Analysis of hydrodynamic models

Table of contents :
cover......Page 1
title......Page 5
copyright......Page 6
Contents......Page 7
Preface......Page 8
Chapter 1
Introduction......Page 9
Chapter 2
Lagrangian and Eulerian
descriptions of
hydrodynamic systems......Page 12
Chapter 3
Hydrodynamic models......Page 30
Chapter 4
Spaces and operators......Page 40
Chapter 5
The Lagrangian-Eulerian
existence theorems......Page 49
Chapter 6
Critical dissipative active
scalars......Page 56
Bibliography......Page 64
Index......Page 67
back......Page 68

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Analysis of Hydrodynamic Models

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CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and published by SIAM. Garrett Birkhoff, The Numerical Solution of Elliptic Equations D. V. Lindley, Bayesian Statistics, A Review R. S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis R. R. Bahadur, Some Limit Theorems in Statistics Patrick Billingsley, Weak Convergence of Measures: Applications in Probability J. L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems Roger Penrose, Techniques of Differential Topology in Relativity Herman Chernoff, Sequential Analysis and Optimal Design J. Durbin, Distribution Theory for Tests Based on the Sample Distribution Function Sol I. Rubinow, Mathematical Problems in the Biological Sciences P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves I. J. Schoenberg, Cardinal Spline Interpolation Ivan Singer, The Theory of Best Approximation and Functional Analysis Werner C. Rheinboldt, Methods of Solving Systems of Nonlinear Equations Hans F. Weinberger, Variational Methods for Eigenvalue Approximation R. Tyrrell Rockafellar, Conjugate Duality and Optimization Sir James Lighthill, Mathematical Biofluiddynamics Gerard Salton, Theory of Indexing Cathleen S. Morawetz, Notes on Time Decay and Scattering for Some Hyperbolic Problems F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics Richard Askey, Orthogonal Polynomials and Special Functions L. E. Payne, Improperly Posed Problems in Partial Differential Equations S. Rosen, Lectures on the Measurement and Evaluation of the Performance of Computing Systems Herbert B. Keller, Numerical Solution of Two Point Boundary Value Problems J. P. LaSalle, The Stability of Dynamical Systems D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications Peter J. Huber, Robust Statistical Procedures Herbert Solomon, Geometric Probability Fred S. Roberts, Graph Theory and Its Applications to Problems of Society Juris Hartmanis, Feasible Computations and Provable Complexity Properties Zohar Manna, Lectures on the Logic of Computer Programming Ellis L. Johnson, Integer Programming: Facets, Subadditivity, and Duality for Group and Semi-Group Problems Shmuel Winograd, Arithmetic Complexity of Computations J. F. C. Kingman, Mathematics of Genetic Diversity Morton E. Gurtin, Topics in Finite Elasticity Thomas G. Kurtz, Approximation of Population Processes Jerrold E. Marsden, Lectures on Geometric Methods in Mathematical Physics Bradley Efron, The Jackknife, the Bootstrap, and Other Resampling Plans M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis D. H. Sattinger, Branching in the Presence of Symmetry R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis Miklós Csörgo, Quantile Processes with Statistical Applications J. D. Buckmaster and G. S. S. Ludford, Lectures on Mathematical Combustion R. E. Tarjan, Data Structures and Network Algorithms

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Paul Waltman, Competition Models in Population Biology S. R. S. Varadhan, Large Deviations and Applications Kiyosi Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces Alan C. Newell, Solitons in Mathematics and Physics Pranab Kumar Sen, Theory and Applications of Sequential Nonparametrics László Lovász, An Algorithmic Theory of Numbers, Graphs and Convexity E. W. Cheney, Multivariate Approximation Theory: Selected Topics Joel Spencer, Ten Lectures on the Probabilistic Method Paul C. Fife, Dynamics of Internal Layers and Diffusive Interfaces Charles K. Chui, Multivariate Splines Herbert S. Wilf, Combinatorial Algorithms: An Update Henry C. Tuckwell, Stochastic Processes in the Neurosciences Frank H. Clarke, Methods of Dynamic and Nonsmooth Optimization Robert B. Gardner, The Method of Equivalence and Its Applications Grace Wahba, Spline Models for Observational Data Richard S. Varga, Scientific Computation on Mathematical Problems and Conjectures Ingrid Daubechies, Ten Lectures on Wavelets Stephen F. McCormick, Multilevel Projection Methods for Partial Differential Equations Harald Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods Joel Spencer, Ten Lectures on the Probabilistic Method, Second Edition Charles A. Micchelli, Mathematical Aspects of Geometric Modeling Roger Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, Second Edition Glenn Shafer, Probabilistic Expert Systems Peter J. Huber, Robust Statistical Procedures, Second Edition J. Michael Steele, Probability Theory and Combinatorial Optimization Werner C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, Second Edition J. M. Cushing, An Introduction to Structured Population Dynamics Tai-Ping Liu, Hyperbolic and Viscous Conservation Laws Michael Renardy, Mathematical Analysis of Viscoelastic Flows Gérard Cornuéjols, Combinatorial Optimization: Packing and Covering Irena Lasiecka, Mathematical Control Theory of Coupled PDEs J. K. Shaw, Mathematical Principles of Optical Fiber Communications Zhangxin Chen, Reservoir Simulation: Mathematical Techniques in Oil Recovery Athanassios S. Fokas, A Unified Approach to Boundary Value Problems Margaret Cheney and Brett Borden, Fundamentals of Radar Imaging Fioralba Cakoni, David Colton, and Peter Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering Adrian Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis Wei-Ming Ni, The Mathematics of Diffusion Arnulf Jentzen and Peter E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations Fred Brauer and Carlos Castillo-Chavez, Mathematical Models for Communicable Diseases Peter Kuchment, The Radon Transform and Medical Imaging Roland Glowinski, Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems Bengt Fornberg and Natasha Flyer, A Primer on Radial Basis Functions with Applications to the Geosciences Mike Steel, Phylogeny: Discrete and Random Processes in Evolution Peter Constantin, Analysis of Hydrodynamic Models

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Peter Constantin Princeton University Princeton, New Jersey

Analysis of Hydrodynamic Models

SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS PHILADELPHIA

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Copyright © 2017 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. Publisher David Marshall Acquisitions Editor Elizabeth Greenspan Developmental Editor Gina Rinelli Harris Managing Editor Kelly Thomas Production Editor Kelly Thomas Copy Editor Kelly Thomas Production Manager Donna Witzleben Production Coordinator Cally Shrader Compositor Scott Collins Graphic Designer Lois Sellers Library of Congress Cataloging-in-Publication Data Names: Constantin, P. (Peter), 1951Title: Analysis of hydrodynamic models / Peter Constantin, Princeton University, Princeton, New Jersey. Description: Philadelphia : Society for Industrial and Applied Mathematics, [2017] | Series: CBMS-NSF regional conference series in applied mathematics ; 90 | Includes bibliographical references and index. Identifiers: LCCN 2017004002 (print) | LCCN 2017004721 (ebook) | ISBN 9781611974799 (print) | ISBN 9781611974805 (electronic) Subjects: LCSH: Hydrodynamics--Mathematical models. Classification: LCC QA911 .C648 2017 (print) | LCC QA911 (ebook) | DDC 532/.05--dc23 LC record available at https://lccn.loc.gov/2017004002

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Contents Preface

ix

1

Introduction

1

2

Lagrangian and Eulerian descriptions of hydrodynamic systems 2.1 Incompressible Euler equations . . . . . . . . . . . . . . . . . .

5 11

3

Hydrodynamic models 3.1 The Lagrangian-Eulerian structure 3.2 SQG . . . . . . . . . . . . . . . . . . . . 3.3 Porous medium . . . . . . . . . . . . . 3.4 Boussinesq . . . . . . . . . . . . . . . . 3.5 Oldroyd-B . . . . . . . . . . . . . . . .

. . . . .

23 23 26 27 28 31

4

Spaces and operators 4.1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34

5

The Lagrangian-Eulerian existence theorems 5.1 Local existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Small data global existence with damping . . . . . . . . . . . .

43 43 47

6

Critical dissipative active scalars 6.1 Lower bounds on the fractional Laplacian . . . . . . . . . . . 6.2 Hölder regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Higher regularity . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 54 56

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Bibliography

59

Index

63

vii

Preface The subject of these notes is the existence, regularity, uniqueness, and continuous dependence on initial data of classical solutions of certain partial differential equations of physical origin. The equations themselves arise in a variety of different physical contexts and disciplines. The diversity of physical situations modeled stands in contrast with the relative economy of expression: there are many phenomena but few equations. Their main common features are transport by divergence-free vector fields, nonlocality, and nonlinearity. The lectures present a unified mathematical approach for local existence and uniqueness based on solution paths, and global regularity results for nonlocal dissipative active scalar equations. These notes are based on CBMS-NSF Regional Research Conference lectures delivered at the Mathematics Department of the Oklahoma State University in the summer of 2014. I thank the NSF for its support, the Mathematics Department for its hospitality, and especially Professor Jiahong Wu for his kind and effective help in organizing and hosting the lectures. Peter Constantin Princeton November 2016

ix

Chapter 1

Introduction

In recent years there have been substantial developments in the mathematical study of hydrodynamic models. These notes are devoted to two subjects: the local existence and uniqueness of smooth solutions for conservative equations, and the global regularity of solutions for dissipative equations. The local existence theory is presented in a unified Lagrangian-Eulerian framework of minimal regularity. The framework is suitable for the incompressible Euler equations, the surface quasi-geostrophic equation (SQG), the porous medium equation, the Boussinesq system, the Oldroyd-B system, and many more. These equations are basic partial differential equation (PDE) models, originating from different areas of science but sharing some common features. They are equations with a divergence-free vector field transport structure and with nonlinear and nonlocal terms. The global-in-time existence of smooth solutions starting from smooth initial data is not known for these equations, even in two space dimensions. Related to the important issue of inviscid formation of small scales, this is an important area of research. The main advantage of the Lagrangian-Eulerian approach is that both the system and the solution map are Lipschitz continuous in low regularity path spaces, and in particular there is Lipschitz dependence of solutions on initial data in these spaces. The organization of these notes is as follows. In the second chapter we recall the Lagrangian-Eulerian kinematic description of flows. We describe oneparameter deformations of diffeomorphisms and the differential relationships between flows and deformations which are subsequently central to the analysis. We use this language to derive the incompressible Euler equations and other equations from action principles. This is of course a well-known result [2]. We continue by recalling properties of solutions of Euler equations, including the derivation of the Weber formula [13]. We conclude the chapter with sufficient conditions for regularity: the classical Beale–Kato–Majda criterion [3] and a criterion based on the Hessian of the pressure. Chapter 3 is devoted to the description of the common Lagrangian-Eulerian structure of several models. The salient feature of this structure is the emergence of commutators from deformations of Lagrangian-Eulerian path variables. The commutators allow a low regularity framework. In addition to incompress1

2

Chapter 1. Introduction

ible Euler equations, we describe the SQG, porous medium, Boussinesq, and Oldroyd-B models. The equations model different physical situations, and the versions we present here are maximally simplified, leaving only the essential nonlinear elements. Thus, for example, in the Boussinesq system we omit viscosity, thermal diffusivity, and, most important, inhomogeneous boundary conditions, leaving only the 2D Euler equations driven by an active scalar thermal buoyancy. This system was extensively studied. Regularity can be obtained if there is Laplacian dissipation in one of the variables (either the temperature or the fluid) [6], [35]. More recently, sub-Laplacian dissipation was considered [34], [46]. In the Oldroyd-B model we consider a creeping flow situation in which the viscous forces instantaneously balance the non-Newtonian stress and inertia is neglected. The system becomes an active matrix equation, in which a symmetric matrix representing the non-Newtonian stress is transported by a velocity it creates instantaneously via an equation of state. The equation is well posed in spaces of relatively low regularity (C α ∩L p ) for any α > 0, p > 1, but uniqueness in this class is not obvious by Eulerian methods. This was the original motivation for the introduction of the Lagrangian-Eulerian framework presented here [20], [14]. The global regularity for large data is an open problem. Laplacian dissipation added to the non-Newtonian stress evolution leads to global regularity, even in the full system retaining time dependence and inertia, i.e., coupling with the Navier–Stokes equations, in two dimensions [18]. There is no known result for regularization with sub-Laplacian dissipation. The SQG model appeared as an equation for frontogenesis in meteorology, but its mathematical study was developed because of analogies with 3D incompressible Euler equations [19], [33]. The 2D SQG equations have a form of weak continuity of the nonlinearity that permits the construction of weak solutions in L2 from arbitrary initial data [42]. In fact, local existence of smooth solutions and global existence of weak solutions hold for inviscid equations with more singular constitutive laws [10]. Chapter 4 deals with the analytical scaffolding: spaces and operators. We prove several key commutator results. We take this opportunity to correct a minor error in the proof [14] of Lemma 3. In Chapter 5 we prove the Lagrangian-Eulerian local existence theorem. The solution paths belong to L∞ (0, T ; C α ∩ L p ) for 0 < α < 1, 1 < p < d . The nonlinearity is Lipschitz in these path spaces, and the solution depends in Lipschitz manner on initial data. The proof is adapted to show global existence of solutions with small initial data in damped versions of the equations. The smallness is measured in low regularity spaces and is explicitly set by the size of the damping. Chapter 6 discusses critical dissipative active scalars. The motivation here comes from work on SQG. The critical dissipative SQG is obtained from SQG by adding square-root Laplacian dissipation. This was proved to have global smooth solutions independently by Caffarelli and Vasseur [5] and by Kiselev, Nazarov, and Volberg [36]. The proofs are different in spirit. The proof of [5] uses a harmonic extension and a de Giorgi methodology. The proof of [36] uses an invariant family of moduli of continuity. Other proofs exist [37]. An extension of an inequality of Córdoba and Córdoba [25] providing a nonlinear lower bound for the fractional Laplacian [22] was used for yet a different proof. The proof we describe in these notes is based on the one in [21], where it was used to study long time behavior of forced critical SQG. Global regularity can be ob-

Chapter 1. Introduction

3

tained also for critical modified SQG equations [17] and for slightly supercritical SQG equations [28], [29], [48]. The problem of global existence of smooth solutions for supercritical SQG is open. Solutions of supercritical drift-diffusion s equations whereby the dissipation is produced by a power 2 of the Laplacian with s < 1 are Hölder continuous with small exponent if the advecting velocity is u ∈ C α , with α = 1 − s [24], [43]. This condition is sharp in the sense that there exist linear drift diffusion equations with drift of lower regularity than C 1−s for which the solutions loose continuity in finite time [45]. Higher regularity is obtained if u ∈ C α with α > 1 − s [23], [31]. Thus, in the critical case s = 1, any C α regularity with α > 0 implies full regularity. All weak solutions the supercritical SQG become regular after a finite time [30], [38], [44], [27]. In these notes we present a unified proof of the unconditional regularity of critical and conditional regularity of supercritical active scalars using pointwise lower bounds on the fractional Laplacian.

Chapter 2

Lagrangian and Eulerian descriptions of hydrodynamic systems

In this chapter we review basic descriptions of hydrodynamic models, introduce notation, and derive general kinematic relationships valid for smooth solutions. The subject is classical; among the many reference texts we mention [2], [9], and [40]. The Lagrangian description is based on the flow maps X (a, t ), a ∈ Rd , t ∈ R, whereby the particle paths t 7→ X (a, t )

(2.1)

give rise to time-dependent diffeomorphisms a 7→ X (a, t ) at fixed time, X (·, t ) : Rd → Rd ,

(2.2)

starting at t = 0 from the identity map, X (a, 0) = a.

(2.3)

The time derivative along a particle path X (a, t ) is the Lagrangian velocity, v(a, t ) = ∂ t X (a, t ),

(2.4)

and it is related to the Eulerian velocity u(x, t ), u(·, t ) : Rd → Rd ,

(2.5)

by composition with the flow map v(a, t ) = u(X (a, t ), t ).

(2.6)

We make a distinction between objects defined locally in terms of the flow maps X (a, t ), and objects determined in terms of the Eulerian velocity u(x, t ). The former are Lagrangian: they trace back information to the original positions of the particles, the labels a. The latter are Eulerian: they represent properties recorded at a fixed position x at different times. The family of maps X obeys ordinary differential equations (ODEs) at each fixed label a: d X (a, t ) = u(X (a, t ), t ) dt 5

(2.7)

6

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

with initial data given by (2.3). Differentiating this basic ODE at fixed label a, we obtain d (2.8) (∇a X (a, t )) = (∇u(X (a, t ), t ))(∇a X (a, t )), dt which is also an ODE at fixed label a. We denote g (a, t ) = (∇u)(X (a, t ), t ).

(2.9)

From (2.8) it follows that the Jacobian j (a, t ) = det ∇X (a, t ) obeys d j (a, t ) = (div u(X (a, t ), t )) j (a, t ), dt

(2.10)

div u = T r (∇u),

(2.11)

where

and integrating in time we have therefore j (a, t ) = e

Rt 0

d i v u(X (a,s),s )d s

.

(2.12)

This shows that if div u ∈ L1 (d t ; L∞ (d x)), then det ∇X is positive and controlled a priori, and the map a 7→ X (a, t ) is locally injective. We recall here the elementary but fundamental transport lemma (see, for instance, [9]). Lemma 1. Let u(x, t ) ∈ L∞ ([0, T ],W 1,∞ (Rd )). Let f ∈ L∞ ([0, T ], L1 (Rd )) ∩ C 1 (Rd ×R) be given. Let Ω0 ⊂ Rd , and let Ω t = {x | ∃a ∈ Ω0 , x = X (a, t )}. Then Z Z d (∂ t f + ∇ · (u f ))(x, t )d x. f (x, t )d x = d t Ωt Ωt Proof. We start by changing variables in the integral Z Z f dx = ( f ◦ X ) det(∇a X )d a, Ωt

Ω0

and then we differentiate in time, R d R f d x = Ω [∂ t f (X (a, t ), t ) + ∂ t X (a, t ) · ∇ x f (X (a, t ))] det ∇a X (a, t )d a d t Ωt 0 R R d + Ω f d t det(∇a X )d a = Ω [(∂ t f + u · ∇ f + (∇ · u) f ) ◦ X ] det(∇a X )d a 0 R 0 = Ω [∂ t f + ∇ · (u f )] d x, t

and this ends the proof of the lemma. Global invertiblity of X (·, t ) can be proved as well. Lemma 2. If u ∈ L1 ([0, T ],W 1,∞ (R3 )), then X (a, t ) is invertible for t ∈ [0, T ]. Proof. First of all, clearly, from the implicit function theorem, because ∇a X is invertible, it follows that the flow map is locally injective. Now consider R > RT kukL∞ (R3 ) d t . Let K b be the closed ball of radius R around b . Notice that if 0 a∈ / K b , then the equation X (a, t ) = b does not have any solution for 0 ≤ t ≤ T ,

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

simply because X (a, t ) starts at a and travels at most It follows that the number

RT 0

7

kukL∞ (R3 ) d t far from a.

#{a | X (a, t ) = b } = #{a ∈ K b | X (a, t ) = b } = n b (t ) is finite, continuous in t , and locally constant in t . Indeed, the finiteness follows at fixed t from the implicit function theorem, because the set of solutions is discrete. In order to show continuity, we consider any fixed time t0 and a fixed solution X (a0 , t0 ) = b . Again, by the implicit function theorem, there exists a unique path α(t ) defined for |t − t0 | small, passing at t = t0 through a0 , and satisfying X (α(t ), t ) = b . This shows that n b (t ) is continuous in time. Therefore, it always takes the value that it takes at t = 0; i.e., n b (t ) = 1 for any b . Note that we did not need div u = 0. Alternatively, we may look at the PDE obeyed by X −1 (x, t ). Because a 7→ X (a, t ) is a map from Lagrangian coordinates called labels to Eulerian coordinates x, it is convenient to denote, when no confusion can arise, X −1 (x, t ) = A(x, t )

(2.13)

and call it the “back-to-labels” map. The back to labels map obeys the PDE

with initial data

∂ t A + u · ∇A = 0

(2.14)

A(x, 0) = x.

(2.15)

This is easily seen by differentiating A(X (a, t ), t ) = a in time and reading at x = X (a, t ). We consider

and note that

`(x, t ) = A(x, t ) − x

(2.16)

∂ t ` + u · ∇` + u = 0

(2.17)

`(x, 0) = 0.

(2.18)

with initial data

The solution of the PDE locally exists and is unique by the method of characteristics, and this can be used to define the inverse when u ∈ C 1 . We now consider variations of X : we take a family Xε of flow maps dependending smoothly on a real parameter ε. We denote by Aε their inverses, Aε = Xε−1 . We denote uε = ∂ t Xε ◦ Xε−1 (2.19) and Xε0 = We start by computing 

d X, dε ε

d X −1 . dε ε

ηε = Xε0 ◦ Xε−1 .

Differentiating a = Xε−1 (Xε (a)) we obtain

‹ d −1 Xε (Xε (a)) + (∇ x Xε−1 )(Xε (a))Xε0 (a) = 0, dε

(2.20)

8

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

and reading at a = Xε−1 (x) we have d −1 X = −ηε · ∇Xε−1 . dε ε

(2.21)

d u: dε ε

Armed with this we compute d u dε ε


d = d ε ∂ t Xε ◦ Xε−1 d = ∂ t Xε0 ◦ Xε−1 + (∇a (∂ t Xε ◦ Xε−1 )) d ε Xε−1 = ∂ t Xε0 ◦ Xε−1 − (∇a (∂ t Xε ◦ Xε−1 ))ηε · ∇Xε−1 = ∂ t Xε0 ◦ Xε−1 − ηε · ∇uε . Now we note that ∂ t Xε0 ◦ Xε−1 = (∂ t + uε · ∇)ηε .

(2.22)

Indeed, this follows because (∂ t + uε · ∇)(Xε0 ◦ Xε−1 ) = ∂ t Xε0 ◦ Xε−1 + ∇a Xε0 ◦ Xε−1 (∂ t + uε · ∇) (Xε−1 ) and (∂ t + uε · ∇)Xε−1 = 0, which is just (2.14), the time derivative of a = Xε−1 (Xε (a)) read at a = Xε−1 (x). In our notation we thus have d A + ηε · ∇Aε = 0 dε ε

(2.23)

and, introducing the notation D tε = ∂ t + uε · ∇,

(2.24)

d u = D tε ηε − ηε · ∇uε . dε ε

(2.25)

This equation can be used to recover the variations Xε from knowledge of the d Eulerian family uε . Indeed, given uε (x, t ) and d ε uε and solving (2.25) as a PDE with initial data ηε (x, 0) = 0 we obtain ηε (x, t ). Solving the ODE (2.20) d (X (a, t )) = ηε (Xε (a, t )) dε ε

(2.26)

with initial data Xε (a, t )| ε=0 = X0 (a, t ) we obtain the variations Xε , and solving the PDE (2.23) with initial data Aε (x, t )| ε=0 = A0 (x, t ) we obtain the variations Aε = Xε−1 . Note that differentiating in a the ODE (2.26) we have d (∇ X ) = (∇ x ηε ◦ Xε )(∇a Xε ) dε a ε and consequently d det(∇a Xε ) = (div ηε ) det(∇a Xε ). dε

(2.27)

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

9

We see thus that Xε are volume preserving for all ε if one of them is, if and only if ηε is divergence-free. Now differentiating the ODE d X (a, t ) = uε (Xε (a, t ), t ), dt ε

(2.28)

which is just (2.19), it follows that d (∇ X ) = (∇ x uε ◦ Xε )(∇a Xε ), dt a ε

(2.29)

which is the analogue of (2.8), and then d det (∇a Xε ) = (div uε ) det (∇Xε ) , dt

(2.30)

which is the analogue of (2.10). In view of (2.30) it follows that Xε is volume preserving if and only if uε is divergence-free. These were purely kinematic considerations, true for any smooth family of diffeomorphisms. We can use them to formally derive dynamical equations from action principles. Let us review briefly here the derivation of the incompressible Euler equations ∂ t u + u · ∇u + ∇ p = 0, (2.31) ∇· u =0 in Rd . Let Aε =

1 2

Z

t2 t1

Z Rd

(2.32)

|uε |2 d x d t

be the action on a flow map Xε . We fix t1 < t2 and impose the condition that the end points are fixed: Xε (a, ti ) = X0 (a, ti ), i = 1, 2. This is so that we do not incur boundary terms when we integrate by parts in t . Note, however, that this also means that we are not discussing all initial value problems but rather maps which transform one fixed diffeomorphism into another fixed diffeomorphism. We make the assumption that the base flow is volume preserving, det ∇a X0 = 0, but the deformations Xε need not be volume preserving for ε 6= 0. The action principle states that the Euler equations are obtained by seeking least action among all volume preserving diffeomorphisms [2]. We treat “volume preserving” as a side constraint in a variational principle. Note that div u = 0 is also necessary and not only sufficient for det(∇a X ) = 1 in view of (2.10). We use a Lagrange multiplier q that is independent of ε, enlarge the action, Aε =

1 2

Z

Z

t2

t2

Z Rd

t1

1 |u |2 + q(div uε )d x d t , 2 ε

compute the variation d dε

t1

Z Rd

1 |u |2 + q(div uε )d x d t , 2 ε

10

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

and require it to vanish at ε = 0. Stationarity with respect to q simply means div uε| ε=0 = 0. The calculation thus gives, in view of (2.25), d dε

R t2 R

1

|uε |2 + q(div uε )d x d t d Š R tt21 R R €2 d d = t Rd uε · d ε uε − ∇q · d ε uε d x d t R t1 R = t 2 Rd (uε − ∇q)(D tε ηε − ηε · ∇uε )d x d t . 1

Now − = +

R t2 R Rtt12 RR

d d

(∇q(ηε · ∇uε ) + ηε · ∇D tε q − ηε · (∇uε )∗ ∇q)d x d t

d

(div uε )(ηε · ∇q)d x d t

R tt12 R R

=

Rtt12 RR t1

∇q(D tε ηε − ηε · ∇uε )d x d t

Rd

[ηε · ∇D tε q + (div uε )(ηε · ∇q)] d x d t ,

and therefore 1 d R t2 R |u |2 + q(div uε )d x d t d ε t1 R d 2 ε R t2 R = t Rd [uε · (D tε ηε − ηε · ∇uε ) + ηε · ∇D tε q + (div uε )(ηε · ∇q)]d x d t R t1 R = t 2 Rd [ηε · (−D tε uε − (∇uε )∗ uε + ∇D tε q) + (div uε )(ηε · (∇q − uε ))] d x d t 1R t R = − t 2 Rd ηε · [(D tε uε + ∇ pε ) + (div uε )(ηε · (uε − ∇q)))] d x d t , 1

where

1 pε = |uε |2 − D tε q. 2 Now we can set ε = 0, use div u0 = 0, let η0 be arbitrary, and deduce by equating the ε derivative of the action to zero for all η0 that u must obey the Euler equations (2.31). In order to derive the Euler equations using deformations Xε which are themselves volume preserving, we do not need the Lagrange multiplier q, and div uε = |u|2 0 is then true for nonzero ε. It follows then that D t u +∇ 2 is perpendicular on all divergence-free vectors, which implies that it is a gradient, and (2.31) follows. Let us use this simple strategy for the action: Z 1 t2 〈Luε , uε 〉d t Bε = 2 t1

with a self-adjoint operator L, with 〈 , 〉 the L2 (Rd ) scalar product. We consider volume-preserving deformations. Because div uε = 0 it follows that D tε is formally skew symmetric under both time and space integration, (D tε )∗ = −D tε , and we obtain, using (2.25), Rt d d Bε = t 2 〈Luε , d ε uε 〉d t dε 1 R t2 Rt = t 〈Luε , D tε ηε − ηε · ∇uε 〉d t = t 2 〈(D tε )∗ (Luε ) − (∇uε )∗ Luε , ηε 〉d t 1R 1 t = − t 2 〈(D tε + (∇uε )∗ )Luε , ηε 〉d t . 1

The equation resulting from stationarity of Bε to volume-preserving deformations is therefore ∂ t Lu + u · ∇Lu + (∇u)∗ Lu + ∇ p = 0,

div u = 0.

2.1. Incompressible Euler equations

11

If L = I, we recover the Euler equations (2.31); if L = −∆ + α2 , this is the “α 1 model” [32], [41]. If L = −(−∆)− 2 , we obtain SQG in d = 2, and if L is a more general fractional operator, we obtain generalized SQG-type equations [7], [10]. Denoting v = Lu and using the fact that L commutes with differentiation we have explicitly the equations ∂ t vi + uk ∂k vi + (∂i uk )vk + ∂i p = 0,

∂i vi = 0

(2.33)

(where i = 1, . . . , d and we use summation convention). The equations are coupled with the equation of state u = L−1 v. The Euler equations and the α model are written in this form. In order to recognize 2D SQG in (2.33) we consider the antisymmetric matrix Š 1€ Ji j = ∂ j vi − ∂i v j 2 and observe that it follows from (2.33) that ∂ t J + u · ∇J + J (∇u) + (∇u)∗ J = 0. In two dimensions J is a scalar multiple of the antisymmetric matrix ‹  0, −1 , I= 1, 0 1

J = 2 θI with θ(x, t ) = ∂1 v2 − ∂2 v1 . Also, in two dimensions, the term J (∇u) + (∇u)∗ J vanishes identically. (This only uses the fact that the matrix ∇u is traceless.) This leaves a pure transport equation, ∂ t θ + u · ∇θ = 0,

(2.34)

which is 2D SQG. Indeed, because v = −Λ−1 u and u = ∇⊥ ψ, it follows that 1 u = ∇⊥ (−∆)− 2 θ.

2.1 Incompressible Euler equations Classical solutions of the incompressible Euler equations (2.31) in d = 3 have some properties which we briefly review below. Most of the results in this section are well known. We refer to [40], [13] for some of them and more references. The simple result on control of blow up by the Hessian of the pressure has not appeared elsewhere, to our knowledge. Smooth solutions of the Euler equations conserve kinetic energy, Z d |u(x, t )|2 d x = 0, 2d t R3 as is easily seen by noting that (u · ∇u + ∇ p) · u is a divergence and assuming enough regularity and decay for the integral of this divergence to vanish. The vorticity ω(x, t ) = ∇ × u(x, t ) is an important field associated with the velocity. The gradient of velocity can be decomposed in its symmetric and anti-symmetric parts, ∇u = S + J ,

12

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

where S=

1 ((∇u) + (∇u)∗ ) 2

J=

1 ((∇u) − (∇u)∗ ) 2

is the rate of strain and

is given by the vorticity; i.e., for any vector v, the matrix J applied to v yields 2J v = ω × v. Taking the gradient of (2.31) we obtain the equation for the gradient matrix: (∂ t + u · ∇) (∇u) + (∇u)2 + (∇∇ p) = 0.

(2.35)

The rate of strain obeys (∂ t + u · ∇) S + S 2 + J 2 + (∇∇ p) = 0, where we can find that

Ò= with ω

ω |ω|

1 ⊥ J 2 = − |ω|2 PωÒ 4

(2.36)

(2.37)

and where Pξ⊥ v = v − (v · ξ )ξ

for ξ ∈ S 2 and v ∈ R3 . The equation obeyed by J is (∂ t + u · ∇) J + SJ + J S = 0,

(2.38)

(∂ t + u · ∇) ω − ω · ∇u = 0.

(2.39)

and in terms of ω this is

This equation is equivalent to the commutator equation [∂ t + u · ∇, ω · ∇] = 0.

(2.40)

The fact that the commutator vanishes is the essence of a basic hydrodynamic fact, Ertel’s theorem. Ertel’s theorem says that if k is a constant of motion, i.e., D t k = 0, then ω · ∇k is also a constant of motion, D t (ω · ∇k) = 0, a fact that follows immediately from the commutation (2.40). A consequence of this is the fact that ω3 = u2,1 − u1,2 is conserved under 2D flow, i.e., flow for which D t x3 = 0.

2.1. Incompressible Euler equations

13

One of the most important conservation laws in fluid mechanis is the conservation of circulation. This says that I u ·dx (2.41) γt

is constant in time, where γ t = X (γ0 , t ) is a closed loop transported by the flow. The proof of this fact is easy in Lagrangian coordinates. If γ0 is given by a parameterization a = α(s ) with s ∈ [0, 1], with α(0) = α(1), then γ t is given by x = X (α(s), t ). The integral is I γt

u ·dx =

Z

1

∂ t X j (α(s), t )(∇a X j (α(s ), t ))

0

dα d s. ds

The Euler equations in Lagrangian form are just d2 X (a, t ) = −∇ x p(X (a, t ), t ), d t2

(2.42)

where ∇ x is the Eulerian gradient. Differentiating we obtain Š d € 1 ∂ t X · ∂ak X = −∇ x p · ∂ak X + ∂ak |∂ t X |2 = ∂ak q dt 2 with q = −p ◦X + Therefore d dt

I γt

u ·dx =

Z

1

0

1 |∂ X |2 . 2 t

d q(X (α(s)), t )d s = 0 ds

because the end points coincide. The Cauchy formula is ω(X (a, t ), t ) = (∇a X (a, t ))ω0 (a).

(2.43)

This is easily verified by checking that e t ) = ω(X (a, t ), t ) ω(a, and

ζ (a, t ) = ∇a X (a, t )ω0 (a)

obey the same ODE (in view of (2.8) and (2.39)) and have the same initial data, so they must coincide. An immediate and important consequence of the Cauchy formula is the Helmholtz theorem, which states that vortex lines are material. Vortex lines are integral curves of the vector field ω(·, t ) at fixed time; that is, they are curves in space, such that the tangent at each point on the curve is parÒ at that point. The fact that they are material means that allel to the direction ω they are transported by the flow X (·, t ); i.e., the image of a vortex line under X is again a vortex line. This is clear from the Cauchy formula because if γ0 is curve a = α(s) parameterized by some parameter s ∈ [0, 1] and if it is a vordα tex line, then d s = c(s)ω0 (α(s )) with some constant c(s). At later time γ t is

14

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems dα

given by x = X (α(s), t ), and therefore its tangent is given by τ(s) = (∇a X ) d s . This is then τ(s ) = c(s)(∇a X )ω0 (α(s)), and by the Cauchy formula τ(s ) = c(s)ω(X (α(s), t ), t ). The Cauchy invariants are εi j k X˙jl Xkl = ω0i ,

i = 1, 2, 3,

(2.44)

where εi j k is the signature of the permutation (1, 2, 3) 7→ (i, j , k), taken to be zero if two indices are the same, and we denote for graphical ease X˙kl = ∂ t ∂ak X l (a, t ),

Xkl = ∂ak X l (a, t ),

ω0 = ω0 (a, t ).

The Cauchy invariants are a direct consequence of the Cauchy formula. Here is the very short proof. We start with the useful formula 1 p r Ans , Xk = ε p mn εk r s Am 2

(2.45)

where we denote Aim the matrix elements of (∇ x X −1 ) ◦ X , which is the inverse of the matrix ∇a X . The identity above uses the fact that the determinant equals one. From this we obtain the formula j

p

ε mn p Aim An = εi j k Xk . Indeed,

p

(2.46)

1

r εi j k Xk = 2 ε p mn εi j k εk r s Am Ans € Š 1 r = 2 ε p mn δi r δ j s − δi s δ j r Am Ans j

= ε p mn Aim An . Now, after applying the inverse matrix Aim to both sides of the Cauchy formula, we have p j ω0i = Aim ω m = Aim ε mn p ∂n (u p ) = Aim ε mn p X˙ j An p p = εi j k X˙ j Xk by (2.46), and that gives the Cauchy invariant. The helicity is the integral Z (u · ω)d x. R3

This is constant in time. Indeed, using the transport lemma, (2.31), and (2.39) we have    Z Z |u|2 d (u · ω)d x = div ω − p + d x = 0. d t R3 2 R3 There exist local versions of this (on vortex tubes, i.e., on regions whose boundaries are foliated by vortex lines). A magnetization variable is a vector w that obeys ∂ t w + u · ∇w + (∇u)∗ w = 0. For any such variable, the scalar k = w · ω is conserved: (∂ t + u · ∇)(w · ω) = 0.

2.1. Incompressible Euler equations

15

The Weber formula is u = P((∇A)∗ u0 (A))

(2.47)

Here the Leray–Hodge matrix of operators P is given by P j l = δ j l − ∂ j ∆−1 ∂ l = δ j l + R j R l , with

(2.48)

1

R j = ∂ j (−∆)− 2

the Riesz operators. Note that P satisfies the basic property that P j l ∂ l f = 0. The derivation of the Weber formula is as follows. We start with (2.42) and apply (∇a X )∗ : ∂ 2 X (a, t ) (2.49) (∇a X (a, t ))∗ = −(∇a ˜p )(a, t ) ∂ t2 or, on components, ∂ ˜p (a, t ) ∂ 2 X j (a, t ) ∂ X j (a, t ) =− , 2 ∂t ∂ ai ∂ ai

(2.50)

˜p (a, t ) = p(X (a, t ), t ).

(2.51)

where Pulling out a time derivative in the left-hand side we obtain   ˜ t) ∂ q(a, ∂ ∂ X j (a, t ) ∂ X j (a, t ) =− , ∂t ∂t ∂ ai ∂ ai where ˜ t ) = ˜p (a, t ) − q(a,

1 ∂ X (a, t ) 2 . 2 ∂t

(2.52)

(2.53)

We integrate (2.52) in time, fixing the label a: ∂ X j (a, t ) ∂ X j (a, t ) ∂ n˜(a, t ) i = u(0) , (a) − ∂t ∂ ai ∂ ai where n˜(a, t ) =

(2.54)

t

Z

˜ s)d s q(a,

(2.55)

0

and u(0) (a) =

∂ X (a, 0) ∂t

(2.56)

is the initial velocity. Note that n˜ has dimensions of circulation or of kinematic viscosity (length squared per time). The conservation of circulation I I ∂ X (γ , t ) ∂ X (γ , 0) · dγ = · dγ ∂t ∂t γ γ

16

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

follows directly from the form (2.54). Applying [(∇a X )∗ ]−1 to (2.54) and reading at a = A(x, t ), we obtain the formula € j Š ∂ A j (x, t ) ∂ n(x, t ) − , u i (x, t ) = u(0) (A(x, t )) ∂ xi ∂ xi

(2.57)

n(x, t ) = n˜(A(x, t )).

(2.58)

where

Because u is divergence-free, the Weber formula follows. The equation (2.57) shows that the general Eulerian velocity can be written in a form that generalizes the Clebsch variable representation: u = (∇A)∗ B − ∇n,

(2.59)

where B = u(0) (A(x, t )) and, consequently, D t B = 0, because of the basic (2.14). Conversely, and somewhat more generally, if one is given a pair of M -uples of active scalars A = (A1 (x, t ), . . . , AM (x, t )) and B = (B 1 (x, t ), . . . , B M (x, t )) of arbitrary dimension M , such that the active scalar equations D t Ai = D t Bi = 0 hold, and if u is given by u(x, t ) =

M X

B k (x, t )∇ x Ak (x, t ) − ∇ x n

(2.60)

k=1

with some function n, then it follows that u solves the Euler equations ∂u + u · ∇u + ∇π = 0, ∂t where

1 π = D t n + |u|2 . 2 Indeed, the only thing one needs is the kinematic commutation relation D t ∇ f = ∇D t f − (∇u)∗ ∇ f ,

(2.61)

which holds for any scalar function f . The kinematic commutation relation (2.61) is a consequence of the chain rule, so it requires no assumption other than smoothness. Differentiating (2.60) and using the active scalar equations it follows that M X ((∇ x u)∗ ∇ x Ak )B k − ∇ x (D t n) + (∇ x u)∗ ∇n D t (u) = − k=1 ∗

= −∇ x (D t n) − (∇ x u)

– M X

™ k

k

(∇ x A )B − ∇ x n

k=1 ∗

= −∇ x (D t n) − (∇ x u) u = −∇ x (π). Clebsch variables are obtained for M = 1. Note that for Clebsch variables (B 1 , A1 ) the vorticity is given by ω = ∇B 1 × ∇A1 ,

2.1. Incompressible Euler equations

17

and thus the helicity vanishes. Not all flows have zero helicity, and thus Clebsch variables do not represent all flows. We now discuss briefly the blow up issue. We start by expressing the gradient of velocity in terms of the vorticity. Let u be divergence-free and smooth enough. ω =∇× u (2.62) implies that ∇ × ω = −∆u because ∇ · u = 0, and therefore Z Š 1 € 1 ∇y × ω(y) d y, u(x) = 4π R3 |x − y|

(2.63)

and integrating by parts we obtain 1 u(x) = − 4π

Z R3

x−y × ω(y)d y, |x − y|3

(2.64)

which is the Biot–Savart law. We differentiate (2.63) carefully: 1 R 1 ∂ j ui (x) = εi k l 4π R3 ∂ x j |x−y| ∂yk ω l (y)d y R x −y 1 j j = −εi k l 4π R3 |x−y| 3 ∂ y ω l (y)d y k R x j −y j 1 = − limε→0 εi k l 4π |x−y|≥ε |x−y| 3 ∂ ω (y)d y ” € xyk−y l — Š € x −y Š 1 R j j j j ω l (y) d y = − limε→0 εi k l 4π |x−y|≥ε ∂yk |x−y| 3 ω l (y) − ∂ y |x−y|3 k 1 R = −εi k l ω l (x) 4π |x−y|=1 (xk − yk )(x j − y j )d S R σk j ( xÔ −y) 1 +εi k l 4π PV R3 |x−y| 3 ω l (y)d y,

where σi j (ξ ) = 3ξi ξ j − δi j and we used the notation b x=

(2.65)

x . |x|

Thus, we have 1 1 ∂ j ui (x) = − εi j l ω l + εi k l PV 3 4π

Z

σk j ( Ö x − y)

R3

|x − y|3

ω l (y)d y.

(2.66)

Using ε m j i εi j l = −2δ m l and ε m j i εi k l σk j = σ m l − δ m l σ j j = σ m l , we check 2 1 ω m (x) = ε m j i ∂ j ui = ω m (x) + PV 3 4π

Z R3

σm l (Ö x − y) ω l (y)d y, |x − y|3

which is true because 1 PV 4π

Z R3

σm l (Ö x − y) 1 ω l (y)d y = ω m (x), |x − y|3 3

(2.67)

18

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

a fact that follows from Pω = ω where P is the projector on divergence-free functions, and 2 1 Pi j v j = δi j v j + PV 3 4π

Z R3

σi j ( Ö x − y) |x − y|3

v j (y)d y.

The rate of strain

Š 1€ ∂i u j + ∂ j ui (2.68) 2 is expressed in terms of ω using (2.66): ˜    R • xÔ −y 3 xÔ −y Si j = 8π PV R3 |x−y|3 × ω(y) ( Ö x − y) j + |x−y|3 × ω(y) ( Ö x − y)i d y. Si j =

i

j

(2.69) In particular (Sω · ω)(x) =

3 PV 4π

Z R3

(ω(x) · b z ) det[b z , ω(x − z), ω(x)]

dz |z|3

(2.70)

is driving the evolution of |ω|2 : 1 D |ω|2 = Sω · ω. 2 t We have thus a stretching factor Z |ω(x − z)|d z 3 Ò ·b Ò − z), ω(x)] Ò α(x) = PV (ω(x) z ) det[b z , ω(x 4π |z|3 R3

(2.71)

(2.72)

for the evolution of the magnitude of vorticity D t |ω| = α|ω|.

(2.73)

ω

Ò = |ω| is regular (Hölder continuous), then α is not a very singular Note that if ω Ò is constant, α vanishes. The stretching is quasiintegral of |ω|. In particular, if ω local, meaning that the contributions to (2.70) coming from distances farther away than some r cannot lead to finite time singularities: Z  ‹ 5 |z| d z φ (ω(x) · b z ) det[b z , ω(x − z), ω(x)] ≤ C r − 2 kukL2 (R3 ) |ω(x)|2 |z|≥r r |z|3 pointwise, as is easily seen by taking a smooth cutoff φ and integrating by parts using the fact that ω(x −z) is the curl of u. Because kukL2 (R3 ) is bounded a priori, it follows that this term can lead at most to exponential growth in (2.71). In fact, 5 r can depend on time, and as long as r − 2 is time integrable, this term cannot lead to finite time blow up of |ω|. Thus, if the vortex lines are locally aligned, then blow up is prevented: a straight narrow vortex tube is not a blow up candidate. The blow up requires the underlying geometry of vortex lines to be nontrivial. We refer to this as geometric depletion of nonlinearity [11], [12]. Singularity formation in the Euler equation requires the magnitude of vorticity to become infinite. (Of course, the gradient of vorticity needs to diverge

2.1. Incompressible Euler equations

19

as well, as can be seen from (2.70).) In order to consider higher derivatives we recall an extrapolation estimate. Theorem 1. Let ω ∈ C γ (R3 ) ∩ L p (R3 ) with 0 < γ < 1 and p < ∞, and let the matrix G be defined by the right-hand side of (2.66), i.e., 1 1 PV Gi j (x) = − εi j l ω l + εi k l 3 4π

σk j ( Ö x − y)

Z R3

|x − y|3

ω l (y)d y.

(2.74)

We introduce the notation [ω]γ = sup x6=y

|ω(x) − ω(y)| |x − y|γ

(2.75)

for 0 < γ < 1. Then §  p 1 ‹ª 1 p − − kGkL∞ (R3 ) ≤ C kωkL∞ (R3 ) 1 + log 1 + [ω]γγ kωkL3 p (R3 ) kωkL∞3 (Rγ3 ) .

(2.76)

3

In particular, if ω ∈ W 1, p (R3 ), p > 3, or if ω ∈ W s ,2 (R3 ), s > 2 , then we have kGkL∞ (R3 ) ≤ C kωkL∞ (R3 ) (1 + log (2 + kωkB )),

(2.77)

where kωkB refers to the norm in either W 1, p (R3 ) or W s ,2 (R3 ). Proof. We split the integral in (2.74) in three pieces: an inner piece I1 on |x −y| ≤ a, a medium piece I2 on a ≤ |x − y| ≤ b , and an outer piece I3 on |x − y| ≥ b . For the inner piece we use the fact that the averages of σk j on the unit sphere vanish to write Z σk j (b z) 1 I1 (x) = εi k l PV (ω l (x − z) − ω l (x)))d y, 3 4π |z|≤a |z| and we deduce that

|I1 (x)| ≤ C1 [ω]γ a γ .

For I2 we have obviously |I2 (x)| ≤ C2 kωkL∞ log

 ‹ b . a

For the last piece we obtain 3

|I3 (x)| ≤ C3 b − p kωkL p . −

1

1

p

−

p

Now we choose a = [ω]γ γ kωkLγ ∞ and b = kωkL3 p kωkL∞3 + a. This gives   p 1 1 ‹‹ p − − |I (x)| ≤ kωkL∞ C1 + C3 + C2 log 1 + [ω]γγ kωkL3 p kωkL∞3 γ . This proves (2.76), which is the nondimensional version of the inequality. The 3 inequalities (2.77) follow from the fact that [ω]γ ≤ C kωkB with γ = 1 − p , 3

respectively, γ < s − 2 . This concludes the proof.

20

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems

Let us denote H s = W s,2 (R3 ). Let us observe that for divergence-free functions k∇ukH s = kωkH s (2.78) because

ub(ξ ) = ξb × (ξb × ub(ξ )) = Pξ⊥ (b u (ξ ))

and

|ξb × v| = |v|

if v ·ξ = 0. The celebrated Beale–Kato–Majda theorem [3], which states that the magnitude of vorticity alone controls singularity formation, is as follows. 3

Theorem 2. Assume ω0 ∈ H s , s > 2 . Assume that ω ∈ C ([0, T ), H s ) solves the 3D Euler equation (2.39) with (2.64) and that Z

T

kωkL∞ d t = A(T ) < ∞.

0

Then there exists a constant C depending only on the initial data such that sup kω(·, t )kH s ≤ kω0 kH s exp (C exp(C A(T ))). t ≤T

Proof. The proof starts by showing that   Zt kω(·, t )kH s ≤ C kω0 kH s exp c [k∇ukL∞ + kωkL∞ ] . 0

This is obtained by bounding the evolution of the square of the H s norm using well-known calculus inequalities k f g kH s ≤ C [k f kL∞ k g kH s + k g kL∞ k f kH s ]

(2.79)

and k∂ α ( f g ) − f ∂ α g kL2 ≤ C [k∇ f kL∞ k g kH s−1 + k g k∞ k f kH s ],

(2.80)

|α| ≤ s . Then use of the extrapolation inequality (2.77) leads to  Z t C b (s) log(2 + N (s))d s , N (t ) ≤ C N0 exp 0

where

N (t ) = kω(·, t )kH s

and b (t ) = kω(·, t )kL∞ . The theorem follows from the Grönwall inequality for log(2 + N ). RT Obviously, now that it is clear that 0 kωkL∞ d t controls blow up, it follows RT from (2.43) that 0 k∇X kL∞ d t controls blow up.

2.1. Incompressible Euler equations

21

Proposition 1. If u is a smooth solution of (2.31) with initial data u0 ∈ H s (R3 ), 5 u ∈ C ([0, T ), H s (R3 ), with s > 2 , and if Z

T

kω0 kL∞

k∇X kL∞ d t = B(T ) < ∞,

0

then there exists a constant C depending only on the initial data such that sup ku(·, t )kH s +1 ≤ ku0 kH s +1 exp (C exp(C B(T ))). t ≤T

Similarly, if the Hessian of the pressure satisfies Z

T

k∇∇ pkL∞ d t = C (T ) < ∞,

0

then

sup ku(·, t )kH s+1 ≤ ku0 kH s+1 E(C (T )), t ≤T

with E(C (T )) a triple exponential. Proof. The fact that the Hessian of the pressure controls blow up is not entirely obvious if one looks at (2.35) or (2.36). But the Hessian of the pressure controls ∇X , and this follows by differentiating the basic Lagrangian equations (2.42), ∂ t2 (∇X ) = −((∇∇ p) ◦ X )∇X , multiplying by ∂ t ∇X , and taking the trace:
d ˙ 2 ∇X = −T r ((∇∇ p) ◦ X )(∇X )(∇X˙ ) , 2d t

(2.81)

where ∇X˙ = ∂ t ∇X . This results in |∂ t ∇X (a, t )| ≤ |∇u0 (a)| +

Z

t

P (a, s)|∇X (a, s)|d s,

0

where

P (a, s ) = |∇∇ p(X (a, s), s)|,

and after one time integration we have   Zt |∇X (a, t )| ≤ t |∇u0 (a)| + P (a, s)|∇X (a, s)|d s + |I|.

(2.82)

0

Let us take for simplicity the maximum norm, so that |I| = 1. (It really does not matter which matrix norm we choose.) Then, denoting by F = F (a, t ) the term in parentheses in the right-hand side of (2.82), we have |∇X (a, t )| ≤ t F + 1 and

dF ≤ P (t )(t F + 1), dt

22

Chapter 2. Lagrangian and Eulerian descriptions of hydrodynamic systems Rt

and using the integrating factor e − 0 s P (s )d s we obtain   R Zt t |∇X (a, t )| ≤ 1 + t |∇u0 (a)| + |∇∇ p(X (a, s), s )|d s e 0 s|∇∇ p(X (a,s),s)|d s . 0

(2.83) This then bounds ω ◦X using (2.43) and we conclude the proof. Note that (2.83) depends locally on the Lagrangian history of the Hessian of pressure.

Chapter 3

Hydrodynamic models

In this chapter we present hydrodynamic models sharing a common LagrangianEulerian structure with the Euler equations. We start by describing the structure and its main features and then give examples of hydrodynamic models sharing this structure: the surface quasi-geostrophic equation of geophysics [33], the ideal incompressible porous medium equation using Darcy’s law [4], the ideal Boussinesq system of Rayleigh–Bénard convection [8], and the Oldroyd-B system of non-Newtonian fluid mechanics [39]. We chose to highlight these models because they are important: the global existence of smooth solutions for large data is not known for them, and they represent examples which illustrate phenomena of small scale creation in inviscid incompressible flows even in two dimensions. The Lagrangian-Eulerian formalism described below can be extended to apply to other models [14], in which time history integrals are present. The formalism is applicable in any dimension. The range of applicability of the method is limited to systems in which there is vector field transport coupled to nonlocal and nonlinear effects and possibly to diffusion. Some compressible systems can be addressed, but not general conservation laws.

3.1 The Lagrangian-Eulerian structure The Euler equations and related models can be presented as an evolution system §

∂ t X = U (X , τ), ∂ t τ = T (X , τ)

(3.1)

in terms of Lagrangian variables X and τ or as the fixed point equations 

Rt X (a, t ) = a + 0 U (X (s), τ(s))d s , Rt τ(a, t ) = τ0 (a) + 0 T (X (s), τ(s))d s .

(3.2)

The Eulerian variables are u = ∂ t X ◦ X −1

(3.3)

σ = τ ◦ X −1 .

(3.4)

and

23

24

Chapter 3. Hydrodynamic models

The maps X (·, t ) : Rd → Rd are diffeomorphisms of ambient space. Depending on the hydrodynamic model, τ and, correspondingly, σ could be vector or matrix valued functions of space and time. The variable τ obeys an ODE dτ = F (g , τ) dt

(3.5)

with g = ∇u ◦ X (see (2.9)). The Eulerian velocity is given in terms of σ by a fixed (model-dependent) operator

and the Eulerian gradient by

u = U(σ),

(3.6)

∇u = G(σ).

(3.7)

The Lagrangian nonlinearities are

and

U (X , τ) = U(τ ◦ X −1 ) ◦ X

(3.8)



T (X , τ) = F G τ ◦ X −1 ◦ X , τ .

(3.9)

We consider a differentiable one-parameter family of paths Xε , τε , with Eulerian form σε = τε ◦Xε−1 and initial data uε (0) and σε (0). We recall the notations (2.20) and we denote d τε , (3.10) τε0 = dε with Eulerian counterpart

δε = τε0 ◦ Xε−1

(3.11)

d uε (0) . dε

(3.12)

and 0 = uε,0

Differentiating U in (3.8) with respect to ε results in U 0 = ((∇U) (σε ) ◦ Xε ) Xε0 − U(ηε · ∇σε ) ◦ Xε + U(δε ) ◦ Xε .

(3.13)

The calculation uses the fact that, at fixed τ, d (τ ◦ Xε−1 ) = −∇(τ ◦ Xε−1 )(Xε0 ◦ Xε−1 ), dε which in turn follows from d Xε−1 = −(∇Xε−1 )(Xε0 ◦ Xε−1 ). dε Composing with Xε−1 from the right and dropping ε for ease of notation, we deduce from (3.13)

Here

U 0 ◦ X −1 = [η · ∇, U] (σ) + U(δ).

(3.14)

[η · ∇, U](σ) = η · ∇U(σ) − U(η · ∇σ)

(3.15)

3.1. The Lagrangian-Eulerian structure

25

is the commutator. Note that X 0 (0) = η(0) = 0, and therefore

δ(0) = τ00 = σ00

(3.16)

U 0 (0) = u00 .

(3.17)

Differentiating T in (3.9) with respect to ε we obtain where

T 0 = D1 F (g , τ)g 0 + D2 F (g , τ)τ 0 ,

(3.18)

g 0 = [∇G(σ) ◦ X ]X 0 − G(η · ∇σ) ◦ X + G(δ) ◦ X

(3.19)

is the ε derivative of Composing with X where

g = G(τ ◦ X −1 ) ◦ X .

−1

(3.20)

we obtain g 0 ◦ X −1 = [η · ∇, G](σ) + G(δ),

(3.21)

[η · ∇, G](σ) = η · ∇G(σ) − G(η · ∇σ).

(3.22)

Summarizing we have   U 0 ◦ X −1 = [η · ∇, U](σ) + U(δ), T 0 = D F (g , τ)g 0 + D2 F (g , τ)τ 0 ,  g 0 ◦ X −11 = [η · ∇, G](σ) + G(δ), where

η = X 0 ◦ X −1 ,

(3.23)

δ = τ 0 ◦ X −1 .

(3.24)

Differentiating U with respect to the Lagrangian independent variable (label) a we have
(∇a U )(a, t ) = G(τ ◦ X −1 ) ◦ X (∇X ) = g (a, t )(∇X )(a, t ), (3.25) and using the fact that

d dε

and label derivatives commute we have

0

∇a U (a, t ) = g 0 (a, t )(∇a X (a, t )) + g (a, t )(∇a X 0 (a, t )),

(3.26)

with g given in(3.20) above and g 0 given by (3.19). Remark 1. The presence of the commutators in (3.23) is the main reason why the equations are well behaved in path space. In the case of the Euler equations we take σ = ω, τ = ω ◦ X , and we see that Z x−y 1 U(ω)(x) = − × ω(y)d y (3.27) 4π R3 |x − y|3 and 1 1 G(ω)i j (x) = − εi j l ω l + εi k l PV 3 4π The ODE for τ is and thus

Z R3

σk j ( Ö x − y) |x − y|3

ω l (y)d y.

(3.28)

dτ = gτ dt

(3.29)

F (g , τ) = g τ.

(3.30)

26

Chapter 3. Hydrodynamic models

3.2 SQG The surface quasi-geostrophic equation (SQG) is an active scalar equation ∂ t θ + u · ∇θ = 0,

(3.31)

where θ(x, t ) is a real valued function of x ∈ Rd and t ∈ R. The constitutive relation for u in this active scalar equation is given by u = R⊥ θ,

(3.32)

where R is the vector of Riesz transforms 1

−2

Ri f (x) = ∂i (−∆)

f (x) = c PV

Z Rd

(xi − yi ) f (y)d y, |x − y|d +1

(3.33)

and R⊥ = M R, where M is a constant coefficient antisymmetrix matrix. The velocity is therefore automatically divergence-free. We take d = 2 for simplicity and v ⊥ = (−v2 , v1 ). The rotated gradient of θ obeys (∂ t + u · ∇) (∇⊥ θ)) = (∇u)(∇⊥ θ).

(3.34)

This is similar to the vorticity equation (2.39), and the same commutation relation as (2.40) holds: ” — ∂ t + u · ∇, ∇⊥ θ · ∇ = 0. (3.35) Its meaning is the same as in the Euler equation. The integral curves of ∇⊥ θ are material: they are carried by the flow. These integal curves are the level sets of θ. The system (3.34), the commutation (3.35), and their interpretation are true for all active scalars. The problem of global existence of smooth solutions is open. Conservation of the L2 norm of u holds, and a geometric depletion of nonlinearity operates. The Beale–Kato–Majda theorem holds, with k∇⊥ θkL∞ replacing kωkL∞ . The Cauchy formula is valid for all active scalars, appropriately reinterpreted: (∇⊥ θ) ◦ X = (∇X )(∇⊥ θ0 ).

(3.36)

For the Lagrangian formulation we take τ(a, t ) = (∇⊥ θ)(X (a, t ), t ),

(3.37)

which means that we took σ(x, t ) = ∇⊥ θ(x, t ).

(3.38)

U(σ)(x) = Λ−1 σ

(3.39)

The operators U and G are

with and

1

Λ = (−∆) 2

(3.40)

G(σ) = Rσ.

(3.41)

3.3. Porous medium

27

The ODE for τ is the same,

dτ = g τ, dt and therefore, as in the Euler equation case (3.30), F (g , τ) = g τ.

(3.42)

(3.43)

The SQG has been extensively studied, as mentioned in the introduction.

3.3 Porous medium The porous medium equation discussed here is an active scalar equation (3.31) with constitutive equation for u,  ‹ θ + R21 θ, u= , (3.44) R1 R2 θ which corresponds to u1 = θ+∂1 p, u2 = ∂2 p, and the divergence-free condition. The equation originates from a coupled system in which a scalar θ is transported by an incompressible velocity in a porous medium. The system u = ∇ p + θe1 ,

div u = 0

(3.45)

with e1 = (1, 0) represents the instantaneous response of the flow to friction due to the porous medium using Darcy’s law [4] and to gravity forces proportional to θ. All dimensional constants having to do with gravity and porosity have been set to 1. This system is not to be confused with the porous medium equation which is obtained in the compressible case, when an equation of state for the pressure yields via Darcy’s law a degenerate nonlinear diffusion equation for the density. The system (3.34) is valid; the commutation (3.35) is valid as well. The Cauchy formula (3.36) holds. We take Lagrangian τ, τ = (θ0 , ∇⊥ θ ◦ X ),

(3.46)

σ = (θ, ∇⊥ θ).

(3.47)

which means

The operators U and G are given by

and

The ODE for τ is

U1 (σ) = σ1 + Λ−1 R1 σ3 , U2 (σ) = −Λ−1 R1 σ2

(3.48)

G11 (σ) = σ3 + R21 σ3 , G12 (σ) = −σ2 + R1 R2 σ3 , G21 (σ) = −R21 σ2 , G22 (σ) = −R1 R2 σ2 .

(3.49)

d τ1 dt d τ2 dt d τ3 dt

= 0, = g11 τ2 + g12 τ3 , = g21 τ2 + g22 τ3

(3.50)

28

Chapter 3. Hydrodynamic models

with corresponding F 

 0 F (g , τ) =  g11 τ2 + g12 τ3  . g21 τ2 + g22 τ3

(3.51)

The problem of global existence of smooth solutions or finite time blow up is completely open. The Beale–Kato–Majda theorem holds, with ∇θ replacing ω. The system has been widely studied ([26] and references therein).

3.4 Boussinesq The Boussinesq system we discuss is   ∂ t u + u · ∇u + ∇ p = θe2 , ∇ · u = 0,  ∂ θ + u · ∇θ = 0 t in d = 2, with e2 = (0, 1), which leads to the vorticity formulation   (∂ t + u · ∇)ω = ∂1 θ, u = ∇⊥ (∆)−1 ω,  ∂ t θ + u · ∇θ = 0.

(3.52)

(3.53)

The equations (3.34), (3.35), and (3.36) hold. For the Lagrangian τ we take

which means

τ = (ω ◦ X , ∇⊥ θ ◦ X ),

(3.54)

σ = (ω, ∇⊥ θ).

(3.55)

U(σ) = −Λ−1 R⊥ σ1

(3.56)

Gσ = −(R ⊗ R⊥ )σ1 .

(3.57)

The operators U and G are

and The ODE for τ is

d τ1 dt d τ2 dt d τ3 dt

= τ3 , = g11 τ2 + g12 τ3 ,

(3.58)

= g21 τ2 + g22 τ3

with corresponding F  τ3 F (g , τ) =  g11 τ2 + g12 τ3  . g21 τ2 + g22 τ3 

(3.59)

The system (3.53) is a simplified version of the full system [8]: we consider the 2D situation and omit dissipative mechanisms. The global existence of smooth solutions of (3.53) is open. The system was studied by many authors ([6], [34],

3.4. Boussinesq

29

[35], [46] and references therein). It is relatively easy to to prove a Beale–Kato– Majda conditional regularity result, based on the assumption that Z

T

0

k∇θ(t )kL∞ (R2 ) d t = C (T )

(3.60)

is finite. This follows from an extrapolation inequality and calculus inequalities. The extrapolation inequality is based on the relation between u and ω, which can be expressed in Biot–Savart form: Z (x − y)⊥ 1 ω(y)d y. (3.61) u(x, t ) = 2π R2 |x − y|2 Differentiation of this relation leads to ω(x) 1 (∂ j ui )(x) = εji + P.V . 2 2π where

Ö Ó σ i j ( x − y)

Z R2

|x − y|2

ω(y)d y,

b Ó σ i j (ξ ) = ε j i − 2εk i ξ j ξk ,

(3.62)

(3.63)

x−y Ö x − y = |x−y| , and εi j is the sign of the permutation (1, 2) 7→ (i, j ) taken to be zero if i = j . This is the 2D analogue of (3.28). The extrapolation inequality, the analogue of Theorem 1, is as follows.

Theorem 3. Let ω ∈ C γ (R2 ) ∩ L p (R2 ) with 0 < γ < 1 and 1 < p. Then the matrix (3.62) obeys §  p ‹ª 1 1 p − − k∇ukL∞ (R2 ) ≤ C kωkL∞ (R2 ) 1 + log 1 + [ω]γγ kωkL2 p (R2 ) kωkL∞γ (R22 ) , (3.64) where [ω]γ is the Hölder seminorm (2.75). In particular, if ω ∈ W 1, p (R2 ), p > 2, or ω ∈ W s ,2 (R2 ), s > 1, then k∇ukL∞ (R2 ) ≤ C kωkL∞ (R2 ) [1 + log(2 + kωkB )],

(3.65)

where kωkB refers to the norm in either W 1, p (R2 ) or W s ,2 (R2 ). Proof. The proof is almost identical to the proof of Theorem 1 and is omitted. Calculus inequalities (2.79) and (2.80) then imply that N (T ) = ku(T )kH s (R2 ) + kθ(T )kH s (R2 )

(3.66)

obeys log(2 + N (T )) ≤ log(2 + N (0)) RT +C 0 (k∇u(t )kL∞ (R2 ) + kω(t )kL∞ (R2 ) + k∇θ(t )kL∞ (R2 ) + kθ(0)kL2 (R2 ) )d t , (3.67) and then because of Zt k∇θ(s)kL∞ (R2 ) d s (3.68) kω(t )kL∞ (R2 ) ≤ kω(0)kL∞ (R2 ) + 0

30

Chapter 3. Hydrodynamic models

and (3.64) we have from a Grönwall inequality for log(2 + N (T )) that N (T ) is controlled a priori in terms of initial data and C (T ) if s > 2. We thus have the following. Theorem 4. Let u0 ∈ H s (R2 ) be divergence-free and let θ0 ∈ H s (R2 ) with s > 2. Assume that a solution (u, θ) of (3.53) with initial data (u0 , θ0 ) exists and is smooth on the time interval [0, T ), and assume that (3.60) holds. Then ku(t )kH s (R2 ) + kθ(t )kH s (R2 ) ≤ F (T , C (T ), ku0 kH s (R2 ) , kθ0 kH s (R2 ) )

(3.69)

holds, with F an explicit function depending only on the parameters indicated. It is interesting to note that from (3.53) we immediately deduce that Z Z ω(x, t )Φ(θ(x, t ))d x = ω0 (a)Φ(θ0 (a))d a (3.70) R2

R2

for any reasonable function Φ such that the right-hand side is finite. For instance, if Φ(θ) = θ, in view of the conservation of the norms of θ we obtain the consequence Z (3.71) ω (a)θ (a)d a kω(t )kL p (R2 ) ≥ kθ0 k−1 0 0 Lq (R2 ) 2 R

with 1 ≤ p < ∞, p −1 + q −1 = 1. A purely Lagrangian approach (as opposed to an Lagrangian-Eulerian one) to local existence is possible. We describe it briefly below by first recalling some general kinematic properties for incompressible flows in two dimensions. Because the map a 7→ X (a, t ) is volume preserving, we have the general property that J ( f , g ) ◦ X = J ( f ◦ X , g ◦ X ), (3.72) where f , g are C 1 (R2 ) functions and J ( f , g ) = (∂1 f )(∂2 g ) − (∂2 f )(∂1 g )

(3.73)

is the usual Poisson bracket. In particular, we have that ∇⊥ f ◦ X = J ( f ◦ X , X ).

(3.74)

ζ =ω◦X

(3.75)

Introducing the notation

and recalling the notations (2.4), (2.6) v = ∂ t X = u ◦ X , the 2D Boussinesq system (3.53 reduces to ∂ t ζ = J (θ0 , X2 ) (3.76) dX

coupled with the ODE (2.7) d t = u(X , t ) and the linear equaltion relating u and ω in (3.53). Indeed, the equation for θ in (3.53) is solved by θ = θ0 ◦ X −1 , where θ0 is the initial datum for θ, and therefore θ ◦ X = θ0 .

(3.77)

The purely Lagrangian approach is based on the observation that, if we denote λ(a, t ) = X (a, t ) − a,

(3.78)

3.5. Oldroyd-B

31

then the equation det ∇a X (a, t ) = 1 becomes div a λ = J (λ2 , λ1 )

(3.79)

and the equation ∇⊥ · u = ω becomes curl v = ∇⊥ a · v = ζ − J (v1 , λ1 ) − J (v2 , λ2 ).

(3.80)

The Boussinesq system in Lagrangian coordinates is therefore closed. Denoting by λ˙ = ∂ t λ, ζ˙ = ∂ t ζ we have   div λ = J (λ2 , λ1 ), curl λ˙ = ζ − J (λ˙1 , λ1 ) − J (λ˙2 , λ2 ), (3.81)  ˙ ζ = J (θ0 , a2 + λ2 ). The div-curl system is elliptic, and thus the Boussinesq system becomes a fully nonlinear evolution equation. As long as |∇λ| is small compared to 1, bounds are easily obtained. This will happen for short time of the order kζ k−1 because L∞ (R2 ) λ(0) = 0. Incidentally, the 2D incompressible Euler system is just the system above with ζ˙ = 0, and the fact that it has global existence follows essentially from the conservation of the magnitude of ζ . The Boussinesq system can be viewed as an equation for the history of particle trajectories. Denoting Zt h(a, t ) = λ(a, s)d s (3.82) 0

we note that

ζ (t ) = ζ0 + t ∂1 θ0 + J (θ0 , h2 ),

(3.83)

and therefore the initial data ω0 = ζ0 and θ0 enter in the equation only as parameters.

3.5 Oldroyd-B The creeping flow Oldroyd-B system in two dimensions is a system coupling a Stokes equation to a system for the evolution of a symmetric matrix of added stresses:   −∆u + ∇ p = div σ, ∇ · u = 0, (3.84)  ∂ σ + u · ∇σ = (∇u)σ + σ(∇u)∗ . t This is a limit case of a more complete Oldroyd-B system [20] in which the stress σ evolves according to (∂ t + u · ∇)σ = (∇u)σ + σ(∇u)∗ − 2kσ + 2kI

(3.85)

and u solves the Navier–Stokes equations driven by div σ. We ignored the damping and driving term by setting k = 0 and took the limit of low Reynolds number (no fluid inertial term u · ∇u) and instantaneous relaxation (no time derivative in the Stokes system). The time-dependent case is more complicated but was solved along the lines we describe below in [14]. The time-dependent Oldroyd-B equations themselves are obtained by exact closure of a kinetic system, coupling

32

Chapter 3. Hydrodynamic models

Navier–Stokes equations to a probability density of particles. The global existence of smooth solutions of the Oldroyd-B system in two dimensions is open, even in the case (3.84). Adding Laplacian diffusion to the evolution of stress regularizes the system [18]. Lagrangian uniqueness was proved in [20]. In the system (3.84) τ = σ ◦X, (3.86) and the operators U and G are

and The ODE for τ is

with F given by

Ui (σ) = Λ−1 Rk (σi k + δi k R l R m σ l m )

(3.87)

Gi j (σ) = R j Rk (σi k + δi k R l R m σ l m ).

(3.88)

dτ = gτ + τg∗ dt

(3.89)

F (g , τ) = g τ + τ g ∗ .

(3.90)

Chapter 4

Spaces and operators

We describe below the choices of path spaces and we prove the basic commutator estimates which form the core of the proof of the local existence and Lipschitz dependence of paths in the Lagrangian-Eulerian description.

4.1 Spaces We use L p to denote L p (Rn ) or L p (Ω). We consider function spaces C α, p = C α (Rd ) ∩ L p (Rd ) with norm

k f kα, p = k f kC α (Rd ) + k f kL p (Rd )

for α ∈ (0, 1), p ∈ (1, ∞), C 1+α (Rd ) with norm k f kC 1+α (Rd ) = k f kL∞ (Rd ) + k∇ f kC α (Rd ) . We need also spaces of paths, L∞ (0, T ; C α, p ), with the usual norm, k f kL∞ (0,T ;C α, p ) = sup k f (t )kα, p ; t ∈[0,T ]

spaces Li p(0, T ; C α, p ) with norm k f kLi p(0,T ;C α, p ) =

k f (t ) − f (s)kα, p

sup

|t − s|

t 6= s,t ,s ∈[0,T ]

+ k f kL∞ (0,T ;C α, p ) ;

spaces L∞ (0, T ; C 1+α (Rd )) with norm k f kL∞ (0,T ;C 1+α (Rd )) = sup k f (t )kC 1+α (Rd ) ; t ∈[0,T ]

and spaces Li p(0, T ; C 1+α ) with norm k f kLi p(0,T ;C 1+α ) =

sup t 6= s,t ,s ∈[0,T ]

k f (t ) − f (s)k1+α |t − s| 33

+ k f kL∞ (0,T ;C 1+α (Rd )) .

34

Chapter 4. Spaces and operators

4.2 Operators We start with a classical lemma. Lemma 3. Let 0 < α < 1, 1 < p < ∞. Let η ∈ C 1+α (Rd ) and let Z (Kσ)(x) = P.V . k(x − y)σ(y)d y Rd

be a classical Calderón–Zygmund operator with kernel k which is smooth away from the origin, homogeneous of degree −d , and with mean zero on spheres about the origin. Then the commutator [η · ∇, K] can be defined as a bounded linear operator in C α, p and k[η · ∇, K]σkα, p ≤ C kηkC 1+α (Rd ) kσkα, p . (4.1) Remark 2. The conclusion of the lemma holds also for operators H which are products of classical Calderón–Zygmund operators. This follows from successive applications of the lemma. We provide here the proof of [14]. We take this opportunity to correct a minor error in that proof. Proof. Let us note that both terms in the commutator, η · ∇Kσ and Kη∇σ are well defined and Hölder continuous if σ is smooth. We first compute Z [η · ∇, K]σ(x) = k(x − y)(η(x) − η(y)) · ∇y σ(y)d y. Rd

Now we introduce a smooth cutoff χ (|x −y|) identically equal to 1 for |x −y| ≤ 1 and compactly supported. The conclusion of the lemma holds for Z Co u t (x) = (1 − χ (|x − y|)) k(x − y)(η(x) − η(y)) · ∇y σ(y)d y Rd

by integration by parts and inspection, using the L p bound for σ. We concentrate our attention on Z Ci n (x) = χ (|x − y|)k(x − y)(η(x) − η(y)) · ∇y σ(y)d y. Rd

We first write Ci n (x) =

Z Rd

χ (|x − y|)k(x − y)(η(x) − η(y)) · ∇y (σ(y) − σ(x))d y,

and then we integrate by parts: Ci n (x) = C (x) + C1 (x) + C2 (x) with C (x) =

Z Rd

χ (|x − y|)∇ x k(x − y)(η(x) − η(y))(σ(y) − σ(x))d y,

4.2. Operators

35

C1 (x) =

Z Rd

and C2 (x) =

∇ x χ (|x − y|))k(x − y)(η(x) − η(y))(σ(y) − σ(x))d y, Z Rd

χ (|x − y|)k(x − y)(∇y η(y))(σ(y) − σ(x))d y.

It is easy to see that C1 and C2 are Hölder continuous and satisfy the bound (4.1). For instance, for C2 we may add and subtract ∇ x η(x), and we obtain two terms which are typical terms in classical singular integral theory; the proof we present below for C (x) can be used as well. We now investigate C (x) and write η(y) − η(x) = (y − x) ·

Z

1

∇η(x + λ(y − x))d λ.

0

We also write

K(z) = z∇ z k(z)

and note that it is homogeneous of order −d and smooth away from the origin. The averages on spheres might not vanish. So, with these preparations, C (x) is C (x) =

Z

1

dλ

Z

χ (|x − y|)K(x − y)(∇η(x + λ(y − x)))(σ(y) − σ(x))d y.

Rd

0

We now write

C (x) = A(x) + B(x),

where A(x) =

Z

1

0

dλ

Z

χ (|x −y|)K(x −y)(∇η(x +λ(y − x))−∇η(x))(σ(y)−σ(x))d y

Rd

and B(x) = ∇η(x)

Z

χ (|x − y|)K(x − y)(σ(y) − σ(x))d y.

Rd

Now B ∈ C α (Rd ) and obeys (4.1). It is obviously enough to check that Z I (x) = χ (|x − y|)K(x − y)(σ(y) − σ(x))d y Rd

is in C α (Rd ) and its norm is bounded by that of σ. To check this we take the difference I (x + h) − I (x) = I1 + I2 + I3 , where I1 is R I1 = |x−y|≤4|h|, |x+h−y|≥4|h| χ (|x + h − y|)K(x + h − y)(σ(y) − σ(x + h))d y R − |x−y|≤4|h|, |x+h−y|≥4|h| χ (|x − y|)K(x − y)(σ(y) − σ(x))d y, I2 is R I2 = |x−y|≥4|h|, |x+h−y|≤4|h| χ (|x + h − y|)K(x + h − y)(σ(y) − σ(x + h))d y R − |x−y|≥4|h|, |x+h−y|≤4|h| χ (|x − y|)K(x − y)(σ(y) − σ(x))d y,

36

Chapter 4. Spaces and operators

and R I3 = |x−y|≥4|h|, |x+h−y|≥4|h| χ (|x + h − y|)K(x + h − y)(σ(y) − σ(x + h))d y R − |x−y|≥4|h|, |x+h−y|≥4|h| χ (|x − y|)K(x − y)(σ(y) − σ(x))d y. For I1 and I2 we note that both |x − y| ≤ 5|h| and |x + h − y| ≤ 5|h|, and we use the straightforward inequality Z 5h r −1 r α d r ≤ C h α . 0

The integral I3 is split into two pieces, I3 = I4 + I5 , with I4 =

1 2

Z

[χ (|x+h−y|)K(x+h−y)−χ (|x−y|)K(x−y)](2σ(y)−σ(x+h)−σ(x))d y

|x−y|≥4|h|, |x+h−y|≥4|h|

and σ(x) − σ(x + h) I5 = 2

Z

[χ (|x+h−y|)K(x+h−y)+χ (|x−y|)K(x−y)]d y.

|x−y|≥4|h|, |x+h−y|≥4|h|

For I4 we use the smoothness of the kernel, the intermediate value theorem, and the Hölder bounds to obtain Z∞ |I4 | ≤ C kσkC α (Rd ) |h|r −2 (r α + h α )d r ≤ C kσkC α (Rd ) |h|α . 3|h|

For I5 we recall that K(z) = z∇k(z). We claim that integrals of Z χ (|z|)K(z)d z |z|≥4|h|,|z±h|≥4|h|

are bounded uniformly, independently of h. Indeed, integrating by parts, R χ (|z|)z j ∂i k(z)d z |z|≥5|h| R R = − |z|≥5|h| δi j χ (|z|)k(z)d z − |z|≥5|h| ∂i (χ (|z|))z j k(z)d z R zi + |z|=5|h| z j χ (5|h|) 5|h| k(z)d S(z) = 0 + bounded. Here we used that k has mean zero on spheres. On the other hand, on the annular regions we use simply the homogeneity of K and Z 5h 1 d r ≤ C. 4h r The integral A(x) is treated in a similar fashion. We write Z A(x) = χ (|x − y|)K(x − y)φ(x, x − y)(σ(y) − σ(x))d y, Rd

4.2. Operators

37

where φ(x, x − y) =

Z

1

(∇η(x + λ(y − x)) − ∇η(x))d λ.

0

We consider

A(x + h) − A(x) = A1 + A2 + A3 ,

where A1 and A2 , like I1 and I2 above, are differences of integrals on |x −y| ≤ 4|h| and |x + h − y| ≥ 4|h| and, respectively, on |x − y| ≥ 4|h| and |x + h − y| ≤ 4|h|, while A3 is the difference of integrals corresponding to both |x − y| ≥ 4|h| and |x + h − y| ≥ 4|h|. As before, using the triangle inequality, the regions of integration for A1 and A2 are regions where both |x − y| ≤ 5|h| and |x + h − y| ≤ 5|h|, and therefore, the integrals are small separately, without need to take the difference. Using the fact that |φ(x, x − y))| ≤ kηkC 1+α (Rd ) |x − y|α we obtain that

|A1 | + |A2 | ≤ C |h|2α kσkC α (Rd ) kηkC 1+α (Rd )

We treat A3 as we treated I3 : we split A3 = A4 + A5 , where 1R A4 = 2 |x−y|≥4|h|, |x+h−y|≥4|h| [χ (|x + h − y|)K(x + h − y) − χ (|x − y|)K(x − y)] ×[φ(x + h, x + h − y)(σ(y) − σ(x + h)) + φ(x, x − y)(σ(y) − σ(x))]d y, and using the smoothness of the kernel and the bounds on φ and σ, this leads to an integral inequality, Z1 h r −2 (r α + h α )2 ≤ C h 2α , 3h

so

|A4 | ≤ C |h|2α kσkC α (Rd ) kηkC 1+α (Rd ) .

Finally we treat 1R A5 = 2 |x−y|≥4|h|, |x+h−y|≥4|h| [χ (|x + h − y|)K(x + h − y) + χ (|x − y|)K(x − y)] ×[φ(x + h, x + h − y)(σ(y) − σ(x + h)) − φ(x, x − y)(σ(y) − σ(x))]d y. We note by polarization that |φ(x + h, x + h − y)(σ(y) − σ(x + h)) − φ(x, x − y)(σ(y) − σ(x)) | ≤ C kσkC α (Rd ) kηkC 1+α (Rd ) (|x − y| + |h|)α |h|α , and therefore A5 is bounded directly using Z1 r −1 (h α (r α + h α )d r ≤ C h α , 4h

|A5 | ≤ C kσkC α (Rd ) kηkC 1+α (Rd ) |h|α . This concludes the proof of the fact that kCi n kC α (Rd ) ≤ C kσkα, p kηkC 1+α (Rd ) .

38

Chapter 4. Spaces and operators

The proof of the L p bound in Lemma 3 is done using the observation that Z C (σ) = [η · ∇, K]σ = PV K(x, y)σ(y)d y − K((∇ · η)σ), Rd

where

K(x, y) = (η j (x) − η j (y)))∂ j k(x − y)

(not to be confused with K(z) = z∇ z k(z)!). Now the operator K is bounded in L p spaces and the operator T is given by Z σ 7→ PV K(x, y)σ(y)d y = (T σ)(x) Rd

is a Calderón–Zygmund operator; that is, the kernel K is smooth away from the diagonal and obeys 1 |K(x, y)| ≤ C |x − y|d and |K(x + h, y) − K(x, y)| + |K(x, y + h) − K(x, y)| ≤ C

|h| |x − y|d +1

for 2|h| ≤ |x −y|, and T is bounded in L2 (Rd ). The boundedness in L2 is verified below. (This part of the proof had a minor error in [14]). It follows that T is bounded in L p (Rd ), 1 < p < ∞ (see, for instance, [47]). For the bound in L2 we need to verify that Z (T f )(x)g (x)d x ≤ C k f kL2 (Rd ) k g kL2 (Rd ) . Rd

We write R R R (T f )(x)g (x)d x = Rd d x PV |z|≤1 K(x, x − z) f (x − z)g (x)d z RRd R + Rd d x |z|≥1 K(x, x − z) f (x − z)g (x)d z = T1 + T2 . Clearly |T2 | ≤ C kηkL∞ (Rd ) k f kL2 (Rd ) k g kL2 (Rd ) because |K(x, y)| ≤ C kηkL∞ (Rd )

1 |x − y|d +1

in view of the homogeneity of k. We write T1 as T1 = T11 + T12 with (T11 f )(x) =

Z |z|≤1

(η j (x) − η j (x − z) − z · ∇η j (x))∂ j k(z) f (x − z)d z

and (T22 f )(x) = PV

Z |z|≤1

(z · ∇η j (x))∂ j k(z) f (x − z)d z.

4.2. Operators

39

For T11 we use the fact that we have |(η(x) − η(x − z) − z∇η(x))∇k(z)| ≤ C |z|−d +α for |z| ≤ 1, uniformly in x, and so Z (T11 f )(x)g (x)d x ≤ C k f kL2 (Rd ) k g kL2 (Rd ) , Rd

using Fubini’s theorem. For T12 we use the fact that (T12 f )(x) = L l j (∂ l η j f ), where (L l j f )(x) = PV

Z |z|≤1

z l ∂ j k(z) f (x−z)d z = PV

It is easily verified that Z R1 0 small, depending only on d , such that from the inequality |∇λ(a, t )| ≤ 2ρ it follows that 1 3 ≤ | det ∇X (a, t )| ≤ . 2 2

(5.6)

Note that by Lemma 2 X −1 (x, t ) exists for X ∈ PρT , |∇a X (a, t )| + |∇ x X −1 (x, t )| ≤ C

(5.7)

and the uniform chord-arc condition C −1 ≤

|X (a, t ) − X (b , t )| ≤C |a − b | 43

(5.8)

44

Chapter 5. The Lagrangian-Eulerian existence theorems

hold for all X ∈ PρT , 0 ≤ t ≤ T and a, b , x ∈ Rd with an absolute constant C , depending via ρ only on the dimension d of space. We consider τ ∈ ΣTγ , (5.9) where γ , kτkL∞ (0,T ;C α, p ) ≤ γ } 2 (5.10) with α ∈ (0, 1) and with 1 < p < d . The positive constant γ is arbitrary and fixed. ΣTγ = {τ ∈ Li p(0, T ; C α, p ); τ(a, 0) = τ0 (a), kτ0 kα, p ≤

Proposition 6. There exists T (ρ, γ ) > 0 such that, for T ≤ T (ρ, γ ), the map F maps PρT × ΣTγ to itself, F : PρT × ΣTγ → PρT × ΣTγ .

(5.11)

Proof. Because of (5.7) and (5.8) we have that kσkL∞ (0,T ;C α, p ) ≤ C γ

(5.12)

with fixed C of order one, depending only on d , α, p, uniformly for all σ of the form σ(x, t ) = τ(X −1 (x, t ), t ), (5.13) where τ ∈ ΣTγ and X ∈ PρT . From (3.20), (4.4), and (5.12) we obtain k g kL∞ (0,T ;C α (Rd )) ≤ C γ

(5.14)

with fixed C of order one, depending only on d , α, p, uniformly for all X ∈ PρT , τ ∈ ΣTγ . Because k g τkL p (Rd ) ≤ C k g kL∞ (Rd ) kτkL p (Rd ) (5.15) and the fact that C α is a Banach algebra, it follows that kT (X , τ)kL∞ (0,T ;C α, p ) ≤ C γ (1 + γ )

(5.16)

holds with fixed C of order one, uniformly for all X ∈ PρT , τ ∈ ΣTγ . We used here the fact that F (g , τ) is at least linear and at most quadratic in all our examples. We use this fact in order to give explicit dependence of the time of existence on γ . This is not a limitation of the method; obviously higher nonlinear behavior does not hurt. Zero order terms would correspond to forced, inhomogeneous equations, but they would not affect the local existence theorem either. From (4.2) with q = ∞ and (5.12) we obtain kU (X , τ)kL∞ (0,T ;L∞ (Rd )) ≤ C γ

(5.17)

with C of order one, uniformly for all X ∈ PρT , τ ∈ ΣTγ . Using (3.25) and (5.14) we obtain kU (X , τ)kL∞ (0,T ;C 1+α (Rd )) ≤ C γ (5.18)

5.1. Local existence

45

with C of order one, uniformly for all X ∈ PρT , τ ∈ ΣTγ . From (5.16) and (5.18) it is clear that a choice ρ T (ρ, γ ) = (5.19) M (1 + γ ) with M large enough serves the purpose: kX ne w − IkL∞ (0,T ;C 1+α (Rd )) ≤ C

ρ M

(5.20)

and kτ ne w kL∞ (0,T ;C α, p ) ≤ γ .

(5.21)

The map F has range in (I + Li p(0, T ; C 1+α (Rd ))) × Li p(0, T ; C α, p ), and the norms satisfy kX ne w − IkLi p(0,T ;C 1+α (Rd )) ≤ C γ

(5.22)

kτ ne w kLi p(0,T ;C α, p ) ≤ C γ (1 + γ )

(5.23)

and with C of order one in view of (5.18) and (5.16). This concludes the proof of Proposition 6. Let us now consider two pairs (X1 , τ1 ) and (X2 , τ2 ), Xi = I + λi , in T Sρ,γ = PρT × ΣTγ

(5.24)

Xε = (2 − ε)X1 + (ε − 1)X2

(5.25)

τε = (2 − ε)τ1 + (ε − 1)τ2

(5.26)

T (Xε , τε ) ∈ Sρ,γ

(5.27)

Xε0 = λ2 − λ1

(5.28)

τε0 = τ2 − τ1

(5.29)

and the segment (Xε , τε ),

and with ε ∈ [1, 2]. Note that and and do not depend on ε but

ηε = Xε0 ◦ Xε−1

and

δε = τε0 ◦ Xε−1

do. Let us observe that

sup kηε kL∞ (0,T ;C 1+α ) ≤ C kλ1 − λ2 kL∞ (0,T ;C 1+α (Rd )) ≤ C

ε∈[1,2]

ρ M

(5.30)

because of (5.20) and the uniform chord-arc condition. We also have sup kδε kα, p ≤ C kτ1 − τ2 kL∞ (0,T ;C α, p ) .

ε∈[1,2]

(5.31)

46

Chapter 5. The Lagrangian-Eulerian existence theorems

Next we estimate gε0 defined in (3.19). In view of the representation (3.21), gε0 ◦ Xε−1 = [ηε · ∇, G](σε ) + G(δε ),

(5.32)

and our bounds (4.6), (5.30), (4.4), (5.31), we have ” — sup k gε0 kL∞ (0,T ;C α, p ) ≤ C γ kλ1 − λ2 kL∞ (0,T ;C 1+α (Rd )) + kτ1 − τ2 kL∞ (0,T ;C α, p ) . ε∈[1,2]

(5.33) Now examining Uε0 using (3.14) and bounding it first in L∞ using (4.5), (5.30), (4.2), (5.31) and then, using the expression (3.26) and bounding using (5.33), we obtain ” — sup kUε0 kL∞ (0,T ;C 1+α (Rd )) ≤ C γ kλ1 − λ2 kL∞ (0,T ;C 1+α (Rd )) + kτ1 − τ2 kL∞ (0,T ;C α, p ) . ε∈[0,1]

(5.34) The term Tε0 defined in (3.18) is bounded using (5.33): supε∈[0,1] kTε0 kL∞ (0,T ;C α, p ) ” — ≤ C (1 + γ ) γ kλ1 − λ2 kL∞ (0,T ;C 1+α (Rd )) + kτ1 − τ2 kL∞ (0,T ;C α, p ) .

(5.35)

ρ

Proposition 7. There exists M such that if T ≤ M (1+γ ) , then the map F defined in (5.1), T T F : Sρ,γ → Sρ,γ , (5.36) obeys 1 k|F (X1 , τ1 ) − F (X2 , τ2 )k| ≤ k|(X1 , τ1 ) − (X2 , τ2 )k| + kτ1 (0) − τ2 (0)kα, p (5.37) 2 for the norm k|(X , τ)k| = kλkL∞ (0,T ;C 1+α (Rd )) + kτkL∞ (0,T ;C α, p ) .

(5.38)

Proof. This follows from (5.34) and (5.35) because Z t Z 2  Zt Z2 0 0 F (X2 , τ2 ) − F (X1 , τ1 ) = dt Uε d ε, τ2 (0) − τ1 (0) + dt Tε d ε . 0

1

0

1

(5.39)

Remark 4. The nonlinearities are Lipschitz continuous: kU (X1 , τ1 ) − U (X2 , τ2 )kL∞ (0,T ;C 1+α (Rd )) + kT (X1 , τ1 ) − T (X2 , τ2 )kL∞ (0,T ;C α, p ) ” — ≤ C (1 + γ ) γ kλ1 − λ2 kL∞ (0,T ;C 1+α (Rd )) + kτ1 − τ2 kL∞ (0,T ;C α, p ) (5.40) T on Sρ,γ . Indeed, this follows from U (X2 , τ2 ) − U (X1 , τ1 ) =

Z

2

1

Uε0 d ε,

5.2. Small data global existence with damping

T (X2 , τ2 ) − T (X1 , τ1 ) =

47

Z

2

1

Tε0 d ε,

and (5.34), (5.35). Theorem 5. Let τ0 ∈ C α, p , with 0 < α < 1, 1 < p < d . There exists T > 0 depending on kτ0 kC α, p such that Lagrangian-Eulerian system § ∂ t X = U (X , τ), (5.41) ∂ t τ = T (X , τ) with U (X , τ) = U(τ ◦X −1 )◦X and T (X , τ) = F (G(τ ◦X −1 )◦X , τ) has a unique solution  X (a, t ) = a + λ(a, t ), λ ∈ Li p(0, T ; C 1+α (Rd )), λ(a, 0) = 0, (5.42) τ(a, t ), τ ∈ Li p(0, T ; C α, p ), τ(a, 0) = τ0 (a) in each of the cases of the Euler equations (3.27), (3.28), (3.30); SQG (3.39), (3.41), (3.43); porous medium equations (3.48), (3.49), (3.51); Boussinesq system (3.56), (3.57), (3.59); and Oldroyd-B (3.87), (3.88), (3.90). The solution depends in a Lipschitz manner in the norm k| · k| on initial data in C α, p : if (Xi , τi ), i = 1, 2, are two solutions with initial data τi (0), then k|(X1 , τ1 ) − (X2 , τ2 )k| ≤ 2kτ1 (0) − τ2 (0)kα, p .

(5.43)

Proof. We take γ = 2kτ0 kα, p and ρ as before. The existence follows from the contraction mapping principle in the subset T T = {(X , τ) ∈ Sρ,γ ; τ(0) = τ0 } (X , τ) ∈ Sρ,τ 0

(5.44)

because for τ1 (0) = τ2 (0) the map F is a contraction in the norm k| · k|, in view of (5.37). The Lipschitz dependence of solutions on initial data follows also from (5.37) because for solutions (X , τ) = F (X , τ).

5.2 Small data global existence with damping In this section we consider the system ( dX = U (X , τ), dt dτ dt

= T1 (X , τ),

(5.45)

with U given in (3.8) and

T1 (X , τ) = F G τ ◦ X −1 ◦ X , τ − τ

(5.46)

corresponding to the damped Lagrangian ODE dτ = F (g , τ) − τ, dt

(5.47)

or, in other words, to adding a damping term in the corresponding PDE evolution equation. We took without loss of generality the damping constant to be 1,

48

Chapter 5. The Lagrangian-Eulerian existence theorems

and the smallness of initial data below will refer to smallness compared to this damping constant and dimension of space. Theorem 6. There exists γ0 depending on the damping constant, the dimension of space, and α, p, such that if τ0 ∈ C α, p , with 0 < α < 1, 1 < p < d , and kτ0 kC α, p ≤ γ0 , then the Lagrangian-Eulerian system (5.45) with U (X , τ) = U(τ ◦ X −1 ) ◦ X 2 and T1 (X , τ) = F (G(τ ◦ X −1 ) ◦ X , τ) − τ has a unique global solution X (a, t ) = a + λ(a, t ), λ(a, 0) = 0, τ(a, t ), τ(a, 0) = τ0 (a),

§

(5.48)

with λ ∈ Li p(0, ∞; C 1+α (Rd )) and τ ∈ Li p(0, ∞; C α, p ), in each of the cases corresponding to the damped Euler equations   ∂ t ω + u · ∇ω = (∇u)ω − ω, divu = 0,  ∇ × u = ω;

(5.49)

damped SQG equation, ∂ t θ + (R⊥ θ) · ∇θ = −θ;

(5.50)

damped porous medium equations ∂ t θ + u · ∇θ = −θ,

u = (θ, 0) + ∇q,

divu = 0;

(5.51)

damped Boussinesq system,  ∂ ω + u · ∇ω = ∂1 θ − 2ω,    ∂ t θ + u · ∇θ = −2θ t divu = 0,    ⊥ ∇ u = ω;

(5.52)

and damped Oldroyd-B system   −∆u + ∇ p = divσ, divu = 0,  ∂ σ + u · ∇σ = (∇u)σ + σ(∇u)∗ − σ. t

(5.53)

Proof. We start by choosing ρ depending on d such that (5.7) and (5.8) hold on PρT . We recall from the proof of Proposition 6 that (5.14) holds with a constant C independent of γ . In fact, kG(τ ◦ X −1 ) ◦ X kα, p ≤ C kτkα, p

(5.54)

holds pointwise in time, with C of order one, depending only on α, p, d , uniformly for X ∈ PρT , τ ∈ ΣTγ , for any T and any γ . It follows that kF (g , τ)(t )kα, p ≤ CF kτ(t )k2α, p

(5.55)

5.2. Small data global existence with damping

49

holds uniformly for all X ∈ PρT , τ ∈ ΣTγ with CF depending on the nonlinearity F and on α, p, d but not on T nor γ . This is true for all models except for the Boussinesq system, where there is a linear term. In that case we have kF (g , τ)(t )kα, p ≤ CF kτ(t )k2α, p + kτ(t )kα, p . We now choose γ such that

2CF γ < 1

(5.57)

From the ODE (5.47) we have, for a solution in τ(t ) = e

−t

τ(0) +

Z

(5.56)

T Sρ,γ ,

t

e −(t −s) F (g (s), τ(s))d s ,

0

and using (5.55) we have kτ(t )kα, p ≤ e −t kτ(0)kα, p + CF Introducing

Z 0

t

e −(t −s ) kτ(s)k2α, p d s.

M (t ) = sup kτ(s)kα, p 0≤s≤t

we thus have

M (t ) ≤ e −t kτ(0)kα, p + CF M (t )2 (1 − e −t )

for t ≤ T . Now we claim that M (t ) ≤ kτ(0)kα, p

(5.58)

holds for t ≤ T . Indeed, as long as CF M (t )2 ≤ kτ(0)kα, p we have M (t ) ≤ e −t (kτ(0)kα, p − CF M (t )2 ) + CF M (t )2 ≤ kτ(0)kα, p . If we assume by contradiction that there exists some ε ∈ (0, 1) and a first time t ∗ ≤ T such that M (t ∗ ) = (1 + ε)kτ(0)kα, p , then, in view of (5.57) and the fact γ that τ ∈ ΣTγ implies kτ(0)kα, p ≤ 2 , we have γ CF M (t ∗ )2 ≤ CF (1 + ε)2 kτ(0)kα, p ≤ kτ(0)kα, p , 2 and we arrive at the contradiction M (t ∗ ) ≤ kτ(0)kα, p . Because ε > 0 was arbitrary, we deduce M (T ) ≤ kτ0 kα, p . This implies that γ kτ(T )kα, p ≤ 2 , and we can extend the solution for another interval of time T = T (ρ, γ ) with the same γ and ρ, resetting the initial τ to be τ(T ) and starting again from λ = 0. The time increments are constant, and because of the uniqueness, this provides a global solution for all time. The calculation for the Boussinesq case is similar. This is why we took in that case the damping constant to be 2. In fact, any damping r > 1 would stabilize the Boussinesq Lagrangian ODE near zero, and a similar argument would work. A similar result for damped Boussinesq equations in a slightly larger space is in [1].

Chapter 6

Critical dissipative active scalars

We consider critical dissipative active scalars ∂ t θ + u · ∇θ + Λθ = 0, where

u = M θ + Kθ,

(6.1)

div u = 0,

(6.2)

where M is a constant vector and K is a translation-invariant singular integral operator, Z (Kθ)(x) = PV k(x − y)|x − y|−d θ(y)d y, (6.3) Rd

where the vector valued kernel k : Rd \ {0} → Rd is homogeneous of degree 0 and smooth away from the origin and has mean zero on spheres about the origin. The two typical examples are the critical SQG where M = (0, 0) and k(x) = c

x⊥ , |x|

(6.4) 1

and the critical porous medium equation where M = ( 2 , 0) and   2x 2 2x x k(x) = c 1 − 1 , 1 2 |x|2 |x|2

(6.5)

in d = 2. These and related sytems have been studied very widely, as briefly mentioned in the introduction.

6.1 Lower bounds on the fractional Laplacian The fractional Laplacian has an explicit kernel in Rd , Z f (x) − f (y) Λ s f (x) = c PV d y, d +s Rd |x − y|

(6.6)

0 < s < 2, and it is this explicit form that was used in [22] to prove the lower bound that we are discussing. Similar bounds can be obtained in bounded domains with smooth boundaries for the fractional Dirichlet Laplacian. This is 51

52

Chapter 6. Critical dissipative active scalars

defined in terms of the eigenfunction expansion, and the kernel is not explicit in general. In view of the identity Z∞ α λ = cα (1 − e −t λ )t −1−α d t , (6.7) 0

with 1 = cα

Z

∞

(1 − e −s )s −1−α d s,

0

valid for 0 ≤ α < 1, we have the representation Z∞   f (x) − e t ∆ f (x) t −1−α d t ((−∆)α f ) (x) = cα

(6.8)

0

for f smooth. The representation holds in the whole space using the Fourier transform and in bounded domains using eigenfunction expansion. The kernel of the heat semigroup in the whole space is explicit, d

|z|2

Gt (z) = (4πt )− 2 e − 4t , R and this, together with the fact that Rd Gt (z)d z = 1, gives Λ s f (x) = c

Z

∞

0

s

t −1− 2

Z Rd

(6.9)

Gt (z)( f (x) − f (x − z))d z,

which yields (6.6). It is known that the kernel of the Dirichlet heat semigroup in bounded domain is positive and nonsingular for t > 0, and this is enough to prove the analogue of the Córdoba–Córdoba inequality [25] in the case of bounded domains as well [15]. Using more information about the heat kernel, the following proposition was proved in [16]. Proposition 8. Let Φ be a C 2 convex function satisfying Φ(0) = 0. Let f ∈ C0∞ (Ω), where Ω is a bounded domain with smooth boundary, and let Λ s , 0 ≤ s ≤ 2 be the fractional Dirichlet Laplacian. Then
Φ0 ( f )Λ s f − Λ s (Φ( f )) ≥ c d (x)−s f Φ0 ( f ) − Φ( f ) (6.10) holds pointwise in Ω, with d (x) = d i s t (x, ∂ Ω) and c > 0. The Córdoba–Córdoba inequality in the whole space corresponds to the case d (x) = ∞. In order to go beyond this inequality, more information about the 1 kernel is needed. Let us explain the case of Rd , Φ( f ) = 2 f 2 , and s = 1. We define D(g ), 1 D(g )(x) = g (x)Λ g (x) − Λ g 2 (x), (6.11) 2 and estimate it for a scalar valued function g = ∂1 f , where ∂1 is a partial derivative and f is a bounded function. We use the explicit representation (6.6) and compute Z (g (x) − g (y))2 c D(g )(x) = d y. (6.12) 2 Rd |x − y|d +1

6.1. Lower bounds on the fractional Laplacian

53

We take a smooth radial cutoff function ψ(r ) obeying 0 ≤ ψ(r ) ≤ 1 with ψ(r ) = 1 0 for r ∈ [0, 2 ] ψ(r ) = 1 on r ∈ [1, ∞). We take an arbitrary length ` (to be chosen later) and write Z  ‹ |x − y| (g (x) − g (y))2 c D(g )(x) ≥ ψ d y. (6.13) 2 Rd ` |x − y|d +1 We open brackets and ignore one positive term: € |x−y| Š g (y) € |x−y| Š 1 R R c D(g )(x) ≥ 2 g 2 (x) Rd ψ ` |x−y|d +1 d y − c g (x) Rd ψ ` |x−y|d +1 d y = G(x) − B(x). (6.14) Now, using g = ∂1 f , we integrate by parts in the term B(x) and bound from above: |B(x)| ≤ C1 | g (x)|`−2 k f kL∞ . We bound the term G(x) below: G(x) ≥ C2 g 2 (x)`−1 . 1

Now we choose ` so that |B(x)| ≤ 2 G(x), i.e., `−1 ≤ C3 | g (x)|k f k−1 . L∞ This proves [22]

D(g )(x) ≥ C k f k−1 | g (x)|3 . L∞

(6.15)

This is effectively a cubic lower bound for a quadratic form, given the existing information on f . The fact that the kernel was precisely a power was not important. What we used was the translation invariance (the kernel is a function of x − y), the positivity of the kernel, the fact that the kernel is not integrable near the origin, and the fact that the kernel is integrable at infinity. The translation invariance requirement can be relaxed. In fact, a similar lower bound can be obtained in the case of the fractional Laplacian with Dirichlet boundary conditions [15]: D(g )(x) ≥ C k f k−1 | g (x)|3 . (6.16) L∞ d kf k

L∞ and gd = 0 otherwise. The proof of where gd (x) = g (x) if |g (x)| ≥ d (x) this fact requires a different treatment, because in general there are no explicit representations of the kernel of the fractional Laplacian. We use instead the heat kernel representation (6.8) and precise lower bounds on the heat kernel and upper bounds on its gradient. There are many possible variants of the arguments above and lower bounds, corresponding to the available information on g . A useful variant concerns finite differences, when

g (x) = (δ h f )(x) = f (x + h) − f (x), where h is vector in Rd . Then in the case of Rd we obtain D(δ h f )(x) ≥ C |h|−1 k f k−1 |δ h f (x)|3 . L∞

(6.17)

Another variant suitable for the case g = ∂1 f with f ∈ C α (Rd ) is [22] s

−

s

D(g )(x) ≥ C g 2+ 1−α k f kC α1−α , (Rd )

(6.18)

54

Chapter 6. Critical dissipative active scalars

where c 1 D(g )(x) = g (x)Λ s g (x) − (Λ s (g 2 ))(x) = 2 2

Z Rd

(g (x) − g (y))2 d y. |x − y|d +s

(6.19)

The proof of the inequalities above, in the case of Rd , is very similar to the proof of (6.15). The corresponding inequalities in bounded domains are modified due to the presence of boundaries, and their proofs are more technical [16], but we will not pursue this subject here.

6.2 Hölder regularity The velocity advecting the scalar θ in (6.1) is given explicitly by Z u(x, t ) = M θ(x, t ) + c PV k(x − y)|x − y|−d θ(y, t )d y.

(6.20)

Rd

We take a finite difference g = δ h θ and compute its evolution: (∂ t + u · ∇ x + δ h (u) · ∇ h + Λ) g = 0.

(6.21)

We used here the translation invariance of Λ and the fact that δ h (u)·∇ x θ(x+h) = δ h (u) · ∇ h (δ h u)(x). Let us denote by L the operator L = (∂ t + u · ∇ x + δ h (u) · ∇ h + Λ)

(6.22)

and note that it has a weak maximum principle. The easiest way to see this is by time-splitting: the short time evolution under the pure transport term does not add size, and the short time evolution under the dissipative semigroup does not add size either. We multiply (6.21) by g in order to have nonnegative quantities, obtain 1 L(g 2 ) + D(g ) = 0, (6.23) 2 and then divide by |h|2α :
1 1 L(|h|−2α g 2 ) + |h|−2α D(g ) = δ h (u) · ∇|h|−2α g 2 . 2 2 The right-hand side is bounded by
1 δ (u) · ∇|h|−2α g 2 ≤ α|δ (u)||h|−2α−1 g 2 . h 2 h

(6.24)

(6.25)

The task is to show that |h|−2α D(g ) is larger than this bound of the right-hand side. As usual in critical cases, constants do matter. Nevertheless, we use the same name C for all constants; they are explicitly computable and universal, and the order in which they are computed can be easily unraveled by the interested reader. We know from (6.17) that |h|−2α D(g ) ≥ C kθk−1 |h|−2α−1 g 3 . L∞

(6.26)

We use the representation (6.20) and split δ h u = δ h ui n + δ h uo u t + M g

(6.27)

6.2. Hölder regularity

55

with ‹‹   |x − y| k(x − y) (g (y) − g (x))d y δ h ui n = c P.V . 1−ψ ` |x − y|d Rd Z

and δ h uo u t = c P.V .

   ‹ |x − y| k(x − y) δ−h ψ θ(y)d y ` |x − y|d Rd

(6.28)

Z

(6.29)

with the ψ we used before and with an ` we’ll choose shortly. We used translation invariance, and in (6.28) we used the fact that g = δ h θ and the vanishing of the spherical averages of the kernel, while in (6.29) we moved the finite difference onto the kernel. We bound δ h ui n using the expression (6.12), |δ h ui n (x)| ≤ C

Æ

`D(g ),

(6.30)

|h| kθkL∞ . `

(6.31)

and we bound δ h uo u t by |δ h uo u t (x)| ≤ C

These bounds are easily obtained using Schwarz inequalities in the first and the homogeneity and smoothness of the kernel in the second. The term α|h|−2α−1 g 2 |δ h ui n | 1

in (6.25) can be hidden in 4 |h|−2α D(g ) (using Young’s inequality), and the price is C α2 `|h|−2−2α g 4 . Let us choose ` = |h| g −1 kθkL∞ ,

(6.32)

and so the price is C α2 |h|−1−2α g 3 kθkL∞ , i.e., 1 α|h|−2α−1 g 2 |δ h ui n | ≤ |h|−2α D(g ) + C α2 |h|−1−2α g 3 kθkL∞ . 4

(6.33)

The term α|h|−2α−1 g 2 |δ h uo u t | in (6.25) is bounded with our choice (6.32) of ` by α|h|−2α−1 g 2 |δ h uo u t | ≤ C α|h|−2α−1 g 3 . (6.34) The term originating from M g is bound by α|h|−2α−1 g 2 |M g | ≤ C α|h|−2α−1 g 3 .

(6.35)

Putting together the bounds (6.33), (6.34), and (6.35) and using (6.26) in (6.24) we have
≤ C α + α2 kθkL∞ g 3 |h|−2α−1 . (6.36) L(|h|−2α g ) + C |h|−2α−1 g 3 kθk−1 L∞

56

Chapter 6. Critical dissipative active scalars

The right-hand side and dissipation have the same order of magnitude, g 3 |h|−2α−1 , as befits a critical case. There are no adjustable parameters, except one: α itself. If this is chosen small enough,

then we obtain

αkθkL∞ ≤ c,

(6.37)

L(|h|−2α−1 g ) ≤ 0

(6.38)

and consequently sup sup h6=0 x,t

|δ θ (x)| |δ h θ(x, t )| ≤ sup sup h 0 . 2α |h| |h|2α h6=0 x

(6.39)

We thus proved, similar to [21], the following theorem. Theorem 7. Let θ0 ∈ L∞ (Rd ). There exists α depending only on kθ0 kL∞ such that, if θ(x, t ) solves (6.1), then kθ(·, t )kC α ≤ kθ0 kC α

(6.40)

holds for all t ≥ 0. In the case of critical SQG in bounded domains we can obtain global existence of weak solutions [15] and global Hölder bounds [16].

6.3 Higher regularity The proof of higher regularity of solutions of critical SQG given in [23] is done using the Littlewood–Paley decomposition. Here we present a different proof based on a version of the nonlinear lower bound on the fractional Laplacian. In this section we consider the supercritical case ∂ t θ + u · ∇θ + Λ s θ = 0,

(6.41)

with u still given by (6.20) with 0 < s ≤ 1, and assume we are given a solution on a time interval [0, T ] and that the solution is bounded in C α (Rd ), with α > 1− s : sup kθkC α (Rd ) = Γ < ∞.

(6.42)

t ∈[0,T ]

We differentiate (6.41), denote g = ∇θ, and multiply by ∇θ: 1 (∂ + u · ∇ + Λ s ) |g |2 + D(g ) = − g (∇u)g . 2 t

(6.43)

Notice that our assumption α > 1 − s makes the situation subcritical: the lower bound (6.19) is better than cubic, s

D(g ) ≥ C Γ − 1−α | g |3+

s+α−1 1−α

.

(6.44)

Now in order to bound the right-hand side of (6.43) we split ∇u = ∇ui n + ∇u me d + ∇uo u t + M g ,

(6.45)

6.3. Higher regularity

57

where ∇ui n (x) = c P.V . ∇u me d (x) =

Z Rd

Z Rd

and ∇uo u t =

k(x − y) (g (y) − g (x))d y, |x − y|d

(6.46)

k(x − y) ∇ (θ(y) − θ(x))d y, |x − y|d y

(6.47)

χ1 (|x − y|)

χ2 (|x − y|)

Z Rd

χ3 (|x − y|)

k(x − y) ∇θ(y)d y. |x − y|d

(6.48)

We employed here a radial partition of unity χ1 (r ) + χ2 (r ) + χ3 (r ) = 1, where χ1 is supported on [0, 2ρ), χ2 supported on [ρ, 2), and χ3 supported on (1, ∞). We choose χi so that 0 ≤ χi (r ) ≤ 1 and |χ20 (r )| ≤ C ρ−1 , |χ30 (r )| ≤ C . (For r r example, χ1 (r ) = φ( ρ ), χ2 (r ) = −φ( ρ ) + φ(r ), and χ3 (r ) = 1 − φ(r ) with φ smooth, nonincreasing, 0 ≤ φ(r ) ≤ 1, identically equal to 1 on [0, 1] and 1 compactly supported in [0, 2).) We’ll choose ρ < 2 below. We use (6.19) and a Schwarz inequality for ∇ui n , s p |∇ui n (x)| ≤ C ρ 2 D(g ). (6.49) For ∇u me d we integrate by parts and use the assumption on θ: |∇u me d (x)| ≤ C Γ ρ−1+α .

(6.50)

For ∇uo u t we just integrate by parts: |∇uo u t (x)| ≤ C kθkL∞ .

(6.51)

We choose ρ to balance the first two terms: 1 ” 1— s ρ = C Γ D(g )− 2 1−α+ 2 .

(6.52)

1

If ρ < 2 , we get the upper bound h i s 1−α | g (∇u)g | ≤ C kθkL∞ + Γ s+2(1−α) D(g ) s +2(1−α) | g |2 + C | g |3 .

(6.53)

Hiding the term involving D(g ) results in s 1−α 2(s +2(1−α)) s 1 C Γ s +2(1−α) D(g ) s +2(1−α) | g |2 ≤ D(g ) + C Γ 1−α+s | g | 1−α+s . 4

2(s+2(1−α)) , is strictly smaller than the exponent of 1−α+s s +α−1 (6.44), 3 + 1−α if and only if s + α − 1 > 0, which is

The exponent of | g | above,

| g | in the lower bound precisely our situation. This allows us to hide again the right-hand side, s 1−α 3s 1 C Γ s +2(1−α) D(g ) s +2(1−α) | g |2 ≤ D(g ) + C Γ s +α−1 . 2

Similarly, from (6.44) and Young’s inequality, 3s 1 C |g |3 ≤ D(g ) + C Γ s+α−1 4

(6.54)

58

Chapter 6. Critical dissipative active scalars

holds. Putting these considerations together results in the bound 3s 1 1 (∂ t + u · ∇ + Λ s ) | g |2 + D(g ) ≤ C kθkL∞ | g |2 + C Γ s +α−1 . 2 4

(6.55)

1

This inequality was obtained if ρ defined in (6.52) obeys ρ < 2 . In the opposite 1 situation we have D(g ) ≤ C Γ 2 . In this case, using (6.49), (6.50), (6.51) with ρ = 4 we obtain that |∇u(x)| ≤ C [kθkL∞ + Γ + g ], and using the fact that (6.54) is still true, we have in this case 3s 3 1 (∂ t + u · ∇ + Λ s ) | g |2 + D(g ) ≤ C [kθkL∞ + Γ ]|g |2 + C Γ s+α−1 . 2 4

(6.56)

In view of (6.55) and (6.56), we have in either case 3s 1 1 (∂ t + u · ∇ + Λ s ) | g |2 + D(g ) ≤ C [kθkL∞ + Γ ]|g |2 + C Γ s+α−1 2 4

(6.57)

and deduce from it uniform Lipschitz bounds. We thus obtain the following. Theorem 8. Let θ be a solution of (6.41) obeying the bound (6.42) on [0, T ]. Then there exists a constant C depending on Γ , kθ0 kL∞ and T such that sup k∇θkL∞ ≤ C [k∇θ0 kL∞ + 1].

(6.58)

Passing now to C 1+α (Rd ) bounds using a similar approach is straightforward, and still higher regularity can then be obtained by calculus inequalities.

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Index action principle, 9 active scalar, 16, 26, 27, 51–58 added stress, 31 back-to-labels map, 7 Biot–Savart law, 17, 29 Boussinesq system, 23, 28–31, 47–49 Córdoba–Córdoba inequality, 52 Calderón–Zygmund operator, 34, 38–40 Cauchy formula, 13, 14, 26, 27 Cauchy invariant, 14 chord-arc condition, 43, 45 Clebsch variables, 16, 17 commutator, 12, 25, 34, 40 commutator estimates, 33

conservation of circulation, 13, 15

Lagrangian velocity, 5 magnetization, 14

damping term, 47 Ertel’s theorem, 12 Eulerian variables, 23 Eulerian velocity, 5, 16, 24 finite differences, 53 flow map, 5–7, 9 fractional Laplacian, 51, 53, 56 helicity, 14, 17 Hessian of the pressure, 21

Oldroyd-B system, 23, 31–32, 47, 48 path spaces, 33 Poisson bracket, 30 rate of strain, 12, 18 Riesz operators, 15

incompressible porous medium equation, 23, 27–28

stretching factor, 18 surface quasi-geostrophic equation (SQG), 11, 23, 26–27, 47, 48, 51, 56

label, 5–7, 15

Weber formula, 15, 16

63