2 f(x, t, ξ) dξ− ρ+v·m = f(x, t, ξ) dξ+ρ . 2 2 2 2 R3
ρE(x, t) ≡
= R3
Thus we have
|ξ − v|2 f(x, t, ξ) dξ, internal energy, 2 R3 |v|2 , total energy = internal energy + kinetic energy. ρE = ρe + ρ 2
(4.2) ρe ≡
Physically, the internal energy ρe is the energy of the gas, per unit volume, in a frame moving with the average velocity v of the gas. In the same moving frame, the stress pij (x, t) measures the flux of the j-momentum in the xi direction. Thus pij (x, t) is the jth component of the force experienced by an internal surface moving with the gas flow with normal in the direction of xi . The most important equation in the kinetic theory for gases is the Boltzmann equation, (4.3)
∂t f(x, t, ξ) + ξ · ∇x f(x, t, ξ) =
1 Q(f, f)(x, t, ξ). k
The left-hand side of the equation, ∂t f + ξ · ∂x f, is the transport term, measuring the temporal rate of change in a frame moving with the velocity of the particles being monitored. The Boltzmann equation says that the change is due to the collision operator Q(f, f), which takes the form of the binary
336
11. Relaxation
collision (4.4) Q(g, h) ≡
1 2
R3 ×S 2
−g(ξ)h(ξ∗ ) − h(ξ)g(ξ∗ )
(ξ−ξ∗ )·Ω≥0
+ g(ξ )h(ξ ∗ ) + h(ξ )g(ξ∗ ) B(ξ − ξ ∗ , Ω) dξ∗ dΩ,
(4.5)
ξ = ξ − (ξ − ξ ∗ ) · Ω Ω, ξ ∗ = ξ ∗ + (ξ − ξ ∗ ) · Ω Ω.
The collision operator Q(f, f) = f(ξ )f(ξ∗ ) − f(ξ)f(ξ∗ ) B(ξ − ξ ∗ , Ω) dξ ∗ dΩ consists of the loss part f(ξ)f(ξ∗ ), which is due to the collision of particles with the given speed ξ and other particles with varying speed ξ ∗ , and the gain part f(ξ )f(ξ∗ ), which comes from particles with speed ξ colliding with particles with speed ξ ∗ to yield particles with the given speed ξ; see (4.5). Thus the integration with respect to ξ ∗ is over R3 . The detail of each collision depends on the relative position Ω ∈ S 2 of the interacting particles, and the integration with respect to Ω is over the relevant half, (ξ − ξ ∗ ) · Ω ≥ 0, of the unit sphere S 2 . The cross-section B(ξ − ξ ∗ , Ω) depends on the intermolecular forces between the particles. For hard sphere models, (4.6)
B(ξ − ξ∗ , Ω) = (ξ − ξ ∗ ) · Ω.
4.1. Conservation Laws The relationships (4.5) hold for some Ω on the unit sphere S 2 if and only if the following conservation laws hold: ⎧ ⎪ ⎨1 + 1 = 1 + 1, conservation of mass, (4.7) ξ + ξ ∗ = ξ + ξ ∗ , conservation of momentum, ⎪ ⎩ 2 |ξ| + |ξ∗ |2 = |ξ |2 + |ξ ∗ |2 , conservation of energy. Here the first trivial relation 1 + 1 = 1 + 1 means that the collision of two particles yields two particles. By a simple change of variables, using the fact that the transformation (4.5) has a Jacobian of 1. Ψ(ξ)Q(g, h)(ξ) dξ (4.8) R3 1 (Ψ(ξ) + Ψ(ξ ∗ ) − Ψ(ξ ) − Ψ(ξ ∗ ))Q(g, h)(ξ) dξ. = 4 R3 A function Ψ(ξ) is called a collision invariant if (4.9)
Ψ(ξ) + Ψ(ξ ∗ ) = Ψ(ξ ) + Ψ(ξ ∗ ).
4. The Boltzmann Equation in Kinetic Theory
337
It is called a collision invariant because, from (4.8) and (4.9), the average of the collision operator with the weight of a collision invariant is zero: (4.10) Ψ(ξ)Q(g, h) dξ = 0. R3
From (4.7), the functions (4.11)
Ψ0 (ξ) ≡ 1,
Ψi (ξ) ≡ ξi , i = 1, 2, 3,
Ψ4 (ξ) ≡ |ξ|2 /2
are collision invariants. Take, for instance, Ψ = Ψ0 ; then (4.10) yields Q(g, h) dξ = 0. R3
This means that although the collision operator Q(f, f) redistributes the density function f(x, t, ξ) on the microscopic level, it does not contribute to the change in mass on the macroscopic level. Integrating Ψj , j = 0, 1, . . . , 4, times Q yields the properties of conservation of mass, momentum, and energy of the collision operator: ⎛ ⎞ ⎛ ⎞ 1 conservation of mass ⎝ ξ ⎠ Q(g, h) dξ = 0 ⎝conservation of momentum⎠ . (4.12) 1 2 R3 conservation of energy 2 |ξ| This yields the conservation laws for the Boltzmann equation. Take, for instance, the integration of 1 1 1 Ψ4 = |ξ|2 = (ξ − v) · (ξ − v) − |v|2 + ξ · v 2 2 2 times the Boltzmann equation: 1 1 ∂ (ξ − v) · (ξ − v) − |v|2 + ξ · v f(x, t, ξ) dξ ∂t R3 2 2 1 1 (ξ − v) · (ξ − v) − |v|2 + ξ · v (ξ − v) + v f(x, t, ξ) dξ + ∇x · 2 R3 2 1 2 |ξ| Q(f, f) dξ = 0. = R3 2 We have from (4.1) that 1 1 (ξ − v) · (ξ − v) − |v|2 + ξ · v f(x, t, ξ) dξ 2 R3 2 1 1 = ρe − |v|2 ρ + ρv · v = ρe + ρ |v|2 = ρE; 2 2 1 1 (ξ − v) · (ξ − v) − |v|2 + ξ · v vf(x, t, ξ) dξ 2 R3 2 1 1 (ξ − v) · (ξ − v) − |v|2 + ξ · v f(x, t, ξ) dξ = ρEv; =v 2 R3 2
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11. Relaxation 1 1 (ξ − v) · (ξ − v) − |v|2 + ξ · v (ξ − v)f(x, t, ξ) dξ 2 R3 2 1 1 = (ξ − v) · (ξ − v) + |v|2 + (ξ − v) · v (ξ − v)f(x, t, ξ) dξ 2 R3 2 = q + Pv.
This yields the equation for the conservation of energy. The other calculations are similar and we obtain, by integrating 1, ξ, and |ξ|2 /2 times the Boltzmann equation (4.3), the conservation laws for the Boltzmann equation in terms of the macroscopic variables introduced in (4.1): (4.13) ∂t ρ + ∂x · (ρv) = 0, conservation of mass, ∂t (ρv) + ∂x · (ρv ⊗ v + P) = 0, conservation of momentum, ∂t (ρE) + ∂x · (ρvE + Pv + q) = 0, conservation of energy. There is a basic theorem of Boltzmann saying that the span of Ψj (ξ), j = 0, 1, . . . , 4, contains all the collision invariants. Thus the above are the only conservation laws resulting from the Boltzmann equation. There are 5 conservation laws: 1 for the mass, 3 for the momentum, and 1 for the energy. On the other hand, there are 14 unknowns: 1 for the density ρ, 3 for the fluid velocity v, 6 for the symmetric stress tensor P, 3 for the heat flux q, and 1 for the internal energy e. Thus the conservation laws are not a system of self-contained partial differential equations. The kinetic theory treats the distribution function f(x, t, ξ), for each fixed pair of space and time variables (x, t), as a function of the microscopic variables ξ. The space of functions in ξ is infinite dimensional. Therefore the Boltzmann equation can be regarded as a system of an infinite number of partial differential equations. The stress tensor and heat flux are third moments. The equations for third moments then involves fourth moments, and so on. There are expansions to obtain approximations through finite moments closure. We will present the Chapman-Enskog expansion for the derivation of the Navier-Stokes equations in gas dynamics later. 4.2. H-Theorem The Boltzmann equation has the so-called molecular chaos hypothesis built into the collision operator in that the loss term −f(ξ)f(ξ∗ ) and the gain term f(ξ )f(ξ∗ ) in the collision operator Q(f, f) are supposed to be the two particles’ distributions before the collision, taken here to be the products of the single-particle distributions f. This is the hypothesis of no correlation before collision. An important consequence of this is the H-Theorem on the irreversibility of the Boltzmann process. Integrating the collision operator
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Q times 1 + log f and using (4.8) yields 1 (1 + log f)Q(f, f) dξ = log f + log f∗ − log f − log f∗ Q(f, f) dξ 4 R3 R3 1 log ff∗ − log f f∗ Q(f, f) dξ. = 4 R3 Plug in the expression (4.4) for Q to get an inequality 1 ff∗ (1+log f)Q(f, f) dξ = log f f∗ −ff∗ B dΩ dξ ∗ dξ ≤ 0. 4 R3 R3 S+2 f f∗ R3 The inequality results from the fact that log f is an increasing function of f . Integrate the Boltzmann equation times 1 + log f to obtain the H-Theorem: 1 ff∗ log f f∗ − ff∗ B dΩ dξ ∗ dξ ≤ 0, (4.14) Ht + ∇x · H = 4k R3 R3 S+2 f f∗ f log f dξ, H ≡ ξf log f dξ. H≡
R3
R3
From the above application of (4.8), R3 log fQ(f, f) dξ = 0 if and only if log f is a collision invariant and, by the Boltzmann theory, log f is in the span of 1, ξ, and |ξ|2 /2. In other words, f is Gaussian in ξ. On the other hand, if log f is an collision invariant, log f + log f∗ = log f + log f∗ , or f f∗ = ff∗ , then from the expression (4.4), Q(f, f) = 0. Thus Q(f, f) = 0 if and only if f is Gaussian in the microscopic velocity ξ. The inequality Ht + ∇x · H ≤ 0 becomes an equality when the state is on the equilibrium manifold. The pair (H, H) is an analogue of the entropy pair for conservation laws; see Definition 2.1 of Chapter 6 and Section 5 of Chapter 10. The second law of thermodynamics holds near the equilibrium manifold. When f is Gaussian in the microscopic velocity ξ, it can be written in the form f(x, t, ξ) = A(x, t)e−B(x,t)|ξ−v(x,t)|
2
for some functions A(x, t) and B(x, t) that are independent of the microscopic velocity ξ. From this we have, for η = ξ − v, π 3 1 A π 32 2 2 2 Ae−B|η| dη = A , p= |η|2 Ae−B|η| dη = . ρ= B 3 R3 2B B R3 We now introduce the macroscopic quantity θ, the temperature, through the ideal gas relation: (4.15)
p = Rρθ.
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11. Relaxation
The above relations yield A = ρ/(2πRθ)3/2 and B = 1/(2Rθ), and the distribution function is explicitly given in terms of the macroscopic variables (ρ, v, θ): (4.16)
f(x, t, ξ) =
|ξ−v(x,t)|2 ρ(x, t) − 2Rθ(x,t) e ≡ M(ρ,v,θ) . (2πRθ(x, t))3/2
The states M(ρ,v,θ) are called the Maxwellian distributions. They are the thermal equilibrium states in that (4.17)
Q(M, M) = 0.
The thermal equilibrium manifold is a five-dimensional manifold of Maxwellian distributions (4.18)
equilibrium manifold ≡ {M(ρ,v,θ) : ρ > 0, v ∈ R3 , θ > 0}.
Remark 4.1. With the definition of the stress pij (x, t) in (4.1) and internal energy ρe in (4.2), the kinetic theory gives concrete physical meaning to these thermodynamics quantities in terms of the motion according to Newtonian mechanics. In the expression (4.16) for a Maxwellian, the density ρ is a simple multiplicative factor. The temperature θ plays a more interesting role. For lower temperatures, the distribution is more concentrated and closer to a multiple of the delta function δ(ξ − v). Expressing the Maxwellian in terms of the density and temperature has the advantage of this interpretation. We will see later that this specific expression of Maxwellians has the important implication that the viscosity and heat conductivity coefficients are functions of the temperature and not of the density. In classical thermodynamics, there is the thermodynamic hypothesis that among the thermodynamic variables, such as the density ρ, pressure p, internal energy e, temperature θ, and entropy s, only two of them are independent, and the rest are given functions, through the constitutive relations, of the two chosen variables. From the viewpoint of the kinetic theory, classical thermodynamics applies only near the Maxwellian states; see figure 11.01. In the above expression (4.16) for Maxwellian states, we see that the state depends only on two thermodynamics states, the density ρ and the temperature θ. The other thermodynamic states can be computed by taking moments of the Maxwellian distribution (4.16). The kinetic theory therefore justifies the thermodynamics hypothesis. By the H-Theorem, a Boltzmann solution has the tendency to reach the five-dimensional thermal equilibrium manifold of Maxwellian states; see figure 11.01. Thus the dynamics of the Boltzmann equation is irreversible. A
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341
state on the equilibrium manifold has a normal distribution in the microscopic variable |ξ − v|, and the stress tensor and the heat flux become (4.19)
P = pI, stress tensor = the pressure;
q = 0, zero heat flux.
Consequently, assuming that the distributional function f stays on the equilibrium manifold, the conservation laws (4.13) become the Euler equations in gas dynamics: (4.20) ∂t ρ + ∇x · (ρv) = 0, conservation of mass, ∂t (ρv) + ∇x · (ρv ⊗ v + pI) = 0, conservation of momentum, ∂t (ρE) + ∇x · (ρvE + pv) = 0, conservation of energy. On the equilibrium manifold, the pressure function p = pjj , j = 1, 2, 3, is related to the internal energy by the pressure law for monatomic gases, p = 23 ρe. By direct calculation, at equilibrium the H-function is the negation of physical entropy function: M(ρ,v,θ) log M(ρ,v,θ) = −ρs. (4.21) R3
The second law of thermodynamics on increase of the entropy is therefore consistent with the H-Theorem on decrease of the H-function. Solutions of the Boltzmann equation in general do not stay on the equilibrium manifold, even if the initial distribution f(x, 0, ξ) is on the equilibrium manifold. The Euler equation can only be viewed as an approximation. The fluid dynamics phenomena occur near the equilibrium manifold, figure 11.01.
Figure 11.01. H-Theorem and equilibrium manifold.
On the boundary, initial, and shock layers, the Boltzmann equation cannot be approximated by fluid dynamics. Away from these layers, the behavior of the gas around the five-dimensional equilibrium manifold can be
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11. Relaxation
approximated by the fluid dynamics equations. The suitable fluid dynamics equations, which accurately approximate the Boltzmann equation, depend on the physical situation as defined, for instance, by the Knudsen number k, the Mach number, the time range, and the extent of deviation from the equilibrium manifold. 4.3. Boltzmann Shocks A Boltzmann shock profile is a traveling wave of the Boltzmann equation which depends only on one spatial dimension, say x1 , x1 − σt , f(x, t; ξ) = φ(η, ξ), η ≡ k and we have from the Boltzmann equation that −σφη + ξ1 φη = Q(φ, φ).
(4.22)
This equation is independent of the mean free path k, and so the shock width is proportional to k. As x1 → ±∞, the profile tends to the limiting states, which are constant solutions of the Boltzmann equation. As we have seen from the H-Theorem, the only constant solutions of the Boltzmann equation are the Maxwellians. Thus φ(x1 − σt, ξ) → M± (ξ) as x1 → ±∞, for some Maxwellians M± (ξ). Multiply (4.22) by the collision invariants ψ(ξ) = 1, ξ1 , |ξ|2 /2 and integrate with respect to ξ to get the conservation laws along the shock profile φ(η, ξ)ψ(ξ) dξ + ξ1 φ(η, ξ)ψ(ξ) dξ = constant, −∞ < η < ∞. −σ R3
R3
At η = ±∞, φ = M± , and the first integral is the density of mass, momentum, and energy; the second integral is the Euler flux of these densities, ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ρ− v− ρ+ ρ+ v+ ρ− −σ ⎝ ρ− v− ⎠+⎝ ρ− (v− )2 + p− ⎠ = −σ ⎝ ρ+ v+ ⎠+⎝ ρ+ (v+ )2 + p+ ⎠ . ρ− E− ρ− E− v− + p− v− ρ+ E+ ρ+ E+ v+ + p+ v+ Thus the end states of the Boltzmann profile satisfy the same RankineHugoniot condition as that for the Euler equations in gas dynamics, (5.4) of Chapter 7. Integrating (4.22) times log φ yields the inequality following the analysis of the H-Theorem: d (−σ + v1 ) φ log φ dξ ≤ 0. (4.23) dη Thus we have from (4.23) and (4.21) that (4.24)
(−σ + v1− )ρ− s− < (−σ + v1+ )ρ+ s+ .
From conservation of mass, m ≡ (−σ + v1− )ρ− = (−σ + v1+ )ρ+ , (4.24) becomes m(s+ − s− ) > 0. For a 1-shock, the momentum m > 0 and we
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343
conclude that s− < s+ , which is the entropy condition; see (5.14) of Chapter 7. The same holds for 3-shocks. Thus a Boltzmann shock satisfies the same Rankine-Hugoniot condition and entropy condition as those for the Euler equations in gas dynamics. The Boltzmann shock profile, however, differs from the shock profile for fluid equations such as the Navier-Stokes equations of gas dynamics. 4.4. Chapman-Enskog Expansion The Chapman-Enskog expansion is used to derive the Navier-Stokes equations in gas dynamics from the Boltzmann equation. A simpler version of the expansion for the relaxation model (1.1) has been described in Subsection 1.3 of Section 1. For the Boltzmann equation, it is based on similar principles. The first principle is that we start with the Euler equations (4.20) for gas dynamics and the solution is a perturbation of a local Maxwellian M: (4.25)
f(x, t, ξ) = M(x, t, ξ) + G(x, t, ξ).
We call M(x, t, ξ) the macro component and G(x, t, ξ) the micro component of the solution f. The macro component M(x, t, ξ) is a local Maxwellian M(x, t, ξ) = M(ρ,v,θ) =
|ξ−v(x,t)|2 ρ(x, t) − 2Rθ(x,t) e , (2πRθ(x, t))3/2
with the macroscopic variables computed from the given solution f(x, t, ξ) of the Boltzmann equation by (4.1): ρ(x, t) = f (x, t, ξ) dξ, ρv(x, t) = ξf (x, t, ξ) dξ, R3 R3 |ξ|2 f (x, t, ξ) dξ, p = Rρθ. ρE(x, t) = R3 2 Therefore the micro component G carries no mass, momentum, and energy: ⎞ ⎛ 1 G(x, t, ξ) ⎝ ξ ⎠ dξ = 0. (4.26) R3 |ξ|2 /2 From (4.25), the Boltzmann equation becomes 1 1 (4.27) (M + G)t + ∇x · (ξ(M + G)) = Q(G, G) + L(G), k k where L is the collision operator linearized around the local Maxwellian M: (4.28)
L(h) = LM (h) ≡ 2Q(M, h), linearized collision operator.
The collision operator vanishes on the equilibrium manifold Q(M, M) = 0. Consequently, the linearized collision operator L vanishes on the tangent
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11. Relaxation
space to the equilibrium manifold at the local Maxwellain M; see figure 11.01. The tangent vectors are obtained by varying the parameters ρ, θ, and v in the Maxwellian M(x, t, ξ). When ρ varies, ∂ 1 M(x, t, ξ) = M(x, t, ξ). ∂ρ ρ This gives one tangent vector in the direction of M. When v varies, ξi − vi ∂ M(x, t, ξ), i = 1, 2, 3. M(x, t, ξ) = ∂vi Rθ This and the above give four tangent vectors M and ξM. By varying the temperature θ, we conclude that the tangent plane is spanned by M times the collision invariants (4.11): (4.29)
kernel of L = LM = span{M, ξM, |ξ|2 M}.
With the decomposition (4.25) and the property (4.26), the conservation laws (4.13) become (4.30) ∂t ρ + ∇x · (ρv) = 0, ∂t (ρv) + ∇x · (ρv ⊗ v + pI) +
ξ(ξ · ∇x G) dξ = 0, |ξ|2 ∂t (ρE) + ∇x · (ρvE + pv) + (ξ · ∇x G) dξ = 0. R3 2 R3
The conservation laws are the Euler equations in gas dynamics, (4.20), when the micro part is set to zero, G = 0. To obtain the second-order approximation by the Navier-Stokes equations of gas dynamics, we make the following assumptions: |∇(x,t) G| |G| |M|, Chapman-Enskog hypothesis. This says that the solution varies slowly in space and time and that it stays close to the equilibrium manifold. Consider the macro projection P0 of a function in the microscopic velocity ξ onto the tangent plane of the fivedimensional equilibrium manifold through the local Maxwellian M. In other words, P0 projects a function in ξ to the kernel of the linearized collision operator L = LM . From (4.29), P0 is a projection onto the five-dimensional space spanned by the collision invariants times the local Maxwellian. The micro projection P1 is the orthogonal projection of P0 , such that P0 + P1 = I. Here the projection is with respect to the inner product 1 h(ξ)g(ξ) dξ. (h, g) ≡ R3 M The whole space is the space of functions in ξ and is therefore infinite dimensional. Thus the micro projection P1 is a projection onto a space of infinite dimension. The range of the micro projection P1 is orthogonal to the kernel
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345
of L, and L is invertible on the range of P1 . The projections are local here in the sense that for the given solution f(x, t, ξ) of the Boltzmann equation, the local Maxwellian M = M(x, t, ξ) is formed, and the macro projection P0 is the projection onto the kernel of the linearized collision operator LM with respect to the local Maxwellian M = M(x, t, ξ). The macro projection of the Boltzmann equation is (4.31)
Mt + P0 (ξ · ∇x (G + M)) = 0,
and the micro projection of the Boltzmann equation is (4.32)
Gt + P1 (ξ · ∇x (G + M)) =
1 1 Q(G, G) + L(G). k k
Using the Chapman-Enskog hypothesis, the micro part (4.32) of the Boltzmann equation is simplified to 1 L(G). k This can be solved for the micro part of the Boltzmann solution in terms of the gradient of the macro part of the Boltzmann solution: P1 (ξ · ∇x M) =
(4.33)
G = kL−1 (P1 (ξ · ∇x M)), Chapman-Enskog relation.
Note that G is in the infinite-dimensional range of the micro projection P1 , and the Chapman-Enskog relation postulates that the micro part G of the Boltzmann solution is related to the gradient of the local Maxwellian, which is in the macro part of the Boltzmann solution. This is one of the finitedimensional closures needed for the derivation of the fluid equations. Plug this into the macro part of the Boltzmann equation (4.31) to obtain the Navier-Stokes equations in gas dynamics in the kinetic form: (4.34)
Mt + P0 (ξ · ∇x (M)) = −P0 (ξ · ∇x (kL−1 (P1 (ξ · ∇x M)).
Integrate the above equation times the collision invariants to obtain the conservation laws, the Navier-Stokes equations in gas dynamics form: (4.35)
∂t ρ + ∇x · (ρv) = 0, mass, ∂t (ρv) + ∇x · (ρv ⊗ v + pI) ξ ⊗ ξk(−L)−1 (P1 (ξ · ∇x M)) dξ, momentum, = ∇x · R3
(4.36)
∂t (ρE) + ∇x · (ρvE + pv) |ξ|2 ξk(−L)−1 (P1 (ξ · ∇x M)) dξ, energy. = ∇x · R3 2
Equation (4.34) is for the evolution of the five-dimensional local Maxwellian and is equivalent to the system (4.35) of five equations. The equations (4.35)
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11. Relaxation
are the same as the Navier-Stokes equations with shear viscosity coefficient λ and heat conductivity coefficient κ given as follows: √ √ (4.37) λ = −k P1 ξ1 ξ2 M, L−1 [P1 ξ1 ξ2 M] , √ √ 1 κ = − k P1 ξ1 (|ξ|2 − 3) M, L−1 [P1 ξ1 (|ξ|2 − 3) M] . 6 The dissipation parameters are proportional to the Knudsen number k and result from the combined effect of the convection ξ · ∇x and the collision L−1 . In the above, g(ξ)h(ξ) dξ, g, h ≡ R2
the standard inner product in the function space L2 (ξ). Here we have done the translation so that the fluid velocity is set to zero and the Maxwellian is given as |ξ|2 ρ − 2Rθ e . M = M(ξ) = (2πRθ)3/2 From the definition (4.28) of the linearized collision operator L, a multiplicative factor ρ in M can be factored out and so the above expressions (4.37) do not depend on the density ρ. These dissipation parameters depend only on the temperature θ. The kinetic theory yields this surprising, basic result, which was subsequently confirmed experimentally. The compressible Navier-Stokes equations, as an approximation of the Boltzmann equation, are accurate as waves dissipate and approach local Maxwellians time-asymptotically. Thus the Navier-Stokes equations in gas dynamics describe the time-asymptotic behavior of the Boltzmann equation in the situation where the initial data is a perturbation of a global Maxwellian. The Boltzmann equation, the Navier-Stokes equation and the Euler equations in gas dynamics are equivalent time-asymptotically on the level of rarefaction waves.
5. Notes The presentation of the simple relaxation model follows basically that of Liu [87]. See Chen-Levermore-Liu [24] for analysis of general systems. See also Boillat-Ruggeri [11, 12] for analysis by the extended thermodynamics approach. The study of the relaxation model has motivated the so-called macro-micro decomposition to bridge the shock wave theory and the kinetic theory for the Boltzmann equation by Liu-Yu in 2004; see references in [97]. There is the Jin-Xin model [65] of numerical schemes for conservation laws based on the zero relaxation limit. The Chapman-Enskog expansion for the Boltzmann equation yields the Navier-Stokes equations in gas dynamics. The more systematic Hilbert expansion yields several fluid dynamics
6. Exercises
347
equations, depending on the physical situation under consideration. The Hilbert expansion is done for small mean free paths. An example of the Hilbert expansion is carried out for the relaxation model in Subsection 1.4 of this chapter. The study of the Boltzmann equation is here only briefly mentioned in the general framework of relaxation. The authoritative book by Yoshio Sone [118] discusses the rich relationship between the Boltzmann equation and the gas dynamics. Further mathematical study of the Boltzmann equation from the point of view of shock wave theory is anticipated; see e.g. [97] and references therein.
6. Exercises 1. Consider the general form of a 2 × 2 relaxation system, (2.1), ut + f (u, v)x = 0, vt + g(u, v)x = h(u, v), under the assumptions fv < 0 and hv < 0. Find an additional condition for the existence of equilibrium states and find the equilibrium conservation law. 2. Show that if the initial data u(x, 0) for the relaxation model (1.1) is sufficiently small and smooth, then the smooth solution u(x, t) exists globally in time. 3. Derive the Navier-Stokes equations for the shallow water equations (2.3). 4. Apply the Hilbert expansion to the shallow water equations (2.3). 5. Derive the Navier-Stokes equations for (2.11) and check with (2.10). 6. Derive the Navier-Stokes equations for the Broadwell model (2.12). 7. Find a sufficient condition for the sub-characteristic condition for the elastic model (2.11). 8. Find the kernel of the linearized collision operator L, (4.29), by a direct algebraic method similar to (4.8). 9. Check that the dissipative hypothesis (5.2) in Section 5 of Chapter 10 is not satisfied for (3.16). 10. Consider the initial value problem for the systems (3.4) and (3.16). Find solutions which are constant in time. (There is a natural extension of the theory mentioned in Section 5 of Chapter 10 for viscous conservation laws to the relaxation models. For such theories, (3.4) and (3.16) are similar; see [128].)
10.1090/gsm/215/12
Chapter 12
Nonlinear Resonance
The phenomenon of resonance occurs when two of the speeds characterizing a situation are close to each other. In ordinary differential equations, there is the simple model y + c2 y = K cos(ωt), y = y(t). Without the source, the equation y + c2 y = 0 has oscillating solutions y(t) = A cos ct + B sin ct with natural frequency c. When the frequency ω of the source K cos(ωt) differs from the characteristic frequency ±c of the differential operator y + c2 y, c2 = ω 2 , the general solutions possess oscillating terms of both frequencies; when they are equal, c2 = ω 2 , resonance occurs and solutions grow in time t:
K 2 2 A cos ct + B sin ct + c2 −ω 2 cos ωt when ω = c ; y(t) = K A cos ct + B sin ct + 2ω t sin ωt when ω 2 = c2 . The above simple example illustrates the resonance phenomenon for the linear ordinary differential equation. We are interested in nonlinear resonance for partial differential equations when the solutions contain shock waves. This is a rich field. In general, resonance can enforce stability or cause instability and result in bifurcation phenomena. When the solutions contain shock waves, the shock wave speed is one of the speeds characterizing the situation. Shock wave speed is defined globally and nonlinearly by the Rankine-Hugoniot condition. As a consequence, nonlinear resonance in the presence of shock waves takes several interesting and distinct forms. For a given situation, the first step in the study of resonance is to identify the speeds characterizing the situation. These speeds can be various characteristic speeds, the speed of the source, the speed of the boundary, the 349
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numerical C-F-L speed, and others. The study of resonance gives rise to several novel analytical issues. We illustrate these by considering several distinct physical and analytical situations in the following sections.
1. Moving Source Consider a system of hyperbolic conservation laws with a moving source: ut + f (u)x = S(x − αt). The system has three sets of speeds, the characteristic speeds λi (u), i = 1, 2, . . . , n, of the hyperbolic conservation operator on the left-hand side, given in (0.2) of Chapter 7, the shock speed defined by the Rankine-Hugoniot condition, and the speed α of the source S(x − αt). The moving source can come from various physical effects; here we consider the geometric effect of the solid boundary of a duct, through which the gas flows. Consider the following quasi-one-dimensional model of gas flows in a duct of variable cross-section A(x): (1.1) (1.2) (1.3)
A (x) ρv, A(x) A (x) 2 ρv , (ρv)t + (ρv 2 + p)x = − A(x) A (x) (ρE)t + (ρEv + pv)x = − (ρEv + pv). A(x) ρt + (ρv)x = −
The left-hand side is the Euler operator of gas dynamics, and the righthand side is the geometric source. The duct is assumed to be still, and so the speed of the source is zero. The Euler operator of gas dynamics has three characteristics, λ1 = v − c, λ2 = v, and λ3 = v + c, given in (5.10) of Chapter 7. When one of the characteristic speeds equals the zero speed of the duct, resonance occurs. The characteristic speed λ2 is linearly degenerate, and the characteristic speeds λ1 and λ3 are related to the shock waves; see Proposition 5.1 of Chapter 7. Thus the interesting case is where either λ1 = v − c or λ3 = v + c is close to zero. This happens when the fluid speed |v| is close to the sound speed c, that is, when the gas flow is transonic. Thus nonlinear resonance occurs for transonic gas flows. For definiteness, assume that the gas flow has positive speed v > 0 so that when the flow speed v is around the sound speed c, λ1 = v − c is around zero and resonance occurs. To illustrate the resonance phenomenon, consider the single equation (1.4)
ut + f (u)x = c(x)h(u), f (u) > 0, f (0) = 1, f (0) = f (0) = 0.
The scalar equation can be obtained from (1.1) by carrying out the asymptotic expansion along the 1-characteristic direction r 1 with the function
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351
u ≡ λ1 = v − c. Thus we call (1.5) u = 0 the sonic state, {u : u > 0} supersonic states, and {u : u < 0} subsonic states. The term c(x) is related to the cross-section A(x) of the duct by c(x) = −A (x)/A(x) so that (1.6) {x : c(x) > 0}, converging duct; {x : c(x) < 0}, diverging duct. The function h(u) represents the coupling of the gas motion with the geometry and satisfies (1.7)
h(u) > 0, h (u) > 0.
It is known from experiments that a stationary transonic shock is stable in the diverging portion of a nozzle, while it is unstable in the converging portion of the duct. The following analysis for the scalar equation (1.4) can be generalized to the system (1.1) to verify this basic fact. The phenomenon of resonance occurs when some of the following three speeds characterizing the situation are equal to each other: (1.8) λ(u) = f (u), characteristic speed; σ = σ(u− , u+ ) =
f (u+ ) − f (u− ) , shock speed; u+ − u− 0, speed of the source.
The source c(x)h(u) is induced by the duct, which is non-moving and therefore has zero speed. The characteristic speed λ(u) and the shock speed σ(u− , u+ ) depend nonlinearly on the solution u(x, t) and so this is a situation of nonlinear resonance. Assume that the duct is uniform, A(x) = constant or c(x) = 0, outside a bounded region 0 < x < 1. For 0 < x < 1, the duct is not uniform, c(x) = 0, and the stationary solutions satisfying the following differential equation are non-constant: (1.9)
f (u)x = c(x)h(u), 0 < x < 1.
The stationary solutions correspond to the zero speed of the source. The differential equation becomes degenerate when f (u) → 0 as the flow becomes sonic, u → 0. A stationary solution can contain a stationary shock wave. In order to study the resonance phenomenon, it is crucial to analyze the dependence of a stationary solution on the location of the stationary shock contained therein. The following basic lemma says that the situation for the converging duct, c(x) > 0, is different from that for the diverging duct, c(x) < 0, 0 < x < 1.
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12. Nonlinear Resonance
Lemma 1.1. Consider a supersonic state u0 > 0. Let uy (x) (or uz (x)) be two stationary flows with the same initial state uy (0) = uz (0) = u0 and containing a stationary shock at x = y (or x = z). Then, for 0 < y < z < x < 1, as shown in figure 12.01, f (uz (x)) < f (uy (x)), uy (x) < uz (x) < 0, when the duct is diverging; f (uz (x)) > f (uy (x)), uz (x) < uy (x) < 0, when the duct is converging. u
u
u0 uy
=
uy = uz
uz
u0 0
y
z
1
x
0
z
y
1 uz
uy
u
uz
Diverging duct
Converging duct
x
y
Figure 12.01. Stationary flow with a transonic shock wave.
Proof. From the Rankine-Hugoniot and entropy conditions, a stationary shock (u− , u+ ) satisfies the continuity condition on the flux, f (u− ) = f (u+ ), and is transonic, u− > 0 > u+ . Thus, for y < x < z, uy (x) is subsonic while uz (x) is supersonic, as defined in (1.5); in particular, uy (x) < uz (x). From the assumption (1.7), h (u) > 0, and we have, in the case of a converging duct c(x) > 0 defined in (1.6), f (uy (x))x = c(x)h(uy (x)) < c(x)h(uz (x)) = f (uz (x))x , y < x ≤ z. Across a stationary shock the flux is continuous, and so the above yields f (uy (x)) < f (uz (x)) for x > z. By (1.4), this implies that uz (x) < uy (x) < 0 for x > z; see left diagram of figure 12.01. The case of a divergent duct is similar. This completes the proof of the lemma. To study possible bifurcation phenomena for (1.4) due to resonance, one considers the time-asymptotic states for solutions with given end states u → ul as x → −∞ and u → ur as x → ∞. A wave pattern is a timeasymptotic state if it solves the generalized Riemann problem in the following sense.
1. Moving Source
353
Definition 1.2. A wave pattern W solves the generalized Riemann problem (ul , ur ) for (1.4) if 1. W consists of a shock or rarefaction wave with negative (or positive) speed for the conservation law ut + f (u)x = 0 for x < 0 (or for x > 1), and consists of a stationary wave for 0 < x < 1; and 2. W attains the state ul as x → −∞ and the state ur as x → ∞. Remark 1.3. The restriction on the wave pattern in the definition is to ensure its non-interacting property. Waves outside the region 0 < x < 1 are for the conservation law ut + f (u)x = 0, as c(x) = 0, i.e. the duct is uniform there. We know from the theory for convex conservation laws, presented in Chapter 3, that the only non-interacting wave pattern is either a single shock or a rarefaction wave. Thus, for both the region x < 0 and the region x > 1, only one shock or rarefaction wave is allowed in W. Waves in the region x < 0 with positive speed would interact with the stationary wave in 0 < x < 1 and therefore are not allowed in W. Similarly, only a wave of positive speed is allowed for x > 0. In conclusion, for x < 0 (or x > 0), only one shock or rarefaction wave with negative (or positive) speed is allowed in W. A general solution of the equation (1.4) would, time-asymptotically, become non-interacting, and so these wave patterns represent the timeasymptotic states of general solutions for (1.4) that attain state ul at x = −∞ and state ur at x = ∞. As with the Riemann problem for hyperbolic conservation laws, solutions to the generalized Riemann problem is much easier to construct than the solution to the general initial value problem. For conservation laws, the Riemann problem has a unique solution. However, generalized Riemann problem for equations with a source such as (1.4) may have more than one solution, indicating that there are bifurcation and stability issues. Consider first a stationary wave u(x), 0 < x < 1, consisting of a smooth part solving f (u)x = c(x)h(u) and possible stationary shocks. For a converging duct, c(x) > 0 for 0 < x < 1, f (u)x = c(x)h(u) > 0 and so f (u(x)) increases in x. Consequently a downstream state u(x) is further away from the sonic state u = 0 than an upstream state u(y), 0 < y < x < 1. In particular, given an upstream state at x = 0 there always exists a stationary flow for 0 < x < 1. Consider a stationary flow u(x), 0 < x < 1, containing a stationary shock at x = y. By the entropy condition and the Rankine-Hugoniot condition, u− > 0 > u+ , f (u− ) = f (u+ ), u− ≡ u(y − 0), u+ ≡ u(y + 0). Thus the upstream state of the shock is supersonic, u(x) > 0 for 0 < x < y, and downstream state of the shock is subsonic, u(x) < 0 for y < x < 1. For
354
12. Nonlinear Resonance
a given supersonic state u(0), we can always construct the stationary flow u(x) = uy (x), 0 < x < 1, with a stationary shock at x = y for any given y ∈ (0, 1). Proposition 1.4. Consider a converging duct, c(x) < 0 for 0 < x < 1, and a stationary solution uy (x) connecting ul at x = 0 and ur at x = 1, with a stationary shock at x = y, as depicted in figure 12.02. Then the generalized Riemann problem (ul , ur ) has three solutions.
u
u1(x) ul
y
u2(x)
x
ur
Figure 12.02. Stationary flow with a transonic shock wave.
Proof. Let u1 (x) be the smooth stationary wave in uy (x) extended from x = 0 to x = 1. u1 (x) connects ul at x = 0 to u1 (1) at x = 1. Similarly, consider the smooth stationary solution u2 (x), which is the extension of the smooth stationary solution in uy (x) extended from x = 1 to x = 0. u2 (x) connects u2 (0) at x = 0 to ur at x = 1; see figure 12.02. We now construct two solutions to the generalized Riemann problem (ul , ur ) besides the stationary solution uy (x). The first solution is the smooth stationary solution u1 (x), 0 < x < 1, followed by the wave (u1 (1), ur ). By Lemma 1.1, f (u1 (1)) > f (uy (1)) = f (ur ). As u1 (1) > 0 > ur , the wave (u1 (1), ur ) forms a transonic shock wave with positive speed: σ(u1 (1), ur ) =
f (ur ) − f (u1 (1)) > 0. ur − u1 (1)
1. Moving Source
355
t
ul
t
u2(0)
0
u1(1) x
1
ur x
Figure 12.03. Smooth stationary flow with a subsonic or supersonic shock wave.
Consequently, the wave pattern consisting of the smooth stationary wave u1 (x) and the shock (u1 (1), ur ), as depicted in figure 12.02 and the right diagram of figure 12.03, is non-interacting and forms a solution of the generalized Riemann problem (ul , ur ). The third wave pattern consists of a shock wave (ul , u2 (0)) followed by the smooth stationary wave u2 (x). Again by Lemma 1.1, the shock wave (ul , u2 (0)) has negative speed and so we have the third solution of the generalized Riemann problem (ul , ur ); see figure 12.02 and left diagram of figure 12.03. Proposition 1.5. There exists at most one solution to the generalized Riemann problem for the diverging duct, c(x) > 0 for 0 < x < 1. Proof. This follows from the first part of Lemma 1.1 and analysis similar to the above for the converging duct. When the generalized Riemann problem has only one solution, it is expected to be stable. On the other hand, when there are multiple solutions, some of them must be unstable, as we now analyze. Proposition 1.6. A stationary flow with a shock is stable when the duct is diverging and unstable when the duct is converging. Proof. Consider a small perturbation of the stationary solution with a stationary shock at x = x0 :
u1 (x), a smooth stationary solution for x < x0 , (1.10) u(x, 0) = u2 (x), a smooth stationary solution for x > x0 , (u1 (x0 − 0), u2 (x0 + 0)) a shock with small, nonzero speed.
356
12. Nonlinear Resonance
As time evolves, the shock (u− (t), u+ (t)) = (u1 (x(t), t), u2 (x(t), t)) moves either to the right, x (t) > 0, or to the left, x (t) < 0, and the smooth stationary solution u1 (x) extends or retreats, respectively. The shock speed x (t) is given by the Rankine-Hugoniot condition: x (t) =
f (u2 (x(t))) − f (u1 (x(t))) . u2 (x(t)) − u1 (x(t))
The shock acceleration is f (u2 (x(t)))x − f (u1 (x(t)))x u2 (x(t))x − u1 (x(t))x x (x (t))2 . (t) − x (t) = u2 (x(t)) − u1 (x(t)) u2 (x(t)) − u1 (x(t)) Since u1 (x) and u2 (x) satisfy the stationary equation f (u)x = c(x)h(u), the above yields c(x(t)) x (t) = 2 x (t) u (x(t)) − u1 (x(t)) h(u2 (x(t))) h(u1 (x(t))) 2 1 − . · h(u (x(t))) − h(u (x(t))) − x (t) 2 f (u (x(t))) f (u1 (x(t)))
t
t
Stable shock
Unstable shock
x
x
Figure 12.04. Perturbation of a stationary shock wave.
For the converging duct, c(x) > 0, and for a nearly stationary shock, small, the above implies that
x (t)
x (t) > 0. x (t) Thus the shock is accelerating and so the stationary shock is unstable; see left diagram of figure 12.04. The same analysis shows that the stationary shock for the diverging duct decelerates when perturbed and therefore is stable; see right diagram of figure 12.04.
2. Sub-shocks
357
A common design is the so-called Devore nozzle with a converging and a diverging duct, e.g. c(x) > 0 for 0 < x < 1/2 and c(x) < 0 for 1/2 < x < 1. For a converging or a diverging duct, a smooth stationary flow must be either supersonic or subsonic, and not transonic. For a Devore nozzle, a smooth transonic flow is possible, with the flow being sonic at the throat x = 1/2, where the duct has minimum cross-section. A Devore nozzle can be used to trap the stable shock in the diverging part of the nozzle, 1/2 < x < 1. The nozzle provides a useful tool for capturing shock waves in a gas flow for experimental observation. There are other resonance phenomena for flows with a moving source. For instance, when a rarefaction wave reaches the sonic state u = 0, it turns back and transforms into compression waves, which eventually form shock waves.
2. Sub-shocks Discontinuous shock waves exist in general solutions of systems of hyperbolic conservation laws. For parabolic conservation laws, a discontinuous shock wave is smoothed out to become a viscous shock profile. There are physical systems that are dissipative but not parabolic. For such a system, a shock wave can be neither a jump discontinuity nor a smooth profile, but a combination of smooth profiles and discontinuity waves. This results from the nonlinear resonance phenomenon. The relaxation models are basic examples that exhibit such a phenomenon. For the relaxation models studied in Chapter 11, the speeds characterizing the situation are the frozen speeds, the equilibrium speeds, and the shock speed. When the shock speed is close to one of the frozen characteristic speeds, resonance occurs and sub-shocks arise within a shock profile. Since a relaxation model is hyperbolic, there are shock waves corresponding to the associated frozen conservation laws. These shock waves usually decay due to the dissipation mechanism induced by the relaxation. What we are interested in are non-decaying shocks. Sub-shocks are non-decaying as a result of the compression of the states at the far field and the nonlinear resonance. The end states of the shock profile satisfy the entropy condition for the equilibrium conservation laws, and a sub-shock is a jump discontinuity for frozen conservation laws. We illustrate this for the simple relaxation model (1.1) studied in Section 1 of Chapter 11: ut + f (u, v)x = 0, , vt + g(u, v)x = V (u)−v τ
358
12. Nonlinear Resonance
or
ut + f (u)x =
0 V (u)−v τ
.
Recall that there are frozen speeds λ1 and λ2 and the equilibrium speed Λ; see (1.3) and (1.4) in Section 1 of Chapter 11. We have assumed the coupling condition fv = 0; for definiteness, assume ∂ f (u, v) < 0. ∂v As we have seen, the mechanism of relaxation has a smoothing effect and there exist smooth shock profiles. Without ambiguity in notation, we write u(x, t) = u(ξ), ξ = (x − σt)/τ , for a traveling wave connecting two equilibrium states u− and u+ , and let v± = V (u± ): 0 d ) + f (u)(ξ) − f (u ) = , −σ(u(ξ) − u − − dξ V (u(ξ)) − v(ξ) (2.1) u(−∞) = u− , u(∞) = u+ , ξ ∈ R. Direct computations yield (2.2)
−fv d u(ξ) = (V (u) − v), dξ (λ1 − σ)(λ2 − σ) fu − σ d v(ξ) = (V (u) − v), dξ (λ1 − σ)(λ2 − σ) (Λ − σ)V (u) d (V (u) − v)(ξ) = (V (u) − v). dξ (λ1 − σ)(λ2 − σ)
For a weak shock, with |u+ − u− | small, its speed σ is close to the equilibrium speed Λ, which lies between the frozen speeds λ1 and λ2 by the sub-characteristic condition, (1.5) of Chapter 11, and so the above differential equations are non-singular. In such a case, a smooth profile exists if the end states satisfy the strict Oleinik entropy condition, as shown in Proposition 1.1 in Chapter 11. The equations become singular when the shock speed σ approaches either of the frozen speeds λi , i = 1, 2, which can happen when |u+ − u− | is large. In this case, resonance occurs and the traveling wave contains discontinuous sub-shocks. The combination of smooth profiles and sub-shocks is constructed as follows. The first equation in (2.1) is a conservation law, which can be integrated to yield an algebraic relation −σ(u(ξ) − u− ) + f (u(ξ), v(ξ)) − f (u− , v− ) = 0. This and the coupling assumption fv = 0 imply that the second variable v can be written as a given function of the first variable u, v = v∗ (u). With
2. Sub-shocks
359
this, the second equation in (2.1) becomes d −σ(v∗ (u) − v− ) + g(u, v∗ (u)) − g(u− , v− ) = V (u) − v∗ (u). dξ From the first equation in (2.2), this can be viewed as a differential equation for the independent variable u: d d −σv + g(u, v) = −σ(v∗ (u) − v− ) + g(u, v∗ (u)) − g(u− , v− ) du du (σ − λ1 )(λ2 − σ) . = fv Thus −σv + g(u, v) has an extremum as a function of u where the shock speed σ crosses either of the frozen characteristic speeds λi , i = 1, 2. Suppose that u− < u+ . For a smooth profile, −σv + g(u, v) is strictly increasing for u− < u < u+ . The construction of the profile with sub-shocks is illustrated graphically as follows: Consider the case when the function g(u, v) − σv is strictly increasing, reaches a maximum at u = u∗ , and then decreases from u = u∗ to u = u+ , as shown in the left diagram of figure 12.05. Draw an horizontal line from the point on the graph at u = u+ that meets the graph at u = u1 for some u1 between u− and u∗ . The profile connecting u− and u+ consists of a smooth profile for u− < u < u1 followed by a sub-shock, the discontinuous jump (u1 , u+ ) shown in the right diagram of figure 12.05.
u(ξ )
g(u, v) − σv
u+
u∗ u1 u−
u+
u− u
u1 ξ
Figure 12.05. One sub-shock.
There can be more than one sub-shock when the function −σv + g(u, v) has several extrema between u− and u+ . In figure 12.06, the profile consists of a smooth profile (u− , u1 ), a sub-shock (u1 , u2 ), another smooth profile (u2 , u3 ), and another sub-shock (u3 , u+ ).
360
12. Nonlinear Resonance
u(ξ )
g(u, v) − σv
u+ u3 u1 u−
u2 u2
u3
u+ u
u−
u1 ξ
Figure 12.06. Two sub-shocks.
The sub-shocks (u1 , u2 ) and (u3 , u4 ) are jump discontinuities and so need to satisfy the Rankine-Hugoniot condition as well as the Liu entropy condition (E) for the frozen system ut + f (u)x = 0. We now check these for (u1 , u2 ). By the above construction, g(u1 ) − σv1 = g(u2 ) − σv2 . In writing v = v∗ (u), the conservation law is observed and so the Rankine-Hugoniot condition holds: f (u1 ) − σu1 = f (u2 ) − σu2 . This establishes the RankineHugoniot condition for (u1 , u2 ). Note that the shock speed σ(u1 , u2 ) is the same as the speed σ of the traveling profile. Next we check the entropy condition (E) for (u1 , u2 ) by showing that σ ≡ σ(u1 , u2 ) < σ(u1 , u) for all u on the Hugoniot curve H(u1 ) between u1 and u2 . If this does not hold, then there is a state u on the Hugoniot curve H(u1 ) between u1 and u2 with the property that σ(u1 , u) = σ:
σ(u − u1 ) = f (u, v) − f (u1 , v1 ), σ(v − v1 ) = g(u, v) − g(u1 , v1 ). The first identity and the conservation law σ(u1 −u− ) = f (u1 , v1 )−f (u− , v− ) imply the conservation law σ(u − u− ) = f (u, v) − f (u− , v− ). This implies that the state u is on the graph depicted in figure 12.06. However, by the construction, the second identity cannot hold as the graph of g − σv does not touch the horizontal line between u1 and u2 . We have thus shown that (u1 , u2 ) is an admissible shock. In the simplified situation where both the frozen system and the equilibrium equation are genuinely nonlinear, there can be at most one sub-shock in a profile, as in figure 12.05.
3. Non-strict Hyperbolicity
361
For the thermal non-equilibrium Euler equations for gas dynamics, (3.4) of Chapter 11, a shock profile x − σt , u(±∞) = u± u(x, t) = u(ξ), ξ ≡ κ0 satisfies the differential equations ⎧ dτ dv ⎪ ⎪−σ dξ + dξ = 0, ⎪ ⎪ ⎨−σ dv + dp = 0, dξ dξ (2.3) dpv d v2 ⎪ (e + q + −σ 0 ⎪ dξ 2 ) + dξ = 0, ⎪ ⎪ ⎩−σ dq = κ0 (Q − q). dξ
κ
The reference relaxation time κ0 is chosen so that the ratio κ0 /κ is of order 1 and the profile is independent of the scaling of the relaxation time. There are two obstacles to the existence of a smooth profile. The first is that the differential equations have a critical state between the end states u− and u+ . This does not happen by the entropy condition for the end states with respect to the equilibrium system, as noted in Chapter 11. The second is that the equations become degenerate. Rewrite (2.3)in matrix form: du κ0 = (0, 0, 0, (Q − q))T . dξ κ The eigenvalues of the flux matrix A are the frozen characteristics −cf , 0, 0, and cf . Consider the forward shock, σ > 0, of moderate strength; then σ is close to the equilibrium sound speed Ce at the end states. By the subcharacteristic condition, Ce < cf , (3.14) of Chapter 11, the matrix −σI + A is invertible and the differential equations are non-degenerate. A smooth shock profile exists in such a case. On the other hand, as the shock strength increases, the shock speed σ may cross the frozen sound speed cf and the differential equations become degenerate. In such a situation, by analysis similar to the above for the simple relaxation model, it can be shown that the shock profile contains a sub-shock. For the case of one non-equilibrium mode as considered here, and for the case of polyatomic gases for which the acoustic modes are genuinely nonlinear, there exists at most one sub-shock in a shock profile. In general, more than one sub-shock is possible. (2.4)
(−σI + A)
Sub-shocks are observed in high-temperature gas dynamics flows as the gases are in general in thermal non-equilibrium. Experimentally, the subshocks are essential signals reflecting some of the basic thermodynamic properties of the gases.
3. Non-strict Hyperbolicity In Chapter 5 on scalar laws and Chapter 7 on systems, the shock wave theory for hyperbolic conservation laws is complete with the entropy condition,
362
12. Nonlinear Resonance
which is obtained by considering viscous shock profiles for associated viscous conservation laws. This works for the Euler equations in gas dynamics, for instance. However, for a more complex physical situation, there may be resonance caused by non-strict hyperbolicity and consideration of the existence of viscous shock profiles may not be sufficient for the formulation of a well-posed theory for hyperbolic conservation laws. Compared to the strictly hyperbolic case, the dissipation parameters play a more subtle role in defining the inviscid theory when the system is non-strictly hyperbolic. Models for nonlinear elasticity and magnetohydrodynamics can be nonstrictly hyperbolic. In the theory of magnetohydrodynamics, there is the issue of physical admissibility of the intermediate shocks, which result from combining an Alf´ ven wave with either a fast or a slow shock; see (7.2) in Section 7 of Chapter 7. These waves seem to exist in astronomical observations of the solar wind around Earth. Moreover, there are viscous profiles for intermediate shocks. On the other hand, the inviscid model becomes ill-posed when the intermediate shocks are admitted; the Riemann problem can have more than one solution. This paradox is resolved by recognizing that for the non-strictly hyperbolic model there is a strong resonance and the inviscid model, with consideration of the existence of shock profiles, is not sufficient to resolve this issue of physical admissibility. The situation depends sensitively on the strength of the viscosity. The following simple rotational symmetry models illustrate this situation; the first is the inviscid model and the second is the viscous model having a small viscosity coefficient 0 < ε 1: ut + u(u2 + v 2 ) x = 0, (3.1) vt + v(u2 + v 2 ) x = 0; (3.2)
ut + u(u2 + v 2 ) x = εuxx , vt + v(u2 + v 2 ) x = εvxx .
The inviscid system (3.1) can also be written as ut + (u|u|2 )x = 0. The characteristic values and characteristic directions for the inviscid model are v 2 2 2 , λ1 u) = |u| = u + v , r 1 (u) = −u (3.3) u . λ2 (u) = 3|u|2 = 3(u2 + v 2 ), r 2 (u, v) = v The characteristic curves Rj , the integral curves for r j , j = 1, 2, are circles and lines through the origin: R1 (u0 ) = {u : |u| = |u0 |},
R2 (u0 ) = {u : u = αu0 , −∞ < α < ∞}.
3. Non-strict Hyperbolicity
363
From (3.3), the first characteristic field is linearly degenerate and the second one is genuinely nonlinear except at the origin, as defined in Definition 4.1 of Chapter 7: ∇λ2 · r 2 = 6|u|2 > 0 for u = 0,
∇λ1 · r 1 ≡ 0.
The second characteristic field is genuinely nonlinear away from the origin. Thus we may form classical shock and rarefaction waves (u0 , u1 ) that take values on R2 (u0 ) without crossing the origin according to the theory in Section 4 of Chapter 7: (3.4) (u0 , u1 ) is a classical shock wave if u1 = αu0 , 0 < α < 1; (u0 , u1 ) is a classical rarefaction wave if u1 = αu0 , α > 1. The first characteristic field is linearly degenerate. Any two states on the same circle u2 + v 2 = C are connected by a contact discontinuity (3.5)
ven waves. (u− , u+ ), |u− | = |u+ |, Al´
The classical shock theory in Chapter 7 can be applied to obtain a unique solution to the Riemann problem for the inviscid model (3.1). Proposition 3.1. The Riemann problem (ul , ur ) is solved uniquely within the class of classical shock, rarefaction, and Al´ven waves, (3.4) and (3.5). Proof. For the given ul , choose the unique um with the property that um = αur for some scalar α > 0. When |ur | > |ul |, 0 < α < 1 and the ven wave (ul , um ) and the Riemann problem (ul , ur ) is solved by the Al´ classical rarefaction wave (um , ur ). In the case of |ur | ≤ |ul |, α ≥ 1 and ven wave (ul , um ) and the the Riemann problem (ul , ur ) is solved by the Al´ classical shock wave (um , ur ); see figure 12.07. An intermediate shock (u− , u+ ) is a jump discontinuity for (3.1) when the states cross the origin: (3.6)
(u− , u+ ) is an intermediate shock if u− = αu+ , α < 0.
We now consider an intermediate shock (ul , ur ). Because of rotational symmetry, we may consider, without loss of generality, the case where ul = (ul , 0) and ur = (ur , 0), with ul < 0 < ur < −ul ; see figure 12.08. The above inviscid theory does not include the intermediate shocks. In the above inviscid theory the Riemann problem is resolved by the Al´ ven wave (ul , um ) and the shock (um , ur ). There is a particular viscous profile with v ≡ 0 for the intermediate shock (ul , ur ) satisfying −σ(u(ξ) − ul ) + (u(ξ))3 − (ul )3 = u (ξ),
ξ≡
(ur )3 − (ul )3 x − σt , σ= . ε ur − ul
364
12. Nonlinear Resonance
v Al´ ven wave
ul ur
shock
rarefaction ur um
u
Figure 12.07. Classical Riemann solution.
v Al´ ven wave
ul intermediate shock
ur
um
u
shock
Figure 12.08. Intermediate shock and classical waves.
The situation is the same as for the scalar conservation law ut +(u3 )x = εuxx in Chapter 5. The shock profile exists if the strict Oleinik entropy condition
3. Non-strict Hyperbolicity
365
holds: (ur )3 − (ul )3 u3 − (ul )3 < for all u such that ul < u < ur . ur − ul u − ul In this case, the right state ur needs to be bigger than the state u∗ of tangency to the graph of f (u) = u3 through the left state ul : ur > u∗ , u∗ > 0, 3(u∗ )2 ≡
(ul )3 − (u∗ )3 ul , or u∗ = − . ul − u∗ 2
We denote the profile (u(ξ), 0) by φ0 (ξ). Proposition 3.2. The Riemann problem (ul , ur ) can have more than one solution if any shock wave with a viscous shock profile is allowed; see Definition 3.1 in Section 3 of Chapter 4. Proof. Take the case where ul , 2 so that (ul , ur ) has a viscous profile. In this case, the Riemann problem (ul , ur ) is solved by an intermediate shock with a shock profile. On the other hand, from Proposition 3.1, there is another solution consisting of an Al´ ven wave and a classical wave. Thus this Riemann problem has two solutions. ul = (ul , 0), ur = (ur , 0), ul < 0 < ur , ur > −
It turns out that there are infinitely many shock profiles u(ξ) for an intermediate shock (ul , ur ) according to the viscous system (3.2). All the profiles φ(ξ) of (3.2) satisfy the system of differential equations (3.7)
u = −σ(u − ul ) + u|u|2 − ul |ul |2 .
Note that this shock is over-compressive: λ1 (ur ) < λ2 (ur ) < σ < λ1 (ul ) < λ2 (ul ). It is not a Lax shock as in Definition 4.3 of Chapter 7. Because of this, in the (u, v) phase plane, the system of ordinary differential equations (3.7) has an unstable node at ul as ξ → −∞ and a stable node at ur as ξ → ∞. All the trajectories leave ul and enter ur . Thus there are infinitely many shock profiles besides the φ0 given above like those shown in figure 12.09. According to the classical approach, the solution of the Riemann probven wave (ul , −um ) and the shock (−um , ur ). lem (ul , ur ) consists of the Al´ Thus the intermediate shock (ul , ur ) can be viewed as the combination of an Alf´ ven wave and a classical shock wave. The infinitely many shock profiles are between the limiting circle |u| = |ul | and the classical shock profile for (−um , ur ), as shown in figure 12.09. The circle represents contact discontinuities. States on the circle |u| = |ul | are critical points of the differential
366
12. Nonlinear Resonance
v Al´ ven wave
ul intermediate shock
ur
um
u
shock
Figure 12.09. Intermediate shock profiles.
equations (3.7). Thus near the circle, the profile approaches the end states at slower rates. If we generalize the theory for strictly hyperbolic conservation laws by admitting any shock with a viscous profile, we would regard (ul , ur ) as an admissible inviscid shock. This leads to a paradox, as we have solved the Riemann problem uniquely without admitting this over-compressive intermediate shock, and now we have another solution, which is the intermediate shock. A stronger admissibility criterion would be to admit only shocks with time-asymptotic stable viscous profiles. We now analyze the stability of these viscous profiles. For a classical Lax shock, a perturbation of its viscous profile gives rise to a shift of the profile plus diffusion waves propagating away from the profile; see (3.1) of Chapter 14. In the present case, a perturbation x ¯ (x, 0) u(x, 0) = φ( ) + u ε of the over-compressive shock profile φ(x − σt) does not give rise to diffusion waves propagating away from the shock, as the shock is over-compressive and there is no characteristic direction leaving the shock. Instead, since there are infinitely many profiles, a perturbation of one profile φ will in ¯ with a certain shift x0 of the profile, as general tend to another profile φ
3. Non-strict Hyperbolicity
367
t → ∞:
x + x0 − σt ¯ x + x0 − σt ) − φ( x − σt ). ¯ (x, t) → φ( ), or u ε ε ε There is a one-parameter family of profiles and the shift x0 is another parameter. Thus there are two degrees of freedom in choosing the time-asymptotic profile and shift. These two degrees are determined by the two conservation laws ∞ ∞ ¯ (x, t) dx = ¯ (x, 0) dx, t > 0. u u ¯ u(x, t) → φ(
−∞
−∞
¯ (x, 0), The question is reduced to whether for a given initial perturbation u ¯ and a shift x0 can be found so that a profile φ ∞ ∞ x + x0 − σt x − σt ¯ ¯ (x, 0) dx = ) − φ( ) dx φ( u ε ε −∞ −∞ ∞ ¯ + x0 − σt) − φ(x − σt) dx. =ε φ(x −∞
The differential equations for φ are independent of the viscosity coefficient ε, and so are the profiles φ. Consider, for instance, a fixed perturbation ¯ (x, 0) having positive integral for its second component: u ∞ v¯(x, 0) dx > 0. A≡ −∞
¯ needs to be chosen closer to the upper-half Then, as ε → 0, the profile φ ¯ grows at the rate circle so that the width in the variable x of the profile φ −1 of ε in order to satisfy the above identity: ∞ ¯ + x0 − σt) − φ(x − σt) dx = O(1)A . φ(x ε −∞ ¯ (x, 0) is allowed to vanish, say A = O(1)ε → 0 as If the perturbation u ¯ ε → 0, then the profile φ(ξ) stays away from the upper half-circle and tends to the jump discontinuity as ε → 0. The profiles are nonlinearly stable for a fixed viscosity ε. However, the perturbation needs to be in a O(1)ε neighborhood, which vanishes as the viscosity coefficient tends to zero. In other words, these profiles are nonlinearly stable, but not uniformly with respect to the strength ε of the viscosity. Thus the intermediate shock is a stable viscous wave, but not an admissible inviscid wave. Under favorable physical conditions, the intermediate shocks can be observed, even though the dissipation parameters are small in gas dynamics. This resolves the paradox. We summarize the above in the following proposition. Proposition 3.3. A shock profile for an intermediate shock is stable when the perturbation is of the same order as the viscosity. In other words, the
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shock profile is not uniformly stable as the viscosity tends to zero. Consequently, an intermediate shock is not an admissible inviscid shock.
4. Vacuum Vacuum represents the physical boundary of gas motion. In astro-physics, it is natural to study gas motion around vacuum; it occurs around gaseous stars, for instance. The Euler equations become non-strictly hyperbolic and degenerate at vacuum. As a consequence, the solutions develop a certain type of singularity near vacuum. Consider the one-dimensional isentropic Euler equations for polyatomic gases:
ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p(ρ))x = 0, p(ρ) = ργ , γ > 1. As before, set u ≡ (ρ, ρv)T . The characteristic and Hugoniot curves as studied in Chapter 7, (5.3), are, as depicted in figure 12.10, (v − v0 ) = −
p(ρ) − p(ρ0 ) (ρ − ρ0 ) along the Hugoniot curve H 1 (u0 ); ρρ0 (ρ − ρ0 )
p(ρ) − p(ρ0 ) (ρ − ρ0 ) along the Hugoniot curve H 2 (u0 ); ρρ0 (ρ − ρ0 ) ρ 1 p (η) dη along the characteristic curve R1 (u0 ); v − v0 = − η ρ0 ρ 1 v − v0 = p (η) dη along the characteristic curve R2 (u0 ). ρ0 η
(v − v0 ) =
ρ
t
Ul = (ρl, vl )
R1(ul ) v
Ur = (0, vr)
Ul
Ur
x
Figure 12.10. Rarefaction wave bordering vacuum.
Proposition 4.1. For polyatomic gases p(ρ) = ργ , γ > 1, only a rarefaction wave, and not a shock wave, can connect to the vacuum state. A Riemann problem with one initial state being a vacuum state is solved by one rarefaction wave.
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369
Proof. The characteristic curves Ri intersect the vacuum line ρ = 0, because for γ > 1, ρ0 ρ0 1 √ − 3−γ lim p (η) dη = γη 2 dη < ∞. ρ→0 ρ η 0 On the other hand, the Hugoniot curves H i do not intersect the vacuum line: 1 1 ( − )(p(ρ) − p(ρ0 )) → ∞ as ρ → 0. ρ ρ0 Consider the Riemann problem (ul , ur ) with ur a vacuum state, ρr = 0. Physically, the velocity vr of the vacuum state ur should not be specified. The velocity vr of the right state is determined from the intersection of the wave curve R1 (ul ) with the vacuum line ρ = 0, as shown in figure 12.10, and the solution consists of only one wave, the 1-rarefaction wave discussed in Section 1 and Section 3 of Chapter 7: ⎧ x ⎪ ⎨(λ1 )t + λ1 (λ1 )x = 0, u ∈ R1 (ul ) for λ1 (ul ) < t < λ1 (ur ), (4.1) u(x, t) = ul for λ1 (ul ) < xt , ⎪ ⎩ ρ(x, t) = 0 for xt > λ1 (ur ). An interesting property of vacuum is that it has a smoothing effect: Proposition 4.2. A shock wave vanishes in finite time as it propagates toward the rarefaction wave bordering the vacuum state. Proof. Consider the case of a 2-shock wave (u0 , ul ) propagating toward the above 1-rarefaction wave (ul , ur ) with vacuum right state ur . Let the states of the shock be (u− , u+ ) = (u− (t), u+ (t)) as it enters the rarefaction wave; see right diagram of figure 12.11.
ρ H2
t
ul
R1
u0
R1
u+
u−
H2 u+ (T )
ur
v
u0 ul
u− u+
u+(T ) ur
x
Figure 12.11. Shock vanishing at vacuum.
The right state u+ (t) is part of the rarefaction wave (ul , ur ) in (4.1). Behind the shock, the solution is another 1-simple wave and so the left state
370
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u− is determined as the intersection of R1 (u0 ) and H 2 (u+ (t)); see diagram figure of figure 12.11. By Galilean transformation, the shape of the wave curves Ri and the Hugoniot curves H i remains unchanged under translation in the velocity direction. This is also seen as a direct consequence of the relations given above for these curves. Thus in the (ρ, v) coordinates R1 (u0 ) and R1 (ul ) are translations of each other in the v direction; in particular, vl − v0 is of the same order as v+ (t) − v− (t). From the Rankine-Hugoniot relation, ρ+ ρ+ (ρ− )γ ˜ l − v0 )2 . 1− 1 − ( )γ = (v− − v+ )2 =(v (4.2) ρ+ ρ− ρ− We claim that
ρ+ → 0 as ρ± → 0. ρ− If not, then ρ+ /ρ− is less than 1 and greater than a positive constant C. Since γ > 1, this would imply that (ρ− )γ /ρ+ tends to zero, contradicting (4.2). This implies that σ(u− , u+ ) − λ1 (u+ ) =
ρ+ v+ − ρ− v− ρ− (v+ − v− ) − (v− − c− ) = c− + ρ+ − ρ− ρ+ − ρ− → c− + v− − v+ → v− − v+ as ρ± → 0.
Since v− − v+ is a positive number of the order vl − v0 , the shock propagates through the rarefaction wave with rate bounded from below and therefore reaches the vacuum state in finite time. The shock strength ρ− − ρ+ also vanishes in finite time. The proposition indicates that near vacuum, shock waves are not the main concern in the analysis of solution behavior. It turns out that a different kind of singularity occurs around vacuum. The singularity comes from the fact that in approaching the vacuum, the Euler equations become nonstrictly hyperbolic, because λ1 = λ2 = v at vacuum. In fact, the Euler equations become degenerate and parabolic at vacuum and the phenomenon of resonance occurs. Within the centered 1-rarefaction wave described above, u ∈ R1 (ur ) and the characteristic speed λ1 = x/t. Thus ρ x 2 5−γ 1 c, λ1 = v − c = vr + c= . p (η) dη = v − vr = γ−3 γ−3 t 0 η Consequently, the sound speed c is linear in x: γ−3 x ( − vr ). c= 5−γ t With shocks not a main concern, we write the Euler equations in nonconservative form, with the shock speed and the fluid speed as the basic
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371
dependent variables: 2 v c + 1 v t γ−1
(γ − 1)c2 v
2 c = 0. v x
At vacuum, c = 0, the flux matrix becomes non-diagonizable, and so the system is parabolic. For such a degenerate situation, the lower-order term becomes important in determining the solution behavior near vacuum. To illustrate this, we consider the Euler equations with damping by adding the source −ρv to the momentum equation: (4.3)
ρt + (ρv)x = 0, (ρv)t + (ρv 2 + p(ρ))x = −ρv.
When the second equation is replaced by Darcy’s law, the first equation becomes the porous media equation: (4.4)
p(ρ)x = −ρv, Darcy’s law, ρt = p(ρ)xx , porous media equation.
The porous media equation is parabolic away from vacuum, and at vacuum it is hyperbolic; the opposite is true for the Euler equations, which are hyperbolic away from vacuum and parabolic at vacuum. Nevertheless, the solutions of the damped Euler equations are expected, time-asymptotically, to tend to the solution of the porous media equation for the following reason. From the second equation in (4.3), the momentum is damped, and from the first equation, the mass is conserved. Thus the velocity v is expected to decay faster than the density ρ. The combination of damping and hyperbolic propagation induces dissipation and makes the first two terms in the momentum equation in (4.3) dissipate faster than the last two terms. This leads to the approximation of the second equation in (4.3) by the first equation in (4.4), Darcy’s law. For a more quantitative understanding, we consider particular solutions to the damped Euler equations. Same as with the Euler equations above, rewrite the system in terms of v and c2 :
(c2 )t + v(c2 )x + (γ − 1)c2 vx = 0, 1 (c2 )x = −v. vt + vvx + γ−1 From the second equation, the acceleration with damping along the particle path vt + vvx + v would slow down the flow exponentially fast, and so the inertia is not the main force pushing the vacuum boundary. The force from the pressure should eventually be finite to push the gas toward the vacuum at x = x0 (t): 1 (c2 )x = O(1), or c = O1 (1) |x − x0 (t)|, − γ−1 for some smooth positive O1 (1). Thus the sound speed is of the order of the square root of the distance |x − x0 (t)| from the vacuum boundary at
372
12. Nonlinear Resonance
x0 (t). The square root singularity of the sound speed around the vacuum boundary is therefore called the physical singularity for the damped Euler equations. To illustrate the time-asymptotic relation with the porous media equation, consider spherical solutions of the damped Euler equations in n spatial dimensions:
ρt + ∇x (ρv) = 0, (ρv)t + ∇x · (ρv ⊗ v + p(ρ)I) = −ρv, v, x ∈ Rn , x r ≡ |x|, v = v, (ρ, v) = (ρ, v)(r, t). r In spherical coordinates, the equations become
2 2 (c2 )t + v(c2 )r + n−1 r c v + (γ − 1)c vr = 0, 1 vt + vvr + γ−1 (c2 )r = −v. For the porous media equation ρt = Δx p(ρ) in spherical coordinates, the second equation is replaced by Darcy’s law, 1 (c2 )r = −v. γ−1 In light of the form of the physical singularity stated above, consider particular solutions of the form c2 (r, t) = e(t) − b(t)r2 ,
v(r, t) = a(t)r.
This form is chosen so that the coefficients a(t), b(t), and e(t) are functions of time only and, from the damped Euler equations, satisfy the ordinary differential equations ⎧ 2 2 ⎪ ⎨a + a + a − γ−1 b = 0, b + (nγ − n + 2)ab = 0, ⎪ ⎩ e + n(γ − 1)ea = 0.
For the porous media equation, Darcy’s law yields 2 b(t), Darcy line. a(t) = γ−1 This class of solutions can be analyzed using the phase plane diagram for (a, b) and shown to approach Darcy’s line as t → ∞. Thus the above particular solutions of the damped Euler equations approach Barenblatt’s selfsimilar solutions of the porous media equation; see the Exercises. For the damped Euler equations with initial values having a singularity of Holder type at vacuum, c2 (x, 0) = O(1)|x − x0 (0)|α , the solution would eventually evolve into the physical singularity α = 2. For a sufficiently smooth situation, α > 3, the solution maintains the same
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373
smoothness, α = constant, for a short time and then, at a certain waiting time, would switch to the physical singularity α = 2. For a higher degree of singularity, α < 2, the solution would evolve to α = 2 immediately at t = 0+. For the case of 2 < α < 3, the solution would also immediately switch to the case of α = 2. The phenomenon of waiting time exists also for the porous media equation (4.4). It would be interesting to study this and other issues for the damped Euler equations. The Euler equations with an additional Poisson equation for taking into account the gravitational force are used to model gaseous stars in astronomy. The lower-order term also induces a similar singularity for the solutions near vacuum.
5. Boundary In the theory of partial differential equations, the analysis of the boundary is a central issue, both in terms of formulation of the boundary condition and behavior of the solutions near the boundary. There is rich wave behavior induced by the presence of a boundary. We focus on resonance phenomena for hyperbolic conservation laws when the speeds of the waves and the boundary speed are close to each other. Consider the simplest setting of a scalar convex conservation law in one space dimension with the boundary fixed at x = 0. In the first subsection, we consider the resolution of boundary discontinuities, and in the second subsection we consider boundary resonance phenomena. 5.1. Boundary Riemann Problem Consider the initial-boundary value problem with given initial values ui (x) and boundary values ub (t): ⎧ ⎪ ⎨ut + f (u)x = 0, x, t > 0, (5.1) u(x, 0) = ui (x), x > 0, ⎪ ⎩ u(0, t) = ub (t), t > 0. The situation is clear for the linear hyperbolic equation ut +λ(x, t)ux = 0. When the characteristic speed λ(0, t) at the boundary is positive, the characteristic direction dx/dt = λ(0, t) points away from the boundary and toward the interior region x > 0. In this case, information propagates from the boundary to the interior and there is a need to prescribe the boundary value ub (t). On the other hand, when λ(0, t) is negative, information propagates toward the boundary from the interior. The solution in the interior provides the boundary value u1 (t) ≡ u(0+, t). In this case, there is no need to prescribe the boundary value ub (t). If a boundary value ub (t) is posed when
374
12. Nonlinear Resonance
λ(0, t) < 0, then it is not assumed by the solution u(x, t) as x → 0+, and there is a discontinuity (ub (t), u1 (t)) at the boundary x = 0. For nonlinear equations, the limiting boundary characteristic value λ(t) ≡ f (u(0+, t)) depends on the limiting value u(0+, t) from the interior of the solution, which is not known a priori. The linear thinking does not apply; it is not known a priori which part of the boundary needs the posing of a boundary value. One way to define the admissibility of solutions to the initial-boundary value problem is to prescribe the boundary value for all time and to allow the possibility of a boundary discontinuity for the solutions. Similar to discontinuous shocks, a boundary discontinuity should correspond to a boundary layer for the associated viscous equation uκt + f (uκ )x = κuκxx . A boundary discontinuity for the inviscid equation ut + f (u)x = 0 is admissible if it is the limit of a boundary layer for uκ as κ → 0+. Definition 5.1. A boundary discontinuity (ub , u1 ) is called admissible for the hyperbolic conservation law ut + f (u)x = 0 if there exists a stationary wave for the viscous conservation law uκt + f (uκ )x = κuκxx connecting the state ub at x = 0 and the state u1 at x = ∞: x (5.2) f (φ(ξ))ξ = φ(ξ)ξξ , φ(0) = ub , φ(∞) = u1 , ξ ≡ . κ There is a simple characterization of admissible boundary discontinuities. Proposition 5.2. Suppose that the Riemann problem (ub , u1 ) for the scalar conservation law ut +f (u)x = 0 is solved by waves with negative speed. Then (ub , u1 ) is an admissible boundary discontinuity for the hyperbolic conservation law. Proof. Integrate the stationary equation (5.2) with state u1 at ξ = ∞ to get φξ = f (φ) − f (u1 ) ≡ g(φ). For definiteness, consider the case ub > u1 so that one uses the upper envelope of the graph of f (u) in figure 5.02 of Chapter 5 to solve the Riemann problem (ub , u1 ). Since all the waves in the solution of the Riemann problem (ub , u1 ) have negative speed, the horizontal line through (u1 , f (u1 )) is above the graph of f (u) for u1 < u < ub . Thus g(u) < 0 for u1 < u < ub . Consequently, there is a solution of φ = g(φ) connecting ub at ξ = 0 to u1 at ξ = ∞. This proves the proposition. Note that in the zero dissipation limit κ → 0+, the stationary solution u(x) = φ(ξ) = φ(x/κ) tends to a discontinuity (ub , u1 ). The solution process for the hyperbolic laws can be studied explicitly using the elementary waves in the Riemann solutions; see Chapter 3 and Chapter 5. To solve the initial-boundary value problem (5.1), it remains to
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375
construct the boundary waves. Based on the above analysis of admissible boundary discontinuities, we now prescribe the solution algorithm near the boundary. For a given interior solution u(x, t) next to the boundary, the goal is to solve the Riemann problem ut + f (u)x = 0, ub (t), x < 0, u(x, 0) = u(0+, t), x > 0.
(5.3)
Proposition 5.3. The boundary Riemann problem (5.3) has a unique solution by elementary waves for the conservation law ut + f (u)x = 0, as described in Section 2 of Chapter 5, with non-negative speed and an admissible boundary discontinuity according to Definition 5.1. Proof. There are several cases. First we solve the Riemann problem (ub (t), u(0+, t) for the conservation law.
t
t
t
u1 ub(t)
u(0+, t)
no laye r
ub(t) u (0+, t) laye r
ub(t)
u(0+, t)
sonic boundary
Figure 12.12. Existence of boundary layer.
When the waves in the Riemann solution have positive speed, the boundary Riemann solution assumes the boundary value ub (t) and there is no boundary layer; see left diagram in figure 12.12. When the Riemann solution consists of waves with negative speed, the boundary Riemann solution contains an admissible boundary discontinuity (ub (t), u(0+, t)) and the boundary value is not assumed, ub (t) = u(0+, t), as shown in the middle diagram of figure 12.12. When the Riemann solution consists of a wave (ub (t), u1 ) with negative speed and another wave (u1 , u(0+, t)) with positive speed, the Riemann solution contains an admissible boundary discontinuity (ub (t), u1 ) plus (u1 , u(0+, t)) with positive speed; see right diagram of figure 12.12. Here u1 is the sonic state, f (u1 ) = 0.
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12. Nonlinear Resonance
In the construction, the admissible boundary discontinuity in each case corresponds to waves with negative speed and therefore has a corresponding viscous boundary layer by Proposition 5.2. Note that when the boundary value has negative characteristic speed, f (ub (t)) < 0, it is clear that the boundary value is not assumed by the solution and there is a boundary layer, as should be the case from linear hyperbolic theory. However, if the boundary value has positive characteristic speed, f (ub (t)) > 0, there is still a possibility that the boundary value is not assumed. This happens, for instance, when the limiting value u(x − 0, t) of the interior is such that (ub (t), u(x − 0, t)) is a shock with negative speed. In another situation, where the boundary shock has positive speed and the right characteristic speed is negative, f (ub (t)) > σ(ub (t), u(0+, t)) > 0 > f (u(0+, t)), the algorithm chooses the shock and there is no boundary layer. In this case, the algorithm disallows the choice of the boundary layer (ub (t), u(x − 0, t). Thus the algorithm has the effect of minimizing the appearance of a boundary layer. This is analogous to the entropy condition, which disallows the rarefaction shocks. As the solution of the Riemann problem depends globally on its end states, the existence of the boundary layer is a global consideration. 5.2. Propagation of Stationary Shocks In the setting of (5.1), the boundary location is fixed at x = 0 and so the boundary has zero speed. The algorithm for construction of the boundary layer in the preceding subsection requires that the boundary waves have negative speed. There is the borderline case where there is a shock (u− , u+ ) next to the boundary also with zero speed. In this case, the resonance phenomenon occurs and there is a subtle process of the zero dissipation limit. From the Rankine-Hugoniot condition, we have f (u− ) = f (u+ ) for a shock (u− , u+ ) with zero speed. For the inviscid theory, the distance of the shock to the boundary can be arbitrary, as the shock is parallel to the boundary. With the presence of viscosity, however, the shock drifts away from the boundary. The degree of the resonance, i.e. the speed of the drifting, depends on the behavior of the shock profile at x = −∞ and the strength of the viscosity. To understand this, we consider the initialboundary value problem (5.4) ut + f (u)x = κuxx , x > 0, u(0, t) = u− for t > 0, u(x, 0) → u+ as x → ∞,
with the end states corresponding to a stationary shock profile φ(x): (5.5)
f (φ(x)) − f (u± ) = κφ (x), φ(−∞) = u− , φ(∞) = u+ .
For definiteness, suppose that u+ < u− so that the profile is decreasing, φ (x) < 0. The shock speed is zero and so by the Oleinik entropy condition,
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377
Definition 1.8 of Chapter 5, the profile exists if f (u− ) = f (u+ ) and f (u) < f (u− ) for u+ < u < u− ; see figure 12.13.
f (u)
u+
f (u)
u−
f (u−) > 0
u+ u
u−
f (u−) = 0
u
Figure 12.13. End state of a shock wave with speed zero.
Since the left state u− is attained by the profile φ(x) at the far end x = −∞, the boundary condition in (5.4) is not satisfied by any translation of the profile φ(x − x0 ) for x0 ∈ R. For a large shift, x0 0, the boundary condition is almost satisfied, 0 < u− − φ(−x0 ) 1. This suggests that, time-asymptotically, the solution is close to the profile drifting toward x = ∞. The question is then how to determine the drifting speed. The main consideration is the conservation law d ∞ (u(x, t) − u+ ) dx + f (u(∞, t)) − f (u(0, t)) − κux (0, t) = dt 0 d ∞ d ∞ (u(x, t) − u+ ) dx + f (u+ ) − f (u− ) = (u(x, t) − u+ ) dx, or = dt 0 dt 0 d ∞ (u(x, t) − u+ ) dx = −κux (0, t), t > 0. (5.6) dt 0 Suppose that the solution at time t is well approximated by the shock profile shifted to x = X(t), u(x, t) = ˜ φ(x − X(t)); then ∞ ∞ ∞ d (u(x, t) − u+ ) dx = ut (x, t) dx = ˜ −X (t) φ (x − X(t)) dx. dt 0 0 0 To analyze the asymptotic behavior of the drifting of the shock profile, consider the situation where the profile is far from the boundary, X(t) 1. In such a situation, we have ∞ φ (x − X(t)) dx = u+ − φ(−X(t)) = ˜ u+ − u− , ux (0, t) = ˜ φ (−X(t)). 0
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Therefore the conservation law (5.6) yields (5.7)
X (t)(u+ − u− ) = ˜ κφ (−X(t)).
This relates the speed X (t) of the drifting profile to the slope φ (−X(t)) of the profile at −X(t). The profile drifts to the right: X(t) → ∞ as t → ∞. Thus the time-asymptotic behavior of the solution is determined by the tail behavior of the profile, the asymptotic behavior of φ (x) as x → −∞. There are two cases: Case 1 (λ− ≡ f (u− ) > 0). This is the generic case, shown in the left diagram of figure 12.13. For convex laws, which have f (u) = 0 for all u, this is always the case. The differential equation κφ (x) = f (φ(x)) − f (u− ) linearized around φ = u− is κφ (x) = f (u− )φ(x), and so the behavior of the profile φ(x) at x = −∞ is f (u− ) f (u− ) f (u− ) x e κ , as x → −∞. ˜ e κ x , φ (x) = ˜− 0 < u− − φ(x) = κ Thus (5.7) yields f (u− ) − f (u− ) X(t) κ e . κ This is solved to obtain the speed of the order of t−1 and the distance of the order of log t: κ 1 κ log t, X (t) = , as t → ∞. ˜ (5.8) X(t) = ˜ f (u− ) f (u− ) t ˜ (u− − u+ )X (t) =
Note that the speed is proportional to the strength of the dissipation κ and inversely proportional to the shock strength. In the zero dissipation limit κ → 0, it tends to the stationary shock. For a strong shock, f (u− ) is large and the speed is small. Case 2 (λ− = f (u− ) = 0). This case can happen only if f (u) changes signs; see right diagram of figure 12.13. For this case we consider the generic situation of f (u− ) < 0 shown in the right figure of figure 12.13. Since f (u− ) = 0, the differential equation κφ (x) = f (φ(x)) − f (u− ) around x = −∞ is approximated by κφ (x) =
f (u− ) (φ(x) − u− )2 , 2
which is solved to obtain f (u− ) x, or 2κ 2κ 2κ , φ (x) = ˜ , as x → −∞. φ(x) = ˜ u− − f (u− )x f (u− )x2 ˜ (u− − φ(x))−1 =
6. Kinetic Boundary Layers and Fluid-like Waves
379
From this, (5.7) yields ˜ (u+ − u− )X (t) =
2κ f (u− )
1 . (X(t))2
This is solved to yield that the speed is of the order of t−2/3 and the distance is of the order of t1/3 : (5.9) X(t) = ˜
1 1 6κ 3 t3 , (u− − u+ )f (u− ) ˜ X (t) =[
1 2 2κ ] 3 t− 3 , as t → ∞. (u− − u+ )f (u− )
Consider the scaling vt + f (v)x = vxx , v(x, t) ≡ u(κx, κt). The equation for v(x, t) is independent of the viscosity κ. Thus, in the zero dissipation limit κ → 0, along the line x = αt, with α any positive constant, u(x, t) = v(x/κ, t/κ) tends to the time-asymptotic state limt→∞ v(αt, t). In both of the above two cases, (5.8) and (5.9), the shock location grows sublinearly in time t, i.e. X(t)/t → 0 as t → ∞. Consequently, in the zero dissipation limit, the shock becomes stationary. This is as expected. As we have seen in the above two cases, the exact sub-linear rates result from the coupling the of nonlinearity of the flux and the strength of the dissipation. From the above analysis we see that the zero dissipation limit and timeasymptotic limit do not commute: lim lim u(x, t) = u− , lim lim u(x, t) contains shock (u− , u+ ) at finite x.
κ→0 t→∞
t→∞ κ→0
6. Kinetic Boundary Layers and Fluid-like Waves For non-equilibrium dynamics, there are the equilibrium and frozen characteristic speeds, as well as the equilibrium and frozen shock speeds. We have seen in Section 2 that, as a consequence of the resonance, there exist sub-shocks. With the presence of the boundary, there are additional resonance phenomena. We consider the Boltzmann equation in kinetic theory, discussed in Section 4 of Chapter 11, for which there are rich resonance phenomena. Various fluid dynamics equations can be derived from the Boltzmann equation to accurately describe the physical situation away from the boundary, shock, and initial layers. Near the boundary, the Boltzmann equation needs to be solved to provide the suitable boundary condition for the fluid equations. Consider a fixed boundary with normal n pointing toward the gas region. Let n be in the x1 direction and consider the plane
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12. Nonlinear Resonance
stationary solution f(x1 , ξ) of the Boltzmann equation: 1 Q(f, f), x1 > 0. k The equation is normalized to be independent of the mean free path k: x1 (6.1) ξ1 φη = Q(φ, φ), η ≡ > 0, φ(η, ξ) ≡ f(x1 , ξ). k This is solved with boundary conditions at the boundary η = 0 and at the far end η = ∞. A standard boundary condition at x1 = 0 is the so-called complete condensation boundary condition, which prescribes the value of particles entering the gas region, ξ1 > 0: ξ1 fx1 =
(6.2)
φ(0, ξ) = b+ (ξ), for a given boundary function b+ (ξ), ξ1 > 0.
At the far end, η = ∞, the gas approaches an equilibrium Maxwellian state by the H-Theorem, e.g. (4.21) in Section 4 of Chapter 11: (6.3)
φ(η, ξ) → M∞ as η → ∞, for a given Maxwellian M∞ = M(ρ,v,θ) .
For a given far-field state M∞ , the boundary function b+ (ξ) needs to satisfy a certain compatibility condition for the problem (6.1) to be solvable. The solution is called a Knudsen boundary layer. The boundary layer has width k, as the above scaling implies. Thus, for small k, it is confined to a small region near the boundary. Outside the boundary layer, the Boltzmann equation can be accurately approximated by fluid dynamics equations. The construction of the Knudsen boundary layer allows for the identification of suitable boundary conditions for the fluid dynamics equations. A central issue is the coupling of the Knudsen-like boundary layer and the fluid-type waves when their speeds are close to each other. The basic speeds of the fluid-like interior waves are the characteristic speeds of the Euler equations, λ1 = v − c, λ2 = v, and λ3 = v + c, given in (5.10) of Chapter 7. Thus resonance occurs between the Knudsen-type boundary layer and the fluid-like interior waves when the Mach number |v|/c is near 1 in the case of transonic condensation or transonic evaporation cases, or near zero in the case of condensation/evaporation. Nonlinear resonance induces nonlinear coupling of the kinetic Knudsen-type boundary layer and interior fluid-like waves and gives rise to striking bifurcation phenomena.
7. Shock Profiles for Difference Schemes There is a subtle resonance phenomenon for shock wave propagation in a finite difference scheme resulting from the rationality property of the C-F-L speed of the shock. It is caused by the waves crossing and occurs only for the system of hyperbolic conservation laws ut + f (u)x = 0, u ∈ Rn , n ≥ 2.
7. Shock Profiles for Difference Schemes
381
Consider a finite difference scheme with um (j) the value of an approximate solution at (x, t) = (jΔx, mΔt). A conservative finite difference scheme is of the following form: F [um ](j + 12 ) − F [um ](j − 12 ) um+1 (j) − um (j) = , Δt Δx for m = 0, 1, 2, . . . , and j = 0, ±1, ±2, . . . . Here the numerical flux F [um ] depends on the specific finite difference under consideration and depends on 2k grid points: 1 (7.2) F [um ](j − ) = F (um (j − k), . . . , um (j + k − 1)). 2 For consistency, the numerical flux reduces to the flux of the differential ¯: equations at a constant state u (7.1)
¯ ) = f (¯ F (¯ u, . . . , u u). Choose spatial and temporary grid sizes (Δx, Δt) for a numerical scheme that satisfy the C-F-L condition Δx > max{λi (u), i = 1, 2, . . . , n, } C-F-L condition, Δt for all states u under consideration. The C-F-L condition ensures that the domain of dependence of the numerical approximate solutions is greater than that of the exact solutions, e.g. (1.1) of Chapter 9 and (1.5) of Chapter 11. There are three relevant speeds for the present consideration: the characteristic speeds λi (u), i = 1, . . . , n, for the system, the C-F-L speed Δx/Δt of the difference scheme, and the shock speed. Consider a shock wave (u− , u+ ) pertaining to the ith genuinely nonlinear characteristic, so that its speed s satisfies the Lax entropy condition, (4.4) in Chapter 7: λi (u− ) > s > λi (u+ ). In calculating the shock wave using a numerical scheme, the ratio of the shock speed s and the C-F-L speed Δx/Δt, the C-F-L shock speed, is the relevant speed: Δt , C-F-L shock speed. Δx The nonlinear resonance phenomenon occurs when the C-F-L shock speed is close to a rational number. Resonance occurs when waves pertaining to the other characteristic family cross the shock, and therefore it occurs only for systems and not for scalar laws. Thus the characteristic speeds λj (u), j = i, are also participating speeds in the resonance phenomena. (7.3)
s˜ ≡ s
A shock wave (u− , u+ ) satisfies the Rankine-Hugoniot condition s(u+ − u− ) = f (u+ ) − f (u− ).
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12. Nonlinear Resonance
For stability and to ensure the entropy condition, a finite difference scheme is often dissipative. For such a scheme, corresponding to a discontinuous shock wave of the system of conservation laws, one looks for a continuous shock profile for the finite difference scheme. Definition 7.1. A continuous function φ(x − st) is a shock profile for the ¯), for m = 0, 1, 2, . . . and difference scheme (7.1) if um (j) ≡ φ(j − sm + x j = 0, ±1, ±2, . . . , satisfy (7.1) and (7.4)
∞
[φ(¯ x + j) − φ(j)] = x ¯(u+ − u− )
j=−∞
for all x ¯ such that 0 ≤ x ¯ ≤ 1. The requirement (7.4) mirrors the similar property of the exact shock profile for hyperbolic and viscous conservation laws, (1.6) of Chapter 10. The goal is to construct these profiles and understand their propagation properties. There are the following two cases. Case 1 (Rational C-F-L shock speed). Consider the case where the C-F-L speed of the shock is rational: p p s˜ = for some rational number , q > 0, p and q relatively prime. q q Thus we have sqΔt = pΔx. For a fixed x ¯ and any given m, the values um (j) ≡ φ(j − sm + x ¯), j = 0, ±1, ±2, . . . , repeat themselves after q time steps: (7.5)
um+q (j) = um (j + p), j = 0, ±1, ±2, . . . .
With this, discrete shock profiles can be constructed as fixed points for the qth iterations of the difference operator (7.1) as follows: Denote the difference scheme (7.1) by um+1 (j) = F[um ](j). Then (7.5) yields (7.6)
um+q (j) = F[F[. . . F[um ] . . . ]](j) = um (j + p), j = 0, ±1, ±2, . . . .
This is an equation for um (j) on the discrete locations j = 0, ±1, ±2, . . . . It can be solved as a fixed point problem to yield the discrete shock profile. The compressibility property of the shock waves, λi (u− ) > s > λi (u+ ), and the dissipative property of the scheme provide the necessary stability requirements for the construction of the fixed point. For a given shock wave (u− , u+ ), a continuum set of discrete profiles exist. These discrete profiles form a unique continuous profile when they are parametrized according to (7.4). Case 2 (Irrational C-F-L shock speed). In the above analysis for rational C-F-L speeds, we note that the qth iteration of the difference operator
7. Shock Profiles for Difference Schemes
383
F[F[. . . F[·] . . . ]] becomes more nonlinear as q gets larger and the construction of discrete shock profiles applies to ever weaker shocks. The fixed point construction based on standard nonlinear analysis methods such as the implicit and inverse function theorems has this property. A series of rational numbers p/q approaching an irrational s˜ has the property that p → s˜ ⇒ q → ∞. q Thus when the C-F-L speed s˜ is irrational, there is an obvious obstacle in using the above fixed point construction by approximating an irrational speed by rational speeds. This analytical difficulty has two possible sources. One possibility is that the difficulty is purely analytical, due to the particular approach adopted for the construction of discrete shock profiles when the C-F-L speed is rational. Another possibility is that the difficulty is real and reflects some rich behavior of shock propagation in the solutions of difference schemes. For scalar laws, the L1 (x) well-posedness theory and the maximum principle apply to standard first-order difference schemes. The discrete profiles for rational C-F-L speeds converge to a shock profile for an irrational speed as the rational speeds converge to the irrational speed. For systems, however, the coupling of waves pertaining to distinct characteristic families gives rise to rich wave phenomena and the second possibility may be the reality, as some numerics seem to indicate. As the approximation of the irrational C-F-L speed of the shock by a series of rational numbers is not feasible, one can try to use time-asymptotic analysis to capture the shock profile. Take a sufficiently accurate approximate profile as initial data. If the exact profile is stable, then the numerical solution would approach the exact profile time-asymptotically. As with the analysis for systems of hyperbolic conservation laws in Chapter 9 and for viscous conservation laws in Chapter 10, the approximate solution is diagonalized for the study of wave interactions. To illustrate the effect of crossing waves on the structure of the shock profile, consider a three-point scheme, k = 1 in (7.2). The time-asymptotic analysis leads to a core equation of the following form: (7.7)
W (x − s˜) = aW (x − 1) + bW (x) + cW (x + 1) + S(x).
This is related to the equation for the shock profile pertaining to the characteristic mode to which the shock profile belongs. The function S(x) represents the coupling with waves of other characteristic families. The constants a, b, and c depend on the scheme; a + b + c = 1 for conservative schemes. Take the Fourier transform of the identity (7.7) to obtain (7.8)
ˆ (η) = W
e−i˜sη
ˆ S(η) . − (ae−iη + b + ceiη )
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12. Nonlinear Resonance
There is the issue of a small divisor when the denominator e−i˜sη − (ae−iη + b + ceiη ) in (7.8) can be arbitrarily small for some Fourier mode η. This difficulty arising from the resonance between the shock speed, the C-F-L speed, and the crossing characteristics is resolved by requiring the C-F-L shock speed s˜ to be far away from being rational. That is, the C-F-L shock speed is required to satisfy the following Diophantine property: p 1 p (7.9) |˜ s − | > C 2 for some constant C = C(˜ s) > 0 and any rational . q q q For a C-F-L shock speed satisfying the Diophantine property, it can be shown that (7.10)
|e−i˜sη − (ae−iη + b + ceiη )| ≥ C(|η|2 + 1)−2
for some positive constant C and for all |η| > 1. A lower-bound property of the form (7.10) resolves the small-divisor difficulty and the continuum profile can be constructed for shocks whose C-F-L speeds satisfy the Diophantine property. It should be noted that almost all irrational numbers have the Diophantine property. However, as a shock propagates, its speed varies over rational, irrational non-Diophantine, and irrational Diophantine speeds. The resonance phenomenon indicates that the propagation of a numerical shock is interesting and highly subtle.
8. Notes Geometric effects on the gas flow are rich and the study of multi-dimensional flows with boundary is a difficult subject; see Chapter 13. For the study of the simple model (1.4) given in Section 1, and also the study of the quasi-one-dimensional model (1.1), see [86] and references therein. These studies yield non-uniqueness of stationary flows for quasi-one-dimensional flows. Non-uniqueness of stationary solutions is a common occurrence for continuum models. For instance, multiple solutions for the buckling of an elastic rod have been known since the time of Euler. There are several well-known examples of non-uniqueness of solutions to the boundary value problem for the incompressible fluid equations. For compressible fluids, nonuniqueness has been observed experimentally; however, exact analysis is not available for the truly multi-dimensional situation. Sub-shocks are observed in high-temperature gas flows, which are generally in thermal non-equilibrium; see Vicenti-Kruger [122] and Zel’dovichRaizer [127]. The system (3.1) for modeling intermediate shocks in magnetohydrodynamics flows was proposed by Freistuhler; see [46]. The analysis given here is taken from [89]. There is a class of conservation laws, called the Temple class, after Temple [121], for which the Hugoniot and characteristic curves coincide as for the system (3.1).
8. Notes
385
The study of the Euler equations with damping and the nature of singularities near vacuum were initiated in Liu [90]. For subsequent work see the recent paper by Luo-Zeng [101] on convergence to Barenblatt self-similar solutions and references therein. The interesting problems of the waiting time, formation of physical singularities near vacuum, and large-time behavior for multi-dimensional flows remain to be fully understood. There is a conjecture that if the initial data for the Euler equations does not give rise immediately to the vacuum state, then the solution will not contain the vacuum state at later times; see Lin [81]. We know from the study in Section 4 that there are non-vacuum initial values which can give rise to two rarefaction waves with vacuum in between. The conjecture is that if the vacuum state does not occur immediately, then it will not occur at later times. If the conjecture is valid then one can avoid the degeneracy of the Euler equations at vacuum for most initial data. The study of boundary phenomena is a core issue in the theory of partial differential equations. For the specific considerations in Section 5, there is the well-posedness theory for scalar equations [2]. Propagation of stationary shock profiles as described in Section 5 has been analytically verified only for the Burgers equation [96]. As pointed out in Section 5, one needs to study viscous systems for the proper consideration of boundary phenomena. For systems, the admissible boundary condition depends on the form of the viscosity matrix; see, for instance, [10]. The general study of boundary phenomena remains largely an open research direction. In the spirit of explicit geometric understanding here, see [98] for a description of multidimensional propagation over a Burgers shock. The resonance phenomena mentioned toward the end of Section 6 on the coupling of the Knudsen-type boundary layer and fluid-type interior waves can be found in [118] and [97]. General research on the interplay of shock wave theory and kinetic theory is an exciting and mostly open research field. The full analysis of shock profiles for difference schemes and the issue of small divisors discussed in Section 7 can be found in Liu-Yu [95]. This analysis may be related to studies showing numerical oscillations of shock propagation in the numerical approximations [1]. The general issue of accuracy and stability of finite difference solutions remains a challenging problem. Other interesting resonance phenomena not included in this chapter include the appearance of under-compressive shocks, such as the weak detonation waves in combustion, e.g. [32]. Such under-compressive waves cannot be studied within hyperbolic theory and depend on the dissipation mechanism. There are also studies of propagation of waves with dispersion effects, such as the surface effect, or waves not satisfying the Oleinik entropy condition (1.7) of Chapter 5, e.g. [7, 77].
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12. Nonlinear Resonance
9. Exercises 1. Construct a nozzle, according to the model (1.4), which is converging for −1 < x < 0 and diverging for 0 < x < 1. Construct a solution of the nozzle consisting of stationary solutions with a stationary shock in the converging portion of the nozzle. Find a small perturbation of this stationary solution such that, time-asymptotically, the perturbed solution tends to a stationary shock in the diverging portion of the nozzle. 2. Construct an explicit flux function f (u) and a stationary shock (u− , u+ ) such that there is a higher order of degeneracy at the left state, f (u) − f (u− ) = O(1)(u − u− )k with k > 2. Carry out the analysis, as in (5.4) and (5.7), of the rate in t of the location and the speed of the viscous profile as it propagates away from the boundary. 3. Construct the solution to the Riemann problem (ul , ur ) for the isentropic Euler equations when the gas is polyatomic, p(ρ) = Aργ with 1 < γ ≤ 5/3, and when the left state is vacuum, ρl = 0. Compute the rate at which the sound speed approaches zero as the gas approaches vacuum to the left. Compare this to the physical singularity discussed in Section 4. 4. Construct Barenblatt’s solution of the porous media equation: m (ργ )xi xi , x ∈ Rm , t > 0; ρ(x, 0) = Aδ(x), ρt = i=1
where A is a positive constant. Relate this solution to the particular solutions for the Euler equations with damping and the Darcy line relation. (Hint: Use scaling arguments similar to those in the construction of the heat and Burgers kernels in Chapter 4.) 5. Consider the initial-boundary value problem (5.1). Take f (u) = u3 /3 and the boundary value ub (t) = 0. Let the initial function be
0 for 0 < x < 2, ui (x) = k for x > 2. Solve the initial-boundary value problem for different values of the constant k.
10.1090/gsm/215/13
Chapter 13
Multi-Dimensional Gas Flows
One of the most important physical models of the form of hyperbolic conservation laws ut + ∇x · F(u) = 0 is the system of Euler equations in gas dynamics, (0.5) of Chapter 1, ⎛ (0.1)
⎞ ⎛ ⎞ ρ ρv ⎝ ρv ⎠ + ∇x · ⎝ρv ⊗ v + pI⎠ = 0. ρE t ρEv + pv
The study of the Euler equations in gas dynamics inaugurated shock wave theory in the mid-nineteenth century. Though there remain important unsolved problems, there is a satisfactory theory for the case of one spatial dimension, m = 1; see e.g. Chapter 7, Chapter 8, and Chapter 9. We consider in this chapter multi-dimensional gas flows, with m ≥ 2. There are hard analytical issues arising from the study of multi-dimensional flows, such as dealing with nonlinear equations of mixed types and nonlinear free boundaries. We will start with the derivation of various simplified systems from the Euler equations and then focus on shock waves. Besides the ideas gained from the one-dimensional theory, direct physical interest is also an important guide to the study of the present more complicated situation. Toward the end of this chapter we will illustrate by one basic example of the Prandtl paradox a continuation of the one-dimensional theory in some general sense. The conserved quantities u for the Euler equations are the mass, momentum, and energy per unit volume, along with the corresponding flux 387
388
13. Multi-Dimensional Gas Flows
F(u): (0.2)
⎛
⎞ ⎛ ⎞ ρ ρv u = ⎝ ρv ⎠ , F(u) = ⎝ρv ⊗ v + pI⎠ . ρE ρEv + pv
Here ρ is the density and v is the velocity of the gas. The total energy per unit mass, E = |v|2 /2 + e, is the sum of the kinetic energy |v|2 /2 and the internal energy e. The first equation, the conservation of mass with the mass flux ρv, is exact. The same continuity equation holds for the Boltzmann equation and the Navier-Stokes equations in gas dynamics. The momentum flux ρv ⊗ v + pI consists of the convection ρv ⊗ v of the momentum and the momentum flux pI of the pressure force, ignoring other effects such as the viscosity effect. The energy flux ρEv + pv consists of the energy convection ρEv and the work pv done by the pressure force, ignoring effects such as the heat conductivity and the work done by the viscous stress. For an explanation of the derivation of the Euler equations, see Chapter 1. The Euler equations are the most basic equations in gas dynamics. They are derived from the Boltzmann equation with the hypothesis of thermal equilibrium; see (4.19) and (4.20) in Section 3 of Chapter 11. For m space variables x = (x1 , x2 , . . . , xm ) ∈ Rm , the velocity is ⎛ 1⎞ v ⎜ v2 ⎟ ⎜ ⎟ v = ⎜ . ⎟ ∈ Rm , ⎝ .. ⎠ vm and there are n = m +2 equations for u ∈ Rn . The Euler system is closed by the constitutive relation, the equation of state, by prescribing the pressure p as a given function of the density ρ and the internal energy e: (0.3)
p = p(ρ, e).
Mostly we will carry out calculations for polyatomic gases by prescribing the pressure p as a given function of the specific volume τ and the entropy s: p¯(τ, s) = AeCs τ −γ , γ > 1, τ ≡ 1/ρ. We have seen in Chapter 6 that for polyatomic gases the Euler equations are hyperbolic because p¯τ (τ, s) < 0 holds. For one spatial dimension, the acoustic modes are genuinely nonlinear because p¯τ τ (τ, s) > 0.
1. Linear Waves We study the linear three-dimensional waves, where x ∈ R3 . Consider the Euler equations (0.1) linearized around the base state u0 = (ρ0 , ρ0 v 0 , ρ0 E0 )t .
1. Linear Waves
389
By a Galilean transformation, we may take the base velocity to be zero, v 0 = 0. Direct calculations yield the linear Euler equations ⎧ ⎪ ⎨ρt + ∇x · (ρv) = 0, (ρv)t + p¯ρ0 − e0 p¯ρe00 ∇x ρ + p¯ρe00 ∇x (ρE) = 0, ⎪ ⎩ (ρE)t + e0 + pρ00 ∇x · (ρv) = 0,
(1.1)
where p = p¯(ρ, e),
p0 = p¯(ρ0 , e0 ),
p¯e0 ≡
∂ p¯ ∂ p¯ (ρ0 , e0 ), p¯ρ0 ≡ (ρ0 , e0 ). ∂e ∂ρ
For the classification of the wave types for the solutions of (1.1), we rewrite the system in terms of the basic dependent variables of pressure p, velocity v, and entropy s:
(1.2)
⎧ 2 ⎪ ⎨pt + ρ0 (c0 ) ∇x · v = 0, v t + ρ10 ∇x p = 0, ⎪ ⎩ st = 0.
Here c0 = pρ (ρ0 , s0 ) is the sound speed at the base state. The first two equations in (1.2) yield the wave equation for the pressure, (1.3)
ptt = (c0 )2 Δx p.
There is the classical Kirchhoff’s formula for the solution of the wave equation in R3 : (1.4) utt = c2 Δu, u(x, 0) = g(x), ut (x, 0) = h(x), x ∈ R3 , 1 t h(x + cty) dS(y) + h(x + cty) dS(y) u(x, t) = 4π 4π |y|=1 |y|=1 ct ∇g(x + cty) · y dS(y), Kirchhoff’s formula. + 4π |y|=1 With the pressure given by Kirchhoff’s formula, the velocity is solved by integrating the second equation in (1.2):
t
v(x, t) = v(x, 0) + 0
∇x p(x, s) ds.
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13. Multi-Dimensional Gas Flows
These expressions yield the explicit solution formula for the initial value problem of the linear Euler equations (1.2): 1 p(x, t) = (1.5) p(x + c0 ty, 0) dS(y, 0) 4π |y|=1 c0 t ∇p(x + c0 ty, 0) · y dS(y) + 4π |y|=1 2 t ∇ · v(x + c0 ty, 0) dS(y), − ρ0 (c0 ) 4π |y|=1 t 1 ∇ − p(x + c0 sy, 0) dS(y) v(x, t) =v(x, 0) + 4πρ0 0 |y|=1 c0 t − ∇p(x + c0 sy, 0) · y dS(y) 4πρ0 |y|=1 2 t ∇ · v(x + c0 sy, 0) dS(y) ds, + (c0 ) 4π |y|=1 s(x, t) =s(x, 0). There are three types of wave propagation in the solution formulas (1.5): The pressure follows the wave equation and therefore satisfies Huygens’ principle and propagates along the edge of the acoustic cone. The corresponding nonlinear waves in the one-dimensional case are the shock and rarefaction waves for λi , i = 1, 3; see Section 5 of Chapter 7. The velocity propagation contains two types of waves, the term v(x, 0) in the second equation in (1.5) is the vortex wave along the particle path, and the rest of the terms in the second equation represent waves coupled with the acoustic waves and fill the acoustic cone as the time integrals of the Huygens waves. The entropy wave propagates only along the particle path and represents the inviscid thermal wave. This corresponds to the contact discontinuities in one-dimensional case. The vortex wave has no one-dimensional analogue. The choice of the basic dependent variables (p, v, s), which allows for the simple classification of wave types, is consistent with the consideration of Riemann invariants for the one-dimensional waves considered in Section 5 of Chapter 7.
2. Discontinuity Waves From the general theory in Chapter 6, a jump discontinuity (u− , u+ ) for the system of hyperbolic conservation laws ut + ∇x · F(u) = 0 propagating in the direction ξ ∈ Rm , |ξ| = 1, with speed σ,
u− for x · ξ − σt < 0, (2.1) u(x, t) = u+ for x · ξ − σt > 0,
2. Discontinuity Waves
391
satisfies the Rankine-Hugoniot condition (2.2) σ(u+ − u− ) = ξ · F(u+ ) − F(u− ) . For the Euler equations (0.2), we have ⎛ ⎞ ρ+ − ρ− (2.3) σ ⎝ ρ+ v + − ρ− v − ⎠ ρ+ E+ − ρ− E− ⎞ ⎛ ρ+ (v + · ξ) − ρ− (v − · ξ) ⎠. ρ+ (v + · ξ)v + + p+ ξ − ρ− (v − · ξ)v − − p− ξ =⎝ ρ+ E+ (v + · ξ) + p+ (v + · ξ) − ρ− E− (v − · ξ) − p− (v − · ξ) Decompose the flow velocity v into the velocity v n normal to the discontinuity front and the velocity v τ tangential to the discontinuity front: v n ≡ (v · ξ)ξ,
(2.4)
vτ ≡ v − vn ,
v n ≡ v · ξ.
The projection of the multi-dimensional Euler equations (0.1) in the direction ξ of the propagation of the discontinuity is the system of onedimensional Euler equations ⎛ ⎞ ⎛ ⎞ ρv ρ ⎝ ρv ⎠ + ⎝ ρv 2 + p ⎠ = 0, x = x · ξ, v = v · ξ. (2.5) ρEv + pv x ρE t There are two situations. The first situation is that the jump vanishes for the one-dimensional projection. Such a jump represents a vortex sheet and is considered in Case 1 below. The second situation is that the jump persists in the one-dimensional projection. The one-dimensional discontinuity can be either a contact discontinuity or a shock wave; see Proposition 5.1 of Chapter 7. We consider the contact discontinuity in Case 2 and the shock wave in Case 3. Case 1 (Vortex sheet). Consider the situation where the jump discontinuity, (2.1) and (2.3), becomes null for the one-dimensional projection, (2.5): ρ− = ρ+ ,
ρ− v− = ρ+ v+ ,
ρ− E− = ρ+ E+ .
In particular, (2.6)
v− = v+ , or v − · ξ = v + · ξ,
and the thermodynamic variables have no jump, (2.7)
ρ− = ρ+ , p− = p+ , s− = s+ , e− = e+ , θ− = θ+ .
From (2.6) and (2.7), the second equation in (2.3) yields that the discontinuity has the same speed as v · ξ: (2.8)
n n = v− . σ = v + · ξ = v − · ξ, or σ = v+
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13. Multi-Dimensional Gas Flows
Thus the speed of propagation of the jump discontinuity equals the normal component of the flow velocity. In other words, there is no mass flux across the jump discontinuity. However, the tangential velocity v τ can jump by an arbitrary amount across the discontinuity. This is a vortex sheet. Summarizing the above, we have (2.9)
n n σ = v+ = v− , v τ+ = v τ− , p+ = p− , s+ = s− , vortex sheet.
See left diagram of figure 13.01, pictured here for the two-dimensional case x = (x, y).
y
y x
x Shock Wave
Vortex sheet
Figure 13.01. Flow velocity around discontinuity waves.
Case 2 (Thermal wave). When the one-dimensional wave is a contact discontinuity, the velocity is constant across the discontinuity, v− = v+ , and so, as in Case 1, (2.8) holds. For a contact discontinuity, the pressure does not jump and all the other thermodynamic variables jump. Thus we have (2.10)
n n σ = v+ = v− , p− = p+ , s− = s+ , thermal wave.
Like in Case 1, the tangential velocity can have an arbitrary jump. Case 1 and Case 2 together yield n n = v− , p+ = p− , (2.11) σ = v+
combination of vortex sheet and thermal wave. Case 3 (Shock wave). When the one-dimensional shock is of non-zero strength, the velocity and the thermodynamic variables jump across the shock. This has been considered in Section 5 of Chapter 7. For the multidimensional shock, we take the direction ξ in the direction of the flow: (2.12)
v · ξ > 0, u− upstream state, u+ downstream state.
3. Potential Flows
393
The straightforward embedding of the one-dimensional shock into the multidimensional setting assumes the special property that the flow velocity is normal to the shock: (2.13)
v ± ≡ v± ξ, v n± = v ± , v τ± = 0, normal shock.
A normal shock has zero tangential velocity, v τ = 0. The general multidimensional shock with any given tangential velocity v τ , v τ · ξ = 0, can be obtained from the normal shock by a Galilean transformation x → x + tv τ . This observation yields that the flow velocity tangential to a shock does not change across the shock: v τ+ = v τ− = v τ . This can also be verified by direct computation from the jump condition (2.3). As a multi-dimensional shock wave is obtained from one-dimensional shocks, the one-dimensional theory, Proposition 5.1 in Chapter 7, applies. With the orientation of (2.12), we have that for the multi-dimensional shock wave (u− , u+ ), n n n n > v+ > 0, ρ+ − ρ− = ˜ p+ − p− = ˜ v− − v+ , (2.14) v τ− = v τ+ = v τ , v− n n 3 ˜ − − v+ ) . s+ − s− =(v
We summarize the above three cases in the following proposition: Proposition 2.1. A jump discontinuity for the Euler equations (2.3) has two possibilities: Type 1. The first kind of jump discontinuity is a combination of vortex sheet and thermal wave with the property that the gas flow is tangential to the jump and the pressure is constant across the jump, (2.11). Within this type, there are vortex sheets with all the thermodynamic variables constant across the jump, (2.9), and thermal waves with no jump in the tangential velocity but with a jump in the entropy, (2.10). Type 2. The second kind of jump discontinuity corresponds to the one-dimensional shock waves. There is no jump in the velocity component tangential to the shock. The upstream state u− and downstream state u+ , (2.12), satisfy the direct generalization (2.14) of the properties of onedimensional shocks.
3. Potential Flows As we have seen, for the Euler equations there are three types of discontinuities, the shock waves, the thermal waves, and the vortex sheets. The study of incompressible flows focuses on the vorticity. The vortex sheets are
394
13. Multi-Dimensional Gas Flows
highly unstable. A thermal wave moves with the gas flow and carries a jump in the entropy. To focus on the shock waves and not consider the thermal waves and vortex sheets, the Euler equations are simplified to the potential flow equation by assuming that the gas is isentropic and irrotational. This is done in two steps. First, for isentropic flows, the energy equation is not considered by letting the pressure depend only on the density, p = p(ρ). The isentropic flow does not carry thermal waves but carries both acoustic waves and vorticity. To obtain a system carrying only acoustic waves, the isentropic Euler equations are further simplified by assuming that the flow is irrotational, ∂xj v i = ∂xi v j for i, j = 1, . . . , m, so that the flow velocity is the gradient of the potential function φ: v(x, t) = ∇x φ(x, t).
(3.1)
It follows directly from the isentropic Euler equations that for smooth solutions of the equations, (3.2) (v i )t + ∇x v i · v + ∂xi Π(ρ) = 0, i = 1, . . . , m,
Π (ρ) ≡
c2 p (ρ) = . ρ ρ
Therefore, in terms of the potential function, (3.3) (φxi )t +
m
(φxi )xj φxj + (Π(ρ))xi = [φt +
j=1
|∇x φ|2 + Π(ρ)]xi = 0, 2 i = 1, . . . , m.
This yields the Bernoulli equation (3.4)
φt +
|∇x φ|2 + Π(ρ) = A, 2
|∇x φ|2 ρ = Π−1 A − φt − , 2
for some constant A. Substituting this into the first equation of the Euler equations, the continuity equation ρt + ∇x · (ρ∇x φ) = 0, one obtains a second-order equation for the potential φ: |∇x φ|2 |∇x φ|2 ∇x φ = 0. + ∇x · Π−1 A − φt − (3.5) Π−1 A − φt − 2 2 t This conservative equation is used to define the weak solutions for the potential flow equation. For smooth solutions we multiply the equation by Π (ρ) and rewrite it as (Π(ρ))t + Π (ρ)ρ
n i=1
φxi xi +
n (Π(ρ))xi φxi = 0, i=1
4. Self-Similar Flows and the Ellipticity Principle
395
or, from Π (ρ) = c2 /ρ and the Bernoulli equation Π(ρ) = A − φt − |∇x φ|2 /2, we arrive at the standard form of the potential flow equation, (3.6)
φtt + 2∇x φt · ∇x φ +
m
φxi φxj φxi xj − c2 Δx φ = 0,
i,j=1
in terms of the sound speed c = p (ρ), which is a given function of the gradient of φ through the Bernoulli equation (3.4). For a stationary flow, φt = 0, it reduces to the stationary potential flow equation: (3.7)
m
φxi φxj φxi xj − c2 Δx φ = 0.
i,j=1
In terms of the gas dynamics variables, the coefficients are the flow velocity v and the sound speed c: m
(v i v j − c2 δij )φxi xj φxi xj = 0.
i,j=1
The stationary potential equation is elliptic (or hyperbolic) when the gas flow is subsonic, |v| < c, (or supersonic, |v| > c.)
4. Self-Similar Flows and the Ellipticity Principle In many physical situations, one considers the self-similar flows that occur when the gas variables are invariant under the self-similar scaling x → αx and t → αt, for α any positive constant. For the solution to have the selfsimilarity property, there are four conditions to satisfy. The first condition is that the underlying differential equations need to be invariant under the self-similar scaling x → αx and t → αt. This is obviously so for the Euler equations (0.1) and for the potential flow equation (3.6). The second condition is that the initial data should not change under x → αx. This condition is satisfied by multi-dimensional Riemann data, which consists of a finite number of constant states and is a function of x/|x|. The third condition is that the physical boundary, if any, is such that if x is in the gas region then αx, α > 0, is also in the gas region. A gas region in a wedge in two dimensions or a cone in three dimensions satisfies the third condition if the tip is set at the origin. The boundary of the gas region therefore is self-similar. Finally, the boundary condition needs to be invariant under the self-similar scaling. When the boundary of the gas region is a solid, the usual boundary condition for inviscid flows is that the gas flow is parallel to the boundary: (4.1)
v · n = 0,
396
13. Multi-Dimensional Gas Flows
where n is the unit normal to the boundary of the gas region. This boundary condition satisfies the fourth condition when the boundary is self-similar. For self-similar flows, the gas variable satisfies u(x, t) = u(αx, αt) and therefore is a function of the self-similar variables ξ ≡ x/t; see Proposition 1.1 in Section 1 of Chapter 3. For potential flows, we introduce the new potential function x ψ(ξ) ≡ tφ(x, t), ξ ≡ , t so that the flow velocity in the self-similar variable is the gradient of the new potential function ψ(ξ): v = ∇x φ = ∇ξ ψ. The potential flow equation becomes m
(ψxi − ξi )(ψξi − ξj ) − c2 Δξ ψ = 0.
i,j=1
This equation is non-autonomous; the coefficients depend on the independent variables ξ. Define another potential function |ξ|2 , ∇ξ χ(ξ) = v − ξ pseudo velocity, 2 and the equation for χ becomes autonomous, i.e. its coefficients do not explicitly depend on the variable ξ: (4.2)
χ(ξ) ≡ ψ(ξ) −
(4.3)
m
χξi χξj χξi ξj − c2 Δξ χ = −|∇ξ χ|2 + mc2 .
i,j=1
The self-similar potential equation (4.3) and the stationary potential flow equation (3.7) have the same form for their left-hand sides. Therefore these two equations are classified similarly. (4.3) is of mixed type, elliptic when the flow is pseudo-subsonic, |v − ξ| < c, and hyperbolic when the flow is pseudo-supersonic, |v − ξ| > c. The Mach number is defined differently for these equations: |v| , Mach number for stationary flow, c |∇ξ χ| |v − ξ| = , pseudo Mach number for self-similar flow. L≡ c c Note that the pseudo Mach number L depends not only on the flow variables v and c, but also on the self-similar location ξ. At the far field, |ξ| → ∞, the flow is pseudo-supersonic and the self-similar equation (4.3) is hyperbolic.
(4.4) M ≡
The self-similar potential equation (4.3) and the stationary potential flow equation (3.7) differ on their right-hand sides in the lower-order terms. The right-hand side of (3.7) is zero; while the right-hand side of (4.3) is non-zero.
4. Self-Similar Flows and the Ellipticity Principle
397
This reflects two essential differences between stationary and self-similar flows from physical and analytical considerations. Physically, the self-similar potential equation is a time-dependent equation and should have a unique solution for a given condition at infinity |ξ| = ∞, while the time-independent stationary flows may not be unique for given boundary conditions. The second essential difference is analytical: The self-similar potential equation (4.3) obeys the following ellipticity principle, which does not hold for the stationary potential equation (3.7). Theorem 4.1 (Ellipticity principle). A pseudo-supersonic bubble cannot emerge inside a pseudo-subsonic region upon a continuous variation of the gas flow. Proof. We will prove this for the two-dimensional case so that the PDE (4.3) and the Bernoulli equation (3.4) can be written as, with χj = χxj for j = 1, 2, (4.5) (χ21 − c2 )χ11 + 2χ1 χ2 χ12 + (χ22 − c2 )χ22 = 2c2 − χ21 − χ22 , c2 = (1 − γ)(χ +
χ21 + χ22 ). 2
We will show that a maximum of the pseudo Mach number inside a pseudosubsonic region cannot exist if the state is almost pseudo-sonic. This would prove the theorem by contradiction. Suppose that the pseudo Mach number L ≡ |∇χ|/c has a local maximum at the location ξ 0 in the interior and that the state is pseudo-subsonic and close to pseudo-sonic there: χ21 + χ22 = 1 − δ, ∇L2 = 0, ∇2 L2 ≤ 0 at the location ξ 0 c2
L2 ≡
for some small positive constant δ with 0 < δ 1. By a suitable rotation, we may assume that χ1 > 0, χ2 = 0 at the location ξ 0 . From these, L2 =
χ21 = 1 − δ, χ1 = c + O(1)δ at the location ξ0 . c2
For the computation of differentials of L, we will use the general Leibniz product rule: A≡
B − AC B − 2A C − AC B , A = , A = . C C C
398
13. Multi-Dimensional Gas Flows
The Bernoulli equation yields |∇χ|2 1 2 (L )j = = 2 (|∇χ|2 )j − L2 (c2 )j 2 c c j 1 = 2 2χ1 χ1j + 2χ2 χ2j + L2 (γ − 1)(χj + χ1 χ1j + χ2 χ2j ) , j = 1, 2. c At the maximum location L = 1 − δ we have ∇L = 0, χ1 = c + O(1)δ and χ2 = 0 and so 0 = c2 (L2 )1 = 2χ1 χ11 + L2 (γ − 1)(χ1 + χ1 χ11 ) = 2cχ11 + (γ − 1)(c + cχ11 ) + O(1)δ, and we obtain χ11 =
1−γ + O(1)δ at the location ξ 0 . 1+γ
This and (4.5) yield, at the location ξ 0 , χ22 = −1 + O(1)δ. Similarly, from the Leibniz product rule we have (χ )2 + (χ )2 1 1 2 = 2 (|∇χ|2 )j − 2(L2 )j (c2 )j − L2 (c2 )jj , (L2 )jj = 2 c c jj and from χ22 = −1 + O(1)δ and ∇2 L2 ≤ 0 we conclude that at the location ξ0 , 0 ≥ c2 (L2 )22 = (γ + 1)(2γ − 1)(χ12 )2 + 2 + (γ + 1)cχ122 + O(1)δ. From the ∂1 derivative of the PDE (4.5), 2χ1 χ11 − (c2 )1 χ11 + (χ1 )2 − c2 χ111 + 2χ11 χ2 + 2χ1 (χ12 )2 + χ1 χ2 χ112 + 2χ2 χ12 − (c2 )1 χ22 + (χ2 )2 − c2 χ122 = 2(c2 )1 − 2χ1 χ11 − χ2 χ12 . Evaluating this at the maximum location, we have, at the location ξ 0 , cχ122 = 2(χ12 )2 + O(1)δ. Plug this expression for χ122 into the above inequality to obtain 0 ≥ (γ + 1)(2γ + 1)(χ12 )2 + 2 + O(1)δ, which is impossible for small δ. We have shown that a maximum of the pseudo Mach number inside a pseudo-subsonic region cannot exist if it is almost pseudo-sonic. This completes the proof of the theorem.
5. Characteristics and Simple Waves
399
Shock Sonic M =1
Supersonic bubble M >1
Subsonic M = 0.82
Figure 13.02. Stationary subsonic flow with supersonic bubbles.
Remark 4.2. It has been known that for stationary flow around an airfoil, a supersonic bubble can emerge inside a subsonic region as the Mach number of the upstream flow increases; see figure 13.02. The ellipticity principle disallows this for self-similar flows. Physically, stationary flows with given boundary conditions at infinity may not be unique; an example is the flow around the airfoil. We see this also for nozzle flows, discussed in Section 1 of Chapter 12. On the other hand, a time-dependent flow with a given initial value (in the case of the time-dependent self-similar flow it is with a given boundary condition at infinity) is expected to have a unique solution. The study of uniqueness is not straightforward because both the stationary and the self-similar potential equations are of mixed type. The equations are nonlinear and therefore the regions of hyperbolicity and ellipticity depend on the solution and are not known a priori. This is the essential difficulty of the subject. In this regard, general statements such as the ellipticity principle are useful.
5. Characteristics and Simple Waves In Chapter 7, particular solutions, such as the simple waves, are constructed for the case of one space dimension. Similarly, particular solutions for multidimensional flows can be constructed. Note, however, that unlike in the one-dimensional case, the list of particular solutions is not sufficient to use as a set of building blocks for constructing general solutions for the initialboundary value problem. Nevertheless, the construction of particular solutions is useful for the study of various physical situations. There is a class of simple waves that arise when a solution u, instead of depending on the whole set of independent variables, u = u(x, t), depends on a certain combination ξ = ξ(x, t) of the independent variables, u = U (ξ). As an illustration of this, consider the stationary potential flow equation in two
400
13. Multi-Dimensional Gas Flows
space dimensions: (5.1)
(φ2x − c2 )φxx + 2φx φy φxy + (φ2y − c2 )φyy = 0, φx = u, φy = v.
Assume that (u, v)(x, y) = (U, V )(ξ(x, y)) for a scalar function ξ(x, y). Write U = dU/dξ and V = dV /dξ. Then (5.1) becomes (U 2 − c2 )U ξx + 2U V U ξy + (V 2 − c2 )V ξy = 0. From the irrotational constraint, uy = vx , of the potential flows, we have U ξy = V ξx , or U = V η, η ≡
ξx , ξy
and the above becomes [(U 2 − c2 )η 2 + 2U V η + (V 2 − c2 )]V = 0 and so −U V ± c2 (U 2 + V 2 − c2 ) −U V ± U 2 V 2 − (U 2 − c2 )(V 2 − c2 ) = . η= U 2 − c2 U 2 − c2 This induces a differential equation for the simple wave: −U V ± c2 (U 2 + V 2 − c2 ) dU =η= . (5.2) dV U 2 − c2 Note that the simple waves are constructed locally and therefore the potential flow equation should be hyperbolic, not elliptic. This agrees with fact that for the above expression to be real the flow needs to be supersonic, U 2 + V 2 > c2 .
x y
= c¯
( u¯, 0) μ1 μ2
θ0
(u1 , v 1 )
Figure 13.03. Prandtl-Meyer simple wave.
We now use this general idea to construct a supersonic flow around a ramp with an angle greater than π. Take the corner to be the origin and
6. Hodograph Transformation
401
one of the sides to be horizontal. The gas region Ω is, as shown in figure 13.03, Ω = {(x, y) = r(sin θ, cos θ), 0 < r < ∞, −θ0 < θ < π}, for some constant 0 < θ0 < π/2. The physical situation starts with a upstream velocity (¯ u, 0) at the far left and heads toward the corner, u ¯ > 0. As we just noted, this upstream state has to be supersonic, u ¯ > c¯, where c¯ is the upstream sound speed provided by the Bernoulli law. At the far right of the gas region, the flow will be parallel to the left portion of ∂Ω; that is, the flow velocity (u1 , v1 ) at the far right, though to be determined, needs to be parallel to the right boundary of Ω, v1 /u1 = − tan θ0 . This can be achieved for θ0 sufficiently small. We now construct such a simple wave, choosing one of the branches in (5.2) and taking ξ(x, y) = y/x and η = ξx /ξy = −y/x = −ξ: −U V − c2 (U 2 + V 2 − c2 ) y dU =η=− = . dV x U 2 − c2 For the upstream state (U, V ) = (¯ u, 0), we have from the above that y/x = c¯ and so the flow equals the upstream state until y/x = c¯. After this, one needs to solve the above differential equation with initial values U = u ¯ and V = 0 , v and with V decreasing to negative values until (U, V ) = (u1 1 ) at the far u, 0) to right. For a moderate value of θ0 , the value y/x can decrease from (¯ (u1 , v1 ) for some end state satisfying v1 /u1 = − tan θ0 .
6. Hodograph Transformation The hodograph transformation transforms the quasilinear stationary potential flow equation (3.7) to a linear equation. For the stationary potential flow equation in two space dimensions, (5.1), view the variables (x, y) as dependent variables and the unknown velocity components (u, v) as independent variables: (x, y) = (x(u, v), y(u, v)). The new function Φ(u, v) would satisfy the property (6.1)
Φu = x, Φv = y,
analogous to φx = u and φy = v for the potential function φ(x, y). Thus Φ is the Legendre transform of the potential φ: (6.2)
Φ(u, v) = xu + yv − φ(x, y).
Here it is understood that (u, v) = (u(x, y), v(x, y)) for the potential equation, and (x, y) = (x(u, v), y(u, v)) for the transformed variables. Thus we
402
13. Multi-Dimensional Gas Flows
have from (6.2) together with φx = u and φy = v that Φu = x + u
∂y ∂x ∂x ∂y +v − φx − φy = x, ∂u ∂u ∂u ∂u
∂y ∂x ∂x ∂y +v − φx − φy = y, ∂v ∂v ∂v ∂v which confirms the property (6.1). Write Φv = y + u
u = u(x, y) = u(x(u, v), y(u, v)), v = v(x, y) = v(x(u, v), y(u, v)), and perform the partial differentiations ∂u and ∂v of these to obtain 1 = ux xu + uy yu , 0 = ux xv + uy yv , 0 = vx xu + vy yu , 1 = vx xv + vy yv , or (6.3)
ux uy vx vy
xu xv yu yv
= I.
Here I is the 2 × 2 identity matrix. From this we have −1 1 ux uy vy −uy xu xv = , A≡ =A . yu yv vx vy −vx ux ux vy − uy vx Therefore from (2.5) we have Φuu = xu = Avy = Aφyy , Φuv = xv = A(−uy ) = −Aφxy , Φvv = yv = Aux = Aφxx , and the hodograph transformation of the potential flow equation (5.1) is (6.4)
(u2 − c2 )Φvv − 2uvΦuv + (v 2 − c2 )Φuu = 0.
In the above it is required that the transformation be invertible, or that A be finite and nonzero, i.e. from (6.3), (6.5)
ux vy − uy vx = φxx φyy − (φxy )2 = 0.
This is the convexity condition on the potential function φ for invertibility of the Legendre transform. Note that the coefficients of the transformed equation (6.4) are functions of the new independent variables (u, v), as c2 is a given function of (u, v) by the Bernoulli law. In other words, the transformed equation (6.4) is linear, while the original potential equation (5.1) is quasilinear. The linearity of the equation (6.4) makes it easier to construct solutions. However, it is not easy to carry out the inverse transform to go back to the physical domain of (u(x, y), v(x, y)). In particular, the physical boundary cannot be prescribed, but rather depends on the solution of (6.4) by resorting to a boundary condition such as (4.1).
7. The Shock Polar
403
7. The Shock Polar For a given state u− , consider all the states u+ with which it forms a shock wave (u− , u+ ) for the Euler equations. By a Galilean transformation, without loss of generality, we may consider only the stationary shock waves, σ(u− , u+ ) = 0. From the continuity equation in the Rankine-Hugoniot condition (2.3), ρ+ (v + · ξ) = ρ− (v − · ξ), the component of the velocity normal to the shock, v · ξ, does not change sign across the shock. We fix the propagation direction ξ so that the normal component is positive, v · ξ > 0. With this orientation, the gas flows across the shock from the state u− to the state u+ . Thus we call u− the upstream state and u+ the downstream state. For simplicity of notation, we denote the normal and tangential speeds by N and L: N± ≡ v ± · ξ > 0,
L± ≡ |v τ± | ≥ 0.
The entropy condition implies that the shock is compressed, i.e. according to (2.14), (7.1)
0 < N+ < N− ,
ρ+ − ρ− = ˜ p+ − p− = ˜ |N− − N+ |,
s+ − s− = ˜ |N− − N+ |3 .
(7.2)
The Rankine-Hugoniot condition (2.3) for the stationary shock, σ = 0, is ⎧ ⎪ ⎨ρ+ N+ = ρ− N− ≡ m, (7.3) −ρ+ (N+ )2 + p+ = ρ− (N− )2 + p− ≡ P, L+ = L− ≡ L, ⎪ ⎩ ρ+ E+ N+ + p+ N+ = ρ− E− N− + p− N− . From the first two equations, we have p+ − p− = −m(N+ − N− ) = −(ρ− N− N+ − ρ+ N+ N− ) = (ρ+ − ρ− )N+ N− , and so N+ N− =
(7.4)
p+ − p− . ρ+ − ρ+
By introducing the enthalpy i ≡ e + p/ρ and the speed of the flow q ≡ |v| = √ N 2 + L2 and recalling that E = q 2 /2 + e, the energy equation can be rewritten as (7.5)
(q− )2 (q+ )2 + i+ = + i− ≡ qˆ2 , energy equation. 2 2
For the polyatomic gases, (7.6)
p = A(s)ργ , c2 = γA(s)ργ−1 =
p γp γp , e= , i= , ρ (γ − 1)ρ (γ − 1)ρ
404
13. Multi-Dimensional Gas Flows
so that the Rankine-Hugoniot conditions assume a simpler form. First, the energy equation becomes μ2 (q+ )2 + (1 − μ2 )(c+ )2 = μ2 (q− )2 + (1 − μ2 )(c− )2 = (μˆ q )2 ≡ (c∗ )2 , γ−1 , μ2 ≡ γ+1 or c∗ )2 . μ2 (N+ )2 + (1 − μ2 )(c+ )2 = μ2 (N− )2 + (1 − μ2 )(c− )2 = (c∗ )2 − μ2 L2 ≡ (˜ From this and the momentum equation, we have c∗ )2 = ρ+ [μ2 (N+ )2 +(1−μ2 )(c+ )2 ] = μ2 [P −p+ ]+(1−μ2 )γp+ = μ2 P +p+ . ρ+ (˜ Similarly, we have c∗ )2 = μ2 P + p− . ρ− (˜ Subtracting the above two identities gives p+ − p− = (˜ c∗ )2 . ρ+ − ρ− We conclude from this and (7.3) that (7.7)
c∗ )2 , Prandtl relation. N+ N− = (˜
This basic Prandtl relation for polyatomic gases involves velocity and the thermodynamics of the upstream state. We now take the upstream velocity to be two-dimensional and pointing to the right, v − = (u0 , 0) with u0 > 0. Let the shock be oblique, making angle β with the x-axis, i.e. the shock is parallel to (cos β, sin β). Let the downstream velocity v + = (u, v) have speed q and angle θ with the x-axis, (u, v) = q(sin θ, cos θ), as shown in figure 13.04.
(u,v) (u0 ,0)
β
θ y x
Figure 13.04. Flow turning across a shock.
7. The Shock Polar
405
The components parallel to the shock are the same, i.e. (2.14); thus L = L0 = q cos(β − θ) = u0 cos β, and so (u, v) − (u0 , 0) is perpendicular to the shock: (u − u0 ) cos β + v sin β = 0. The components normal to the shock are N0 = u0 sin β and N = q sin(β −θ), and the Prandtl relation becomes u2 u0 q sin β sin(β − θ) = μ2 0 + i0 − (u0 cos β)2 . 2 This and the relation q cos(β − θ) = u0 cos β yield the speed q as a function of the angle θ and the upstream state. By varying the turning angle β of the shock, the possible states form the shock polar; see figure 13.05.
v
(u, v)
(¯ u, v¯) γ
β
θ
(u0, 0)
u
Figure 13.05. Shock polar.
The shock polar can be used to construct various gas flows involving shock reflection. Take a physical situation where a supersonic upstream flow (u0 , 0) hits a ramp at an angle θ; see figure 13.06 and figure 13.07. The upstream horizontal flow needs to turn and become parallel to the ramp, which causes the air to compress and produce a shock. When the angle θ is not too large, there are two possible shock reflections, as shown in figure 13.05: a weak one S1 with supersonic downstream flow (u, v), depicted in u, v¯), figure 13.06, and a strong one S2 with subsonic downstream flow (¯ depicted in figure 13.07. For the given upstream state (u0 , 0), in order for the solution to be a single reflected shock, there is a limiting angle θ0 for the range of θ. When
406
13. Multi-Dimensional Gas Flows
y S1 (u,v) (u0 , 0) β
x
θ
Figure 13.06. Weak shock reflection off a ramp.
y S2 (¯u, v¯)
(u0 , 0) γ θ
x
Figure 13.07. Strong shock reflection off a ramp.
the two states are equal, (u, v) = (¯ u, v¯), θ attains its maximum value θ0 ; see figure 13.08. Next we use the shock polar analysis to construct the reflection of the shock off a flat plane, taken to be the x-axis in figure 13.09. The upstream flow (u0 , 0) parallel to the solid boundary and the incident shock Si are given. The incident shock Si turns the flow toward the boundary. Shock polar analysis is used to construct the reflected shock Sr to make the downstream flow parallel to the boundary again, so that the boundary condition (4.1) for the Euler equations holds along the solid boundary. The configuration of a reflection involving two shocks is called a regular reflection, as shown
8. Prandtl Paradox
407
v
(u, v) = (¯ u, v¯)
(u0, 0)
θ0
u
Figure 13.08. Shock polar, with maximum turning angle.
in figure 13.09. For a given incident shock Si , the reflected shock Sr may not exist. In such a situation, the regular reflection configuration does not exist, and more complex configurations, such as the Mach reflection with more than two shocks, occur.
y Si
Sr α
β
x
Figure 13.09. Regular reflection.
8. Prandtl Paradox The study of multi-dimensional gas flows with shock waves is physically interesting and mathematically challenging. It is not practical to aim at the same level of well-posedness theory as for the one-dimensional flows in Chapter 9. Instead, the aim is to focus on experimentally realizable
408
13. Multi-Dimensional Gas Flows
situations involving the interaction of the physical boundary with shock waves. One of the simplest, and therefore most basic, problems is the socalled Prandtl paradox, concerning shock reflections off the tip of a ramp. Consider a two-dimensional ramp y Ω ≡ {(x, y) : x < 0, y > 0} ∪ {(x, y) : x > 0, > tan θ}, x with boundary condition v · n = 0 on its boundary y = tan θ}. {(x, 0), x < 0} ∪ {(x, y) : x, y > 0, x There are two possible reflected shocks, depicted in figure 13.06 and figure 13.07. Both reflections satisfy the Euler equations, the entropy condition, and the boundary condition. On the other hand, when a pointed projectile is accelerated to a supersonic speed (u0 , 0), it is the weak shock reflection, figure 13.06, that is eventually observed. This problem illustrates the challenge facing mathematical analysis: As the projectile is accelerated, the gas flow in front of it is compressed and shock waves begin to form and propagate away from the projectile. As the projectile becomes supersonic, some shocks stay around it. Supposedly these shock waves will combine and eventually a single weak shock reflection will emerge to dominate the flow. The process is clearly complex. To understand the situation analytically, it is necessary to focus only on the end result of the physical process, namely the time-asymptotic state of the accelerated projectile. Remark 8.1. We have seen the central role played by the resolution of the Riemann problem in the one-dimensional theory, covered in Chapter 9. The main reason is that a Riemann solution consists of non-interacting waves. Thus the wave pattern in a Riemann solution represents time-asymptotic behavior of a general solution. This leads to the idea of using the generalized Riemann problem, e.g. Subsection 1.6 of Chapter 11 for relaxation models and Section 1 of Chapter 12 for a moving source, to identify the timeasymptotic wave patterns. There is also the notion of interaction measures, discussed in Section 4 of Chapter 5 and Section 2 of Chapter 9, for measuring the distance of a general solution to the corresponding Riemann solution. The search for time-asymptotic wave patterns can be a guiding principle for the analysis of multi-dimensional gas flows. Proposition 8.2. Consider a trajectory gradually accelerated from zero speed 0 = (0, 0)T to the supersonic speed u0 = (u0 , 0)T during the time span T0 . Suppose that the solution of the initial-boundary value problem for the gas dynamics equations is unique and depends on its initial data continuously. Let u(x, t) be the velocity field for the gas flow. Dilate space and time by a factor of α > 0 and set v(x, t) ≡ u(αx, αt).
8. Prandtl Paradox
409
Then, for large α, the dilated velocity field v(x, t) is close to the velocity ˆ (x, t) corresponding to instantaneous acceleration, field u ˆ (x, 0) = u0 . u Proof. During the time period (0, T0 ), the acceleration creates a perturbation of 0 for a region around the origin of size O(1)T0 due to the finite speed of propagation of the solutions of the Euler equations. Outside this finite region around the origin and the ramp, the velocity field, viewed from the moving frame of the trajectory, is the supersonic velocity u0 , and so the velocity field u(x, t) for the Euler flow satisfies u(x, T0 ) = u0 for {x = (x, y) : |x| > O(1)T0 , y > x tan θ + O(1)T0 }. This and the above definition for v(x, t) yield O(1)T0 O(1)T0 T0 ) = u0 for {x = (x, y) : |x| > , y > x tan θ + }. α α α Taking the limit α → ∞, we obtain v(x,
v ∞ (x, 0) = u0 for x ∈ Ω; v ∞ (x, t) ≡ lim u(αx, αt). α→∞
From the assumption of well-posedness of the solution operator for the gas ˆ (x, t) and v(x, t) is close to u ˆ (x, t) for dynamics equations, v ∞ (x, t) = u large α. This completes the proof of the proposition.
y (u, v)(x, y, 0) = (u0 , 0)
θ
Figure 13.10. Initial data for instantaneous acceleration.
x
410
13. Multi-Dimensional Gas Flows
With constant initial data, the solution becomes self-similar. This leads to consideration of the self-similar potential equation (4.3). For large |ξ|, the pseudo velocity v−ξ has large amplitude and therefore is psuedo-supersonic, and the equation (4.3) is hyperbolic. The initial value problem for the potential flow equation turns into the boundary value problem for the selfsimilar equation with boundary value posed at the far field |ξ| = ∞. It has been proved for some range of the angle θ not far from π/2 that the solution consists of a weak shock S1 connecting the upstream state (u0 , 0) to (u, v) at the tip of the ramp, a one-dimensional shock parallel to the boundary near |ξ| = ∞, and a pseudo-subsonic region in between. The pseudo-subsonic region is bounded by a curved shock, the solid boundary, and two pseudosonic circles; see figure 13.11.
ξ2
1-D shock (u ¯, v¯) Curved shock (u,v) (u0 ,0)
Pseudo-sonic Pseudo-subsonic
S1 θ β
ξ1
Figure 13.11. Self-similar flow with weak shock reflection.
At the far field, the flow is affected only by the flat solid boundary and so the solution is found by solving the one-dimensional Riemann problem with boundary; its solution is denoted by u, v¯)T , (u0 , 0)T ). (¯ u, u0 ) = ((¯ It turns out that the linear theory of acoustic waves in Section 1 applies to the determination of the boundary of the two constant states (u, v) and (¯ u, v¯). The boundary consists of pseudo-sonic circles induced by these states: |ξ − u| = c,
¯ | = c¯, |ξ − u
8. Prandtl Paradox
411
where c is the sound speed of the state u and c¯ is the sound speed of the ¯ , calculated by Bernoulli’s law. state u The curved shock connecting the one-dimensional shock and the weak reflection S1 needs to be constructed. This is done along with the solution in the pseudo-subsonic region. The global topological method and the ellipticity principle are used for the construction. By Proposition 8.2, the solution of the original problem for gradual acceleration of the projectile would tend to the self-similar solution under the scaling v(x, t) ≡ u(αx, αt), as α → ∞. Consider a bounded region ΩC ≡ {x : |x| < C}. For large times t 1 and for x ∈ ΩC , the self-similar variable belongs to a small region, ξ = (x/t, y/t) ∈ ΩC /t. The self-similar flow consists of the weak shock reflection for ξ around the origin. We thus conclude that for sufficiently large time, the flow u(x, t) for the gradual acceleration of the projectile consists of the weak shock reflection for any finite domain ΩC around the origin. The Prandtl paradox is resolved by noting that the weak shock reflection is the one eventually produced by a projectile accelerated to the supersonic velocity (u0 , 0). To obtain a complete analysis of the problem, it is necessary to consider the time-dependent problem.
y
(u 0 ,0)
(0, 0)
θ
x
Figure 13.12. Initial data for a shock impinging on a ramp.
Another classical experiment is that initiated by Mach on the shock wave impinging on a ramp; see figure 13.12. The flow is self-similar, and there
412
13. Multi-Dimensional Gas Flows
y
θ
x
Figure 13.13. Regular shock reflection.
y
θ
Figure 13.14. Mach reflection.
x
10. Exercises
413
are several possibilities at later times. When the angle θ is close to π/2, the shock polar analysis shows that it is possible for the local configuration around the meeting of the original shock and the ramp to be a regular reflection, as in figure 13.08 and figure 13.09. This is the necessary condition, the so-called geometric condition, for the global flow to be a regular reflection; see figure 13.13. For a smaller angle θ more complex shock configurations arise; one possibility is the Mach reflection, depicted in figure 13.14.
9. Notes The classical book by Courant-Friedrichs [32] remains an excellent reference on the mathematical results for multi-dimensional gas flows. Classical analytical techniques of analytic function theory and special functions combined with the hodograph transformation are useful for the explicit construction of solutions; see Bers [6] and Ferrari-Tricomi [44]. See Whitham [126] for a general discussion of nonlinear waves. The ellipticity principle is covered in Elling-Liu [41] and is used in Elling-Liu [42] to study the physicality of Prandtl weak reflections and in Chen-Feldman [27] for regular reflections. For discussions on two-dimensional Riemann problems, see Zheng [129]. The research on multi-dimensional gas flows is ongoing; see for instance [28], [123], and references therein. See [4] and references therein for the experimental evidence of Mach and other reflections. Remark 9.1. Von Neumann chaired a panel discussion in 1949 and raised the Prandtl paradox, which then became the focus of exchanges among the leading fluid dynamicists of the day. The panel article has been reprinted as [125]; see also [91] for historical perspectives. The above consideration of the Prandtl paradox draws from the one-dimensional theory in looking for time-asymptotic wave patterns. The presence of a physical boundary helps to stabilize and limit the possible time-asymptotic wave patterns for the solutions of the Euler equations. The wave pattern as depicted in figure 13.11 contains only one free boundary, the curved shock. The pseudo-sonic circles as well as the weak shock at the tip and the far-field one-dimensional shock are explicitly calculated from the initial data. The construction of the curved shock and the solution in the pseudo-subsonic region are the main efforts in [42]. The subsequent study of regular reflections in [27] deals with the analogous situation. It would be interesting to identify and study other time-asymptotic wave patterns.
10. Exercises 1. From the Rankine-Hugoniot condition show that the tangential velocity does not change across a shock wave.
414
13. Multi-Dimensional Gas Flows
2. Show that the reflected shock S2 is weaker than the incident shock S1 and that the incident angle α is larger than the reflected angle β in figure 13.09. 3. Derive the self-similar equations u(x, t) = φ(ξ), ξ ≡ x/t, from the Euler equations (0.1). 4. Show that the flows corresponding to figure 13.10 for instantaneous acceleration and figure 13.12 for a shock impinging on a ramp are self-similar. Go through analysis similar to that for one space dimension in the proof of Proposition 1.1 in Section 1 of Chapter 3. Notice that, besides the differential equations and initial data as in Proposition 1.1, here one needs to check the self-similarity of the boundary as well as the boundary condition. 5. Prove the ellipticity principle for x ∈ Rm , m > 2. 6. Consider the two-dimensional Euler equations in gas dynamics, with x = (x, y). Formulate a Riemann problem and show that the solution is self-similar. Write down the Euler equations in the self-similar variables (ξ, η) ≡ (x/t, y/t).
10.1090/gsm/215/14
Chapter 14
Concluding Remarks
The strongly nonlinear nature of shock waves induces rich physical phenomena, the study of which calls for distinctive mathematical techniques and ideas. The present volume has aimed at illustrating these features through the development of general theory and the analysis of specific examples. This concluding chapter presents additional selected topics with possible future research directions.
1. Development of Singularities In the development of shock wave theory, it is natural to consider the development of shock waves from smooth solutions. In fact, Stokes [119] already considered the problem for simple waves for the Euler equations in gas dynamics. His analysis is the same as for scalar conservation laws, with shocks resulting from the compression of characteristic lines, as described in Section 2 of Chapter 2. The simplest approach to the study of systems is to use the coordinates of Riemann invariants, covered in Section 2 of Chapter 8, to study the formation of singularities for 2 × 2 systems in one space dimension; see Lax [75]. This approach was generalized to genuinely nonlinear systems by John [66] and to Euler equations in gas dynamics by Liu (1979); see references in [92]. For one spatial dimension, it is possible to identify necessary and sufficient conditions on the initial data for the development of singularities. In the book [29], a sufficient condition for the development of singularities in multiple spatial dimensions is found and analyzed. The mechanism considered for multi-dimensional cases mirrors that for the one-dimensional situation; see [29], [30], and references therein. Truly multi-dimensional singularity formation can be studied by considering the Euler equations with spherical symmetry, where x ∈ Rm , m ≥ 2, and 415
416
14. Concluding Remarks
r ≡ |x|: (1.1)
ρt + (ρv)x = − m−1 r ρv, 2 (ρv)t + (ρv 2 + p)x = − m−1 r ρv , (ρE)t + (ρEv + pv)x = − m−1 r (ρEv + pv).
The singular factor (m−1)/r makes the right-hand side of the equations not a finite source—its integral has a logarithmic singularity both at the origin and at the far field: b m−1 dr → ∞ as either a → 0 + or b → ∞. r a In particular, the approach used for conservation laws with a finite source in Section 1 of Chapter 12 does not apply. It is interesting to study the behavior of waves propagating toward r = ∞, a multi-dimensional analogue of the N -waves for one spatial dimension; see Section 6 of Chapter 3 and Section 9 of Chapter 9. As waves propagate toward r = ∞, some are reflected back toward the origin r = 0. Similarly, as waves propagate toward the origin r = 0, some are reflected and move toward the direction of r = ∞. The above logarithmic singularities would indicate that these reflected waves are strong. In particular, waves propagating toward the origin r = 0 become stronger and even of infinite strength. There is no exact analysis of this singular behavior; see [126] for some heuristic analysis and [53] for its usage in producing industrial diamonds. Historically, there was another development by Riemann [112] around the time of Stokes. Unlike Stokes, Riemann considered solutions with shock waves at the initial time. One notes that the study of the development of singularities from smooth solutions differs markedly, in terms of both analytical techniques and physical interest, from the study of wave patterns containing shocks; see also Remark 9.1 of Chapter 13. In the next section we consider time-asymptotic wave patterns containing shock waves.
2. Local and Global Behavior for Gas Flows with Shock Waves For one-spatial-dimension case, the Riemann problem plays a central role. This can be understood in the following basic sense: For smooth solutions, the method of characteristics is used to study the wave propagation. For a piecewise smooth weak solution, besides the method of characteristics, there is the need to analyze the local structure of waves when two discontinuities meet. This is done by solving the Riemann problem. The success of the theory for one space dimension based on the Riemann solutions is by now well established; see Chapter 7, Chapter 8, and Chapter 9. This success points to the possibility that a key step in the establishment of multi-dimensional
2. Local and Global Behavior for Gas Flows with Shock Waves
417
well-posedness theory is resolution of the Riemann problem, expressed here for the two-dimensional case u = u(x, y, t) as (2.1) ut + f 1 (u)x + f 2 (u)y = 0, y < tan(θj ), j = 1, 2, . . . , l, x for some constant states uj , j = 1, 2, . . . , l, and angles θ0 = 0 < θ1 < · · · < θl < 2π. Here, for definiteness, we take the starting angle to be zero. As with the one-dimensional case the solution is self-similar: y x u(x, y, t) = v(ξ, η), ξ = , η = . t t This simplifies the problem by reducing the number of independent variables from three to two. However, the original conservation laws are hyperbolic, while the system for the self-similar solution v(ξ, η), u(x, y, 0) = uj for tan(θj−1 )
m/2 + 1, where m is the spatial dimension. Such a local theory is used for studies of the local-in-time stability of simple wave patterns containing shocks, e.g. [102] and the book [5] on local-in-time existence of a curved shock wave, [28] on the local stability of Mach configurations, and [25] on the local stability of shock waves in a cylinder. Some particular Riemann problems can be solved; see [129]. There are few global-in-time results. This is not surprising, considering the complexity of wave patterns that can arise after the formation of a singularity. On the other hand, the recognizable wave patterns are those which start with simple and experimentally realizable initial conditions and survive for a definite period of time. Physically, this often involves the solid
418
14. Concluding Remarks
boundary in order to select stable wave patterns. The classical book by Courant-Friedrichs [32] remains a valuable reference. For self-similar flows, the global-in-time behavior is converted to a global-in-space boundary value problem. A canonical setting takes the solid to be a cone. The simplest selfsimilar problem is the one with initial condition being a constant state, i.e. the Prandtl paradox [42], as sketched in Section 7 of Chapter 13. Another simple initial state is a plane shock, for which regular reflection can hold [26]. The self-similar formulation gives rise to equations of mixed type and free boundaries. The ellipticity principle, discussed in Section 3 of Chapter 13 and [41], and the application of the Leray-Schauder fixed point theorem in [42] are global analysis techniques and keys to large data studies in [27]. It is of central interest to identify general global statements such as the ellipticity principle. The above studies are for potential flows. It would be interesting to consider the full Euler equations, for which there is the formation of vorticity; see Section 1 and Section 2 of Chapter 13. Without the vorticity, the criterion for transition from the regular reflection is reduced to a geometric one, e.g. [27]. The general study of transitions between various wave patterns is a challenging problem. There are several wave patterns known from experiments and preliminary analysis; see [4]. The study of multi-dimensional gas flows with shocks is an area of ongoing research; see also references in Chapter XVIII in [35].
3. Nonlinear Waves for Viscous Conservation Laws It is only natural to extend the shock wave theory for hyperbolic conservation laws to other partial differential equations and to connect it with other fields of scientific interest. For this, a crucial step is to extend the theory to viscous conservation laws of the form (0.1) in Chapter 10, ut + ∇x · F(u) = ∇x · B(u, ε)∇x g(u) , x ∈ Rm , u ∈ Rn . The system can be viewed as a nonlinear perturbation of the system linearized around a constant state u0 : ut + ∇x · F (u0 )u = ∇x · B(u0 , ε)g (u0 )∇x u . This perspective is useful for the study of dissipation of solutions around the constant state u0 in multiple dimensions, m ≥ 2, as the dispersion would make the nonlinearity dissipate at a faster rate than the heat kernel. For the study of nonlinear waves for viscous conservation laws, it is natural to view the viscous conservation laws as a perturbation of the hyperbolic conservation laws ut +∇x ·F(u) = 0, and to make use of the shock wave theory for the hyperbolic system. The study of nonlinear waves for viscous conservation laws is essential for the understanding of systems modeling non-equilibrium
3. Nonlinear Waves for Viscous Conservation Laws
419
phenomena, such as the Boltzmann equation in kinetic theory, discussed in Section 4 of Chapter 11. For viscous conservation laws, the coupled effect of the nonlinear flux and dissipation term induces rich phenomena, including the formation of initial and boundary layers, interaction of nonlinear waves, and, in the case of systems with physical viscosity, propagation of singularities. In Chapter 4, this key subject is addressed for the Burgers equation using the Hopf-Cole
n transformation. In Chapter 10, nonlinear diffusion waves θ(x, t) ≡ j=1 θj (x, t)r j (u0 ) for systems around a constant state u0 are described and the coupling of these waves is analyzed in Lemma 2.1 and Theorem 2.2. See [88] for the analysis of coupling of waves in the simplest setting. For the perturbation of a constant state in the one-dimensional case, the amount of diffusion waves is determined from the conservation laws; see (2.12) of Chapter 10. Similarly, the diffusion waves can be used for the study of viscous shock profiles in determining the time-asymptotic shock location. Suppose that a shock profile φ(x − st), with φ(±∞) = u± , and s = σ(u− , u+ ), for a system of viscous conservation laws is perturbed: ¯ (x). u(x, 0) = φ(x) + u The conservation laws hold:
∞ −∞
u(x, t) − φ(x − st) dx =
∞
¯ (x) dx, t > 0. u −∞
Suppose that the shock belongs to the ith characteristic field, (4.4) of Chapter 7: λi (u+ ) < s < λi (u− ). Time-asymptotically, the perturbation causes the shock to have a shift x0 : u(x, t) → φ(x + x0 − st) as t → ∞. This is in the pointwise sense. In the integral sense, there are diffusion waves θj (x, t)rj (u− ), j < i, propagating to the left of the shock around the left state u− , and θj (x, t)rj (u+ ), j > i, propagating to the right of the shock around the right state u+ : u(x, t) → φ(x + x0 − st) +
i−1
θj (x, t)rj (u− ) +
j=1
n
θj (x, t)rj (u+ ) as t → ∞.
j=i+1
From these we deduce that, time-asymptotically,
∞
∞
¯ (x) dx = u −∞
−∞
φ(x + x0 − st) − φ(x − st) dx
∞
+ −∞
i−1 j=1
θj (x, t)rj (u− ) +
n j=i+1
θj (x, t)rj (u+ ) dx.
420
14. Concluding Remarks
The integral
A(x0 ) ≡
∞ −∞
φ(x + x0 − st) − φ(x − st) dx
has the properties A(0) = 0 and ∞ A (x0 ) = φ (x + x0 − st) dx = u+ − u− , −∞
and so A(x0 ) = x0 (u+ − u− ). Let cj =
∞
(3.1) −∞
∞
¯ (x) dx = x0 (u+ − u− ) + u
−∞ θj (x, t)
i−1 j=1
and we conclude that
cj r j (u− ) +
n
cj r j (u+ ).
j=i+1
The shock shift x0 and the strength of diffusion waves cj , j = i, are thus ex¯ (x). Analysis plicitly expressed in terms of the integral of the perturbation u of the behavior of the waves propagating around the shock profile is done in Liu-Zeng [100]. Besides the types of waves for the coupling of diffusion waves described in Theorem 2.2 of Chapter 10, there are new wave types around the shock profile. The stability analysis for systems with physical viscosity needs to take account of stronger coupling than that for systems with artificial viscosity. The method is based on explicit construction of the Green’s function for the viscous conservation laws linearized around a constant state; see Liu-Zeng (1997) in the references of [100], and compare with Section 4 of Chapter 10. The study of wave patterns containing several wave types remains largely open; see [63] and references therein for such studies, particularly for the study of contact discontinuities. The Riemann problem for viscous conservation laws is such an open problem. It is also interesting to study the interaction of elementary waves; such a study is done for hyperbolic conservation laws in Chapter 8. Multi-dimensional wave propagation around shock profiles is a subject that has been studied by spectral methods; see, for instance, [51] for the study of linear stability of shock profiles. It would be desirable to study the propagation in a quantitative, pointwise sense. For scalar laws, the propagation along the shock profile is governed by Hamilton-Jacobi equation; see Goodman [59]. For multi-dimensional wave propagation over a shock profile for systems, there are richer wave patterns, including Rayleigh-type waves propagating along the profile; see [98] for the analysis of a simple model in two spatial dimensions based on the Burgers shock. Multi-dimensional propagation over rarefaction waves exhibits additional complexity in that the rarefaction waves for viscous conservation laws are not exactly constructed, Section 1 of Chapter 10. Consequently, the perturbation cannot be local and is necessarily infinite in global norms.
4. Well-Posedness Theory for Weak Solutions
421
4. Well-Posedness Theory for Weak Solutions In Chapter 9, well-posedness theory for systems of hyperbolic conservation laws was presented. The significance of this theory is that it is for weak solutions. The standard Hadamard well-posedness theory for the initial value problem requires that there exists a topological space such that: 1. There exists a solution in the space. 2. There is at most one solution in the space. 3. The solution depends continuously on its initial data with respect to the chosen topology. For linear partial differential equations, the requirements 2 and 3 are usually shown through some a priori estimates, as the difference of two solutions satisfies the homogeneous equations by the linear superposition principle. For nonlinear equations, this works for solutions that are sufficiently smooth, essentially by the mean value theorem of calculus. For general hyperbolic systems, there is uniqueness of weak solutions to the initial value problem when there exists a smooth solution, the so-called weak-strong uniqueness. This is established by the entropy method; see Section 5.2 in [35]. There is a serious difficulty in establishing the well-posedness theory for weak solutions of nonlinear evolutionary equations. In general, weak solutions are constructed using certain compactness properties. The difficulty is due to insufficient information on the time evolution of the weak solutions constructed through the compactness arguments. There are some well-known constructions, chief among them the construction of weak solutions for the Navier-Stokes equations by Leray [79]. There is also the Diperna-Lions [39] construction of weak solutions for the Boltzmann equation. There is no well-posedness theory for these weak solutions. It is not known if there should be such a theory for the Navier-Stokes equations or the Boltzmann equation. Whether smooth initial data would give rise to smooth solutions at later times is an important unsolved problem for both the Navier-Stokes equations and the Boltzmann equation. There is a striking non-uniqueness construction by convex integration of weak solutions for incompressible Euler equations; see Scheffer [113]. The non-uniqueness construction has been greatly extended and is applicable to the compressible Euler equations [36] and to the incompressible Navier-Stokes equations [18]. The convex integration construction does not extend to the potential flow equation, indicating that the source of the nonuniqueness is vorticity and not shock waves. In other words, it does not seem to be caused by the deficiency of the entropy condition for shock waves. It is
422
14. Concluding Remarks
not clear how to strengthen the notion of weak solutions, in terms of energy dissipation, for instance, so as to regain the well-posedness. The well-posedness of weak solutions is therefore a subtle subject. As mentioned in Remark 4.5 of Chapter 3, even for scalar conservation laws, the question of L1 (x) well-posedness is not straightforward. For systems in one space dimension, we also have the L1 (x) well-posedness theory studied in Chapter 9. However, what it essentially entails is that there is a solution procedure, through the Glimm scheme or some other equivalent procedure such as the front tracking method, that is stable in the L1 (x) norm. As the Glimm scheme is based on the solutions of the Riemann problem, this procedure produces a so-called Riemann semigroup. The well-posedness theory is on the existence of the Riemann semigroup, and does not apply to solutions obtained by the method of compensated compactness; see [114] for discussion of the method. The Bianchini-Bressan paper [9] shows that the Glimm solution operator for hyperbolic conservation laws can be generalized to the system with artificial viscosity, ut + f (u)x = εuxx , and therefore induces an L1 (x) stability theory for the inviscid system in the zero dissipation limit ε → 0+. The zero relaxation limit is also singular; see Bianchini [8] for a result on the zero relaxation limit of the Jin-Xin model from [65]. For systems with physical viscosity of the form ut + f (u)x = B(ε, g(u)ux x , such as the Navier-Stokes equations in gas dynamics, this remains to be understood. The situation with the shock wave theory prompts one to propose a notion of well-posedness in the following sense: It is to find a constructive solution algorithm and to show that solutions obtained by this algorithm depend continuously on their initial data. An important example of well-posedness theory in this sense is the Glimm solution algorithm for hyperbolic conservation laws. Such an algorithm would be useful for the study of solution behavior. The Glimm algorithm has been used for the study of regularity and time-asymptotic behavior of solutions in Chapter 9. For viscous conservation laws with physical viscosity, non-smooth initial data will give rise to weak and not strong solutions; see Section 4 of Chapter 10. For the Navier-Stokes equations in gas dynamics, there is well-posedness theory for classical solutions in the L2 (x) norms; see [106]. No well-posedness theory has been established for the weak solutions constructed by compactness analysis in [60], [82], and [43]. There is an example
5. Kinetic Theory and Fluid Dynamics
423
of well-poseness theory for the constructive solution algorithm for isentropic Navier-Stokes equations in [99].
5. Kinetic Theory and Fluid Dynamics Fluid equations can be derived from the Boltzmann equation through Hilbert and Chapman-Enskog expansions, e.g. Section 4 of Chapter 11. The authoritative text by Sone [118] carries out the expansions for various physical situations. Classical as well as new fluid dynamics equations and their associated boundary conditions are derived. Also, besides the classical viscosity and heat conductivity coefficients, new dissipation parameters, such as the thermal transpiration coefficient, have been derived for registering phenomena not explained by classical fluid dynamics; see Chapter 5 of [118]. There is a modern fluid dynamics resulting from the theory for the Boltzmann equation. Around the physical boundary, the gases are far from thermal equilibrium and the Boltzmann equation needs to be solved exactly to obtain the suitable boundary condition for the fluid equations in the given physical situation. The kinetic boundary layer of length proportional to the Knudsen number is called the Knudsen layer. In the case where the Mach number of the fluid is near zero or one, there is nonlinear coupling of the Knudsen-type boundary layer and fluid-like interior waves. The coupling of fluid dynamics waves and the kinetic boundary layer is a fully nonlinear phenomenon and induces interesting bifurcation phenomena. The computational and asymptotic analysis is highly intricate; see Chapter 6 of Sone [118]. The analytical study in [97] requires explicit construction of the Green’s function for the Boltzmann equation; see reference Liu-Yu (2006) in [97]. Just as for viscous conservation laws, the one-dimensional Riemann problem for the Boltzmann equation is an interesting problem waiting to be resolved. For the one-dimensional Boltzmann equation the distribution function depends on one space variable x, time t, and three microscopic variables ξ = (ξ1 , ξ2 , ξ3 ), and satisfies (4.3) in Chapter 11,
(5.1)
∂t f(x, t, ξ) + ξ1 ∂x f(x, t, ξ) =
1 Q(f, f)(x, t, ξ). k
Since only global Maxwellians are constant solutions of the Boltzmann equation, the Riemann initial data consists of two Maxwellian states, defined in (4.16) of Chapter 11,
M(ρl ,vl ,θl ) for x < 0, (5.2) f(x, t, ξ) = M(ρr ,vr ,θr ) for x > 0.
424
14. Concluding Remarks
The macroscopic variables (ρl , vl , θl ) and (ρr , vr , θr ) are constant states. Set (5.3)
g(x, t, ξ) ≡ f(αx, αt, ξ)
for some constant α > 0. Then g(x, t, ξ) satisfies the same initial data (5.2) and the Boltzmann equation α (5.4) ∂t g(x, t, ξ) + ξ1 ∂x g(x, t, ξ) = Q(g, g)(x, t, ξ). k For fixed (x, t), as α → ∞, the new function g represents the large-time behavior of the original solution f. The equation (5.4) has a new mean free path of k/α, which tends to zero as α → ∞. As the mean free path tends to zero, the collision operator is expected to tend to zero almost everywhere, Q(g, g) → 0, and so the solution g(x, t, ξ) would tend to the local Maxwellian; see Subsection 4.2 of Chapter 11. We know also that the local Maxwellian solution of the Boltzmann equation satisfies the Euler equations in gas dynamics. In other words, the large-time behavior of the solution f of the Riemann problem for the Boltzmann equation, with scaling (5.3), would tend to the solution of the Riemann problem for the Euler equations in gas dynamics. The Riemann problem for the Euler equations consists of elementary waves of shock, rarefaction, and contact discontinuity type; see Chapter 7. In summary, we expect the solution f(x, t, ξ) to tend to these elementary waves time-asymptotically. Before the scaling (5.3), the shock wave in the time-asymptotic state for f(x, t, ξ) is a Boltzmann shock profile. Boltzmann shock profiles have been studied, e.g. in [97]. Time-asymptotically, the contact discontinuities are thermal waves for the Boltzmann equation; see also [97]. The rarefaction waves for the Boltzmann equation and the Euler equations approach each other time-asymptotically. The above scaling arguments on the convergence of the Riemann problem for the Boltzmann equation to that of the Euler equations in gas dynamics requires exact analysis of the resolution of the Riemann problem for the Boltzmann equation. The solution of the Riemann problem contains initial and intermediate layers, as well as large-time wave propagation. So far the Riemann problem for the Boltzmann equation has been studied only when the corresponding problem for the Euler equations consists of one shock wave; see Yu [124]. In general, study of the Riemann problem for dissipative systems, such as the Navier-Stokes equations in gas dynamics analyzed by Hoff-Liu [61] and the Boltzmann equation in kinetic theory studied in [124], requires analysis of the initial layer. An important problem in the study of the Boltzmann equation in kinetic theory is to consider the zero mean free path limit. There are several ways to view this problem, depending on the limiting Mach number. The most basic one is to keep the Mach number at order one in the zero mean free time limit. In such a limit, the solution of the Boltzmann equation would
6. Multiple Effects
425
tend to the solution of the Euler equations in gas dynamics. The study of the Riemann problem should shed some light on this open problem.
6. Multiple Effects The connection of shock wave theory with other fields of partial differential equations is already strong. Shock waves are considered for both hyperbolic and viscous conservation laws due to natural physical considerations. The study of transonic flows requires the theory for elliptic equations, as equations for transonic flows are of mixed type. In the study of combustion, there are strong and weak detonation waves, as well as deflagration waves. Deflagration waves are slow-moving combustion waves, with speed a few meters per second, and are often accurately analyzed in the setting of reaction diffusion equations. The detonation waves model the high-speed, in the range of thousands of meters per second, explosive combustion process. Detonation waves are often modeled by the Chapman-Jouguet theory. Strong detonation waves are similar to classical shock waves. Weak detonation waves are under-compressive and depend sensitively on the relative strengths of the reaction and the dissipation. The study of the transition from deflagration waves to detonation waves involves several distinct types of partial differential equations, and is therefore interesting and challenging. The theory of hyperbolic partial differential equations is an essential element in the modeling of astrophysical phenomena. For instance, it has been recognized that the unusually fast speed observed in galaxies is not the speed of movement of materials; rather, it is the acoustic speed that the movement generates. With vastly different scales and the occurrence of drastic changes, shock wave theory should play a key role in the study of cosmology. Relating the classical shock wave theory to other formulations, such as the kinetic formulation, in order to address several physical considerations is an important task. There are works on gaseous stars using the Euler equations in gas dynamics coupled with the Poisson equation, e.g. [104]. The book [21] is a classical reference on gaseous stars. Gas dynamics and kinetic formulations are used for modeling the evolution of galaxies. The issue of vacuum naturally arises in such studies. Evolutionary partial differential equations in general relativity are necessarily hyperbolic due the fact that no information travels faster than the speed of light. Many of the analytical studies of general relativity focus on geometric effects, with the base state of vacuum. The study of material effects requires combining shock wave theory and the geometric considerations.
426
14. Concluding Remarks
There is a rich variety of phenomena for wave propagation in solids. Besides the pressure waves that are similar to acoustic waves in gas dynamics, there are shear and torsional waves. In the study of earthquakes in geophysics, the pressure waves induce surface, Rayleigh-type waves that do much damage. The coupling of various waves and the generation of surface waves are basic topics for further research in shock wave theory.
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Index
acoustic speed, 310, 425 balance laws, 2 Boltzmann equation, 334, 423 boundary layer, 374, 380, 423 Burgers equation, 24, 71, 151, 293 C-F-L condition, 208, 272, 381 cancellation of waves, 29, 186, 212 Chapman-Enskog expansion, 319, 343 characteristic curve, 5, 149 characteristics, method of, 5–7, 150, 399–401 conservation laws hyperbolic, 2, 5, 123 viscous, 71, 103, 279 constitutive relation, 4, 138, 340 contact discontinuity, 162 delta function, 11–12, 50, 72, 308–311 development of singularity, 7, 136, 415 diffusion waves, 72, 81, 293 Duhamel’s principle, 77, 87, 291, 300 elasticity, 175, 327 435
energy estimate, 59, 77, 113, 128, 133, 285, 311 entropy condition Lax, 159 Liu, 170 Oleinik, 105 entropy pair, 57, 113, 126, 139, 313, 339 equi-distributed sequences, 210 equilibrium, 318, 339 Euler equations in gas dynamics Eulerian and Lagrangian coordinates, 136–138 full, 4, 136, 192, 338 isentropic, 164, 183 potential flow, 393–395 existence theory scalar laws, 31, 35–38, 80, 120 systems, 132, 230, 315 Fourier transform, 308 front tracking, 274 generalized characteristics, 40, 243 generalized entropy functional, 59–68, 237 generalized Riemann problem, 325, 353
436
INDEX
genuinely nonlinear, 151 Glimm functional, 213–214 Glimm scheme, xiv Green’s function, 72, 86, 310, 315
Navier-Stokes equations, 301, 320, 333, 344, 421 nozzle flows, 351 numerical schemes, 207, 272, 380
heat equation, 71, 294, 310 Hilbert expansion, 320, 346 Hopf equation, 24, 151 Hopf-Cole transformation, 78, 84 Hugoniot curve, 152 hyperbolic, 2, 5, 124, 147, 395–396
Oleinik entropy condition, 105
initial layer, 83, 306, 310, 424 interaction measure, 109, 111–113, 200, 212–213, 216–218 irreversible, 25, 27, 74, 338 kinetic theory, 329, 334, 423 L1 stability theory, 32, 119, 233, 276 L2 stability theory, 69, 133 large-time behavior hyperbolic conservation laws, 249–254, 260–272 scalar laws, 45–59, 94–100, 111–113 viscous conservation laws, 280–301 with boundary, 376–379, 410 Lax entropy condition, 159 linearly degenerate, 158 Liu entropy condition, 170 Liu-Yang functional, see also generalized entropy functional magnetohydrodynamics, 176, 302, 362 metastable, 98 N -wave, 50, 68, 99, 267
p-system, 153, 302 polyatomic gases, 164–165, 192–194 porous media equation, 101, 371 potential flows, see also Euler equations in gas dynamics propagation of discontinuity, 11, 305, 360 quasilinear, 7, 305 Rankine-Hugoniot condition, 10, 142, 152, 322, 403 rarefaction wave, 14, 83, 152, 346 regularity of solutions, 42, 254, 275 relaxation, 317 resonance, 349 Riemann invariants, 168, 187 Riemann problem, 22, 106, 159, 170, 325, 424 second law of thermodynamics, 139, 302, 339, 341 self-similar, 22, 73, 395 semilinear, 7 shallow water wave, 327 shock polar, 403 shock profiles, 82, 103, 303, 323, 342, 366, 382 shock wave, see also Rankine-Hugoniot condition simple wave, 149, 399 small divisor, 384 sound speed, see also acoustic speed
INDEX
stability of waves, see also large-time behavior diffusion waves, 54, 98, 260, 281, 300 rarefaction waves, 47, 289–292 shock waves, 45, 284–286, 419 sub-characteristic condition, 319, 333 sub-shocks, 357 symmetric systems, 126–128 thermal equilibrium, 330, 340 thermal wave, 392 uniqueness, 16, 37–38, 275, 421–423 vacuum, 165, 368, 385, 425 variation, 41, 68, 205 viscosity matrix, 279–280, 302, 345–346
437
vortex sheet, 391 wave coupling, 373–380 wave curves, 160–161, 171 wave interactions Euler equations, 183–194 hyperbolic conservation laws, 194–201 multi-dimensional Euler equations, 407–413 scalar laws, 27–31, 109 viscous conservation laws, 292–301 wave tracing, 218 weak solutions, 9, 118, 119, 306, 421 well-posedness theory, 32, 120, 205, 421 zero dissipation limit, 71, 276, 424 zero mean free path, 424
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215 214 211 210
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193 192 191 190
Martin Lorenz, A Tour of Representation Theory, 2018 Tai-Peng Tsai, Lectures on Navier-Stokes Equations, 2018 Theo B¨ uhler and Dietmar A. Salamon, Functional Analysis, 2018 Xiang-dong Hou, Lectures on Finite Fields, 2018
189 188 187 186
I. Martin Isaacs, Characters of Solvable Groups, 2018 Steven Dale Cutkosky, Introduction to Algebraic Geometry, 2018 John Douglas Moore, Introduction to Global Analysis, 2017 Bjorn Poonen, Rational Points on Varieties, 2017
185 Douglas J. LaFountain and William W. Menasco, Braid Foliations in Low-Dimensional Topology, 2017 184 Harm Derksen and Jerzy Weyman, An Introduction to Quiver Representations, 2017 183 Timothy J. Ford, Separable Algebras, 2017 182 Guido Schneider and Hannes Uecker, Nonlinear PDEs, 2017 181 180 179 178
Giovanni Leoni, A First Course in Sobolev Spaces, Second Edition, 2017 Joseph J. Rotman, Advanced Modern Algebra: Third Edition, Part 2, 2017 Henri Cohen and Fredrik Str¨ omberg, Modular Forms, 2017 Jeanne N. Clelland, From Frenet to Cartan: The Method of Moving Frames, 2017
177 Jacques Sauloy, Differential Galois Theory through Riemann-Hilbert Correspondence, 2016 176 Adam Clay and Dale Rolfsen, Ordered Groups and Topology, 2016 175 Thomas A. Ivey and Joseph M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition, 2016
For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/gsmseries/.
This book presents the fundamentals of the shock wave theory. The first part of the book, Chapters 1 through 5, covers the basic elements of the shock wave theory by analyzing the scalar conservation laws. The main focus of the analysis is on the explicit solution behavior. This first part of the book requires only a course in multi-variable calculus, and can be used as a text for an undergraduate topics course. In the second part of the book, Chapters 6 through 9, this general theory is used to study systems of hyperbolic conservation laws. This is a most significant well-posedness theory for weak solutions of quasilinear evolutionary partial differential equations. The final part of the book, Chapters 10 through 14, returns to the original subject of the shock wave theory by focusing on specific physical models. Potentially interesting questions and research directions are also raised in these chapters. The book can serve as an introductory text for advanced undergraduate students and for graduate students in mathematics, engineering, and physical sciences. Each chapter ends with suggestions for further reading and exercises for students.
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