*488*
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*English*
*Pages 337
[351]*
*Year 2018*

- Author / Uploaded
- Boling Guo
- Chunxiao Guo
- Yaqing Liu
- Qiaoxin Li

- Categories
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- Mechanics: Fluid Mechanics

*Table of contents : IntroductionContents1. Non-Newtonian fluids and their mathematical model2. Global solutions to the equations of non-Newtonian fluids3. Global attractors of incompressible non-Newtonian fluids4. Global attractors of modified Boussinesq approximation5. Inertial manifolds of incompressible non-Newtonian fluids6. The regularity of solutions and related problems7. Global attractors and time-spatial chaos8. Non-Newtonian generalized fluid and their applicationsBibliographyIndex*

Boling Guo, Chunxiao Guo, Yaqing Liu, and Qiaoxin Li Non-Newtonian Fluids

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Stochastic PDEs and Dynamics Boling Guo, Hongjun Gao, Xueke Pu, 2016 ISBN 978-3-11-049510-2, e-ISBN (PDF) 978-3-11-049388-7, e-ISBN (EPUB) 978-3-11-049243-9 Rigid Body Dynamics Alexey V. Borisov, Ivan S. Mamaev, 2018 ISBN 978-3-11-054279-0, e-ISBN (PDF) 978-3-11-054444-2, e-ISBN (EPUB) 978-3-11-054297-4

Boling Guo, Chunxiao Guo, Yaqing Liu, and Qiaoxin Li

Non-Newtonian Fluids |

A Dynamical Systems Approach

Mathematics Subject Classification 2010 35x, 76xx Authors Prof. Boling Guo Institute of Applied Physics and Computational Mathematics No. 6 Huayuan Road Haidian District Beijing 100088 China [email protected]

Yaqing Liu Beijing Information Science and Technology University No. 12 Qinghe Xiaoying East Road Haidian District Beijing 100192 China [email protected]

Chunxiao Guo China University of Mining and Technology (Beijing) Ding No. 11 Xueyuan Road Haidian District Beijing 100083 China [email protected]

Qiaoxin Li Institute of Physics Chinese Academy of Sciences No. 8 South-Three Street ZhongGuanCun Beijing 100190 China [email protected]

ISBN 978-3-11-054923-2 e-ISBN (PDF) 978-3-11-054961-4 e-ISBN (EPUB) 978-3-11-054940-9 Library of Congress Cataloging-in-Publication Data Names: Guo, Boling, author. Title: Non-Newtonian fluids : a dynamical systems approach / Boling Guo [and three others]. Description: Berlin ; Boston : De Gruyter, [2018] | Includes bibliographical references and index. Identifiers: LCCN 2018026144 (print) | LCCN 2018032642 (ebook) | ISBN 9783110549614 (electronic Portable Document Format (pdf) | ISBN 9783110549232 (print : alk. paper) | ISBN 9783110549614 (e-book pdf) | ISBN 9783110549409 (e-book epub) Subjects: LCSH: Non-Newtonian fluids. | Fluids. | Fluid dynamics. | Matter--Properties. | Viscosity. Classification: LCC QA929.5 (ebook) | LCC QA929.5 .N67 2018 (print) | DDC 532/.053--dc23 LC record available at https://lccn.loc.gov/2018026144 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Cover image: Chong Guo Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Introduction Non-Newtonian flow phenomenon exists in the fields of biology, physics and aviation industry. Newtonian fluids reflect the linear constitutive relation between stress tensor and the velocity gradient tensor, and the motion of incompressible viscous fluids can be described by famous Navier–Stokes equation. By contrast, the non-Newtonian fluids reflect that the stress tensor and the velocity gradient tensor of fluid movement no longer satisfy the linear relationship in a given temperature and pressure. The scientific studies have shown that the viscous fluid of the production and life generally belongs to non-Newtonian fluids; the main forms are suspended as colloid and high molecular fluid, such as various solution of polymer, blood of the human, fruit pulp and egg white are non-Newtonian fluids that satisfy these properties, because they all have the high molecular. The nature of the non-Newtonian fluids are summarized as follows by Rajagopal in 1993; the non-Newtonian fluids in the shear fluid exhibits an ability to differ from other fluids, it has the ability to cut into thin or thick fluid and the non-Newtonian has a nonzero standard stress difference in the shear fluid. In addition, it can produce creep under the action of stress. Non-Newtonian characteristics are exhibited by numerous fluids including physiological liquids, geological suspensions, industrial tribological liquids and biotechnological liquids. To describe the viscoelastic properties of such fluids, recently, constitutive equations with ordinary and fractional time or space derivatives have been introduced. The starting point of the fractional derivative model of viscoelastic fluids is usually a classical differential equation which is modified by replacing the time derivative of an integer order with fractional order and may be formulated both in the Riemann–Liouville or Caputo sense. With the development of fractional calculus, fractional derivatives and fractional partial differential equations have been applied to the numerical solution of the complex problems in fluids and continuum mechanics. In recent years, we collected and summarized the mathematical theories of non-Newtonian fluids evolution equations. This book introduces the latest research achievements, with particular emphasis on various modern approaches and recent advances. The specific content is not only concerned with the existence, uniqueness and stability of weak solutions to the initial boundary value problems, that is, strong solutions, periodic solutions and so on, but also with stability analysis, regularity or partial regularity analysis, the existence and dimensional estimates of global attractors, inertial manifold, approximated inertial manifolds as well. In particular, we give some numerical results for non-Newtonian generalized fluid, and a brief introduction of viscoelastic fluid with fractional derivative models. It should be noted that this includes the achievements in cooperation with Professor Yadong Shang and Professor Guoguang Lin. The aim of this book is to give a basic understanding of recent development in this field for researchers as well as for senior students and graduate students. Our https://doi.org/10.1515/9783110549614-201

VI | Introduction expectation is that the readers who work in the related areas can access the frontier of this study based on the reading of this book. Due to the time and knowledge limited, errors and inadequacies of the book are inevitable. Any suggestions and comments are welcome. We express our thanks to the seminar members of the Institute of Applied Physics and Computational Mathematics. We also express our gratitude to all those unnamed here. 20. August 2017

Boling Guo

Contents Introduction | V 1 1.1 1.2 1.3 1.4 2 2.1 2.1.1 2.1.2 2.2

2.3 2.3.1 2.3.2 2.4

2.4.1 2.4.2 2.4.3 2.4.4 2.5 2.6

3 3.1

Non-Newtonian fluids and their mathematical model | 1 Non-Newtonian fluids and their characteristics | 1 Incompressible and isothermal bipolar non-Newtonian fluids models | 3 Isothermal compressible viscous fluids models | 8 Other related models | 9 Global solutions to the equations of non-Newtonian fluids | 15 Global solutions to the periodic initial value problems for the incompressible non-Newtonian fluids | 15 Existence and uniqueness of global solutions to the incompressible bipolar fluids | 15 Existence of weak solutions to the incompressible monopolar fluids | 23 Global solutions to the compressible non-Newtonian fluids – Existence and uniqueness of weak solution to the bipolar compressible non-Newtonian fluids | 24 Time-periodic solutions to the incompressible bipolar fluids | 29 Time-periodic weak solutions to the incompressible bipolar fluids | 29 Existence and uniqueness of strong time-periodic solutions to the incompressible bipolar fluids | 33 Existence and uniqueness and stability of global solutions to the initial boundary value problems for the incompressible bipolar viscous fluids | 43 Existence | 44 Regularity | 50 Uniqueness | 53 Stability | 53 The periodic initial value problem and initial value problem for the modified Boussinesq approximation | 55 Periodic initial value problem and initial value problem for the non-Newtonian–Boussinesq approximation | 67 Global attractors of incompressible non-Newtonian fluids | 89 Global attractors of incompressible non-Newtonian fluids on bounded domain | 93

VIII | Contents 3.1.1 3.1.2 3.1.3 3.2 3.3 3.3.1 3.3.2

Existence of Absorbing Sets | 93 Consistently differentiability for the solution semigroup | 98 For μ1 > 0, the upper bounded estimates of dH (Aμ1 ) and dF (Aμ1 ) of attractor Aμ1 | 105 Global attractors of incompressible non-Newtonian fluids on unbounded domain | 107 Exponential attractors of incompressible non-Newtonian fluids | 119 Estimates for the nonlinear terms | 120 Compressibility on L2 (Ω) | 128

4

Global attractors of modified Boussinesq approximation | 133

5 5.1

Inertial manifolds of incompressible non-Newtonian fluids | 159 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 159 Lipschitz property | 160 The squeezing property | 163 Fixed-point theorem | 167 Inertial manifolds | 177 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids | 179 The analyticity in time and behavior of higher order modes | 180 Approximated inertial manifolds | 185

5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2 6 6.1 6.2 6.3 6.4 6.5

The regularity of solutions and related problems | 191 Stationary solutions of the incompressible bipolar non-Newtonian fluids | 191 Decay estimates of one kind of incompressible monopolar non-Newtonian fluid | 195 Partial regularity of one kind of incompressible monopolar non-Newtonian fluid | 202 The convergence of solution and attractors between one kind of incompressible non-Newtonian fluid and the Newtonian fluids | 211 Other decay estimates of incompressible non-Newtonian fluids | 215

7 7.1 7.2

Global attractors and time-spatial chaos | 227 Global attractor of low regularity | 227 Attractors and their spatial complexity of reaction-diffusion equations on bounded domain | 244

8 8.1

Non-Newtonian generalized fluid and their applications | 293 An inverse problem of a heated generalized second grade fluid | 293

Contents | IX

8.1.1 8.1.2 8.1.3 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5 8.3.6 8.3.7

Formulation | 293 Outline of the optimization method | 294 Illustrative examples | 298 A numerical study of a generalized Maxwell fluid through a porous medium | 304 Mathematical model | 305 HPM solutions | 308 Numerical results and discussion | 309 Conclusions | 313 Viscoelastic fluid with fractional derivative models | 313 Preliminaries | 314 Eigenfunction expansion of the solution and properties of the time-dependent components | 316 Finite difference approximation | 321 Duhamel-type representation of the solution | 322 Numerical experiments | 324 Two-dimensional problem | 330 Conclusion | 332

Bibliography | 333 Index | 339

1 Non-Newtonian fluids and their mathematical model In mathematics, there is twofold important significance in non-Newtonian fluid mechanics equations. On the one hand, we need to consider the more general Navier– Stokes equations when researching the uniqueness of weak solutions of three-dimensional Navier–Stokes equations in fluid mechanics. All of the research gives rise to the study of various incompressible or compressible non-Newtonian fluid dynamic equations with nonlinear constitutive equations. On the other hand, for non-Newtonian fluid, there have been various applications in chemical industry, biological engineering, glaciology, geology, hemorheology and many others, so that considerable attention has been devoted to it. In this chapter, we will show a brief introduction for the characteristics of non-Newtonian fluid, and reveal the derivation of multipolar viscous non-Newtonian fluid mechanics equations, and make some statements outlining some non-Newtonian fluid mechanics equations in hemorheology, glaciology and earth covering dynamics.

1.1 Non-Newtonian fluids and their characteristics In the past 30 years, the research of non-Newtonian fluid mechanics got great development, and has become an important branch of fluid mechanics. Schwalter [81], Huilgol [48], Böhme [14] and Rajagopal [78] have shown that the flow of high molecular weight fluid exhibit non-Newtonian flow behavior, such as polymer solution, viscoelastic fluid or viscoplastic fluids and colloidal suspension, the vast majority of biological fluids. The physical properties of non-Newtonian fluid are reflected in: this kind of material has the ability to flow generally, which can be seen as a fluid; however, it also has some properties of a solid, such as elasticity as the most typical materials may be silicon rubber. In 1993, Rajagopal summed up the main characteristics of non-Newtonian fluid: (1) In the shear flow, the fluid has the ability for shear thinning or shear thickening; (2) In the shear flow, the fluid has a nonzero normal stress difference; (3) This fluid has the ability to produce stress; (4) The fluid has the ability of present stress relaxation; (5) The fluid has the ability to spread. Then non-Newtonian fluid may have one or all of the above listed characteristics. Blood rheology research has shown that for the blood protein polymer, the flow of blood exhibits non-Newtonian flow characteristics, such as shear thinning, spread https://doi.org/10.1515/9783110549614-001

2 | 1 Non-Newtonian fluids and their mathematical model phenomenon, stress relaxation and others. The study of glaciology reveals a spreading flow of glaciers and also have non-Newtonian flow behavior. As is known to all, Newtonian fluid is a kind of fluid with constant viscosity μ0 , and the basis of its constitutive equation is τ v = 2μ0 e(u)

(1.1.1)

where τ v is partial stress tensor; e is partial strain rate tensor; u is velocity. The generalized Newton formula follows from equation (1.1.1), which reflects the linear constitutive relation between each component of partial stress tensor and the local velocity gradient tensor. With the Stokes’ assumption of zero inflation (or second) viscosity coefficient, we combine the generalized Newton formula and the momentum equation, and then obtain the (also suitable for incompressible) viscous fluid Navier– Stokes equation. It is generally believed that air and water, other general low molecular weight gases and most liquids follow the generalized Newtonian formula and the Navier–Stokes equation. In view of mathematics, the fluid, the relationship between the stress tensor and the strain rate tensor cannot be described by generalized Newtonian fluid formula, usually called the non-Newtonian fluid. For non-Newtonian fluids, even in a steady shear flow, where normal stress difference might not be equal, cause many interesting phenomena different from a Newtonian fluid, for example: (1) Weissenberg effect of viscoelastic fluid steady shear flow (Weissenberg’s lecture in the British Imperial College, London, 1994). When filled with a fluid container rotating around the vessel axis, the Newtonian fluid along the vessel wall climbs and the liquid surface is concave down, as shown in Figure 1.1.1; whereas the viscoelasticity of non-Newtonian fluid is up along the axis of rotation, as shown in Figure 1.1.2. (2) Viscoelastic fluid jet expansion effect of steady shear flow. This effect is also referred to as the Barus effect or Merrington effect. When the viscoelasticity of non-Newtonian fluid flow from a big container to a capillary, and

Figure 1.1.1: Newtonian fluid.

1.2 Incompressible and isothermal bipolar non-Newtonian fluids models | 3

Figure 1.1.2: Non-Newtonian fluid.

again flow out of capillary, we can observe the viscoelasticity of the non-Newtonian fluid from the capillary mouth begin to slowly getting bigger and bigger, and endeavor to increase to the size of the container, then it slowly got smaller and smaller. There is also an explanation: non-Newtonian fluid has memory function, and this memory is fading, as shown in Figure 1.1.3.

Figure 1.1.3: Memory function of non-Newtonian fluid.

The reader is referred to [19, 73] for more details about other properties of nonNewtonian fluid.

1.2 Incompressible and isothermal bipolar non-Newtonian fluids models The fluid flow requirement generalized Newton equation has a linear relationship between partial stress tensor and the velocity gradient tensor. Thus, by relaxing the constraints of a generalized Newton’s formula, to build mathematical models of the nonNewtonian fluid, is reflected in the following three areas: (1) nonlinear constitutive relations between the viscous part of the stress tensor and velocity gradients; (2) dependence of the viscous stress tensor on velocity gradients of order two or higher; (3) constitutive relations for higher order partial stress tensors and higher order velocity gradients tensor.

4 | 1 Non-Newtonian fluids and their mathematical model From a mathematical point of view, Ladyzhenskaya [55, 56], Kaniel [51], Du and Gunzburger [23] and others have studied the constitutive relations of various viscous fluid models with nonlinear and higher order velocity gradient properties. For perturbation of Navier–Stokes equation with a high order velocity gradient, one can refer to Lions [59], Ou and Sritharan [73, 74] and others works. In the applications, Green and Rivlin [33, 32] first studied the multipolar continuum theories; they considered an elastic nonstick material constitutive equation. For multipolar fluid models, please refer to Bleustein and Green’s thesis [9]. Necas [6] and Silhavy [71], under the theoretical framework of Green and Rivlin, established thermodynamics theory of the multipolar viscous fluid constitutive equation. Generalized constitutive principles and material frame indifference principles of their development are constant with the second law of thermodynamics is presented by the Clausius–Duhem inequality. With special emphasis on nonlinear, isothermal and incompressible, Bellout et al. [6] in the case of bipolar, extend some conclusions of the multipolar fluid mode. Let u be the velocity field, θ is temperature, ρ is density, E is the energy per unit mass of the material, η is entropy, f is the outer physical per unit mass, q is heat flux vector, r is outer radiative heat exchange rate, τi,i1 ,...,ik ,j is k = 0, 1, . . . , N − 1 space multipolar stress tensor. xi , i = 1, 2, . . . , n is the Euclidean space coordinate, and suppose τi,i1 ,...,ik ,j are symmetry about indicators i1 , . . . , ik . Taken

εi,j,k

1, if i, j, k are an even arrangement of 1, 2, 3 { { { { = {0, if i, j, k are the true repeatedly arrangement of 1, 2, 3 { { { {−1, if i, j, k are an odd arrangement of 1, 2, 3.

Using the material derivative mark, dξ 𝜕ξ 𝜕ξ = + vj . dt 𝜕t 𝜕xj Followed by the mass conservation law, the law of conservation of momentum, conservation of energy, conservation of angular momentum and the second law of thermodynamics (Clausius–Duhem inequality), we can get the above functions to satisfy the following equations: 𝜕uj dρ +ρ =0 dt 𝜕xj

(1.2.1)

ρ

(1.2.2)

ρ

dui 𝜕τi,j = + ρfi dt 𝜕xj

N−1 𝜕k+1 uj d |u|2 𝜕 (E + )= (−qi + ∑ τji1 ...ik i ) + ρfi ui + ρr dt 2 𝜕xj 𝜕xj1 ⋅ ⋅ ⋅ 𝜕xjk 𝜕xi k=0

(1.2.3)

1.2 Incompressible and isothermal bipolar non-Newtonian fluids models | 5

ρ ρ

d 𝜕 (εjkl xk ul ) = (ε x τ + εjkl τlki ) + ρεjkl xk fl dt 𝜕xi ijk k li dη 𝜕 q r ≥ − ( i)+ρ . dt 𝜕xi θ θ

(1.2.4) (1.2.5)

If E, η, q and τii1 ...ik j , k = 0, 1, 2, . . . , N − 1 are functions of ρ, ∇u, . . . , ∇k u, θ and ∇θ, and when k = N − 1, the fluid complies with the above constitutive assumptions and are called the N-polar fluid. Traditionally, this requires a sufficiently smooth process, and the constitutive relation satisfies the Clausius–Duhem inequality. Set the frame transformation of the form xi = Qij (t)xi + Ci (t)

(1.2.6)

and Qij (t)Qik (t) = δik , then the material frame indifference theory requires a transformation of equation (1.2.6) and θ, η, ρ, E and r are invariant, while qi , τii1 ...ik j change under the usual tensor form. In constitutive relations, we introduce the rate of deformation tensor e as follows, 𝜕uj 1 𝜕u ). e = (eij ) = ( i + 2 𝜕xj 𝜕xi Set τii01 ...ik j = τii1 ...ik j (ρ, 0, . . . , θ, 0) which represents the balance part of multipolar

stress τii1 ...ik j = τii1 ...ik j (ρ, e, . . . , ∇k u, θ, ∇θ), and τiiv 1 ...ik j = τii1 ...ik j − τii01 ...ik j represents the adhesive portion of the stress tensor. Let τii1 ...ik j = 0. Under the condition of isothermal incompressible, assume k ≥ 1, τii01 ...ik j ≡ 0, τij = τji ,

𝜕τlki 𝜕τ = 𝜕xkli ; furthermore, assume 𝜕xi i τiiu1 ...ik j = τiiu1 ...ik j (e, . . . , ∇k u);

(1) (2) q = −K∇θ, K > 0.

k = 0, 1, . . . , N − 1:

From equation (1.2.5), we have N−1

∑ (τjj1 ...jk i +

k=0

and

𝜕k+1 uj 𝜕 τjj1 ...jk jil ) ≥0 𝜕xl 𝜕xj1 ⋅ ⋅ ⋅ 𝜕xjk 𝜕xi

(1.2.7)

τji0 = −p(ρ, θ)δji

where p is pressure. Let Helmholtz free energy be ψ(ρ, θ) = E(ρ, θ) − θη(ρ, θ); then condition (1.2.7) and the generalized Gibb equation ρ

𝜕u dψ dθ = −ρη − p(ρ, θ) i dt dt 𝜕xi

are equivalent to equation (1.2.5), by equation (1.2.5), we know that E and η don’t depend on the gradient of u and θ, and η=−

𝜕ψ , 𝜕θ

p = ρ2

𝜕ψ . 𝜕ρ

6 | 1 Non-Newtonian fluids and their mathematical model Assuming that the fluid in constant temperature, incompressible, Bellout, Bloom and Necas investigated three basic steady flows: (1) plane Poiseuille flow between a fixed parallel plates; (2) Poiseuille flow of cylindrical pipes; (3) plane Couette flow above the moving plate with constant velocity. The constitutive equation is τij = −pδij + 2μ0 (ε + |e|2 ) τijk = 2μ1

p−2 2

eij − 2μ1 Δeij

𝜕eij

(1.2.8) (1.2.9)

𝜕xk

where, τij is the stress tensor component; τijk is the first multipolar stress tensor; eij is the rate of deformation tensor component, that is, 𝜕uj 1 𝜕u ) e = (eij ) = ( i + 2 𝜕xj 𝜕xi there ε, μ0 , μ1 and ρ are constitutive parameters. In addition, there is a multipolar stress tensor (which affects only the higher order boundary conditions), higher order velocity gradient and also involves nonlinear viscous: γ(u) = μ0 (ε + |e|2 )

p−2 2

.

So to obtain the isothermal, nonlinear bipolar incompressible viscous fluid mathematical model: ρ

𝜕u + ρ(u ⋅ ∇)u = −∇(p) + ∇ ⋅ (2γe) − 2μ1 ∇ ⋅ (Δe) + ρf 𝜕t

∇ ⋅ u = 0.

(1.2.10) (1.2.11)

In equation (1.2.8), if p = 2, μ1 = 0, the constitutive relation becomes τij = −pδij + 2μ0 eij .

(1.2.12)

This is the generalized Newton formula under the Stokes’ assumption. Then equations (1.2.10)–(1.2.11) are ρ

𝜕u + ρ(u ⋅ ∇)u − μ0 Δu = −∇(p) + ρf 𝜕t

∇ ⋅ u = 0.

(1.2.13) (1.2.14)

They are the Navier–Stokes equation and continuity equation of incompressible flow.

1.2 Incompressible and isothermal bipolar non-Newtonian fluids models | 7

If p = 2, μ1 > 0, the constitutive relation equation (1.2.8) becomes τij = −pδij + 2μ0 eij − 2μ1 Δeij .

(1.2.15)

Equations (1.2.10)–(1.2.11) are regularization Navier–Stokes equations, which Ou and Sritharan [74] discussed: ρ

𝜕u + ρ(u ⋅ ∇)u + ∇(p) + μ1 Δ2 u − μ0 Δu = ρf 𝜕t

(1.2.16)

and ∇ ⋅ u = 0.

(1.2.17)

If p ≠ 2, ε = 0, μ1 = 0, the constitutive relation becomes τij = −pδij + 2μ0 |e|p−2 eij .

(1.2.18)

Equations (1.2.10)–(1.2.11) are modified Navier–Stokes equations that Lions [59] investigated ρ

𝜕u 𝜕 + ρ(u ⋅ ∇)u + ∇(p) − 2μ0 (|e|p−2 eij ) = ρf 𝜕t 𝜕j

(1.2.19)

and ∇ ⋅ u = 0.

(1.2.20)

If p ≠ 2, ε ≠ 0, μ1 = 0, the constitutive relation becomes τij = −pδij + 2μ0 (ε + |e|2 )

p−2 2

eij .

(1.2.21)

Equations (1.2.10)–(1.2.11) are general monopolar fluid model equations ρ

p−2 𝜕u 𝜕 + ρ(u ⋅ ∇)u + ∇(p) − 2μ0 ((ε + |e|2 ) 2 eij ) = ρf 𝜕t 𝜕j

(1.2.22)

and ∇ ⋅ u = 0.

(1.2.23)

Definition. The incompressible fluid is claimed as a Newtonian fluid, if its behavior can be portrayed by the generalized Newton formula; Otherwise, it is called a nonNewtoian fluid; Furthermore, the bipolar incompressible fluid can be described by constitutive relation equations (1.2.8) with p ≠ 2, ε > 0, μ0 > 0, μ1 > 0. The incompressible mono-polar fluid can be described by the constitutive relation equation (1.2.8) with p ≠ 2, ε > 0, μ0 > 0, μ1 = 0.

8 | 1 Non-Newtonian fluids and their mathematical model

1.3 Isothermal compressible viscous fluids models Let Ω = ⋃Tt=0 Ωt , there Ωt ⊂ Rn is the region that the substance at time t ∈ [0, T), T > 0, and then we have the following equations: 𝜕ρ 𝜕(ρuj ) + =0 𝜕t 𝜕xj

(𝜕ρui ) 𝜕(ρui uj ) 𝜕(Tij ) + = + ρfi , 1 ≤ i ≤ n 𝜕t 𝜕xj 𝜕xj

(𝜕ρe) 𝜕(ρeuj ) 𝜕(Tkj uk ) 𝜕qj + = − + ρr + ρfj uj 𝜕t 𝜕xj 𝜕xj 𝜕xj

(1.3.1) (1.3.2) (1.3.3)

represent the local form of the conservation of mass, the law of conservation of momentum and the law of conservation of energy, respectively. There ρ is density; u is velocity field; E is material energy per unit mass; T is symmetric stress tensor; q is heat flux vector; r is heat exchange with the outside world by radiation; f is outer physical 2 per unit mass and e is E + |u|2 . Here, all variable assignments are in (x, t) ∈ Ωt × [0, T). In the following, for all t ≥ 0, consider Ωt = Ω. p is pressure; θ is temperature; η is the entropy of fluid; E is internal energy. Because these variables describe the fluid thermodynamic state and only two are independent, if we give ρ, θ as independent variables, then p = p(ρ, θ)

(1.3.4)

η = η(ρ, θ)

(1.3.6)

E = E(ρ, θ) e = e(ρ, θ, u).

(1.3.5) (1.3.7)

If we consider that the fluid viscous effect is quite large, the stress tensor T is not only dependent on pressure p, but also with other relevant amount, such as ∇u, ∇θ, then ̂ θ, ∇θ, ∇u). T = T(ρ, If the fluid movement is isothermal, namely the temperature θ = θ0 > 0 is constant, the tensor function T̂ does not rely on θ and ∇θ, so ̂ ∇u). T = T(ρ,

(1.3.8)

Thus, equations (1.3.1), (1.3.2) are not coupled with equation (1.3.3) and can be considered independent. In other words, if we determine ρ, u from equations (1.3.1), (1.3.2), then we can calculate the rest of the thermodynamic variables from equation (1.3.3). Taking into account the principle of material frame indifference, equation (1.3.8) can be simplified ̂ e) T = −p(ρ)I + T(ρ, where 2e = 2e(u) ≡ ∇u + (∇u)T is a symmetric part of the velocity gradient ∇u.

(1.3.9)

1.4 Other related models | 9

Then consider the special case of equation (1.3.9) T = −p(ρ)I + τE .

(1.3.10)

τE = τ(e)

(1.3.11)

τE is given by

2

2

2

, where τ : Rnsym → Rnsym is given a continuous function, and Rnsym ≡ {M ∈ Rn × Rn ; Mij = Mji , i, j = 1, 2, . . . , n}. If we consider that the gas complies with the state equation p = Rρθ,

(1.3.12)

R is universal gas constant. In the case of isothermal, the pressure is a linear function of density ρ, namely p(ρ) = βρ,

β = Rθ0 > 0.

(1.3.13)

Under the above assumptions of equations (1.3.10)–(1.3.13), equations (1.3.1)–(1.3.2) become 𝜕ρ 𝜕(ρuj ) + =0 𝜕t 𝜕xj

(1.3.14)

𝜕(ρui ) 𝜕(ρui uj ) 𝜕ρ 𝜕τij (e) + = −β + + ρfi , 𝜕t 𝜕xj 𝜕xi 𝜕xj 𝜕u

The left-hand side of equation (1.3.15) is ρ 𝜕ti + ρuj

𝜕(ui ) . 𝜕xj

i = 1, . . . , n.

(1.3.15)

If τE = τ(e) is not a linear

function of e, equations (1.3.14)–(1.3.15) are compressible non-Newtonian fluid mathematical model equations. In order to facilitate and establish global solution theory, we assume that there exists constants C1 > 0, C2 > 0, and parameters p > 1 and 2 q ∈ [p − 1, p) make for all η ∈ Rnsym and satisfy the p-mandatory condition τ(η) ⋅ η ≥ C1 |η|p

(1.3.16)

|τ(η)| ≥ C2 (1 + |η|)q

(1.3.17)

and q-growth conditions

2

where |η| = (ηij ηij )1/2 and for τ, η ∈ Rnsym , τ ⋅ η = τij ηij .

1.4 Other related models For nonlinear isothermal incompressible non-Newtonian fluid, from Section 1.3 it is known T = −pI + τE

(1.4.1)

10 | 1 Non-Newtonian fluids and their mathematical model where p is pending pressure; τE is extra stress tensor. The dynamic equations are ρ0

E 𝜕ui 𝜕u 𝜕p 𝜕τij + ρ0 uj i = − + + ρ0 fj 𝜕t 𝜕xj 𝜕xj 𝜕xj

div u = 0.

(1.4.2) (1.4.3)

Generally assume that the stress tensor τE = τ satisfies equations (1.3.16), (1.3.17). Sometimes because of mathematical theory, extra stress tensors are given by the sum of two symmetric tensor functions, that is to say for all (x, t) ∈ Ω × [0, T), τE = τ(e(x, t)) + σ(e(x, t))

(1.4.4)

τE = τ(e) + σ(e)

(1.4.5)

or

For τ, σ, adding the extra conditions: p > 1, q ∈ [p − 1, p), C1 > 0, C2 > 0, and the following estimates hold τ(η) ⋅ η ≥ C1 |η|p

σ(σ) ⋅ η ≥ 0

|τ(η) + σ(η)| ≥ C2 (1 + |η|)

(1.4.6) (1.4.7)

q

(1.4.8)

2

for all η ∈ Rnsym . The equation (1.4.5) is only used when it needs the stronger assumptions for τ and σ, for example, assuming that there exists a strictly convex potential U for τ, such that 𝜕U(e) = τij (e), 𝜕eij

i, j = 1, . . . , n

and τ is controlled by σ. If τ dependence on e is linear, that is, τ(e) = 2νe,

ν>0

and σ ≡ 0, then equations (1.4.2)–(1.4.3) become the well-known Navier–Stokes equations: ρ0

𝜕ui 𝜕u 𝜕p + ρ0 uj i = − + νΔvi + ρ0 fi , 𝜕t 𝜕xj 𝜕xi

div u = 0. There

τ(η) ⋅ η = 2ν|η|2 that is, equation (1.4.6) is satisfied for p = 2.

i = 1, 2, . . . , n

(1.4.9) (1.4.10)

1.4 Other related models | 11

The fluid characterized by equation (1.4.5) can reveal the characteristic of nonNewtonian fluid shear thinning and shear thickening. In order to describe the nature of the fluid, we consider the steady shear flow, and the velocity field is u = (v1 (x2 ), 0, 0). Let κ ≡ | dxd v1 (x2 )|, define the generalized viscosity μg , 2

μg (x) ≡

τ12 (κ) + σ12 (κ) . κ

(1.4.11)

Definition. In equation (1.4.11), if the definition of generalized viscosity μg is a monotone increasing function of κ, we call equation (1.4.5) that depicts the shear thickening fluid; if μg is a monotone decreasing function of κ, equation (1.4.5) describes a shear thinning fluid. Example 1.4.1 (The generalized Newtonian fluid and power-law fluid). given by ̃ τ(e) = 2μ(|e|2 )e = 2μ(e)e

Assume τ is (1.4.12)

and σ ≡ 0, the potential function U is defined as follows: |e|2

U(e) = ∫ μ(s)ds.

(1.4.13)

0

Let r

μ(s) = ν0 s 2 , so

ν0 > 0

(1.4.14)

2ν0 r+2 |e| r+2 τ(η) ⋅ τ = 2ν0 |η|r η ⋅ η = 2ν0 |η|r+2 U(e) =

then in equation (1.4.6) p = r + 2. So p ∈ (1, +∞), if and only if r ∈ (−1, +∞). Furtherdμ dμ more, for shear fluid μg (κ) = μ(κ), if r ∈ (−1, 0), then dκ < 0; if r > 0, then dκ > 0. In other words, if r ∈ (−1, 0) (or p ∈ (1, 2)), the model equations (1.4.12), (1.4.14) describe shear thinning fluid; if r > 0 (or p > 2), the model equation depicts a shear thickening fluid; r = 0 (or p = 2) the model equation portrays the Newtonian fluid. Example 1.4.2 (Various modifications of power-law fluids). Equation (1.4.12) contains various submodels, considering τ(1) (e) = 2ν0 |e|r e { { { { { {τ(2) (e) = 2ν0 (1 + |e|r )e { {τ(3) (e) = 2ν0 (1 + |e|2 )r/2 e { { { { (3+i) (e) = 2ν∞ e + τi (e), i = 1, 2, 3 {τ

(a) (b) (c) (d)

(1.4.15)

12 | 1 Non-Newtonian fluids and their mathematical model where v0 and v∞ are positive numbers and related with the limit of μg (κ) in κ → 0 and κ → ∞. When r ∈ (−1, 0), equation (1.4.15) depicts the model of a shear thinning fluid; If r > 0, equation (1.4.15) describes the model of a shear thickening fluid. Equation (1.4.15) can be understood by equation (1.4.14), and for example, (1.4.15b) can be decomposed into τ(4) (e) = τ(e) + σ(e). There τ(e) = 2ν∞ e, ν∞ > 0, and σ(e) = 2ν0 |e|r e, ν0 > 0. If (1.4.15d) (r > 0) is τ(e) = 2ν∞ e + 2ν0 |e|r e,

ν0 > 0,

ν∞ > 0,

(1.4.16)

which was introduced by Ladyzhenskaya in mathematical literature in 1969; equations (1.4.2), (1.4.3), (1.4.16) are also called the modified Navier–Stokes equations. Using equations (1.4.12), (1.4.13), easily obtain that tensor τ(i) of equation (1.4.15) corresponds the potential U (i) when r > −1 and i = 1, 2, . . . , 6 as follows: 2ν0 r+2 |e| , r+2 2ν0 2ν0 2ν0 (1 + |e|)r+2 − (1 + |e|)r+1 + , U (2) (e) = r+2 r+2 (r + 1)(r + 2) r+2 2ν0 [(1 + |e|2 ) 2 − 1], U (3) (e) = r+2 (3+i) U (e) = ν∞ |e|2 + U (i) (e), i = 1, 2, 3. U (1) (e) =

(1.4.15a ) (1.4.15b ) (1.4.15c ) (1.4.15d )

Example 1.4.3 (The spread of the glacier flow). In order to keep consistent with the experimental results of Kjartanson et al. [53, 54] and Van Der Veen and Whillans, Man and his collaborators (see Man and Sun [67] and their references) proposed the extra stress tensor τE as follows: τE = μ|A1 |r A1 + α1 A2 + α2 A21 .

(1.4.17)

This is the reasonable model for the spread of the glacier. Tensors A1 , A2 are the first two in Rivlin–Ericken tensors; r, μ, α1 , α2 are material constants, μ > 0, r ≈ − 32 . According to Truesdell [86], A1 = 2e and A2 =

d A + A1 (∇u) + (∇u)T A1 . dt 1

Let α1 = α2 = 0 of equation (1.4.17) to obtain equations (1.4.12), (1.4.14). On the other hand, tensor equation (1.4.17) with parameters α1 , α2 cannot be described by equations d (1.4.5)–(1.4.18), that is, τE not only depends on A1 , but also on dt A1 . In fact, r = 0, and equation (1.4.17) depicts a second grade fluid.

1.4 Other related models | 13

Example 1.4.4 (The blood flow). Experimental test revealed that a blood exhibits nonNewtonian behavior such as shear thinning, motility and stress relaxation, etc. In order to include all of these features in the model, Yeleswarapu and others proposed the generalized Oldroyd-B model in 1994. The constitutive equation is as follows; d E τ − (∇u)τE − τE (∇u)T ] dt de − (∇u)e − e(∇u)T ) = μ(e)e + λ2 ( dt

τE + λ1 [

(1.4.18)

there μ(e) = ν∞ + (ν0 − ν∞ )[

1 + ln(1 + λ|e|) ] 1 + λ|e|

(1.4.19)

where λ is material constant; λ1 λ2 are relaxation and delay coefficients; v0 > v∞ > 0 are the limit of μ(e) when |e| → 0 and |e| → ∞, respectively. Equations (1.4.18), (1.4.19) are contained in equations (1.4.5)–(1.4.8) only if λ1 = λ2 = 0. Example 1.4.5 (Earth’s mantle dynamics). The study of flows in the Earth’s mantle consists of thermal convection in a highly viscous fluid. As Malevsky and Yuen noted, experimental research indicates the nonlinear dependence between τE and e. It is used the so-called Boussinesq approximation regarding power-law fluid in planetary physics for describing processes in body interior: ρ

𝜕τijE 𝜕p 𝜕u 𝜕ui + ρuj i = − + ρθen + ρfi , 𝜕t 𝜕xj 𝜕xj 𝜕xi

div u = 0 𝜕θ 𝜕θ ρ + ρuj − Δθ = g(x, t) 𝜕t 𝜕xj

i = 1, 2, . . . , n

(1.4.20) (1.4.21) (1.4.22)

there τE is a nonlinear function of e. If θ is constant, equations (1.4.20)–(1.4.22) become equations (1.4.2)–(1.4.3). For the derivation of equations (1.4.20)–(1.4.22), one can refer to the articles of Padula [75] and Hills and Roberts [47].

2 Global solutions to the equations of non-Newtonian fluids 2.1 Global solutions to the periodic initial value problems for the incompressible non-Newtonian fluids 2.1.1 Existence and uniqueness of global solutions to the incompressible bipolar fluids Consider the bipolar fluid equation 𝜕ui 𝜕u 𝜕 𝜕p 𝜕 + (γ(u)eij ) − 2μ1 (Δeij ) + fi + ui i = − 𝜕t 𝜕xj 𝜕xi 𝜕xj 𝜕xj

∇ ⋅ u(x⋅, t) = 0,

(x, t) ∈ Ω × [0, T]

(2.1.1) (2.1.2)

and the initial boundary conditions u(x, 0) = u0 (x)

(2.1.3)

ui (x, t) = ui (x + Lej , t),

t≥0

(2.1.4)

where {ei }ni being the canonical basis of Rn , and ∫Ω u(x, t)dx = 0, t ≥ 0, Ω = [0, L]n , L > 0, γ(u) = 2μ0 (ϵ + |e|2 )

p−2 2

n

𝜕uj 1 𝜕u )) , e = (eij )ni=1 = ( ( i + 2 𝜕xj 𝜕xi i=1

(2.1.5) (2.1.6)

where p > 2, ϵ > 0, μ0 > 0, μ1 > 0 are constitutive parameters. Defining the potential function of γ(u) to be eij eij

Γ(eij eij ) = ∫ μ0 (ϵ + s)

p−2 2

ds,

(2.1.7)

0

satisfying 𝜕eij dp = γ(u)eij . dt 𝜕t

(2.1.8)

Let H(Ω) = {u ∈ L2per (Ω) : ∇ ⋅ u = 0, ∫ udx = 0} Ω

https://doi.org/10.1515/9783110549614-002

16 | 2 Global solutions to the equations of non-Newtonian fluids 2 V(Ω) = {u ∈ Hper (Ω) : ∇ ⋅ u = 0, ∫ udx = 0}. Ω

We use (⋅, ⋅) and ‖ ⋅ ‖ to denote the inner product and norm of H, respectively, and ‖ ⋅ ‖1 the norm of H 1 (Ω), ‖ ⋅ ‖2 the norm of V(Ω). We consider the following fundamental linear problem associated with (2.1.1)– (2.1.4): ∇ ⋅ (Δe) + ∇p = f ,

∇ ⋅ u = 0,

x∈Ω

x ∈ Ω.

(2.1.9) (2.1.10)

With (2.1.9)–(2.1.10) associate the linear operator A which is defined as the positive definite V-elliptic bilinear form a(⋅, ⋅) : V × V → R

(2.1.11)

1 a(u, v) = (Δu, Δv). 2

(2.1.12)

given by

Therefore, A : V → V is a linear operator, ⟨Au, Av⟩ = a(u, v) = ⟨f , v⟩,

v ∈ V, f ∈ V ,

(2.1.13)

where V is the dual of V. The domain D(A) of A is given by D(A) = {u ∈ V : a(u, v) = (f , v), f ∈ H, v ∈ V}.

(2.1.14)

Thus, the positive, self-adjoint operator A : D(A) → H is also a linear operator, by the Rellich lemma, A−1 is a compact map in H, then there exists accumulation real eigenvalues λj such that 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ≤ λj → ∞,

j→∞

(2.1.15)

and corresponding eigenfunctions {ϕj }∞ j=1 forms orthogonal base of H, that is, Aϕj = λj ϕj ,

ϕj ∈ D(A),

j = 1, 2, . . . .

(2.1.16)

Let u ∈ D(A), i = 1, . . . , n: (Au)i =

𝜕 (δeij ) 𝜕xj

p−2 𝜕 [(ϵ + |e|2 ) 2 eij ] 𝜕xi B(u, v) = u ⋅ ∇v, u, v ∈ D(A).

(Ap u)i =

Then the problem (2.1.1)–(2.1.4) is equivalent to the abstract problem posed in V ut + 2μ1 Au − 2μ0 Ap u + B(u, u) = f

u(0) = u0 .

(2.1.17) (2.1.18)

2.1 Global solutions for the incompressible non-Newtonian fluids | 17

Theorem 2.1.1. Assuming f ∈ L2 (0, T; V ), u0 ∈ H. Then the problem (2.1.17)–(2.1.18) exist the uniquely weak solution u ∈ L2 (0, T; V) ∩ C(0, T; H). And for any t ∈ (0, T) satisfying the following energy equation: t

t

‖u‖2 + 2μ1 ∫ ‖u(s)‖22 ds + μ0 ∫ ∫(ϵ + |e(u)|2 ) 0

p−2 2

|e(u)|2 dxdt

0 Ω

t

= ‖u(0)‖2 + 2 ∫⟨f , u⟩dt.

(2.1.19)

0

Proof. We prove the existence of the solution by using the Galerkin method. Assuming Em = span{ϕ1 , . . . , ϕn }, for each m, an approximation solution of equation (2.1.17) will be defined as follows: m

um = ∑ Cim (t)ϕi

(2.1.20)

i=1

and

1

1

(um , ϕj ) + 2μ1 (A 2 um , A 2 ϕj ) + μ0 (γ(um e(um )), e(ϕj )) + b(um , um , ϕj ) = ⟨f , ϕj ⟩ t ∈ [0, T], j = 1, . . . , m.

(2.1.21)

We will choose the initial data for this system in the following special way. Let um (0) = u0m ∈ Em , where u0m is the orthogonal projection of u0 onto Em . Substituting (2.1.20) into (2.1.21), we obtain m

m

i=1

i=1

1

m

1

(t) + 2μ1 ∑(A 2 ϕi , A 2 ϕj )Cim (t) + ∑ b(ϕi , ϕl ), ϕj Cim (t)Clm (t) ∑(ϕi , ϕj )Cim m

m

i=1

i=1

i,l=1

2

+ μ0 ∑([ϵ + (∑ e(ϕi )Clm (t)) ]

p−2 2

e(ϕi ), e(ϕj ))

Cim (t) = ⟨f , ϕj ⟩.

Inverting the nonsingular matrix with elements (ϕi , ϕj ), 1 ≤ i, j ≤ m, we write the differential equation in the form m

m

m

j=1

j,k=1

j,k=1

Cim (t) + ∑ αij Cjm (t) + ∑ γijk Cjm (t)Ckm (t) + ∑ βijk [ϵ + Cjm (t)2 ] m

p−2 2

Ckm (t)

= ∑ δij ⟨f , ϕj ⟩, j=1

i = 1, . . . , m, where αij , βijk , γijk , δij ∈ R and Cim (0) is the component of u0m . That is, d C (t) = Φi (t, Clm (t), . . . , Cmm (t)). dt im Cim (0), i = 1, . . . , m, is given, and ⟨f , ϕj ⟩ are square integrable.

18 | 2 Global solutions to the equations of non-Newtonian fluids For a fixed t, Φi are continuous to Cim and Φi is square integrable in time. Hence due to the Carathéodory theorem, there exists a maximal solution on [0, Tm ]. In addition, this Galerkin solution is unique. We will now derive certain a priori estimates independent of m and t. We multiply equation (2.1.20) by Cjm (t) and add for j = 1, . . . , m to get p−2 1 d ‖um (t)‖2 + μ1 ‖um ‖22 + μ0 ∫(ϵ + |e(um )|2 ) 2 |e(um )|2 dx 2 dt

Ω

= ⟨f , um ⟩ ≤ ‖f ‖V ‖um ‖2 . Applying the Young inequality to get p−2 ‖f ‖2V d‖um ‖2 + μ1 ‖um ‖22 + 2μ0 ∫(ϵ + |e(um )|2 ) 2 |e(um )|2 dx ≤ . dt μ1

(2.1.22)

Ω

We drop the term 2μ0 ∫Ω (ϵ + |e(um )|2 )

p−2 2

|e(um )|2 dx of t equation (2.1.20):

‖f ‖2V d‖um ‖2 . + μ1 ‖um ‖22 ≤ dt μ1

(2.1.23)

Dropping the term μ1 ‖um ‖22 of equation (2.1.23) and integrate the inequality from 0 to s to obtain s

s

0

0

1 1 ‖um (s)‖ ≤ ‖u0m ‖ + ∫ ‖f (t)‖2V dt ≤ ‖u0 ‖2 + ∫ ‖f (t)‖2V dt, μ1 μ1 2

2

(2.1.24)

since u0m is the orthogonal projection of u0 . Thus sup ‖um (s)‖2 ≤ ‖u0 ‖2 +

s∈[0,T]

T

1 ∫ ‖f (t)‖2V dt. μ1 0

Since u0 ∈ H and f ∈ L2 (0, T; V ), the right-hand side is bounded independent of m for each t ∈ [0, T]. We write um ∈ bounded set of L∞ (0, T; H). We come back to equation (2.1.23) and integrate from 0 to T to get T

∫ ‖um (t)‖22 dt ≤ 0

T

1 1 [‖u0 ‖2 + ∫ ‖f (t)‖2V dt]. μ1 μ1

(2.1.25)

0

This implies that um ∈ bounded set of L∞ (0, T; V). Similarly, the sequence um remains in a bounded set of L∞ (0, T; H), hence we can extract a subsequence um ⇀ u∗ in the weak topology of L∞ (0, T; H). That is, as m →

2.1 Global solutions for the incompressible non-Newtonian fluids | 19

T

∞, ∫0 (um − u∗ , v)dt → 0, ∀v ∈ L1 (0, T; H). In particular, we have T

∀v ∈ L2 (0, T; V ).

lim ∫(um − u∗ , v)dt = 0,

m →∞

(2.1.26)

0

Since this last convergence result is valid for v ∈ L2 (0, T; H), we also have T

∀v ∈ L2 (0, T; H).

lim ∫(um − u, v)dt = 0,

m→∞

(2.1.27)

0

By subtracting (2.1.25) from (2.1.27), we have T

∀v ∈ L2 (0, T; H).

∫(u − u∗ , v)dt = 0,

(2.1.28)

0

This implies that u = u∗ ∈ L2 (0, T; V) ∩ L∞ (0, T; H). Thus we can extract a subsequence um from um such that as m → ∞, um → u in L2 (0, T; V) um → u weakly, um → u in L∞ (0, T; H) weak-star. In fact, we have (um , ϕj ) = ⟨f − 2μ1 Aum − 2μ0 Ap (um ) − B(um , um ), ϕj ⟩, which implies um = f − 2μ1 Aum − Ap (um ) − B(um , um ) ∈ L2 (0, T; V ). Lemma 2.1.1. Let {uk } be a sequence that converge weakly in L2 (0, T; V), weak-star in L∞ (0, T; H) and strongly in L2 (0, T; H). Then for any ω ∈ X = {ω ∈ C(0, T; V); ω ∈ L2 (0, T; H)} the following limits are satisfied: T

(1)

lim ∫(uk (t), ω (t))dt = ∫(u(t), ω (t))dt

k→∞

0

T

(2)

k→∞

1

1

T

1

lim ∫((ϵ + |e(uk )|2 )

k→∞

0

0

0

T

(4)

0

1

lim ∫(A 2 uk (t), A 2 ω(t))dt = ∫ t0T (A 2 u(t), A 2 ω(t))dt T

(3)

T

p−2 2

T

e(uk ), e(ω))dt = ∫((ϵ + |e(u)|2 ) T

p−2 2

e(u), e(ω))dt

0

lim ∫ b(uk (t), uk (t), ω(t))dt = ∫ b(u(t), u(t), ω(t))dt.

k→∞

0

0

Proof. (1) This result follows from the fact that uk → u is strongly in L2 (0, T; H) and ω ∈ L2 (0, T; H).

20 | 2 Global solutions to the equations of non-Newtonian fluids 1

1

(2) The relation ⟨u, Aω⟩ = (A 2 u, A 2 ω), ∀u, ω ∈ V implies T

T

1 2

1 2

lim ∫(A uk (t), A ω(t))dt = lim ∫⟨uk (t), Aω(t)⟩dt

k→∞

k→∞

0

T

0

T

1

1

= ∫⟨u(t), Aω(t)⟩dt = ∫(A 2 u(t), A 2 ω(t))dt. 0

0

(3) The difference T

∫((ϵ + |e(uk )|2 ) 0

p−2 2

T

e(uk ), e(ω))dt − ∫((ϵ + |e(ω)|2 ) 0

T

≤ ∫((ϵ + |e(uk )|2 ) 0 2

−(ϵ + |e(u)| )

p−2 2

p−2 2

p−2 2

e(u), e(ω))dt

T

e(uk − u), e(ω))dt + ∫[((ϵ + |e(uk )|2 ) 0

)e(u), e(ω)]dt = I1 + I2 .

We use the Hölder inequality, notice 2 < p ≤ 6, n ≤ 3, 2 < p < 2 + the Sobolev embedding inequality T

2

I1 ≤ C0 ∫[(∫(ϵ + |e(uk )| ) 0

Ω T

≤ C1 (∫(ϵ

p−2 2

p−2 2

2

3(p−2) 4

1 2

2 ≤ n < 10 and

2 3

dx) ‖uk − u‖W 1,4 (Ω) ‖ω‖W 1,3 (Ω) ]dt T

+ ‖uk ‖2 ) dt) (∫ ‖uk −

0

4 , n−2

0

u‖2W 1,4 (Ω) dt)

1 2

→ 0 (k → ∞)

where C0 , C1 are constants dependent of Ω and independent of k. Using the Aubin– Lions lemma, Y = {uk ∈ L2 (0, T; V), uk ∈ L2 (0, T; V )} →→ L2 (0, T; W 1,q (Ω)) compact embedding, so when 21 ≥ q1 > 21 − n1 , Ω ∈ Rn , then uk → u strongly converges in L2 (0, T; W 1,q (Ω)). Therefore, I1 → 0 (k → ∞). We prove I2 → 0 (k → ∞) in the following two cases. Assuming |e(uk )| < |e(u)|, then by the mean value theorem, there exists |e(uk )| < s∗ < |e(u)| such that T

p−4 p−2 I2 ≤ ∫ ∫(ϵ + s∗ ) 2 (|e(uk )| + |e(ω)|)|e(uk ) − u| ⋅ |e(ω)|dxdt. 2

0 Ω

For (ϵ + s∗ )

p−4 2

≤ p−4 2 , { ϵ p−4 ≤ (ϵ + |e(ω)|) 2 ,

p |e(u)| also can be obtained. So I2 → 0 (k → 0). Therefore, (3) is proved. (4) The difference T

T

∫ b(uk (t), uk (t), ω(t))dt − ∫ b(u(t), u(t), ω(t))dt 0

0

T

T

= ∫ b(uk (t) − u(t), uk (t), ω(t))dt + ∫ b(u(t), uk (t) − u(t), ω(t))dt = I1 + I2 0

0

T

T

I1 ≤ ∫ ∫ |uk (t) − u(t)||∇uk ||ω(t)|ddt ≤ ∫ ∫ ‖ω‖‖uk (t) − u(t)‖ ⋅ ‖∇uk ‖dxdt 0 Ω

0 Ω T

T

0

0

1 2

T

≤ C4 ∫ ‖uk (t) − u(t)‖ ⋅ ‖∇uk ‖ ⋅ ‖ω‖2 dt ≤ C5 (∫ ‖uk (t) − u(t)‖2 dt) (∫ ‖uk ‖22 dt) →0

0

(k → 0)

by the condition uk (t) → u(t) strongly converges in L2 (0, T; H). T

T

I2 ≤ ∫ ∫ |u(t)| ⋅ |∇(uk − u)||ω|dxdt ≤ ∫ ‖ω‖∞ ‖u‖ 0 ω

T

2

1 2

0 T

≤ C6 (∫ ‖u‖ dt) (∫ ‖uk − 0

0

u‖2W 1,4 (Ω)dt )

4

L3

1 2

→0

‖uk − u‖W 1,4 (Ω) dt

(k → 0).

Similar to the proof of (3), using the Aubin–Lions lemma, we get Lemma 2.1.1.

1 2

22 | 2 Global solutions to the equations of non-Newtonian fluids Let ψ ∈ C(0, T), ψ(T) = 0, ψ(t). Multiplying (2.1.21) by ψ(t) and integrating by parts T

T

1

T

1

− ∫(um (t), ψ (t)ϕj )dt + 2μ ∫(A 2 um , A 2 ψ(t)ϕj ) + μ0 ∫(γ(um )e(um ), e(ψ(t)ϕj ))dt 0

0

T

0

T

+ ∫ b(um , um , ψ(t)ϕj )dt = (u0m , ϕj )ψ(0) + ∫⟨f , ψ(t)ϕj ⟩dt. 0

0

Using the conclusion of Lemma 2.1.1, let m → ∞: T

T

1 2

I(ϕj ) = − ∫(u(t), ψ (t)ϕj )dt + 2μ ∫(A u, A

0

0

T

1 ψ(t)ϕj 2

T

)dt + μ0 ∫(γ(u)e(u), e(ψ(t))ϕj )dt 0

T

+ ∫ b(u, u, ψ(t)ϕj )dt − (u0 , ϕj )ψ(0) + ∫⟨f , ψ(t)ϕj ⟩dt 0

0

I(⋅) is continuous in V, span{ϕ1 , . . . , ϕm } is dense in V, so I(ν) = 0,

∀ν ∈ V.

(2.1.29)

Now we assume ψ(t) ∈ D(0, T), T

T

1

1

T

− ∫(u, ν)ψ (t)dt = −2μ1 ∫(A 2 u, A 2 ν)ψ(t)dt + ∫{−μ0 (γ(u)e(u), e(ν)) 0

0

0

− b(u, u, ν) + ⟨f (t), ν⟩}ψ(t)dt,

∀ν ∈ V

then in D(0, T) , 1 1 d (u, ν) + 2μ1 (A 2 u, A 2 ν) + μ0 (γ(u)e(u), e(ν)) + b(u, u, ν) = ⟨f , ν⟩. dt

Multiplying (2.1.30) by ψ(t) and integrating T

T

1

1

T

− ∫(u, ν)ψ (t)dt + 2μ1 ∫(A 2 u, A 2 ψ(t)ν)dt + μ0 ∫(γ(u)e(u), e(ψ(t)ν))dt 0

T

0

T

0

+ ∫ b(u, u, ψ(t)ν)dt = ∫⟨f , ψ(t)ν⟩dt + (u(0), ν)ψ(0). 0

0

Comparing the above equation with (2.1.29), we obtain (u(0) − u0 , ν)ψ(0) = 0.

(2.1.30)

2.1 Global solutions for the incompressible non-Newtonian fluids | 23

Taking ψ(0) ≠ 0, then (u(0) − u0 , ν) = 0, ∀ν ∈ V, so u(0) = u0 . d Because u ∈ L2 (0, T; V ), u ∈ L2 (0, T; V), then u ∈ C(0, T; H) and dt ‖u‖2 = 2⟨u , u⟩ are established in D(0, T) . Taking inner product on both sides of equation (2.1.17) with u, and integrating from 0 to t, then equation (2.1.19) is obtained. Therefore, the global solution of problem (2.1.17)–(2.1.18) is obtained. Now we prove the uniqueness of the solution. Actually, let u, v be two weak solutions of problem (2.1.17)–(2.1.18), denote ω(t) = u(t) − ν(t), u(0) = ν(0), and then by equation (2.1.17) we have ω (t) + 2μ1 Aω + 2μ0 (Ap (u) − Ap (ν)) + B(u, ω) + B(ω, u) = 0 ω(0) = 0

(2.1.31) (2.1.32)

Taking inner product of the equation (2.1.31) with ω, d‖ω‖2 + 2μ1 ‖ω‖22 + 2μ0 (Ap (u) − Ap (ν), ω) ≤ |b(ω, u, ω)| dt

(2.1.33)

computing, we get (Ap (u) − Ap (ν), ω) ≥ 0 b(ω, u, ω) ≤ ‖ω‖∞ ‖u‖2 ‖∇u‖ ≤ C0 ‖ω‖2 ‖u‖2 ‖ω‖ ≤ μ1 ‖ω‖22 +

C0 ‖u‖22 ‖ω‖2 . 4μ1

Therefore, by equation (2.1.33), we have C d‖ω‖2 + μ1 ‖ω‖22 ≤ 0 ‖u‖22 ‖ω‖2 , dt 4μ1 dropping the left-hand term μ1 ‖ω‖22 of the above equation and applying the Gronwall inequality, T

C ‖ω‖ ≤ ‖ω(0)‖ exp{ 0 ∫ ‖u‖22 dt} = 0 4μ1 2

2

0

then u ≡ ν; the uniqueness is proved. Therefore, Theorem 2.1.1 is obtained. 2.1.2 Existence of weak solutions to the incompressible monopolar fluids Under the conditions of the above section, and assuming μ1 = 0, by equation (2.1.1) we can obtain the unipolar flow equation ut − 2μ0 Ap u + B(u, u) = f

u(0) = u0 .

(2.1.34) (2.1.35)

24 | 2 Global solutions to the equations of non-Newtonian fluids Redefining the space H(Ω) = {u ∈ L2per : ∇ ⋅ u = 0, ∫ udx = 0} Ω

n

1,p V(Ω) = {ν|ν ∈ (Wper (Ω)) , ∇ ⋅ ν = 0, ∫ dx = 0}. Ω

Selecting the base: assuming s > 1 + n2 , u ∈ H s (Ω), then Di u ∈ H s−1 (Ω) ⊂ L∞ (Ω) for 1 − s−1 < 0 so we can get Vs ⊂ V ⊂ H ⊂ H ⊂ V ⊂ Vs where Vs denote the closure of 2 n D(Ω) in H s (Ω)n . For the eigen-problem (ωj , ν)Vs = λj (ωj , ν) where the eigenfunctions {ωj }∞ j=1 selected the base, similar to the method of the above section, and using the Galerkin method, the following theorem is obtained. 2n . Then problem (2.1.34)– Theorem 2.1.2. Assume f ∈ Lp (0, T; V ), u0 ∈ H and p ≥ 1+ 2+n (2.1.35) exists a weak solution u,

u ∈ Lp (0, T; V) ∩ L∞ (0, T; H). Remark. For the Cauchy problem of the incompressible non-Newtonian fluids, the conclusions are obtained. But the uniqueness of unipolar flow (p > 2) has not been solved yet.

2.2 Global solutions to the compressible non-Newtonian fluids – Existence and uniqueness of weak solution to the bipolar compressible non-Newtonian fluids Considering the dipolar compressible non-Newtonian fluids equation of constant temperature 𝜕ρ 𝜕 + (ρνi ) = 0, 𝜕t 𝜕xi 𝜕 𝜕 𝜕 𝜕p (ρν ) + (ρνi νj ) − τ (ν) = − + ρbi 𝜕t i 𝜕xj 𝜕xj ij 𝜕xi

(2.2.1) i = 1, . . . , n

(2.2.2)

where ρ is density; ν = (ν1 , . . . , νn ) is velocity vector; b = (b1 , . . . , bn ) is the external force density; p = βρ is pressure, β is the constant. Denoting ΩT = I × Ω, I = (0, T), Ω ⊂ Rn be the boundary domain, having an infinitely differentiable boundary 𝜕Ω. Let b ∈ L∞ (QT )

(2.2.3)

stress tensor τij is τij =

𝜕V(e) 𝜕 𝜕W(De) − [ ] 𝜕eij 𝜕xk 𝜕( 𝜕eij ) 𝜕xk

(2.2.4)

2.2 Existence and uniqueness of weak solution |

25

where De = (𝜕eij /𝜕xk )ni,j,k=1 𝜕W(De)

τijk =

𝜕e

𝜕( 𝜕xij )

(2.2.5)

.

k

Moreover, assume the potential function V, W satisfying C1 (1 + |De|)q−2 |ξ |2 ≤ C3 (1 + |e|)q−2 |ξ |2 ≤

𝜕2 W(De) ξ k ξ g ≤ C2 (1 + |De|)q−2 |ξ |2 𝜕(𝜕eij /𝜕xk )𝜕(𝜕elm /𝜕xg ) ij lm

(2.2.6)

𝜕2 V(e) ξ ξ ≤ C4 (1 + |e|)q−2 |ξ |2 𝜕eij 𝜕elm ij lm

(2.2.7)

W(0) = 0, 𝜕W = 0, 𝜕(𝜕eij /𝜕xk )

V(0) = 0

(2.2.8)

𝜕V =0 𝜕eij

(2.2.9)

where C1 , C2 , C3 , C4 are positive constants; q > n. The initial value condition ν(0) = ν0 ,

ρ0 = ρ0 ,

ρ0 > 0,

x∈Ω

(2.2.10)

and boundary condition, τijk νj νk = 0,

(t, x) ∈ (0, T) × 𝜕Ω,

(2.2.11)

where ν is the external normal vector of 𝜕Ω: ν = 0,

(t, x) ∈ (0, T) × 𝜕Ω.

(2.2.12)

The basic space: Lp (Ω) and W 1,p (Ω), 0 ≤ q, l < ∞ are the usual Lebesgue and Sobolev spaces. Let k,p

W 0 (Ω) = {ϕ ∈ W01,p (Ω) : ∫ ϕdx = 0} k,p ∗ (W 0 (Ω))

Ω

=

−k,p W 0 (Ω),

p > 1, k = 1, 2, . . . .

Assuming Q1,2 = I1,2 × Ω = (t1 , t2 ) × Ω, W01 W0s,2 (Q1,2 ) = C0∞ (Q1,2 ) W01 W0s−2

0 ≤ t1 < t2 ≤ +∞

‖⋅‖

‖u‖W 1 W s,2 (Q1,2 ) 0

1

2 𝜕m u 2 = [ ∑ ] 𝜕x ⋅ 𝜕xi W s−2 (Q1,2 ) m=0 i1 m 0

s

and assume ψ(t) = (1 + t) ln(1 + t) − t, Lψ is the Orliz space.

26 | 2 Global solutions to the equations of non-Newtonian fluids Definition 2.2.1. The pair of functions (ρ, ν) is called to be the weak solution of the problem (2.2.1), (2.2.2), (2.2.10)–(2.2.12), if (ρ, ν) satisfy the following conditions: (1) (2) (3) (4)

ρ ∈ L∞ (I; W 1,q (Ω)); 𝜕ρ ∈ L∞ (I; Lq (Ω)); 𝜕t ν ∈ L∞ (I; W 2,q (Ω) ∩ W01,2 (Ω)); 𝜕ν ∈ L2 (QT ); 𝜕t

(5) the continuous equation is in the sense of distribution and is also true; (6)

∫ Ω

𝜕ϕ 𝜕ϕ 𝜕 (ρν )ϕ dx − ∫ ρνi νj i dx − β ∫ ρ i dx 𝜕t i i 𝜕xj 𝜕xi Ω

Ω

𝜕V 𝜕W(De(ν)) 𝜕eij +∫ (e(ν))eij (ϕ)dx + (ϕ)dx = ∫ ρbi ϕi dx, 𝜕eij 𝜕(𝜕eij /𝜕xk ) 𝜕xk Ω

Ω

for a, e, t ∈ I, any φ = (φ1 , . . . , φn ) ∈ W 2,q (Ω) ∩ W01,2 (Ω) ∩ C ∞ (Ω) hold; (7) initial condition ρ0 ∈ C 1 (Ω), ν0 ∈ W 2,q (Ω) ∩ W01,2 (Ω). Theorem 2.2.1. Assume ρ0 ∈ C 1 (Ω), ρ0 > 0, ν0 ∈ W 2,q (Ω) ∩ W01,2 (Ω), and we assume (2.2.6)–(2.2.9) are true. Then problems (2.2.1), (2.2.2), (2.2.10)–(2.2.12) exist and the uniquely weak solution (ρ, ν), 𝜕ρ ∈ L∞ (I; Lq (Ω)) 𝜕t 𝜕ν ν ∈ L∞ (I; W 2,q (Ω) ∩ W01,2 (Ω)), ∈ L2 (QT ). 𝜕t

ρ ∈ L∞ (I; W 1,q (Ω)),

(2.2.13) (2.2.14)

2,2 1,2 Proof. Existence. We choose the orthogonal system {z k }∞ k=1 in W (Ω) ∩ W (Ω). By the following eigenvalue problem,

≪ ν, z k ≫ = λk (ν, z k ),

ν ∈ W 2,2 (Ω) ∩ W01,2 (Ω)

(2.2.15)

where ≪ ⋅, ⋅ ≫ denotes the inner product of W 2,2 (Ω). (⋅, ⋅) is the L2 (Ω) inner product. Selecting the mapping Pm , m

Pm ν = ∑ λk (ν, z k ) ⋅ z k , k=1

ν ∈ L2 (Ω),

(2.2.16)

is the orthogonal projection from L2 to span{z 1 , . . . , z m } and from W 2,2 (Ω) to span{z 1 , . . . , z m }. Letting the approximate solution m

νm (t, x) = ∑ Ck (t)z k (x) k=1

(2.2.17)

2.2 Existence and uniqueness of weak solution |

27

where C = (C1 , . . . , Cm ) ∈ C 1 (I), solving ρm ∈ C 1 (QT ) such that 𝜕ρm 𝜕 + (ρ νm ) = 0 𝜕t 𝜕xi m i

(2.2.18)

ρm (0, x) = ρ0 (x) ∈ C 1 (Ω),

ρ0 (x) > 0,

∀x ∈ Ω.

By the characteristics of the method, t

ρm (t, x) = ρ0 (y) exp(− ∫ 0 m

m

𝜕 m ν (τ, x m (τ))dτ), 𝜕xi i

(2.2.19)

m

where y = x (0), x = x (t) and ν of (2.2.17) exist C = (C1 (t), . . . , Cm (t)), t ∈ I satisfying ∫(ρm Ω

𝜕νim 𝜕νm 𝜕ρ 𝜕V (e(νm )) ⋅ eij (z l )dx + ρm νjm i + β m )zil dx = − ∫ 𝜕t 𝜕xj 𝜕xi 𝜕eij Ω

𝜕eij l 𝜕W −∫ (De(νm )) ⋅ (z )dx + ∫ ρm bi zil dx, 𝜕(𝜕eij /𝜕xk ) 𝜕xk Ω

l = 1, 2, . . . , m

(2.2.20)

Ω

by (2.2.18) and (2.2.20), we have ∫ ρm dx = ∫ ρ0 dx = m0 Ωt

(2.2.21)

Ω

𝜕 𝜕 1 ∫ (ρm |νm |2 )dx + ∫ (ρm ln ρm − ρm )dx 2 𝜕t 𝜕t Ωt

Ωt

+ ∫[ Qt

∫ ρm ( Qt

𝜕eij m 𝜕W 𝜕V (e(νm )) ⋅ eij (νm ) + ⋅ (ν )]dx = ∫ ρm bi νim dx 𝜕eij 𝜕(𝜕eij )/𝜕xk 𝜕xk

(2.2.22)

Qt

2

𝜕νim 𝜕νm 𝜕νm ) dx + ∫ ρm i νjm i dx + ∫ [V(e(νm (0))) + W(De(νm ))]dx 𝜕 𝜕xj 𝜕t Qt

Qt

𝜕νm 𝜕ρ 𝜕νm + β ∫ m i dx = ∫ [V(e(νm (0))) + W(De(νm (0)))]dx + ∫ ρm bi i dx. (2.2.23) 𝜕xi 𝜕t 𝜕t Ωt

Qt

Qt

Applying (2.2.21), we have ‖ρm ‖L∞ (I;L1 )(Ω) ≤ m0 = ∫ ρ0 dx.

(2.2.24)

Ω

Using (2.2.22), we get ‖ρm |νm |2 ‖L∞ (I;L1 (Ω)) ≤ C

‖ρm ln ρm ‖L∞ (I;L1 (Ω)) ≤ C.

(2.2.25) (2.2.26)

28 | 2 Global solutions to the equations of non-Newtonian fluids 𝜕V By the ellipticity of ((ν, u)) = ∫Ω [ 𝜕e (e(ν))eij (u) + ij

𝜕e 𝜕W (De(ν)) 𝜕xij (u)]dx, 𝜕(𝜕eij /𝜕xk ) k

((ν, u)) ≥ C1 ‖ν‖qW 2,q (Ω) + C2 ‖ν‖2W 2,2 (Ω) ,

C1 > 0, C2 > 0.

Therefore, ‖νm ‖Lq (I;W 2,q (Ω)) ≤ C,

‖νm ‖L2 (I;W 2,2 (Ω)) ≤ C.

(2.2.27)

Combining (2.2.27) and (2.2.19), ‖ρm ‖L∞ (QT ) ≤ C

(2.2.28)

‖ρm ‖L∞ (I;W 1,q (Ω)) ≤ C

(2.2.29)

and by (2.2.18) we have

and 𝜕ρ m ≤ C. 𝜕t Lq (QT ) Computing (2.2.23), we get 𝜕νm ≤C 𝜕t L2 (QT )

(2.2.30)

‖νm ‖L∞ (I;W 2,q (Ω)∩W 1,2 (Ω)) ≤ C

(2.2.31)

𝜕ρ m ≤ C. 𝜕t L∞ (I;Lq (Ω))

(2.2.32)

0

and

So the priori estimates are obtained. Therefore, there exists a subsequence (ρm , νm ) such that ρm → ρ, νm → ν by the limit process, the existence is obtained. Now we prove the uniqueness. Let (ρ, ν), (ρ, ν) be two solutions of problem (2.2.1), (2.2.2), (2.2.10)–(2.2.12), denoting ξ = ρ − ρ, ω = ν − ν, and by (2.2.1), we get 𝜕ωj 𝜕ξ 𝜕νj 𝜕ξ 𝜕ρ = −ξ −ρ − ν − ω. 𝜕t 𝜕xj 𝜕xj 𝜕xj j 𝜕xj j

(2.2.33)

Multiplying (2.2.33) by ξ and integrating over QT , applying the priori estimates of ρ, ρ, ν, ν and the Young inequality, we have t

1 ∫ ξ 2 dx ≤ C1 K1 (ϵ) ∫ ξ 2 dxdt + C2 ϵ ∫ ‖ω‖2W 2,2 (ω) dt 2 QT

QT

where, K1 (ϵ) > 0, ϵ > 0 are positive constants.

0

(2.2.34)

2.3 Time-periodic solutions to the incompressible bipolar fluids | 29

By (2.2.2), we can imply ∫ρ QT

𝜕ωi 𝜕ν 𝜕ν 𝜕dxdt + ∫ ξ i ωdxdt + ∫ ξνi i ωj dxdt 𝜕t 𝜕t 𝜕xj QT

+ ∫ (ρωj QT

QT

𝜕νi 𝜕ω ω + ρνj i ωi )dxdt 𝜕xj i 𝜕xj

= β ∫ [(ρ − ρ)

𝜕ωi − ((ν, ω)) + ((ν, ω))]dxdt + ∫ ξbi ωi dxdt 𝜕xi

(2.2.35)

QT

QT

t

β ∫ (ρ − ρ)

𝜕ωi dxdt ≤ C‖ξ ‖L2 (QT ) ∫ ‖ω‖W 2,2 (Ω) dt 𝜕xi 0

QT

T

≤ C1 K(ϵ) ∫ ξ 2 dxdt + C2 ϵ ∫ ‖ω‖2W 2,2 (Ω) dt.

(2.2.36)

0

QT

The other terms can be obtained directly by the priori estimates, and combining (2.2.34) and (2.2.36), we get T

1 ∫ (|ω|2 + |ξ |2 )dx ≤ C ∫ ∫(|ω|2 + |ξ |2 )dxdt. 2

(2.2.37)

0 Ω

QT

Applying the Gronwall inequality, we get ω = 0, ξ = 0, a. e. (t, x) ∈ ΩT , therefore, Theorem 2.2.1 is proved.

2.3 Time-periodic solutions to the incompressible bipolar fluids 2.3.1 Time-periodic weak solutions to the incompressible bipolar fluids In this section, the dipolar fluids equation and the assuming conditions are the same as in Section 1.1. Definition 2.3.1. Let f (t) ∈ L2loc (−∞, +∞; V ) is periodic with period T. Then u(t) is said to be a T-time periodic solution of non-Newtonian fluids if it satisfies u(t) ∈ L∞ (−∞, +∞; H) ∩ L2 (−∞, +∞; V), T-periodic in time and +∞

1

1

∫ {−(u(t), φ (t)) + 2μ1 (A 2 u(t), A 2 φ(t)) − μ0 (γ(u)e(u), e(φ(t)))}dt −∞ +∞

= ∫ {−b(u, u, φ(t)) + ⟨f (t), φ(t)⟩}dt, −∞

(2.3.1)

30 | 2 Global solutions to the equations of non-Newtonian fluids for any φ ∈ C 0 (−∞, +∞; V), φ ∈ L2 (−∞, +∞). Here, C 0 (−∞, +∞; V) denotes the set of continuous V-valued functions that vanish outside a compact time interval. We obtain the main results. Theorem 2.3.1. (1) Let u(t) ∈ L2 (0, T; V) ∩ C(0, T; H) be a weak solution, with initial data u0 ∈ H, f (t) ∈ L2 (0, T; V ) with period T. Then there exists a T-time periodic solution for the non-Newtonian fluids (2.1.17), (2.1.18). (2) A function u(t) that is T-periodic in time would be a weak solution of non-Newtonian fluids if and only if u(t) is a weak solution in the interval (0, T). In order to define the solution operator FT such that u(T) = FT (u(0)), we need the following lemmas. Lemma 2.3.1. Let u be a weak solution of the non-Newtonian fluids and initial data u(0) with ‖u(0)‖ ≤ R. Then, for fixed T and large enough R, the operator FT maps the ball of radius R in H into itself. Proof. Taking inner product of equation (2.1.17) with u, p−2 𝜕 ‖u‖2 + 2μ1 ‖u‖22 + 2μ0 K0 ϵ 2 ‖u‖22 ≤ 2|⟨f , u⟩|. 𝜕t

(2.3.2)

For ‖u‖22 ≥ λ1 ‖u‖2 2|⟨f , u⟩| ≤ 2‖f ‖V ‖u‖2 ≤ μ1 ‖u‖22 +

‖f ‖2V , μ1

implying ‖f ‖2V 𝜕 ‖u‖2 + μ1 λ1 ‖u‖2 ≤ , 𝜕t μ1 therefore, ‖u(T)‖2 ≤ ‖u(0)‖2 exp{μ1 λ1 T} + C, where C =

1 μ21

∀T > 0,

T ∫0 ‖f ‖2V dt. So when R ≥ √C(1 − exp{−μ1 λ1 T}) 2 by ‖u(0)‖ ≤ R, we get −1

‖u(T)‖ ≤ R. So FT : BR → BR , then Lemma 2.3.1 is proved.

Lemma 2.3.2. Let u ∈ L2 (0, T; V)∩L∞ (0, T; H) be a weak solution of the non-Newtonian fluids (2.1.17), (2.1.18). Then the time T-map FT : u0 ∈ H → u(T) ∈ H is continuous in the weak topology of H. Proof. Consider a sequence {an } ⊂ H that converges weakly to a ∈ H. This means lim (an , ω) = (a, ω),

n→∞

∀ω ∈ H.

2.3 Time-periodic solutions to the incompressible bipolar fluids | 31

Let {un (t)} be a sequence of weak solutions with initial datum un (0) = an and let u(t) be a weak solution corresponding to the initial data u(0) = a. We will prove that for fixed t ≥ 0, un (t) converges weakly to u(t) ∈ H. Recalling a priori estimates established in the proof of the existence theorem, we see that the sequence {un (t)} is contained in a bounded set of L∞ (0, T; H) ∩ L2 (0, T; V). Moreover, as before we can argue that un (t) is contained in a bounded set of L2 (0, T; V ). This implies that we can extract a subsequence {unk (t)} from {un (t)} such that as nk → ∞, unk → u∗

in L2 (0, T; V) weakly;

unk → u∗

in L2 (0, T; V) strongly.

unk → u∗

in L∞ (0, T; H) weak-star;

The strong convergence is due to the compactness of the previously stated embedding Y → L2 (0, T; H). Now, let ψ be such that ψ ∈ C(0, T; V), and ψ ∈ L2 (0, T; H), and by (2.1.17), we get T

T

1

1

(uT , ψ(T)) − (an , ψ(0)) − ∫(un (t), ψ (t))dt + 2μ1 ∫(A 2 un (t), A 2 ψ(t))dt T

0

0

T

T

+ μ0 ∫(γ(un )e(un ), e(ψ(t)))dt + ∫ b(un , un , ψ(t))dt = ∫⟨f (t), ψ(t)⟩dt.

(2.3.3)

0

0

0

Using the Lemma 2.1.1, for (2.3.3), when n → ∞, we have T

T

1

1

(u∗ (T), ψ(T)) − (a, ψ(0)) − ∫(u∗ (t), ψ (t))dt + 2μ1 ∫(A 2 u∗ (t), A 2 ψ(t))dt 0

T

0

T

T

+ μ0 ∫(γ(u∗ (t))e(u∗ (t)), e(ψ(t)))dt + ∫ b(u∗ (t), u∗ (t), ψ(t))dt = ∫⟨f (t), ψ(t)⟩dt. 0

0

0

Therefore, u (T) is the weak solution for the initial value u(0) = a, using the uniqueness of the weak solution in Theorem 2.1.1, u∗ (T) = u(T) and when n → ∞, we have ∗

(an , ψ(0)) → (a, ψ(t)),

∀ψ(0) ∈ V,

that is, (un (T), ψ(T)) → (u(T), ψ(T)),

∀ψ(T) ∈ V.

V ⊂ H is dense, so when n → ∞, we have (an , ψ(0)) → (a, ψ(0)),

∀ψ(0) ∈ H

32 | 2 Global solutions to the equations of non-Newtonian fluids that is, (un (T), ψ(T)) → (u(T), ψ(T)),

∀ψ(T) ∈ H.

So FT : u0 ∈ H → u(T) is continuous in the weak topology of H. From Lemma 2.3.1, we know the time mapping is weakly compact. From Lemma 2.3.2, we know the time mapping is weakly continuous, and then by Tikonov’s fixed-point theorem, there at least exist one fixed point u∗ (0) ∈ BR , such that (1) of Theorem 2.3.1 is true. Now we prove Theorem 2.3.1 (2): (Necessity) If u(t) is a periodic solution with period T of problem (2.1.17), (2.1.18) and satisfies (2.3.1), then we can choose a test function ψ(t) so that it vanishes outside the compact interval (kT; (k + 1)T): (k+1)T

1

1

∫ {−u(t), ψ (t) + 2μ1 (A 2 u(t), A 2 ψ(t)) + μ0 (γ(u)e(u), e(ψ(t)))}dt kT (k+1)T

∫ {−b(u, u, ψ(t)) + ⟨f (t), ψ(t)⟩}dt,

=

k ∈ Z,

(2.3.4)

kT

with u(t) ∈ L∞ (kT, (k + 1)T; H) ∩ L2 (kT, (k + 1)T; V). In particular, if we let k = 0, we get the weak solution in the interval (0, T). (Sufficiency) Let u(t) be a weak solution in the interval (0, T). Then T

1

1

(u(T), ψ0 (T)) − (u0 , ψ0 (0)) + ∫{−(u(t), ψ (t)) + 2μ1 (A 2 u, A 2 ψ0 (t))}dt T

0

T

+ ∫{μ0 (γ(u)e(u), e(ψ0 (t))) + b(u, u, ψ0 (t))}dt = ∫⟨f (t), ψ0 (t)⟩dt. 0

0

Due to the periodicity of u(t) and f (t), if follows that (u((k + 1)T), ψk ((k + 1)T)) − (u(kT), ψk (kT)) (k+1)T

1

1

+ ∫ {−(u(t), ψk (t)) + 2μ1 (A 2 u, A 2 ψk (t))}dt kT (k+1)T

+ ∫ {μ0 (γ(u)e(u), e(ψk (t))) + b(u, u, ψk (t))}dt = kT

(k+1)T

∫ ⟨f (t), ψk (t)⟩dt kT

for each subinterval (kT, (k + 1)T) and where ϕk (t) = ϕ0 (t − kT). This means u(t) is a weak solution in each subinterval. Now, by choosing a suitable ψ(t) and summing the above equation over k, we get u(t) satisfying (2.3.1), so Theorem 2.3.1 (2) is proved.

2.3 Time-periodic solutions to the incompressible bipolar fluids | 33

2.3.2 Existence and uniqueness of strong time-periodic solutions to the incompressible bipolar fluids In this section, we consider the incompressible dipolar fluids equation satisfying the periodic boundary condition and every term assumptions are the same as in Section 2.1, ut = 2μ1 Au − 2μ0 Ap (u) + B(u, u) = f

(2.3.5)

u(t + T) = u(t),

(2.3.6)

obtaining the following results. Theorem 2.3.2. (Existence) Assuming f ∈ H 1 (T; H), T > 0, then problem (2.3.5), (2.3.6) has a T-time periodic strong solution and satisfying u ∈ H 2 (T; H) ∩ H 1 (T; D(A)) ∩ L∞ (T; D(A)) ∩ W 1,∞ (T; H 2 ). (Uniqueness) Assuming M = sup ‖f (t)‖ < K

(2.3.7)

0≥t≤T

where K = min{1, [

2μ41 λ13 K03

−1

12

3

C0 (4μ1 K0 λ1 T + 1)(k0 (2C0 λ1 4 ) 4−n + μ1 λ12 )

]}

then the time periodic strong solution in (1) is unique. Before proving Theorem 2.3.2, we first consider under the conditions of Theorem 2.3.2, the existence of the approximate solution to problem (2.3.5), (2.3.6). Using the Galerkin method to construct approximate solutions, m

um = ∑ Cim (t)ϕi i=1

(umt + 2μ1 Aum − 2μ0 Ap (um ) + B(um , um ), ϕj ) = (f , ϕj ), um (t + T) = um (t).

j = 1, . . . , m

(2.3.8) (2.3.9)

1 It is well known that for any νm (t) = ∑m i=1 dim (t)ϕi ∈ C (T; Em ), in the linear equation

(umt + 2μ1 Aum , ϕi ) = (2μ0 Ap (νm ) − B(νm , νm ) + f , ϕi ),

i = 1, . . . , m

(2.3.10)

1 there exists the unique T-time periodic solution um (t) = ∑m i=1 Cim (t)ϕi ∈ C (T; Em ) and the mapping F : νm → um is compactly continuous in C 1 (T; Em ). For this purpose, we give some boundary estimates. Applying the Leray–Schauder fixed-point theorem, we can obtain the existence of the solution to the problem (2.3.8), (2.3.9).

34 | 2 Global solutions to the equations of non-Newtonian fluids For all of the possible solutions um (t) to problems (2.3.8), (2.3.9), only use θB(um , um ) and θAp (um ), (0 ≤ θ ≤ 1), respectively, and substitute the nonlinear term B(um , um ) and Ap (um ) in (2.3.8), (2.3.9). And sup ‖um (t)‖ ≤ C1

(2.3.11)

0≤t≤T

where the constant C1 = ( μ21λ2 + 1 1

2T )M 2 μ1 λ1

is independent in θ.

Actually, multiplying (2.3.8) with Cjm (t) and summing over j = 1, . . . , m, p−2 1 d ‖um ‖2 + μ1 ‖um ‖22 + μ0 ∫(ϵ + |e(um )|2 ) 2 |e(um )|2 dx 2 dr

≤ ‖f ‖‖um ‖ ≤ λ

− 21

Ω

‖f ‖‖um ‖2 ,

Young inequality p−2 d‖um ‖2 ‖f ‖2 . + μ1 ‖um ‖22 + 2μ0 ∫(ϵ + |e(um )|2 ) 2 |e(um )|2 dx ≤ dt μ1 λ1

(2.3.12)

Ω

p−2

Because of the term (ϵ + |e(um )|2 ) 2 ≥ |e(um )|p−2 , when p ≥ 2 and when 1 < p < 2 by the upper bound of ‖um ‖22 , we contain 2μ0 ∫(ϵ + |e(um )|2 )

p−2 2

|e(um )|2 dx

Ω

is surely boundary, so in the following we only consider the case p ≥ 2, that is, d‖um ‖2 M2 + μ1 ‖um ‖22 ≤ dt μ1 λ1

(2.3.13)

d‖um ‖2 M2 + 2μ0 ‖um ‖22 ≤ . dt μ1 λ1

(2.3.14)

Using the periodicity of um (t), integrating (2.3.13) (2.3.14), respectively, from 0 to T, we obtain T

∫ ‖um ‖22 dt ≤ T

0

∫ ‖e(um )‖pLp dt ≤ 0

M2T μ21 λ1

(2.3.15)

M2T . 2μ0 μ1 λ1

(2.3.16)

By (2.3.15), there exists t ∗ ∈ [0, T] such that ‖um (t ∗ )‖2 ≤

2

1 M ‖u (t ∗ )‖22 ≤ ( ). λ1 m μ1 λ1

2.3 Time-periodic solutions to the incompressible bipolar fluids | 35

And then integrating (2.3.13) from t ∗ to t + T(t ∈ [0, T]), sup ‖um (t)‖ ≤ C1 ,

0≤t≤T

then (2.3.13) is established. In the following, we deduce some estimates of a higher derivative. Lemma 2.3.3. Assuming um (t) are the solutions of (2.3.8), (2.3.9), then: (1)

sup ‖um (t)‖1 ≤ C2

(2)

sup ‖um (t)‖2 ≤ C3

(3)

sup ‖um t(t)‖ ≤ C4

(4)

sup ‖um (t)‖4 ≤ C5

(5)

sup ‖um t(t)‖2 ≤ C6

t

t

t

t

T

t

∫ ‖um (t)‖24 dt ≤ C7

(6)

0

T

∫ ‖um (t)‖2 dt ≤ C8

(7)

0

where Ci (i = 2, . . . , 8) are constants independent of m: T

1 2

2

M0 = (∫ ‖f (t)‖ dt) ,

T

2

1 2

M1 = (∫ ‖ft (t)‖ dt) .

0

0

Proof. (1) Differentiating (2.3.5) about t and taking the inner produce with −Δum : d ‖∇um ‖2 + (4μ1 Aum − 4μ0 Ap (um ) − 2f + 2B(um , um ) − Δum ) = 0 dt

(2.3.17)

Korn inequality 4μ1 (Aum , −Δum ) = 4μ1 ∫ Ω

−4μ0 (Ap (um ), Δum ) = 2 ∫ Ω

𝜕2 eij 𝜕2 eij ⋅ dx ≥ 4μ1 K0 ‖um ‖23 𝜕xk2 𝜕xk2

𝜕2 Γ 𝜕ekl 𝜕eij dx ≥ 0 𝜕ekl 𝜕eij 𝜕xs 𝜕xs

Hölder inequality 2(B(um , um ), −Δum ) = 2|b(um , um , −Δum )| ≤ 2 ∫ |um ||∇um ||Δum |dx Ω

(2.3.18) (2.3.19)

36 | 2 Global solutions to the equations of non-Newtonian fluids 10−n

−1

8+n

≤ 2‖um ‖L4 ‖∇um ‖L4 ‖Δum ‖ ≤ 2C0 C1 6 λ1 4 ‖um ‖3 6 ≤ μ1 K0 ‖um ‖23 10−n 6

+ (2C0 C1

− 1 12 λ1 4 ) 4−n (

−1

8+n

4−n M 2 λ1 2 1 ) 2|(f , −Δum )| ≤ μ1 K0 ‖um ‖23 + μ1 K0 μ1 K0

(2.3.20)

then (2.3.17) can be written as −1

10−n M 2 λ1 2 − 1 12 d ‖∇um ‖2 + 2μ1 K0 ‖um ‖23 ≤ (2C0 C1 6 λ1 4 ) 4−n + . dt μ1 K0

(2.3.21)

Integrating (2.3.21) from 0 to T, T

∫ ‖um ‖23 0

−1

10−n M 2 λ1 2 − 1 12 T [(2C0 C1 6 λ1 4 ) 4−n + ]. ≤ 2μ1 K0 μ1 K0

(2.3.22)

By (2.3.22), there exists t∗ ∈ [0, T] such that ‖um (t∗ )‖21

≤

λ1−1 ‖um (t∗ )‖23

−1

10−n M 2 λ1 2 − 1 12 1 [(2C0 C1 6 λ1 4 ) 4−n + ] ≤ 2μ1 K0 λ1 μ1 K0

from t∗ to t + T, t ∈ [0, T], integrating (2.3.21), −1

1

‖um (t)‖21

10−n 2 M 2 λ1 2 − 1 12 1 + 2T) [(2C0 C1 6 λ1 4 ) 4−n + ] ≤( 2μ1 K0 λ1 μ1 K0

that is, sup ‖um (t)‖1 ≤ C2 t

where −1

1

10−n 2 M 2 λ1 2 − 1 12 1 C2 = ( + 2T) [(2C0 C1 6 λ1 4 ) 4−n + ]. 2μ1 K0 λ1 μ1 K0

(2) Differentiating (2.3.8) about t and taking the inner produce with umt , 1 1 1 1 d ‖um ‖2 + {∫ Γ(eij eij )dx + μ1 (A 2 um , A 2 um )} ≤ |b(um , um , um )| + M 2 2 dt 2

Ω

and by |b(um , um , umt )| ≤ ∫ |um ||∇um ||umt |dx ≤ Ω

1 ‖u ‖2 + ∫ |um |2 |∇um |2 dx 4 mt Ω

(2.3.23)

2.3 Time-periodic solutions to the incompressible bipolar fluids | 37 4

4

4−n n 1 C 4−n C 2 C 4−n 1 1 ≤ ‖um ‖2 + C0 C1 2 C2 ‖um ‖ 2 ≤ ‖umt ‖2 + μ1 ‖A 2 um ‖2 + 0 1 2 . 4 4 4μ1 (2.3.24)

Using (2.3.23), we get 1 d 1 ‖u ‖2 + {∫ Γ(eij eij )dx + μ1 ‖A 2 um ‖2 } 4 m dt

Ω

4

4

1 C 4−n C 2 C 4−n 1 ≤ M 2 + 0 1 2 + μ1 ‖A 2 um ‖2 . 2 4μ1

(2.3.25)

Integrating (2.3.25) from 0 to T and applying (2.3.15) T

∫ ‖umt ‖2 dt ≤ C92

(2.3.26)

0

where 4

4

1

2 1 K M 2 1 2 C04−n C1 C24−n 2 + M + ] ⋅T2. C9 = 2[ 0 μ1 λ1 2 4μ1 K0

For ∫ Γ(eij eij )dx ≤ Ω

p p 2μ0 M p+2 p ∫(ϵ + |e(um )|2 ) 2 dx ≤ 0 2 2 (|Ω|ϵ 2 + ‖e(um )‖Lp ) p p

ω

(2.3.15), (2.3.16), we have T

∫(∫ Γ(eij eij )dx + K0 μ1 ‖um ‖22 )dt ≤ C10 T 0

Ω

where C10 =

p μ0 p+2 K M2 M2 2 2 (ϵ 2 |Ω| + + 0 ). p 2μ0 μ1 λ1 μ1 λ1

Therefore, there exists t ∗ ∈ [0, T] such that ∫ Γ(eij (um (t ∗ ))eij (um (t ∗ )))dx + K0 μ1 ‖um (t ∗ )‖22 ≤ C10 .

Ω

Integrating (2.3.25), t ∈ [0, T] from t∗ to t + T, ∫ Γ(eij eij )dx + K0 μ1 ‖um (t)‖22 ≤ K0 μ1 C32

Ω

38 | 2 Global solutions to the equations of non-Newtonian fluids that is, sup ‖um (t)‖2 ≤ C3 t

where 4

4

1

2 2 K M 2 1 2 C04−n C1 C24−n 1 [C10 + 2( 0 + M + )T]} . C3 = { K0 μ μ1 λ1 2 4μ1 K0

(3) Differentiating (2.3.5) about t and taking the inner produce with umt , d ‖u ‖2 + 2μ1 K0 ‖um ‖22 − 4μ0 (Ap (um )t , umt ) + 2b(umt , um , umt ) ≤ 2‖ft ‖‖umt ‖ dt m 𝜕2 p 𝜕ekl 𝜕eij dx ≥ 0, − 4μ0 (Ap (um )t , umt ) = 2 ∫ 𝜕ekl 𝜕eij 𝜕t 𝜕t Ω

2|b(umt , um , umt )| ≤ 2‖um ‖1 ‖um ‖2L4 ≤ C2 ‖umt ‖

8−n 4

4

≤

μ1 ‖umt ‖22

(2.3.27)

n

‖umt ‖24

4 1 8−n + ( ) (2C2 ) 8−n ‖umt ‖2 . μ1

Therefore, d 1 ‖u ‖2 + μ1 K0 ‖umt ‖22 ≤ ‖ft ‖2 + [1 + dt mt μ1

n 8−n

8

(2C2 ) 8−n ]‖umt ‖2 .

(2.3.28)

Integrating from 0 to T, T

∫ ‖umt ‖22 dt ≤ 0

where

2 C11 μ1 K0

(2.3.29)

n

C11 =

{2M12

1

2 8 1 8−n + [1 + ( ) ⋅ (2C2 ) 8−n ]C92 } . μ1

There exists t∗ ∈ [0, T] such that ‖umt (t∗ )‖2 ≤

2

C11 1 ‖u (t )‖2 ≤ . λ1 mt ∗ 2 μ1 K0 λ1 T

Integrating (2.3.28) from 0 to T, t ∈ [0, T], we get sup ‖um (t)‖ ≤ C4 t

1

2 2 C11 2 C4 = { + 2C11 } . μ1 K0 λ1 T

2.3 Time-periodic solutions to the incompressible bipolar fluids | 39

(4) By equation (2.3.5), we have 2μ1 ‖Aum ‖ ≤ ‖B(um , um )‖ + 2μ0 ‖Ap (um )‖ + ‖umt ‖ + ‖f ‖

(2.3.30)

‖B(um , um )‖ ≤ C0 ‖um ‖2 ‖um ‖1 ≤ C0 C2 C3

2μ0 ‖Ap (um )‖ ≤ K0 (p − 1) p−2

(∫(ϵ + |∇um |2 )

1 2

‖Δum ‖2 dx) ≤ 4μ0 K0 (p − 1)ϵ

p−2 2

‖um ‖2 + 4μ0 K0 C0 (p − 1)

Ω

⋅ ‖Δum ‖L4 ‖∇um ‖p−2 ≤ 4μ0 K0 (p − 1)(ϵ L4

p−2 2

8(p−1)−n 2

n

‖um ‖2 + C0 ‖Aum ‖ 8 ‖um ‖2

)

≤ μ1 ‖Aum ‖ + C12

where C12 = 4μ0 K0 (p − 1)ϵ

p−2 2

C3 + [4μ0 K0 C0 (p − 1)]

8 8−n

8(p−1)−n 8−n

C3

n

1 8−n ( ) . μ1

By the conclusion of (3) and (2.3.30), we deduce sup ‖um (t)‖ ≤ C3 . t

(5) Differentiating (2.3.5) and taking the inner produce with Δ2 um , d ‖u ‖2 + 4μ1 K0 ‖umt ‖22 dt mt 2 ≤ (4μ0 ‖Ap (um )t ‖ + 2‖ft ‖)‖umt ‖4 + 2|b(umt , um , Δ2 umt )|

(2.3.31)

where 4μ0 ‖Ap (um )t ‖‖umt ‖4 ≤ 4μ0 K0 (p − 1) 2 p−2

[∫(ϵ + |∇um | ) Ω

1 2

2

|Δumt | dx] ‖umt ‖4 + 4μ0 K0 (7 − p)(p − 2) 1 2

[∫ |∇um |2 |Δum |2 |∇umt |2 dx] ‖umt ‖4 ≤ 4μ0 K0 (p − 1) Ω

[ϵ

p−2 2

‖umt ‖2 + ‖∇um ‖p−2 4 ‖Δumt ‖4 ]‖umt ‖4 + 4μ0 K0 (7 − p)(p − 2)‖∇um ‖6 ‖∇umt ‖6 ‖umt ‖4

≤ 4μ0 K0 (p − 1)ϵ

p−2 2

−

+ 4μ0 K0 (7 − p)(p − 2)‖um ‖2 ‖um ‖3 ‖umt ‖2 ‖umt ‖4

≤ 4μ0 K0 [(p − 1)ϵ

n

4+n

4 16 ‖umt ‖2 ‖umt ‖4 + 4μ0 K0 (p − 1)C0 ‖um ‖p−2 2 λ1 ‖umt ‖4

p−2 2

−1

+ (7 − p)(p − 2)C3 C5 λ1 4 ] n

‖umt ‖2 ‖umt ‖4 + 4μ0 K0 (p − 1)C0 C3p−2 λ1 16 −

40 | 2 Global solutions to the equations of non-Newtonian fluids 4−n

4+n

‖umt ‖2 4 ‖umt ‖4 4 ≤ μ1 K0 ‖umt ‖24 +{

8μ20 K02 [(p − 1)ϵ

p−2 2

−1

+ (7 − p)(p − 2)C3 C5 λ1 4 ]2 μ1 K0 4+n

+ [4μ0 K0 (p −

4−n 2 −n 8 ) }‖umt ‖22 . 1)C0 C3(p−2) λ1 16 ] 4−n ( μ1 K0

Therefore, by (2.3.31) we get ‖f ‖2 d ‖umt ‖22 + μ1 K0 ‖umt ‖24 ≤ C13 ‖umt ‖22 + t + C14 dt μ1 K0

(2.3.32)

where C14 = C13 =

(C0 C4 C5 )2 μ1 K0 8μ20 K02 [(p − 1)ϵ + [4μ0 K0 (p −

p−2 2

−1

+ (7 − p)(p − 2)C3 C5 λ1 4 ]2 μ1 K0

−n 8 1)C0 C3p−2 λ1 16 ] 4−n ( μ

4+n

4−n 2 ) . 1 K0

Integrating (2.3.32) from 0 to T, T

∫ ‖umt ‖24 dt ≤ C7 0

where C7 =

C C2 M12 1 ( 13 11 + + C14 T). μ1 K0 μ1 K0 μ1 K0

By (2.3.33), there exists t ∈ [0, T] such that C 1 ∗ 2 ∗ 2 umt (t ) ≤ umt (t )4 ≤ 7 . λ1 λ1 T Integrating (2.3.32) from t to t + T, t ∈ (0, T), ‖umt ‖22 ≤ (

1 + 2μ1 K0 )C7 . λ1 T

Therefore, sup ‖umt ‖2 ≤ C6 t

(2.3.33)

2.3 Time-periodic solutions to the incompressible bipolar fluids | 41

where

1

2 1 C6 = {( + 2μ1 K0 )C7 } . λ1

(6) By the process of proving (2.3.33), we can obtained (6). (7) Differentiating each side of the equation (2.3.5) about t and applying the above estimates, we have ‖umt ‖ ≤ 2μ1 K0 ‖umt ‖4 + 2μ0 ‖Ap (um )t ‖ + ‖B(um , um )‖ + ‖B(um , um )‖ + ‖ft ‖ ≤ 3μ1 K0 ‖umt ‖4 + C13 C6 + C14 + 2C0 C4 C5 + ‖ft ‖.

Therefore, T

‖umtt ‖2 ≤ 18μ21 K02 ‖umt ‖24 + 4(C6 C13 + 2C0 C4 C5 + C14 )2 + 4‖ft ‖2

∫ ‖umtt ‖2 dt ≤ C8 0

where

C8 = 18μ21 K02 C7 + 4(C6 C7 + 2C0 C5 C4 + C14 )2 T + 4M12 .

Now we prove Theorem 2.3.2. Proof. (1) using the above obtained approximate solution um (t) and Lemma 2.3.3, by the standard compactly assumption, there exists um (t) convergence to function u(t) : m → ∞, um → u in L∞ (T; D(A)) weak-star

(2.3.34)

um → u in L∞ (T; D(A )) strongly

(2.3.35)

1 2

umt → ut

1 2

in L∞ (T; D(A )) weak-star

(2.3.36)

um → u in L (T; H) strongly

(2.3.37)

∞

and u(t) ∈ H 2 (T; H) ∩ H 1 (T; D(A)) ∩ L∞ (T; D(A)) ∩ W 1,∞ (T; H 2 ). Obviously, (2.3.34)–(2.3.36) are the direct deduction. Now we only prove (2.3.37). Actually, for all h > 0, 0 ≤ t ≤ t + h ≤ T, the sequence |(umt (t), ϕi )|, (m = i, i + 1, . . .) are uniformly boundary and equicontinuous: t+h (umt (t + h) − um (t), ϕi ) ≤ ∫ (umtt (s), ϕi )ds t 1 2

T

1 2

1

1

≤ ‖ϕi ‖h (∫ ‖umtt (t)‖2 dt) ≤ C82 h 2 ‖ϕi ‖ 0

42 | 2 Global solutions to the equations of non-Newtonian fluids where ϕi (i = 1, 2, . . .) are the orthonormal basis of H, that is, the eigenfunction giving by (2.3.16). Therefore, adopting focusing principle choosing a subsequence (umt → ut (t)) weakly convergence about t ∈ [0, T] in H, applying the estimates of Lemma 2.3.3 we can get (2.3.37). Meanwhile, by Lemma 2.3.3 we obtain B(u, u) and Ap (u) are well-posed: ‖B(um , um ) − B(u, u)‖ ≤ ‖B(um , um − u)‖ + ‖B(um − u, um )‖ ≤ C0 (‖um ‖2 ‖um − u‖1 + ‖um − u‖‖um ‖3 ) → 0

‖Ap (um ) − Ap (u)‖ ≤ [(p − 2)2p−2 ϵ

p−2 2

(m → ∞), ∀t ∈ [0, T]

(p−2)(4−n) 4

K0 n2 + (p − 2)2p−2 K0 C0 n2 ‖u‖1

+ (p − 2)(7 − p)ϵ

p−4 2

(2.3.38) n(p−2) 4

‖um ‖3

K0 C0 n2 (‖um ‖2 + ‖u‖2 )‖u‖3 ]‖um − u‖2 → 0

(m → ∞), ∀t ∈ [0, T].

(2.3.39)

Therefore, (ut + 2μ1 Au − 2μ0 Ap (u) + B(u, u), ϕi ) = (f , ϕi )

i = 1, 2, . . . , t ∈ [0, T]

(2.3.40)

and (2.3.40) is true for any ϕ ∈ H, so we get (2.3.5), (2.3.6), ut + 2μ1 Au − 2μ0 Ap (u) + B(u, u) = f , u(t + T) = u(t)

∀t ∈ R1

so Theorem 2.3.2 (1) is proved. (2) Assuming u, ν are two solutions of (2.3.5), (2.3.6), let ω = u − ν, then d ‖ω‖2 + 2μ1 K0 ‖ω‖22 − 4μ0 (Ap (u) − Ap (ν), ω) ≤ 2|b(ω, u, ω)|. dt Because −4μ0 (Ap (u) − Ap (ν), ω) ≥ C‖ω‖pW 1,p , where C is a positive constant only dependent on p and Ω. |b(ω, u, ω)| ≤ C0 ‖u‖1 ‖ω‖22 ≤ C0 C1 ‖ω‖22 . Then d ‖ω‖2 + 2(μ1 K0 − C0 C1 )‖Ω‖22 ≤ 0 dt i. e. d ‖ω‖2 + L‖ω‖2 ≤ 0. dt

(2.3.41)

2.4 Existence and uniqueness and stability of global solutions | 43

By the assuming (2.3.7), L = 2(μ1 K0 − C0 C2 λ1 ) > 0 and Gronwall inequality, ‖ω(t)‖2 ≤ ‖ω(0)‖ exp(−Lt),

∀t > 0,

for ω(t) is the T-time periodic function, for ∀t ∈ R, there exists positive integer N0 , such that t + N0 T > 0 and ‖ω(t)‖2 = ‖ω(t + N0 T)‖2 , so for any N ≥ N0 , ‖ω(t)‖2 ≤ ‖ω(0)‖2 exp(−LNT) that is, ‖ω(t)‖ ≡ o, then Theorem 2.3.2 (2) is proved, then Theorem 2.3.2 is proved.

2.4 Existence and uniqueness and stability of global solutions to the initial boundary value problems for the incompressible bipolar viscous fluids Considering the initial boundary value problem of bipolar viscous incompressible fluid equation: 𝜕u 𝜕ui 𝜕p 𝜕 𝜕 − μ1 Δe (u) + (γ(u)eij (e)) + fi + uj i = − 𝜕t 𝜕xj 𝜕xi 𝜕xj ij 𝜕xj div u = 0,

u(0) = h(x), u = g,

τijk (u)uj uk − τijk (u)uj uk ul ui = Mi

(x, t) ∈ Ω × (0, T) x∈Ω

(x, t) ∈ 𝜕Ω × (0, T)

(x, t) ∈ 𝜕Ω × (0, T),

(2.4.1) (2.4.2) (2.4.3) (2.4.4) (2.4.5)

where u is the velocity field associated with the flow of an incompressible bipolar fluid; p is the pressure; fi , g, h and Mi are given functions, the smoothness of which will be specified later; eij is the usual rate of deformation tensor, that is, 𝜕uj 1 𝜕u eij = ( i + ) 2 𝜕xj 𝜕xi

(2.4.6)

γ(u) is the nonlinear viscosity γ(u) = μ0 (ϵ + eij (u)eij (u))

−α/2

(2.4.7)

the τijk (u) are the components of the first multipolar stress tensor τijk (u) = μ1

𝜕 e (u) 𝜕xk ij

(2.4.8)

μ0 , μ1 and ϵ are positive constants; and α ∈ (0, 1). Assuming there exists a vector field ũ such that ũ = g,

(x, t) ∈ 𝜕Ω × (0, T)

(2.4.9)

44 | 2 Global solutions to the equations of non-Newtonian fluids ̃ j uk − τijk (u)u ̃ j uk ul ui = Mi τijk (u)u div ũ = 0,

(x, t) ∈ Ω × (0, T).

(x, t) ∈ 𝜕Ω × (0, T)

(2.4.10)

n

(2.4.12)

(2.4.11)

We introduce the space H = {u ∈ [W 2,2 (Ω) ∩ W01,2 (Ω)] : div u = 0}.

2 ̃ ∈ L∞ Definition 2.4.1. The function u is called a weak solution if u − u loc (0, ∞; L (Ω)) ∩ −1,2 L2loc (0, ∞; H) ∩ Wloc (0, ∞; W 4,2 (Ω)), and satisfies ∞

∞

−∫∫ 0 Ωt

∞

𝜕eij (u) 𝜕eij (ϕ) 𝜕ϕi 𝜕ϕ ui dxdt + ∫ ϕi (0)hi (x) + ∫ ∫ dxdt = ∫ ∫ uj i dxdt 𝜕t 𝜕xk 𝜕xk 𝜕xj 0 Ωt

Ω

∞

− ∫ ∫ γ(u)eij (u) 0 Ωt

0 Ωt

∞

∞

0 Ωt

0 𝜕Ωt

𝜕ϕ 𝜕ϕi dxdt + ∫ ∫ fi ϕi dxdt + ∫ ∫ Mi i dsdt 𝜕xj 𝜕ν

(2.4.13)

∀ϕ ∈ W 1,2 (0, ∞; H), with compact support in [0, ∞). 2.4.1 Existence It will be more convenient, for our purposes, to work with the function ω = u−u.̃ If u is a 2 2 weak solution to the problem (2.4.1)–(2.4.5), then ω ∈ L∞ loc (0, ∞; L (Ω))∩Lloc (0, ∞; H)∩ −1,2 4,2 Wloc (0, ∞; W (Ω)), and satisfies ̃ )uj uk − τijk (u ̃ )uj uk ul ui = 0 τijk (u

(2.4.14)

and ∫ Ωt

𝜕eij (ω) 𝜕eij (ψ) 𝜕u 𝜕ψ 𝜕ωi ψi dx + ∫ dx = ∫ uj i ψi dx − ∫ γ(u)eij (u) i dx 𝜕t 𝜕xk 𝜕xk 𝜕xj 𝜕xj Ωt

Ωt

Ωt

𝜕eij (u) 𝜕eij (ψ) 𝜕ψ 𝜕u dx + ∫ Mi i ds, + ∫ fi ψi dx − ∫ ψi dx + ∫ 𝜕t 𝜕xk 𝜕xk 𝜕ν Ωt

Ωt

Ωt

(2.4.15)

𝜕Ωt

∀ψW 1,2 (0, ∞; H), supp ψ ⊂⊂ [0, ∞). We will use the Galerkin method to prove the existence of the function ω. In order to introduce the needed basis, we define in H the scalar product ((ω, ψ)) = ∫ Ω

𝜕 𝜕 e (ω) e (ψ)dx 𝜕xk ij 𝜕xk ij

and denote by (ω, ψ) the usual L2 scalar product.

2.4 Existence and uniqueness and stability of global solutions | 45

Lemma 2.4.1. The eigenvalue problem ((ω, ψ)) = λ(ω, ψ),

∀ψ ∈ H

(2.4.16)

has a sequence of solutions W 1 ∈ H ∩ C ∞ (Ω) corresponding to a sequence of positive eigenvalues λl . Furthermore: (1) the sequence W l is a basis for the closure of H under the L2 norm. (2) the sequence W l is a basis of H. (3) (W l , W k ) = δlk . This is a standard consequence of the Korn inequality. Lemma 2.4.2 (Korn inequality). There exists the constant c > 0 such that ∫

𝜕eij (u) 𝜕eij (u)

Ω

𝜕xk

𝜕xk

dx ≥ c‖u‖2W 2,2 (Ω) ,

∀u ∈ W 2,2 (Ω) ∩ W01,2 (Ω).

(2.4.17)

Now, for l fixed, let ωl ∈ El = Span{W l , . . . , W l }, l

ωl (x, t) = ∑ Cl,k (t)W k (x), k=1

is the solution of ∫ Ωt

𝜕eij (ωl ) 𝜕eij (ψ) 𝜕ul 𝜕ωli 𝜕ψ dx = ∫ ulj i ψi dx − ∫ γ(ul )eij (ul ) i dx ψi dx + ∫ 𝜕t 𝜕xk 𝜕xk 𝜕xj 𝜕xj Ωt

Ωt

+ ∫ fi ψi dx − ∫ Ωt

Ωt

Ωt

𝜕eij (u) 𝜕eij (ψ) 𝜕ψ 𝜕ui ψ dx + ∫ dx + ∫ Mi i ds, 𝜕t i 𝜕xk 𝜕xk 𝜕ν Ωt

(2.4.18)

Ωt

∀ψ ∈ El , where ul = wl + u.̃ The nonlinear system of ordinary differential equations for the coefficients Clk (t) generated by (2.4.18) and satisfies the conditions of Picard’s theorem because of the regularity of γ(u); hence this system along with the initial conditions ̃ l dx Cl,k (0) = ∫(h − u)W

∀(l, k)

(2.4.19)

Ω

has a unique local solution on [0, Tl ]. We will now proceed with proving some a priori estimates for ωl . In order to state our first a priori estimate, we introduce the quantity E l (t) defined by t

1 E (t) = ‖wl ‖2L2 (Ωt ) + σ ∫ ‖wl ‖2W 2,2 (Ωt ) dt, 2 l

0

σ > 0.

(2.4.20)

46 | 2 Global solutions to the equations of non-Newtonian fluids Lemma 2.4.3. There exists c > 0 such that for all l, E l (t) ≤ cect , ∀t ≥ 0.

(2.4.21)

Proof. Set ψ = ωl in (2.4.18) and sum over i; it then follows that 𝜕eij (wl ) 𝜕eij (wl ) 1 𝜕 2 dx = −b(ul , vl , wl ) − ∫ γ(ul )eij (ul )eij (wl )dx ∫ (wl ) dx + ∫ 2 𝜕t 𝜕xk 𝜕xk Ωt

Ωt

+ ∫ f ⋅ wl dx − ∫ Ωt

Ωt

l

Ωt

𝜕eij (u)̃ 𝜕eij (w ) 𝜕wl 𝜕ũ ⋅ wl dx − ∫ dx + ∫ Mi i ds 𝜕t 𝜕xk 𝜕xk 𝜕ν Ωt

(2.4.22)

𝜕Ωt

where b(ul , vl , wl ) = ∫Ω uj 𝜕x i ui dx. Since ul = wl + u,̃ 𝜕w

j

t

b(ul , vl , wl ) = b(wl , wl , wl ) + b(u,̃ wl , wl ) + b(u,̃ u,̃ wl ) + b(wl , u,̃ wl ).

(2.4.23)

Since ωl is divergence free, and ωl = 0 on 𝜕Ωt , an easy calculation shows that b(wl , wl , wl ) = 0. Similarly, b(u,̃ wl , wl ) = 0. Hence 𝜕eij (wl ) 𝜕eij (wl ) 1 d 2 dx + ∫ γ(ul )eij (wl )eij (wl )dx ∫ (wl ) dx + ∫ 2 dt 𝜕xk 𝜕xk Ωt

Ωt

Ωt

̃ ij (wl )dx + ∫ f ⋅ wl dx − b(u,̃ u,̃ wl ) − b(wl , u,̃ wl ) − ∫ = − ∫ γ(ul )eij (u)e Ωt

−∫ Ωt

l

𝜕eij (u)̃ 𝜕eij (w ) 𝜕xk

𝜕xk

Ωt

dx + ∫ Mi 𝜕Ωt

Ωt

𝜕wil 𝜕ν

𝜕ũ ⋅ wl dx 𝜕t

ds.

(2.4.24)

Thus 𝜕eij (wl ) 𝜕eij (wl ) 1 d 2 dx + ∫ γ(ul )eij (wl )eij (wl )dx ∫ (wl ) dx + ∫ 2 dt 𝜕xk 𝜕xk Ωt

Ωt

Ωt

≤ (c‖u‖̃ W 1,2 (Ωt ) + ‖f ‖W −1,2 (Ωt ) + ‖ũ t ‖W 1,2 (Ωt ) )‖w‖W 1,2 (Ωt ) + (c‖u‖̃ W 2,2 (Ωt ) + ‖Mi ‖W −1/2,2 (𝜕Ωt ) )‖w‖W 2,2 (Ωt ) + b(u,̃ u,̃ wl ) + b(u,̃ u,̃ wl ). Using ‖u‖̃ L∞ (Ωt ) ≤ c‖u‖̃ W 2,2 (Ωt ) , we find that ̃ ̃ l l b(u, u, w ) ≤ C‖u‖̃ W 2,2 (Ωt ) ⋅ ‖w ‖W 1,2 (Ωt ) . Also

(2.4.25)

2.4 Existence and uniqueness and stability of global solutions | 47

l l l l l l b(w , u,̃ w ) ≤ ‖u‖̃ L∞ (Ωt ) ∫ |w | ⋅ |∇w |dx ≤ C‖u‖̃ W 2,2 (Ωt ) ‖w ‖L2 (Ωt ) ‖w ‖W 2,2 (Ωt ) .

(2.4.26)

Ωt

Inequality (2.4.25) then yields 𝜕eij (wl ) 𝜕eij (wl ) 1 d 2 dx + ∫ γ(ul )eij (wl )eij wl dx ∫ (wl ) dx + ∫ 2 dt 𝜕xk 𝜕xk Ωt

Ωt

Ωt

≤ (c‖u‖̃ W 1,2 (Ωt ) + ‖f ‖W −1,2 (Ωt ) + C‖u‖̃ W 2,2 (Ωt ) ‖wl ‖L2 (Ωt ) )‖wl ‖W 2,2 (Ωt ) .

(2.4.27)

Using the inequality |xy| ≤ 21 (x2 /σ + y2 σ) on the term on the right-hand side of (2.4.27), and (2.4.17) to estimate the second term on the left-hand side, and dropping the third term on the left-hand side, we find that (2.4.27) yields 1 d 2 ∫ (wl ) dx + σ‖wl ‖2W 2,2 (Ωt ) ≤ θ−1 (c‖u‖̃ W 1,2 (Ωt ) + ‖f ‖W −1,2 (Ωt ) + ‖ũ t ‖W 1,2 (Ωt ) 2 dt Ωt

+ c‖u‖̃ W 2,2 (Ωt ) + ‖Mi ‖W −1/2,2 (𝜕Ωt ) + C‖u‖̃ W 2,2 (Ωt ) ‖wl ‖L2 (Ωt ) )2

≤ 2θ−1 (c‖u‖̃ W 1,2 (Ωt ) + ‖f ‖W −1,2 (Ωt ) + ‖ũ t ‖W 1,2 (Ωt )

2

+ c‖u‖̃ W 2,2 (Ωt ) + ‖Mi ‖W −1/2,2 (𝜕Ωt ) )2 + 2θ−1 (C‖u‖̃ W 2,2 (Ωt ) ‖wl ‖L2 (Ωt ) ) .

(2.4.28)

Therefore, dE l (t) ≤ aE l (t) + b dt

(2.4.29)

where C sup(‖u‖̃ W 2,2 (Ωt ) ) (2.4.30) σ t≥0 C b = sup(c‖u‖̃ W 1,2 (Ωt ) + ‖f ‖W −1,2 (Ωt ) + ‖ũ t ‖W 1,2 (Ωt ) + c‖u‖̃ W 2,2 (Ωt ) + ‖Mi ‖W −1/2,2 (𝜕Ωt ) )2 . σ t≥0 (2.4.31) a=

The lemma is then a direct consequence of Gronwall’s inequality. We will now derive the second a priori estimate. Lemma 2.4.4. The norm of

dωl dt

in L3/2 (0, T; W −2,2 ) is bounded independently of l.

Proof. Let u ∈ Ls (0, T; W02,2 (Ω)), s > 2 (s will be specified later), and set ul = Pl (u) where Pl is the projection operator onto the space El = Span{W l ⋅ ⋅ ⋅ W l }. We then have that T

∫∫ 0 Ωt

T

T

0 Ωt

0 Ωt

dwl dwl l dũ l ⋅ νdxdt = ∫ ∫ ⋅ ν dxdt ≡ − ∫ ∫ ⋅ ν dxdt dt dt dt

48 | 2 Global solutions to the equations of non-Newtonian fluids T

+

T

𝜕ul ∫ ∫ ulj i vil dxdt 𝜕xj 0 Ωt

−∫∫

𝜕eij (ul ) 𝜕eij (νl ) 𝜕xk

0 Ωt

𝜕xk

T

dxdt − ∫ ∫ γ(u)eij (ul ) 0 Ωt

𝜕vil dxdt 𝜕xj

T

+ ∫ ∫ fi vil dxdt ≡ I1 + I2 + I3 + I4 + I5 . 0 Ωt

Using Lemma 2.4.3, it is easy to see that |I1 + I3 + I4 + I5 | ≤ C‖νl ‖L2 (0,T;W 2,2 (Ω)) ≤ C‖νl ‖Lp (0,T;W 2,2 (Ω)) . 0

(2.4.32)

Let p be the conjugate of p; then T T p p 𝜕vl |I2 | = − ∫ ∫ ulj uli i dxdt ≤ ∫[(∫ |ul |2p dx) (∫ |∇ν|p dx) ]dt. 𝜕xj 1

0

0 Ωt

Ωt

(∫ (u )

2n . n−2

l 2

dx) ≤ (∫ (u ) dx)

Ωt

(2.4.33)

Ωt

Recall that ‖Dul ‖Lp (Ω) ≤ c‖u‖W 2,2 (Ω) , for p ≤ l 2p

1

Also

1 2p

l 4p −2

⋅ (∫ (u )

Ωt

dx)

1 2p

.

(2.4.34)

Ωt

By Lemma 2.4.3, we have that ∫Ω (ul )2 dx is uniformly bounded in L∞ (0, T), hence t

T

l 4p −2

|I2 | ≤ C(∫ (u )

dx)

0

T

l 4p −2

≤ C[∫(∫ (u ) 0

1 2p

Ωt

⋅ ‖u‖W 2,2 (Ω) dt

dx)

s sp

dt]

1 s

⋅ ‖u‖Ls (0,T;W 2,2 (Ω)) .

Now we are interested in estimating T

s

2p 4p −2 dx) dt. ∫(∫ (ul )

0

Ωt

1

For this purpose, we use the inequality (∫Ω (ul )4p −2 dx) 4p−2 ≤ ‖ul ‖W 2,2 (Ω) , which is valid

provided p >

n 2

t

− 1; from Lemma 2.4.3, it then follows that T

1

4p −2 4p −2 dx) dt ≤ cecT . ∫(∫ (ul )

0

Ωt

(2.4.35)

2.4 Existence and uniqueness and stability of global solutions | 49

Hence for

s 2p

=

2 , 4p −2

we have that T

s

2p 4p −2 dx) dt ≤ cecT . ∫(∫ (ul )

0

(2.4.36)

Ωt

For n ≤ 3, p = 3 satisfies all the restrictions listed above and yields s = 32 , s = 3. Hence T

∫∫ 0 Ωt

and

dwl ⋅ udxdt ≤ ‖u‖L2 (0,T;W 2,2 ) 0 dt

dwl ≤ C. dt L3/2 (0,T;W −2,2 (Ω))

(2.4.37)

(2.4.38)

Theorem 2.4.1. Assume that: (1) ũ ∈ L∞ (0, ∞; W 2,2 (Ω)) (2) ũ t ∈ L2loc ([0, ∞); W −1,2 (Ω)) (3) f ̃ ∈ L2loc ([0, ∞); W −1,2 (Ω)) (4) Mi ∈ L2loc ([0, ∞); W −1/2,2 (𝜕Ωt )) (5) h ∈ L2 (Ω) (6) there exists c > 0 such that t

∫(‖f ‖W −1,2 (Ωt ) + ‖u‖̃ W −1,2 (Ωt ) + ‖Mi ‖W −1/2,2 (𝜕Ωt ) )2 dt ≤ cect .

(2.4.39)

0

Then the problems (2.4.1)–(2.4.5) have a weak solution u in the sense of (2.4.13), which satisfies t

1 ‖u‖ 2 + σ ∫ ‖u‖W 2,2 (Ωt ) dt ≤ cect . 2 L (Ωt )

(2.4.40)

0

Proof. From Lemma 2.4.3, it follows that the sequence ωl has a convergent subsequence, denoted again by ωl , which is convergent in L2loc ([0, ∞); W 2,2 (Ω)) to a function ω. Furthermore, t

1 ‖w‖L2 (Ωt ) + σ ∫ ‖u‖W 2,2 (Ωt ) dt ≤ cect 2

(2.4.41)

0

Using the results of Lemmas 2.4.3 and 2.4.4 of [59], we then have that the sequence ωl is compact in L2 (0, T; W s,2 (Ω)), s < 2. Hence there exists a subsequence, denoted again by ωl , which converges, strongly, to ω in L2 (0, T; W s,2 (Ω)). From the compactness of the

50 | 2 Global solutions to the equations of non-Newtonian fluids embedding of Sobolev spaces, it then follows that (wl + u)̃ j (wl + u)̃ i converges weakly in L2loc ([0, ∞); L2 (Ω)) and strongly in L1loc ([0, ∞); L2 (Ω)); thus its limit is (w + u)̃ j (w + u)̃ i . Similarly, γ(ul )eij (ul ) converges strongly in L2 ((0, T) × Ω) to γ(u)eij (u). Letting l go to infinity in (2.4.18), it then follows that supp ψ ⊂⊂ [0, ∞) ψ ∈ W 1,2 (0, ∞; H), ∞

∞

𝜕eij (u) 𝜕eij (ψ) 𝜕ψ −∫∫ ui dxdt + ∫ ψi (0)hi (x)dx + ∫ ∫ dxdt 𝜕t 𝜕xk 𝜕xk 0 Ωt

0 Ωt

Ω

∞

∞

∞

∞

0 Ωt

0 Ωt

0 Ωt

0 𝜕Ωt

𝜕ψj 𝜕ψ 𝜕ψ dxdt + ∫ ∫ fi ψi dxdt + ∫ ∫ Mi i dsdt = ∫ ∫ uj i dxdt − ∫ ∫ γ(u)eij (u) 𝜕xj 𝜕xj 𝜕ν where u = w + u.̃ Remark. The above result is still valid, and the proof essentially unchanged, if instead of assuming (2.4.7) one requires that γ be a positive decreasing function and that lims→0 sγ(s) < ∞. In particular, one can set ϵ = 0 in (2.4.7).

2.4.2 Regularity In this section, we will establish two regularity results for the solution whose existence was proven in the above section; we will also make precise the sense in which the solution of the variational problem satisfies the boundary value problem. Our result is obtained in the following. Theorem 2.4.2. Assume that: (1) ũ ∈ L∞ (0, ∞; W 2,2 (Ω)) ∩ W −1,2 (0, ∞; W 4,2 (Ω)) (2) ũ t ∈ L2loc ([0, ∞); L2 (Ω)) (3) f ∈ L2loc ([0, ∞); L2 (Ω)) 1,2 (4) Mi ∈ Wloc ([0, ∞); W −1/2,2 (Ω)). Then problems (2.4.1)–(2.4.5) has a weak solution u in the sense of (2.4.13) which satisfies 1,2 (1) u ∈ Wloc ((0, ∞); L2 (Ω)) ∞ (2) u ∈ Lloc ((0, ∞); H). Proof. Set ψ = wtl in (2.4.17). After integration by parts over (t1 , T) and summing over i, it follows that T

∫ ∫(

t1 Ωt

2 l l 𝜕ul 1 𝜕eij (u ) 𝜕eij (u ) ) dxdt + ∫ γ(ul )dx + ∫ dx 𝜕t 2 𝜕xk 𝜕xk Ωt

Ωt

2.4 Existence and uniqueness and stability of global solutions | 51 T

= − ∫ ∫ ulj t1 Ωt

𝜕eij (ul ) 𝜕eij (ul ) 𝜕uli l 1 ̃ l )dx + ∫ (ui )t dxdt + ∫ γ(u dx 𝜕xj 2 𝜕xk 𝜕xk Ωt1

T

T

+ ∫ ∫ f ⋅ ult dxdt − ∫ ∫ t1 Ωt

̃ where γ(u) =

1 (ϵ 2−a

t1 𝜕Ωt

Ωt1

𝜕ul 𝜕ul 𝜕Mi 𝜕ult dsdt + ∫ Mi t ds − ∫ t ds 𝜕t 𝜕ν 𝜕ν 𝜕ν

(2.4.42)

Ωt1

𝜕ΩT

+ eij eij )(2−a)/2 . Hence

T

t

t

t1

t1

̄ ‖2L2 (Ωt ) dτ + ∫ ‖ul Dul ‖2L2 (Ωt ) dτ ∫ ‖ult ‖2L2 (Ωt ) + ‖ul ‖2W 2,2 (Ωt ) ≤ (‖ul ‖2W 2,2 (Ωt ) + ∫ ‖f 1

t1

t

2

𝜕M dτ + ‖Mi ‖2W −1/2,2 (𝜕Ω ) + ‖Mi ‖2W −1/2,2 (𝜕Ω ) ) + ∫ i t t1 𝜕t W −1/2,2 (𝜕Ωt ) t1

t

≤ A(T) + C‖ul ‖2W 2,2 (Ωt ) + C ∫ ‖ul Dul ‖2L2 (Ωt ) dτ 1

(2.4.43)

t1

where T 𝜕M 2 A(T) = C(‖f ‖2L2 ((0,T);L2 (Ω)) + ∫ i dτ + ‖Mi ‖2W 1,2 ((0,T);W −1/2,2 (𝜕Ω)) ). 𝜕t W −1/2,2 (𝜕Ωt )

(2.4.44)

0

Next, we estimate the last term on the right-hand side of (2.4.43), t

∫ ‖u

l

t1

Dul ‖2L2 (Ωt ) dτ

t

≤

C ∫ ‖ul ‖2W 2,2 (Ωt ) ‖Dul ‖2L2 (Ωt ) dτ t1

t

≤ ϵ ∫ ‖ul ‖4W 2,2 (Ω ) dτ t1

t

t

+ C ∫ ‖ul ‖2W 2,2 (Ωt ) ‖ul ‖2L2 (Ωt ) dτ, t1

where standard embedding results and the estimate ‖Dul ‖2L2 (Ωt ) dτ ≤ ϵ‖ul ‖2W 2,2 (Ωt ) + C(ϵ)‖ul ‖2L2 (Ωt ) were used. Hence, ∀ϵ > 0, we have t

t

∫ ‖ul Dul ‖2L2 (Ωt ) dτ ≤ ( sup ‖ul ‖2W 2,2 (Ωt ) ) ∫ ‖ul ‖2W 2,2 (Ωt ) dτ

t1

t1 ≤t≤T

t1

t

+ C(ϵ)( sup ‖ul ‖2L2 (Ωt ) ) ∫ ‖ul ‖2W 2,2 (Ωt ) dτ. t1 ≤t≤T

t1

(2.4.45)

52 | 2 Global solutions to the equations of non-Newtonian fluids Using Lemma 2.4.3 with a small enough ϵ, and (2.4.43), (2.4.45) we find that T

∫ ‖ult ‖2L2 (Ωt ) dt + ‖ul ‖2W 2,2 (Ωt ) ≤ 0

1 sup ‖ul ‖2W 2,2 (Ωt ) + A(T) + C‖ul ‖2W 2,2 (Ωt ) 1 2 t1 ≤t≤T

(2.4.46)

where A(T) depends only on f , Mi , g. Taking the sup, in the above inequality we then find T

∫ ‖ult ‖2L2 (Ωt ) dt + sup ‖ul ‖2W 2,2 (Ωt ) ≤ 2A(T) + C‖ul ‖2W 2,2 (Ωt ) . t1 ≤t≤T

t1

1

(2.4.47)

After integration with respect to t1 over the interval (s, 2s) we then have T

∫ ‖ult ‖2L2 (Ωt ) dt s

+ sup

2s≤t≤T

‖ul ‖2W 2,2 (Ωt )

T

C ≤ 2A(T) + ∫ ‖ul ‖2W 2,2 (Ωt ) dt. s

(2.4.48)

0

Using Lemma 2.4.3, and letting l go to ∞ completes the proof. The content of our second regularity result is expressed in the following. Theorem 2.4.3. Assume that: (1) ũ ∈ L∞ (0, ∞; W 2,2 (Ω)) ∩ W −1,2 (0, ∞; W 4,2 (Ω)) (2) ũ t ∈ L2loc ([0, ∞); L2 (Ω)) (3) f ∈ L2loc ([0, ∞); L2 (Ω)) 1,2 Mi ∈ Wloc ([0, ∞); W −1/2,2 (Ω)) ̃ 0)) ∈ W01,2 (Ω). h ∈ H (H − u(x, Then problems (2.4.1)–(2.4.5) have a weak solution u in the sense of (2.4.13), which satisfies: 1,2 (1) u ∈ Wloc ([0, ∞); L2 (Ω)) ∞ (2) u ∈ Lloc ([0, ∞); H). Proof. Set t1 = 0 in the proof of Theorem 2.4.2. Finally, we relate the solution of the variational problem to the boundary value problem. Theorem 2.4.4. Under the assumptions of Theorem 2.4.3, the weak solution u of problem (2.4.13) that was constructed in Theorem 2.4.1 satisfies: 1,2 (1) u ∈ Wloc ([0, ∞); H ) and (2) there exists p(x, t) ∈ L2loc (0, ∞); W −1,2 such that (u, p) is a solution of (2.4.1)–(2.4.5). Proof. There are standard consequence of Theorems 2.4.1, 2.4.2 and (2.4.13).

2.4 Existence and uniqueness and stability of global solutions | 53

2.4.3 Uniqueness We establish in this section the uniqueness of the weak solution constructed in the above section; specifically, we will prove the following. Proposition 2.4.1. Under the assumptions of Theorem 2.4.1 the weak solution of problems (2.4.1)–(2.4.5) is unique. Proof. Assume that we have two weak distinct solution u, v and let ω = u − v. Taking the difference of the equations satisfied by u and v, multiplying the resulting equation by ω and integrating by parts over Ωt we find ∫ Ωt

𝜕w dx + ∫ (γ(u)eij (u) − γ(u)eij (ν))(eij (u) − eij (ν))dx 𝜕t Ωt

+∫

𝜕eij (w) 𝜕eij (w) 𝜕xk

Ωt

𝜕xk

dx = ∫ (uj Ωt

𝜕ui 𝜕v − vj i )wi dx. 𝜕xj 𝜕xj

(2.4.49)

From the monotonicity of γ, it follows that the second term on the left is positive and can be dropped. A simple calculation shows that ∫ (uj Ωt

𝜕ui 𝜕v 𝜕w − vj i )wi dx = ∫ wj i ui dx. 𝜕xj 𝜕xj 𝜕xj

(2.4.50)

Ωt

Since 2 1/2 1/2 𝜕wi 𝜕wi 2 ∞ u dx ) dx) (∫ w dx) ≤ ‖u‖ (∫ ( w ∫ L j j 𝜕xj i 𝜕xj Ωt

Ωt

Ωt

≤ ϵC‖u‖2W 2,2 ‖w‖W 2,2

1 + ‖w‖L2 ϵ

(2.4.51)

choosing ϵ small enough it then follows from (2.4.49), and the Korn-type inequality, that d‖w‖L2 (ΩT ) dt

≤ ‖w‖L2 (ΩT )

since ω = 0 at t = 0, it follows that ω(x, t) = 0, ∀t > 0. 2.4.4 Stability Finally, we will prove some estimates here which establish the asymptotic stability of the solution u under an appropriate set of conditions on b, f , u. We begin with the following.

54 | 2 Global solutions to the equations of non-Newtonian fluids Lemma 2.4.5. ∀δ > 0 there exists a function Gδ such that: (1) Gδ = g,

∀(x, t) ∈ 𝜕Ω × (0, ∞)

(2)

|b(u, Gδ , u)| ≤ δ‖u‖2W 2,2 (Ω) ,

(3)

div Gδ = 0,

∀u ∈ W01,2 (Ω)

0

‖Gδ ‖W 2,2 (Ω) ≤ C‖g‖W 2,2 (Ω) .

Let u be the solution to the problem (2.4.1)–(2.4.3) and set wδ = u − Gδ ; we will drop the subscript δ when there is no confusion. Taking ψ = w, and for the sake of convenience, dropping the δ subscript on Gg we find that ∫ Ωt

𝜕eij (w) 𝜕eij (w) 𝜕wi 𝜕u 𝜕w dx = ∫ uj i wi dx − ∫ γ(u)eij (u) i dx w dx + ∫ 𝜕t i 𝜕xk 𝜕xk 𝜕xj 𝜕xj Ωt

Ωt

+ ∫ fi wi dx − ∫ Ωt

Ωt

𝜕ũ w dx + ∫ 𝜕t i Ωt

Ωt

𝜕eij (G) 𝜕eij (w) 𝜕xk

𝜕xk

dx + ∫ Mi 𝜕Ωt

𝜕wi ds. 𝜕ν

(2.4.52)

Hence 𝜕eij (w) 𝜕eij (w) 1 d dx + ∫ γ(u)eij (w)eij (w)dx ∫ |w|2 dx + ∫ 2 dt 𝜕xk 𝜕xk Ωt

Ωt

Ωt

≤ (c‖G‖W 1,2 (Ωt ) + ‖f ‖W −1,2 (Ωt ) + ‖Gt ‖W 1,2 (Ωt ) )‖w‖W 1,2 (Ωt )

+ (c‖G‖W 2,2 (Ωt ) + ‖Mi ‖W −1/2,2 (𝜕Ωt ) )‖w‖W 2,2 (Ωt ) + |b(u, u, w)|.

(2.4.53)

Since b(u, u, w) = b(w, G, w) + b(G, G, w) and Gi Gj ∈ L2 (Ωt ), we have that |b(u, u, w)| ≤ |b(w, G, w)| + |b(G, G, w)| ≤ δ‖u‖2W 1,2 (Ω) + c‖Gi Gj ‖L2 (Ω) ‖u‖2W 1,2 (Ω) 0

0

(2.4.54)

where Lemma 2.4.5 was used. It then follows from (2.4.54) that 𝜕eij (w) 𝜕eij (w) 1 d dx + ∫ γ(u)eij (w)eij (w)dx ∫ |w|2 dx + ∫ 2 dt 𝜕xk 𝜕xk Ωt

Ωt

Ωt

1 ≤ (c‖G‖W 1,2 (Ωt ) + ‖f ‖W −1,2 (Ωt ) ‖Gt ‖W 1,2 (Ωt ) + c‖G‖W 2,2 (Ωt ) 2δ + ‖Mi ‖W −1/2,2 (𝜕Ωt ) + c‖Gi Gj ‖L2 (Ω) )2 + 2δ‖w‖2W 2,2 (Ωt ) .

(2.4.55)

Using Lemma 2.4.2 and dropped the positive third term ∫Ω γ(u)eij (w)eij (w)dx we find t that, for δ small enough, d σ ‖w‖2L2 (Ωt ) + ‖w‖2W 2,2 (Ωt ) ≤ cB(t) dt 2 where σ > 0 and B(t) = (‖f ‖W −1,2 (Ωt ) + ‖gt ‖W 1,2 (Ωt ) + ‖Mi ‖W −1/2,2 (𝜕Ωt ) )2 . As an immediate consequence of (2.4.56), we have the following.

(2.4.56)

2.5 The modified Boussinesq approximation | 55

Proposition 2.4.2. Assume that B(t) ∈ L∞ (0, ∞); then the solution u of problems (2.4.1)–(2.4.5) are in L∞ (0, ∞; L2 (Ω)); furthermore, if B(t) decays exponentially to 0, then ‖u‖L2 (Ω) also decays exponentially to 0.

2.5 The periodic initial value problem and initial value problem for the modified Boussinesq approximation The study of flows in the Earth’s mantle consists of thermal convection in a highly viscous fluid. For a description of dynamics of flows of an incompressible fluid in processes where the thermal effects play an essential role; the Boussinesq approximation is a reasonable model to present essential phenomena of such flows and it is used, for example, in planetary physics for describing processes in body interior [10, 13]. Let n = 2 or 3. We denote u = (u1 , u2 , . . . , un ) the velocity field associated with the flow of an incompressible bipolar viscous fluid, π the pressure and θ the temperature. The Boussinesq approximation in nondimensional form is described by 𝜕u + ρ(u ⋅ ∇)u = −∇π + div τν + ρen θ + f 𝜕t 𝜕u div u = i = 0 𝜕xi 𝜕θ ρ + ρ(u ⋅ ∇)θ − κΔθ = g 𝜕t

ρ

(2.5.1) (2.5.2) (2.5.3)

where ρ = const > 0 is the density; κ > 0 is a positive constant coefficient (thermometric conductivity), en is a unit vector in Rn and f (x, t), g(x, t) are given vector value and scalar functions, respectively. For the sake of simplicity, we put ρ = 1 and κ ≥ 1. In order to make the system of equations complete, it is necessary to prescribe the constitutive relation for the viscous part of the stress tensor. In the present work, we will assume that the constitutive laws have the form τν = τ(e) − 2μ1 Δe,

(2.5.4)

where the τ(e) is given via a scalar potential U of the symmetrized velocity gradient e, that is τij (e) =

𝜕U(e) , 𝜕eij

e(u) = (eij (u)),

i, j = 1, 2, . . . , n

(2.5.5)

𝜕uj 1 𝜕u eij (u) = ( i + ) 2 𝜕xj 𝜕xi

(2.5.6)

U(⋅) is twice continuously differentiable in Rn . U ≥ 0, U(0) = 1, 2, . . . , n, such that

|e|p−2 |ξ |2 , 𝜕2 U(e) ξij ξkl ≥ c1 { (1 + |e(u)|)p−2 |ξ |2 , 𝜕eij 𝜕ekl

p 0, ξ ∈ Rnsym ≡ {M ∈ Rn ; Mij = Mji , i, j = 1, 2, . . . , n}, |e| = (eij eij )1/2 . From the hypothesis of U above, we can show that p−1

|τij (e)| ≤ c3 (1 + |e(u)|) p−1

|τij (e)| ≤ c3 |e(u)|

,

p≥2

,

(2.5.9)

1≤p 0 such that ‖u‖s,p ≤ c5 ‖u‖as1 ,p ‖u‖1−a s2 ,p

(2.5.20)

where s = as1 + (1 − a)s2 , a ∈ (0, 1). Lemma 2.5.2 (embeddings). Let 1 < p, q < ∞, 0 ≤ s2 ≤ s1 < ∞ be an integer or noninteger. Then W s1 ,p (Ω) → W s2 ,q (Ω) if 1 1 s − s2 = − 1 . q p n Lemma 2.5.3 (see [77]; generalized Korn inequality). Let φ ∈ W 1,q (Ω)n ∩ W 1,2 (Ω)n , q > 1. Then q

1 q

(∫ |e(φ)| dx) ≥ Kq |φ|1,q

(2.5.21)

Ω

where Kq > 0, 2eij (φ) =

𝜕φi 𝜕xj

+

𝜕φj . 𝜕xi

Lemma 2.5.4 (see [3]). Let QT ⊂ Rn+1 be bounded. Let fN : QT → R be integrable for every N and let: (i) limN→∞ fN (y) exist and be finite for a. e. y ∈ QT (ii) ∀ϵ > 0, ∃δ > 0 such that sup ∫ |fN (y)|dy < ϵ, N

H

∀H ⊂ QT ;

|H| < δ.

58 | 2 Global solutions to the equations of non-Newtonian fluids Then lim ∫ |fN (y)|dy = ∫ lim fN (y)dy.

N→∞

QT

QT

N→∞

Lemma 2.5.5 (Lions–Aubin). Let X be a Banach space, and X0 , X1 separable, reflexive Banach spaces. Provided that X0 →→ X →→ X1 , then {v ∈ Lα (I; X0 );

dv ∈ Lβ (I; X1 )} →→ Lα (I; X) dt

where 1 < α, β < +∞. Consider the following initial value problem: 𝜕ui 𝜕u 𝜕π 𝜕τij 𝜕 + − 2μ1 Δe + en θ + fi + uj i = − 𝜕t 𝜕xj 𝜕xi 𝜕xj 𝜕xj ij

(2.5.22)

𝜕ui =0 𝜕xi 𝜕θ 𝜕θ − Δθ = g + uj 𝜕t 𝜕xj

(2.5.23)

div u =

u|Γj = u|Γj+n ,

(2.5.24)

θ|Γj = θ|Γj+n

u(x, 0) = u0 (x),

(2.5.25)

θ(x, 0) − θ0 (x)

(2.5.26)

where the nonlinear tensor function τ(⋅) satisfies the conditions (2.5.5)–(2.5.11), fi , g ∈ L2 (I; L2 (Ω)) are given scalar valued functions and u0 (x), θ0 (x), fi , g are given vector valued and scalar valued functions satisfying the periodic boundary condition. 2,2 Definition 2.5.1. Let u0 ∈ Wper (Ω)n ∩ H, θ0 ∈ M = {θ̃ ∈ L2per (Ω); T1 ≤ θ̃ ≤ T0 }, fi , g ∈ 2 2 L (I; Lper (Ω)) The pare of the functions (u, θ) is called a weak solution of the problems (2.5.22)–(2.5.26) if

∫ Ω

𝜕u 𝜕ui φ dx + ∫ uj i φi dx + ∫ τij (e(u))eij (φ)dx 𝜕t i 𝜕xj Ω

+ 2μ1 ∫ Ω

∫ Ω

Ω

𝜕eij (u) 𝜕eij (φ) 𝜕xk

𝜕xk

dx = ∫ en θφdx + ∫ fi φdx Ω

𝜕θ 𝜕θ ψdx + ∫ uj φdx + ∫ ∇θ ⋅ ∇φdx = ∫ gφdx 𝜕t 𝜕xj Ω

Ω

(2.5.27)

Ω

(2.5.28)

Ω

is satisfied a. e. for every φ = (φ1 , φ2 , . . . , φn ) ∈ Y = W 2,2 (Ω)n ∩ Vp ; ψ ∈ Vθ in I. We will show the existence of a weak solution to the problems (2.5.22)–(2.5.26) via Galerkin approximations: ((w, ψ)) = ∫ Ω

𝜕 𝜕 e (w) ⋅ e (ψ)dx 𝜕xk ij 𝜕xk ij

2.5 The modified Boussinesq approximation | 59

(w, ψ) L2 ((w, ψ)) = λ(w, ψ),

∀ψ ∈ V1

(2.5.29)

wl ∈ V1 ∩ C ∞ (Ω) λ1 wl V1 (wk , wl ) = δkl . 2

Lemma 2.5.6. Let u ∈ L2 (Ω), D2 (u) ∈ L2 (Ω)n . Then Du ∈ L2 (Ω)n and ‖Du‖22 ≤ c6 ‖u‖2 ‖D2 u‖2 .

(2.5.30)

Lemma 2.5.7. Let u ∈ W 2,2 (Ω), n ≤ 3, then u ∈ L∞ (Ω), that is, there exists c7 > 0 such that |u|∞ = ess ⋅ sup |u(x)| ≤ c7 ‖u‖22,2 . x∈Ω

(2.5.31)

Definition 2.5.2. We say that (uN , θN ), uN = ∑Nk=1 ckN (t)wk (x), θN = ∑Nk=1 dkN (t)wk (x) is the Galerkin approximation of the solution of the problems (2.5.22)–(2.5.26) if ∫(∑ Ω

𝜕c1N (t) l wi (x))wik (x)dx + ∫ τij (e(uN (x, t))eij (wk (x)))dx 𝜕t

+ 2μ1 ∫

𝜕eij (u (x, t)) 𝜕eij (w (x, t)) 𝜕xl

Ω

= ∫ en θ

Ω k

N

N

Ω

wik (x)dx

𝜕xl +

dx + ∫(∑ ckN (t)wjk (x))(∑ ckN (t) Ω

∫ fi (x, t)wik (x)dx,

∀wk , k = 1, 2, . . . , N

𝜕wik (x) k )wi (x)dx 𝜕xj (2.5.32)

Ω

𝜕dN (t) l 𝜕θN k ω (k)dx ω (x))ωk (x) + ∫(∑ clN (t)ωlj (x)) ∫(∑ l 𝜕t 𝜕xj

Ω

Ω

N

+ ∫ ∇θ ⋅ ∇ω (x)dx = ∫ gωk (x)dx, Ω

k

∀ωk , k = 1, 2, . . . , N.

(2.5.33)

Ω

Lemma 2.5.8. Let u0 ∈ H, θ0 ∈ M, f ∈ L2 (I; L2 (Ω)n ), g ∈ L2 (I; L2 (Ω)). Then the sequence of Galerkin approximations satisfies the following uniform estimates: ‖uN ‖L∞ (I;H) ≤ c8

(2.5.34)

‖u ‖Lp (I;Vp ) ≤ c9

(2.5.35)

N

‖uN ‖L2 (I;W 2,2 (Ω)n ) ≤ c10

(2.5.36)

‖θ ‖L∞ (I;L2 (Ω)) ≤ c11

(2.5.37)

‖θ ‖L2 (I;Vθ ) ≤ c11

(2.5.38)

N

N

where ci (i = 8, . . . , 11) is independent on the constant N.

60 | 2 Global solutions to the equations of non-Newtonian fluids Proof. Multiplying (2.5.32) by ckN (t) and summing up the equations we get (using the

fact that ∫Ω uNj

𝜕uNi 𝜕xj

uNi dx = 0, for divergence free functions)

1 d ∫ |uN |2 dx + ∫ τij (e(uN )eij (uN ))dx 2 dt Ω

+ 2μ1 ∫

Ω N

𝜕eij (u ) 𝜕eij (uN )

Ω

𝜕xk

𝜕xk

dx = ∫ en θN ⋅ uN dx + ∫ fi uNi dx. Ω

(2.5.39)

Ω

Multiplying (2.5.33) by dkN (t) and summing up the equations we get (also using the fact 𝜕uNi 𝜕xj

that ∫Ω uNj

uNi dx = 0, for divergence-free functions)

1 1 d N 2 ‖θ ‖2 + ‖∇θN ‖22 = ∫ gθN dx ≤ ∫(|g|2 + |θN |2 )dx. 2 dt 2 Ω

(2.5.40)

Ω

Using the Gronwall inequality, we get ‖θN ‖22

t

+ ∫ ‖∇θN ‖22 dt ≤ C(T, ‖g‖L2 (QT ) ),

∀t ∈ (0, T).

(2.5.41)

0

Using the coercivity condition (2.5.11) and the Korn inequality (2.5.21), we obtain N

‖u

(t)‖22

+

t p 2c4 Kp ∫ |uN |p1,p dt

t

+ 4μ1 K

0

∫ |uN |22,2 dt 0

t N ≤ ∫ ∫ fi ui dxdt 0 Ω

t t t t 1 + ∫ ∫ en θN ⋅ uN dxdt ≤ [∫ ‖f ‖22 dt + ∫ ‖θN ‖22 dt] + ∫ ‖uN ‖22 dt. 2 0 Ω 0 0 0

(2.5.42)

Because of the assuming conditions u0 , f and the estimates (2.5.37), we obtain t

t

‖uN (t)‖22 +2c4 Kpp ∫ |uN |p1,p dt + 4μ1 K ∫ |uN |22,2 dt 0

t

0

≤ C(u0 , θ0 , f , g, T) + ∫ ‖uN (τ)‖22 dτ.

(2.5.43)

0

Employing the Gronwall inequality, the other three estimates we also can get (2.5.34)–(2.5.36). Lemma 2.5.9. Let n ≤ 3, u0 ∈ Y, θ0 ∈ Vθ , f ∈ L2 (I; L2 (Ω)n ), g ∈ L2 (I; L2 (Ω)), then ‖uN ‖L∞ (I;W 2,2 (Ω)n ) ≤ c12

(2.5.44)

2.5 The modified Boussinesq approximation

‖uN ‖L∞ (I;Vp ) ≤ c12 𝜕uN ≤ c12 𝜕t L2 (I;L2 (Ω)n )

| 61

(2.5.45) (2.5.46)

‖θN ‖L∞ (I;Vθ ) ≤ c13

(2.5.47)

‖θ ‖L2 (I;W 2,2 (Ω)) ≤ c14 𝜕θN ≤ c15 𝜕t L2 (I;L2 (Ω))

(2.5.48)

N

(2.5.49)

where ci (i = 12, . . . , 15) is independent on the constant N. Proof. Multiplying (2.5.33) by λk dkN (t) and summing up the equations, we get 1 d 𝜕θN N ‖∇θN ‖22 + ‖ΔθN ‖22 ≤ ∫ uNj Δθ dx + ∫ gΔθN dx. 2 dt 𝜕xj

(2.5.50)

Ω

Ω

Using the Hölder and Young inequalities, we have ∫ |g∇θN |dx ≤ ‖g‖22 +

Ω

1 ‖ΔθN ‖22 4

𝜕θN 1 ΔθN dx ≤ ∫ |uN |2 |∇θN |2 dx + ‖ΔθN ‖22 . ∫uNj 𝜕xj 4

Ω

Ω

Substituting two estimates above into (2.5.50) and integrating over (0, t), we obtain t

t

1 ‖∇θN ‖22 + c16 ∫ ‖D2 θN ‖22 dt ≤ ‖∇θN (0)‖22 + ‖g‖2L2 (Qt ) + ∫ ∫ |uN |2 |∇θN |2 dxdt. 2 0

(2.5.51)

0 Ω

The last term on the right-hand side of (2.5.51) can be estimated by means of Lemmas 2.5.7 and 2.5.8: t

N 2

N 2

t

∫ ∫ |u | |∇θ | dxdt ≤ c7 ∫ ‖uN ‖22,2 ‖∇θN ‖22 dt 0 Ω

0

t

≤ c6 c7 ∫ ‖uN ‖22,2 (ϵ‖D2 θN ‖22 + λ(ϵ)‖θN ‖22 )dt. 0

Combining (2.5.30) with (2.5.31), we get ‖∇θN ‖22

T

+ c16 ∫ ‖D2 θN ‖22 dt ≤ ‖∇θN (0)‖22 + ‖g‖2L2 (QT ) 0

(2.5.52)

62 | 2 Global solutions to the equations of non-Newtonian fluids

+

c6 c7 ϵ sup ‖uN ‖22,2 (0,T)

T

⋅

∫ ‖D2 θN ‖22 dt 0

T

+ c6 c7 λ(ϵ) ∫ ‖uN ‖2 ‖θN ‖22 dt.

(2.5.53)

0

By virtue of (2.5.15) and (2.5.16), with a small coefficient ϵ, we infer c6 c7 ϵ sup(0,T) ‖uN ‖22,2 < 21 c16 : ‖∇θN ‖22

T

+ c16 ∫ ‖D2 θN ‖22 dt ≤ c13 .

(2.5.54)

0

From (2.5.54), we can get the estimates (2.5.47) and (2.5.48). Multiplying (2.5.32) by t ∈ (0, T] it follows:

𝜕ckN (t) , 𝜕t

summing the equations and integrating over (0, t),

𝜕uN 𝜕eij (uN (t)) 2 N N dxdt + U(e(u (t)))dx − U(e(u (0)))dx + μ ∫ ∫ ∫ ∫ dx 1 𝜕t 𝜕x k

Qt

Ω

Ω

Ω

N N 𝜕eij (uN (0)) 2 𝜕uN 𝜕uN N 𝜕ui 𝜕ui − μ1 ∫ dxdt = ∫ en θN ⋅ dxdt + ∫ fi i dxdt dx + ∫ uj 𝜕xk 𝜕xj 𝜕t 𝜕t 𝜕t Qt

Qt

Ω

Qt

(2.5.55)

where U(e(uN )) is the scalar potential given by (2.5.5). From (2.5.9)–(2.5.11) and the generalized Korn inequality, we have p

c17 ‖DuN ‖pp ≤ c18 ‖e(uN )‖pp ≤ ‖U(e(uN ))‖1 ≤ c19 (1 + |e(uN )|) .

(2.5.56)

The Hölder and Young inequalities yield N N N 𝜕ui 𝜕ui dxdt ≤ c20 ∫ |uN |2 |DuN |2 dxdt + ∫ uj 𝜕xj 𝜕t Qt

Qt

N N 𝜕u dxdt ≤ c21 ‖θN ‖2L2 (Qt ) + ∫ en θ ⋅ 𝜕t Q

𝜕uN 2 ∫ fi i dxdt ≤ c22 ‖f ‖L2 (Qt ) + 𝜕t Qt

2

1 𝜕uN 6 𝜕t L2 (Qt )

2

1 𝜕uN 6 𝜕t L2 (Qt ) 2

1 𝜕uN . 6 𝜕t L2 (Qt )

(2.5.57) (2.5.58) (2.5.59)

The assumptions on u0 , f and using the generalized Korn inequality yield 𝜕uN 2 + c17 ‖DuN ‖pp + c23 ‖D2 uN ‖22 𝜕t L2 (Qt )

≤ C(u0 , θ0 , f , g, T) + c20 ∫ |uN |2 |DuN |2 dxdt. Qt

(2.5.60)

2.5 The modified Boussinesq approximation

| 63

Applying the embedding Lemma 2.5.7 and the standard interpolation inequality, ‖v‖2W 1,2 (Rn ) ≤ δ‖v‖2W 2,2 (Rn ) + λ(δ)‖v‖2L2 (Rn ) .

(2.5.61)

For c24 > 0, we have t

∫ |u | |Du | dxdt ≤ c24 ∫ ‖u‖22,2 [δ‖uN ‖22,2 + λ(δ)‖uN ‖22 ]dt. N 2

N 2

(2.5.62)

0

Qt

From (2.5.60) and (2.5.62), we get 𝜕uN 2 + c17 ‖DuN ‖pp + c23 sup ‖D2 uN ‖22 ≤ C(u0 , θ0 , f , g, T) 𝜕t L2 (Qt ) (0,T) T 2 N 2 c20 c24 δ sup ‖D u ‖2 ∫ ‖uN ‖22,2 dt (0,T) 0

T

+ c20 c24 λ(δ) ∫ ‖uN ‖22,2 ‖uN ‖22 dt

(2.5.63)

0

using the estimates (2.5.14) and (2.5.15), choosing δ small enough, we can obtain the estimates (2.5.44)–(2.5.46). Now we prove the estimate (2.5.49). Multiplying (2.5.12) by dj (t), summing the equations and integrating over (0, t), t ∈ (0, T], we have 𝜕θN 2 1 + ∫ |∇θN (t)|2 dx 𝜕t L2 (Qt ) 2 Ω

𝜕θiN 𝜕θN 𝜕θN ≤ ‖∇θN (0)‖22 + ∫ uNj dxdt + ∫ g dxdt . 𝜕xj 𝜕t 𝜕t Qt

(2.5.64)

Qt

The assumptions on θ0 , g and using the Hölder and Young inequalities yield 𝜕θN 2 + ‖∇θN ‖22 ≤ C(θ0 , g) + ∫ |uN |2 |∇θN |2 dxdt. 𝜕t L2 (Qt )

(2.5.65)

Qt

By virtue of Lemma 2.5.7, the estimates (2.5.44) and (2.5.47), we know the last term on the right-hand side of (2.5.65) is finite. So from (2.5.65), the desired estimate (2.5.49) can be obtained. 2n 2n Theorem 2.5.1. Let n ≤ 3 n+2 < p < n−2 , u0 ∈ W 2,2 (Ω)n ∩ H, θ0 ∈ Vθ , f ∈ L2 (I; L2 (Ω)n ), 2 2 g ∈ L (I; L (Ω)) Then there exists a unique weak solution (u, θ) of the problems (2.5.22)– (2.5.26). Moreover,

u ∈ L∞ (I; H) ∩ Lp (I; Vp ) ∩ L2 (I; W 2,2 (Ω)n ) ∩ C(I; H)

(2.5.66)

u ∈ L∞ (I; Vp ) ∩ L∞ (I; W 2,2 (Ω)n )

(2.5.67)

𝜕u ∈ L2 (I; H) 𝜕t

64 | 2 Global solutions to the equations of non-Newtonian fluids θ ∈ L∞ (I; L2 (Ω)) ∩ L2 (I; Vθ ) ∩ C(I; L2 (Ω))

(2.5.68)

θ ∈ L∞ (I; Vθ ) ∩ L2 (I; W 2,2 (Ω)) and

(2.5.69)

𝜕θ ∈ L2 (I; L2 (Ω)). 𝜕t

Proof. Existence. From Lemmas 2.5.8 and 2.5.9, we can select the subsequence of {uN , θN }, and also denote {uN , θN }, {uN } is uniformly bounded in L∞ (I; H) ∩ Lp (I; Vp ) ∩ N

L2 (I; W 2,2 (Ω)n ), {θN } is uniformly bounded in L∞ (I; L2 (Ω)) ∩ L2 (I; Vθ ) and so { 𝜕u𝜕t } N

is uniformly bounded in L2 (I; H), { 𝜕θ𝜕t } is uniformly bounded in L2 (I; L2 (Ω)). By the Lemma 2.5.5, we get that uN → u strongly in L2 (I; W 1,p̃ (Ω)n ), θN → θ strongly in L2 (I; W 1,2 (Ω)). As u ∈ L2 (I; W 2,2 (Ω)) ∩ L2 (I; H) and 𝜕u ∈ L2 (I; H), it follows from Lemma 1.2, Chap𝜕t ter III in [70] that u ∈ C(I; H). Similarly, we get that θ ∈ C(I; L2 (Ω)). Uniqueness. Let (u1 , θ1 ), (u2 , θ2 ) be two couples of solutions for the same initial values. Let us denote (u, θ) = (u1 − u2 , θ1 − θ2 ), then we have 𝜕ul 1 d 𝜕θ dx (‖u‖22 + ‖θ‖22 ) + C(|u|22,2 + |θ|21,2 ) ≤ C‖u‖22 + ‖θ‖22 + ∫ ui i ui dx + ∫ uj θl 2 dt 𝜕xj 𝜕xj Ω

Ω

(2.5.70)

𝜕ul |I1 | = ∫ ui i ui dx ≤ ‖u‖24 |ul |1,2 ≤ ‖u‖2 ‖u‖∞ |ul |1,2 ≤ c‖u‖2 ‖u‖2,2 |ul |1,2 𝜕xj Ω

≤ ϵ1 ‖u‖22,2 + λ(ϵ1 )‖u‖22 |ul |21,2 𝜕θ 1 θ dx ≤ ϵ2 |θ|21,2 + λ(ϵ2 )‖θ1 ‖22,2 ‖u‖22 . |I2 | = ∫ uj 𝜕xj

(2.5.71) (2.5.72)

Ω

Substituting the above two estimates (2.5.71), (2.5.72) into (2.5.70) and integrating over (0, t), t ∈ (0, T] using (u, θ)(0) = (0, 0), it follows from the Gronwall inequality that ‖u‖22 + ‖θ‖22 = 0 a. e. in I and, therefore, (u1 , θ1 ) = (u2 , θ2 ) a. e. in QT . 2n 2n Theorem 2.5.2. Let n ≤ 3 n+2 < p < n−2 , u0 ∈ H, θ0 ∈ M, f ∈ L2 (I; L2 (Ω)n ), g ∈ 2 2 L (I; L (Ω)), Iδ = [δ, T], δ > 0. Then there exists a unique weak solution (u, θ) of the initial value problems (2.5.22)–(2.5.26). Moreover,

u ∈ L∞ (I; H) ∩ Lp (I; Vp ) ∩ L2 (I; W 2,2 (Ω)n ) ∩ C(Iδ ; V2 ) u ∈ L∞ (Iδ ; Vp ) ∩ L∞ (Iδ ; W 2,2 (Ω)n ) and θ ∈ L∞ (I; L2 (Ω)) ∩ L2 (I; Vθ ) ∩ C(Iδ ; Vθ ) θ ∈ L∞ (Iδ ; Vθ ) ∩ L2 (Iδ ; W 2,2 (Ω)) and

(2.5.73)

𝜕u ∈ L2 (Iδ ; H) 𝜕t

(2.5.74)

𝜕θ ∈ L2 (Iδ ; L2 (Ω)). 𝜕t

(2.5.76)

(2.5.75)

If g = 0 has the maximum property, T1 ≤ θ(x, t) ≤ T0

or

θ ∈ L∞ (I; M)

(2.5.77)

2.5 The modified Boussinesq approximation

| 65

Proof. By the proof of Theorem 2.5.1, we only prove the regularity (2.5.74), (2.5.76) and (2.5.77). In order to prove (2.5.77), let ψ = (θ − T0 )+ in (2.5.28) and integrating over (0, t), we have 1 2 ∫[(θ − T0 )+ ] dx + ∫ |∇(θ − T0 )+ |2 dxdt = 0. 2 QT

Ω

So θ ≤ T0 in QT . Similarly, let ψ = (θ − T0 )+ in (2.5.7), and we obtain T1 ≤ θ in QT . The maximum property is obtained. Since u0 ∈ H not in Y, θ0 ∈ M not in Vθ , we only use the standard cutting function method to obtain the local estimates (2.5.73), (2.5.75). Multiplying (2.5.33) by λk dkN (t), summing over, applying Hölder and Young inequalities, d ‖∇θN ‖22 + ‖ΔθN ‖22 ≤ ‖g‖22 + ∫ |uN |2 |∇θN |2 dx. dt

(2.5.78)

Rn

For δ > 0, we define the cutting function ξ ∈ C 1 (I) such that ξ (t) ∈ [0, 1] in I and 0, ξ (t) = { 1,

t ∈ [0, δ2 ] t ∈ [δ, T].

Multiplying (2.5.78) by ξ (t), dξ d [ξ (t)‖∇θN ‖22 ] + ξ (t)‖ΔθN ‖22 ≤ ξ (t)‖g‖22 + ‖∇θN ‖22 + ξ ∫ |uN |2 |∇θN |2 dx. dt dt

(2.5.79)

Rn

By the definition of the function ξ (t), we have dξ 2 ≤ . dt δ So integrating (2.5.79) on (0, t), t ∈ [δ, T], ‖∇θN ‖22

t

+

∫ ‖δθN ‖22 dt 0

≤

‖g‖2L2 (Qt )

t

+

C ∫ ‖∇θN ‖22 dt 0

t

+ ∫ ∫ |uN |2 |∇θN |2 dxdt.

(2.5.80)

0 Ω

The last term of the right-hand side to (2.5.80) can be obtained by Lemma 2.5.9, hence θN ∈ L∞ (Iδ ; Vθ ) ∩ L2 (Iδ ; W 2,2 (Ω)). Similarly, we have 𝜕θN ∈ L∞ (Iδ ; L2 (Rn )). 𝜕t

66 | 2 Global solutions to the equations of non-Newtonian fluids 𝜕ckN (t) , 𝜕t

Multiplying (2.5.32) by

summing over k from 1 to N, we get N

𝜕uN 2 𝜕eij (uN ) 𝜕eij ( 𝜕u𝜕t ) 𝜕uN N )dx + 2μ1 ∫ dx + ∫ τij (e(u ))eij ( 𝜕t 𝜕xl 𝜕xl 𝜕t 2 Ω

Ω

𝜕uNi 𝜕uNi 𝜕uN 𝜕uN + ∫ uNj dx = ∫ en θN dx + ∫ fi i dx. 𝜕xj 𝜕t 𝜕t 𝜕t Rn

Rn

(2.5.81)

Rn

The Hölder and Young inequalities yield N

𝜕uN 2 𝜕eij (uN ) 𝜕eij ( 𝜕u𝜕t ) 𝜕uN N )dx + 2μ1 ∫ dx + ∫ τij (e(u ))eij ( 𝜕t 2 𝜕t 𝜕xl 𝜕xl n n R

R

≤

‖θN ‖22

+

‖f ‖22

N 2

N 2

+ ∫ |u | |Du | dx.

(2.5.82)

Rn

Multiplying (2.5.82) by ξ (t), 𝜕uN 2 𝜕eij (uN ) 2 d N ξ (t) dx + ξ (t) ∫ U(eij (u ))dx + μ1 ξ (t) ∫ 𝜕t 2 𝜕xk dt n n R

R

≤ ξ (t)(‖θN ‖22 + ‖f ‖22 ) + ξ (t) ∫ |uN |2 |DuN |2 dx.

(2.5.83)

Rn

dξ

Because of dt ≤ δc , t ∈ [0, T], integrating (2.5.83) on [δ, T], using the generalized Korn inequality, we have t 𝜕uN 2 N p 2 N 2 N 2 2 N 2 N 2 + C‖Du ‖p + C‖D u ‖2 ≤ ‖θ ‖L2 (Qt ) + ‖f ‖L2 (Qt ) + ∫ ∫ |u | |Du | dxdt. (2.5.84) 𝜕t 2 n 0R

For the initial value problem for the Boussinesq approximation equations, 𝜕ui 𝜕u 𝜕π 𝜕τij 𝜕 + uj i = − + − 2μ1 Δe + en θ + fi 𝜕t 𝜕xj 𝜕xi 𝜕xj 𝜕xj ij div u =

𝜕ui =0 𝜕xi

(2.5.86)

𝜕θ 𝜕θ + ui − Δθ = g 𝜕t 𝜕xj u(x, 0) = u0 (x),

(2.5.85)

(2.5.87) θ(x, 0) = θ0 (x),

(2.5.88)

where the nonlinear tensor function τ(⋅) satisfies the conditions (2.5.5)–(2.5.11), fi , g ∈ L2 (I; L2 (Ω)) be given scalar valued functions, u0 (x), θ0 (x), fi , g be given vector valued, noticing that all the above estimates are independent on L. So in the problems (2.5.22)– (2.5.26), let L → 0 and we obtain the following.

2.5 Periodic initial value problem and initial value problem | 67

2n 2n Theorem 2.5.3. Let n ≤ 3 n+2 < p < n−2 , u0 ∈ W 2,2 (Rn )n ∩ H, θ0 ∈ Vθ , f ∈ L2 (I; L2 (Rn )n ), 2 2 n g ∈ L (I; L (R )). Then there exists a unique weak solution of the initial value problems (2.5.85)–(2.5.88). Moreover, n

u ∈ L∞ (I; H) ∩ Lp (I; Vp ) ∩ L2 (I; W 2,2 (Rn ) ) ∩ C(I; H) n

u ∈ L∞ (I; Vp ) ∩ L∞ (I; W 2,2 (Rn ) )

𝜕u ∈ L2 (I; H) 𝜕t

(2.5.90)

𝜕θ ∈ L2 (I; L2 (Rn )). 𝜕t

(2.5.92)

and

θ ∈ L∞ (I; L2 (Ω)) ∩ L2 (I; Vθ ) ∩ C(I; L2 (Rn )) θ ∈ L∞ (I; Vθ ) ∩ L2 (I; W 2,2 (Rn ))

and

(2.5.89)

(2.5.91)

2n 2n Theorem 2.5.4. Let n ≤ 3 and n+2 < p < n−2 , u0 ∈ H, θ0 ∈ M, f ∈ L2 (I; L2 (Rn )n ), 2 2 n g ∈ L (I; L (R )), Iδ = [δ, T], δ > 0. Then there exists a unique weak solution of the initial value problems (2.5.85)–(2.5.88). Moreover, n

u ∈ L∞ (I; H) ∩ Lp (I; Vp ) ∩ L2 (I; W 2,2 (Rn ) ) ∩ C(Iδ ; V2 ) n

u ∈ L∞ (Iδ ; Vp ) ∩ L∞ (Iδ ; W 2,2 (Rn ) ) and 2

n

2

θ ∈ L (I; L (R )) ∩ L (I; Vθ ) ∩ C(Iδ ; Vθ ) ∞

θ ∈ L∞ (Iδ ; Vθ ) ∩ L2 (Iδ ; W 2,2 (Rn ))

and

(2.5.93)

𝜕u ∈ L2 (Iδ ; H) 𝜕t

(2.5.94)

𝜕θ ∈ L2 (Iδ ; L2 (Rn )). 𝜕t

(2.5.96)

(2.5.95)

If g = 0 has the maximum property, then T1 ≤ θ(x, t) ≤ T0

or

θ ∈ L∞ (I; M).

(2.5.97)

2.6 Periodic initial value problem and initial value problem for the non-Newtonian–Boussinesq approximation Let n = 2 or n = 3, and we denote u = (u1 , u2 , . . . , un ), the velocity field associated with the flow of an incompressible bipolar viscous fluid, π the pressure and θ the temperature. The Boussinesq approximation in nondimensional form is described by ρ

ρ

𝜕u + ρ(u ⋅ ∇)u = −∇π + div τν + ρen θ + f 𝜕t 𝜕u div u = i = 0 𝜕xi

𝜕θ + ρ(u ⋅ ∇)θ − κΔθ = g 𝜕t

(2.6.1) (2.6.2) (2.6.3)

where ρ = const > 0 is the density, κ > 0 is a positive constant coefficient (thermometric conductivity), en is a unit vector in Rn and f (x, t), g(x, t) are given vector value and scalar functions, respectively. For the sake of simplicity, we put ρ = 1 and κ = 1.

68 | 2 Global solutions to the equations of non-Newtonian fluids In the present work, we will assume that the constitutive laws have the form τν = τ(e),

𝜕uj 1 𝜕u ) eij (u) = ( i + 2 𝜕xj 𝜕xi

e(u) = (eij (u)),

(2.6.4)

where the τ(e) is given via a scalar potential U of the symmetrized velocity gradient e, that is, τij (e) =

𝜕U(e) , 𝜕eij

i, j = 1, 2, . . . , n.

(2.6.5)

U(⋅) is twice continuously differentiable in Rn , U ≥ 0, for all i, j = 1, 2, . . . , n, U(0) = 𝜕U(0) = 0 and such that 𝜕e ij

|e|p−2 |ξ |2 , 𝜕2 U(e) ξij ξkl ≥ c1 { (1 + |e(u)|)p−2 |ξ |2 , 𝜕eij 𝜕ekl 𝜕2 U(e) p−2 ≤ c2 (1 + |e|) 𝜕eij 𝜕ekl 2

p 0, c2 > 0, ξ ∈ Rnsym ≡ {M ∈ Rn ; Mij = Mji , i, j = 1, . . . , n},

|e| = (eij eij )1/2 . From the hypothesis of U above, we can show that p−1

|τij (e)| ≤ c3 (1 + |e(u)|) p−1

|τij (e)| ≤ c3 |e(u)|

,

p≥2

,

(2.6.8)

1≤p 2, ϵ > 0

(2.6.11)

and τij (e) = 2μ0 |e(u)|p−2 eij ,

(2.6.12)

the potentials |e|2

U(e) = μ0 ∫ (ϵ + s(p−2)/2 )ds,

p>2

(2.6.13)

0

and

|e|2

U(e) = μ0 ∫ s 0

satisfy the assumptions (2.6.5)–(2.6.7).

p−2 2

ds,

1 0, QT = Ω × I. The standard notation is used for both scalar (u : Ω → R or (QT :→ R)) and vector-valued functions (u :→ Rn or (QT :→ Rn )). The space D(Ω) is defined as the space of functions from C ∞ (Ω) with compact support in Ω. The space of distributions on Ω denoted by D (Ω), consists of all continuous linear functional on D(Ω). Let q > 1 and q be the dual index to q, that is, 1/q + 1/q = 1. The Lebesque spaces of scalar and vector valued functions are denoted by Lq (Ω) and Lq (Ω)n , respectively. The spaces equipped with the standard norm are denoted by ‖ ⋅ ‖q . The Sobolev spaces W m,p (Ω) and W m,p (Ω)n are the sets of all measurable functions, for which the functions and all their generalized derivatives up to the order m belong to Lp (Ω) and Lp (Ω)n , respectively. The spaces are equipped with the standard norms and semi-norms denoted by ‖ ⋅ ‖m,p and | ⋅ |m,p . We also use the following spaces: ∞ V ≡ {ϕ ∈ Cper (Ω)n ; ∇ ⋅ ϕ = 0, ∫ ϕdx = 0}

(2.6.15)

H ≡ closure of V in the L2 (Ω)n -norm;

(2.6.16)

Ω

2

Vq ≡ closure of D(Ω) in the Lq (Ω)n -norm of gradients;

(2.6.17)

Vq ≡ closure of D(Ω) in the L (Ω)-norm of gradients

(2.6.18)

2

where the Lq -norm of the gradient of φ by |φ|1,q . Hereafter, u ∈ Lq (I; Vp ) means that 2

Du ∈ Lp (I; Lp (Ω)n ) and |u|Lp (I;Vp ) = ‖Du‖Lp (I;Lp (Ω)n2 ) .

Lemma 2.6.1 (see [65]). Let u ∈ Lp1 (Ω) ∩ Lp2 (Ω) and ∞ ≥ p1 ≥ p ≥ p2 ≥ 1. Then ‖u‖p ≤ ‖u‖αp1 ‖u‖1−α p2 , where

1 p

=

α p1

+

1−α , p2

α ∈ [0, 1].

(2.6.19)

70 | 2 Global solutions to the equations of non-Newtonian fluids Lemma 2.6.2 (see [2]; interpolation in k). Let k1 ≥ k2 > 0 with k1 , k2 not necessarily the integer. Then there exists a constant c such that for all u ∈ W k1 ,p (Ω) k2 k

k

1− k2

‖u‖k2 ,p ≤ c‖u‖k 1,p ‖u‖p 1

1

.

(2.6.20)

Lemma 2.6.3 (see [28]; generalized Korn inequality). Let φ ∈ W 1,q (Ω)n ∩ W 1,2 (Ω)n , q > 1. Then 1 q

(∫ |e(φ)|q dx) ≥ Kq |φ|1,q

(2.6.21)

Ω

where Kq > 0. Lemma 2.6.4 (see [65]). Let QT ⊂ Rn+1 be bounded. Let fN : QT | → R be integrable for every N and let: (1) limN→∞ fN (y) exist and be finite for a. e. y ∈ QT (2) ∀ϵ > 0, ∃δ > 0 such that sup ∫ |fN (y)|dy < ϵ, N

∀H ⊂ QT ; |H| < δ.

H

Then limN→∞ ∫Q fN (y)dy = ∫Q limN→∞ fN (y)dy. T

T

Lemma 2.6.5 (Lions–Aubin). Let X be a Banach space, and X0 , X1 be separable, reflexive Banach spaces. Provided that X0 →→ X →→ X1 , then {v ∈ Lα (I; X0 );

dv ∈ Lβ (I; X1 )} →→ Lα (I; X) dt

where 1 < α, β < +∞. Considering the initial value problem of the Boussinesq approximation (2.6.1)– (2.6.3) 𝜕ui 𝜕u 𝜕π 𝜕τij + uj i = − + + en θ + fi , 𝜕t 𝜕xj 𝜕xi 𝜕xj

(2.6.22)

div u = 0

(2.6.23)

𝜕θ 𝜕θ + uj − Δθ = g 𝜕t 𝜕xj

(2.6.24)

u(x − ej L, t) = u(x + ej L, t), u(x, 0) = u0 (x),

θ(x − ej L, t) = θ(x + ej L, t)

θ(x, 0) = θ0

(2.6.25) (2.6.26)

where the nonlinear tensor function τ(⋅) satisfies the conditions (2.6.4)–(2.6.10), {ei }ni=1 being the canonical basis of Rn , u0 (x), θ0 (x) be given vector valued and scalar valued

2.5 Periodic initial value problem and initial value problem

| 71

functions satisfy periodic boundary condition (2.6.25). Hereafter, we will assume the following: u0 ∈ V2 , θ0 ∈ Vθ , f ∈ L2 (I; L2per (Ω)n ), if p ≥ 2, ∫Ω fdx = 0, g ∈ L2 (I; L2per (Ω)),

∫ gdx = 0.

Definition 2.6.1. Let u0 , θ0 , f , g satisfy (2.6.27), and let p ≥ 1 + called a weak solution of the problems (2.6.22)–(2.6.26) if

∫ Ω

2n . n+2

A couple (u, θ) is

𝜕u ∈ L2 (I; H) 𝜕t

1,2 u ∈ Lp (I; Vp ) ∩ C(I; H) ∩ L2 (I; Wper (Ω)n ),

θ ∈ L2 (I; Vθ ) ∩ C(I; L2per (Ω)),

(2.6.27)

Ω

(2.6.28)

𝜕θ ∈ L2 (I; L2per (Ω)) 𝜕t

(2.6.29)

𝜕ui 𝜕u φ dx + ∫ uj i φi dx + ∫ τij (e(u))eij (φ)dx = ∫ en θφdx + ∫ fi φi dx 𝜕t i 𝜕xj Ω

Ω

Ω

Ω

𝜕θ 𝜕θ ψdx + ∫ ∇θ ⋅ ∇ψdx = ∫ gψdx ∫ ψdx + ∫ uj 𝜕t 𝜕xj Ω

Ω

Ω

(2.6.30) (2.6.31)

Ω

is satisfied a. e. in I for every φ = φ1 , φ2 , . . . , φn ∈ W 1,2 (Ω)n ∩ Vp ; ψ ∈ Vθ . Definition 2.6.2. Let u0 , θ0 , f , g satisfy (2.6.27), and let p ≥ 2. A couple (u, θ) is called a weak solution of the problems (2.6.22)–(2.6.26) if u ∈ Lp (I; Vp ) ∩ L∞ (I; H), θ ∈ L∞ (I; L2per (Ω)) ∩ L2 (I; Vθ ) − ∫ ui ΩT

𝜕u 𝜕φ dxdt + ∫ uj i φi dxdt + ∫ τij (e(u))eij (φ)dxdt 𝜕t 𝜕xj ΩT

ΩT

= ∫ eij θφdxdt + ∫ fi ⋅ φi dxdt + ∫ u0i ⋅ φi (0)dx ΩT

−∫θ ΩT

(2.6.32)

Rn

ΩT

𝜕ψ 𝜕θ dxdt + ∫ ∇θ ⋅ ∇ψdxdt + ∫ uj ψdxdt = ∫ gψdxdt + ∫ θ0 ψ(0)dx (2.6.33) 𝜕t 𝜕xj ΩT

ΩT

ΩT

1

Ω

n

is satisfied for every φ = (φ1 , . . . , φn ) ∈ C (I; D(Ω) ) with φ(T) = 0, ψ ∈ C 1 (I; D(Ω)), ψ(T) = 0. Theorem 2.6.1. Let u0 , θ0 , f , g satisfy (2.6.27), and let u0 ∈ Vp . Let p ≥ 11/5. Then there exists a unique weak solution of the problems (2.6.22)–(2.6.26) in the sense of Definition 2.6.1. Moreover, the solution is regular, that is, u ∈ L∞ (I; W 1,2 (Ω)3 )∩L2 (I; W 2,2 (Ω)3 )∩ Lp (I; W 1,3p (Ω)3 ); θ ∈ L∞ (I; W 1,2 (Ω)) ∩ L2 (I; W 2,2 (Ω)). Theorem 2.6.2. Let u0 , θ0 , f , g satisfy (2.6.27), and let p ≥ 2. Then there exists a unique weak solution of the problem (2.6.22)–(2.6.26) in the sense of Definition 2.6.2. Moreover, 2β(p−r) 1+σ,p u ∈ Lr (I; Wper (Ω)n ) where r ∈ (1, p) and σ = s r(p−2β) where 4p−8

β = { 13p−5 , 2

,

p>2 , p=2

s = 6 − p/2p.

72 | 2 Global solutions to the equations of non-Newtonian fluids Analogous to the Theorems 2.6.1 and 2.6.2, for the two-dimensional situation, we have the following theorems. Theorem 2.6.3. Let u0 , θ0 , f , g satisfy (2.6.27), and let u0 ∈ Vp , p ≥ 2. Then there exists a unique weak solution of the problem (2.6.22)–(2.6.26) in the sense of Definition 2.6.1. Moreover, the solution is regular, that is, u ∈ L∞ (I; W 1,2 (Ω)2 ) ∩ L2 (I; W 2,2 (Ω)2 ), θ ∈ L∞ (I; W 1,2 (Ω)) ∩ L2 (I; W 2,2 (Ω)). The next two simple lemmas will be useful for the a priori estimates. 2

Lemma 2.6.6. Let u ∈ L2 (Ω), D2 u ∈ L2 (Ω)n . Then Du ∈ L2 (Ω)n and ‖Du‖22 ≤ c6 ‖u‖2 ‖D2 u‖2 .

(2.6.34)

Lemma 2.6.7. Let u ∈ W 2,2 (Ω), n ≤ 3. Then u ∈ L∞ (Ω), that is, there exists c7 > 0 such that |u|∞ = ess ⋅ sup |u(x)| ≤ c7 ‖u‖2,2 . x∈Ω

(2.6.35)

We will show the existence of a weak solution to the problems (2.6.22)–(2.6.26) (Theorem 2.6.1) via Galerkin approximations. For this purpose, we can take the set k {ωk }∞ k=1 formed by the eigenvectors ω , k = 1, 2, . . . of the Stokes’ operator denoted N N k k by A. Define P u ≡ ∑k=1 (u, ω )ω . By Lemma 4.26 in the Appendix of [65], we know that P N are continuous uniformly with respect to the norm of W 2,2 (Ω)n . Because of 2n the imbedding W 2,2 (Ω)n → W 1,p (Ω)n , valid for p ≤ n−2 , we consider henceforth the problems (2.6.22)–(2.6.26) only for such p. Definition 2.6.3. We called that (uN , θN ), uN = ∑Nk=1 ckN (t)ωk (N)(t)ωk (x), θN = ∑Nk=1 dkN (t)ωk is the Galerkin approximation of the solution of the problems (2.6.22)– (2.6.26), if ∫(∑

𝜕ckN (t) k ωi (x))ωki (x)dx + ∫ τij (e(uN (x, t))eij (ωk (x)))dx 𝜕t Ω

Ω

+ ∫(∑ ckN (t)ωkj (x))(∑ ckN (t) Ω

𝜕ωki (x) k )ωi (x)dx 𝜕xj

= ∫ en θN ωki (x)dx + ∫ fi (x, t)ωki (x)dx, Ω k

Ω

k = 1, 2, . . . , N

∀ω , N

∫ Ω

(2.6.36) N

𝜕θ k 𝜕θ k ω dx + ∫(∑ ckN (t)ωkj (x)) ω dx 𝜕t 𝜕xj Ω

N

+ ∫ ∇θ ⋅ ∇ωk dx = ∫ g(x, t)ωk dx, Ω

uN (0) = P N u0 ,

∀ωk , 1 ≤ k ≤ N

(2.6.37)

Ω

θN (0) = P N θ0 .

(2.6.38)

2.5 Periodic initial value problem and initial value problem

| 73

Here, P N is the orthogonal continuous projector of H(Vθ ) onto the linear hull of the first N eigenvectors ωk (ωk ), k = 1, 2, . . . , N. Using the Caratheodory theorem, we get the existence of the Galerkin approximation locally in time. To obtain the existence on each time interval (0, T), T < ∞, we need the following a priori estimates. Lemma 2.6.8. Let u0 ∈ H, θ0 ∈ M = {θ̃ ∈ L2per (Ω) : T1 ≤ θ̃ ≤ T0 }, f ∈ L2 (I; L2per (Ω)n ), g(x, t) ∈ L2 (I; L2per (Ω)). Then the sequence of Galerkin approximation satisfies the following uniform estimates: ‖uN ‖L∞ (I;H) ≤ c8 ,

(2.6.39)

‖u ‖Lp (I;Vp ) ≤ c9 ,

(2.6.40)

N

‖θN ‖L∞ (I;L2 (Ω)) ≤ c10 ,

(2.6.41)

N

‖θ ‖L2 (I;Vθ ) ≤ c11

(2.6.42)

where ci (i = 8, 9, 10, 11) are constants independent of N. Proof. Multiplying (2.6.36) by ckN (t) and summing up the equations, we get (using the fact that ∫Ω uNj

𝜕uNi 𝜕xj

uNi dx = 0 for divergence-free functions):

1 d ∫ |uN |2 dx + ∫ τij (e(uN ))eij (uN )dx = ∫ en θN ⋅ uN dx + ∫ fi uNi dx. 2 dt Ω

Ω

Ω

(2.6.43)

Ω

Multiplying (2.6.37) by drN (t) and summation, we get (using the fact that ∫Ω uNj

𝜕uNi 𝜕xj

uNi dx = 0 also for divergence-free functions):

1 d 1 ∫ |θN |2 dx + ‖∇θN ‖22 ≤ (‖θN (t)‖22 + ‖g‖22 ). 2 dt 2

(2.6.44)

Ω

Applying the Gronwall inequality, we obtain N

t

‖θ ‖2 + 2 ∫ ‖∇θN ‖22 dt ≤ C(θ0 , g, T),

∀t ∈ [0, T].

(2.6.45)

0

Thus (2.6.41) and (2.6.42) follow from (2.6.45). For p ≥ 2, using the coercivity condition (2.6.10) and Korn’s inequality (2.6.21), we have 1 d N 2 ‖u ‖2 + c4 Kpp ‖uN ‖p1,p ≤ ∫ en θN ⋅ uN dx + ∫ fi uNi dx ≤ ‖θN (t)‖2 ‖uN ‖2 2 dt Ω

Ω

1 + ‖f ‖2 ‖u ‖2 ≤ C(‖θN (t)‖22 + ‖f ‖22 ) + ‖uN ‖22 . 2 N

(2.6.46)

74 | 2 Global solutions to the equations of non-Newtonian fluids Integrating (2.6.46) between 0 and t, t ∈ I, the assumptions on u0 , f and the estimate (2.6.45) yield t

t

‖uN (t)‖22 + 2c4 Kpp ∫ |uN |p1,p dτ ≤ C(u0 , θ0 , f , g, T) + ∫ ‖uN ‖22 dt. 0

(2.6.47)

0

Applying the Gronwall inequality, we obtain the estimates (2.6.39)–(2.6.40) from (2.6.47). In order to get the existence of a weak solution of the problems (2.6.22)–(2.6.26), the key point is to justify the limiting process in the nonlinear τ. In the next part, we will try to find new estimates of solution of the problems (2.6.22)–(2.6.26), which make the limiting passage in the nonlinear term possible. In fact, the estimates will 2 guarantee that ∇uN → ∇u in Lp̃ (I; Lp̃ (Rn )n ), 1 ≤ p̃ ≤ p, that is, ∇uN → ∇u a. e. in QT . Then using Lemma 2.6.4 we will get the desired limiting passage. Multiplying the (2.6.36) by λk ckN (t) where λk are the corresponding eigenvalues, using λk (ωk , uN ) = (Aωk , uN ) = (∇ωk , ∇uN ) and summing over k = 1, 2, . . . , N, we obtain N

𝜕ui 1 d N 2 (AuN )i dx + ∫ τij (e(uN ))eij (AuN )dx ‖u ‖2 + ∫ uNj 2 dt 𝜕xj Ω

Ω

N

N

N

= ∫ en θ ⋅ Au dx + ∫ f ⋅ Au dx. Ω

(2.6.48)

Ω

Because of the periodicity of ωk , AuN = −ΔuN . 𝜕2 uN 𝜕uNi j k 𝜕xk

Hence, using ∫Ω uNj 𝜕x 𝜕xi ∫ uNj

Ω

dx = 0, we have N

𝜕uj 𝜕uNi 𝜕uNi 𝜕uNi (AuN )i dx = ∫ dx 𝜕xj 𝜕xk 𝜕xj 𝜕xk Ω

∫ τij (e(uN )eij (AuN ))dx = ∫ Ω

Ω

N 𝜕2 U(e(uN ) 𝜕eij (u )) 𝜕ekl (uN ) dx. 𝜕ekl 𝜕eij 𝜕xs 𝜕xs

Using (2.6.49) and (2.6.50) together with (2.6.6), we obtain N N 1 d p−2 𝜕eij (u ) 𝜕ekl (u ) ‖∇uN ‖22 + C1 ∫(1 + |e(uN )|) dx 2 dt 𝜕xk 𝜕xk Ω

(2.6.49) (2.6.50)

2.5 Periodic initial value problem and initial value problem

≤ ‖∇uN ‖33 + ∫ f ⋅ ΔuN dx + ∫ en θN ⋅ ΔuN dx,

| 75

(2.6.51)

Ω

Ω

for p ≥ 2. Let us denote by N N p−2 𝜕eij (u ) 𝜕ekl (u ) dx, Ip (u) ≡ ∫(1 + e(uN )) 𝜕xk 𝜕xk

p ≥ 2.

(2.6.52)

Ω

In order to handle the right-hand side of (2.6.51), we need estimates from below of the integral Ip (u) by means of some norms of the first and second derivatives. This is provided by the following result. 2 Lemma 2.6.9. Let u ∈ Cper (Ω)n , then there exists a constant c depending only on Ω, p and n such that 1

‖D2 u‖2 ≤ c(Ip (u)) 2 ,

p ≥ 2.

(2.6.53)

For 1 ≥ q ≥ 2, q ≠ n, q

‖∇uN ‖ np ≤ c(Ip (u)) 2p (1 + ‖∇uN ‖p )

2−q 2

n−q

T

, p > 1.

(2.6.54)

T

Proof. In order to estimate the term ∫0 |(f , ΔuN )|dt, ∫0 |(en θN , ΔuN )|dt, which appears on the right-hand side of (2.6.41) after integrating it with respect to time, we use Lemma 2.6.8 and Lemma 2.6.9. If p ≥ 2, we have T

T

∫ |(f , ΔuN )|dt ≤ ∫ ‖f ‖2 ‖D2 uN ‖2 dt 0

0

T

N

1 2

≤ c ∫ ‖f ‖2 (Ip (u )) dt ≤ 0

c‖f ‖2L2 (QT )

T

C + 1 ∫ Ip (u)dt. 4

(2.6.55)

0

Similarly, we have T

T

∫ |(en θ , Δu )|dt ≤ ∫ ‖θN ‖2 ‖D2 uN ‖2 dt 0

N

N

0

T

T

C ≤ c ∫(Ip (u ) )dt ≤ C(θ0 , g, T) + 1 ∫ Ip (uN )dt. 4 N

1 2

(2.6.56)

0

0

T

In both cases, the term C41 ∫0 Ip (uN )dt can be moved to the left-hand side of (2.6.51) integrated with respect to time. The remaining terms come from the estimate of T T ∫0 |(f , ΔuN )|dt, ∫0 |(en θN , ΔuN )|dt and are uniformly bounded.

76 | 2 Global solutions to the equations of non-Newtonian fluids In the next part, we will prove for p ≥ 2 that L2 -norm of the second derivatives of u is uniformly summable with some fractional exponent. Let us consider two cases: (1) p ≥ 3 (2) p < 3 N

Ad 1. Integrating (2.6.51) with respect to time between 0 and t, t ≤ T, we obtain N

‖∇u

(t)‖22

t

+ C1 ∫ Ip (uN )dt ≤ ‖∇u0 (t)‖22 + C(θ0 , f , g, T) + ‖∇uN ‖3L3 (Q

T)

0

≤ C(u0 , θ0 , f , g, T) + c‖∇uN ‖3Lp (QT ) ≤ C due to (2.6.55), (2.6.56), (2.6.27) and (2.6.40). Particularly, t

∫ Ip (uN )dt ≤ C. 0

Ad 2. Let us consider p < 3 in the sequel. By Lemma 2.6.1, we have the following two interpolation inequalities: 2(np+3q−3n)

np

(2.6.57)

‖v‖3 ≤ ‖v‖23(np+2q−2n) ‖v‖ 3(np+2q−2n) np n−q

np+3q−3n 3q

‖v‖3 ≤ ‖v‖p for q ≥

n(3−p) . 3

n(3−p)

(2.6.58)

‖v‖ np3q n−q

Since for α ∈ (0, 1), we have . ‖∇uN ‖33 = ‖∇uN ‖3(1−α)+3α 3

(2.6.59)

We obtain from (2.6.57)–(2.6.59): 2(1−α) (np+3q−3n) (np+2q−2n)

‖∇uN ‖33 ≤ ‖∇uN ‖2

α np+3q−3n q

‖∇uN ‖p

(1−α)np α n(3−p) + np+2q−2n

‖∇uN ‖ np q n−q

.

(2.6.60)

Set np + 3q − 3n np + 3q − 3n , Q2 ≡ α np + 2q − 2n q n(3 − p) np Q3 ≡ α + (1 − α) . q np + 2q − 2n Q1 ≡ (1 − α)

Then, by the inequalities (2.6.53), (2.6.55) from (2.6.51), we have 1 1 d Q Q Q ‖∇uN ‖22 + C1 Ip (uN ) ≤ (‖∇uN ‖22 ) 1 (1 + ‖∇uN ‖p ) 2 ‖∇uN ‖ np3 + c‖f ‖2 (Ip (uN )) 2 2 dt nq

2.5 Periodic initial value problem and initial value problem 1

)Q3 Q2 +( 2−q 2

Q

+ c‖θN ‖2 (Ip (uN )) 2 ≤ C(‖∇uN ‖22 ) 1 (1 + ‖∇uN ‖p ) q

× (Ip (uN )) 2p

Q3

| 77

1

1

+ c‖f ‖2 (Ip (uN )) 2 + c‖θN ‖2 (Ip (uN )) 2 .

(2.6.61)

By Young’s inequality, we obtain d Q δ (Q +( 2−q )Q )δ ‖∇uN ‖22 + C1 Ip (uN ) ≤ C(‖∇uN ‖22 ) 1 (1 + ‖∇uN ‖p ) 2 2 3 dt 1

1

+ C‖f ‖2 (Ip (uN )) 2 + C‖θN ‖22 (Ip (uN )) 2

(2.6.62)

provided that 1 1 + = 1, δ δ

and

q Q δ = 1, 2p 3

δ > 1, δ > 1.

(2.6.63)

Requiring (Q2 + (

2−q )Q3 )δ = p, 2

(2.6.64)

we have from (2.6.63), (2.6.64) p(np + 2q − 3n) 2(np + 3q − 3n) (3 − p)(np + 2q − 2n) 1−α= 2(np + 3q − 3n) α=

and δ =

4 . np − 3n + 4

Notice that δ does not depend on the choice of q. Inserting α, 1 − α and δ into the inequality (2.6.62), we obtain 2(3−p) d p ‖∇uN ‖22 + C1 Ip (uN ) ≤ C(‖∇uN ‖22 ) np−3n+4 (1 + ‖∇uN ‖p ) dt 1

+ C(‖f ‖2 + ‖θN ‖2 )(Ip (uN )) 2

(2.6.65)

or d λ p ‖∇uN ‖22 + C1 Ip (uN ) ≤ C(‖∇uN ‖22 ) (1 + ‖∇uN ‖p ) dt

1

+ C(‖f ‖2 + ‖θN ‖2 )(Ip (uN )) 2 ,

(2.6.66)

where λ=

2(3 − p) . np − 3n + 4

(2.6.67)

78 | 2 Global solutions to the equations of non-Newtonian fluids Dividing (2.6.66) by (1 + ‖∇uN ‖22 )λ , we obtain for λ ≠ 1: 1 d 1−λ −λ (1 + ‖∇uN ‖22 ) + C1 (1 + ‖∇uN ‖22 ) Ip (uN ) 1 − λ dt 1

p

≤ C(1 + ‖∇uN ‖p ) + C(‖f ‖2 + ‖θN ‖2 )(Ip (uN )) 2 (1 + ‖∇uN ‖22 )

−λ

(2.6.68)

d ln(1+‖∇uN ‖22 ). Integrating for λ = 1, the first term on the left-hand side is replaced by dt (2.6.68) between 0 and t, t ∈ (0, T], and using Hölder and Young inequalities, we get for λ ≠ 1: t

1 −λ 1−λ (1 + ‖∇uN (t)‖22 ) + C1 ∫(1 + ‖∇uN (τ)‖22 ) Ip (uN (τ))dτ 1−λ t

0

t

p

+ C ∫(1 + ‖∇u (τ)‖p ) dτ + C ∫ ‖f (τ)‖22 (1 + ‖∇uN (τ)‖22 ) dτ N

−λ

0

0

t

+ C ∫ ‖θN (τ)‖22 (1 + ‖∇uN ‖22 ) dτ ≤ C(u0 , θ0 , f , g, T), ∀t ∈ (0, T] −λ

(2.6.69)

0

for λ = 1, the first term of (2.6.69) is replaced by ln(1 + ‖∇uN ‖22 ). From (2.6.69), we obtain the estimate ‖∇uN ‖L∞ (I;L2per (Ω)) ≤ C.

(2.6.70)

Consequently, T

∫ Ip (uN (t))dt ≤ C

(2.6.71)

0

for λ ≤ 1, that is, p ≥ 1 +

2n . n+2

Using (2.6.54), we finally have ‖uN ‖L2 (I;Wper 2,2 (Ω)n ) ≤ C

for p ≥ 1 +

(2.6.72)

2n . n+2

Thus for n = 2, we have the following. Lemma 2.6.10. Let u0 , f satisfy (2.6.27), and p ≥ 2. Then uN are uniformly bounded in the following norms: ‖uN ‖L∞ (I;W 1,2 (Ω)2 ) ≤ C,

(2.6.73)

‖u ‖L2 (I;Wper 2,2 (Ω)2 ) ≤ C.

(2.6.74)

N

| 79

2.5 Periodic initial value problem and initial value problem

For 2 ≤ p < 115 , n = 3. Because λ > 1, the first term in (2.6.69)is negative. However, 1 it can be moved across to the right-hand side and estimated by λ−1 . Therefore, we can dispose of T

∫(1 + ‖∇uN (τ)‖22 ) Ip (uN (τ))dτ ≤ C.

(2.6.75)

−λ

0

We first show that (2.6.75) implies T

2β

∫ ‖D2 uN ‖2 dt ≤ C,

with β =

0

1 4p − 8 for p > 2, β = for p = 2. 3p − 5 3

(2.6.76)

Indeed, using (2.6.53) we obtain for some β < 1, T

2β ∫ ‖D2 uN ‖2 dt 0

T

−λ β

λβ

≤ c ∫{Ip (uN )(1 + ‖∇uN (τ)‖22 ) } × (1 + ‖∇uN (τ)‖22 ) dτ 0

β

T

T

≤ c(∫(1 + ‖∇uN (τ)‖22 ) Ip (uN (τ))dτ) × (∫(1 + ‖∇uN (τ)‖22 ) −λ

0

T

β

λ 1−β

≤ c(∫(1 + ‖uN ‖2 ‖D2 uN (τ)‖2 )

1−β

β

λ 1−β

1−β

dτ)

0

dτ)

0

where Lemma 2.6.6 and (2.6.75) are used. As we know that u is bounded in L∞ (I; H), we can put 2β = λ and get β T ϵ ∫0

=

2β ‖D2 uN ‖2 dt

4p−8 . 3p−5

β 1−β

Now we use the Young inequality and transfer the term

with ϵ sufficiently small to the left-hand side. For p = 2, we can β

use directly the estimate in L2 (I; V2 ) and get λ 1−β = 1, that is, β = 31 .

Having (2.6.76), we use the imbedding W 2,2 (Ω)n → W 1+s,p (Ω)n which holds for T 2β s= (s ∈ [ 19 , 1], p ∈ [2, 115 )), we see that ∫0 ‖u‖1+s,p dt ≤ C. By Lemma 2.6.8, we have 22 the following interpolation inequality: 6−p 2p

1− σ

σ

s ‖u‖1+σ,p ≤ c‖u‖1,p s ‖u‖1+s,p

(2.6.77)

with σ ∈ (0, s). We prove that for chosen r ∈ (1, p) there exists a σ ∈ (0, s) such that T

∫ ‖u‖r1+σ,p dt ≤ C. 0

(2.6.78)

80 | 2 Global solutions to the equations of non-Newtonian fluids Indeed, T

∫ ‖uN ‖r1+σ,p dt 0

T

s ‖uN ‖1+s,p dt

r(1− σs )

‖uN ‖2,2s dt

0

T

rσ

r(1− σs )

≤ c ∫ ‖uN ‖1,p ≤ c ∫ ‖uN ‖1,p 0

rσ

T

≤ where

1 δ

+

1 δ

r(1− σ )δ c(∫ ‖uN ‖1,p s dt) 0

1 δ

T

r σ δ (∫ ‖uN ‖2,2s dt) 0

1 δ

= 1. Both terms on the right-hand side are bounded when r(1 −

σ )δ = p, s

σ r δ = 2β. s

Solving the system above, we get σ=s So we get for 2 ≤ p

2 , p=2

s=

6−p . 2p

Lemma 2.6.11. Let u0 , θ0 , f , g satisfy (2.6.27), and let p ≥ 1 +

2n . n+2

Then

‖θN ‖L∞ (I;W 1,2 (Ω)) ≤ C

(2.6.80)

‖θ ‖L2 (I;W 2,2 (Ω)) ≤ C.

(2.6.81)

N

Proof. Multiplying (2.6.37) by λr drN (t) and summing up the equations, we have 1 d 𝜕θN N ‖∇θN ‖22 + ‖ΔθN ‖22 ≤ ∫ uNj Δθ dx + ∫ gΔθN dx . 2 dt 𝜕xj Ω

Ω

By the Hölder and Young inequalities, we get ∫ |gΔθN |dx ≤ ‖g‖22 +

Ω

1 ‖ΔθN ‖22 4

(2.6.82)

2.5 Periodic initial value problem and initial value problem

| 81

𝜕θN 1 ΔθN dx ≤ ∫ |uN |2 |∇θN |2 dx + ‖ΔθN ‖22 . ∫uNj 𝜕xj 4

Ω

Ω

Substituting two estimates above into (2.6.82) and integrating over (0, t), we obtain t

t

‖∇θN (t)‖22 + c ∫ ‖D2 θN ‖22 dτ ≤ ‖∇θN (0)‖22 + ‖g‖2L2 (QT ) + ∫ ∫ |uN |2 |∇θN |2 dxdτ. 0

(2.6.83)

0 Ω

The last term on the right-hand side of (2.6.83) can be estimated by means of Lemmas 2.6.8 and 2.6.9: t

N 2

t

N 2

∫ ∫ |u | |∇θ | dxdτ ≤ c ∫ ‖uN ‖22,2 ‖∇θN ‖22 dτ 0 Ω

0

t

≤ c ∫ ‖uN ‖22,2 (ϵ‖D2 θN ‖22 + λ(ϵ)‖θN ‖22 )dτ.

(2.6.84)

0

Combining (2.6.83) with (2.6.84), we get T

‖∇θN (t)‖22 + c ∫ ‖D2 θN ‖22 dτ ≤ C(θ0 , g, T) + ϵc sup ‖uN ‖22,2 (0,T)

0

T

T

0

0

× ∫ ‖D2 θN ‖22 dτ + c ∫ ‖uN ‖22,2 λ(ϵ)‖θN ‖22 dτ.

(2.6.85)

By virtue of (2.6.41) and (2.6.72), and transfer the second term on the right-hand side of (2.6.85) with a small coefficient e to the left-hand side of (2.6.85), we infer T

‖∇θN (t)‖22 + c ∫ ‖D2 θN ‖22 dτ ≤ C

(2.6.86)

0

and the desired result follows. Lemma 2.6.12. Let p ≥

11 5

and let u0 , θ0 , f , g satisfy (2.6.27). Then we have ‖uN ‖Lp (I;W 1,3p (Ω)3 ) ≤ C.

(2.6.87)

Proof. By Lemma 2.6.10, we have for q = 2, n = 3, p > 1, 1

‖∇uN ‖3p ≤ c(Ip (uN )) p . Using (2.6.71) and (2.6.88), we obtain T

T

0

0

∫ ‖∇uN ‖p3p dt ≤ c ∫ Ip (uN )dt ≤ C holds for n = 3, p ≥

11 , 5

and the desired result follows.

(2.6.88)

82 | 2 Global solutions to the equations of non-Newtonian fluids 2n . n+2

Lemma 2.6.13. Let u0 , θ0 , f , g satisfy (2.6.27), and let u0 ∈ Vp , p ≥ 1 + 𝜕θN 𝜕t

2

uniformly bounded in L (I; H)

Proof. Multiplying (2.6.36) by (0, T), we have

2

is uniformly bounded in L

𝜕cαN (t) , 𝜕t

(I; L2per (Ω)).

Then

𝜕uN 𝜕t

is

summing up the equations and integrating over

N N 𝜕uN 2 N N N 𝜕ui 𝜕ui dxdt ∫ dxdt + ∫ U(e(u (t)))dx − ∫ U(e(u (0)))dx + ∫ uj 𝜕t 𝜕xj 𝜕t

QT

Ω

= ∫ en θN ⋅

QT

Ω

𝜕uNi

N

𝜕u dxdt + ∫ fi dxdt 𝜕t 𝜕t

(2.6.89)

QT

QT

where U(e(uN )) is the scalar potential given by (2.6.5). From (2.6.8)–(2.6.11) and the generalized Korn inequality, we have p p c‖DuN ‖pp ≤ ce(uN )p ≤ U(e(uN ))1 ≤ c(1 + e(uN )p ) . The Hölder and Young inequalities yield ∫ uNj

QT

𝜕uNi 𝜕uNi dxdt ≤ c ∫ |uN |2 |DuN |2 dxdt + 𝜕xj 𝜕t QT

N N 𝜕u dxdt ≤ c‖θN ‖2L2 (QT ) + ∫ en θ ⋅ 𝜕t QT

𝜕uN 2 ∫ fi i dxdt ≤ c‖f ‖L2 (QT ) + 𝜕t QT

2

1 𝜕uN 6 𝜕t L2 (QT )

2

1 𝜕uN 6 𝜕t L2 (QT ) 2

1 𝜕uN . 6 𝜕t L2 (QT )

Substituting the all estimates above into (2.6.89), we obtain 2

1 𝜕uN + c‖DuN ‖pp ≤ C(u0 , θ0 , f , g, T) + c ∫ |uN |2 |DuN |2 dxdt. 2 𝜕t L2 (QT )

(2.6.90)

QT

Using (2.6.70) and (2.6.72) and Lemma 2.6.7, we obtain N 2

N 2

T

∫ |u | |Du | dxdt ≤ c ∫ ‖uN ‖22,2 ‖DuN ‖22 dt 0

QT

≤ c‖DuN ‖L∞ (I;L2 (Ω)n ) ‖uN ‖L2 (I;W 2,2 (Ω)n ) ≤ C.

(2.6.91)

Combining (2.6.90) and (2.6.91), the desired result follows. Multiplying (2.6.36) by summing the equations and integrating over (0, T), it follows:

𝜕drN 𝜕t

𝜕θN 2 1 1 𝜕θN 𝜕θN 𝜕θN + ∫ |∇θN (t)|2 dx ≤ ‖∇θN (0)‖22 + ∫ uNj dxdt + ∫ g dxdt . 2 𝜕xj 𝜕t 𝜕t 𝜕t L2 (QT ) 2 Ω

QT

QT

,

2.5 Periodic initial value problem and initial value problem

| 83

The assumptions on θ0 , g and using the Hölder and Young inequalities yield 𝜕θN 2 + ‖∇θN ‖22 ≤ C(θ0 , g, T) + ∫ |uN |2 |∇θN (t)|2 dxdt 𝜕t L2 (QT ) QT

≤ C(θ0 , g, T) + c‖DθN ‖2L∞ (I;L2 (Ω)) ‖uN ‖2L2 (I;W 2,2 (Ω)n ) ≤ C

(2.6.92)

where we have used Lemma 2.6.7, the estimates (2.6.72) and (2.6.80). The desired result follows from (2.6.92). Lemma 2.6.14. Let u0 , θ0 , f , g satisfy (2.6.27), and let 2 ≤ p

1. Similarly, we have T

T

0

0 Ω

T T N N ∫ I4 dt ≤ ∫∫ f ⋅ P φdxdt ≤ ∫ ∫ |P φ||f |dxdt ≤ C ∫ ‖P N φ‖2,2 ‖f ‖2 dt 0 Ω

T

0

≤ C ∫ ‖φ‖2,2 ‖f ‖2 dt ≤ C‖φ‖L2 (I;Y) ‖f ‖L2 (I,L2per (Ω)3 ) ≤ C. 0

We now estimate I1 . By Hölder’s inequality, we have T

T

0

0 Ω

T N 𝜕(P N φ)i N 𝜕ui N (P φ)i dxdt = ∫∫ uNj uNi dx dt ∫ I1 dt ≤ ∫∫ uj 𝜕xj 𝜕xj 0 Ω

T

≤ ∫ ‖∇P N φ‖6 ‖uN ‖212 dt. 0

5

2.5 Periodic initial value problem and initial value problem | 85

By using the interpolation inequality, λ ‖φ‖ 12 ≤ ‖φ‖1−λ 2 ‖v‖ 3p , 5

with λ =

3−p

p 2(5p − 6)

and (2.6.39), (2.6.40), (2.6.95) and imbedding W 2,2 (Ω)3 → W 1,p (Ω)3 for p < 6, 3p

W 1,p (Ω)3 → L 3−p (Ω)3 , p < 3, we get T

T

0

0

9p−12

T

p

9p−12

p

N 5p−6 N ‖∇uN ‖p5p−6 ∫ I1 dt ≤ ∫ ‖uN ‖22p−6 ‖uN ‖ 5p−6 3p ‖∇P φ‖6 dt ≤ C ∫ ‖u ‖2 3−p

9p−12 5p−6 L∞ (I;H)

⋅ ‖P N φ‖2,2 dt ≤ C‖uN ‖ 9p−12 5p−6 L∞ (I;H)

≤ C‖uN ‖

p 5p−6 p L (I;V

‖uN ‖

p)

T

0

1

∫(‖∇uN ‖pp ) 5p−6 ‖φ‖2,2 dt 0

‖φ‖Lδ (I;Y) ≤ C

δ and δ = 5p − 6. Putting γ = min(2, p, δ), we obtain from previous calcuwhere δ = δ−1 lations the desired estimate (2.6.93). Similarly, we can derive the estimate (2.6.94).

Now we give the proof of the basic theorem. Proof of Theorem 2.6.1. Existence. From Lemma 2.6.9, (2.6.72), Lemmas 2.6.12, 2.6.13 and 2.6.14, we see that 2,2 {uN } is uniformly bounded in L∞ (I; H) ∩ Lp (I; Vp ) ∩ L∞ (I; W 1,2 (Ω)3 ) ∩ L2 (I; Wper (Ω)3 ) ∩ p 1,3p 3 L (I; W (Ω) ); N 1,2 {θ } is uniformly bounded in L∞ (I; L2per (Ω)) ∩ L2 (I; Vθ ) ∩ L∞ (I; Wper (Ω)) ∩ N

N

2,2 L2 (I; Wper (Ω)) { 𝜕u𝜕t } L2 (I; H) { 𝜕θ𝜕t } L2 (I; L2per (Ω));

and consequently N { 𝜕u𝜕t } is uniformly bounded in L2 (I; H); N

{ 𝜕θ𝜕t } is uniformly bounded in L2 (I; L2per (Ω)); which allow us to find a subsequence still {uN , θN } such that uN → u weakly in Lp (I; Vp ); uN → u weakly* L∞ (I; H); uN → u strongly in Lq (I; H), q ≥ 1 arbitrary; ∇uN → ∇u strongly in Lr (I; Lrper (Ω)9 ) r ∈ (1, p) arbitrary; θN → θ weakly * in L∞ (I; L2per (Ω)); θN → θ strongly in L2 (I; W 1,2 (Ω)). Thus the existence of weak solutions is demonstrated. 2,2 As u ∈ L2 (I; Wper (Ω)) ∩ L2 (I; H) and 𝜕u ∈ L2 (I; H), it follows from Lemma 1.2, Chap𝜕t ter III in [70] that u ∈ C(I; H). Similarly, we get that θ ∈ C(I; L2per (Ω)).

86 | 2 Global solutions to the equations of non-Newtonian fluids Uniqueness. Let (u1 , θ1 ), (u2 , θ2 ) be two couples of solutions for the same initial values. Let us denote (u, θ) = (u1 − u2 , θ1 − θ2 ). Then we have 𝜕u1 1 d 𝜕θ (‖u‖22 + ‖θ‖22 ) + c(|u|21,2 + |θ|21,2 ) ≤ C‖u‖22 + ‖θ‖22 + ∫ uj i ui dx + ∫ uj θ1 dx 2 dt 𝜕xj 𝜕xj Ω

Ω

(2.6.96)

1 1 𝜕u1 |I1 | = ∫ uj i ui dx ≤ ‖u‖26 |u1 |1,2 ≤ ‖u‖22 ‖u‖62 |u1 |1,2 𝜕xj Ω

1

1

2 |u1 |1,2 ≤ c(ϵ)‖u‖22 + ϵ|u|21,2 ≤ c‖u‖22 |u|1,2 𝜕θ 1 |I2 | = ∫ uj θ dx ≤ |θ|1,2 ‖θ1 ‖∞ ‖u‖2 ≤ c(ϵ)‖u‖22 + ϵ|θ|21,2 . 𝜕xj

Ω

Substituting the two estimates above into (2.6.96) and integrating over (0, t), t ∈ (0, T] using (u, θ)(0) = (0, 0), it follows from the Gronwall inequality that ‖u‖22 + ‖θ‖22 = 0

a. e. in I

QT and, therefore, (u1 , θ1 ) = (u2 , θ2 ) a. e. in QT . Remark. When u0 ∈ Vp , we do not have the information about the time derivative from Lemma 2.6.14. Nevertheless, we can get the estimates of the time derivative in L2 (I; Y ∗ ), which implies that u belongs to C(I; H). So we can get the existence and uniqueness of the weak solution of the problems (2.6.22)–(2.6.26). However, we have to assume (2.6.30) instead of (2.6.32) with test functions φ ∈ L2 (I; Vp ∩ W 1,2 (Ω)n ) with 𝜕φ 𝜕t

∈ L2 (I; L2 (Ω)n ).

Proof of Theorem 2.6.2. Because of the estimates (2.6.41), (2.6.42), (2.6.79), (2.6.93), and (2.6.94), we get from Lemma 2.6.5 that uN → u in Lr (I; W 1,r (Ω)n ), θN → θ in L2 (I; W 1,2 (Ω)). From Lemma 2.6.4, we get for φ ∈ C 1 (I; D(Ω)n ) with φ(T) = 0, ψ ∈ C 1 (I; D(Ω)), ψ(T) = 0: − ∫ ui QT

𝜕u 𝜕φi dxdt + ∫ uj i φi dxdt + ∫ τij (e(u))eij (φ)dxdt = ∫ en θφdxdt 𝜕t 𝜕xj QT

QT

QT

𝜕ψ + ∫ fi ⋅ φi dxdt + ∫ u0i ⋅ φi (0)dx − ∫ θ dxdt + ∫ ∇θ ⋅ ∇ψdxdt 𝜕t QT

+ ∫ uj QT

Rn

QT

QT

𝜕θ ψdxdt = ∫ gψdxdt + ∫ θ0 ψ(0)dx. 𝜕xj QT

Ω

Proof of Theorem 2.6.3. The proof is analogous to the proof of Theorem 2.6.1. Only in 1

1

2 the uniqueness part, we use ‖u‖4 ≤ C‖u‖22 |u|1,2 .

2.5 Periodic initial value problem and initial value problem

| 87

Theorem 2.6.4. Let u0 ∈ H, θ0 ∈ M, f , g satisfy (2.6.27) and let p ≥ 115 . Then there exists a unique weak solution of the problems (2.6.22)–(2.6.26) in the sense of Definition 2.6.1. Moreover, the solution is regular, that is, u ∈ L∞ (Iδ ; W 1,2 (Ω)3 ) ∩ L2 (Iδ ; W 2,2 (Ω)3 ) ∩ Lp (Iδ ; W 1,2 (Ω)) ∩ L2 (Iδ ; W 2,2 (Ω)) where Iδ = [δ, T), δ > 0. Proof. Since u0 ∈ H and not in Vp θ0 ∈ M and not in Vθ , we cannot immediately integrate (2.6.68), (2.6.82) between (0, t), t ≤ T. Let us define for δ > 0 a cut-off function ξ ∈ C 1 (I) such that ξ (t) ∈ [0, 1] on I and 0, ξ (t) = { 1,

t ∈ [0, δ2 ] t ∈ [δ, T]

and multiply (2.6.68), (2.6.82) by ξ (t) we can get the same estimates as (2.6.72), (2.6.79), (2.6.80) and (2.6.81) only in [δ, T] with δ > 0 arbitrary. Similarly, we have 𝜕θN ∈ L∞ (Iδ ; L2 (Ω)). 𝜕t

𝜕u ∈ L2 (Iδ ; H), 𝜕t

Theorem 2.6.5. Let u0 ∈ H, θ0 ∈ M, f , g satisfy (2.6.27). Let p ≥ 2. Then there exists a unique weak solution of the problems (2.6.22)–(2.6.26) in the sense of Definition 2.6.1. Moreover, the solution is regular, that is, u ∈ L∞ (Iδ ; W 1,2 (Ω)2 ) ∩ L2 (Iδ ; W 2,2 (Ω)2 ), θ ∈ L∞ (Iδ ; W 1,2 (Ω)) ∩ L2 (Iδ ; W 2,2 (Ω)), for δ > 0, Iδ = [δ, T]. Theorem 2.6.6. Let u0 ∈ H, θ0 ∈ M, f , g satisfy (2.6.27). Let p ≥ 2. Then there exists a unique weak solution of the problems (2.6.22)–(2.6.26) in the sense of Definition 2.6.2. 2β(p−r) 1+σ,p Moreover, u ∈ Lr (Iδ ; Wper (Ω)3 ) where r ∈ (1, p) and σ = s r(p−2β) where 4p−8

β = { 13p−5 , 2

,

p>2 , p=2

s=

6−p . 2p

Now we consider the initial value problem of the Boussinesq approximation, 𝜕ui 𝜕u 𝜕π 𝜕τij + uj i = − + + en θ + fi 𝜕t 𝜕xj 𝜕xi 𝜕xj

(2.6.97)

div u = 0

(2.6.98)

𝜕θ 𝜕θ + ui − Δθ = g 𝜕t 𝜕xj

(2.6.99)

u(x, 0) = u0 (x),

θ(x, 0) = θ0 (x)

(2.6.100)

where the nonlinear tensor function τ(⋅) satisfies the condition (2.6.4)–(2.6.10), f , g ∈ L2 (I; L2 (Rn )) are given real functions and u0 (x), θ0 (x) are given functions; we noted that the a priori estimates stated above to the periodic initial value problems (2.6.22)– (2.6.26) are independent of the periodic L so we can get the following similar results by using the technique developed in [35, 94], that is, we can instead of Ω with Rn in Theorems 2.6.1–2.6.6.

3 Global attractors of incompressible non-Newtonian fluids The estimate for the global attractor and its Hausdorff-fractal dimensions is the research of long time behavior for the solution of the deterministic problem. They are important mathematical theory and research methods for infinite dimensional dynamical systems. Teman et al. in [83] give the basic theoretical framework as follows. Definition 3.0.1. Assume E is a Banach space; S(t) is a semigroup operator, S(t) : E → E, S(t + s) = S(t)S(s), ∀t, s ≥ 0, S(0) = I (identity operator). If the compact set A ⊂ E satisfies: (1) Invariance: S(t)A = A, ∀t ≥ 0; (2) Attractivity: A attracts all bounded sets of E, that is to say for any bounded set B ⊂ E, we have dist(S(t)B, A) = sup ‖S(t)x − y‖E → 0, x∈B,y∈A

t → ∞,

especially, every trajectory S(t)u0 departing from u0 converges to A, that is, dist(S(t)u0 , A) → 0,

t → ∞,

then we call the compact set A the global attractor of the semigroup operator S(t). Definition 3.0.2. For the bounded set B0 ⊂ E, and any bounded set B ⊂ E there exists t0 (B0 ) > 0; when t ≥ t0 , we have S(t)B ⊂ B0 , then we call B0 is the bounded attracted sets in E. Two common theorems for the existence of the attractor are the following. Lemma 3.0.1. Assume E is a Banach space; {S(t), t ≥ 0} is semigroup operator, and satisfying: (1) Semigroup operator S(t) in E is uniformly bounded, that is to say for any ℝ > 0 there exists a constant C(ℝ) > 0, such that when ‖u‖E ≤ ℝ we have ‖S(t)u‖E ≤ C(ℝ); (2) There exists the bounded attracted sets B0 in E; (3) When t > 0, S(t) is a completely continuous operator. Then semigroup operator S(t) has the compactly global attractor A = w(B0 ) = ⋂s≥0 ⋃t≥s S(t)B0 , that is, the limit set w of attractor B0 , ‘−’ represents the closure of E. Lemma 3.0.2. Assume E is a Banach space, semigroup operator S(t) is continuous, and assume there exists an open set B ⊂ E; we have bounded attracted sets B0 in B, and satisfying: https://doi.org/10.1515/9783110549614-003

90 | 3 Global attractors of incompressible non-Newtonian fluids (1) Operator S(t) is consistently compact for enough large t, that is, for every bounded set B0 , there exists t0 = t0 (B0 ) > 0 such that ⋃t≥t0 S(t)B0 is relatively compact in E; (2) S(t) = S1 (t) + S2 (t), where the operator S1 (t) is consistently compact for enough large t, operator S2 (t) is a continuous mapping, S2 (t) : E → E, and for every bounded set D ⊂ E, rD (t) = sup ‖S2 (t)u‖E → 0, u∈D

then w limit set A = W(B0 ) of B0 is compact attractor, which attracts the bounded set of B, and which is a maximum bounded attractor in B, and when B is convex and connectivity, A is the connectivity. Note. If we modify condition (2) attract bounded sets B0 of Lemma 3.0.1 as a compact attracting set B0 , then the condition of Lemma 3.0.1 full continuity S(t) weakened into S(t) is a continuous operator, then Lemma 3.0.1 also holds. Definition 3.0.3. The Hausdorff measure of set X is μH (X, d) = lim μH (x, d, ε) = sup μH (x, d, ε) ε→0

ε>0

where

∞

μH (x, d, ε) = ∑ rid i

where ∞ is taken by the number of ball whose radius is ri < ε of all covering X. There exists a number d = dH (X) ∈ [0, ∞), such that μH (x, d) = 0,

μH (x, d) = ∞,

d > dH (X) d < dH (X)

then we call this number dH (X) is a Hausdorff dimension of the set. Definition 3.0.4. The fractal dimension of set X is dF (X) = lim sup ε>0

log nX (ε) log ε1

where nX (ε) is the minimum number of all balls that the radius of covering X is ri ≤ ε. Clearly, dF (X) = ∞{d > 0, μF (X, d) = 0} and μF (X, d) = lim sup nX (ε) ε→0

μF (X, d) ≥ μH (X, d) so we have DH (X) ≤ dF (X).

3 Global attractors of incompressible non-Newtonian fluids | 91

Considering the initial value problem, {

dut dt

= F(u(t)), u(0) = u0

t > 0,

(3.1)

where F(u) is a given function, F(u) : E → E, E is a Banach space, and assume that for any u0 ∈ E, there exists the global solution ut ∈ E, and the mapping S(t) : E → E u(t) = S(t)u0 is the semigroup operator of the initial value problem (3.1). Assume F is Frechet differentiable, F : E → E. Linear initial value problems {

dU(t) dt

= F (S(t)u0 ) ⋅ U(t) U(0) = ξ

(3.2)

for every u0 and ξ are solvable. Finally, assume S(t) is differentiable, and has derivative L(t, u0 ). We define L(t, u0 ) ⋅ ξ = U(t),

∀ξ ∈ E

(3.3)

and U(t) is the solution of problem (3.2). For fixed u0 ∈ L2 , assume ξ1 , ξ2 , . . . , ξJ are the J elements of L2 , U1 (t), U2 (t), . . . , UJ (t) are the solutions of linear problem (3.2) with initial data U1 (0) = ξ1 , U2 (0) = ξ2 , . . . , UJ (0) = ξJ . Computing d ‖U (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)‖2L2 − 2tr(F (u(t) ⋅ QJ )) dt 1 ‖U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)‖2L2 = 0

(3.4)

where F (u(t)) = F (S(t)u0 ) : U → F (u(t))U is a linear mapping, u(t) = S(t)u0 is the solution of problem (3.1); ‘∧’ is outer product; ‘tr’ is the operator track; ‘QJ ’ is the orthogonal projection of L2 to all subspaces constructed by U1 (t), U2 (t), . . . , UJ (t). J dimension volume ⋀Jj=1 ξj is wj (t) = sup sup ‖U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)‖2∧j . u0 ∈A ξi ∈L2

L2

(3.5)

Easily verifying that wj (t) is subexponential, a. m. wj (t + t ) ≤ wj (t)wj (t ),

∀t, t ≥ 0

(3.6)

therefore, 1

lim wj (t) t = Πj ,

t→∞

∀1 ≤ j ≤ J,

(3.7)

by (3.4) we can obtain Πj ≤ exp{qJ }

(3.8)

92 | 3 Global attractors of incompressible non-Newtonian fluids qJ = lim sup qJ (t)

(3.9)

t→∞

t

1 qJ (t) = sup sup { ∫ tr(F (S(τ)u0 ))QJ (τ)dτ, i = 1, 2, . . .}. 2 u0 ∈A ξ ∈L ,|ξi |≤1 t 0

Definition 3.0.5. A set of series Λ1 , Λ2 , . . . , Λm defining as Λ1 = Π1 ,

Λ1 Λ2 = Π2 ,

Λ1 ⋅ ⋅ ⋅ Λm = Πm

...,

or Λ1 = Π1 ,

Λm =

Πm , Πm−1

m≥2

Λm = lim (

wm (t) 2 ) , wm−1 (t)

m≥2

1

t→∞

then we call Λm the uniform Lyapunov number in set A, and call μm = log Λm the corresponding Lyapunov index. By (3.8), we have μ1 + μ2 + ⋅ ⋅ ⋅ + μJ ≤ qJ . Lemma 3.0.3. Under the solvability assumptions of initial value problem equations (3.1)–(3.2), assume for m and t0 > 0 have qJ ≤ −δ < 0, ∀t ≥ t0 . Then when t → ∞, volume element ‖U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)‖∧J is exponential decay, that is, for u0 ∈ A, ξ1 , ξ2 , . . . , ξJ ∈ L2 L2

have

‖U1 (t) ∧ ⋅ ⋅ ⋅ ∧ UJ (t)‖∧m2 ≤ ‖U1 (t0 ) ∧ ⋅ ⋅ ⋅ ∧ Um (t0 )‖∧J exp(−δ(t − t0 )). L2

L

Lemma 3.0.4. Assume there exists the global attractor A for the initial value problem (3.1), and it is bounded in H 1 (Ω), the linear problem equation (3.2) is solvable, and the solution operator S(t)u0 of (3.1) is differentiable, and assuming for some j, such that the qj < 0 defined by (3.9), then the Hausdorff dimension and fractal dimension of the global attractor A are finite, and DH (A) ≤ j

(ql )+ ). 1≤l≤j−1 |qj |

dF (A) ≤ j(1 + max

Assume H is a Hilbert space, X ⊂ H is a compact set, S : X → H is a nonlinearly continuous mapping, and SX = X. For every u ∈ X, there exists the linear operator L(u) ∈ £(H), and sup

|u,v∈X,0 0

vi (x, t) = vi (x + Lej , t),

t≥0

(3.1.5)

where {ei }ni=1 ing the canonical basis of ℝn , and ∫Ω v(x, t)dx = 0, t ≥ 0. 3.1.1 Existence of Absorbing Sets Only considering the space dimension n = 2, and periodic initial boundary equations (3.1.1)–(3.1.3) and equation (3.1.5) for the parameter p > 2, two cases: μ1 > 0 and μ1 = 0. Theorem 3.1.1. Let n = 2, p > 2, v0 (x) ∈ L2 (Ω), f ∈ L∞ (0, T; L2 (Ω)), then the problem equations (3.1.1)–(3.1.3) and equation (3.1.5) have the absorbing sets in Hper (Ω) and W 1,2 (Ω) for Sμ1 (t), μ1 ≥ 0 which are independent of μ1 > 0. When μ1 > 0, there exists the absorbing sets for Sμ1 (t) in W 2,2 (Ω). Proof. Taking the inner product of equation (3.1.1) with v in L2 (Ω), we get 𝜕eij 𝜕eij 1 d ‖v‖2 + ∫ γ(v)eij eij dx + μ1 ∫ dx ≤ ‖f ‖∞ ‖v‖ 2 dt 𝜕xk 𝜕xk Ω

Ω

(3.1.6)

94 | 3 Global attractors of incompressible non-Newtonian fluids for p > 2, ε > 0, ∫ γ(v)eij eij dx = μ0 ∫(ε + |e|2 ) Ω

p−2 2

|e|2 dx ≤ μ0 ∫ |e|p dx

Ω

(3.1.7)

Ω

and by the Korn inequality, there exists a constant K1 (Ω) such that ∫ γ(v)eij eij dx ≥ μ0 ε Ω

p−2 2

∫ |e|2 dx ≥ μ0 ε

p−2 2

Ω

K1 ‖v‖2W 1,2 (Ω) .

(3.1.8)

Combining (3.1.6) and (3.1.7), we get 1 d ‖v‖2 + μ0 ‖e‖pLp ≤ ‖f ‖∞ ‖v‖. 2 dt

(3.1.9)

While dropping the expression involving μ1 in (3.1.6) and using (3.1.8), we obtain p−2 1 d ‖v‖2 + μ0 ε 2 K1 ‖v‖2W 1,2 (Ω) ≤ ‖f ‖∞ ‖v‖. 2 dt

(3.1.10)

Both (3.1.9) and (3.1.10) hold for any μ1 > 0, and by the boundary condition, there exists a constant C0 > 0 such that ‖v‖2W 1,2 (Ω) ≥ C0 ‖v‖2 ‖f ‖2∞

d ‖v‖ + K1 ‖v‖2 ≤ dt K2

(3.1.11) (3.1.12)

p−2

where K2 = C0 K1 ε 2 . By using the Gronwall inequality in (3.1.12), we obtain ‖v(t)‖2 ≤ exp(−K2 t)‖v0 ‖2 +

‖f ‖2∞ . K22

(3.1.13)

Thus, for some bounded set B0 ⊂ L2 (Ω), with B0 ⊂ BR0 (0) a ball of radius R0 centered at 0 of the ball, then there exists t0 = t0 (R) for any t ≥ t0 , any μ1 ≥ 0, and ‖v(t)‖2 ≤

2‖f ‖2∞ K22

(3.1.14)

is satisfied whenever v0 ∈ B0 . √ Let ρ = K 2 . Then from (3.1.9) and (3.1.14), for v0 ∈ B0 , t ≥ t0 , and all μ1 ≥ 0, we find 2 that t+r

∫ ‖e‖pLp dt ≤ t

1 1 2 ( ρ + ρr )‖f ‖2∞ , μ0 2

r > 0, p > 2.

(3.1.15)

3.1 Global attractors of incompressible non-Newtonian fluids on bounded domain

| 95

If v0 ∈ B0 , t ≥ t0 , any r > 0 and all μ1 ≥ 0, p > 2, from (3.1.10) and (3.1.14) we can deduce that t+r

C0 1 2 ( ρ + ρr )‖f ‖2∞ K2 2

(3.1.16)

1 dt ≤ ( ρ2 + ρr )‖f ‖2∞ . 2

(3.1.17)

∫ ‖v‖2W 1,2 (Ω) dt ≤ t

from (3.1.6)and (3.1.14) we also get t+r

μ1 ∫ t

𝜕eij 𝜕eij 𝜕xk 𝜕xk

For the existence of absorbing sets in W 1,2 (Ω), we take the inner product of (3.1.1) with 𝜕vi to obtain 𝜕t 𝜕eij 𝜕eij 1 𝜕v d dx} + {∫ Γ(eij eij )dx + μ1 ∫ 2 𝜕t dt 𝜕xk 𝜕xl Ω

𝜕v 𝜕v 1 ≤ ∫ vj i i dx + ‖f ‖2∞ 𝜕xj 𝜕t 2

Ω

(3.1.18)

Ω

which holds for any μ1 > 0 and all t > 0. We note that 𝜕v 𝜕v 1 𝜕v 2 2 ∫ vj i i dx ≤ ‖v‖∞ ‖v‖H 1 (Ω) + 𝜕xj 𝜕t 4 𝜕t Ω

and W 1,p (Ω) → L∞ (Ω), p > 2 = n and Korn inequality, there exists a constant K3 > 0 such that 𝜕v 𝜕v 1 𝜕v 2 2 ∫ vj i i dx ≤ K3 ‖e‖ ‖v‖H 1 (Ω) + . 𝜕xj 𝜕t 4 𝜕t

(3.1.19)

Ω

Combining (3.1.18) and (3.1.19), we find that p > 2, t > 0 and μ1 ≥ 0, 𝜕eij 𝜕eij d {∫ Γ(eij eij )dx + μ1 ∫ dx} dt 𝜕xk 𝜕xl Ω

≤

Ω

K3 ‖e‖2Lp (Ω) ‖v‖2H 1 (Ω)

1 + ‖f ‖2∞ 2 t+r

for v0 ∈ B0 , ∀r > 0, t ≥ t0 the estimate for the integral ∫t ∫Ω Γ(eij eij )dxds. Recall p > 2, then p2 > 1 and using the inequality we have (a + b)m ≤ 2m (am + bm )

(3.1.20)

96 | 3 Global attractors of incompressible non-Newtonian fluids where m ≥ 1, a ≥ 0, b ≥ 0 are real numbers, we infer that eij eij

Γ(eij eij ) = μ0 ∫ (ε + s)

p−2 p

ds

0

p p 2 = μ0 {(ε + |e|2 ) 2 − ε 2 } p p 2 ≤ μ0 (ε + |e|2 ) 2 p p p 2 ≤ μ0 2 2 (ε 2 + |e|p ) p

(3.1.21)

therefore, ∫ Γ(eij eij )dx ≤ C1 + C2 ‖e‖pLp (Ω) ,

(3.1.22)

Ω

where p p 2 μ0 2 2 ε 2 |Ω| p p 2 C2 = μ 0 2 2 p

C1 =

and by (3.1.15), we have t+r

∫ ∫ Γ(eij eij )dxds ≤ C3

(3.1.23)

t Ω

where C3 = C1 r + C2 (

1 1 2 ( ρ + ρr )‖f ‖2∞ ). μ0 2

By the Kron-type inequality and ‖v‖2W 1,2 (Ω) ≤ C4 (∫ Γ(eij eij )dx + μ1 ∫ Ω

Ω

𝜕eij 𝜕eij 𝜕xk 𝜕xl

dx)

(3.1.24)

where C4 = (K1 μ0 ε

p−1 2

−1

)

Combining equation (3.1.20) and equation (3.1.24), we get dy ≤ a(t)y(t) + b(t), dt

t>0

(3.1.25)

3.1 Global attractors of incompressible non-Newtonian fluids on bounded domain

| 97

where y = ∫ Γ(eij eij )dx + μ1 ∫ Ω

Ω

a(t) =

K3 C4 ‖e‖2Lp (Ω) ,

𝜕eij 𝜕eij 𝜕xk 𝜕xl

dx

1 b(t) = ‖f ‖2∞ . 2

By applying the uniform Gronwall lemma, for v0 ∈ B0 , t ≥ t0 , all μ1 ≥ 0, t+r

t+r

t

t

∫ a(s)ds = K3 C4 ∫ ‖e‖2Lp (Ω) ds ≤ K3 C4 r

p−2 p

t+r

( ∫ ‖e‖2Lp (Ω) ds)

p 2

t

≤ K3 C4 r t+r

= C5

p−2 p

p

2 1 1 [ ( ρ2 + ρr )‖f ‖2∞ ] μ0 2

1 ∫ b(s)ds = ‖f ‖2∞ ⋅ r = C6 2

(3.1.26) (3.1.27)

t

t+r

1 ∫ y(t)dt = C3 + ( ρ2 + ρr )‖f ‖2∞ = C7 2

(3.1.28)

y(t) ≤ (C6 + C7 ) exp(C3 ) = K4 ,

(3.1.29)

t

then t ≥ t0 + r.

Using (3.1.24) again, one also obtains ‖v‖2W 1,2 (Ω) ≤ K5 ,

t ≥ t0 + r

(3.1.30)

where K5 = K4 C4 . Thus, if v0 ∈ B0 ⊂ BR0 (0), then there exists ρ1 > 0, t ≥ t0 = 1, which are independent of μ1 ≥ 0 such that for p > 2, ρ

Sμ1 (t)v0 ∈ BW1 1,2 (Ω) ,

t ≥ t1 .

(3.1.31)

So the ball B of radius ρ1 in W 1,2 (Ω) is the absorbing set for Sμ1 (t), which is independent of μ1 ≥ 0. From (3.1.29), it also follows that for all μ1 ≥ 0, t ≥ t0 , p > 2, μ1 ∫ Ω

𝜕eij 𝜕eij 𝜕xk 𝜕xk

dx|t+r ≤ K4 .

(3.1.32)

98 | 3 Global attractors of incompressible non-Newtonian fluids By the Korn inequality, for v ∈ W01,2 (Ω) ∩ W 2,2 (Ω) there exists a constant K6 > 0 such that 𝜕eij 𝜕eij dx ≥ K6 ‖v‖2H 2 (Ω) (3.1.33) ∫ 𝜕xk 𝜕xk Ω

so that (3.1.32) implies that for t ≥ t1 , μ1 > 0, p > 2 have ‖v‖2H 2 (Ω) ≤

1 K4 . μ1 K6

(3.1.34)

We deduce from (3.1.34) the existence of an absorbing set for Sμ1 (t) for μ1 > 0, of radius ρμ1 , while v0 ∈ B0 ⊂ BR0 (0), ρμ

1 Sμ1 (t)v0 ∈ BW 2,2 , (Ω)

t ≥ t1 (R0 ).

(3.1.35)

Therefore, the proof of Theorem 3.1.1 is complete. About the existence of attractors, for enough large t and fixed μ1 > 0, there exists ρμ1 the absorbing set BW 2,2 , so that the solution semigroup Sμ1 (t) is consistently com(Ω) ρ

pact. Similarly, for μ1 ≥ 0, there exists the absorbing set BW1 1,2 (Ω) , so that the solution semigroup Sμ1 (t) is consistently compact. Applying Lemma 3.0.2, we can define the global attractor for Sμ1 (t), μ1 ≥ 0: When μ1 > 0, ρμ

1 Aμ1 = ⋂ Sμ1 (t)BW 2,2 (Ω)

t≥0

when μ1 = 0

ρ

A0 = ⋂ S0 (t)BW 1,2 (Ω) . t>0

(3.1.36)

(3.1.37)

The definition for Aμ1 , A0 in this way is the compactly global attractor. 3.1.2 Consistently differentiability for the solution semigroup Computing the upper bound of the Hausdorff dimension dH (Aμ1 ) and fractal dimension dF (Aμ1 ) for the compactly global attractor, it is an indispensable condition for the differentiability of solutions semigroup. Since the solutions semigroup S0 (t) in A0 cannot obtain the differentiable condition, we only consider the differentiability for the solutions semigroup Sμ1 (t) in Aμ1 , μ1 > 0. Definition 3.1.1. Assume ∀u0 ∈ Aμ1 , then there exists the linear operator Lμ1 (t; u0 ) ∈ £(Hper , Hper ), such that when ε∗ → 0, we have ∀t > 0, sup

u0 ,v0 ∈Aμ1 ,0 0 is consistently differentiable in Aμ1 .

3.1 Global attractors of incompressible non-Newtonian fluids on bounded domain

| 99

Theorem 3.1.2. The compact global attractor for nonlinear semigroup Sμ1 (t), μ1 > 0 defined in equation (3.1.36) is consistently differentiable. Proof. Assume u0 , v0 ∈ Aμ1 , and ‖v0 − u0 ‖L2 (Ω) ≤ ε∗ , we need to estimate θ(t, μ1 ) = Sμ1 (t)v0 − Sμ1 (t)u0 − Lμ1 (t; u0 )(v0 − u0 )

(3.1.39)

L2 (Ω) norm. Let w(t; μ1 ) = V(t; μ1 ) − u(t; μ1 ),

θ(t; μ1 ) = w(t; μ1 ) − U(t; μ1 )

(3.1.40)

then w(t) satisfies 𝜕wi 𝜕u 𝜕w 𝜕P 𝜕 [γ(v)eij (v) − γ(u)eij (u)] + wj i + uj i = − w + 𝜕t 𝜕xj 𝜕xj 𝜕xi 𝜕xj − eμ1

𝜕 (△eij (w)) 𝜕xj

(3.1.41)

where Pw is the pressure difference of v and u, and ∇ ⋅ w = 0,

(x, t) ∈ Ω × [0, T)

w(0) = v0 − u0 ,

x∈Ω

wi (x, t) = wi (x + Lej , t),

x ∈ Ω, t ≥ 0.

Taking the inner product of wi with equation (3.1.41), 𝜕eij 𝜕eij 𝜕u 1 d ‖w‖2 + ∫ wj i wi dx + 2μ1 ∫ (w) (w)dx 2 dt 𝜕xj 𝜕xk 𝜕xk Ω

Ω

+ ∫[γ(v)eij (v) − γ(u)eij (u)]eij (w)dx = 0 Ω

or, by the definition of Γ in equation (2.1.7), the above equation can be rewritten as 𝜕u 1 d ‖w‖2 + ∫ wj i wi dx + 2μ1 K0 ‖w‖2W 2,2 (Ω) 2 dt 𝜕xj Ω

𝜕Γ 𝜕Γ + ∫( (eij (v)) − (e (w)))eij (w)dx ≤ 0. 𝜕eij 𝜕eij ij

(3.1.42)

Ω

Moreover, it can be rewritten as 𝜕u 1 d ‖w‖2 + ∫ wj i wi dx + 2μ1 K0 ‖w‖2W 2,2 (Ω) 2 dt 𝜕xj 1

+ ∫(∫ Ω

0

Ω

𝜕2 Γ (e(u + τw))dτ)eij (w)ekl (w)dx ≤ 0. 𝜕eij 𝜕ekl

(3.1.43)

100 | 3 Global attractors of incompressible non-Newtonian fluids 2

Potential function Γ for any ξ ∈ ℝn , ξ ≠ 0, p−2 |ξ ⋅ e|2 𝜕2 ξij ξkl = μ0 (ε + |e|2 ) 2 [ξ 2 + (p − 2) ] 𝜕eij 𝜕ekl ε + |e|2

≥ μ0 (ε + |e|2 )

p−2 2

|ξ 2 | ≥ μ0 ε

p−2 2

2

|ξ |ξ ,

p > 2.

Thus, using the Korn inequality, we have 1

∫(∫ Ω

0

𝜕2 Γ (e (u + τw))dτ)eij (w)ekl (w)dx 𝜕eij 𝜕ekl ij

≥ μ0 ε ≥ μ0 ε

p−2 2

p−2 2

∫ eij (w)eij (w)dx Ω

K2 ‖w‖2W 1,2 (Ω) .

(3.1.44)

So, for any μ1 > 0 ‖u(t)‖W 1,2 (Ω) ≤ ρμ1 ,

∀t ≥ 0.

(3.1.45)

For ‖v(t)‖W 1,2 (Ω) and ‖w‖W 1,2 (Ω) , we also have similar results. Using the Young inequality, 1 1 𝜕u 2 ∫ wj i wi dx ≤ C1 ‖w‖W 1,2 (Ω) ‖u(t)‖W 1,2 (Ω) ‖w‖ 2 ‖w‖W 1,2 (Ω) 𝜕xj Ω

3

1

2 2 ≤ C1 ρμ1 ‖w‖W 1,2 (Ω) ‖w‖

≤ μ0 ε

p−2 2

K2 ‖w‖2W 1,2 (Ω) + C2 (μ; Ω)‖w‖2 .

(3.1.46)

By equation (3.1.43), we have p−2 d ‖w‖2 + μ0 ε 2 K2 ‖w‖2W 2,2 (Ω) + 4μ1 K0 ‖w‖2W 1,2 (Ω) ≤ C3 ‖w‖2 . dt

Integrating the above equation, we get ‖v(t) − u(t)‖ ≤ ‖v0 − u0 ‖2 exp(C3 t) and t

∫ ‖w(τ)‖2W 1,2 (Ω) dτ ≤ 0

t

∫ ‖w(τ)‖2W 2,2 (Ω) dτ ≤ 0

p−2

2ε 2 ‖v − u0 ‖2 exp(C3 t) μ0 K2 0

(3.1.47)

1 ‖v − u0 ‖2 exp(C3 t) μ1 K0 0

(3.1.48)

3.1 Global attractors of incompressible non-Newtonian fluids on bounded domain | 101

and for the definition of θ(t) in equation (3.1.40) satisfies 𝜕P 𝜕u 𝜕θ 𝜕w 𝜕θi 𝜕 + θj i + uj i + wj i = − θ + [γ(v)eij (v) − γ(u)eij (u)] 𝜕t 𝜕xj 𝜕xj 𝜕xj 𝜕xi 𝜕xj 𝜕 𝜕 [γ(u)eij (U) − αAijkl (u)ekl (U)] − 2μ1 (△eij (θ)) 𝜕xj 𝜕xj

−

∇ ⋅ θ = 0,

(x, t) ∈ Ω × [0, T)

θ(0) = 0,

x∈Ω

θi (x, t) = θi (x + Lej , t),

(3.1.49)

x ∈ Ω, t ≥ 0

where α = 2 − p Aijk (u) = μ0 (ε + |e(u)|2 )

p−4 2

eij (u)eij (u)

γ(u)eij (U) − αAijkl (u)ekl (U) =

𝜕2 Γ (e (u))ekl (u) 𝜕eij 𝜕ekl ij

(3.1.50)

taking the inner product of θi with (3.1.49),

𝜕u 𝜕w 1 d ‖θ‖2 + ∫ θj i θj dx + ∫ wj i θi dx + ∫[γ(v)eij (v) 2 dt 𝜕xj 𝜕xj Ω

Ω

Ω

− γ(u)eij (u)]eij (θ)dx − ∫[γ(u)eij (U) − αAijkl (u)ekl (U)]eij (θ)dx + μ1 ∫ Ω

𝜕eij 𝜕xk

(θ)

𝜕eij 𝜕xk

Ω

(θ)dx = 0

(3.1.51)

∫[γ(v)eij (v) − γ(u)eij (u)]eij (θ)dx − ∫[γ(u)eij (U) − αAijkl (u)ekl (U)]eij (θ)dx Ω

1

= ∫(∫ 0

Ω

=∫ Ω

Ω 2

𝜕Γ 𝜕2 Γ (e(u + τw))dτ)eij (θ)ekl (θ)dx − ∫ (e(u))eij (θ)ekl (U)dx 𝜕eij 𝜕ekl 𝜕eij 𝜕ekl 2

𝜕Γ (e(u))eij (θ)ekl (θ)dx 𝜕eij 𝜕ekl 1

= ∫(∫[ Ω

≥ε

0 p−2 2

Ω

𝜕2 Γ 𝜕2 Γ (e(u + τw)) − (e(u))]dτ)eij (θ)ekl (w)dx 𝜕eij 𝜕ekl 𝜕eij 𝜕ekl

μ0 K2 ‖θ‖2W 1,2 (Ω) + ∫ Γijklmn eij (θ)ekl (θ)emn (w)dx

(3.1.52)

Ω

where 1 1

Γijklmn = ∫ ∫ 0 0

𝜕3 Γ (e(u + στ)τ)dτdσ. 𝜕eij 𝜕ekl 𝜕emn

(3.1.53)

102 | 3 Global attractors of incompressible non-Newtonian fluids According the definition of Γ, p−2 𝜕3 Γ = μ0 (p − 2)(ε + |e|2 ) 2 𝜕eij 𝜕ekl 𝜕emn

×{

δim δjn ekl + δkm δln eij + δij δkl emn (ε +

|e|2 )

+

p−4 e e e } (ε + |e|2 )2 ij kl mn

(3.1.54)

where δij , δkl , δmn are Kronecker numbers. So we have p−2 1 1 𝜕3 Γ 2 − 2 − 2 ≤ μ0 (p − 2)(ε + |e| ) 2 [(ε + |e| ) 2 + |p − 4| ⋅ (ε + |e| ) 2 ] 𝜕eij 𝜕ekl 𝜕emn p−3 2

≤ Cp ⋅ (ε + |e|2 )

p−3

≤C ε 2 , { p p−3 ≤ Cp (ε + |e|2 ) 2 ,

p≤3

p > 3.

For the case of p ≤ 3, it is relatively easy to estimate, so next we only consider the case of p > 3. 1 For ε + |e|2 ≤ (ε 2 + |e|2 )2 , we only consider 1 1

1

p−3

1

p−3

|Γijklmn | ≤ ∫ ∫ (ε 2 + |e(u + στw)|)

⋅ τdτdσ

0 0

1 1

≤ ∫ ∫ (ε 2 + |e(u + στw)|) 0 0

dτdσ

p−3

1

≤ (ε 2 + |e(u + στw)|)

|e(u)| + |e(w)| + C4 ,

={

2p−3 ⋅ ε

p−3 2

+ 4p−3 (|e(u)|p−3 + |e(w)|p−3 ),

p−3≤1

p − 3 > 1.

(3.1.55)

Similarly, the case of estimating p − 3 > 1 contains the case of p − 3 ≤ 1, so we only consider |Γijklmn | ≤ 2p−3 ε

p−3 2

+ 4p−3 (|e(u)|p−3 + |e(w)|p−3 )

= C5 + C6 (|e(u)|p−3 + |e(w)|p−3 ).

(3.1.56)

Therefore, ∫ Γijklmn eij (θ)ekl (θ)emn (w)dx Ω

≤ C5 ∫ |e(θ)||e(w)|2 dx + C6 ∫ |e(θ)||e(w)|p−1 dx + C6 ∫ |e(θ)||e(w)|2 |e(u)|p−3 dx Ω

≤

C5 ‖θ‖2W 1,2 (Ω) ‖e(w)‖L4 (Ω) +

Ω

p−3

+ C6 ‖θ‖L6 (Ω) ⋅ ‖e(Ω)‖

C6 ‖e(θ)‖L6 ‖e(θ)‖L6 ‖e(w)‖2L6 ‖e(u)‖p−3 3p−9 L 2

L

6p−6 5

Ω

3.1 Global attractors of incompressible non-Newtonian fluids on bounded domain | 103

≤ μ1 ‖θ‖2W 2,2 (Ω) + C7 (‖w‖4W 2,2 (Ω) + ‖w‖2(p−1) + ‖w‖4W 2,2 (Ω) ⋅ ‖u‖2(p−1) ) W 2,2 (Ω) W 2,2 (Ω) ≤ K0 μ1 ‖θ‖2W 2,2 (Ω) + C8 ‖w‖2W 2,2 (Ω)

(3.1.57)

where p < 6 p−2 C8 = C7 (ρ2μ1 + ρ2p−4 ) μ1 )(8 + 4

and u0 , v0 ∈ Aμ1 are the attractors, u(t) = Sμ1 (t)u0 , v(t) = Sμ1 (t)v0 and Sμ1 (t)Aμ1 = Aμ1 , therefore, we have ‖u‖W 2,2 (Ω) ≤ ρμ1 , ‖v‖W 2,2 (Ω) ≤ ρμ1 , ‖w(t)‖W 2,2 (Ω) ≤ 2ρμ1 , ∀t ≥ 0. Note 3.1.1. The estimate for last term of (3.1.57) is the bold promotion for semigroup’s differentiability, and the key of this problem is to build the relation between t t ∫0 ‖w(t)‖pW 2,2 (Ω) dτ ≤ (2ρμ1 )p−2 ∫0 ‖w(t)‖2W 2,2 (Ω) dτ and v0 − u0 . For 𝜕u 2 ∫ θi i θj dx ≤ ‖u‖W 1,2 (Ω) ‖θ‖L4 (Ω) 𝜕xj Ω

≤ C0 ‖u‖W 1,2 (Ω) ‖θ‖‖θ‖2W 2,2 (Ω)

1 2 ≤ K0 μ1 ‖θ‖2W 2,2 (Ω) + C 2 ρ2 ‖θ‖2 2 K0 μ1 0 μ1

(3.1.58)

𝜕w ∫ wj i θi dx ≤ ∫ |w||∇w||θ|dx 𝜕xj Ω

Ω

≤ ‖θ‖ ⋅ ‖w‖L4 (Ω) ‖∇w‖L4 (Ω) dx

≤ C0 ‖θ‖‖w‖W 1,2 (Ω) ‖∇w‖W 1,2 (Ω)

≤ ‖θ‖2 + C02 ρ2μ |‖w‖2W 1,2 (Ω) .

(3.1.59)

Again using Korn inequality: 2μ1 ∫

𝜕eij (θ) 𝜕eij (θ)

Ω

𝜕xk

𝜕xk

dx ≤ 2μ1 K0 ‖θ‖2W 2,2 (Ω) .

(3.1.60)

Combining equations (3.1.51)–(3.1.60), we have p−2 d ‖θ‖2 + μ1 K0 ‖θ‖2W 2,2 (Ω) + 2μ0 K2 ε 2 ‖θ‖2W 2,2 (Ω) dt ≤ C9 ‖θ‖2 + C10 ‖w‖2W 2,2 (Ω) .

Furthermore, d ‖θ‖2 ≤ C9 ‖θ‖2 + C10 ‖w‖2W 2,2 (Ω) dt where C9 =

2C0 ρ2μ1 μ1 K0

+ 1,

C10 = C8 + C02 ρ2μ .

(3.1.61)

104 | 3 Global attractors of incompressible non-Newtonian fluids Noting θ(0) = 0 and equation (3.1.48), then by equation (3.1.61) we can get t

‖θ‖2 ≤ eC9 t ∫ ‖w(t)‖2W 2,2 (Ω) dτ 0

C10 ≤ [exp(C3 + C9 )t] ⋅ ε∗2 , 2μ1 K0

∀t ≥ 0.

So, there exist constants β1 > 0, β2 > 0, which are dependent of ε, μ1 , p, Ω, such that sup

u0 ,v0 ∈Aμ1 ,0 0 is monotonically increasing, and noting (3.1.48) and θ(0) = 0, so integrating the above equation from 0 to t, we have 2

1

1

2

‖θ‖2 ≤ C12 [(β1 eβ2 t ) 5 ε∗ 15 + (β1 eβ2 t ) 3 + β1 eβ2 t ε∗ 3 ] ×ε

∗ 31

≤ C13 ε

e

∗ 31

2 ρ2 2C0 μ μ1 K0

1

t

t

∫ ‖w‖2W 2,2 (Ω) dt 0

‖v0 − u0 ‖2

that is, for ∀t > 0 1 ‖θ‖L2 Ω ≤ C13 (t) ⋅ ε∗ 3 ‖v0 − u0 ‖L2 Ω

(3.1.66)

where C13 (t) =

2 1 1 2 C12 [(β1 eβ2 t ) 5 ε∗ 15 + (β1 eβ2 t ) 3 + β1 eβ2 t ε∗ 3 ] 2μ1 K0

× exp(

2C02 ρ2μ1 μ1 K0

+ C3 )t.

So, Theorem 3.1.2 is proved. 3.1.3 For μ1 > 0, the upper bounded estimates of dH (Aμ1 ) and dF (Aμ1 ) of attractor Aμ1 Write the linear equations of (3.1.1), Ut = F (u)U = −2μ1 AU + 2μ0 Ap (u)U − B(u, U) − B(U, u).

(3.1.67)

Assume m ∈ N, define t

1 qm = lim sup ∫ Tr(F S(s)u0 ) ⋅ Qm (s)ds t→∞ u ∈A ,ξ ∈H,|ξ |≤1 t 0 μ i i 1

(3.1.68)

0

Qm (s) = Qm (s; u0 , ξ1 , . . . , ξm ) from H to Span{L(t; u0 )ξ1 , . . . L(t; u0 )ξm } orthogonal projection, assume u0 ∈ Aμ1 , ξ1 , . . . , ξm ∈ H, u(t) = S(t)u0 , Ui (t) = L(t; u0 )ξi , t ≥ 0. Suppose

106 | 3 Global attractors of incompressible non-Newtonian fluids ϕ1 (t), . . . , ϕm (t) t ≥ 0 is orthonormal basis of Span{U1 (t), . . . Um (t)} in H. So we can assume ϕj (t) ∈ V, for the estimate of the trace for F (S(t)u0 ) ⋅ Qm (t), we have m

Tr(F S(s)u0 ) ⋅ Qm (s) = ∑⟨F (u(s))ϕj , ϕj ⟩ j=1 m

= ∑⟨−2μ1 Aϕj + 2μ0 Ap (u)ϕj − B(u, ϕj ) − B(ϕj , u), ϕj ⟩ j=1 m

= ∑⟨−2μ1 Aϕj + 2μ0 Ap (u)ϕj + B(ϕj , u), ϕj ⟩ j=1

m

m

m

j=1

j=1

∑⟨−2μ1 Aϕj , ϕj ⟩ ≤ −μ1 K0 ∑ ‖ϕj ‖21 ≤ −μ1 K0 λ1 ∑ ‖ϕj ‖21 j=1

m

m

j=1

j=1

2μ0 ∑⟨Ap (u)ϕj , ϕj ⟩ = − 2 ∑⟨γ(u)eim (ϕj ) + (p − 2)μ0 (ε + |e|2 )

(3.1.69) (3.1.70)

p−4 2

× eim (u)ekl (ϕj ), eim (ϕj )⟩ m

= −2 ∑⟨ j=1

𝜕2 Γ (e(u))ekl (ϕj ), eim (ϕj )⟩ 𝜕eim 𝜕ekl

≤0

(3.1.71)

then m

∑⟨B(ϕj , u), ϕj ⟩ ≤ j=1

C 2 ‖u‖2 m μ1 K0 λ1 m ∑ ‖ϕj ‖21 + 0 2 2 2μ1 K0 λ j=1

(3.1.72)

thus Tr(F S(s)u0 ) ⋅ Qm (s) ≤ −

C 2 ‖u‖2 m μ1 K0 λ1 m ∑ ‖ϕj ‖21 + 0 2 . 2 2μ1 K0 λ1 j=1

If λj , j = 1, . . . , m are the jth eigenvector of −△ in Ω corresponding to eigenvector ϕj ∈ 1 H 1 (Ω) ∩ Hper (Ω) satisfying ∇ ⋅ ϕj = 0, then m

∑ ‖ϕj ‖21 ≥ λ1 + ⋅ ⋅ ⋅ + λm

(3.1.73)

j=1

2

λj ≥ C0 (Ω)λ1 j 3 ,

∀j ≥ 1

Tr(F S(s)u0 ) ⋅ Qm (s) ≤ −

μ1 K0 λ1 35 C02 ‖u‖22 m m + . 2 2μ1 K0 λ1

Let 1

1 K3 = lim sup ∫ ‖S(s)u0 ‖22 dx t→∞ u ∈A t 0 μ1 0

(3.1.74)

3.2 Global attractors of incompressible non-Newtonian fluids on unbounded domain |

107

then qm ≤ −

μ1 K0 λ12 5 C02 K 3 m m3 . 2 2μ1 K0 λ1

Let m ∈ N C03

3

K 2 m −1≤ ( 3 ) ≤ m 3 λ2 μ1 K0 1

(3.1.75)

in this time qm < 0. Theorem 3.1.3. Assume m is smallest positive integer satisfying (3.1.75), ∀μ1 > 0, then the global attractor Aμ1 has finite Hausdorff dimension and fractal dimension, that is, 3

C03

K 2 dH (Aμ1 ) ≤ m < 1 + ( 3) 3 λ2 μ1 K0 1

and dF (Aμ1 ) ≤ 2m < 2 +

2C03

( 3

μ1 K0

3

K3 2 ) . λ12

Note 3.1.2. (1) Modifying (3.1.71) and (3.1.74), dH (Aμ1 ) and dF (Aμ1 ) can be independent of μ1 in Theorem 3.1.3; (2) μ1 = 0, dH (Aμ1 ) cannot be estimated using the above method, and the solution semigroup S0 (t) cannot built the consistent differentiability; (3) Attractor Aμ1 → A0 , when μ1 → 0, we have supu∈A0 ∞uμ ∈Aμ d(u, uμ1 ) → 0, it is 1 1 equivalent to sup ∞uμ

u∈A0

1

∈Aμ1 ‖u

− uμ1 ‖L2 (Ω) → 0,

μ1 → 0.

Note 3.1.3. We can similarly get the attractor of monopolar fluid in unbounded region; The bipolar fluid attractor, Hausdorff dimension, Fractal dimension estimate and space dimension in the case of n = 3 can be obtained in the unbounded region.

3.2 Global attractors of incompressible non-Newtonian fluids on unbounded domain Considering the bipolar incompressible non-Newtonian fluid equations with initial boundary value problem v + 2μ1 Av − 2μ0 Ap (v) + B(v, v) = f { t v(0) = v0

(3.2.1)

108 | 3 Global attractors of incompressible non-Newtonian fluids where f ∈ L2 (0, T; H), v0 ∈ H, then there exists the unique solution for (3.2.1). All marks and assumptions in this section are the same with in Section 3.1. Now emphasizing that the parameter p satisfies 1 < p ≤ 2, and note α = 2 − p, then 0 ≤ α < 1, and then α appears the term (ε + |e|2 )− 2 in Ap (u). Ω = ℝ × (−a, a) ⊂ ℝ2 , a > 0 is the unbounded bar domain. The boundary domain 𝜕Ω satisfying v = 0,

τijk vj vk = 0, (x, t) ∈ 𝜕Ω × [0, T) ̄ J(Ω) = {φ ∈ C0∞ (Ω)|φ = 0, x ∈ 𝜕Ω, div φ = 0, x ∈ Ω}

(3.2.2)

V = J(Ω) is in the closure of H 2 (Ω); H = J(Ω) is in the closure of L2 (Ω). Since Ω is unbounded, the embedding H 2 → L2 (Ω) and H 1 (Ω) → L2 (Ω) are not compact. Therefore, we cannot directly use Galerkin method to obtain the existence of solution for problem (3.2.2). Denote {ΩN }, N = 1, 2, . . . be the bounded subdomain sequences of Ω, ΩN → Ω, N → ∞, and 𝜕Ω is C ∞ category Γ+N = {(x, a) | (x, a) ∈ ΩN }

Γ−N = {(x, a) | (x, −a) ∈ ΩN }

J(ΓN ) = {φ ∈ J(Ω) ∩ (D(ΓN )) ∪ Γ+N ∪ Γ−N } VN = J(ΓN ) in H 2 (ΓN ) closure

HN = J(ΓN ) in L2 (ΓN ) closure eij (v) eij (φ) , ), ∀v, φ ∈ VN ⟨ANv , φ⟩ = ( xk xk J(Γ1 ) ⊂ J(Γ2 ) ⊂ ⋅ ⋅ ⋅ ⊂ J(Γ) V1 ⊂ V2 ⊂ ⋅ ⋅ ⋅ ⊂ V

H1 ⊂ H2 ⊂ ⋅ ⋅ ⋅ ⊂ H. The approximate solution of problem equation (3.2.1) is {

vtN + 2μ1 AvN − 2μ0 Ap (vN ) + B(vN , vN ) = f N vN (0) = v0N

(3.2.3)

where f N ∈ L2 (0, T; HN ), v0N ∈ HN , using the Galerkin method, we can obtain the existence of vN and satisfy the energy equation t

‖vN ‖2 + 4μ1 ∫ ∫ 0 ΩN

t

𝜕eij (vN ) 𝜕eij (vN ) ⋅ dxdτ + ∫ ∫ Γ(vN )eij eij dxdτ 𝜕xk 𝜕xk 0 ΩN

t

‖v0 ‖2 + 2 ∫(f N (τ), vN )dτ.

(3.2.4)

0

Similarly, for H 2 (Ω) → H is not compact, we cannot use the previous method. Here, we use the Abergel method to consider the weighting function space.

3.2 Global attractors of incompressible non-Newtonian fluids on unbounded domain |

109

Lemma 3.2.1. Assume that h(x1 , t) is the smoothing function satisfying: (1) h(x1 , t) ≥ 0, ∀t ≥ 0, −∞ < x1 < +∞, h(x, 0) = 0; (2) any derivative of h(x1 , t) for the order ≥ 1 is a bounded function; (3) h(x1 , t) → +∞, |x1 | → ∞. Then there exists a constant C > 0, such that the solution v(x1 , x2 , t) of (3.2.1) satisfying sup ∫ |v(x1 , x2 , t)|2 dx1 dx2 h(x1 , t) ≤ C. t≥t0

(3.2.5)

Ω ρ

For some t0 ≥ 0, and v0 ∈ BH12 is a absorbing set in H 2 (Ω), the solution semigroup S(t) of problem equation (3.2.1) has a compact global attractor Aμ1 . Theorem 3.2.1. Nonlinear semigroup S(t) v0 ∈ H, defined by problem (3.2.1), there exist ρ ρ the absorbing set BH and BH12 in H and H 2 . And for v0 ∈ B0 ⊂ BR0 (0) ⊂ H with radius R0 , ρ center at 0 of the ball, there exists t0 = t0 (R0 ) such that when t ≥ t0 , S(t)v0 ∈ B0 ∩ BH12 . Proof. Assume v0 ∈ H, f ∈ L∞ (0, ∞; H), v0N , f N is projection of v0 and f in HN , vN is the solution of problem (3.2.3), taking the inner product of vN with (3.2.3), noting that b(vN , vN , vN ) = 0, (−Ap (vN ), vN ) ≥ 0, then β 1 d N 2 1 ‖v ‖ + 2μ1 K0 ‖vN ‖2H 2 (Ω) ≤ ‖f ‖∞ ‖vN ‖ ≤ ‖f ‖2∞ + ‖v2 ‖2 2 dt 2 2β d N 2 ‖v ‖ + 2μ1 K0 ‖vN ‖H 2 (Ω) ≤ β‖f ‖2∞ dt

(3.2.6) (3.2.7)

where β = 2μ1K . Using e2μ1 K0 t to multiply (3.2.7) and integrating the resulting equation 1 0 from 0 to t, we get t

‖vN ‖2 ≤ e−2μ1 K0 t [‖vN (0)‖2 + β ∫ e2μ1 Ks ‖f ‖2∞ ds] ≤e

−2μ1 K0 t

N

2

2

0

[‖v (0)‖ + β ‖f ‖2∞ ]

(3.2.8)

then there exists t0 = t0 (μ1 , v(0)) > 0 such that ‖vN ‖2 ≤ 2β2 ‖f ‖2∞ = C1 ,

∀t ≥ t0

(3.2.9)

a. m. R0 , such that when v0 ∈ B0 ⊂ H, we have ‖v‖ ≤ R, v0 ∈ B0 , t ≥ t0 (R), so there exists the absorbing set in H. Next, we prove that there exists the absorbing set in H 2 (Ω). Taking the inner prodN uct of dvdt with (3.2.3), dvN 𝜕eij (vN ) 𝜕eij (dvN )/dt , ) + 2μ1 ( 𝜕xk 𝜕xk dt dvN dvN dvN + ⟨2μ0 Ap (vN ), ⟩ + b(vN , vN , ) = (f N , ). dt dt dt

(3.2.10)

110 | 3 Global attractors of incompressible non-Newtonian fluids By the definition of Γ, ⟨2μ0 Ap (vN ),

𝜕eij dvN d ⟩ = 2 ∫ γ(vN )eij dx = {∫ Γ(eij eij )dx} dt 𝜕t dt Ω

Ω

N 𝜕vN 𝜕vN N N dv ) ≤ ∫ vjN i ⋅ i dx b(v , v , dt 𝜕xj 𝜕t Ω

dv ≤ ‖vN ‖L4 (Ω) ‖∇vN ‖L4 (Ω) dt dv ≤ C(Ω) ⋅ ‖vN ‖H 2 (Ω) dt N 1 dv 2 ≤ ‖ ‖ + C(Ω) ⋅ ‖vN ‖H 2 (Ω) 4 dt N dvN 1 dvN 2 N dv 2 ) ≤ ‖f ‖∞ ≤ + ‖f ‖∞ . (f , dt dt 4 dt For (3.2.10), we can get 2 𝜕eij (vN ) 𝜕eij (vN ) 1 dvN dvN dx + ∫ Γ(eij eij )dx} {μ1 ∫ + 2 dt dt 𝜕xk 𝜕xk

≤ ≤

C(Ω)‖vN ‖4H 2 (Ω)

Ω

Ω

+

‖f ‖2∞

𝜕eij (vN ) 𝜕eij (vN ) C(Ω) N 2 ‖v ‖H 2 (Ω) {μ1 ∫ dx} + ‖f ‖2∞ . μ1 𝜕xk 𝜕xk

(3.2.11)

Ω

Let y(t) = μ1 ∫ Ω

𝜕eij (vN ) 𝜕eij (vN ) 𝜕xk

𝜕xk

dx + ∫ Γ(eij eij )dx.

(3.2.12)

Ω

By (3.2.11), we have dy ≤ a(t)y(t) + b(t) dt C(Ω) N 2 a(t) = ‖v ‖H 2 (Ω) , b(t) = ‖f ‖2∞ . μ1

(3.2.13) (3.2.14)

Integrating (3.2.6) from t to t + r, we get t+r

1 N 1 ‖v (t + r)‖2 − ‖vN (t)‖ + 2μ1 K0 ∫ ‖vN ‖2H 2 (Ω) ds 2 2 t+r

≤ ∫ ‖f ‖∞ ‖vN ‖ds, t

t

(3.2.15)

3.2 Global attractors of incompressible non-Newtonian fluids on unbounded domain

| 111

therefore, t+r

t+r

1 2μ1 K0 ∫ ‖vN ‖2H 2 (Ω) ds ≤ ∫ ‖f ‖∞ ‖vN ‖ds + ‖vN (t)‖2 2 t

(3.2.16)

t

by (3.2.9) and (3.2.16), ∀t ≥ t0 (‖v0N ‖), t+r

t+r

t

t

C(Ω) ∫ ‖vN ‖2H 2 (Ω) ds ∫ a(s)ds = μ1 t+r

C(Ω) ∫ 2μ1 K0 ‖vN ‖2H 2 (Ω) ds 2μ21 K(Ω)

(3.2.17)

C(Ω) 1 (‖f ‖∞ ρr + ρ2 ) = K1 (r) 2 2μ21 K0 (Ω)

(3.2.18)

=

t

or t+r

∫ a(s)ds = t

t+r

t+r

∫ b(s)ds = ∫ ‖f ‖∞ ds = r‖f ‖∞ ≡ K2 (r) t

t

t+r

t+r

∫ y(s)ds = μ1 ∫ ∫ t

𝜕eij (vN ) 𝜕eij (vN ) 𝜕xk

t Ω

𝜕xk

(3.2.19)

dxdτ

t+r

+ ∫ ∫ Γ(eij (vN )eij (vN ))dxdτ ≤ K3 (r).

(3.2.20)

t Ω

Integrating (3.2.6) and (3.2.9) from t to t + r, we have t+r

𝜕eij (vN ) 𝜕eij (vN ) 1 N ‖v (t + r)‖2 + 2μ1 ∫ ∫ dxdτ 2 𝜕xk 𝜕xk t Ω

t+r

1 ≤ ‖f ‖∞ ∫ ‖v(s)‖ds + ‖vN (t)‖2 2 t

1 ≤ ‖f ‖∞ ρr + ρ2 2

(3.2.21)

so t+r

μ1 ∫ ∫ t Ω

𝜕eij (vN ) 𝜕eij (vN ) 𝜕xk

𝜕xk

1 1 dxdτ ≤ ‖f ‖∞ ρr + ρ2 . 2 4

(3.2.22)

112 | 3 Global attractors of incompressible non-Newtonian fluids α

α

For 0 ≤ α < 1, (ε + s)− 2 ≤ ε− 2 , ∀s ≥ 0, eij eij

α

α

Γ(eij eij ) = ∫ μ0 (ε + s)− 2 ds ≤ μ0 ε− 2 eij eij ,

(3.2.23)

0

therefore, ∀t ≥ t0 (‖v0 ‖) t+r

∫ ∫ Γ(eij eij )dxds ≤ t Ω

≤

≤

μ0

t+r α

ε + s2

∫ (∫ eij eij dx)ds t

μ0 C(Ω) ε

α 2

t+r

Ω

∫ ‖vN ‖2H 2 (Ω) ds t

μ1 μ0 C(Ω) α

ε2

t+r

∫ a(s)ds t

≤ μ1 μ0 C(Ω)ε

− α2

⋅ K1

(3.2.24)

combining (3.2.20), (3.2.22) and (3.2.24), ∀t ≥ t0 (‖nu0 ‖) α 1 1 K3 (r) = ‖f ‖∞ ρr + ρ2 + μ1 μ0 C(Ω)ε− 2 K1 . 2 4

(3.2.25)

By the Gronwall lemma, y(t + r) ≤ (

K3 + K2 ) exp(K1 ), r

∀t ≥ t0 (‖v0 ‖)

that is, y(t) ≤ (

K3 + K2 ) exp(K1 ), r

∀t ≥ t0 (‖v0 ‖) + r.

(3.2.26)

Because K1 , K2 , K3 are independent of N, there exists a constant K(r) independent of N satisfying ‖v(t)‖H 2 (Ω) ≤ K(r),

∀t ≥ t0 (‖v0 ‖) + r.

(3.2.27) ρ

The above inequality shows v0 ∈ B0 , ‖v0 ‖ ≤ R0 , S(t)v0 ∈ BH12 t ≥ t0 (R0 ) + r. So Theorem 3.2.1 is proved. In order to obtain the existence of attractor, we will verify (3.2.25). Especially, taking h(x1 , t), φ(x1 ) = ln(2 + x12 )

3.2 Global attractors of incompressible non-Newtonian fluids on unbounded domain

| 113

and h(x1 , t) = φ(x1 )[1 − exp(

−(t − t0 ) )] φ(x1 )

(3.2.28)

where t0 is independent of N, and (1) and (3) of Lemma 3.2.1 hold for the taking of h. Next, we prove (2), 2x1 , 2 + x12

φ (x1 ) =

= exp(−

t − t0 ) ≤ 1, φ(x1 )

φ (x1 ) =

4 − 2x12 (2 + x12 )2

(3.2.29)

−(t − t0 ) 𝜕h 1 = φ(x1 )[− exp( )](− ) 𝜕t φ(x1 ) φ(x1 ) ∀t ≥ t0 , −∞ < x1 < +∞

(3.2.30)

−(t − t0 ) −(t − t0 ) (t − t ) 𝜕h = φ (x1 )[1 − exp( ) + 2 0 ⋅ exp( )] 𝜕x1 φ(x1 ) φ(x1 ) φ (x1 )

(3.2.31)

(t − t ) −(t − t0 ) −(t − t0 ) 𝜕2 h ) + 2 0 ⋅ exp( )] = φ (x1 )[1 − exp( φ(x1 ) φ(x1 ) φ (x1 ) 𝜕x12 +

(t − t0 )2 [φ ]2 −(t − t0 ) ). exp( 3 φ(x1 ) φ (x1 )

(3.2.32)

From (3.2.30)–(3.2.32), there exists a constant C > 0 such that 𝜕h , 𝜕t

𝜕h , 𝜕x1

𝜕2 h 2 ≤ C, 𝜕x 1

∀t ≥ t0 , −∞ < x1 < +∞.

(3.2.33)

Lemma 3.2.2 (Poincaré inequality). Assume f ∈ C 2 (Ω), h(x1 ) ≥ 0, then there exists a constant C2 such that 𝜕2 f 2 2 |f (x , x )| h(x )x x ≤ C (x , x ) ∫ h(x1 )dx1 dx2 . 1 2 1 1 2 2 ∫ 2 1 2 𝜕x 2 Ω

Ω

Proof. f (x1 , x2 ) =

y ∫−a (𝜕f /𝜕y)(x1 , η)dη,

then

a y 𝜕f 2 2 𝜕2 f |f (x1 , x2 )| ≤ ∫ (x1 , η)dη ≤ 2a ∫ 2 (x1 , η) dη 𝜕y 𝜕x 2 −a −a 2

a 2 𝜕2 f 𝜕2 f ∫ 2 (x1 , η) dη = − ∫ f (x1 , η) 2 (x1 , η)dη 𝜕x 𝜕x2 2 −a −a a

(3.2.34)

a

1 2

a

(3.2.35)

1

2 𝜕2 f 2 ≤ ( ∫ |f (x1 , η)|2 dη) ( ∫ 2 dη) 𝜕x 1 −a −a

a 2 1 𝜕2 f ε ≤ ∫ |f (x1 , η)|2 dη + 2 dη, 2 2ε 𝜕x2 −a

∀ε > 0.

(3.2.36)

114 | 3 Global attractors of incompressible non-Newtonian fluids By (3.2.35)–(3.2.36), we have a

𝜕f 2 (x1 , η) dη]h(x1 )dx1 dx2 ∫ |f (x1 , x2 )| h(x1 )dx1 dx2 ≤ 2a ∫[ ∫ 𝜕x2 −a 2

Ω

Ω

a

a 𝜕2 f 2 a ∫[ ∫ 2 (x1 , η) dη]h(x1 )dx1 dx2 𝜕x ε 2

≤ aε ∫[ ∫ |f (x1 , η)|2 dη]h(x1 )dx1 dx2 + Ω −a

Ω −a

= 2a2 ε ∫ |f (x1 , x2 )|2 h(x1 )dx1 dx2 +

2 2a2 𝜕2 f ∫ 2 (x1 , x2 ) h(x1 )dx1 dx2 . ε 𝜕x2

(3.2.37)

Ω

Ω

In equation (3.2.37), let ε =

1 . 4a2

We get

𝜕2 f ∫ |f (x1 , x2 )|2 h(x1 )dx1 dx2 ≤ C2 ∫ 2 (x1 , x2 )h(x1 )dx1 dx2 𝜕x 2 Ω

Ω

where C2 = 16a4 . Lemma 3.2.2 is proved. Lemma 3.2.3. Assume ∀v ∈ V h(x1 ) ≥ 0, then there exists a constant C3 ≥ 0, ∫ |v(x1 , x2 )|2 h(x1 )dx1 dx2 ≤ C3 ∫ Ω

Proof.

𝜕e11 𝜕x1 𝜕v1 By 𝜕x 1

So

= +

𝜕eij (v) 𝜕eij (v)

Ω

2 𝜕2 v1 𝜕e22 , = 𝜕𝜕xv22 𝜕x12 𝜕x1 2 𝜕v2 𝜕e12 = 0, have 𝜕x2 𝜕x1

𝜕eij (v) 𝜕eij (v) 𝜕xk

𝜕xk

2

= 21 ( 𝜕𝜕xv21 − 2

𝜕xk

𝜕xk

h(x1 )dx1 dx2 .

(3.2.38)

𝜕2 v1 ). 𝜕x12

𝜕e 2 𝜕x 2 𝜕e 2 ≥ 11 + 12 + 22 𝜕x1 𝜕x2 𝜕x1 2 𝜕2 v 2 1 𝜕2 v 𝜕2 v1 𝜕2 v2 ≥ 21 + 21 − + 𝜕x 4 𝜕x 𝜕x 2 𝜕x2 1

2

2 3 𝜕2 v 𝜕2 v ≥ 22 + 22 16 𝜕x1 𝜕x1 3 𝜕2 v 𝜕2 v ≥ ( 22 + 22 ). 16 𝜕x1 𝜕x1

1

1

(3.2.39)

By (3.2.39) and Lemma 3.2.2, we can get the proof of Lemma 3.2.3. Considering the approximate solution vN and ∇ ⋅ vN = 0, there exists a function φ(x1 , x2 , t), vN = (

𝜕φ 𝜕φ ,− ) 𝜕x2 𝜕x1

(3.2.40)

3.2 Global attractors of incompressible non-Newtonian fluids on unbounded domain

| 115

and only φ satisfies the boundary problem 𝜕vN

△φ = − 𝜕x2 +

{

1

φ = 0,

𝜕v1N 𝜕x2

x ∈ 𝜕ΩN

(3.2.41)

.

ρ

By (3.2.27), ∀v0N ∈ BH12 (Ω ) there exists t0 ≥ 0 independent of N, such that N

‖vN ‖H 2 (ΩN ) ≤ ρ1 , 𝜕vN

where − 𝜕x2 + 1

𝜕v1N 𝜕x2

∀t ≥ t0 , N = 1, 2, . . .

(3.2.42)

∈ H 1 (ΩN ). So, there exists a constant C4 > 0, ‖φ‖H 2 (ΩN ) ≤ C4 ,

Let w = (w1 , w2 ) = (

∀t ≥ t0 .

(3.2.43)

𝜕(φh) 𝜕(φh) ,− ) 𝜕x2 𝜕x1

(3.2.44)

and w ∈ VN , taking inner product of w with (3.2.3), (

𝜕eij (v) 𝜕eij (w) 𝜕viN , wi ) = 2μ1 ( , ) 𝜕t 𝜕xk 𝜕xk

+ 2⟨γ(viN )eij (viN ), eij (w)⟩ + B(vN , vN , w) = (f N , w).

By (3.2.40) and

𝜕h 𝜕x2

(3.2.45)

= 0, we can obtain w1 = v1N h,

w2 = v1N h − φ

𝜕h . 𝜕x1

(3.2.46)

By (3.2.33) and (3.2.42), we have (

𝜕vN 𝜕viN 𝜕vN , wi ) = ( i , w1 ) + ( 2 , w2 ) 𝜕t 𝜕t 𝜕t

𝜕viN 𝜕vN 𝜕h , vi h) − ( 2 , φ ) 𝜕t 𝜕t 𝜕x1 1 d = ∫ |vN |2 hdx1 dx2 2 dt

=(

ΩN

−

N

𝜕v 𝜕h 1 𝜕h ) ∫ |vN |2 dx1 dx2 − ( 2 , φ 2 𝜕t 𝜕t 𝜕x1 ΩN

≥

dvN 1 d N2 |v | hdx1 dx2 − K1 − K2 . dt L2 (ΩN ) 2 dt

(3.2.47)

However, 𝜕eij (w) 𝜕xk

=h

𝜕eij (vN ) 𝜕xk

+ Rijk

(3.2.48)

116 | 3 Global attractors of incompressible non-Newtonian fluids where Rijk is the sum term of the derivative product containing h and φ. So there exists a constant K3 , such that ‖Rijk ‖L2 (ΩN ) ≤ K3 2μ1 (

(3.2.49)

𝜕eij (vN ) 𝜕eij (w) , ) 𝜕xk 𝜕xk

= 2μ1 (

𝜕eij (vN ) 𝜕xk

,h

𝜕eij (vN ) 𝜕xk

𝜕eij (vN )

) + 2μ1 (

𝜕eij (vN ) 𝜕xk

, Rijk (w))

𝜕eij (vN ) ) − 2μ1 ‖R (w)‖. 𝜕xk ijk

(3.2.50)

𝜕eij (vN ) 𝜕eij (w) 𝜕eij (vN ) 𝜕eij (vN ) , ) ≥ 2μ1 ( ,h ) − K4 . 𝜕xk 𝜕xk 𝜕xk 𝜕xk

(3.2.51)

≥ 2μ1 (

𝜕eij (vN ) 𝜕xk

,h

𝜕xk

So there exists a constant K4 > 0, such that 2μ1 ( Similarly, 2⟨γ(vN )eij (vN ), eij (w)⟩ = 2 ∫ (viN )eij (vN )eij (w)dx1 dx2 ΩN

≥ 2 ∫ γ(vN )eij (vN )eij (vN )hdx1 dx2 − K5 .

(3.2.52)

ΩN

Noting ∇ ⋅ v = 0 b(vN , vN , w) = ∫ vjN ΩN

=

𝜕vN 𝜕viN N 𝜕h vi hdx1 dx2 − ∫ vjN i viN φ dx dx 𝜕xj 𝜕xj 𝜕x1 1 2

1 ∫ vjN h 2 ΩN

=−

1 ∫ 2 ΩN

𝜕(viN viN ) 𝜕xj

𝜕vjN viN viN 𝜕xj

ΩN

dx1 dx2 − ∫ vjN ΩN

hdx1 dx2 −

𝜕vN 𝜕h − ∫ vjN 2 φ dx dx 𝜕xj 𝜕x1 1 2

𝜕viN N 𝜕h v φ dx dx 𝜕xj i 𝜕x1 1 2

1 𝜕h dx dx ∫ viN viN vjN 2 𝜕xj 1 2 ΩN

ΩN

=−

𝜕vN 𝜕h 1 𝜕h dx1 dx2 − ∫ vjN 2 φ dx dx ∫ viN viN vjN 2 𝜕xj 𝜕xj 𝜕x1 1 2 ΩN

ΩN

(3.2.53)

3.2 Global attractors of incompressible non-Newtonian fluids on unbounded domain

| 117

so N N b(v , v , w) ≤

1 N N N 𝜕h dx1 dx2 ∫ vi vi vj 2 𝜕xj ΩN

𝜕vN 𝜕h + ∫ vjN 2 φ dx1 dx2 ≤ K6 . 𝜕xj 𝜕x1

(3.2.54)

ΩN

For ∀η > 0, 𝜕h dx1 dx2 (f N , w) ≤ ∫ fiN viN hdx1 dx2 + ∫ f2N φ 𝜕x1 ΩN

ΩN

2

≤ ( ∫ |fN | h(x1 , t)dx1 dx2 )

1 2

ΩN 1 2

2

+ ( ∫ |vN | h(x1 , t)dx1 dx2 ) + K7 ΩN 1

2 η ≤ ( ∫ |fN |2 hdx1 dx2 ) 2

ΩN

1

2 1 + ( ∫ |vN |2 hdx1 dx2 ) + K7 . 2η

(3.2.55)

ΩN

Therefore, combining the above estimates, 𝜕eij (vN ) 𝜕eij (vN ) 1 d ( ∫ |vN |2 hdx1 dx2 + 2μ1 ∫ hdx1 dx2 2 dt 𝜕xk 𝜕xk ΩN

ΩN

+ 2 ∫ γ(vN )eij (vN )eij (vN )hdx1 dx2 ) ΩN

dvN η 2 ≤ K + K8 + ∫ |fN | hdx1 dx2 2 dt ΩN

1 + ∫ |vN |2 hdx1 dx2 . 2η ΩN

By Lemma 3.2.3, let η = 2μ1 C3 , getting rid of term ∫Ω γ(vN )eij (vN )eij (vN )hdx1 dx2 , N

dvN 2 d ∫ |vN |2 hdx1 dx2 + 2μ1 C3 ∫ |vN |2 hdx1 dx2 ≤ 2K + K9 . dt dt ΩN

ΩN

(3.2.56)

118 | 3 Global attractors of incompressible non-Newtonian fluids For ∀t ≥ t0 dvN 2 d ∫ |vN |2 hdx1 dx2 + β1 ∫ |vN |2 hdx1 dx2 ≤ β2 + β3 . dt dt ΩN

(3.2.57)

ΩN

Multiplying (3.2.57) by eβ1 t and integrating it from t0 to s, ∀s ≥ t0 , eβ1 s ∫ |vN |2 hdx1 dx2 − eβ1 t0 ∫ |vN |2 hdx1 dx2 ΩN

ΩN

s

s

N 2

dv ≤ β2 ∫ eβ1 t dt t0

βt dt + β3 ∫ e 1 dt.

(3.2.58)

t0

For h(x1 , t0 ) = 0, from (3.2.58) we can obtain ∫ |vN |2 h(x1 , s)dx1 dx2 ΩN

s dvN 2 β βs βt −β s ≤ β2 e−β1 s ∫ eβ1 t dt + 3 (e 1 − e 1 0 )e 1 dt β1

(3.2.59)

t0

there exists β4 > 0, such that β3 β1 s (e − eβ1 t0 )e−β1 s ≤ β4 , β1

∀s ≥ t0 .

(3.2.60)

And by the estimate of (3.1.18), ∀t ≥ 0, there exists β5 > 0 dvN 2 𝜕eij (vN ) 𝜕eij (vN ) d hdx1 dx2 + {2μ1 ∫ 𝜕xk 𝜕xk dt L2 (ΩN ) dt ΩN

+ 2 ∫ Γ(eij eij )dx1 dx2 } ≤ β5

(3.2.61)

ΩN

therefore, dvN 2 d + z(t) ≤ β6 , ∀t ≥ t0 dt dt 𝜕eij (vN ) 𝜕eij (vN ) z(t) = 2μ1 ∫ hdx1 dx2 + 2 ∫ Γ(eij eij )dx1 dx2 . 𝜕xk 𝜕xk ΩN

(3.2.62) (3.2.63)

ΩN

Multiplying eβ1 t in both sides of (3.2.62) and adding β1 eβ1 t z(t), dvN d βt βt βt βt eβ1 t + [e 1 z(t)] ≤ β1 e 1 z(t) + β5 e 1 ≤ β6 e 1 . dt dt

(3.2.64)

3.3 Exponential attractors of incompressible non-Newtonian fluids | 119

Integrating (3.2.64) from t0 to s, and then multiplying e−β1 s on both sides, e

−β1 t

s

dvN 2 ∫ eβ1 t dt + z(t) dt

t0

t

≤ e−β1 (s−t0 ) z(t0 ) + β6 e−β1 s dt ∫ eβ1 t dt β6 β6 −β1 (s−t0 ) − e β1 β1 β ≤ z(t0 ) + 6 ≡ β7 . β1

t0

≤ z(t0 ) +

Therefore, by (3.2.59), there exists β8 > 0, such that ∫ |vN |2 h(x1 , s)dx1 dx2 ≤ β8 ,

∀s ≥ t0

ΩN

where β8 is independent of N, ∫ |v|2 h(x1 , s)dx1 dx2 ≤ β8 , ∀s ≥ t0 ΩN

that is, equation (3.2.5) of Lemma 3.2.1 holds. And by Lemma 3.2.1, we can get the following conclusion. There exists the compact global attractor Aμ1 for problem (3.2.1) of Theorem 3.2.1, and attracts all bounded sets of H.

3.3 Exponential attractors of incompressible non-Newtonian fluids The authors in [24] have studied the exponential attractors for dissipative evolution equations. Now, we consider the following incompressible bipolar fluid equation with p > 2, and all assumptions and marks are the same with in Section 3.1: ut + 2μ1 Au − 2μ0 Ap (u) + B(u, u) = f u(0) = u0

(3.3.1) (3.3.2)

Aϕj = λj ϕj , ϕj ∈ D(A). The eigenvalues λj of A satisfying λj =

8π 4 , L4

λN ∼ (

J = (j1 , j2 ), n = 2, 3 1 n

8π 4 ) , L4

N ∈ Z + , n = 2, 3.

(3.3.3) (3.3.4)

120 | 3 Global attractors of incompressible non-Newtonian fluids For the case 1 < p ≤ 2, it is easier than p > 2. In this section, we consider the case p > 2 all of the time. Definition 3.3.1. Assume u, v ∈ B, P is orthogonal projection of the rank N0 , then there exists t∗ > 0 for the continuous semigroup operator S(t), such that S∗ = S(t∗ ), and if ‖P(S∗ u − S∗ v)‖ ≤ ‖(I − P)(S∗ u − S∗ v)‖, then ‖(S∗ u − S∗ v)‖ ≤ 81 ‖u − v‖. We call the continuous semigroup {S(t), t ≥ 0} satisfying compressibility in B. Definition 3.3.2. A set M can be called the exponential attractor for ({S(t), t ≥ 0}, B), if the following conditions are satisfied: (1) Aμ ⊆ M ⊆ B; (2) S(t)M ⊂ M, ∀t ≥ 0; (3) u0 ∈ B, distH (S(t)u0 , M) ≤ K1 exp(−K2 t), ∀t ≥ 0, K1 > 0, K2 > 0, are not independent of u0 ; (4) M has the finite fractal dimension. Lemma 3.3.1. Assume ({S(t), t ≥ 0}, B) satisfying compressibility in B, S∗ = S(t∗ ), and it is Lipschitz continuous in B with the Lipschitz constant LB , then there exists the exponential attractor M for ({S(t), t ≥ 0}, B) and dF (M) ≤ N0 max(1,

ln(16LB + 1) ). ln 2

3.3.1 Estimates for the nonlinear terms Lemma 3.3.2. Assume Ω ⊂ Rn , n = 2, 3, u, v, w ∈ H 1 (Ω), then there exists a constant C1 > 0, independent of u, v, w, such that 1 2

(∫ u2 v2 w2 dx) ≤ C1 ‖u‖1 ‖v‖1 ‖w‖1 .

(3.3.5)

Ω

Proof. By the Sobolev embedding theorem, we have ‖u‖L6 (Ω) ≤ C0 ‖u‖1 and applying the Hölder inequality, we get 1 2

1 6

(∫ u2 v2 w2 dx) ≤ (∫ |u|dx) (∫ |v|dx) Ω

1 6

Ω

Ω

(∫ |w|dx) ≤ C1 ‖u‖1 ‖v‖1 ‖w‖1 Ω

where C1 = C03 . So Lemma 3.3.2 is proved.

1 6

(3.3.6)

3.3 Exponential attractors of incompressible non-Newtonian fluids | 121

Lemma 3.3.3. There exist constants C2 > 0, C3 > 0, C4 > 0, such that the two solutions u, v for problem (3.3.1) satisfying ‖Ap (u) − Ap (v)‖ ≤ (C2 + C3 ‖u‖p−2 4 + C4 (‖u‖2 + ‖v‖2 )‖v‖3 )‖u − v‖2

(3.3.7)

where C2 = (p − 1)2p−2 ε

p−2 2

K0 n2 |Ω|

C3 = (p − 1)2p−2 K0 C0 n2 (p − 1)(7 − p) p−2 ε 2 K0 C1 n2 . C4 = 2

Proof. Fix i, j,

p−2 𝜕 [(ε + |e(u)|2 ) 2 eij (u)] 𝜕xj

𝜕 2 [(ε + ∑ ek,l (u)) = 𝜕xj k,l = (ε +

2 (u)) ∑ ek,l k,l

+ (p − 2)(ε +

p−2 2

⋅

p−2 2

eij (u)]

𝜕eij (u) 𝜕xj

2 (u)) ∑ ek,l k,l

p−2 2

eij (u)(∑ ek,l

𝜕xkl (u) ). 𝜕xj

(3.3.8)

eij (v)(∑ ek,l

𝜕xkl (v) ). 𝜕xj

(3.3.9)

k,l

Similarly, we have p−2 𝜕 [(ε + |e(v)|2 ) 2 eij (v)] 𝜕xj

= (ε +

2 (v)) ∑ ek,l k,l

+ (p − 2)(ε + Denote

p−2 2

⋅

𝜕eij (v) 𝜕xj

2 (v)) ∑ ek,l k,l

p−2 2

k,l

𝜕 p−2 p−2 𝜕 [(ε + |e(v)|2 ) 2 eij (v)] I = [(ε + |e(u)|2 ) 2 eij (u)] − 𝜕xj 𝜕xj 2 I1 = (ε + ∑ ek,l (u)) k,l

p−2 2

2 I2 = (ε + ∑ ek,l (u)) k,l

p−4 2

2 − (ε + ∑ ek,l (v)) k,l

⋅

𝜕eij (u)

− (ε +

𝜕xj

eij (u)(∑ ek,l k,l

p−4 2

𝜕xkl (u) ) 𝜕xj

eij (v)(∑ ek,l k,l

2 (v)) ∑ ek,l k,l

𝜕xkl (v) ). 𝜕xj

p−2 2

⋅

𝜕eij (v) 𝜕xj

122 | 3 Global attractors of incompressible non-Newtonian fluids Then I ≤ I1 + (p − 2)I2 .

(3.3.10)

For the convenience of the following estimates, let p ≤ 4 and using the mean value theorem, and suppose 2 2 (u) < S∗ < ∑ ek,l (v) ∑ ek,l k,l

k,l

(3.3.11)

p−2 2

𝜕eij (u − v) 𝜕xj 𝜕x (v) p−4 p−2 2 2 (ε + S∗ ) 2 ∑ ekl (u) − ∑ ek,l (v) kl + 𝜕xj 2 k,l k,l

I1 ≤ (ε +

2 (u)) ∑ ek,l k,l

≤ (ε +

2 (u)) ∑ ek,l k,l

+

p−2 2

𝜕eij (u − v) 𝜕xj

𝜕eij (v) p − 2 p−4 ⋅ ε 2 (∑ |ek,l (u + v)||ek,l (u − v)|) 𝜕xj 2 k,l

(3.3.12)

p−4 𝜕e (u − v) I2 ≤ ∑ (ε + |e(u)|2 ) 2 eij (u)ekl (u) kl 𝜕xj k,l + ∑ (ε + |e(u)|2 ) k,l

− (ε + |e(u)|2 )

p−4 2

p−4 2

eij (u)ekl (u)

𝜕e (v) eij (v)ekl (v) ⋅ kl . 𝜕xj

(3.3.13)

Fix i, j, k, l, 1 2 |eij (u)ekl (u)| ≤ (eij2 (u)ekl (u)) ≤ |e(u)|2 . 2

(3.3.14)

Let us assume |e(u)|2 ≤ |e(v)|2 . By the mean value theorem, there exists s∗∗ , |e(u)|2 < s∗∗ < |e(v)|2 ( |e(u)|2 > |e(v)|2 similarly holds) p−4 p−4 2 2 (ε + |e(u)| ) 2 − (ε + |e(v)| ) 2 p−6 p−4 (ε + s∗∗ ) 2 ||e(u)|2 − |e(v)|2 | = 2 p−6 p−4 ||e(u)|2 − |e(v)|2 |(ε + |e(u)|2 ) 2 ≤ 2 p−4 p−4 (ε + |e(u)|2 ) 2 eij (u)ekl (u) − (ε + |e(v)|2 ) 2 eij (v)ekl (v) p−4 p−4 2 2 2 2 ≤ [(ε + |e(u)| ) − (ε + |e(v)| ) ]eij (u)ekl (u)

+ (ε + |e(u)|2 )

p−4 2

|eij (u)ekl (u) − eij (v)ekl (v)|

(3.3.15)

3.3 Exponential attractors of incompressible non-Newtonian fluids | 123

≤

p − 4 p−4 ε 2 ⋅ |e(u)|2 − |e(v)|2 2 p−4 2

⋅ (|eij (u)| ⋅ |ekl (u − v)| + |eij (u − v)| ⋅ |ekl (v)|) 𝜕e (u − v) p−2 I2 ≤ ∑(ε + |e(u)|2 ) 2 ⋅ kl 𝜕x j k,l +ε

+

𝜕e (v) p − 4 p−4 ⋅ ε 2 ∑ ∑ |emn (u + v)| ⋅ |emn (u − v)| kl 𝜕xj 2 k,l m,n

×ε

p−4 2

∑(|eij (u)| ⋅ |ekl (u − v)| k,l

𝜕e (v) + |eij (u − v)| ⋅ |ekl (v)|) kl . 𝜕xj

(3.3.16)

So combining the above estimates, we get I ≤ (ε + |e(u)|2 ) +

p−2 2

𝜕eij (u − v) 𝜕xj

𝜕e (v) p − 2 p−4 ε 2 ∑ |ekl (u + v)| ⋅ |ekl (u − v)| kl 𝜕xj 2 k,l

+ (p − 2){∑(ε + |e(u)|2 ) k,l

+

p−4 ε 2

+ε

p−4 2

p−4 2

p−2 2

𝜕e (u − v) kl 𝜕xj

𝜕e (v) ∑ ∑ |emn (u + v)||emn (u − v)| kl 𝜕x j k,l m,n

∑(|eij (u)| ⋅ |ekl (u − v)|) k,l

𝜕e (v) + |eij (u − v)| ⋅ |ekl (v)| kl } 𝜕xj 2

‖I‖ = [∫(ε + |e(u)| )

p−2 2

Ω

(3.3.17) 1

𝜕eij (u − v) 2 dx] 𝜕xj 1

𝜕eij (v) 2 p − 2 p−4 + ε 2 ∑[∫ |ekl (u + v)|2 ⋅ |ekl (u − v)|2 ⋅ dx] 𝜕xj 2 k,l Ω

1

2 2 − v) dx] 𝜕xj

2 p−2 𝜕ekl (u

+ (p − 2){∑[∫(ε + |e(u)| ) k,l Ω

+

1

𝜕e (v) 2 2 p − 4 p−4 ε 2 ∑ ∑ [∫ |emn (u + v)|2 × |emn (u − v)|2 × kl dx] 𝜕xj 2 k,l m,n Ω

+ε

p−4 2

2

2 2 𝜕ekl (v)

∑[∫ |eij (u)| |ekl (u − v)| dx] 𝜕xj k,l Ω

1 2

124 | 3 Global attractors of incompressible non-Newtonian fluids

+ε

p−4 2

∑[∫ |eij (u − v)| |ekl (v)| 𝜕xj k,l Ω

Ω

≤ C(Ω)(ε ≤ C(Ω)(ε

p−2 2 p−2 2

(3.3.18)

1

2 2 − v) dx] 𝜕xj

2 p−2 𝜕ekl (u

α[∫(ε + |e(u)| )

1

2 2 dx] }

2 𝜕ekl (v)

2

+ ‖e(u)‖p−2 ∞ )‖eij (u − v)‖1

+ ‖e(u)‖p−2 3 )‖eij (u − v)‖2 .

(3.3.19)

Applying Lemma 3.3.2, we have ‖I‖ ≤ 2p−2 (ε + C1

p−2 2

|Ω| + ‖e(u)‖p−2 ∞ )‖eij (u − v)‖1

𝜕eij (v) p − 2 p−4 ε 2 ∑ ‖ekl (u + v)‖1 × ‖ekl (u − v)‖1 𝜕xj 1 2 k,l

+ C1 (p − 2){∑[2p−2 (ε k,l

p−2 2

|Ω| + ‖e(u)‖p−2 ∞ )‖ekl (u + v)‖1 ]

𝜕e (v) ∑ ∑ [‖emn (u + v)‖1 ‖emn (u − v)‖1 kl ] 𝜕x 1 j k,l m,n p−4 𝜕e (v) + ε 2 ∑ ‖eij (u)‖1 ‖ekl (v)‖1 kl 𝜕xj 1 k,l 𝜕e (v) + ‖eij (u − v)‖1 ‖ekl (v)‖1 kl }. 𝜕xj 1

+

4−p ε 2

p−4 2

(3.3.20)

Applying the Korn inequality and the Gagliardo–Nirenberg inequality, we obtain ‖I‖ ≤ {(p − 1)K0 2p−2 ε + K0 C1

p−2 2

|Ω| + (p − 1)K0 C0 2p−2 ‖u‖p−2 3

(p − 2)(7 − p) p−4 ε 2 ‖u‖2 2

+ ‖v‖2 ‖v‖3 }‖u − v‖2 .

(3.3.21)

Summing up i, j in both sides of equation (3.3.21), we can get equation (3.3.7). So Lemma 3.3.3 is proved. Lemma 3.3.4. Let f ∈ L∞ (0, ∞; H 1 (Ω)), u0 ∈ B0 = {u ∈ V : ‖u‖ ≤ ρ0 , ‖u‖1 ≤ ρ1 , ‖u‖2 ≤ ρ2 }, then the solution u(t) of equations (3.3.1)–(3.3.2) satisfying ‖u(t)‖ ≤ ρ3 , C

where ρ3 = 2( 2μ5 + 1

C8 21 ) 2μ1

exp(

C5 C9 ), 2

t ≥ 1.

t≥1

(3.3.22)

3.3 Exponential attractors of incompressible non-Newtonian fluids | 125

𝜕2 u

Proof. Fixed indicator l, multiplying equation (3.3.1) by − 𝜕x2i and summing up i, and l

then integrating in Ω,

∫ Ω

𝜕ui 𝜕u 𝜕2 u dx − ∫ uj i 2i dx 𝜕t 𝜕uj 𝜕xl Ω

𝜕2 ui 𝜕P 𝜕2 ui 𝜕 = ∫ uj (γ(u)e ) dx − dx ∫ ij 𝜕xi 𝜕xl2 𝜕xj 𝜕xl2 Ω

Ω

𝜕2 u 𝜕2 u 𝜕 (△eij ) 2i dx − ∫ fi 2i dx 𝜕xj 𝜕xl 𝜕xl

+ 2μ1 ∫

(3.3.23)

Ω

Ω

while ∫ Ω

𝜕2 u 𝜕ui 1 d 𝜕ui 𝜕ui (− 2i )dx = ( , ) 𝜕t 2 dt 𝜕xl 𝜕xl 𝜕xl

∫ uj Ω

∫ Ω

∫ Ω

𝜕P 𝜕2 ui dx = 0 𝜕xi 𝜕xl2

𝜕eij 𝜕2 u 𝜕 𝜕2 ui 𝜕 𝜕eij 𝜕 (△eij ) 2 dx = ∫ dx = − ∫ ( )⋅ dx 2 𝜕xj 𝜕xj 𝜕xl 𝜕xk 𝜕xk 𝜕xk 𝜕xl 2

Ω

Ω

p−2 𝜕u 𝜕2 u 𝜕 𝜕 (γ(u)eij ) 2i dx = 2μ0 ∫ [(ε + |e(u)|2 ) 2 eij ] 2i dx 𝜕xj 𝜕xj 𝜕xl 𝜕xl

Ω

= 2μ0 ∫(ε + |e(u)|2 )

p−2 2

Ω

p−2 𝜕eij 𝜕eij 𝜕eij 𝜕 ⋅ dx + 2μ0 ∫ [(ε + |e(u)|2 ) 2 ]eij dx 𝜕xl 𝜕xl 𝜕xl 𝜕xl

Ω

≥ 2μ0 (p − 2) ∫(ε + |e(u)|2 )

p−4 2

2 𝜕eij 𝜕eij

|e(u)|

Ω

𝜕xl 𝜕xl

dx ≥ 0

𝜕u 𝜕2 u 2 2 ∫ uj i 2i dx ≤ ‖u‖∞ ‖u‖1 ‖u‖2 ≤ C0 ‖u‖1 ‖u‖2 ≤ C0 ρ1 ρ2 . 𝜕xj 𝜕xl Ω

Therefore, by (3.3.23) we have 1 d 𝜕ui 𝜕ui 𝜕 𝜕eij 𝜕 𝜕eij ( , ) + 2μ1 ∫ ( )⋅ ( )dx ≤ (C0 ρ1 ρ22 + ‖f ‖ρ2 ). (3.3.24) 2 dt 𝜕xl 𝜕xl 𝜕xk 𝜕xk 𝜕xk 𝜕xk Ω

For ‖u‖23 is equivalent to ∫Ω 𝜕x𝜕 ( 𝜕xij ) ⋅ 𝜕x𝜕 ( 𝜕xij )dx, sum up the indicators i, j, k, l of equak k k k tion (3.3.24) and integrate in [t, t + 1], we have 𝜕e

𝜕e

1 1 ‖u(t + 1)‖21 − ‖u(1)‖1 + 2μ1 ‖u‖23 ≤ (C0 ρ1 ρ22 + ‖f ‖ρ2 ) 2 2

126 | 3 Global attractors of incompressible non-Newtonian fluids then t+1

∫ ‖u‖23 ds ≤ C5 ,

t≥0

(3.3.25)

t

where C5 =

1 1 2 ( ρ + C0 ρ1 ρ22 + ρ2 ‖f ‖L∞ (0,T;H) ). 2μ1 2 1 𝜕3 u

On the other hand, for the fixed indicator l, multiplying equation (3.3.1) by − 𝜕x2 𝜕ti , and l

integrating in Ω, (

𝜕ui 𝜕3 ui 𝜕3 ui 𝜕3 ui 𝜕3 ui 𝜕 , ) − u dx − 2μ (△e ) dx ∫ ∫ j 1 ij 𝜕xj 𝜕xl2 𝜕t 𝜕xj 𝜕xl2 𝜕t 𝜕xl2 𝜕t 𝜕xl2 𝜕t = − ∫ fi Ω

3

Ω

3

Ω

𝜕 ui 𝜕 u 𝜕 (γ(u)eij ) 2 i dx dx − ∫ 𝜕xj 𝜕xl2 𝜕t 𝜕xl 𝜕t

(3.3.26)

Ω

but −∫ Ω

𝜕3 u 𝜕3 u 𝜕 𝜕2 𝜕 (△eij ) 2 i dx = − ∫ (∑ 2 eij ) 2 i dx 𝜕xj 𝜕xj k 𝜕xk 𝜕xl 𝜕t 𝜕xl 𝜕t Ω

= −∫ Ω

𝜕 𝜕2 𝜕 𝜕eij (∑ 2 eij ) ( )dx 𝜕xj k 𝜕xk 𝜕xl 𝜕t

(3.3.27)

𝜕 𝜕eij 𝜕 𝜕 𝜕eij =∫ ( ) ( )dx 𝜕xk 𝜕xl 𝜕t 𝜕xk 𝜕xl Ω

=

2 𝜕2 eij 1 d 𝜕 eij ( , ) 2 dt 𝜕xk 𝜕xl 𝜕xk 𝜕xl

𝜕uj 𝜕ui 𝜕2 ui 𝜕u 𝜕3 u 𝜕2 ui + ) dx ∫ uj i 2 i dx = ∫(uj 𝜕xj 𝜕xl 𝜕t 𝜕xj 𝜕xl 𝜕xl 𝜕xj 𝜕xl2 Ω

Ω

1

𝜕2 ui 𝜕2 ui 2 , ) ≤ (‖u‖∞ + ‖u‖2 + ‖∇u‖∞ ‖u‖1 )( 𝜕xl 𝜕t 𝜕xl 𝜕t

(3.3.28)

1 𝜕2 ui 𝜕2 ui , ) ≤ 2C0 ρ22 + 2C0 ρ21 ‖u‖23 + ( 2 𝜕xl 𝜕t 𝜕xl 𝜕t

𝜕3 u 1 𝜕2 ui 𝜕2 ui 2 , ) ∫ fi ⋅ 2 i dx ≤ ‖f ‖L∞ (0,T;H 1 ) + ( 4 𝜕xl 𝜕t 𝜕xl 𝜕t 𝜕xl 𝜕t Ω

(3.3.29)

3.3 Exponential attractors of incompressible non-Newtonian fluids | 127

for 𝜕2 p−2 p−2 𝜕e p−4 ij 𝜕 [(ε + |e(u)|2 ) 2 ] = {(ε + |e(u)|2 ) 2 + (p − 2)(ε + |e(u)|2 ) 2 𝜕xl 𝜕xj 𝜕xl 𝜕xj 𝜕e ⋅ eij (u)(∑ emn mn )} 𝜕xj m,n

2 p−2 𝜕 e p−4 ij = {(ε + |e(u)|2 ) 2 + (p − 2)(ε + |e(u)|2 ) 2 𝜕xl 𝜕xj p−6 𝜕eij 𝜕e (∑ emn mn ) + (p − 2)(p − 4)(ε + |e(u)|2 ) 2 eij ⋅ 𝜕xj m,n 𝜕xj

⋅ (∑ emn m,n

𝜕eij

p−4 𝜕emn 𝜕e )(∑ emn mn ) + (p − 2)(ε + |e(u)|2 ) 2 𝜕xl 𝜕xj m,n

p−4 𝜕emn ) + (p − 2)(ε + |e(u)|2 ) 2 eij 𝜕xl m,n 𝜕xj 𝜕e 𝜕e 𝜕2 emn ⋅ (∑ mn mn + ∑ emn )} 𝜕xj 𝜕xl m,n 𝜕xj 𝜕xl m,n 𝜕eij 𝜕eij p−4 + )|emn | 𝜕emn ≤ (p − 2)ε 2 ∑ ( 𝜕xj m,n 𝜕xj 𝜕xl 𝜕e 𝜕e p−4 + (p − 2)(4 − p)ε 2 ∑ ∑ mn mn |eij | 𝜕x 𝜕x l j m,n m,n 2 p−2 𝜕 emn + (p − 2) ∑ (ε + |e(u)|2 ) 2 𝜕xj 𝜕xl m,n

⋅

(∑ emn

(3.3.30)

and by the Gragliardo–Nirenberg inequality, Korn inequality and Young inequality, we have 𝜕2 p−2 [(ε + |e(u)|2 ) 2 eij ] 𝜕xj 𝜕xl 1

𝜕2 eij 2 2 p−4 dx) + (p − 2)ε 2 𝜕xl 𝜕xj

2 p−2

≤ (∫(ε + |e(u)| ) Ω

1

2 𝜕eij 2 𝜕eij 2 2 2 𝜕e ⋅ ∑ (∫( + )|emn | mn dx) 𝜕xj 𝜕xj 𝜕xl m,n Ω

+ (p − 2)(4 − p)ε

p−4 2

1

𝜕e 2 𝜕e 2 2 ∑ (∫ mn mn |eij |2 dx) 𝜕xl 𝜕xj m,n Ω

+ (p − 2) ∑ (∫(ε + |e(u)| ) m,n

Ω

≤ (p − 1)K0 (ε + |e(u)|2 )

p−2 2

1

2 e 2 2 mn dx) 𝜕xl 𝜕xj

2 p−2 𝜕

‖u‖3 + (p − 2)(7 − p)K0 ε

p−4 2

‖u‖2 ‖u‖23

128 | 3 Global attractors of incompressible non-Newtonian fluids ≤ (p − 1)K0 ε

p−2 2

≤ C6 + C7 ‖u‖43

‖u‖3 + (p − 2)(7 − p)K0 ρ2 ‖u‖23 + (p − 1)K0 C0 ‖u‖p−1 3

(3.3.31)

where 3 |3 − p| (4(p − 1)2 K02 C02 ) |3−p| 2 2 2 2 p−4 2 C7 = p + 1 + 4(p − 2) (7 − p) K0 ε ρ2 .

C6 = 2(p − 1)4 K04 ε2(p−2) +

Therefore, 1

𝜕 p−2 2 𝜕2 ui 𝜕2 ui (γ(u)eij ) dx ≤ 2μ20 {∫ [ (ε + |e(u)|2 ) 2 ]dx} ∫ 𝜕xj 𝜕xl 𝜕t 𝜕xl 𝜕xj Ω

Ω

1 𝜕2 ui 𝜕2 ui + ( , ) 2 𝜕xl 𝜕t 𝜕xl 𝜕t

≤ 2μ20 C6 + 2μ20 C7 ‖u‖43 .

(3.3.32)

Combining (3.3.26)–(3.3.32), we have d‖u‖23 ≤ C8 + C9 ‖u‖43 dt

(3.3.33)

where C8 = C9 =

1 2 [2μ20 C6 + ‖f ‖2L∞ (0,T;H 1 ) + 4C0 (ρ21 + ρ22 ) ] μ1 2μ20 C7 + 1 . μ1

Using the uniform Gronwall lemma for (3.3.25)–(3.3.33), we obtain ‖u(t)‖23 ≤

C5 + C8 ⋅ exp(C5 ⋅ C9 ), 2μ1

t≥1

a. m. ‖u(t)‖3 ≤ ρ3 ,

t ≥ 1.

Lemma 3.3.4 is proved. 3.3.2 Compressibility on L2 (Ω) Assume B0 is defined in Lemma 3.3.4, and it is the absorbing set for (3.3.1)–(3.3.2), let B = ⋃ S(t)B0 t≥1

(3.3.34)

3.3 Exponential attractors of incompressible non-Newtonian fluids | 129

then B is compact set of B0 , and it is a constant flow, then we have u ∈ B,

then ‖u(t)‖3 ≤ ρ3 .

(3.3.35)

Suppose ϕ1 , . . . , ϕN is the first N eigenvectors of A, for (2.1.16) defining PN : H → span{ϕ1 , . . . , ϕN } and the projection operator is QN = I − PN , and suppose u(t), v(t) are two solutions of (3.3.1), (3.3.2) with the initial values u0 , v0 , let w(t) = u(t) − v(t), then dw = 2μ1 AW − 2μ0 (Ap (u) − Ap (v)) + B(u, w) + B(w, u) = 0 dt w(0) = u0 − v0 .

(3.3.36) (3.3.37)

Taking the inner product of w for (3.3.36), we obtain 𝜕u d ‖w‖2 + 4K0 μ1 ‖w‖22 − 4μ0 (Ap (u) − Ap (v), w) ≤ 2 ∫ |w|2 dx 𝜕x dt

(3.3.38)

Ω

here, by (3.1.44) we have 4μ0 (Ap (u) − Ap (v), w) ≥ 2K0 μ0 ε 2 ∫ |w|2 | Ω

p−2 2

‖w‖ ≥ 0

𝜕u 𝜕u |dx ≤ 2‖ ‖‖w‖2 ≤ 2C0 ρ3 ‖w‖2 . 𝜕x 𝜕x

Therefore, d ‖w‖2 + 4K0 μ1 ‖w‖22 ≤ 2C0 ρ3 ‖w‖2 . dt

(3.3.39)

First, eliminate the positive term 4K0 μ1 ‖w‖22 in (3.3.39), and by Gronwall inequality we obtain ‖w(t)‖2 ≤ exp(2C0 ρ3 t)‖w(0)‖2

(3.3.40)

LB = exp(C0 ρ3 t).

(3.3.41)

with the Lipschitz constant

Secondly, let λ(t) =

‖w‖2 , ‖w(t)‖2

by (3.3.39) we obtain

d ‖w‖2 + (4K0 μ1 λ(t) − 2C0 ρ3 )‖w‖2 ≤ 0. dt

(3.3.42)

Similarly, using Gronwall inequality we have ‖w(t)‖ ≤ δ(t)‖w(0)‖

(3.3.43)

where t

δ(t) = exp ( − 2K0 μ1 ∫ λ(s)ds) + C0 ρ3 t. 0

(3.3.44)

130 | 3 Global attractors of incompressible non-Newtonian fluids Theorem 3.3.1. There exist t∗ =

1 η

1

and N0 > πL ( 81 ) 4 (

6η ln 2+2C0 ρ3 η4 ) μ1 K0

− 1, if

‖Pw(t∗ )‖ ≤ ‖(1 − p)w(t∗ )‖ then ‖w(t∗ )‖ ≤ 81 ‖w(0)‖. where η=

1 2 [μ (C + C3 ρp−2 3 + 2C4 ρ2 ρ3 ) + C0 ρ2 ] . 4μ1 0 2

Proof. 2 dλ(t) = (w , Aw − w) dt ‖w‖2 2 (2μ1 Aw − 2μ0 (Ap (u) − Ap (v)), Aw − w) =− ‖w‖2 2 − (B(u, w) + B(w, u), Aw − w) ‖w‖2 where w =

(3.3.45)

dw dt

(λ(t)w, Aw − λ(t)w) = λ(t)(w, Aw) − λ2 (t)(w, w) = 0 then (Aw, Aw − λ(t)w) = ‖Aw − λ(t)‖2

(3.3.46)

and by the Sobolev embedding theorem we obtain ‖B(u, w)‖ + ‖B(w, u)‖ ≤ ‖u‖∞ ‖w‖1 + ‖w‖∞ ‖u‖1 ≤ 2C0 ρ2 ‖w‖2 .

(3.3.47)

By (3.3.7), we have ‖Ap (u) − Ap (v)‖ ≤ (C1 + C3 ρp−2 3 + 2C4 ρ2 ρ3 )‖w‖2 .

(3.3.48)

Therefore, combining the above inequality (3.3.45), we have dλ(t) ≤ ηλ(t) dt

(3.3.49)

λ(t) ≤ λ(t0 ) exp[η(t − t0 )].

(3.3.50)

then

Let t = t∗ , λ(t∗ ) = λ∗ , antisolving (3.3.50), we obtain t∗

∫ λ(t0 )dt0 ≥ 0

λ∗ [1 − exp(ηt∗ )] η

(3.3.51)

3.3 Exponential attractors of incompressible non-Newtonian fluids | 131

let t∗ = η1 , then 1 − exp(ηt∗ ) > 21 , by (3.3.44) we have t∗

δ(t∗ ) = exp(−2K0 μ1 ∫ λ(s)ds + C0 ρ3 t∗ ) 0

−K μ λ + C0 ρ3 ). ≤ exp( 0 1 ∗ η If ‖Pw(t∗ )‖ ≤ ‖(1 − P)w(t∗ )‖, then λ∗ =

‖Pw(t∗ )‖22 + ‖(1 − P)w(t∗ )‖22 ‖Pw(t∗ )‖2 + ‖(1 − P)w(t∗ )‖2

‖(1 − P)w(t∗ )‖22 2‖(1 − P)w(t∗ )‖2 1 ≥ λN0 +1 . 2

≥

For (3.3.4), we obtain 1 λ∗ ≥ λN0 +1 2

η

1

L 1 4 6η ln 2 + 2C0 ρ3 4 ) −1 N0 > ( ) ( π 8 μ1 K0

(3.3.52)

then we have δ(t∗ )

0 is density; κ > 0 is a coefficient(thermal conductivity); en is unit vector in nth direction of Rn ; f(x, t), g(x, t) are the given vector value and scalar functions, respectively. We will assume that the constitutive laws have the form τv = τ(e) − 2μ1 △e

(4.4)

where, μ1 ≥ 0, and τ(e) is given by (p−2)/2

τ(e) = 2μ0 (ε + |e(u)|) e(u) = (eij (u)),

e(u),

p > 1, ε > 0

𝜕uj 1 𝜕u eij (u) = ( i + ). 2 𝜕xj 𝜕xi

(4.5) (4.6)

If p = 2, μ1 = 0, and τ(e) satisfies (4.5), then equations (4.1)–(4.4) turns out to be Newtonian–Boussinesq equations, which can be described as the Bénard flow; one can see [17, 80, 18] for more details. Cannon and Dibenedetto in [16] considered the initial value problem for the Boussinesq equations with data in Lp . Foias, Ghidagliain et al. in [26, 30] proved the existence of global attractors and discussed the fractal dimension of the attractor for the Bénard problem. Boling Guo in [36] investigated the nonlinear Galerkin method of Newtonian–Boussinesq equations. If μ1 ≠ 0, p = 2 and θ = 0, equations (4.1)–(4.4) turn out to be the regularized Navier–Stokes equations; see [73, 74]. Maled, Ruzicka and Thater in [64] studied the asymptotic behaviors of the solution of the Boussinesq approximation where the viscosity depends polynomially on the shear rate. Bloom, Bellout et al. in [8, 10, 11] considered the existence of https://doi.org/10.1515/9783110549614-004

134 | 4 Global attractors of modified Boussinesq approximation global attractor for p > 2 and 1 < p ≤ 2 respectively. Bloom and Hao in [12, 13] studied the squeezing property and the existence of inertial manifolds for incompressible nonlinear bipolar viscous fluids. Boling Guo and Yadong Shang in [41] proved the existence and uniqueness of global solutions of modified Newtonian–Boussinesq equa2n tions, equations (4.1)–(4.4) for p > n+2 . One can refer to [42] for further results. For convenience, we introduce the function space as follows: ∞ V ≡ {ϕ ∈ Cper (Ω)n ; ∇ ⋅ ϕ = 0, ∫ ϕdx = 0}

(4.7)

H ≡ closure of V in L2 (Ω)n -norm

(4.8)

Ω

n2

q

Vq ≡ closure of V in L (Ω) -norm of gradients Vθ ≡ closure of

∞ Cper (Ω)n

2

in L (Ω)-norm of gradients

(4.9) (4.10)

where the Lq -norm of gradient of φ denoted by |φ|1,q . Hereafter, u ∈ Lp (I; Vp ) means 2

Du ∈ Lp (I; Lp (Ω)n ) and |u|Lp (I;Lp ) = ‖Du‖Lp (I;Lp (Ω)n2 ) .

Lemma 4.0.1 (Generalized Korn inequality). Suppose φ ∈ W 1,q (Ω)n ∩ W 1,2 (Ω)n , q > 1, then 1 q

q

(∫ |e(φ)| dx) ≥ kq |φ|1,q ,

kq > 0.

(4.11)

Ω

Lemma 4.0.2 (Uniform Gronwall lemma). Suppose g, h, y are positive local integrable functions on [t0 , ∞), if the following inequalities are satisfied: dy ≤ gy + h, dt

∀t ≥ t0

(4.12)

and t+r

t+r

∫ g(s)ds ≤ a1 ,

∫ h(s)ds ≤ a2

t

t+r

∫ y(s)ds ≤ a3 , t

t

∀t ≥ t0

(4.13)

where r, a1 , a2 , a3 are constants. Then y(t + r) ≤ (

a3 + a2 ) exp(a1 ), r

∀t ≥ t0

(4.14)

Consider the periodic initial value problem as follows: 𝜕ui 𝜕u 𝜕π 𝜕τij 𝜕 + uj i = − + + 2μ1 △e + en θ + fi 𝜕t 𝜕xj 𝜕xj 𝜕xj 𝜕xj ij

(4.15)

4 Global attractors of modified Boussinesq approximation

div u = 0 𝜕θ 𝜕θ + uj − κ△θ = g 𝜕t 𝜕xj

| 135

(4.16) (4.17)

u(x, t) = u(x + Lej , t),

u(x, 0) = u0 (x),

θ(x, t) = θ(x + Lej , t)

θ(x, 0) = θ0 (x)

(4.18) (4.19)

where nonlinear stress tensor functions τ(⋅) satisfy equation (4.5); fi , g ∈ L2 (I; L2 (Ω)) are given real function, u0 (x), θ0 (x), fi , g are periodic function with respect to x. Furthermore, we suppose ∫ f (x)dx = 0, ∫ g(x)dx = 0. Ω

(4.20)

Ω

2,2 Definition 4.0.1. Suppose given u0 ∈ H ∩ Wper (Ω)n , θ̃0 ∈ M = {θ̃ ∈ L2per (Ω); T0 ≤ θ̃ ≤ T1 }, f ∈ L2 (I; L2per (Ω)n ), g ∈ L2 (I; L2per (Ω)). A couple (u, θ) is called a weak solution of equations (4.15)–(4.19), if

∫ Ω

𝜕ui 𝜕u φ dx + ∫ uj i φi dx + ∫ τij (e(u))eij (φ)dx 𝜕t i 𝜕xj Ω

+ 2μ1 ∫ Ω

Ω

𝜕eij (u) 𝜕eij (φ) 𝜕xk

𝜕xk

dx = ∫ en θφdx + ∫ fi φdx Ω

Ω

𝜕θ 𝜕θ ψdx + ∫ ∇θ ⋅ ∇ψdx = ∫ gφdx ∫ ψdx + ∫ uj 𝜕t 𝜕xj

Ω

Ω

(4.21)

Ω

(4.22)

Ω

is satisfied a. e. in I for all φ = (φ1 , φ2 , . . . , φn ) ∈ Y = W 2,2 (Ω)n ∩ Vp , ψ ∈ Vθ . In [41], the following existence of a solution has been proved. 2n 2n 2,2 Theorem 4.0.1. Suppose n ≤ 3 and n+2 < p < n−2 , u0 ∈ H ∩ Wper (Ω)n , θ0 ∈ Vθ , f ∈ 2 2 n 2 2 n L (I; Lper (Ω) ), g ∈ L (I; Lper (Ω) ), Iδ = [δ, T), δ > 0. Then there exists a unique weak solution (u, θ) of the problem (4.23)–(4.27) such that

u ∈ L∞ (I; H) ∩ Lp (I; Vp ) ∩ L2 (I; W 2,2 (Ω)n ) ∩ C(Iδ ; H)

(4.23)

u ∈ L∞ (Iδ ; Vp ) ∩ L∞ (Iδ ; W 2,2 (Ω)n ) and

(4.24)

𝜕u ∈ L2 (Iδ ; H) 𝜕t

θ ∈ L∞ (I; L2 (Ω)) ∩ L2 (I; Vθ ) ∩ C(Iδ ; Vθ ) θ ∈ L∞ (Iδ ; Vθ ) ∩ L2 (Iδ ; W 2,2 (Ω)n ) and

𝜕θ ∈ L2 (Iδ ; L2per (Ω)). 𝜕t

(4.25) (4.26)

2n 2n Suppose u0 ∈ H, θ0 ∈ L2per (Ω), n+2 < p < n−2 and f ∈ L2 (I; L2per (Ω)), g ∈ 2 L (I; Lper (Ω)) satisfying (4.20), according to Theorem 4.0.1, we can define operator families {St }t≥0 : H × L2per (Ω) → H × L2per (Ω) 2

St : (u0 , θ0 ) → (u(t), θ(t)).

(4.27)

136 | 4 Global attractors of modified Boussinesq approximation Lemma 4.0.3. (1) The operator {St }t≥0 is a nonlinear semigroup; (2) For all t ≥ 0, St : H × L2 (Ω) → H × L2per (Ω) is continuous map; (3) St is continuous about t. Now, for p > 2, we consider the 2-dimensional periodic initial value problem equations (4.1)–(4.4), and prove the existence of a global attractor with finite Hausdorff dimension and finite fractal dimension. Definition 4.0.2. Let X is a metric space, the subset 𝒜 ⊂ X is called global attractor of semigroup {S(t)}t≥0 , if the following conditions are satisfied: (1) for all t ≥ 0, S(t)𝒜 = 𝒜; (2) 𝒜 is compact set; (3) for all bounded sets ℬ0 ⊂ X, dist(S(t)ℬ0 , 𝒜)t→∞ → 0. Definition 4.0.3. ℬ ⊂ X is called absorbing set, if and only if ℬ is bounded set and for all bounded sets G ⊂ X, there exists t0 ≡ t0 (G) so that for all t ≥ t0 , S(t)G ⊂ ℬ. Equation (4.17) has a feature is for all t > 0, θ mean conservation m(θ(t)) =

1 1 ∫ θ(x)dx = ∫ θ0 (x)dx = m(θ0 ). |Ω| |Ω| Ω

(4.28)

Ω

So, there does not exist a bounded absorbing set in the whole space E = H × L2 . In order to overcome this difficulty, for a fixed α, we introduce the subset of E = H × L2 (Ω) as follows: Eα = {(u, θ) ∈ E : |m(θ)| ≤ α}. We will prove that there exists bounded absorbing set in Eα . Lemma 4.0.4. Suppose that p > 2, (u0 , θ0 ) ∈ Eα , and fi ∈ L∞ ([0, ∞); H), g ∈ L∞ ([0, ∞); L2 (Ω)) satisfying (4.20). Then, for the solution (u, θ) of two-dimensional periodic initial value problem equations (4.15)–(4.19), the following estimates ‖u‖2 ≤ K1 ,

‖θ‖2 ≤ K1 ,

∀t ≥ t1

hold, where K1 is a constant only depending on data (α, Ω, μ0 , μ1 , ε, p, f , g). While ‖u0 ‖2 ≤ R, and ‖θ‖2 ≤ R, t1 is dependent of data (α, Ω, μ0 , μ1 , ε, p, f , g) and R. Proof. For simplicity, let θ̃ = θ − m(θ)

(4.29)

then m(θ) =

1 ∫ θ(x)dx. |Ω| Ω

(4.30)

4 Global attractors of modified Boussinesq approximation

| 137

Equation (4.30) shows that 1 1 1 ̃ = ∫ θ(x)dx ∫ θ(x)dx − ∫ m(θ)dx = 0. |Ω| |Ω| |Ω|

(4.31)

̃ ̃ = m(θ0 ) ∫ θ(x)dx = 0. ∫ θm(θ)dx

(4.32)

Ω

Ω

Ω

Furthermore,

Ω

Ω

̃ and m(θ(t)) are orthogonal in L2 , and Thus, θ(t) 2 2 ̃ ‖θ(t)‖22 = ‖θ(t)‖ 2 + ‖m(θ(t))‖2 .

(4.33)

From equation (4.28), we have ‖m(θ)‖22 = |Ω||m(θ)|2 ≤ α2 |Ω|.

(4.34)

Equations (4.17), (4.28) and (4.29) imply that 𝜕θ̃ 𝜕θ̃ − △θ̃ = g. + uj 𝜕t 𝜕xj

(4.35)

Taking the inner product of equation (4.35) with θ̃ in L2 , we have 1 d ̃ 2 2 ̃ ̃ ‖θ‖2 + ‖∇θ(t)‖ 2 ≤ |(g, θ)|. 2 dt

(4.36)

According to the following Poincaré inequality, ‖u‖2 ≤ C0 ‖∇u‖2 .

(4.37)

It follows from (4.37) and Young inequality that d ̃ 2 1 ̃ 2 2 2 ‖θ‖2 + 2 ‖θ(t)‖ 2 ≤ C0 |g|∞ dt C0

(4.38)

where |g|∞ = |g|L∞ ([0, ∞); L2 (Ω)). By the Gronwall inequality, we get 2 2 ̃ ̃ ‖θ(t)‖ 2 ≤ ‖θ(0)‖2 e 1 2 − C02 t

≤R e

−

1 C2 0

t

+ C04 |g|2∞ (1 − e

0

∞

1 c2

t

)

+ C04 |g|2∞ ,

≤ 2C04 |g|2∞ , ∀t ≥ t∗ R where t∗ = 2C02 ln C2 |g| .

−

(4.39)

138 | 4 Global attractors of modified Boussinesq approximation Estimates (4.33), (4.34) and (4.39) show that ‖θ(t)‖22 ≤ 2C04 |g|2∞ + α2 |Ω|,

∀t ≥ t∗ .

(4.40)

Taking the inner product of equation (4.15) with ui in H, we obtain 𝜕eij 𝜕eij 1 d ‖u‖22 + ∫ τij (e(u))eij (u)dx + 2μ1 ∫ dx 2 dt 𝜕xk 𝜕xk Ω

Ω

= ∫ θ(en ⋅ u)dx + ∫ fi ui dx Ω

Ω

(4.41)

≤ (‖θ‖2 + |f |∞ )‖u‖2 where |f |∞ = |f |L∞ ([0,∞);H)2 . Noting that for p > 2, ε > 0 ∫ τij (e(u))eij (u)dx = 2μ0 ∫(ε + |e|2 )

(p−2)/2

|e|2 dx

Ω

Ω

≥ 2μ0 ∫ |e|p dx

(4.42)

Ω

there exists positive constant C1 (Ω) = 2μ0 ε(p−2)/2 k1 such that ∫ τij (e(u))eij (u)dx ≥ 2μ0 ε(p−2)/2 ∫ |e|2 dx Ω

Ω

≥ C1 (Ω)|u|21,2 .

(4.43)

Removing the item containing μ1 in (4.41). From (4.42), we have 1 d ‖u‖22 + 2μ0 ‖e‖pp ≤ (‖θ‖2 + |f |∞ )‖u‖2 . 2 dt

(4.44)

Removing the item containing μ1 in equation (4.41). The estimates (4.43) yields 1 d ‖u‖22 + C1 (Ω)|u|21,2 ≤ (‖θ‖2 + |f |∞ )‖u‖2 . 2 dt

(4.45)

It follows from Young inequality that C02 C (Ω) d ‖u‖22 + 1 2 ‖u‖22 ≤ (‖θ‖2 + |f |∞ )2 . dt C1 (Ω) C0 By the Gronwall lemma, we get ‖u(t)‖22 ≤ ‖u0 ‖22 exp(− +

C1 (Ω) t) C02

C02 C (Ω) (‖θ‖2 + |f |∞ )2 [1 − exp ( − 1 2 t)] C1 (Ω) C0

(4.46)

| 139

4 Global attractors of modified Boussinesq approximation

≤ ‖u0 ‖22 exp(− ≤ R2 exp(− ≤2 where t∗∗ =

2C02 C1 (Ω)

C02 C1 (Ω) t) + (‖θ‖2 + |f |∞ )2 , C1 (Ω) C02

C02 C1 (Ω) t) + (‖θ‖2 + |f |∞ )2 C1 (Ω) C02

C02 (‖θ‖2 + |f |∞ )2 , C1 (Ω)

∀t ≥ t∗∗

C1 (Ω)R ]. ln[ C2 (‖θ‖ +|f | ) 2

0

∞

(4.47)

2

C0 Let t = max{t∗ , t∗∗ }, K1 = {(2C04 |g|2∞ + α2 |Ω|) 2 , √2 C (Ω) [(2C04 |g|2∞ + α2 |Ω|) 2 + |f |∞ ]}, 1 and (4.40) and (4.47) imply the conclusion of Lemma 4.0.4. 1

1

Lemma 4.0.5. Suppose that the conditions of Lemma 4.0.4 are satisfied. Then ‖u(t)‖21 , ‖u(t)‖22 , ‖∇θ(t)‖22 ≤ K2 ,

∀t ≥ t2

where K2 only depends on data, t2 is dependent of data and R, ‖u0 ‖2 ≤ R, ‖θ0 ‖21 ≤ R. Proof. From (4.40), (4.44) and (4.47), we get 1 d ‖u‖22 + 2μ0 |e|pp ≤ (K1 + |f |∞ )K1 , 2 dt

∀t ≥ t1

(4.48)

∀r > 0, integrating (4.48) from t to t + r (t ≥ t1 ) yields t+r

1 ‖u(t + r)‖22 + 2μ0 ∫ |e|pp dτ 2 r

1 ≤ ‖u(t)‖22 + r(K1 + |f |∞ )K1 2 1 ≤ ( + r)K12 + r|f |∞ K1 . 2

(4.49)

For ∀r > 0, t ≥ t1 and p > 2, we get t+r

1 1 [( + r)K12 + r|f |∞ K1 ]. 2μ0 2

(4.50)

1 1 [( + r)K12 + r|f |∞ K1 ]. C1 (Ω) 2

(4.51)

∫ ‖e‖pp dτ ≤ r

Similarly, t+r

∫ |u|21,2 dτ ≤ r

It follows from (4.40), (4.44) and equation (4.47) that t+r

2μ1 ∫ ∫ r Ω

𝜕eij 𝜕eij 𝜕xk 𝜕xk

1 dxdt ≤ [( + r)K12 + r|f |∞ K1 ]. 2

(4.52)

140 | 4 Global attractors of modified Boussinesq approximation Taking the inner product of equation (4.15) with

𝜕ui 𝜕t

in H, we have

2 𝜕eij 𝜕eij d 1 𝜕u dx} + {∫ ϑ(e(u))dx + μ1 ∫ 2 𝜕t 2 dt 𝜕xk 𝜕xk Ω

Ω

𝜕u 𝜕u 𝜕u 1 2 ≤ ∫ uj i i dx + ∫ θen ⋅ dx + |f | 𝜕xj 𝜕t 𝜕t 2 ∞ Ω

(4.53)

Ω

where eij eij

ϑ(e(u)) = ∫ μ0 (ε + s)(p−2)/2 ds

(4.54)

0

for any δ > 0, we have 2 𝜕u 𝜕u 2 2 ∫ uj i i dx ≤ ‖u‖∞ ‖u‖1,2 + δ 𝜕xj 𝜕t Ω

2 δ 𝜕u 2 𝜕t 2

(4.55)

and noting that W 1,p (Ω) → L∞ (Ω) for p > 2 = n, there exists C2 (Ω) > 0 such that C 𝜕u 𝜕u 2 2 ∫ uj i i dx ≤ 2 ‖u‖1,p ‖u‖1,2 + 2δ 𝜕xj 𝜕t Ω

2 δ 𝜕u . 2 𝜕t 2

(4.56)

Similarly, 𝜕u 2 2 dx ≤ ‖θ‖ + ∫ θen ⋅ 𝜕t δ 2 Ω

2 δ 𝜕u . 2 𝜕t 2

(4.57)

Choosing δ = 2 in (4.56) and (4.57), and substituting the two estimates into (4.53). Then for p > 2, t > t1 and μ1 ≥ 0, we have 𝜕eij 𝜕eij d {∫ ϑ(e(u))dx + μ1 ∫ dx} dt 𝜕xk 𝜕xk Ω

≤

Ω

C2 (‖u‖21,p ‖u‖21,2

1 + ‖θ‖22 ) + |f |2∞ . 2

(4.58)

For any r > 0, t ≥ t1 and ∀u0 ∈ BR (0), θ0 ∈ BR (0) (radius of the ball is r, the center of t+r the ball is zero), we need to estimate integration ∫r ∫Ω ϑ(e(u))dxdτ. Since p > 2, applying the inequality (a + b)p ≤ 2(p−1) (ap + bp ), we can obtain eij eij

ϑ(e(u)) = ∫ μ0 (ε + s)(p−2)/2 ds 0

p p 2μ = 0 {(ε + s) 2 − ε 2 } p

4 Global attractors of modified Boussinesq approximation | 141

≤ ≤

p 2μ0 (ε + s) 2 p

p 2p/2 μ0 (ε + s) 2 . p

(4.59)

Then ∫ ϑ(e(u))dx ≤ C3 + C4 ‖e‖pp

(4.60)

Ω 2p/2 μ

2p/2 μ

p

where C3 = p 0 (ε + s) 2 |Ω|; C4 = p 0 . Estimates (4.50) and (4.60) show that t+r

∫ ∫ ϑ(e(u))dxdτ ≤ C3 r + r Ω

C4 1 [( + r)K12 + r|f |∞ K1 ]. 2μ0 2

(4.61)

On the other hand, similar to (4.43), we have eij eij

∫ ϑ(e(u))dx = ∫ ∫ μ0 (ε + s)(p−2)/2 dsdx ≥ μ0 ε(p−2)/2 ∫ |e|2 dx Ω 0

Ω

Ω

≥ μ0 ε(p−2)/2 k2 |u|21,2 .

(4.62)

Furthermore, |u|21,2 ≤ C5 (∫ ϑ(e(u))dx + μ1 ∫ Ω

Ω

𝜕eij 𝜕eij 𝜕xk 𝜕xk

dx)

(4.63)

where C5 = (μ0 ε(p−2)/2 k2 )−1 , and μ1 ≥ 0. Applying the following differential inequality to (4.58), dy ≤ a(t)y(t) + b(t), dt

t>0

(4.64)

where y(t) = ∫ ϑ(e(u))dx + μ1 ∫ Ω

a(t) = C2 (kp )−2 ‖e(u)‖2p

1 b(t) = ‖θ‖22 + |f |2∞ 2

for u0 ∈ BR (0), θ0 ∈ BR (0), ∀r > 0, t ≥ t1 , we get t+r

∫ a(s)ds = C2 (kp )

−2

t

t+r

2 ∫ e(u(s))p ds t

Ω

𝜕eij 𝜕eij 𝜕xk 𝜕xk

dx

(4.65) (4.66) (4.67)

142 | 4 Global attractors of modified Boussinesq approximation

−2 p−2/p

≤ C2 (kp ) r

2/p

t+r

p ( ∫ e(u(s))p ds) t

≤ C2 (kp )−2 r p−2/p {

2/p

1 1 [( + r)K12 + r|f |∞ K1 ]} 2μ0 2

= C6

(4.68)

1 1 ∫ b(s)ds = (‖θ‖22 + |f |2∞ )r ≤ (2C04 |g|2∞ + α2 |Ω| + |f |2∞ )r = C7 2 2

(4.69)

t+r t

t+r

t+r

t+r

∫ y(s)ds = ∫ ∫ ϑ(e(u))dxdτ + μ1 ∫ ∫ t

t Ω

t Ω

≤ C3 r +

𝜕eij 𝜕eij 𝜕xk 𝜕xk

dxdt

C4 1 [( + r)K12 + r|f |∞ K1 ] 2μ0 2

1 1 + [( + r)K12 + r|f |∞ K1 ] = C8 . 2 2

(4.70)

It follows from (4.68)–(4.70) that y(t + r) ≤ (

C8 + C7 ) exp(C6 ) = C9 , r

∀t ≥ t1

or ∫{ϑ(e(u)) + μ1 Ω

𝜕eij 𝜕eij 𝜕xk 𝜕xk

}dx ≤ C9 . t+r

(4.71)

For any r > 0, p > 2 and μ1 ≥ 0, the estimate (4.63) shows that |u|21,2 ≤ C5 C9 ,

∀t ≥ t1 + r.

(4.72)

Furthermore, ∫ Ω

C dx ≤ 9 . 𝜕xk 𝜕xk t+r μ1

𝜕eij 𝜕eij

(4.73)

However, since generalized the Korn inequality, we have ∫ Ω

𝜕eij 𝜕eij 𝜕xk 𝜕xk

dx ≤ C10 (Ω)‖u‖22,2

(4.74)

for p > 2, t ≥ t1 + r and all μ1 ≥ 0, the estimate equation (4.73) implies that ‖u‖22,2 ≤

C9 . μ1 C10

(4.75)

4 Global attractors of modified Boussinesq approximation | 143

By the Agmon inequality, we get 1/2 |u|∞ ≤ C1 1|u|1/2 2 |u|2,2 ,

∀u ∈ W 2,2 (Ω).

(4.76)

Then for p > 2, and all μ1 ≥ 0, ‖u‖∞ ≤ C12 ,

∀t ≥ t1 + r.

(4.77)

Combing the equation (4.31) and estimates (4.36)–(4.37), we have

It is also that

1 d ̃ 2 2 ̃ ̃ ̃ ‖θ‖2 + ‖θ(t)‖ 2 ≤ |g|∞ ‖θ‖2 ≤ C0 |g|∞ ‖∇(θ)‖2 2 dt 2 1 ̃ 2 + C0 |g|2 . ≤ ‖∇(θ)‖ 2 ∞ 2 2 d ̃ 2 2 2 2 ̃ ‖θ‖2 + ‖∇θ(t)‖ 2 ≤ C0 |g|∞ . dt

(4.78)

Integrating (4.78) in time from t to t + r yields t+r

2 2 2 ̃ ∫ ‖∇θ(t)‖ 2 dτ ≤ rC0 |g|∞ .

(4.79)

t

Taking the inner product of equation (4.17) with △θ in L2 (Ω), we have 𝜕θ 1 d ‖∇θ‖22 + ‖△θ‖22 ≤ ∫ uj △θdx + ∫ g△θdx 2 dt 𝜕xj Ω

(4.80)

Ω

and 𝜕θ 𝜕θ △θdx ≤ ‖u‖∞ ∫ |△θ|dx ∫ uj 𝜕xj 𝜕xj Ω

Ω

≤ C13 ‖∇θ‖2 ‖△θ‖2 1 2 ‖∇θ‖22 ≤ ‖△θ‖22 + C13 4 1 2 2 ∫ g△θdx ≤ ‖△θ‖2 + |g|∞ . 4

(4.81) (4.82)

Ω

From the above estimates, we have d 2 ‖∇θ‖22 + ‖△θ‖22 ≤ 2C13 ‖△θ‖22 + 2|g|2∞ . dt

(4.83)

2 By the uniform Gronwall inequality (g, h, y replaced by 2C13 , 2|g|2∞ , ‖△θ‖22 respectively), since equation (4.79) obtains

‖∇θ(t)‖22 ≤ (C02 |g|2∞ + 2r|g|2∞ ) exp(2rC13 ),

∀t ≥ t1 + r

(4.72), (4.75) and (4.84) give the conclusion of Lemma 4.0.5.

(4.84)

144 | 4 Global attractors of modified Boussinesq approximation Lemma 4.0.4 and Lemma 4.0.5 show that for Sμ1 (t), μ1 ≥ 0, ball B1 = {(u, θ) ∈ V2 × L2 (Ω) : |u|1,2 ≤ K1 , ‖θ‖2 ≤ K1 } is absorbing set of (V2 × L2 (Ω)) ∩ Eα , which is independent of μ1 > 0, and ball B2 = {(u, θ) ∈ (W 2,2 (Ω) ∩ H) × Vθ : ‖u‖2,2 ≤ K2 , ‖θ‖1,2 ≤ K2 } is the absorbing set of ((W 2,2 (Ω) ∩ H) × Vθ ) ∩ Eα . For fixed μ1 ≥ 0, the existence of absorbing set B1 shows that operator Sμ1 (t) is uniformly compactness for sufficiently large t. Similarly, for fixed μ1 ≥ 0, the existence of the absorbing set B2 shows operator Sμ1 (t) is uniformly compact for sufficiently large t. Define 𝒜μ1 = ⋂ ⋃ Sμ1 (t)B2 , s≥0 t≥s

μ1 > 0

(4.85)

and for μ1 = 0, let 𝒜0 = ⋂ ⋃ S0 (t)B1

(4.86)

s≥0 t≥s

where the closure is chosen in (W 2,2 (Ω)∩H)×Vθ , V2 ×L2per (Ω), respectively. Theorem 1.1.1 in [83] shows the following. Theorem 4.0.2. Suppose that the conditions of Lemma 4.0.4 are satisfied. Then sets 𝒜μ1 , μ1 > 0 and 𝒜0 defined by (4.85) and (4.86) are global attractors of Sμ1 , μ1 > 0 and S0 , respectively. Owing to the Definition 4.0.2, we know that all orbits of solution are attracted into a tight invariant set 𝒜μ1 , whether the orbits of solution can be characterized by finite multiple degrees of freedom at infinity? It is well known, if fractal dimension of 𝒜μ1 (denoted by df (𝒜μ1 )) is finite, then the answer to this question is yes, df (𝒜μ1 ) = lim sup

log nε (𝒜μ1 )

t→0+

log(1/ε)

,

where nε (𝒜μ1 ) is minimum number of balls with a radius less than ε required to cover 𝒜μ1 . The aim is to compute Hausdorff dimension dH (𝒜μ1 ) and fractal dimension df (𝒜μ1 ), μ1 > 0. In order to obtain the upper bound estimates of Hausdorff dimension dH (𝒜μ1 ) and fractal dimension df (𝒜μ1 ), μ1 > 0, consider the following first-order variational equations of (4.15) and (4.17), 𝜕u 𝜕U 𝜕Ui + Uj i + uj i 𝜕t 𝜕xj 𝜕xj =

𝜕 (p−4)/2 [τ(u)eij (U) + (p − 2)μ0 (ε + |e(u)|2 ) |e(u)|2 ekl (U)] 𝜕xj −

𝜕π 𝜕 − 2μ1 △e (U) + en Θ 𝜕xi 𝜕xj ij

(4.87)

4 Global attractors of modified Boussinesq approximation

𝜕Θ 𝜕Θ 𝜕θ + uj + Uj − △Θ = 0 𝜕t 𝜕xj 𝜕xj

| 145

(4.88)

with the initial values U(x, 0) = U0 (x),

Θ(x, 0) = Θ0 (x)

(4.89)

Θ(x, t) = Θ(x + Lej , t)

(4.90)

and periodic boundary conditions U(x, t) = U(x + Lej , t), and incompressible condition ∇⋅U=0

(4.91)

where ω(t) = (u(t), θ(t)) = S(t)(u0 , θ0 ) = S(t)ω0 is the solution of equations (4.15)– (4.19) with ω0 = (u0 , θ0 ) ∈ 𝒜μ1 . For ∀℧0 = (u0 (x), θ0 (x)) ∈ (W 2,2 (Ω) ∩ H) × Vθ , there exists a unique solution for the initial boundary value problem equations (4.87)–(4.91): U ∈ L∞ ([0, T); W 2,2 (Ω) ∩ H),

Θ ∈ L∞ ([0, T); Vθ )

(4.92)

also noting τ(u)eij (U) + (p − 2)μ0 (ε + |e(u)|2 ) =

(p−4)/2

|e(u)|2 ekl (U)]

𝜕ϑ (e(u))ekl (U). 𝜕eij 𝜕ekl

(4.93)

Definition 4.0.4. Nonlinear semigroup Sμ1 (t), μ1 > 0 is said uniformly differentiable in 𝒜μ1 , if ∀ω10 = (u10 , θ01 ) ∈ Aμ1 , Lμ1 (t; ω10 ) ∈ £(H × L2per (Ω)), H × L2per (Ω) such that as ε∗ → 0, sup

‖Sμ1 (t)ω20 − Sμ1 (t)ω10 − Lμ1 (t; ω10 )(ω20 − ω10 )‖H×L2

ω10 ,ω20 ∈𝒜μ1 ,0 0. We will prove Sμ1 (t) is uniformly differentiable in Aμ1 , for ∀μ1 > 0. Theorem 4.0.3. For μ1 > 0, nonlinear semigroup Sμ1 (t) is uniformly differentiable in global attractor Aμ1 defined by equation (4.85). Proof. For (u0 , θ0 ), (v0 , ϕ0 ) ∈ 𝒜μ1 , with ‖u0 −v0 , θ0 −ϕ0 ‖H×L2 ≤ ε∗ , we need to estimate the H × L2 norm of Φ, Φ = Sμ1 (t)(u0 , θ0 ) − Sμ1 (t)(v0 , ϕ0 )

− Lμ1 (t; (u0 , θ0 ))[(v0 − u0 ), (ϕ0 − θ0 )].

(4.95)

146 | 4 Global attractors of modified Boussinesq approximation For this reason, let w(t, μ1 ) = v(t, μ1 ) − u(t, μ1 )

(4.96a)

χ(t, μ1 ) = ϕ(t, μ1 ) − θ(t, μ1 )

(4.96b)

Ψ(t, μ1 ) = χ(t, μ1 ) − Θ(t, μ1 ).

(4.96d)

W(t, μ1 ) = w(t, μ1 ) − U(t, μ1 )

(4.96c)

For simplicity, in the following, assuming μ1 > 0 fixed and no longer show the dependence of u, v, w, θ, ϕ, χ, U, Θ, W, Ψ to μ1 , and w(t), χ(t) satisfy the following equations: 𝜕wi 𝜕u 𝜕w 𝜕π 𝜕 + wj i + vj i = − w − 2μ1 (△eij (w)) + en χ 𝜕t 𝜕xj 𝜕xj 𝜕xi 𝜕xj +

𝜕 [τ (v) − τij (u)], 𝜕xj ij

(x, t) ∈ Ω × [0, T)

𝜕ϕ 𝜕χ 𝜕χ + uj − △χ = 0, + wj 𝜕t 𝜕xj 𝜕xj

(x, t) ∈ Ω × [0, T)

(4.97) (4.98)

where πw is the related pressure difference of v and u. ∇⋅w=0

(4.99a)

w(0) = v0 − u0 ,

χ(0) = ϕ0 − θ0 ,

w(x, t) = w(x + Lei , t),

x∈Ω

χ(x, t) = χ(x + Lei , t)

(4.99b) x ∈ Ω, t ≥ 0.

(4.99c)

Taking the inner product of equation (4.97) with wi , we have 𝜕eij (w) 𝜕eij (w) 𝜕u 1 d ‖w‖22 + ∫ wj i wi dx + 2μ1 ∫ dx 2 dt 𝜕xj 𝜕xk 𝜕xk Ω

Ω

+ ∫[τij (v) − τij (u)]eij (w)dx ≤ ‖χ‖2 ‖w‖2 .

(4.100)

Ω

Taking into account the definition of potential function in equation (4.54) and the generalized Korn inequality (4.74), we get 𝜕u 1 d ‖w‖22 + ∫ wj i wi dx + 2μ1 C10 ‖w‖22,2 2 dt 𝜕xj Ω

+ ∫[ Ω

𝜕ϑ(eij (v)) 𝜕eij

−

𝜕ϑ(eij (u)) 𝜕eij

]eij (w)dx ≤ ‖χ‖2 ‖w‖2 .

For p > 2, the Korn inequality shows that ∫[ Ω

𝜕ϑ(eij (v)) 𝜕eij

−

𝜕ϑ(eij (u)) 𝜕eij

]eij (w)dx

(4.101)

4 Global attractors of modified Boussinesq approximation | 147 1

= ∫∫ ≥

𝜕2 ϑ (e(u + τw)dτ)eij (w)ekj (w)dx 𝜕eij 𝜕ekl

Ω 0 ε(p−2)/2 k22 |w|21,2 .

(4.102)

Substituting (4.102) into (4.101) yields 1 d ‖w‖22 + 2μ1 C10 ‖w‖22,2 + ε(p−2)/2 k22 |w|21,2 2 dt 𝜕u ≤ ∫ wj i wi dx + ‖χ‖2 ‖w‖2 . 𝜕xj

(4.103)

Ω

Now we estimate the convective terms of the right-hand side of (4.103), and we can easily deduce 𝜕u 1/2 1/2 1/2 1/2 ∫ wj i wi dx ≤ C14 ‖w‖2 |w|1,2 |u|1,2 ‖u‖2,2 ‖w‖2 . 𝜕xj

(4.104)

Ω

For a suitable constant C14 , noting that ω0 ∈ 𝒜μ1 , (v0 , ϕ0 ) ∈ 𝒜μ1 , for all t ≥ 0, ω(t), (v(t), θ(t)) are contained in B2 . Then for ∀μ1 > 0, |w|1,2 ≤ K2 ,

‖u‖2,2 ≤ K2 ,

‖∇ϕ‖2 ≤ K2 ,

∀t ≥ 0.

(4.105)

From (4.104), ∀t ≥ 0, we get 𝜕u 3/2 1/2 ∫ wj i wi dx ≤ C14 K2 ‖w‖2 |w|1,2 . 𝜕xj

(4.106)

Ω

Applying the following Young inequality to (4.106) (a, b, δ > 0), ab ≤

δp p bq a + q, p qδ

1 < p, q < ∞, 1/p + 1/q = 1.

3/2 Here, a = |w|1/2 1,2 , b = ‖w‖2 , p = 4.

δ4 𝜕u 3 2 2 ∫ wj i wi dx ≤ C14 K2 |w|1,2 + 4/3 C14 K2 ‖w‖2 . 4 𝜕xj 4δ

(4.107)

Ω

It follows from (4.107) that 1 d ‖w‖22 + 2μ1 C10 ‖w‖22,2 + ε(p−2)/2 k22 |w|21,2 2 dt δ4 3 1 1 ≤ C14 K2 |w|21,2 + 4/3 C14 K2 ‖w‖22 + ‖χ‖22 + ‖w‖22 . 4 2 2 4δ

(4.108)

148 | 4 Global attractors of modified Boussinesq approximation Taking the inner product of equation (4.97) with χ, we have 𝜕ϕ 1 d ‖χ‖22 + ‖∇χ‖22 ≤ ∫ wj χdx. 2 dt 𝜕xj

(4.109)

Ω

For the convective terms in the right-hand side of (4.109), we have 𝜕ϕ 1/2 1/2 1/2 χdx ≤ C14 ‖w‖1/2 ∫ wj 2 |w|1,2 |ϕ|1,2 ‖χ‖2 ‖∇χ‖2 . 𝜕xj

(4.110)

Ω

Furthermore, 𝜕ϕ 1/2 1/2 1/2 χdx ≤ C14 K2 ‖w‖1/2 ∫ wj 2 ‖χ‖2 |w|1,2 ‖∇χ‖2 𝜕xj Ω

1 ≤ C14 K2 (‖w‖2 |w|1,2 + ‖χ‖2 ‖∇χ‖2 ) 2 C Kβ 1 ≤ C K ‖w‖22 + 14 2 |w|21,2 4β 14 2 4 C Kγ 1 C K ‖χ‖2 + 14 2 ‖∇χ‖22 . + 4γ 14 2 2 4

(4.111)

Combing the estimates (4.98), (4.99) and (4.111), we can choose δ, β, γ sufficiently small such that d (‖w‖22 + ‖χ‖22 ) + 4μ1 C10 ‖w‖22,2 + ε(p−2)/2 k22 |w|21,2 + ‖∇χ‖22 dt ≤ C15 (‖w‖22 + ‖χ‖22 ).

(4.112)

Integrating (4.112) over (0, t), we obtain ‖v(t) − u(t)‖22 + ‖ϕ(t) − θ(t)‖22

≤ exp(C15 t)(‖v0 − u0 ‖22 + ‖ϕ0 − θ0 ‖22 ),

(4.113a)

and t

∫ |w(τ)|21,2 dτ ≤ exp(C15 t)[ 0

t

∫ ‖w(τ)‖22,2 dτ ≤ exp(C15 t)[ 0

1 ](‖v0 − u0 ‖22 + ‖ϕ0 − θ0 ‖22 ) ε(p−2)/2 k22

(4.113b)

1 ](‖v0 − u0 ‖22 + ‖ϕ0 − θ0 ‖22 ) 4μ1 C10

(4.113c)

t

∫ ‖∇χ(τ)‖22,2 dτ ≤ exp(C15 t)(‖v0 − u0 ‖22 + ‖ϕ0 − θ0 ‖22 ). 0

(4.113d)

4 Global attractors of modified Boussinesq approximation | 149

W(t), Ψ(t) defined by (4.96c) and (4.96d) satisfy the following equations: 𝜕Wi 𝜕u 𝜕Wi 𝜕U + Wj i + vj + wj i 𝜕t 𝜕xj 𝜕xj 𝜕xj =−

𝜕πw 𝜕 𝜕 − 2μ1 (△eij (W)) + [τ (v) − τij (u)] 𝜕xi 𝜕xj 𝜕xj ij

+ en Ψ −

𝜕 (p−4)/2 [τ(u)eij (U) + (p − 2)μ0 (ε + |e(u)|2 ) 𝜕xj

× |e(u)|2 ekl (U)],

(x, t) ∈ Ω × [0, T)

𝜕ϕ 𝜕χ 𝜕Ψ 𝜕Ψ + uj + Wj + Uj − △Ψ = 0, 𝜕t 𝜕xj 𝜕xj 𝜕xj

(4.114) (x, t) ∈ Ω × [0, T)

(4.115)

with the incompressible condition ∇ ⋅ W = 0,

(x, t) ∈ Ω × [0, T)

(4.116)

and initial values W(0) = 0,

Ψ(0) = 0,

W(x, t) = W(x + Lej , t),

x∈Ω

(4.117)

Ψ(x, t) = Ψ(x + Lej , t)

x ∈ Ω, t ≥ 0.

(4.118)

Taking the inner product of equation (4.114) with Wi in H, we have 𝜕u 𝜕U 1 d ‖W‖22 + ∫ Wi i Wj dx + ∫ wj i Wi dx 2 dt 𝜕xj 𝜕xj Ω

Ω

− ∫[τ(u)eij (U) + (p − 2)μ0 (ε + |e(u)|2 )

(p−4)/2

Ω

× |e(u)|2 ekl (U)]eij (W)dx + ∫[τij (v) − τij (u)]eij (W)dx Ω

+ 2μ1 ∫ Ω

𝜕eij (W) 𝜕eij (W) 𝜕xk

𝜕xk

dx = ∫ en Ψ ⋅ Wdx. Ω

First, we deal with the fourth and fifth terms in the left-hand side of (4.119), ∫[τij (v) − τij (u)]eij (W)dx − ∫[τ(u)eij (U) Ω 2 (p−4)/2

Ω

+ (p − 2)μ0 (ε + |e(u)| ) 1

= ∫(∫ 0

Ω

−∫ Ω

|e(u)|2 ekl (U)]eij (W)dx

𝜕2 ϑ (e(u + τw))dτ)eij (W)ekl (w)dx 𝜕eij 𝜕ekl

𝜕2 ϑ (e(u))ekl (U)eij (W)dx 𝜕eij 𝜕ekl

(4.119)

150 | 4 Global attractors of modified Boussinesq approximation

=∫ Ω

𝜕2 ϑ (e(u))eij (W)ekl (W)dx 𝜕eij 𝜕ekl 1

+ ∫ ( ∫[ Ω

0

𝜕2 ϑ 𝜕2 ϑ (e(u + τw)) − (e(u))]dτ)eij (W)ekl (w)dx 𝜕eij 𝜕ekl 𝜕eij 𝜕ekl

≥ ε(p−2)/2 k22 |W|21,2 + ∫ ϑijklmn eij (W)ekl (w)dx

(4.120)

Ω

where 1 1

ϑijklmn = ∫ ∫ 0 0

𝜕3 ϑ (e(u + στw))τdτdσ. 𝜕eij 𝜕ekl 𝜕emn

(4.121)

After directly calculation, we have 𝜕3 ϑ (p−2)/2 = μ0 (p − 2)(ε + |e(u)|2 ) 𝜕eij 𝜕ekl 𝜕emn ×[

δim δjn ekl + δkm δln eij + δij δkl emn (ε + |e(u)|2 )

+

p−4 e e e ] (ε + |e(u)|2 )2 ij kl mn

where δij is Kronecker of δ symbol. For p < 3, there exists constant cp > 0 such that 𝜕3 ϑ ≤ cp , 𝜕eij 𝜕ekl 𝜕emn

i, j, k, l, m, n = 1, 2, 3.

(4.122)

Combining generalized Korn inequality and estimates (4.120), (4.121), (4.122), it follows that 1 d ‖W‖22 + ε(p−2)/2 k22 |W|21,2 + 2μ1 C10 ‖W‖22,2 2 dt 𝜕u ≤ cp ∫ |eij (W)||eij (w)|2 dx + ∫ Wi i Wj dx 𝜕xj Ω

Ω

𝜕U + ∫ wj i Wi dx + ∫ en Ψ ⋅ Wdx. 𝜕xj Ω

(4.123)

Ω

Taking the inner product of equation (4.115) with Ψ in L2 (Ω), we have 𝜕ϕ 𝜕χ 1 d ‖Ψ‖22 + ‖∇Ψ‖22 ≤ ∫ Wj Ψdx + ∫ Uj Ψdx. 2 dt 𝜕xj 𝜕xj Ω

Ω

For w ∈ B2 , ∀t > 0, we have 𝜕u 1/2 1/2 1/2 1/2 ∫ Wi i Wj dx ≤ C14 ‖W‖2 |W|1,2 |u|1,2 ‖u‖2,2 ‖W‖2 𝜕xj Ω

(4.124)

4 Global attractors of modified Boussinesq approximation

| 151

1/2 ≤ C14 K2 ‖W‖3/2 2 |W|1,2

η4 3 C K |W|21,2 + 4/3 C14 K2 ‖W‖22 4 14 2 4η

≤

(4.125)

𝜕U 1/2 1/2 ∫ wj i Wi dx ≤ C14 ‖w‖2 ‖w‖2,2 |U|1,2 ‖W‖2 𝜕xj Ω

2 C14 1 ‖w‖2 |U|21,2 ‖w‖2,2 + ‖W‖22 2 2

(4.126)

1 2 2 ∫ en Ψ ⋅ Wdx ≤ ‖Ψ‖2 ‖W‖2 ≤ (‖Ψ‖2 + ‖W‖2 ). 2

(4.127)

≤

Ω

Similarly, we obtain 𝜕ϕ Ψdx ≤ C14 ‖Ψ‖2 ‖ϕ‖1,2 ‖W‖1/2 ∫ Wj 2,2 𝜕xj Ω

≤

λ 1 1 2 2 C K ‖Ψ‖22 + ‖W‖22,2 + ‖W‖22 4 14 2 2 2λ

(4.128)

𝜕χ 𝜕Ψ Ψdx ≤ ∫ Uj χdx ∫ Uj 𝜕xj 𝜕xj Ω

Ω

1/2 ≤ C14 ‖Uj ‖1/2 2 ‖Uj ‖2,2 ‖∇Ψ‖2 ‖χ‖2

≤

γ 2 1 C14 ‖Uj ‖2 ‖Uj ‖2,2 ‖∇Ψ‖22 + ‖χ‖22 . 2 2γ

(4.129)

It can also deduce 1/3

∫ |eij (W)||eij (w)|2 dx ≤ (∫ |eij (W)|3 dx) Ω

Ω

2 ̃ ≤ k‖W‖ 1,3 ‖w‖1,3

(∫ |eij (w)|3 dx)

2/3

Ω

2/3 ≤ k∗ ‖W‖2,2 ‖w‖4/3 2 ‖w‖2,2

≤

k2 k 4/3 ‖W‖22,2 + ∗ ‖w‖8/3 2 ‖w‖2,2 2 2κ

(4.130)

where we used H 2 (Ω) → W 1,3 (Ω) and Sobolev interpolation inequality 2 1/3 ‖Du‖3 ≤ C‖u‖2/3 2 ‖D u‖2 ,

∀u ∈ H 2 , Du ∈ L3 , n = 2

combining the estimates (4.125)–(4.130), and we can select η, λ, γ, κ are small enough such that d d ‖W‖22 + ‖Ψ‖22 ≤ C16 (‖W‖22 + ‖Ψ‖22 ) + C17 (T)‖w‖22 ‖w‖22,2 dt dt k2 4/3 + ∗ ‖w‖8/3 2 ‖w‖2,2 . 2κ

(4.131)

152 | 4 Global attractors of modified Boussinesq approximation For any fixed T > 0, 0 < t ≤ T. Let y(t) = C17 (T)‖w‖22 ‖w‖22,2 +

k∗2 4/3 ‖w‖8/3 2 ‖w‖2,2 . 2κ

We have t

t

∫ y(s)ds = C17 (T) ∫ ‖w(s)‖22 ‖w(s)‖22,2 ds + 0

0

t

k∗2 4/3 ∫ ‖w(s)‖8/3 2 ‖w(s)‖2,2 ds 2κ 0

≤ C18 (T)ε∗4 exp(2C15 T)

(4.132)

k2

where C18 (T) = 4μ 1C C17 (T) + 2κ∗ T 1/3 . 1 10 Integrating equation (4.131) from 0 to t and noting W(0) = 0, Ψ(0) = 0, we have ‖W(t)‖22 + ‖Ψ(t)‖22 ≤

C18 (T)ε∗4 exp(2C15 + C16 )T , C16

∀t > 0.

(4.133)

This fact implies that sup

(u0 ,θ0 )∈𝒜μ1 ,(v0 ,ϕ0 )∈𝒜μ1 0 0), we prove that the Fréchet derivativeL(t; w0 ), w0 ∈ 𝒜μ1 of Sμ1 (t) is uniformly bounded in strong operator norm £(H ×L2per (Ω), H ×L2per (Ω)), for all t > 0. Recalling that for any μ1 > 0, the linearization equations of (U, Θ)(t; μ1 ) = £μ1 (t; w0 )(U0 , Θ0 ) are equations (4.87)–(4.88) with initial value (U, Θ)(0) = (U0 , Θ0 ), and |L(t; w0 )|£(H×L2per (Ω),H×L2per (Ω)) =

sup

(U0 ,Θ0 )∈H×L2per (Ω)

|Lμ1 (t; w0 )(U0 , Θ0 )|H×L2per (Ω) |(U0 , Θ0 )|H×L2per (Ω)

.

(4.136)

Next, we prove that there exists l > 0 such that for ∀t ≥ 0, sup |L(t; w0 )|£(H×L2per (Ω),H×L2per (Ω)) ≤ l[t]+1 .

w0 ∈𝒜μ1

(4.137)

4 Global attractors of modified Boussinesq approximation

| 153

Taking the inner product of equation (4.87) with Ui in H, we have 𝜕eij (U) 𝜕eij (U) 1 d ‖U‖22 + 2μ1 ∫ dx 2 dt 𝜕xk 𝜕xk Ω

𝜕u ≤ ∫ Uj i Ui dx + ∫ en Θ ⋅ Udx 𝜕xj

(4.138)

Ω

Ω

where we have removed the nonnegative item in the left-hand side of equation (4.138), ∫[τ(u)eij (U) + (p − 2)μ0 (ε + |e(u)|2 )

(p−4)/2

Ω

=∫ Ω

|e(u)|2 ekl (U)]dx

𝜕2 ϑ (e(u))eij (U)eij (U)dx ≥ 0. 𝜕eij 𝜕ekl

For p > 2, if w0 ∈ 𝒜μ1 , then μ1 > 0, w(t) = (u(t), θ(t)) ∈ B2 . Thus, for all t ≥ 0, we get 𝜕u 𝜕U ∫ Uj i Ui dx = ∫ ui i Uj dx 𝜕xj 𝜕xj Ω

Ω

≤ ‖u‖∞ ‖U‖1,2 ‖U‖2

≤ C‖u‖2,2 ‖U‖2,2 ‖U‖2 ≤ CK2 ‖U‖2,2 ‖U‖2 .

(4.139)

Substituting (4.139) into (4.138), and using the generalized Korn inequality and Young inequality, we get C 2 K22 d 1 1 ‖U‖22 + 2μ1 C10 ‖U‖22,2 ≤ ( + )‖U‖22 + ‖Θ‖22 . dt 4μ0 C10 2 2

(4.140)

Taking inner product of equation (4.88) with Θ in L2 (Ω), we have 1 d 𝜕θ ‖Θ‖22 + ‖∇Θ‖22 ≤ ∫ Uj Θdx 2 dt 𝜕xj Ω

1/2 1/2 1/2 ≤ C‖U‖1/2 2 ‖U‖1,2 ‖∇θ‖2 ‖Θ‖2 ‖∇Θ‖2 1/2 1/2 1/2 ≤ CK2 ‖U‖1/2 2 ‖U‖1,2 ‖Θ‖2 ‖∇Θ‖2

≤

C 4 K24 μ1 C10 1 1 ‖U‖22,2 + ‖∇Θ‖22 + ‖U‖22 + ‖Θ‖22 . 2 2 8μ1 C10 2

(4.141)

The estimates (4.140) and (4.141) show that d d ‖U‖22 + ‖Θ‖22 + μ1 C10 ‖U‖22,2 + ‖∇Θ‖22 ≤ C20 (‖U‖22 + ‖Θ‖22 ), dt dt

∀t ≥ 0.

(4.142)

154 | 4 Global attractors of modified Boussinesq approximation Applying the Gronwall inequality, we obtain ‖U(t)‖22 + ‖Θ(t)‖22 ≤ (‖U(0)‖22 + ‖Θ(0)‖22 ) exp(C20 t),

∀t ≥ 0.

(4.143)

So ∀t ≥ 0, w0 ∈ 𝒜μ1 , μ1 > 0 and (U0 , Θ0 ) ∈ H × L2per (Ω), and we obtain |Lμ1 (t; w0 )(U0 , Θ0 )|H×L2per (Ω) |(U0 , Θ0 )|H×L2per (Ω)

≤ exp(C20 t),

∀t ≥ 0.

(4.144)

Thus sup sup Lμ1 (t; w0 )(U0 , Θ0 )H×L2 (Ω) ≤ eC20 ≡ l. per

0≤t≤1 w0 ∈𝒜μ1

(4.145)

Since Sμ1 (t) = Sμ1 (t − [t])Sμ1 (1)[t], the result of (4.145) shows that of (4.137). And for any μ1 > 0, uniform boundedness of Lμ1 (t; w0 )(U0 , Θ0 ) in 𝒜μ1 is established. Now using standard techniques to compute the upper bounds of dH (𝒜μ1 ) and df (𝒜μ1 ), μ1 > 0, let E = (W 2,2 (Ω) ∩ H) × Vθ

℧(t) = (U(t), Θ(t))

(4.146) (4.147)

and rewrite equation (4.87) and equation (4.88) as follows: ℧t = F (w)℧

(4.148)

where w = Sμ1 (t)w0 , μ1 > 0, w0 = (u0 , θ0 ) ∈ 𝒜μ1 . For ∀T > 0, (4.92) shows the existence and uniqueness of solution ℧ ∈ L∞ ([0, T); E) (℧(0) = ξ ∈ E) of equation (4.148). For ξ = ξ1 , ξ2 , . . . ξm ∈ E, suppose ℧1 , . . . , ℧m are the corresponding solutions of equation (4.148). For all m ∈ N, consider |℧1 (t) ∧ ⋅ ⋅ ⋅ ∧ ℧m (t)|∧mE

t

= |ξ1 (t) ∧ ⋅ ⋅ ⋅ ∧ ξm (t)|∧mE exp ∫ Tr F (S(τ)w0 ) ∘ Qm (τ)dτ 0

where Qm (τ) = Qm (τ, w0 ; ξ1 , ξ2 , . . . , ξm ) is an orthogonal projection operator from E to space Span{℧1 , . . . , ℧m }. At a given moment τ, suppose φk (τ), k ∈ N are an orthogonal basis of E, such that Qm (τ)E = Span[℧1 , . . . , ℧m ], and the trace of F (S(τ)w0 ) ∘ Qm (τ) is given by m

Tr F (S(τ)w0 ) ∘ Qm (τ) = ∑ (F (w(τ))φk (τ), φk (τ)). k=1

Omitting variable τ temporarily, (4.87) and (4.88) yield (F (w(τ))φk (τ), φk (τ))

(4.149)

4 Global attractors of modified Boussinesq approximation

𝜕eij (Uk ) 𝜕eij (Uk )

= −2μ1 ∫

𝜕xl

Ω 2

−∫ Ω

𝜕xl

dx

𝜕 ϑ(e(u)) e (Uk )emn (Uk )dx 𝜕eij 𝜕emn ij

+ ∫ en Θk ⋅ Uk dx − ∫ Ujk Ω

−

| 155

Ω

∫ Ujk

𝜕ui k U dx 𝜕xj i

𝜕θ k Θ dx − ‖∇Θk ‖22 . 𝜕xj

Ω

(4.150)

Since equation (4.74) has −2μ1 ∫ Ω

∫ Ω

𝜕eij (Uk ) 𝜕eij (Uk ) 𝜕xl

𝜕xl

dx ≤ 0

(4.151)

𝜕2 ϑ(e(u)) e (Uk )emn (Uk )dx ≥ ∫ ε(p−2)/2 |e(Uk )|2 dx 𝜕eij 𝜕emn ij Ω

≥ ε(p−2)/2 k22 |Uk |21,2

(4.152)

and 1 ∫ en Θk ⋅ Uk dx ≤ ‖Uk ‖2 ‖Θk ‖2 ≤ (‖Uk ‖22 + ‖Θk ‖22 ) 2

(4.153)

Ω

It follows from the Schwarz inequality that 𝜕u k i k Ui ≤ | grad u||Uk |2 . Uj 𝜕xj Furthermore, m k 1/2 k 𝜕u ∑ ∫ Uj i Ui dx ≤ ∫ | grad u|ρ(x)dx ≤ |u|1,2 |ρ|2 𝜕x j k=1

(4.154)

Ω

Ω

where function ρ is dependent on x and t, given by m

1/2

ρ(x) = ∑ {|Uk |2 + |Θk |2 } . k=1

(4.155)

Now according to the Theorem A.3.1 in [83], there exists a constant c1 dependent of Ω such that m

∫ ρ(x)2 dx ≤ c1 ∑ ∫(| grad Uk |2 + | grad Θk |2 )dx

Ω

i=1 Ω

156 | 4 Global attractors of modified Boussinesq approximation m

≤ c1 ∑ (|Uk |21,2 + ‖∇Θk ‖22 ).

(4.156)

k=1

Estimates (4.155) and (4.156) imply that m 1/2 (c1 ) |u|1,2 { ∑ (|Uk |21,2 k=1

≤

1/2 (c1 ) |u|1,2 {(

1 m ≤ ∑ ‖∇Θk ‖22 + 4 k=1 + c1 (1 +

m

∑

+

1/2 k 2 ‖∇Θ ‖2 )}

|Uk |21,2 )

k=1 ε(p−2)/2 k22

4

1/2

m

+ (∑

k=1

1/2 k 2 ‖∇Θ ‖2 ) }

|Uk |21,2

1 )|u|21,2 . ε(p−2)/2 k22

(4.157)

Similarly, k 𝜕θ k k 1/2 k 1/2 k 1/2 Θ dx ≤ C14 ‖Uk ‖1/2 ∫ Uj 2 |U |1,2 ‖∇θ‖2 ‖Θ ‖2 ‖∇Θ ‖2 𝜕xj Ω

≤ C14 K2 |Uk |1,2 ‖∇Θk ‖2 1 2 2 2 k 2 K2 C0 |U |1,2 ≤ ‖∇Θk ‖22 + C14 4

(4.158)

where we have used (4.37) and the results: for μ1 > 0, and for all t ≥ 0, w0 = (u0 , θ0 ) ∈

𝒜μ1 , w = (u, θ) ∈ B2 .

Combining all these estimates, we obtain m

Tr F (w(τ)) ∘ Qm (τ) ≤ − ∑ [( k=1

3ε(p−2)/2 k22 2 2 2 + C14 K2 C0 )|Uk |21,2 4

m 1 1 + ‖∇Θk ‖22 ] + + c1 (1 + (p−2)/2 2 )|u|21,2 , 2 2 ε k2

(4.159)

1 −1 now applying Chapter 6, Lemma 2.1 in [83] to operator 21 μ−1 1 A × A0 . The operator 2 μ1 A is two-dimensional linear operator defined by [13], according to Bloom and Hao [13], when j → ∞, eigenvalues satisfy λj ∼ c |j|4 . Similarly, A0 is a two-dimensional Laplace operator with periodic boundary conditions, according to Courant and Hilbert’s monograph [22], when j → ∞, eigenvalues satisfy λj ∼ c |j|4 . Thus, the eigenvalue of 1 −1 μ A × A0 is the sum of λj and λj . So, when j → ∞, eigenvalues have lower bounds 2 1 2μ

denoted by c( 1+2μ1 )|j|. Applying Chapter 6, Lemma 2.1 in [83], there exists constant c2 1 dependent of Ω (i. e., L) such that m

∑ (|Uk |21,2 + ‖∇Θk ‖22 ) ≥ c2 m2 ,

k=1

for a. e. τ > 0

(4.160)

4 Global attractors of modified Boussinesq approximation

| 157

Then (4.160) shows that Tr F (w(τ)) ∘ Qm (τ) ≤ − +

2 2 2 2 (3ε(p−2)/2 k22 + C14 K2 C0 )c2 m 2 K 2 C2 + 4 6ε(p−2)/2 k22 + 8C14 2 0

m 1 + c1 (1 + (p−2)/2 2 )|u|21,2 2 ε k2

(4.161)

Define qm (t), qm , m ∈ N as follows: t

qm (t) = sup

w0 ∈𝒜μ1

1 sup ( ∫ Tr F (w(τ)) ∘ Qm (τ)dτ) t ξi ∈E 0

|ξi |≤1 i=1,...,m

qm = lim sup qm (t). t→∞

Let t

σ = lim sup sup t→∞

w0 ∈𝒜μ1

1 ∫ |u(τ)|21,2 dτ. t 0

Then qm ≤ − +

2 2 2 2 (3ε(p−2)/2 k22 + C14 K2 C0 )c2 m 2 K 2 C2 + 4 6ε(p−2)/2 k22 + 8C14 2 0

1 m + c1 (1 + (p−2)/2 2 )σ 2 ε k2

(4.162)

(4.37) and (4.45) yield σ≤

C02 (K1 + |f |∞ )2 . C12 (Ω)

(4.163)

Combining (4.162) and (4.163), we get 1 qm ≤ − κ1 m2 + κ2 2 where κ1 =

2 2 2 (3ε(p−2)/2 k22 + C14 K2 C0 )c2

2 K 2 C2 + 4 6ε(p−2)/2 k22 + 8C14 2 0

κ2 = 2κ1 + c1 (1 +

1

) 2

ε(p−2)/2 k2

Then [83, 21] give the main conclusions.

C02 (K1 + |f |∞ )2 . C12 (Ω)

(4.164)

158 | 4 Global attractors of modified Boussinesq approximation Theorem 4.0.4. Suppose m is the smallest positive integer satisfying the following inequality: m − 1 < 2(

1/2

κ2 ) κ1

≤ m,

(4.165)

then for any μ1 > 0, dH (𝒜μ1 ) ≤ m < 1 + 2(

1/2

κ2 ) κ1

(4.166)

and df (𝒜μ1 ) ≤ 2m. Note. (1) The approach cannot be applied to the case μ1 = 0. For example, the upper bound of dH (𝒜μ0 ) is given by (4.166); the reason is that the concept used here is associated with the uniform differentiability of the nonlinear semigroup; however, this property has not been established for S0 (t). (2) The results of Theorem 4.0.4 show that the upper bound of dH (𝒜μ1 ), df (𝒜μ1 ), μ1 > 0 are dependent of μ1 , which is different from the results of [10] for bipolar incompressible non-Newtonian fluid; (3) Using the method of [10], we can prove easily that as μ1 → 0, 𝒜μ1 → 𝒜μ0 , in the sense of d(̂ 𝒜μ1 , 𝒜μ0 ) → 0, d̂ actually is the seminorm of set.

μ1 → 0

(4.167)

5 Inertial manifolds of incompressible non-Newtonian fluids Inertial manifold is an important channel-linked infinite dimensional dynamical system and finite dimensional dynamical system. In an infinite dimensional system exists inertia manifolds; then in inertia manifolds, an infinite dimensional system state can entirely be determined by a finite system [25, 27]. However, due to a sufficient condition for the existence of inertial manifolds “spectral gap,” many dissipative dynamical systems cannot be satisfied, and the relevant conclusions are very few. So a lot of work is researching the approximate inertial manifold [37].

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids Consider

du + 2μ1 Au + R(u) = 0, dt u(0) = u0

t>0

(5.1.1) (5.1.2)

n

where R(u) = −2μ0 Ap (u) + B(u, u) − f , Ω = [0, L] , n = 2, 3, the parameter 0 ≤ α < 1 ⇔ 1 < p ≤ 2. Definition 5.1.1. A set M is an inertial manifold of equation (5.1.1) and equation (5.1.2), with solution operator sμ1 (t) if: (1) M is a finite dimension Lipschitz manifold; (2) M is invariant in the sense that ∀t ≥ 0, sμ1 (t)M ⊂ M; (3) M attracts exponentially all orbits of sμ1 (t), i. e., dist(sμ1 (t)u0 , M) → 0,

t → ∞.

Because A−1 is compact, we can define the fractional powers of A: ∀α ∈ R ∞

Aα u = ∑ λkα (u, ϕk )ϕk , k=1

∀u ∈ D(Aα )

(5.1.3)

where for α > 0 ∞ D(Aα ) = {u ∈ Hper ∑ λj2α (u, ϕj )2 < ∞} j=1

(5.1.4)

while for α < 0, D(Aα ) is the completion of Hper with respect to the norm D(Aα ) ∞

‖u‖α = https://doi.org/10.1515/9783110549614-005

{∑ λj2α (u, ϕj )2 } j=1

1 2

(5.1.5)

160 | 5 Inertial manifolds of incompressible non-Newtonian fluids which is induced by the scalar product ∞

(u, v)D(Aα ) = ∑ λj2α (u, ϕj )(v, ϕj ),

(5.1.6)

j=1

using Fourier transform, there exist constants k1 > 0, k2 > 0 such that K1 ‖u‖H 4 (Ω) ≤ ‖Au‖L2 (Ω) ≤ K2 ‖u‖H 4 (Ω) ,

∀u ∈ D(A)

1 2

(5.1.7) 1 2

K1 ‖u‖H 2 (Ω) ≤ ‖A u‖L2 (Ω) ≤ K2 ‖u‖H 2 (Ω) ,

∀u ∈ D(A )

(5.1.8)

K1 ‖u‖H 1 (Ω) ≤ ‖A u‖L2 (Ω) ≤ K2 ‖u‖H 1 (Ω) ,

∀u ∈ D(A ).

(5.1.9)

1 4

1 4

Therefore, we have the equivalences, ‖Au‖ ∼ ‖u‖4 { { 1 {‖A 21 u‖ ∼ ‖u‖2 { {‖A 4 u‖ ∼ ‖u‖1 .

(5.1.10)

‖ ⋅ ‖ is L2 (Ω) norm; it is easy to show that the eigenvalues of A, for n = 2, 3, λk =

8π 4 4 |k| , L4

k = (k1 , . . . , kn ).

(5.1.11)

5.1.1 Lipschitz property 1

Lemma 5.1.1. The nonlinear term R(u) is defined on the bounded set of D(A 4 ), and R(u) is a Lipschitz function, i. e., for M > 0, ∃cM > 0 such that 1

1

1

‖A− 4 R(u) − A− 4 R(v)‖ ≤ cM ‖A− 4 (u − v)‖ 1

1

(5.1.12)

1

where ∀u, v ∈ D(A 4 ), ‖A 4 u‖ ≤ M, ‖A 4 v‖ ≤ M. 1

Proof. Let u, v ∈ D(A 4 ) and set w = u − v, (Ap (u) − Ap (v), w) = ∫{ Ω

(p−2) (p−2) 𝜕 𝜕 [ε + |e(u)|2 ] 2 eij (u) − [ε + |e(v)|2 ] 2 eij (v)} ⋅ wi dx. 𝜕xj 𝜕xj

(5.1.13)

For u, v, satisfy space periodic boundary conditions ∫[ε + |e(u)|2 ]

(p−2) 2

eij (u) − (ε + |e(v)|2 )

Ω

Integrating equation (5.1.13) by parts

(p−2) 2

eij (v) ⋅ wi vj ds = 0.

(5.1.14)

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 161

(Ap (u) − Ap (v), w) = ∫[ε + |e(u)|2 ]

(p−2) 2

eij (u) − (ε + |e(v)|2 )

(p−2) 2

eij (v) ⋅

Ω

𝜕wi dx. (5.1.15) 𝜕xj

We now set rp (e) =

p 1 (ε + |e|2 ) 2 p

(5.1.16)

ēij (t) = eij (u) + t(eij (v) − eij (u)),

0 ≤ t ≤ 1.

(5.1.17)

Then 𝜕rp 𝜕eij

= (ε + |e|2 )

(p−2) 2

eij .

(5.1.18)

So that [ε + |e(u)|2 ] 1

=∫ 0

𝜕2 rp 𝜕t𝜕eij =

(p−2) 2

eij (u) − (ε + |e(v)|2 )

(p−2) 2

eij (v)

𝜕 𝜕rp ( (ē (t)))dt 𝜕t 𝜕eij ij

(ēij (t)) = 𝜕2 rp

𝜕eij 𝜕ekl

𝜕2 rp 𝜕eij 𝜕ekl

=

𝜕2 rp 𝜕eij 𝜕ekl

(ēij (t))

(5.1.19) 𝜕ēkl 𝜕t

(ēij (t))(ekl (v) − ekl (u))

(p−4) (p−2) p−2 (ε + |e|2 ) 2 eij ekl + (ε + |e|2 ) 2 δij δjl . 2

(5.1.20) (5.1.21)

For any ξ ≠ 0, η ≠ 0, because 0 < p ≤ 2, 𝜕2 rp ξij ηkl 𝜕eij 𝜕ekl p − 2 (p−4) (p−2) (ε + |e|2 ) 2 eij ekl ξij ηij + (ε + |e|2 ) 2 ξkj ηkj = 2 p − 2 (p−4) (p−2) (ε + |e|2 ) 2 ⋅ |e|2 + (ε + |e|2 ) 2 )|ξ ||η| ≤ ( 2 4 − p (p−2) ≤ ε 2 |ξ ||η|. 2 Combining equations (5.1.15)–(5.1.22), we obtain (Ap (u) − Ap (v), w) 1

𝜕2 rp 𝜕wi = ∫ ∫ (ēij (t))(ekl (v) − ekl (u)) ⋅ dtdx 𝜕eij 𝜕ekl 𝜕xj Ω 0

(5.1.22)

162 | 5 Inertial manifolds of incompressible non-Newtonian fluids 1 𝜕w 2 2 2 4 − p (p−2) 2 ≤ ε ∫ ∫(∑ |ekl (v − u)|2 ) (∑ i ) dtdx 𝜕xj 2 i,j k,l 1

1

Ω 0

(p−2) 4−p ≤ c1 (Ω)ε 2 ⋅ ‖u − v‖1 ‖w‖1 2

(5.1.23)

(B(u, u) − B(v, v), w) 𝜕v 𝜕u = ∫(uj i − vj i )wi dx 𝜕xj 𝜕xj Ω

𝜕u 𝜕 (ui − vi )wi dx ∫(uj − vj ) i wi dx + ∫ vi 𝜕xj 𝜕xj Ω

2

(∫(uj − vj ) Ω

wi2 dx)

1 2

Ω

1

2 𝜕u 𝜕u (∫ i i dx) 𝜕xj 𝜕xj

Ω

1 2

1

𝜕(u − v ) 2 i i + (∫ vj2 wi2 dx) (∫ ∑ dx) 𝜕xj i,j Ω

1 4

Ω

1

1

𝜕u 2 4 ≤ (∫(uj − vj ) dx) (∫ ∑ dx) (∫ ∑ |wi |4 dx) 𝜕x i,j j i 4

Ω

1 4

Ω

Ω

1

1

𝜕(u − v ) 2 4 4 i i + (∫ ∑ |vj | dx) (∫ ∑ dx) (∫ ∑ |wi | dx) . 𝜕xj i i,j i 4

Ω

Ω

1

Ω

4

Using the Sobolev embedding H (Ω) → L (Ω), n = 2, 3, so that for some c2 (Ω) > 0, (B(u, u) − B(v, v), w) = c2 (‖u‖1 + ‖v‖1 )‖u − v‖1 ‖w‖1 .

(5.1.24)

So, equation (5.1.23) and equation (5.1.24) have (R(u) − R(v), w) ≤ 2μ0 (Ap (u) − Ap (v), w) + (B(u, u) − B(v, v), w) ≤ [(4 − p)μ0 ε

p−2 2

c1 + c2 (‖u‖1 + ‖v‖1 )]‖u − v‖1 ‖w‖1 .

(5.1.25)

1

Since the equivalence between the norms ‖u‖H 1 (Ω) = ‖u‖1 and ‖A 4 u‖, for some c3 > 0, (R(u) − R(v), w)

≤ c3 [M0 (4 − p)ε 1

1

p−2 2

1

1

1

1

c1 + c2 (‖A 4 u‖ + ‖A 4 v‖)]‖A 4 (u − v)‖‖A 4 w‖.

(5.1.26)

1

For u, v ∈ D(A 4 ) satisfying ‖A 4 u‖ ≤ M, ‖A 4 v‖ ≤ M, for some M > 0, 1 1 (R(u) − R(v), w) ≤ cM ‖A 4 (u − v)‖‖A 4 w‖

with cM = c3 [μ0 (4 − p)ε

p−2 2

c1 + 2c2 M]. Lemma 5.1.1 is proved.

(5.1.27)

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 163

5.1.2 The squeezing property Let w1 , . . . , wn be first N eigenvectors of the operator A, PN : H → Span{w1 , . . . , wn } the 1 projection operator, QN = 1 − pN , set u0 , v0 ∈ D(A 4 ), f ∈ L2 (QT ), QT = Ω × [0, T), we set w(t) = u(t) − v(t), 1

1

q(t) = ‖A 4 w(t)‖2 /‖A− 4 w(t)‖2 .

(5.1.28)

1 4

Lemma 5.1.2. Let u0 , v0 ∈ D(A ) be the unique solutions of the initial-value problems: { du + 2μ1 Au + R(u) = 0, { dt {u(0) = u0 { dv + 2μ1 Av + R(v) = 0, { dt {v(0) = v0 .

t>0

(5.1.29)

t>0

(5.1.30)

1

1

Let M > 0 and suppose that ∀t ∈ [0, T], ‖A 4 u‖ ≤ M, ‖A 4 v‖ ≤ M, then ∃c4 > 0, such that 1

‖A 4 w(τ)‖2 1

1

‖A− 4 w(τ)‖2

≤

‖A 4 w(t)‖2 1

‖A− 4 w(t)‖2

exp(c4 (τ − t)),

0 γ‖pN A− 4 w(t0 )‖

(5.1.46)

and ‖QN A− 4 w(t0 )‖ ≤ γ‖pN A− 4 w(t0 )‖.

(5.1.47)

166 | 5 Inertial manifolds of incompressible non-Newtonian fluids Then, in view of the statement of Theorem 5.1.1 (i. e., either equation (5.1.37) holds, or equation (5.1.38) does), it is necessary only to consider what happens if equation (5.1.46) applies; in this case, 1

η= =

‖A− 4 w(t0 )‖2 1

‖A− 4 w(t0 )‖2 1

1

1

1

‖pN A− 4 w(t0 )‖2 + ‖QN A− 4 w(t0 )‖2

‖pN A− 4 w(t0 )‖2 + ‖QN A− 4 w(t0 )‖2 1

≥

‖QN A− 4 w(t0 )‖2 1

(1 + γ1 ‖QN A− 4 w(t0 )w(t0 )‖2 ) 1 2

γ γ λN+1 λ = ≥ 1 + γ − 21 1 + γ N+1 λN+1 λN+1 being the (N + l)st eigenvalue of A. Employing equation (5.1.45), we are led to the estimate 1

‖A− 4 w(t0 )‖2 1

1

≤ ‖A− 4 w− 4 (0)‖2 exp{− 1

≤ ‖A− 4 w(0)‖2 exp{−

4c2 γ μ1 λN+1 t0 exp(−c4 t0 ) + M t0 } 1+γ 3μ1

4c2 γ μ1 λN+1 t0 exp(−c4 T) + M T} 1+γ 3μ1

(5.1.48)

as t0 < T, replacing t0 by t in the above equation, we obtain equation (5.1.38) with 4c2 T

γ exp(−c4 T). c1 = exp( 3μM ) and c2 = 1+γ 1 However, by equation (5.1.40), we have

1 1 1 d − 41 2 ‖A w‖ + 4μ1 ‖A 4 w‖2 ≤ 2cM ‖A− 4 w‖‖A 4 w‖ dt 1 1 c2 ≤ 4μ1 ‖A 4 w‖2 + M ‖A− 4 w‖2 μ1

so that 2 d − 41 2 cM − 41 2 ‖A w‖ ≤ ‖A w‖ dt μ1

(5.1.49)

integrating the above equation from 0 to t, then equation (5.1.39) is proved, so Theorem 5.1.1 is proved.

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 167

5.1.3 Fixed-point theorem 1

Equations (5.1.1)–(5.1.12) in H, D(A 4 ) and V exist attract ball, attract all solutions track, 1 consider attract ball Br1 ⊆ D(A 4 ), smooth cut off function θ : R+ → [0, 1], θ(ξ ) = 1, { { θ(ξ { ) = 0, { {|θ (ξ )| ≤ 2,

0≤ξ ≤1 ξ ≥2 ξ ≥ 0.

(5.1.50)

Let θr1 (r) = θ( rr ), then equation (5.1.1) modified equation 1

1 du + 2μ1 Au + θr1 (‖A 4 uw‖)R(u) = 0 dt

(5.1.51)

for u0 ∈ H, there exists unique solution. Define Hb,l (b > 0, l > 0) is ϕ formed space of Lipschitz map 1

1

ϕ : PN D(A 4 ) → QN D(A 4 ) satisfy 1 1 supp ϕ ⊂ {p ∈ PN D(A 4 )‖A 4 p‖ ≤ 2r1 } 1 4

‖A ϕ(p)‖ ≤ b,

1 4

∀p ∈ PN D(A )

(5.1.52) (5.1.53)

1 4

and for ∀p − 1, p2 ∈ PN D(A ), 1

1

1

‖A 4 ϕ(p1 ) − A 4 ϕ(p2 )‖ ≤ l‖A 4 (p1 − p2 )‖,

(5.1.54)

we easily deduce that, Hb,l is a complete metric space, norm is ‖ϕ1 − ϕ2 ‖ = ϕi ∈ Hb,l ,

1

p∈PN

1 D(A 4

1

‖A 4 ϕ1 (p) − A 4 ϕ2 (p)‖,

sup

)

i = 1, 2.

(5.1.55) 1

Consider the map T, ϕ ∈ Hb,l , Tϕ defined in PN D(A 4 ), projection operator PN , QN in equation (5.1.51), and p = PN u, q = QN u,

1

dp { { { dt + 2μ1 Ap + PN F(u) = 0 { { { dq + 2μ Aq + QP F(u) = 0 1 N { dt

where F(u) = θr1 (‖A 4 uw‖)R(u).

(5.1.56)

1

Setting ϕ ∈ Hb,l , P0 ∈ PN D(A 4 ), we have

setting p(t) = p(t; ϕ, t0 ).

{ dp + 2μ1 Ap + PN F(P + ϕ(p)) = 0 { dt {p(0) = P0

(5.1.57)

168 | 5 Inertial manifolds of incompressible non-Newtonian fluids For ∀α ∈ R , σ ∈ L∞ (R ; D(A

α−1 2

)), exist ξ : R1 → D(Aα ) continuous bounded dξ + Aξ = σ. dt

(5.1.58)

{ dξ + Aξ = 0 { dt {ξ (0) = ξ0

(5.1.59)

For the initial value problem,

has a unique solution e−tA : ξ0 → ξ (t) is from D(A unique solution of equation (5.1.58),

α−1 2

) to D(Aα ) a continuous map, the

t

ξ (t) = e−(t−t0 )A ξ (t0 ) + ∫ e−(t−τ)A σ(τ)dτ

(5.1.60)

t0

1

1

and ‖e−tA ‖L(D(Aα )) ≤ exp(−A 4 λt), for a A 4 λ > 0, all α ∈ R1 , t ≥ 0. When t0 → −∞, t

ξ (t) = ∫ e−(t−τ)Aσ(τ) dτ.

(5.1.61)

t0

Combining equation (5.1.57), we obtain p = p(t; ϕ, p0 ), similarly p(t) have dq + 2μ1 Aq + QN F(p + ϕ(p)) = 0. dt

(5.1.62)

1

In this equation, σ = −QN F(p + ϕ(p)) ∈ L∞ (R1 ; D(A− 4 )), so equation (5.1.62) is the specific object of equation (5.1.58), then solution q = q(t; ϕ, p0 ) of equation (5.1.62) is 1 a continuous bounded mapping of R1 to QN D(A 4 ). Specially, 1

q(0) = q(0; ϕ, p0 ) ∈ QN D(A 4 ).

(5.1.63)

Finally, the function mapping Tϕ , ϕ ∈ Hb,l , 1

1

p0 ∈ PN D(A 4 ) → q(0; ϕ, p0 ) ∈ QN D(A 4 ).

(5.1.64)

Similarly with equation (5.1.61), Tϕ : p0 → q(0; ϕ, p0 ) have the special case, 0

Tϕ (p0 ) = − ∫ e2μ1 Aτ QN F(p(τ) + ϕ(p(τ)))dτ ≡ q(0; ϕ, p0 ).

(5.1.65)

−∞

Lemma 5.1.3. Let ϕ ∈ Hb,l , then 1 1 supp Tϕ ⊂ {p ∈ pN D(A 4 )‖A 4 p‖ ≤ 2r1 }.

(5.1.66)

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 169

1

Proof. Let p0 such that ‖A 4 p‖ > 2r1 , setting u0 = p0 + ϕ(p0 ), then 1

1

1

1

1

‖A 4 u‖ = (‖A 4 p0 ‖2 + ‖A 4 ϕ(p0 )‖2 ) 2 ≥ ‖A 4 p0 ‖ > 2r1 1

so θr1 (‖A 4 u0 ‖) = 0, since continuity, for sufficiently small t, 1

θr1 (‖A 4 u(t)‖) = 0,

u(t) = p(t) + ϕ(p(t))

dp + 2μ1 Ap = −PN F(u) dt

(5.1.67) (5.1.68)

so 3 1 d 41 2 ‖A p‖ + 2μ1 ‖A 4 p‖2 = 0 2 dt 1 1 d 41 2 ‖A p‖ + 2μ1 A 4 λ1 ‖A 4 p‖2 ≤ 0 dt

for τ < 0 and |τ| sufficiently small 1

1

1

1

‖A 4 p(τ)‖ ≥ ‖A 4 p0 ‖ exp(−2μ1 A 4 λ1 τ) ≥ ‖A 4 p0 ‖ > 2r1 then 1

‖A 4 p(τ)‖ > 2r1 , ∀τ < 0 1 41 A (p(τ) + ϕ(p(τ))) ≥ ‖A 4 p(τ)‖ > 2r1 ,

∀τ < 0

for t < 0, the corresponding equation of q, dq + 2μ1 Aq = −QN F(u) = 0 dt has a unique solution, when t → −∞ bounded, and in (−∞, 0] is zero; Specially, q(0) = q(0; p0 , ϕ) = Tϕ (p0 ) = 0, so Lemma 5.1.3 is proved. 1

Lemma 5.1.4. Suppose ϕ ∈ Hb,l , p1 , p2 ∈ pN D(A 4 ), ui = pi +ϕ(pi ), i = 1, 2, then ∃M1 , M2 > 0 so that 1

‖A− 4 F(u1 )‖ ≤ M1 1 − 41 A (F(u1 ) − F(u2 )) ≤ M2 (1 + l)‖A 4 (p1 − p2 )‖. 1

1

Proof. Since F(u) is abounded operator of D(A 4 ) to D(A− 4 ), then 1

sup ‖A 4 F(u)‖ ≤

1 u∈D(A 4

)

so equation (5.1.69) is proved. Let

sup

1 ‖A− 4

u‖≤2r1

1

‖A− 4 F(u)‖ = M1

(5.1.69) (5.1.70)

170 | 5 Inertial manifolds of incompressible non-Newtonian fluids 1

θi = θr1 (‖A 4 ui ‖),

i = 1, 2, 1

1

L = ‖θ1 A− 4 R(u1 ) − θ2 A− 4 R(u)‖ 1

(5.1.71)

1

when ‖A 4 u1 ‖ and ‖A 4 u2 ‖ less than or equal to 2r1 , then θi = 0, L = 0, equation (5.1.70) obviously. When 1

1

‖A 4 u1 ‖ ≤ 2r1 ≤ ‖A 4 u2 ‖ then θ2 = 0 1

1

1

L = (θr1 (‖A 4 u1 ‖) − θr1 (‖A 4 u2 ‖))‖A 4 R(u1 )‖ ≤ 1

2M1 41 ‖A (u1 − u2 )‖. r1

1

1

1

For ‖A 4 u2 ‖ ≤ 2r1 ≤ ‖A 4 u1 ‖ have a similar conclusion. Finally, when ‖A 4 u1 | and ‖A 4 u2 ‖ are greater or equal to 2r1 , then 1

1

L ≤ |θ1 − θ2 |‖A− 4 R(u1 )‖ + θ2 ‖A− 4 (R(u1 ) − R(u2 ))‖ and by virtue of Lemma 5.1.1 have L≤

1 2M1 41 ‖A (u1 − u2 )‖ + C2r1 ‖A 4 (u1 − u2 )‖. r1

Combining the above equations, ∃ M2 =

2M1 r1

+ C2r1 , such that

1 41 A (F(u1 ) − F(u2 )) ≤ M‖A 4 (u1 − u2 )‖

(5.1.72)

and 1 1 1 ‖A 4 (u1 − u2 )‖ ≤ ‖A 4 (p1 − p2 )‖ + A 4 (ϕ(p1 ) − ϕ(p2 )) 1

≤ (1 + l)‖A 4 (p1 − p2 )‖

(5.1.73)

so equation (5.1.70) is proved. Lemma 5.1.4 is proved. 1

Lemma 5.1.5. Let p0 ∈ PN D(A 4 ), then 1

Tϕ (p0 ) ∈ QN D(A 4 ), 1

−1

1

41 A [Tϕ (p0 )] ≤ b

2 2 where b = e− 2 μ−1 1 ⋅ λN+1 < b (when λN+1 sufficiently large).

(5.1.74)

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 171

1

1

Proof. Let p0 ∈ PN D(A 4 ), then Tϕ (p0 ) ∈ QN D(A 4 ), by virtue of equation (5.1.65), we have 0

1 1 ‖A 4 (Tϕ )(p0 )‖ ≤ ∫ A 4 e2μ1 Aτ QN F(p(τ) + ϕ(p(τ)))dτ

−∞

0

1 2

1

≤ (2μ1 ) ∫ {‖(2μ1 AQN ) 2 e2μ1 Aτ ‖L(QN H) −∞

1 × A− 4 F(p(τ) + ϕ(p(τ)))}dτ 0

1 2

1

≤ (2μ1 ) M1 ∫ ‖(2μ1 AQN ) 2 e2μ1 Aτ ‖L(QN H) dτ.

(5.1.75)

−∞

Let δ ∈ R1 , τ < 0, k2 (δ) = δδ e−δ 1, e−δ +

K3δ = {

k2 (δ) 1−δ

⋅ δ 1 − δ), (

δ 0, then for ϕ1 , ϕ2 ∈ Hb,l , p0 ∈ PN D(A 4 ) have 1

‖A 4 (Tϕ1 (p0 ) − Tϕ2 (p0 ))‖ ≤ ‖ϕ1 − ϕ2 ‖, where L =

1 −1 M2 2 (2e− 2 λN+1 2μ1

(5.1.95)

−1

− λN 2 l ), ‖ϕ1 − ϕ2 ‖ is given by equation (5.1.55).

Proof. Set pi = p(t, ϕi , p0 ),

ui = pi + ϕi (pi ),

i = 1, 2

(5.1.96)

p̄ = p1 − p2 , make use of Lemma 5.1.4, i. e., of equation (5.1.70) as well as equation (5.1.73), we have 1 1 41 A (F(u1 ) − F(u2 )) ≤ M2 ‖A 4 u1 − A 4 u2 ‖ 1

1

1

1

≤ M2 (‖A 4 (p1 − p2 )‖ + ‖A 4 (ϕ1 (p1 ) − ϕ2 (p2 ))‖) ≤ M2 (‖A 4 (p1 − p2 )‖ + ‖A 4 (ϕ1 (p1 ) − ϕ2 (p1 ))‖ 1

1

+ ‖A 4 ϕ1 (p2 ) − A 4 ϕ2 (p2 )‖) 1

≤ M2 [(1 + l)‖A 4 (p1 − p2 )‖ + ‖ϕ1 − ϕ2 ‖] 1

= M2 [(1 + l)‖A 4 p‖̄ + ‖ϕ1 − ϕ2 ‖] where ‖ϕ1 − ϕ2 ‖ is given by equation (5.1.55). 3

1

(5.1.97) 1

1

1

Using, the elementary estimate ‖A 4 p‖̄ = ‖A 2 ⋅ A 4 p‖̄ ≤ λN2 ‖A 4 p‖̄ and equation (5.1.87), 1 1 d 41 2 ‖A p‖̄ + 2μ1 λN ‖A 4 p‖̄ 2 2 dt 1

1

1

1

≥ −M2 (1 + l)λN2 ‖A 4 p‖̄ 2 − M2 λN2 ‖ϕ1 − ϕ2 ‖‖A 4 p‖̄ 1 1 d 41 ‖A p‖̄ + (2μ1 λN + M2 (1 + l)λN2 )‖A 4 p‖̄ dt 1

≥ −M2 λN2 ‖ϕ1 − ϕ2 ‖

(5.1.98)

̄ p(0) = 0, integrate equation (5.1.98) from 0 to τ > 0, 1

̄ ‖A 4 p(τ)‖ ≤ M2 λN (ξN λN )−1 (exp(−ξN λN τ) − 1)‖ϕ1 − ϕ2 ‖ − 21

ξN = 2μ1 + M2 (1 + l)λN .

(5.1.99)

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 175

From equation (5.1.65), we have 41 A (Tϕ1 (p0 ) − Tϕ2 (p0 )) 0

1 ≤ ∫ A 4 e2μ1 Aτ QN (F(u1 ) − F(u2 ))dτ −∞

1 2

0

1

≤ (2μ1 ) ∫ ‖(2μ1 AQN ) 2 e2μ1 Aτ ‖dτ −∞

1 × A− 4 (F(u1 ) − F(u2 ))dτ 0

1 2

1

× (2μ1 ) M2 ∫ {‖(2μ1 AQN ) 2 e2μ1 Aτ ‖ −∞ 1

̄ × [(1 + l)‖A 4 p(t)‖ + ‖ϕ1 − ϕ2 ‖]}dτ 1

≤ (2μ1 ) 2 M2 ‖ϕ1 − ϕ2 ‖ 0

1

× ∫ ‖(2μ1 AQN ) 2 e2μ1 Aτ ‖[1 + (1 + l) −∞ 0

1

M2 − 21 −τλN ξN λ e ]dτ 2μ1 N 1

≤ (2μ1 ) 2 M2 ‖ϕ1 − ϕ2 ‖ × [ ∫ ‖(2μ1 AQN ) 2 e2μ1 Aτ ‖dτ −∞ 0

1

+ ∫ ‖(2μ1 AQN ) 2 e2μ1 Aτ ‖(1 + l) −∞

M2 − 21 −τλN ξN λ e dτ]. 2μ1 N

Applying equation (5.1.94), we deduce that 41 A (Tϕ1 (p0 ) − Tϕ2 (p0 )) 1 −1 M 2 + λN l )‖ϕ1 − ϕ2 ‖ ≤ 2 (2e 2 λN+1 2μ1 = L ⋅ ‖ϕ1 − ϕ2 ‖

(5.1.100)

where L=

1 −1 M2 2 (2e 2 λN+1 + λN l ) 2μ1

and l is given by equation (5.1.81). So Lemma 5.1.7 is proved. Lemma 5.1.8. For 0 < l < 1, and 1

K1 2μ1 1 K λN2 ≥ 1 2μ1 1

2 λN+1 − λN2 ≥

(5.1.101) (5.1.102)

176 | 5 Inertial manifolds of incompressible non-Newtonian fluids 1

K1 = 2M2 (1 + l)l K2 = 2M2 (2e− 2 + l)

(5.1.103)

then σN > 0, l < l and l ≤ 21 . Proof. By virtue of equation (5.1.101), we have 1

σN = 2μ1 (λN+1 − λN ) − M2 (1 + l)λN2 > 0 is equivalent to the statement that 2μ − 1 − rN ξN > 0;

(5.1.104)

however, −1

1

2 [(2μ1 )−1 + (2μ − 1 − rN ξN )−1 ]e− 2 erN ξN /(4μ1 ) l = M2 (1 + l)λN+1

−1

2 [(2μ1 )−1 + (2μ − 1 − rN ξN )−1 ]. ≤ M2 (1 + l)λN+1

(5.1.105)

Now, equation (5.1.101) can be written in the form 1

2 (2μ1 )−1 K1 ≤ λN+1 ,

K1 = 2M2 (1 + l)l−1

then −1

2 < (2μ1 )−1 M2 (1 + l)λN+1

then from equation (5.1.101), we have

(5.1.106)

l 2

(5.1.107)

−1

1

2 ≤ 2μ1 2μ1 rN2 + K1 λN+1

(5.1.108)

then −1

1

−1

2 2 k1 λN+1 − 2μ1 + 2μ1 rN + M2 (1 + l)λN+1 rN2

−1

−1

1

2 2 − 2μ1 + 2μ1 rN + k1 λN+1 rN2 ≤ k1 λN+1

−1

1

2 ≤ k1 λN+1 − 2μ1 + 2μ1 rN2 ≤ 0

(5.1.109)

so −1

2 K1 λN+1 ≤ 2μ1 − rN ξN .

(5.1.110)

Combining equation (5.1.107), equation (5.1.110), from equation (5.1.105) have l < l

(5.1.111)

by virtue of equation (5.1.100), it suffices to demonstrate that L= 1

1

1 −1 M2 1 −1 (2e− 2 λNH2 + λN 2 l ) < . 2μ1 2 1

−1

(5.1.112)

M2 2 Since l < l, λN+1 ≥ λN2 , l < 2μ (2e− 2 + l)λN 2 < 21 , this is obtained by virtue of equation 1 (5.1.102). So Lemma 5.1.8 is proved.

5.1 Inertial manifolds of incompressible bipolar non-Newtonian fluids | 177

1

1

Theorem 5.1.2. Set Hb,l = {ϕ : PN D(A 4 ) → QN D(A 4 ), satisfying equations (5.1.53)– (5.1.55), b > 0, l > 0} is a Lipschitz mapping space, the mapping Tϕ ∈ Hb,l is given by 1

1

equation (5.1.65), p0 ∈ PN D(A 4 ) and q(0; ϕ, p0 ) ∈ QN D(A 4 ) is only a continuous solution of equation (5.1.62) at t = 0, if equation (5.1.101) and equation (5.1.102) hold, 1

1

2 λN+1 − λN2 ≥ K1 /(2μ1 )

and 1

λN2 ≥

K2 2μ1

then T : Hb,l → Hb,l and which is a strictly compression mapping. Proof. The direct results of Lemmas 5.1.3–5.1.8.

5.1.4 Inertial manifolds Theorem 5.1.3. Set T has a fixed point ϕ∗ ∈ Hb,l , M = graph ϕ∗ is the graph of ϕ∗ , then M is a inertial manifolds of problem equation (5.1.1), equation (5.1.2) and there exist 1 t0 > 0, ∀u0 ∈ D(A 4 ), such that for all t ≥ t0 , dist(Sμ1 (t)u0 , M) ≤ exp(−

t ln 2) dist(u0 , M). 2t0

(5.1.113)

Proof. As a consequence of Theorem 5.1.2, we know that the mapping TT is compression mapping in complete metric space Hb,l , there exists a fixed point ϕ∗ ∈ Hb,l to obtain M = graph ϕ∗ , in order to show that M is the inertial manifold. First, from the definition of Hb,l , it should be clear that M is a finite-dimension Lipschitz manifold. And it is an easy exercise to show directly that M is invariant under the action of the solution operator Sμ1 (t), i. e., Sμ1 (t) ⊂ M. Therefore, it suffices to show that M attracts exponentially. Let the solution of equation (5.1.1) or equation (5.1.2) satisfy equation (5.1.12). Set solutions u(t), v(t) satisfying equation (5.1.51), for some M > 0, if we are given γ > 0, then for ∀t ∈ [0, T], ∀N, exist C̄ i , i = 1, 2 such that either equation (5.1.37) holds or equation (5.1.38) does. Similarly, equation (5.1.39) will also hold, for CM replaced by some C̄ M > 0, and t ∈ [0, T]; thus, setting t0 = min(

μ1 ln 2 T , ) 2 C̄ 2

(5.1.114)

M

we obtain, from equation (5.1.39), the estimate − 41 −1 A (u(t) − v(t)) ≤ 2A 4 (u(0) − v(0)),

t ≤ 2t0 .

(5.1.115)

178 | 5 Inertial manifolds of incompressible non-Newtonian fluids If we set γ = 81 , and choose N > N0 , where N0 satisfies λN0 +1 ≥ (C̄ 1 μ1 t0 )−1 ln(2C)̄

(5.1.116)

then from the modified forms of equation (5.1.37), equation (5.1.38), we will have either 1 −1 −1 QN A 4 (u(t) − v(t)) ≤ PN A 4 (u(t) − v(t)) 8

(5.1.117)

or − 41 A (u(t) − v(t)) ≤

1 − 41 A (u(0) − v(0)), 2

1

1

t ≤ 2t0

(5.1.118)

1

where u0 , v0 ∈ D(A 4 ), ‖A− 4 v(0)‖ ≤ M, ‖A− 4 u(0)‖ ≤ M and t0 ≤ t ≤ 2t0 . 1 We now denote the distance between any point w in the absorbing ball Br2 ⊂ D(A 4 ) and the manifold M by 1

dist(w, M) = inf ‖A− 4 (w − v)‖.

(5.1.119)

v∈M

Let v(0) = v0 ∈ M, v0 = PN v0 + ϕ(PN v0 ), 1 dist(u(0), M) = A− 4 (u(0) − v(0)).

(5.1.120)

Obviously, 1

‖PN A 4 v(0)‖ ≤ r2

(5.1.121)

so that, with b > 0 as in the definition of Hb,l , 1

‖A 4 v(0)‖ ≤ r2 + b

(5.1.122)

in which case, for t ≥ 0 1

1

‖A 4 v(t)‖ = ‖A 4 Sμ1 (t)v(0)‖ ≤ r2 + b,

∀t ≥ 0.

(5.1.123)

In the squeezing property, choosing M = r2 + b, we apply the estimates recorded in equation (5.1.118), for Sμ1 (t)u0 and Sμ1 (t)v0 , with t0 ≤ t ≤ 2t0 , 1

dist(Sμ1 (t1 )u0 , M)‖ ≤ A− 4 (Sμ1 (t1 )u0 − Sμ1 (t1 )v0 ) 1 1 ≤ A− 4 (u(0) − v(0)) 2 1 = dist(u0 , M). 2 On the other hand, from equation (5.1.117) we have

(5.1.124)

5.2 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids | 179

dist (Sμ1 (t1 )u0 , M) 1 ≤ A− 4 (Sμ1 (t1 )u0 − (PN Sμ1 (t1 )v0 + ϕ(PN Sμ1 (t1 )v0 ))) 1 ≤ A− 4 (QN Sμ1 (t1 )u0 − ϕ(PN Sμ1 (t1 )v0 )) 1 ≤ A− 4 (QN Sμ1 (t1 )u0 − QN Sμ1 (t1 )v0 ) 1 ≤ A− 4 (ϕ(PN Sμ1 (t1 )u0 ) − ϕ(PN Sμ1 (t1 )v0 )) 1 1 ≤ (l + )A− 4 (PN Sμ1 (t1 )u0 − PN Sμ1 (t1 )v0 ). 8

(5.1.125)

Taking l = 81 , 1 − 41 A (Sμ1 (t1 )u0 − Sμ1 (t1 )v0 ) 4 1 1 ≤ × 2 ⋅ ‖A− 4 (u0 − v0 )‖ 4 1 ≤ dist(u0 , M) 2

dist (Sμ1 (t1 )u0 , M) ≤

(5.1.126)

for t0 ≤ t ≤ 2t0 . Iterating upon the procedure delineated above, we have, therefore, for t0 ≤ t ≤ 2t0 , as n → ∞, n

1 dist (Sμ1 (nt1 )u0 , M) ≤ ( ) dist(u0 , M) → 0 2

(5.1.127)

so for arbitrary t ≥ t0 , we may write t = nt1 , for some t1 , t0 ≤ t ≤ 2t0 , in which case, n

1 dist (Sμ1 (nt1 )u0 , M) ≤ ( ) dist(u0 , M) 2 t ≤ exp(− ln 2) dist(u0 , M) t1 t ln 2) dist(u0 , M) ≤ exp(− 2t0

(5.1.128)

thus establishing the required exponential convergence of orbits. Hence, the original problem equation (5.1.1), equation (5.1.2) exist inertial manifolds. Theorem 5.1.3 is proved.

5.2 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids This section considers the incompressible bipolar fluid equation (5.1.1), equation(5.1.2); all marks and assumptions are the same with the above section. In Section 5.1, for

180 | 5 Inertial manifolds of incompressible non-Newtonian fluids 1 < p ≤ 2 the existence of an inertial manifold for the model was a known result. However, in the case of p > 2, the existence of inertial manifold is still an open problem. So, for p > 2, we will structure an approximate inertial manifold to approximate the global attractor and inertial manifold. 5.2.1 The analyticity in time and behavior of higher order modes We consider the complexified equation of equation (5.1.1) as du + 2μ1 Au − 2μ0 Ap (u) + B(u, u) = f dξ

(5.2.1)

where u ∈ H, ξ ∈ C, Re ξ > 0; and H, D(A), A, Ap , B denote the complexified extensions of the related spaces and operators. Theorem 5.2.1. If ρ > 0, D0 be a region in complex plane defined as the union of all the open disks, with radius δ, centered at δ + τ ∀τ ≥ 0. δ = δ(ρ) =

C6

2√2C2 (2ρ2 + C6 )

where u(ξ ) is a solution of equation (5.2.1) on a domain containing the positive real axis, such that ‖u(t)‖ ≤ ρ,

∀t > 0

(5.2.2)

then for f ∈ L∞ (0, T; H), the solution u(ξ ) exists in D0 and the mapping u: D0 → D(A) is analytic and satisfies ‖u(ξ )‖2 ≤ 2ρ, ξ ∈ D0 du(t) 4ρ 1 , t> δ ≤ dt 2 δ 2 ‖Au(t)‖ ≤ C1 (ρ)

(5.2.3) (5.2.4) (5.2.5)

where C1 (ρ) =

−1 1 − 21 4ρ (λ1 + 2C0 λ1 4 ρ2 + ‖f ‖L∞ (0,T;H) ) μ1 δ

2μ K (p − 1)ε + 0 0 μ1 1

p−2 2

ρ

+2

12 12−n

36+3n−12p

12p−12−n C 2 λ− 2 1 2 C2 = 0 + ( ) μ1 μ1 √2μ1

C3 =

μ20 k02 (p − 1)2 εp−2 2μ1

12

μ c k (p − 1) 12−n 12p−12−n [ 0 0 0 ] ρ 12−n μ1

5.2 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids | 181

C4 =

48 24+2n ‖f ‖L∞ (0,T;H) + (C0 (p − 1)K0 μ0 ) 36−12p−n 2 36−12p−n μ1

C5 = max{

C3 C ,√ 4} 2C2 C2

C6 = C5 + 1.

Proof. We take the inner product of equation (5.2.1) with Au, multiply the result by exp(iθ) and take the real part to get d‖u(seiθ )‖ ≤ −4μ1 cos θ‖Au‖2 + 2μ0 (Ap (u), Au) ds + 2|b(u, u, Au)| + 2‖f ‖‖Au‖.

(5.2.6)

We infer from 2 < p < 3, Gragliardo–Nirenberg inequality and the Korn inequality that 2μ0 (Ap (u), Au)

p−2 ≤ 2μ0 ∫ ∇ ⋅ [(ε + |e(u)|2 ) 2 e(u)]|Au|dx

Ω

≤ μ0 K0 (p − 1) ∫(ε

p−2 2

+ |e(u)|p−2 )| △ u||Au|dx

Ω

≤ μ0 K0 (p − 1)ε

p−2 2

‖u‖2 ‖Au‖

+ μ0 K0 (p − 1) ∫ |∇u|p−2 | △ u||Au|dx Ω

≤ μ0 K0 (p − 1)ε

p−2 2

‖u‖2 ‖Au‖

+ μ0 K0 (p − 1)‖∇u‖p−2 ‖ △ u‖ L6

≤ μ0 K0 (p − 1)ε

p−2 2

‖u‖2 ‖Au‖

+ μ0 K0 (p − 1)C0 ‖u‖ ≤ μ1 cos θ‖Au‖2 +

6

L 5−p

12p−12−n 12

‖Au‖

‖Au‖

12+n 12

μ20 k02 (p − 1)2 εp−2 ‖u‖22 2μ1 cos θ

36+3n−12p

48 12p−12−n 24+2n 1 ) +( ‖u‖42 + [C0 (p − 1)K0 μ0 ] 36−12p−n 2 36−12p−n μ1 cos θ

(5.2.7)

2|B(u, u, Au)| ≤ 2‖u‖∞ ‖∇u‖‖Au‖ −1

≤ 2C0 λ1 4 ‖u‖22 ⋅ ‖Au‖

−1

C2 λ 2 ≤ μ1 cos θ‖Au‖ + 0 1 ‖u‖42 μ1 cos θ 2

2‖f ‖‖Au‖ ≤ μ1 cos θ‖Au‖2 +

‖f ‖2L∞ (0,T;H) μ1 cos θ

(5.2.8) (5.2.9)

182 | 5 Inertial manifolds of incompressible non-Newtonian fluids so C C C2 d ‖u‖2 ≤ (‖u‖42 + 3 ‖u‖22 + 4 ). ds cos θ C2 C2

(5.2.10)

Let 2 y(s) = u(seiθ )2 ,

y0 = ‖u0 ‖22

then C2 dy ≤ (y + C5 )(y + C6 ). ds cos θ Direct integration leads to y + C5 y + C5 C ≤ exp( 2 s) 0 y + C6 cos θ y0 + C6 seeking an upper bound on S to ensure that y ≤ 2y0 .

(5.2.11)

y+C

We note that since the function y+C5 increasing on (0, ∞), the above equation will hold 6 provided exp(

y + C5 2y0 + C5 C2 S ≤ . s) 0 cos θ y0 + C6 2y0 + C6

It follows that S≤

2(y + C6 ) cos θ log 0 C2 2y0 + C6

and since log(1 + a) ≥ a2 , 0 < a < 1, we see that 2(y + C6 ) cos θ C6 cos θ log 0 = log(1 + ) C2 2y0 + C6 2C2 2y0 + C6 cos θ C6 cos θ C6 ≥ ≥ ≥ √2δ cos θ. 2C2 2y0 + C6 2C2 2ρ2 + C6 π Thus, in the domain D0 = {ξ = seiθ |S = √2δ cos θ, −4 ≤ θ ≤ π4 } relation equation (5.2.11) holds. By assumption, equation (5.2.2), taking the initial value as u(t0 ) at any t0 > 0 concludes that the solution is analytic and satisfies equation (5.2.11) within the domain t0 + D0 . It is easy to check that the union for ∀t0 > 0 contains D0 , so equation (5.2.3) is proved.

5.2 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids | 183

We apply the Cauchy integral formula to the circle S with radius 21 δ, centered at any point on the real axis t > 21 δ, obtain 1 du u(ξ ) 4ρ dξ ≤ ∫ = dt 2 2πi t − ξ 2 δ s which implies equation (5.2.4). Finally, from equation (5.2.1), we have that ‖Au‖ ≤ ≤

1 du ( + 2μ0 ‖Ap (u)‖ + ‖B(u, u)‖ + ‖f ‖L∞ (0,T;H) ) 2μ1 dt

−1 1 − 1 4ρ (λ1 2 + 2C0 λ1 4 ρ2 + ‖f ‖L∞ (0,T;H) ) 2μ1 δ p−2 n M C K (p − 1) 12p−12−n u0 ρ 12 ‖Au‖ 12 . + (p − 1)K0 ε 2 ρ + 0 0 0 μ1 u1

Then by using the Young inequality, we have proved equation (5.2.5). We will show the behavior of higher modes. Let ϕ1 , . . . , ϕm be the first m eigenvector of A, and P = Pm : H → Span{ϕ1 , . . . , ϕm } the projection operator, Q = Qm = I − P. Since P and Q commute with A and its power, we may split equation (5.2.1): dr + 2μ1 Ar − 2μ0 PAp (u) + PB(u, u) = Pf dt dq + 2μ1 Aq − 2μ0 QAp (u) + QB(u, u) = Qf dt

(5.2.12) (5.2.13)

where u is the solution of equation (5.2.1), r = Pu, q = Qu. Since question equation (5.2.1), equation (5.2.2) exist attractor, assume exist t0 = t0 (ρ) > 0, such that when ‖u0 ‖ ≤ ρ, have ‖u(t)‖ ≤ ρ0 ,

‖u(t)‖1 ≤ ρ1 ,

‖u(t)‖2 ≤ ρ2 ,

∀t ≥ t0 .

(5.2.14)

Theorem 5.2.2. Let m be large enough, so that 2

λm+1

2√2C7 3 ≥ max{1, ( ) } C6

(5.2.15)

then every solution u(t) = t(t) + q(t) of equation (5.1.1) with initial data ‖u(0)‖ ≤ ρ satisfies −2

3 ‖q(t)‖2 ≤ ‖q(t0 )‖2 exp[−μ1 λm+1 (t − t0 )] + C7 λm+1

(5.2.16)

for all t ≥ t0 , and dq(t) − 32 , ≤ C8 λm+1 dt

∀t ≥ t1

(5.2.17)

184 | 5 Inertial manifolds of incompressible non-Newtonian fluids where t1 (ρ) = max{t0 +

C7 =

1 log( μ1 λm+1

C7 3 2

ρ0 λm+1

1 ), } 8

3‖f ‖2L∞ (0,T;H) + 3C0 ρ21 ρ22 + 12[μ0 K0 (ε

p−2 2

2μ21

(4−n)(p−1) 4

ρ1 + C0 ρ1

ρn(p−1) )]2 2

C8 = 16C2 √2C7 . Proof. Let t ≥ t0 . Taking the inner product of q and equation (5.2.13) inner product, we have 1 1 d ‖q‖2 + 2μ1 ‖A 2 q‖2 2 dt ≤ ‖f ‖L∞ (0,T;H) ‖q‖ + |b(u, u, q)| + (Ap (u), q)

(Ap (u), q)

≤ 2μ0 ∫(ε + |e(u)|2 )

p−2 2

(5.2.18)

|e(u)||e(q)|dx

Ω

≤ 2μ0 ∫(ε

p−2 2

|e(u)||e(q)| + |e(u)|p−1 |e(q)|)dx

Ω

≤ 2μ0 K0 ∫(ε Ω

≤ 2μ0 K0 (ε ≤ 2μ0 K0 (ε

p−2 2

p−2 2 p−2 2

|∇u||∇q| + |∇u|p−1 |∇q|)dx

‖u‖1 + C0 ‖∇u‖p−1 )‖∇q‖ L4 (4−n)(p−1) 4

ρ1 + C0 ρ1

1 3[μ0 K0 (ε μ ≤ 1 ‖A 2 q‖2 + 3

p−2 2

n(p−1) 4

ρ2

−1

1

4 )λm+1 ‖A 2 q‖ (4−n)(p−1) 4

ρ1 + C0 ρ1 μ1 −1

n(p−1) 4

ρ2

)]2

−1

2 λm+1

(5.2.19)

1

2 ‖f ‖L∞ (0,T;H) ‖q‖ ≤ ‖f ‖L∞ (0,T;H) λm+1 ‖A 2 q‖

≤

2 μ1 21 2 3‖f ‖L∞ (0,T;H) −1 ‖A q‖ + λm+1 3 4μ1

(5.2.20) −1

1

2 |b(u, u, q)| ≤ ‖u‖∞ ‖u‖1 ‖q‖ ≤ C0 ρ1 ρ2 λn+1 ‖A 2 q‖

≤

μ1 21 2 3C0 ρ21 ρ22 −1 ‖A q‖ + λm+1 . 3 4μ1

By equation (5.2.18), we deduce that 1 d − 21 ‖q‖2 + μ1 ‖A 2 q‖2 ≤ C7 μ1 λm+1 dt

(5.2.21)

5.2 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids |

185

−1

d ‖q‖2 dt

2 , using the Gronwall inequality, equation (5.2.16) is + μ1 λm+1 ‖q‖2 ≤ C7 μ1 λm+1 proved.

−3

4 , in particular, then equation (5.2.7) follows that t1 > 0, We set ρ = √2C7 λm+1

‖q‖ ≤ ρ,

∀t ≥ t1 .

By virtue of Theorem 5.2.1, we have dq(t) 4ρ , ≤ dt δ(ρ) Since from equation (5.2.15), we have δ(ρ) ≥

∀t ≥ t1 .

1 ; 4C2

so, equation (5.2.17) is proved.

Corollary. Let m be large enough to satisfy equation (5.2.15). Then for every solution u(t) = r(t) + q(t) on the global attractor, we have −3

2 ‖q(t)‖2 ≤ C7 λm+1 ,

∀t ∈ R+

and dq(t) − 43 , ≤ C8 λm+1 dt

∀t ∈ R+ .

5.2.2 Approximated inertial manifolds Let B = {r ∈ PH : ‖r‖2 ≤ 2ρ2 }

B⊥ = {q ∈ QH : ‖q‖2 ≤ 2ρ2 }. In order to construct a steady approximate inertial manifold ϕs : B → QH satisfying 2μ1 Aϕs (r) − 2μ0 QAp (r + ϕs (r))

+ QB(r + ϕs (r), r + ϕs (r)) = Qf ,

r ∈ B.

(5.2.22)

The graph of ϕs denoted by μs = graph ϕs is an analytic manifold, and it also is an approximate inertial manifold. Theorem 5.2.3. Let m be large enough so that 12

λm+1 ≥ max{1, (

16 C9 6−n ) , (2C10 ) 8−n } 2ρ2

where C9 =

μ0 K0 (p − 1)(ε

p−2 2

−1

4 2 ρ2 + C0 ρp−1 2 ) + C0 λ1 ρ2 + ‖f ‖L∞ (0,T;H) 2μ1

(5.2.23)

186 | 5 Inertial manifolds of incompressible non-Newtonian fluids C10 =

p−2 K0 −1 − (4−n)(p−2) − 8−n [(p − 1)ε 2 + C0 λ1 4 ρp−2 + C0 λ1 4 ρ2 + C0 λ1 16 ρ2 ] 2 2μ1

then there exists a unique mapping ϕs : B → QH, satisfying equation (5.2.22), whose graph defines a μs = graph ϕs analytic manifold. Moreover, − 6−n

12 , ‖ϕs (r)‖2 ≤ C9 λm+1

∀r ∈ B.

Proof. For each r ∈ B, we define Tr (q) =

1 [A−1 (2μ0 )QAp (r + q) 2μ1

− QB(r + q, r + q) + Qf ],

‖Tr (q)‖ =

1 1 [2μ0 ‖A− 2 QAp (r + q)‖ 2μ1 1

∀q ∈ B⊥

(5.2.24)

1

− ‖A− 2 QB(r + q, r + q)‖ + ‖A− 2 Qf ‖]

(5.2.25)

from equation (5.2.7), we infer that 1

2u0 ‖A− 2 QAp (r + q)‖ −1

2 ‖Ap (u)‖ ≤ 2μ0 λm+1

≤ μ0 K0 (p − 1)(ε ≤ μ0 K0 (p − 1)(ε

p−2 2 p−2 2

12p−12−n 12

‖u‖2 + C0 ‖u‖2

− 6−n

n

−1

2 ‖Au‖ 12 )λm+1

12 ρ2 + C0 ρp−1 2 )λm+1

(5.2.26)

1

‖A− 2 QB(r + q, r + q)‖ −1

2 ≤ λm+1 ‖B(u, u)‖

−1

−1

2 ≤ C0 λ1 4 ‖u‖22 λm+1

−1

−1

2 ≤ C0 λ1 4 ρ22 λm+1 1

(5.2.27) −1

2 . ‖A− 2 Qf ‖ ≤ ‖f ‖L∞ (0,T;H) ⋅ λm+1

(5.2.28)

From equations (5.2.25)–(5.2.28), it follows that − 6−n

12 ≤ 2ρ2 . ‖Tr (q)‖2 ≤ C9 λm+1

So Tr : B⊥ → B⊥ . We now show that Tr is a contraction. From the differentiating equation (5.2.24), we obtain 𝜕Tr (q)η 1 −1 𝜕Ap (r + q)η = A {Q η − Q[B(η, r + q) + B(r + q, η)]} 𝜕q 2μ1 𝜕q

5.2 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids | 187

=

p−2 p−2 1 {A−1 Q∇[(ε + |e(u)|2 ) 2 e(η) + (p − 2)(ε + |e(u)|2 ) 2 e(u)e(u)e(η)] 2μ1

− A−1 Q[B(η, r + q) + B(r + q, η)]}

𝜕T (q)η p−2 − 41 1 r (ε + |e(u)|2 ) 2 e(η) {(p − 1)λm+1 ≤ 𝜕q 2μ1 −1

2 (‖B(η, u)‖ + ‖B(u, η)‖)} + λm+1 p−2 K0 − 41 λm+1 (ε 2 ‖η‖1 + ‖∇u‖p−2 ≤ ‖∇η‖L4 ) L4(p−2) 2μ1

−1

2 (‖η‖L4 ‖∇u‖L4 + ‖u‖L4 ‖∇η‖L4 ) + λm+1 1 K − 4 p−2 (ε 2 ‖η‖1 + C0 ‖∇u‖p−2 ≤ 0 λm+1 ‖∇η‖L4 ) L4 2μ1

−1

2 (‖η‖L4 ‖∇u‖L4 + ‖u‖L4 ‖∇η‖L4 ) + λm+1 p−2 − 1 (4−n)(p−2) K0 − 41 − 4−n 4 16 ) λm+1 (ε 2 λm+1 λm+1 ≤ + C0 λ− 4 ρp−2 2 2μ1

−1

−1

− 8−n

− 8−n 16

16 2 (C0 λ1 4 ρ2 λm+1 + λm+1 + C0 λ1 8−n 1 − 16 ‖η‖2 ≤ ‖η‖2 . ≤ C10 λm+1 2

− 4−n

16 ρ2 λm+1 ‖η‖2 )

It follows that Tr is a contraction mapping. thus for ∀r ∈ B there exists a unique fixed point of ϕs (r), which defines Hϕs : B → B⊥ satisfying equation (5.2.22); ϕs is an analytic manifold. Theorem 5.2.4. Let m be large enough so that equation (5.2.15) and equation (5.2.23) hold, and − 8−n(p−2)

≥ μ1

2μ1 − (C11 + C12 )λm+1 16

(5.2.29)

satisfies. Then for every ∀ρ > 0 and for every solution u(t) = r(t) + q(t) with ‖u(0)‖2 ≤ ρ, we have C − 43 s A(q(r) − ϕ (r)) ≤ 8 λm+1 μ1

(5.2.30)

and ‖q(r) − ϕs (r)‖2 ≤

C8 − 45 λ μ1 m+1

for all t ≥ t1 , there t1 and C8 are given in Theorem 5.2.2, C11 = (p − 1)2p−2 ε

p−2 2

+ (p − 1)2p−2 ε

K0 n2 p−2 2

− (p−2)(4−n) 16

K0 C0 n2 λ1

(5.2.31)

188 | 5 Inertial manifolds of incompressible non-Newtonian fluids

× [2ρ2 + (2ρ2 )−

(p−2)(4−n) 4

+ 2ρ2 (p − 2)(7 − p)

p−4 2

− 16−n 16

n

C12 = 2ρC0 [(4ρ2 )− 4 (λ1

C1 (ρ)

n(p−2) 4

2

]

K0 C0 n (2ρ2 + C9 )

+ 1) + λ−

16−3n 16

].

Proof. Let w(t) = q(t) − ϕs (r)(t), t ≥ t1 . From equation (5.2.13) and equation (5.2.22), we have 2μ1 Aw + Q(Ap (u) − Ap (r + ϕs )) + Q[B(w, r + ϕs ) + B(u, w)] +

dq = 0. dt

(5.2.32)

From Theorem 5.2.3 and λm ≤ λm+1 , we infer that 2 s p−2 p−2 Ap (u) − Ap (r + ϕ ) ≤ (p − 1)2 ε 2 K0 n ‖w‖2 + (p − 1)2p−2 ε

× (2ρ2 )−

p−2 2

(p−2)(4−n) 4

− (p−2)(4−n) 4

K0 C0 n2 λ1 n(p−2) 4

‖r + q‖3

+ 2ρ2 (p − 2)(7 − p) + {(p − 1)2p−2 ε

p−2 2

− n(p−2) 16

n(p−2) 4

− 8−n(p−2)

K0 C0 n2 ‖r + ϕs ‖3 ‖w‖2 − (p−2)(4−n) 4

+ (2ρ2 )−

(p−2)(4−n) 4

] + (p − 1)2p−2 ε

+ 2ρ2 (p − 2)(7 − p) ≤ C11 λm+1 16

‖w‖2

K0 C0 n2 λ1

× [(2ρ2 )p−2 λm × C1 (ρ)

p−4 2

p−4 2

p−2 2

− n(p−2)

λm+116

K0 n2

1

K0 C0 n2 (2ρ2 λm4 + C9 )}‖w‖2 (5.2.33)

‖Aw‖

s B(w, r + ϕ ) + ‖B(u, w)‖ ≤ ‖w‖L4 ‖∇(r + ϕs )‖L4 + ‖u‖L4 ‖∇w‖L4 n

− 16−n

− 16−n

− 4−n

16 ≤ 2ρ0 C0 [(4ρ2 ) 4 λm+116 (λm+116 + λm+1 )

− 16−3n − 12−n 16 λm+116 ]‖Aw‖

+λ1

− 8−n(p−2)

≤ C12 λm+1 16

− 12−n

≤ C12 λm+116 ‖Aw‖ (5.2.34)

‖Aw‖

dq − 43 . ≤ C8 λm+1 dt

(5.2.35)

We infer from equations (5.2.32)–(5.2.35) that − 8−n(p−2)

2μ1 ‖Aw‖ − (C11 + C12 )λm+1 16

−3

4 ‖Aw‖ ≤ C8 λm+1

5.2 Approximated inertial manifolds of incompressible bipolar non-Newtonian fluids | 189

C

−3

4 then it follows from equation (5.2.29) that ‖Aw‖ ≤ μ8 λm+1 , equation (5.2.30) is proved. 1 The estimate equation (5.2.31) follows directly from equation (5.2.30), so Theorem 5.2.4 is proved completely.

The mapping ϕs , which is given by the implicit relation equation (5.2.22). In the next theorems, we introduce a sequence {ϕi }, i = 1, 2, . . . of simple explicit approximating functions for ϕs . These approximating functions will also be associated with approximate inertial manifolds, which are better for computations. Theorem 5.2.5. Let m be large enough so that equation (5.2.23) holds, and define ϕ0 (r) = 0,

∀r ∈ B

ϕi+1 (r) = Tr (ϕi (r)),

∀r ∈ B, i = 1, 2, . . .

(5.2.36)

then − 12−n

‖ϕi (r)‖2 ≤ C13 λm+124

(5.2.37)

and − 12−n

‖ϕs (r) − ϕi (r)‖2 ≤ C13 αi λm+124

(5.2.38)

where − 8−n

16 α = C10 λm+1 p−2 1 {2μ0 K0 (p − 1)ε 2 ρ2 + μ0 K0 (p − 1)C0 (2ρ2 )p−1 C13 = 2μ1

−1

+ 4C0 λ1 4 ρ22 + ‖f ‖L∞ (0,T;H) }. Proof. Let r ∈ B be fixed. We find that from equation (5.2.22), ‖ϕ1 (r)‖2 ≤

1 [‖B(r, r)‖ + 2μ0 ‖Ap (r)‖ + ‖f ‖L∞ (0,T;H) ]. 2μ1

Similarly, equation (5.2.7) and equation (5.2.8) get 2μ0 ‖Ap (r)‖ ≤ 2μ0 K0 (p − 1)ε −1

p−2 2

n

ρ2 + μ0 K0 (p − 1)C0 (2ρ2 )p−1 λm24

‖B(r, r)‖ ≤ 4C0 λ1 4 ρ22 . So since λm+1 ≥ λm , we have − 12−n

‖ϕ1 (r)‖2 ≤ C13 λm+124 . Since Tr a contraction mapping, with a contraction constant α, we have ‖ϕk+1 (r) − ϕk (r)‖2 ≤ αk ‖ϕ1 (r) − ϕ0 (r)‖2

190 | 5 Inertial manifolds of incompressible non-Newtonian fluids = αk ‖ϕ1 (r)‖2 ,

k = 0, 1, 2, . . .

∞

ϕs (r) − ϕi (r) = ∑[ϕk+1 (r) − ϕk (r)] k=i

then ∞

‖ϕs (r) − ϕi (r)‖2 ≤ ∑ αk ‖ϕ1 (r)‖2 ≤ 2αi ‖ϕ1 (r)‖2 . k=i

This inequality combines with equation (5.2.37) to get equation (5.2.38). Therefore, Theorem 5.2.5 is proved. Theorem 5.2.6. Let m be large enough so that equation (5.2.15), equation (5.2.23) and equation (5.2.29) hold, then for ∀ρ > 0, ‖u(0)‖ ≤ ρ, and for every solution u(t) = r(t)+q(t) of equation (5.1.1), we have C − 45 − 12−n + 2C13 αi λm+124 . q(r) − ϕi (r(t))2 ≤ 8 λm+1 μ1

(5.2.39)

Proof. This proof is obtained directly from equation (5.2.31) and equation (5.2.38).

6 The regularity of solutions and related problems This chapter contains the existence and uniqueness of a stationary solution of incompressible bipolar non-Newtonian fluids; the decay estimates and partial regularity of solutions of one kind of incompressible monopolar non-Newtonian fluids; the convergence between perturbed equation and original equation.

6.1 Stationary solutions of the incompressible bipolar non-Newtonian fluids Consider the initial boundary value problem of incompressible bipolar non-Newtonian fluids: 𝜕ui 𝜕u 𝜕 𝜕 + uj i = (γ(u)eij ) − 2μ1 (Δeij ) + fi , 𝜕t 𝜕xj 𝜕xj 𝜕xj ∇ ⋅ u = 0,

(x, t) ∈ Ω × [0, T],

ui (x, t) = τijk νj νk = 0,

(6.1.2)

x ∈ 𝜕Ω,

where Ω ⊂ Rn , n = 2, 3, is a bounded open set, τijk = to 𝜕Ω, or the periodic boundary conditions ui (x, t) = ui (x + Lej , t),

𝜕eij 𝜕xk

(6.1.1)

t ≥ 0,

(6.1.3)

and ν is exterior unit normal

t ≥ 0,

(6.1.4)

where Ω = [0, L]n , L > 0, and {ei }ni=1 is the canonical basis of Rn , ∫Ω u(x, t)dx = 0. Let H = {u ∈ L2 (Ω) : ∇ ⋅ u = 0, (6.1.3) or (6.1.4)}, V = {u ∈ H 2 (Ω) : ∇ ⋅ u = 0, (6.1.3) or (6.1.4)}. We use (⋅, ⋅) and ‖ ⋅ ‖ to denote the inner product and norm of H, respectively, and ‖ ⋅ ‖1 – the norm of H 1 (Ω); ‖ ⋅ ‖2 – the norm of V; ‖ ⋅ ‖3 – the norm of H 3 (Ω); ‖ ⋅ ‖X – the norm of Banach space X. We introduce the linear operator A as follows: consider the positive definite V-elliptic symmetric bilinear form a(⋅, ⋅) : V × V → R given by a(u, υ) = ∫ Δu Δυdx,

(u, υ ∈ V).

D

As a consequence of the Lax–Milgram lemma, we obtain an isometry A ∈ L (V, V ), ⟨Au, υ⟩V ×V = a(u, υ) = ⟨f , υ⟩V ×V , https://doi.org/10.1515/9783110549614-006

∀υ ∈ V,

192 | 6 The regularity of solutions and related problems where V is the dual space of V, and the domain of A is D(A) = {u ∈ V : a(u, υ) = (f , υ), f ∈ H ⊂ V , ∀υ ∈ V}. The abstract problem of incompressible bipolar non-Newtonian fluids can be described by ut + 2μ1 Au − 2μ0 Ap (u) + B(u, u) = f ,

u(x, 0) = u0 (x),

where (Au)i = (Ap u)i =

(6.1.5)

𝜕 (Δeij ), 𝜕xj p−2 𝜕 [(ϵ + |e|2 ) 2 eij ], 𝜕xj

B(u, v) = u ⋅ ∇v. We focus on the stationary solution of the following equation [39], 2μ1 Au − 2μ0 Ap (u) + B(u, u) = f .

(6.1.6)

Theorem 6.1.1. Assume that bounded domain Ω ⊂ Rn (n = 2, 3) with the boundary conditions (6.1.3) or (6.1.4), and 2 < p < 3 are satisfied. Then (1) If f ∈ V , there exists at least one solution of (6.1.6); (2) f ∈ H, all the solutions of (6.1.6) belong to D(A); (3) If −5

μ21 − C0 ‖f ‖λ1 4 > 0,

(6.1.7)

the solution of (6.1.6) is unique. Proof. (1) First, we need to prove the existence of the solution. Let Em = Span{ϕ1 , . . . , ϕm }, with ϕ1 , . . . , ϕm being the first m eigenfunctions of A, by using the Galerkin method, we construct an approximate solution of (6.1.6) as follows: m

um = ∑ Cim ϕi , i=1

Cim ∈ ℝ

satisfying (2μ1 Aum − 2μ0 Ap (um ) + B(um , um ), ν) = (f , ν),

(6.1.8)

∀ν ∈ Em . Equation (6.1.8) is equivalent to 2μ1 Aum − 2μ0 Pm Ap (um ) + Pm B(um , um ) = Pm f , where Pm : H → Em is an orthogonal projection.

(6.1.9)

6.1 Stationary solutions of the incompressible bipolar non-Newtonian fluids | 193

The existence of a solution um of (6.1.9) follows from the Brouwer fixed-point theorem. Taking ν = um in (6.1.8), and noting that (B(um , um ), um ) = b(um , um , um ) = 0, we have μ1 ‖um ‖22 + μ0 ∫(ϵ + |e(um )|2 )

p−2 2

|e(um )|2 dx ≤ ‖um ‖2 ‖f ‖V .

(6.1.10)

Ω

Thus ‖um ‖2 ≤

1 ‖f ‖ . μ1 V

(6.1.11)

We extract from um a sequence still denoted by um , which converges to u weakly in V, and since V →→ H is compact, um → u weakly in V, um → u strongly in H. Let m → ∞, by passing to the limit in (6.1.8), we find that u is a solution of (6.1.6). (2) The next thing is to prove that the solution belongs to D(A). Due to (6.1.6), we have μ1 ‖Au‖ ≤ ‖f ‖ + ‖B(u, u)‖ + 2μ0 ‖Ap (u)‖,

(6.1.12)

noting that (Ap (u))i =

p−2 𝜕 [(ϵ + |e(u)|2 ) 2 eij (u)] 𝜕xj

= (ϵ +

2 ) ∑ ekl k,l

p−2 2

⋅

𝜕eij (u) 𝜕xj

+ (p − 2)(ϵ +

2 ) ∑ ekl k,l

p−4 2

eij (u)(∑ ekl (u) k,l

ekl (u) ). 𝜕xj

(6.1.13)

From (6.1.11), we get ‖u‖2 ≤

‖f ‖V 1 (≤ ‖f ‖, μ1 μ1 λ1

f ∈ H).

(6.1.14)

Applying the Sobolev inequality, Korn’s inequality and (6.1.14), we deduce 2 p−2

2μ0 ‖Ap (u)‖ ≤ 2μ0 K0 (∫(ϵ + |∇u| ) Ω

≤ 4μ0 K0 (p − 1)[ϵ ≤ 4μ0 K0 (p − 1)[ϵ μ ≤ 1 ‖Au‖ + C1 , 2

p−2 2 p−2 2

2

|Δu| dx)

1 2

‖u‖2 + C0 ‖Δu‖L4 ‖∇u‖p−2 ] L4 n

8(p−1)−n 8

‖u‖2 + C0 ‖Au‖ 8 ‖u‖2

] (6.1.15)

194 | 6 The regularity of solutions and related problems where C1 =

4μ0 K0 (p − 1)ϵ μ1 λ1

‖B(u, u)‖ ≤ C0 ‖u‖2 ‖u‖1 ≤

p−2 2

‖f ‖

8

+(

−1 C0 λ1 4 ‖u‖22

8μ0 K0 C0 (p − 1) 8−n ‖f ‖ ) ( ) μ1 μ1 λ1

8(p−1)−n 8−n

,

−9

≤

C0 ‖f ‖2 λ1 4 μ21

(6.1.16) (6.1.17)

.

Combining (6.1.12) with (6.1.15)–(6.1.17), we obtain ‖Au‖ ≤ C2 ,

(6.1.18)

where −9

C0 ‖f ‖2 λ1 4 2 C2 = (‖f ‖ + C1 + ). μ1 μ21 (3) Finally, we show the uniqueness. Let u1 , u2 be solutions of (6.1.6), and ν = u1 − u2 , then μ1 ‖u1 − u2 ‖22 − 2μ0 (Ap (u1 ) − Ap (u2 ), ν) ≤ |b(ν, u2 , ν)|,

(6.1.19)

From (3.1.46) in Chapter 3, we have − 2μ0 (Ap (u1 ) − Ap (u2 ), ν) ≥ ϵ

p−2 2

K0 ‖ν‖21 ≥ 0.

(6.1.20)

Combining (6.1.14) with the Sobolev inequality, we get |b(ν, u2 , ν)| ≤

C0 ‖ν‖2L∞ ‖u2 ‖1

≤

−1 C0 λ1 4 ‖ν‖22 ‖u2 ‖2

−5

C0 λ1 4 ‖f ‖ 2 ‖ν‖2 . ≤ μ1

(6.1.21)

It is easy to see that −5

C0 λ1 4 ‖f ‖ (μ1 − )‖ν‖22 ≤ 0. μ1

(6.1.22)

From the assumption (6.1.7), ‖ν‖22 = 0. The proof of uniqueness is completed. Theorem 6.1.2. Assume that f ∈ H, μ1 λ1 − C0 C2 ≥ 0, C2 is given by (6.1.18). Then the solution of (6.1.6) is unique (denoted by u). Assume that u0 ∈ H, f (t) ≡ f , if u(t) is any weak of (6.1.5), then u(t) → u in H, as t → ∞. Proof. Let ω(t) = u(t) − u, then we have d ω(t) + 2μ1 Aω(t) − 2μ0 (Ap (u) − Ap (u)) + B(u, u) − B(u, u) = 0. dt

6.2 Decay estimates of one kind of incompressible monopolar non-Newtonian fluid | 195

Taking the inner product with ω(t), combining with the result of (6.1.20), we get d ‖ω(t)‖2 + 2μ1 ‖ω(t)‖22 ≤ 2|b(ω, u, ω)|. dt

(6.1.23)

By the Sobolev inequality and (6.1.18), we have 2|b(ω, u, ω)| ≤ 2C0 ‖ω‖2 ‖∇u‖L∞ −1

≤ 2C0 λ1 4 ‖Au‖‖ω(t)‖2

≤ 2C0 C2 ‖ω(t)‖2 .

(6.1.24)

Then d‖ω(t)‖2 + (2μ1 λ1 − 2C0 C2 )‖ω(t)‖2 ≤ 0. dt Applying the Gronwall inequality and assumption r = μ1 λ1 − C0 C2 ≥ 0, then ‖ω(t)‖ ≤ ‖ω(0)‖e−rt → 0, t → ∞. u(t) converges to u exponentially. If we let ω(t) = u∗ − u, u∗ is another stationary solution of (6.1.6), then 2r‖u∗ − u‖2 ≤ 0. the condition μ1 λ1 − C0 C2 ≥ 0 in Theorem 6.1.2 ensures that u∗ = u, i. e., is a sufficient condition for uniqueness of a stationary solution. The theorem is proved completely.

6.2 Decay estimates of one kind of incompressible monopolar non-Newtonian fluid The authors in [49] considered the L2 decay of the weak solutions of the Navier-Stokes equations in Rn . We intend to study the L2 decay rates for the solution to a class system of non-Newtonian fluids in Rn [45]. Consider the equation ut − ∇ ⋅ τ + u ⋅ ∇u + ∇P = f ,

∇ ⋅ u = 0,

(6.2.1) (6.2.2)

where τ = 2(μ0 + μ1 |e|r )e, e = (eij ),

p = r + 2,

(6.2.3)

𝜕uj 1 𝜕u eij = ( i + ), 2 𝜕xj 𝜕xi

(6.2.4)

196 | 6 The regularity of solutions and related problems with the initial value u(0) = a(x).

(6.2.5)

We study the L2 decay rate of solution for a non-Newtonian fluids system in the case of ℝn (n ≥ 2), which can be transformed to a Navier–Stokes system, either μ1 = 0 or r = 0 with τ = 2μ2 e. For simplicity, let f = 0, and denoted Lp -norm by ‖ ⋅ ‖Lp , and L2 -norm by ‖ ⋅ ‖. 2 L = {u ∈ L2 , div u = 0} The Fourier transformation of a function f is denoted by f ̂ σ

of ℱ (f ).

Lemma 6.2.1. Let n ≥ 2, p ≥ 3 u(t) be the solution of (6.2.1), then we have t

t

0

0

‖u(t)‖2 exp(∫ d2 (s)ds) ≤ ‖a‖2 + C ∫ +d

n+2

s

d exp(∫ d2 (τ)dτ) × {‖u0 (t)‖2 ds 0

(s) + d

n+2

s

2

2

(s)(∫ ‖u(τ)‖ dτ) }ds,

(6.2.6)

0

where d(t) is radius of a ball. Proof. Taking the inner product of equation (6.2.1) with u, we have d ‖u‖2 + 4μ0 ‖∇u‖2 + 4μ1 ∫ |e(u)|p dx = 0. dt

(6.2.7)

ℝn

Integrating (6.2.7) over (0, t), we get t

‖u‖ + C ∫(‖∇u‖2 + ‖e(u)‖2 + ‖e(u)‖pLp )ds ≤ ‖a‖2 . 2

(6.2.8)

0

Here, we have used the fact that ‖e(u)‖2 ≤ ‖∇u‖2 . Due to p ≥ 3, we have p − 1 ∈ [2, p), using the interpolation inequality, we get t

≤ C. ∫ ‖e(u)‖p−1 Lp−1

(6.2.9)

t r ∫ ∫ |e(u)| eij (u)dxds ≤ C. n

(6.2.10)

0

It follows from p = r + 2 that

0ℝ

Applying the Plancherel theorem to (6.2.7), we have d ̂ |2 dξ + C ∫ |e(u)|p dx ∫ |u dt ℝn

ℝn

6.2 Decay estimates of one kind of incompressible monopolar non-Newtonian fluid | 197

̂ |2 dξ = −C ∫ |ξ |2 |u ℝn

̂ |2 dξ = −C( ∫ + ∫ )|ξ |2 |u s(t)

s(t)C

̂ |2 dξ − Cd2 (t) ∫ |u ̂ |2 dξ ≤ −C ∫ |ξ |2 |u s(t)

s(t)C

̂ |2 dξ − Cd2 (t) ∫ |u ̂ |2 dξ . ≤ C ∫ d2 (t)|u

(6.2.11)

ℝn

s(t)

Hence, d ̂ |2 dξ + Cd2 (t) ∫ |u ̂ |2 dξ + ∫ |e(u)|p dx ≤ C ∫ d2 (t)|u ̂ |2 dξ , ∫ |u dt ℝn

ℝn

ℝn

(6.2.12)

s(t)

where s(t) = {ξ ; |ξ | ≤ d(t)}, the function d(t) decreases monotonously and will be specified later on. Taking the Fourier transform of (6.2.1) to obtain ̂ t + |ξ |2 u ̂ = G(ξ , t), u

(6.2.13)

r

G(ξ , t) = ℱ (u ⋅ ∇u) + ℱ (∇P) − ℱ (2∇ ⋅ |e(u)| e(u)) = I1 + I2 + I3 .

(6.2.14)

Equation (6.2.13) yields t

2

2

̂ + ∫ e−|ξ | (t−s) G(ξ , s)ds. ̂ = e−|ξ | t a u

(6.2.15)

0

Then 2

̂ |2 dξ ̂ |2 dξ ≤ C ∫ e−2|ξ | t |a ∫ |u s(t)

s(t) t

+ C ∫ (∫ e s(t)

−|ξ |2 (t−s)

2

G(ξ , s)ds) dξ .

(6.2.16)

0

We now estimate the right-hand side of (6.2.16) term-by-term, 2

2

2

̂ |2 dξ ≤ ∫ (e−|ξ | t |a ̂ |) dξ = ‖u ̂ (0)‖2 = ‖u0 (t)‖2 . ∫ e−2|ξ | t |a ℝn

s(t)

For the term I1 , due to ℱ (u ⋅ ∇u) = −iξk ℱ (uj uk ), we obtain t

2

2

∫ (∫ e−|ξ | (t−s) ℱ (u ⋅ ∇u)ds) dξ s(t)

0

(6.2.17)

198 | 6 The regularity of solutions and related problems t

2

2

2

≤ C ∫ |ξ | (∫ ‖u‖ ds) dξ = C ⋅ d

n+2

t

(t)(∫ ‖u‖ ds) .

0

s(t)

2

2

(6.2.18)

0

As to I3 , from (6.2.10), we have t

∫ (∫ e s(t)

−|ξ |2 (t−s)

2

r

ℱ (∇ ⋅ |e(u)| e(u))ds) dξ

0 t

2

2

r

≤ C ∫ |ξ | dξ (∫ ∫ ||e(u)| e(u)|dxds) ≤ Cdn+2 (t).

(6.2.19)

0 ℝn

s(t)

The next to deal with I2 , it follows from (6.2.1) and (6.2.2) that ΔP =

𝜕2 (2|e(u)|r eij (u) − ui uj ). 𝜕xi 𝜕xj

Taking the Fourier transform, we have ̂ |ξ |2 P = ξi ξj (ℱ (2|e(u)|r eij (u)) − ℱ (ui uj )), ̂ P=

ξi ξj

|ξ |2

(6.2.20)

(ℱ (2|e(u)|r eij (u)) − ℱ (ui uj )).

(6.2.21)

Equations (6.2.18) and (6.2.19) yield t

∫ (∫ e s(t)

−|ξ |2 (t−s)

2

ℱ (∇p)ds) dξ ≤ Cd

n+2

t

2

2

(t)(1 + (∫ ‖u‖ ds) ).

0

(6.2.22)

0

Combining (6.2.12) and (6.2.16)–(6.2.22), we see that (6.2.6) holds. The proof of Lemma 6.2.1 is completed. Theorem 6.2.1. Let n ≥ 2 and p ≥ 3, and the initial value a ∈ L2 ∩ Lq (ℝn ). Moreover, assume that ‖f (t)‖ ≤ C(1 + t)−

α0 2

−1

,

α0 =

n n − , q 2

(6.2.23)

then ‖u(t)‖2 ≤ C(1 + t)−a0 ,

‖u(t) − u0 (t)‖2 → 0,

as t → ∞.

Especially, for f = 0, n ≥ 2 and p ≥ 3, the initial value a ∈ L2σ ∩ Lq (ℝn ); we have the same conclusion.

6.2 Decay estimates of one kind of incompressible monopolar non-Newtonian fluid | 199

Proof. For the sake of simplicity, we prove only the case f = 0. It is easy to prove the case of f ≠ 0 in a similar manner. In the case of n ≥ 3, from (6.2.6) and (6.2.8), we have t

t

‖u(t)‖2 exp(∫ d2 (s)ds) ≤ ‖a‖2 + C ∫ 0

0

s

d exp(∫ d2 (τ)dτ) ds 0

× {‖u0 (t)‖2 + dn+2 (s) + dn+2 (s)s2 }ds.

(6.2.24)

d2 (t) = α(1 + t)−1 ,

(6.2.25)

Taking

and α is large enough, then t

exp(∫ d2 (s)ds) = (1 + t)α .

(6.2.26)

0

Combining this with (6.2.24), we have t

‖u(t)‖2 (1 + t)α ≤ ‖a‖2 + C ∫(1 + s)α−1 0

× {‖u0 (s)‖2 + (1 + s)− −α0 +α

≤ C(1 + t)

n+2 2

+ (1 + s)−

n−2 2

}ds (6.2.27)

,

, α0 }, we get where α0 = min{ n−2 2 ‖u(t)‖2 ≤ (1 + t)−α0 .

(6.2.28)

The conclusion follows as α0 = α0 . Otherwise, substituting (6.2.28) into the right-hand side of (6.2.6), we obtain a new decay index α 0 satisfying α0 ≥ α 0 ≥ α0 , by at most finite iterations, we obtain ‖u(t)‖2 ≤ (1 + t)−α0 . In the case of n = 2, the decay rate of (6.2.28) is zero. We need two choices of d(t). First, we choose d2 (t) = k(t + e)−1 [ln(t + e)] , −1

k ∈ N,

so that t

k

exp(∫ d2 (s)ds) = [ln(t + e)] . 0

(6.2.29)

200 | 6 The regularity of solutions and related problems Then we can infer from (6.2.6) with n = 2 that k

‖u(t)‖2 [ln(t + e)] t

k−1

≤ ‖a‖2 + C ∫ k[ln(s + e)]

(e + t)−1 × {‖u0 (s)‖2

0 2

−2

s

−2

2

2

+ k [ln(s + e)] (e + s) (1 + ∫ ‖u(τ)‖ dτ) }ds.

(6.2.30)

0

Let k = 1. It follows from (6.2.30) that ln(e + t)‖u(t)‖2 ≤ C. Hence s

∫ ‖u(τ)‖2 dτ ≤ C(s + e) ln−1 (s + e). 0

Let k = 2 in (6.2.30). We get ln2 (e + t)‖u(t)‖2 ≤ C. Consequently, s

∫ ‖u(τ)‖2 dτ ≤ C(s + e) ln−2 (s + e).

(6.2.31)

0

Next, we choose d2 (t) = α(1 + t)−1 . Noting that ‖u0 (t)‖2 ≤ C(1 + t)−α0 , we can take α ∈ (α0 , 1). Then from (6.2.6) with n = 2 and (6.2.31), it follows that 2

α

t

α−α0

‖u(t)‖ (1 + t) ≤ C(1 + t)

α−3

+ C ∫(1 + s) α−α0

≤ C[1 + (1 + t)

0

s 0

]

t

s

0

0

+ C ∫(1 + s)α−2 ln−2 (e + s) ∫ ‖u(τ)‖2 dτds. For t ≥ 1, we define t

y(t) = ∫ ‖u(r)‖2 (1 + r)α dr; t−1

2

[1 + (∫ ‖u(τ)‖ dτ) ]ds

α−2

+ (1 + t)

2

Y(t) = max{y(x) | 1 ≤ x ≤ t}.

(6.2.32)

6.2 Decay estimates of one kind of incompressible monopolar non-Newtonian fluid | 201

It is easy to see that y(t), Y(t) are continuous. Now we estimate the factor s

I(s) = ∫ ‖u(τ)‖2 dτ, 0

due to s − [s] < 1, we have s−[s]

s

2

I(s) = ∫ ‖u(τ)‖ dτ + ∫ ‖u(τ)‖2 dτ s

s−[s]

s

≤ C + ∫ ‖u(τ)‖2 dτ s−[s] [s]−1

s−j

≤ C + ∑ ∫ ‖u(τ)‖2 (1 + τ)α (1 + τ)−α dτ j=0 s−j−1

[s]−1

≤ C + Y(s) ∑ (s − j)−α j=0

≤ C + Y(s)

(1 + s)1−α . 1−α

(6.2.33)

We integrate (6.2.32) over (t − 1, t) to get for t ≥ 1, α−α0

y(t) ≤ Y(t) ≤ C[1 + (1 + t)

+ (1 + t)

α−2

t

] + C ∫(1 + s)−1 ln−2 (e + s)Y(s)ds. 0

t

By the Gronwall inequality, and the fact that ∫0 (1 + s)−1 ln−2 (e + s)ds ≤ C, then we have for all t ≥ 1 that Y(t) ≤ C[1 + (1 + t)α−α0 + (1 + t)α−2 ]. From this and (6.2.33), we obtain a new estimate of I(s) as follows: I(s) ≤ C[1 + (1 + t)1−α + (1 + t)1−α0 + (1 + t)−1 ]. Plugging this new growth rate of I(s) into (6.2.32), we have ‖u(t)‖2 (1 + t)α ≤ C[1 + (1 + t)α−α0 + (1 + t)α−1 ]. Note that α0 < 1 and α can be taken large enough, and we obtain the decay rate for n = 2.

202 | 6 The regularity of solutions and related problems

6.3 Partial regularity of one kind of incompressible monopolar non-Newtonian fluid We intend to study the partial regularity of the generalized solutions to a class of system in the incompressible monopolar non-Newtonian fluids [46]. As we know, the motion of a continuous medium (incompressible, with viscosity) is described by the system ut − ∇ ⋅ τV + u ⋅ ∇u + ∇P = 0, div u = 0,

(6.3.1) (6.3.2)

where τV = 2(μ0 + μ1 |e(u)|r )e(u),

𝜕uj 1 𝜕u ). (eij (u)) = ( i + 2 𝜕uj 𝜕ui

e(u) = (eij (u)),

(6.3.3) (6.3.4)

We assume r is a nonnegative real number. For the sake of completeness, the following boundary and initial conditions should be added: u|𝜕Ω = 0,

(6.3.5)

u|t=0 = a(x).

(6.3.6)

It is proved that the singular points are concentrated on a closed set whose 5 − 2p (for p ≤ 52 ) dimensional Hausdorff measure is zero, and the solution is a regular one (for p > 52 ). The main results are the following. Theorem 6.3.1. There exists at least one suitable weak solution to the problems (6.3.1)– (6.3.6). Moreover, we have (i) For p ≤ 52 , the set of singular points S of the suitable weak solutions is of 5−2p dimensional Hausdorff measure zero. We prove first that 𝒫 5−2p (S) = 0, then ℋ5−2p (S) = 0 follows. (ii) For p > 52 , the weak solution is a regular one. For p < 3, ∀ϕ ∈ C0∞ (Ω × (0, T); R), ϕ ≥ 0, the generalized energy inequality is defined as follows: ∫ |u|2 ϕdx + 2 ∬(|∇u|2 + |e(u)|p )ϕdxdt Ω

≤ ∬ |u|2 (ϕt + Δϕ)dxdt − 2 ∬ |e(u)|r e(u) : (u ⊗ ∇ϕ)dxdt + ∬(|u|2 + 2P)u ⋅ ∇ϕdxdt. (6.3.7) Let f ≡ 0 in (6.2.1), multiplying 2ui ϕ in equation (6.2.1), and integrating the both sides of the equation, we can obtain (6.3.7).

6.3 Partial regularity | 203

Definition 6.3.1. The pair (u, P) is called a suitable weak solution to the system (6.3.1)–(6.3.6), if the following conditions are satisfied on an open D ⊂ ℝ3 × ℝ: (1) u, P are measurable functions on D. For some constants E0 , E1 < ∞, 5

P ∈ L 4 (D), ∫ |u|2 dx ≤ E0 , for almost every t, and ∬ |∇u|p dxdt ≤ E1 ,

(6.3.8)

D

Ω

(2) u, P satisfy (6.2.1) in the sense of distribution on D, (3) For each real-valued function ϕ ∈ C0∞ (D), ϕ ≥ 0, inequality (6.3.7) holds. Assume that D = Ω × (0, T), E0 (u) = sup ∫ |u|2 dx, 0≤t≤T T

Ω

E1 (u) = ∫ ∫ |∇u|p dxdt. 0 Ω

Theorem 6.3.2. Suppose that Ω, u0 satisfy the same conditions as in [65]; i. e., when Ω = R3 , we assume u0 ∈ H. For bounded Ω, we require that Ω is a bounded, open, connected set an R3 , lying locally on one side of its boundary, that 𝜕Ω is a smooth manifold, and that 2

the initial data u0 ∈ H ∩ W 55 . There exists a weak solution (u, P) of system (6.2.1)∼(6.2.5)

satisfying

4

u ∈ L2 (0, T; V) ∩ L∞ (0, T; H), u(t) → u0 ,

weakly in H, t → 0,

5 3

(6.3.9) (6.3.10)

3

∇P ∈ L (D), 5 4

∇P ∈ L (D),

as Ω = ℝ ,

(6.3.11)

as Ω bounded,

(6.3.12)

and the generalized energy inequality is satisfied. Proof. Assume that ψδ (u) is a retarded mollifier defined in [15]. To construct the suitable weak solution, we consider the approximate equation: 𝜕 u + ψ(uN ) ⋅ ∇uN − ΔuN − 2∇ ⋅ |e(uN )|r e(uN ) + ∇P N = 0, 𝜕t N

(6.3.13)

then we have ∫ |uN |2 ϕdx + 2 ∬ |∇uN |2 ϕdxdt = ∫ |u0 |2 ϕ(x, 0)dx + ∬ |uN |2 (ϕt + Δϕ)dxdt Ω

Ω 2

+ ∬(|uN | ψ(uN ) + 2PuN ) ⋅ ∇ϕdxdt + 2 ∬(uN ⋅ fN )ϕdxdt,

(6.3.14)

204 | 6 The regularity of solutions and related problems where fN = ∇ ⋅ |e(uN )|r e(uN ). Combining the estimates found in [65] and the procedure devised in [15], the proof of Theorem 6.3.2 is complete. Assume that a, b, c, d, e are nonnegative constants to be specified later, and let A(r) = r −a ∫ |u|2 dx, Br

G(r) = r

−c

E(r) = r

−e

D(r) = r

−d

∫ ∫ |u|3 dxdt, Qr

∫ ∫ |∇u|p dxdt, Qr 3

∫ |P| 2 dxdt. Qr

Lemma 6.3.1. The following interpolation inequality holds for any q ≥ 2: 3

q ‖u‖Lq (Br ) ≤ ‖∇u‖αLp (Br ) ‖u‖1−α L2 (B ) + C ⋅ r r

where

1 q

− 32

(6.3.15)

‖u‖L2 (Br ) ,

= α( p1 − 31 ) + (1 − α) 21 .

Proof. According to the Nirenberg inequality, we have 1 ‖u‖Lq (B1 ) ≤ C‖∇u‖αLp (B1 ) ‖u‖1−α L2 (B ) + C ‖u‖L2 (B1 ) , 1

where 1 1 1 1 = α( − ) + (1 − α) . q p 3 2 Then 1 q

(∫ |u(x)|q dx) = (r 3 ∫ |u(ry)|q dy) Br

1 q

B1 3 q

α p

p

2

≤ Cr (∫ |∇y (u(ry))| dy) (∫ |u(ry)| dy) B1

≤ Cr

3 α(p−3) 3 + p − 2 (1−α) q

1

3 q

2

+ C r (∫ |u(ry)| dy)

B1 p

2

(∫ |∇x (u(x))| dx) (∫ |u(x)| dx)

1−α 2

1

+C r

3 3 − q 2

2

(∫ |u(x)| dx)

Br α p

= C(∫ |∇x (u(x))|p dx) (∫ |u(x)|2 dx) Br

Thus, Lemma 6.3.1 is established.

1 2

B1 α p

Br

Br

1−α 2

1−α 2

Br 3

+ C1 r q

− 32

1 2

(∫ |u(x)|2 dx) . Br

1 2

6.3 Partial regularity | 205

Lemma 6.3.2. Assume that 1 + 3a , 2 C(5p − 6) − 3a(2p − 3) − 2(5p − 6) 4pC + 30 − (17 + 3a)p , }. e ≥ max{ 3 6

r ≤ ρ ≤ 1, C ≤

(6.3.16) (6.3.17)

Then c

3(4p−6) 3 ρ r G(r) ≤ C( ) A(ρ) 5p−6 E(ρ) 5p−6 + C( ) r ρ

9−3a 2

3

+c

3 3 ρ 2 A(ρ) + C( ) A(ρ) 4 E(ρ) 2p . r 3 2

(6.3.18)

Proof. Applying (6.3.15), let q = 3, and we have α=

p , 5p − 6

4p − 6 , 5p − 6

1−α= 3p 5p−6 p L (B

3

∫ |u| dx ≤ C‖∇u‖ Br

r)

3(4p−6) 5p−6 L2 (Br )

‖u‖

(6.3.19) + Cr

− 32

3 2

2

(6.3.20)

(∫ |u| dx) . Br

On the other hand, applying Poincaré inequality, |u|2ρ =

we have

1 ∫ |Bρ | Bρ

|u|2 dx and

1 q

=

1 2

− p1 ,

∫ |u|2 dx = ∫(|u|2 − |u|2ρ )dx + ∫ |u|2ρ dx

Br

Br

Br

3

r ≤ Cρ ∫ |u||∇u|dx + C( ) ∫ |u|2 dx ρ Bρ

Bρ

1 2

1 p

1 q

3

r ≤ Cρ(∫ |u| dx) (∫ |∇u| dx) (∫ 1 dx) + C( ) ∫ |u|2 dx. ρ 2

p

Bρ

Bρ

q

Bρ

(6.3.21)

Bρ

Therefore, we have 3

∫ ∫ |u| dxdt ≤ Cρ

3a(1−α) 2

A(ρ)

3(1−α) 2

Qr

0

3p

∫ ‖∇u‖L5p−6 p dt

−r 2 9 2

0

3

3 3 3 3a 3 r r 2 3a + 9 + C( ) r − 2 ρ 2 r 2 A(ρ) 2 + C( ) ρ 4 2q A(ρ) 4 ∫ (|∇u|p ) 2p dt ρ ρ

−r 2

α

3

≤ Cρg1 A(ρ)3(1− 2 ) E(ρ) 5p−6 +

Cr 5 ρ

9−3a 2

3 3 3 1 A(ρ) 2 + C( ) ρg2 A(ρ) 4 E(ρ) 2p , (6.3.22) r

where g1 =

3 2

3a(2p − 3) + 3e + 2(5p − 9) , 5p − 6

206 | 6 The regularity of solutions and related problems g2 =

15 + 3a 3e − 9 2p − 3 + + . 4 2p p

If the following inequality holds, 3 g2 ≥ C + , 2

g1 ≥ C,

5−C ≥

9 − 3a , 2

then G(r) = r −C ∫ ∫ |u|3 dxdt Qr

≤ Cr −C ρg1 A(ρ)

3(4p−6) 5p−6

3

3

3

4

E(ρ) 5p−6 + Cρg2 r − 2 −C A(ρ) 3 E(ρ) 2p +

Cr 5−C ρ

(9−3a) 2

3

A(ρ) 2 .

The proof of Lemma 6.3.2 is complete. Lemma 6.3.3. If d ≤ min{

3p2 − 8p − 3(p − 1)e + 15 6e + 3ap + 17p − 30 , , 3}, 2p 4p

(6.3.23)

ρ

there exists a constant θ0 ∈ (0, 1), for ∀r ∈ (θ0 ρ, 2 ), ρ ≤ 1 such that α

α

3−d

3(p−1) 3 3 ρ 2 ρ 1 r D(r) ≤ C( ) A(ρ) 4 E(ρ) 2p + C( ) E(ρ) 2p + ( ) r r ρ

D(ρ),

where 9 − 3e 3a + 9 − + d, 2p 4 3(p − 1) 3(p − 1)e 5(3 − p) α2 = −d+ + . 2 2p 2p α1 =

Proof. It is easy to show that for a. e. t ≥ 0, ΔP = −

𝜕2 (u u − |e(u)|r eij (u)). 𝜕xi 𝜕xj i j

Next, we divide the pressure P into three parts, P = P 0 + P 1 + P 2 , such that ΔP 0 = 0, P 0 = u, ΔP 1 = −

in Bρ , on 𝜕Bρ ,

𝜕2 ((u − ui )(uj − uj )), 𝜕xi 𝜕xj i

P 1 = 0,

on 𝜕Bρ ,

(6.3.24)

6.3 Partial regularity | 207

ΔP 2 =

𝜕2 (|e(u)|r eij (u)), 𝜕xi 𝜕xj

P 2 = 0,

in Bρ ,

on 𝜕Bρ .

For the terms P 0 , P 1 , we treat them in a way similar to [57]. As for the term P 2 , we handle it in a slightly different way. By the Calderon–Zygmond estimate, we have for ρ ≤ R2 , 3

∫ |P 2 | 2 dx ≤ C(R) ∫ |∇u|

3(p−1) 2

dx.

BR

Bρ

By scaling, we can easily find that C(R) = CR 3

∫ |P 2 | 2 dx ≤ CR

3(p−1) 2

Bρ

3(p−1) 2

. Thus

∫ |∇u|

3(p−1) 2

dx.

BR

Whence, 3

D(r) = r −d ∫ ∫ |P| 2 dxdt Qr 3

3

3

≤ Cr −d ∫ ∫ |P 0 | 2 + |P 1 | 2 + |P 2 | 2 dxdt Qr

≤ Cρ−d ∫ ∫ |u − u|3 dxdt Qρ

+ Cρ

3(p−1) −d 2

∫ ∫ |e(u)|

3(p−1) 2

Qρ

≤ C[

3−d

r dxdt + C( ) ρ

Qρ 3−d

2p−3 p

3

ρ−d ∫ ∫ |P| 2 dxdt

3(p−1) 3 3 r r A(ρ) 4 E(ρ) 2p + ρα2 E(ρ) 2p + ( ) ρα1 ρ

D(ρ)],

where 1 1 1 = − , q 2 p

1 3−p = , q1 2p

reminding (6.3.23), we have 2p − 3 9 − 3e 3a + 9 ≥ α1 = − + d, p 2p 4 3(p − 1)e 5(3 − p) 3(p − 1) −d+ + ≥ 0, α2 = 2 2p 2p 3 − d ≥ 0.

The proof is complete.

208 | 6 The regularity of solutions and related problems Lemma 6.3.4. Assume that the constants a, c, e, d satisfy the relations c=

3a + 1 , 2

d=

3a + 5 , 4

e ≥ 5 − 2p,

0 ≤ a ≤ 1,

and the following condition is satisfied: lim sup r −e ∫ ∫ |∇u|p dxdt ≤ ϵ, r→0

(6.3.25)

Qr

ϵ is a small enough constant. Then for 0 < θ0 < 21 , r < 1, we have 3 3 1 A(θ0 r) 2 + D(θ0 r)2 ≤ (A(r) 2 + D(r)2 ), 2

(6.3.26)

or 3

A(r) 2 + D(r)2 ≤ ϵ0 ≪ 1.

(6.3.27)

Proof. Let ϕ ∈ C0∞ (Qρ ), such that 0 ≤ ϕ ≤ 1,

(x, t) ∈ Qρ ,

ϕ(x, t) = 1,

(x, t) ∈ Qr ,

‖∇ϕ‖ ≤

‖ϕt ‖ + ‖Δϕ‖ ≤

C , ρ

C . ρ2

(6.3.28)

Taking ϕp as the testing function, then the general energy inequality becomes ∫ |u|2 ϕp dx + 2 ∬(|∇u|2 + |e(u)|p )ϕp dxdt Ω

≤ ∬ |u|2 ((ϕp )t + Δ(ϕp ))dxdt − 2p ∬ |e(u)|r e(u)ϕp−1 : (u ⊗ ∇ϕ)dxdt + p ∬(|u|2 + 2P)u ⋅ ϕp−1 ∇ϕdxdt. Let

p−1 p

+

1 3

+

1 q

(6.3.29)

= 1, and ∫Q ∫ |u|3 dxdt ≤ C, using Young inequality, we have ρ

r p−1 ∬ |e(u)| e(u)ϕ : (u ⊗ ∇ϕ)dxdt ≤ C ∬ |e(u)ϕ|p−1 |u||∇ϕ|dxdt p

≤ C(∬ |e(u)ϕ| dxdt)

p−1 p

1 3

3

q

(∫ ∫ |u| dxdt) (∫ ∫ |∇ϕ| dxdt) Qρ

1 q

Qρ p

2 3

q 1 ≤ ∬ |e(u)ϕ|p dxdt + C(∫ ∫ |u|3 dxdt) (∫ ∫ |∇ϕ|q dxdt) . 2

Qρ

Qρ

(6.3.30)

6.3 Partial regularity | 209

Then ∫ |u|2 ϕp dx + ∬(|∇u|2 + |e(u)|p )ϕp dxdt Ω

≤ ∬ |u|2 (|(ϕp )t | + |Δ(ϕp )|)dxdt + 2p ∬(|u|2 + 2P)u ⋅ ϕp−1 ∇ϕdxdt 2 3

3

p q

q

+ C(∫ ∫ |u| dxdt) (∫ ∫ |∇ϕ| dxdt) . Qρ

(6.3.31)

Qρ

Thus 2

2

∫ |u|2 dx ≤ Cρ(2C−1)/3 G(ρ) 3 + Cρa1 G(ρ) 3 Br 1

1

1

2

1

+ Cρa2 A(ρ) 2 G(ρ) 3 E(ρ) p + Cρa3 D(ρ) 3 G(ρ) 3 ,

(6.3.32)

where a1 , a2 , a3 satisfy 2 5p C+ , 3 q 3 C ea 2 + , a2 = −1 + + + q 3 p 2 q1 2d + C a3 = − 1, 3 a1 =

and

1 q

=

1 2

− p1 ,

1 q1

=

2p−3 1 ,q 3p 2

=

5p−9 . 15p

Multiplying (6.3.32) by r −a , we have 2

2

A(r) = r −a ∫ |u|2 dx ≤ Cr −a ρ(2C−1)/3 G(ρ) 3 + Cr −a ρa1 G(ρ) 3 Br 1

1

1

2

1

+ Cr −a ρa2 A(ρ) 2 G(ρ) 3 E(ρ) p + Cr −a ρa3 D(ρ) 3 G(ρ) 3 .

(6.3.33)

Assume that the following four inequalities hold: 2C − 1 ≥ a, 3 2 5p a1 = C + ≤ a, 3 q 3 C e a 2 ≥ a, a2 = −1 + + + + + q 3 p 2 q1 2d + C a3 = − 1 ≥ a, 3

(6.3.34) (6.3.35) (6.3.36) (6.3.37)

according to (6.3.34), we have C≥

3a + 1 , 2

(6.3.38)

210 | 6 The regularity of solutions and related problems contrasting (6.3.38) with (6.3.16), we get 3a + 1 . 2

C=

(6.3.39)

From (6.3.39) and q ≥ 0, (6.3.35) is established, and (6.3.36) can be written e ≥ 5 − 2p,

d≥

3(a + 1) − C 3a + 5 = . 2 4

(6.3.40)

On the other hand, it follows from (6.3.39) that e1 , e2 become e1 = e2 =

ap + 10 − 5p . 2

(6.3.41)

Therefore, we may assume that for all p ≥ 2, 1 ≥ a ≥ 0, e = 5 − 2p, e = 0,

if p ≤

5 if p > , 2

5 , 2

then (6.3.23) becomes d ≤ min{

13 − 3p 3a + 5 , }, 2 4

which combined with (6.3.40) yields that d ≤

13−3p . 2

(6.3.42)

Furthermore, d =

3a+5 , 4

a ≤ 7 − 2p.

that is, (6.3.43)

Taking a = 1, from (6.3.33), we have a

a

a

1 2 1 1 2 1 ρ ρ ρ A(r) ≤ C( ) G(ρ) 3 + C( ) A(ρ) 2 G(ρ) 3 E(ρ) p + C( ) D(ρ) 3 G(ρ) 3 . r r r

(6.3.44)

Applying Lemma 6.3.2 with ρ = 2r, combining (6.3.43) and Lemma 6.3.3, taking the constant ϵ suitably small, we have for 0 < θ0 < 21 , r < 1, either 3 3 1 A(θ0 r) 2 + D(θ0 r)2 ≤ (A(r) 2 + D(r)2 ), 2

or

3

A(r) 2 + D(r)2 ≤ ϵ0 ≪ 1. This completes the proof of Lemma 6.3.4. Theorem 6.3.3. Assume lim sup r 2p−5 ∫ ∫ |∇u|p dxdt ≤ ϵ, r→0

Qr

then |u(x, t)| ≤ C. That is, (x, t) is a regular point.

p≤

5 , 2

6.4 The convergence of solution and attractors |

211

Proof. From Lemma 6.3.4, for 0 < θ0 < 21 , r < 1, either 3 3 1 A(θ0 r) 2 + D(θ0 r)2 ≤ (A(r) 2 + D(r)2 ), 2

or 3

A(r) 2 + D(r)2 ≤ ϵ0 ≪ 1. Next, we can prove that for suitably small ϵ0 there exists a constant C0 such that 3

∫ ∫(|u|3 + |P|̄ 2 )dxdt ≤ ϵ0 , Q1

implies |u(x, t)| ≤ C0 . It follows from Theorem 6.3.3, the standard covering argument, and the fact ∫ ∫ |∇u|p dxdt ≤ C Qr

that the singular set of the suitable weak solution is of 5 − 2p dimensional Hausdorff measure zero. From this result, we see that if p > 52 , then the solution is a regular one.

6.4 The convergence of solution and attractors between one kind of incompressible non-Newtonian fluid and the Newtonian fluids Consider the following non-Newtonian fluid equations: uϵt − Δuϵ + uϵ ⋅ ∇uϵ − ϵ∇ ⋅ (|e(uϵ )|p−2 e(uϵ )) + ∇p2 = f , ∇ ⋅ uϵ (x, t) = 0, uϵ (x, t) = 0,

(x, t) ∈ Ω × ℝ+ ,

(x, t) ∈ 𝜕Ω × ℝ+ ,

uϵ (x, 0) = u0 (x),

x ∈ Ω.

ϵ > 0, 1 < p < 2, e = (eij ) = ( 21 ( 𝜕xi + 𝜕u

j

(6.4.1) (6.4.2) (6.4.3) (6.4.4)

𝜕uj )). 𝜕xi

When ϵ = 0, with the disappearance of

nonlinear stress, the Newtonian fluids follows, that is, Navier–Stokes equation: ut − Δu + u ⋅ ∇u + ∇p2 = f , ∇ ⋅ u(x, t) = 0,

(x, t) ∈ Ω × ℝ+ ,

(6.4.5) (6.4.6)

212 | 6 The regularity of solutions and related problems u(x, t) = 0,

(x, t) ∈ 𝜕Ω × ℝ+ ,

u(x, 0) = u0 (x),

x ∈ Ω,

(6.4.7) (6.4.8)

where Ω ⊂ ℝn is bounded open domain, 𝜕Ω is smooth enough. Given the basic space E = {u ∈ C ∞ (Ω) : div u = 0}, H = the closure of E in L2 , Vp = the closure of E in W 1,p . Especially, V = V2 . For u0 ∈ H, 1 < p ≤ 2, it is easy to obtain the existence, uniqueness of solution and global attractor corresponding to the dynamic systems (6.4.1)–(6.4.4). Aϵ = ⋂ ⋃ Sϵ (t)B0 , t≥0 s≥t

B0 = {u ∈ V; ‖u‖2 ≤ ρ0 },

(6.4.9)

where ρ0 is a constant independent of ϵ, 0 < ϵ ≤ ϵ0 . Likewise, the dynamic system (6.4.5)–(6.4.8) exists a unique solution and a global attractor A = ⋂ ⋃ Sϵ (t)B1 , t≥0 s≥t

B1 = {u ∈ V; ‖u‖2 ≤ ρ1 }.

(6.4.10)

Taking the inner product of (6.4.1) with uϵ , and noting the fact that (uϵ ⋅ ∇uϵ , uϵ ) = 0, we obtain 1 d ‖u ‖2 + ‖∇uϵ ‖2 + ϵ‖e(uϵ )‖pLp ≤ ‖f ‖‖uϵ ‖. 2 dt ϵ

(6.4.11)

Using Poincaré inequality, ‖∇uϵ ‖2 ≥ λ1 ‖uϵ ‖2 ,

(6.4.12)

λ1 > 0 is the first eigenvalue of Laplace operator −Δ under the boundary condition (6.4.3), ‖f ‖‖uϵ ‖ ≤

‖f ‖2 λ1 ‖∇uϵ ‖2 + , 2λ1 2

(6.4.13)

from (6.4.11), we have d ‖f ‖2 ‖uϵ ‖2 + λ1 ‖uϵ ‖2 + 2ϵ‖e(uϵ )‖pLp ≤ . dt λ1

(6.4.14)

Then ‖uϵ ‖2 ≤ ‖u0 ‖2 e−λ1 t +

‖f ‖2 . λ12

(6.4.15)

6.4 The convergence of solution and attractors |

213

There exists a constant R0 > 0, uniform estimate in t, such that ‖uϵ ‖ ≤ R0 , t

∫ ‖∇uϵ (s)‖2 ds ≤ 0

t

ϵ ∫ ‖e(uϵ (s))‖2Lp ds ≤ 0

(6.4.16)

‖f ‖2 t + R0 , λ12

(6.4.17)

‖f ‖2 t + R0 . 2λ12

(6.4.18)

Similar to the above estimates, there exists a constant R1 such that the solution u(t) of (6.4.5) satisfies ‖u(t)‖ ≤ R1 , t

∫ ‖∇u(s)‖2 ds ≤ 0

(6.4.19)

‖f ‖2 t + R1 . λ1

(6.4.20)

Theorem 6.4.1 (see [40]). Assume f ∈ H, u0 ∈ H, 1 < p ≤ 2, ϵ → 0, when uϵ (s)(0) = u(0); (1) the solution uϵ (s)(t) of (6.4.1) converges to the solution u(t) of (6.4.5) in H; (2) the global attractor Aϵ of (6.4.1) converges to the global attractor A of (6.4.5). Proof. (1) Assume that (6.4.1) and (6.4.5) have the same initial value u0 (x), and uϵ = Sϵ (t)u0 , u = S(t)u0 are the corresponding solutions. Let w = u − uϵ . We get wt − Δw + u ⋅ ∇w + w ⋅ ∇uϵ − ϵ∇ ⋅ (|e(uϵ )|p−2 e(uϵ )) + ∇(p − pϵ ) = 0, w(0) = 0.

(6.4.21) (6.4.22)

Taking the inner product of (6.4.21) with w, and noticing the orthogonality (u ⋅ ∇w, w) = 0, (∇(p − pϵ ), w) = 0,

1 d ‖w‖2 + ‖∇w‖2 + (w ⋅ ∇uϵ , w) + ϵ(∇(|e(uϵ )|p−2 ⋅ e(uϵ )), w) = 0. 2 dt

(6.4.23)

Using the Holder inequality, we get |(w ⋅ ∇uϵ , w)| ≤ ∫ |w||∇w||uϵ |dx ≤ ‖∇w‖‖w‖L4 ‖uϵ ‖L4 .

(6.4.24)

Ω

For n = 2, by the Gagliardo–Nirenberg inequality, we have 1

1

‖w‖L4 ≤ C0 ‖w‖ 2 ‖∇w‖ 2 .

(6.4.25)

214 | 6 The regularity of solutions and related problems Combining (6.4.16) with Young inequality, we obtain 3

1

1

1

|b(w, uϵ , w)| ≤ C0 ‖∇w‖ 2 ‖w‖ 2 ‖∇uϵ ‖ 2 ‖uϵ ‖ 2 1 ≤ ‖∇w‖2 + 27C0 R20 ‖w‖2 ‖∇uϵ ‖2 , 4

(6.4.26)

and ϵ(∇ ⋅ (|e(uϵ )|p−2 e(uϵ )), w) p−1

≤ ϵ ∫ |e(uϵ )|

2(p−1)

|∇w|dx ≤ ϵ‖∇w‖(∫ |e(uϵ )|

Ω

≤

dx)

1 2

Ω

1 ‖∇w‖2 + ϵ2 ‖e(uϵ )‖2(p−1) , L2(p−1) 4

(6.4.27)

due to 1 < p ≤ 2, we have ∫ |e(uϵ )|

2(p−1)

dx ≤ |Ω|

Ω

≤ |Ω|

2−p p

2−p p

p

(∫ |e(uϵ )| dx)

2(p−1) p

Ω

‖e(uϵ )‖2(p−1) . Lp

(6.4.28)

Combining above inequalities, (6.4.23) yields 2−p d ‖w‖2 ≤ 54C0 R20 ‖w‖2 ‖∇uϵ ‖2 + 2ϵ2 |Ω| p ‖e(uϵ )‖2(p−1) . Lp dt

(6.4.29)

Integrating equation from 0 to t, we have 2

2

‖w‖ ≤ ‖w(0)‖

t 2 exp(54C0 R0 ∫ ‖∇uϵ ‖2 ds) 0

+ 2|Ω|

2−p p

t

t

ϵ2 exp(54C0 R20 ∫ ‖∇uϵ ‖2 dτ) ⋅ ∫ ‖e(uϵ )‖L2(p−1) ds. p 0

(6.4.30)

0

Due to ‖w(0)‖ ≡ 0, ‖e(uϵ )‖Lp is equivalent to ‖∇uϵ ‖Lp , when p = 2, from (6.4.17) and (6.4.30), we get ‖w(t)‖ ≤ a(t)ϵ → 0,

ϵ → 0,

(6.4.31)

where a(t) = √2|Ω|

2−p 2p

t

1 2

(∫ ‖e(uϵ )‖2(p−1) ds) exp(27C0 R20 ( Lp 0

‖f ‖2 t + R0 )). λ1

6.5 Other decay estimates of incompressible non-Newtonian fluids | 215

For 1 < p < 2, we obtain t

ds ≤ t ∫ ‖e(uϵ )‖2(p−1) Lp

2−p p

0

t

(∫ ‖e(uϵ )‖pLp ds)

2(p−1) p

.

(6.4.32)

0

Combining (6.4.30), (6.4.32) and (6.4.18), we have ‖w(t)‖ ≤ b(t)ϵ

2− p1

2−p

b(t) = √2|Ω| 2p t

→ 0,

2−p 2p

(

ϵ → 0,

‖f ‖2 t + R0 ) 2λ12

(6.4.33) p−1 p

exp(27C0 R20 (

‖f ‖2 t + R0 )). λ12

Hence, (6.4.31) and (6.4.33) yield that the result (1) of Theorem 6.4.1 is true for 1 < p ≤ 2, and remembering that the uniformly attractive region of semigroup S(t) is independence of ϵ in H. Then the result (2) of Theorem 6.4.1 is also true, which completes the proof of Theorem 6.4.1.

6.5 Other decay estimates of incompressible non-Newtonian fluids Now consider the L2 decay of the weak solution of the incompressible non-Newtonian fluid in the whole space [72]. The initial value problem is given as follows: div u = 0, { { u + (u ⋅ ∇)u = div T, { { t {u(⋅, 0) = u0 ,

(6.5.1)

where u = (u1 , u2 , u3 ) is the velocity field, T is stress tensor and u0 is initial value of the velocity. The stress tensor can be decomposed as Tij = −πδij + τijν ,

(6.5.2)

where π is pressure, δij is Kronecker delta; τν is the viscous part of stress, and assume stress tensor τν of the form τν = τ(Du),

(6.5.3)

where τ is the nonlinear tensor function, the components of symmetric deformation velocity tensor are given by 𝜕uj 1 𝜕u ). Duij = ( i + 2 𝜕xj 𝜕xi

(6.5.4)

216 | 6 The regularity of solutions and related problems Consider the following growth condition: {

|τij (Du)| ≤ C1 (|Du| + |Du|p−1 ), |τij (Du)| ≤ C|Du|p−1 ,

C1 > 0, p ≥ 2, 1 < p < 2,

(6.5.5)

as well as the strong coercivity condition τij (Du)Duij ≥ C2 (|Du|p + |Du|2 ),

1 < p < ∞, C2 > 0,

(6.5.6)

where |Du| = (Duij , Duij )1/2 . Lemma 6.5.1. Assume u ∈ Lp (I, W 1,p (R3 )3 ) ∩ C(I, H) ∩ L2 (I, W 1,2 (R3 )3 ), u → 0, |x| → ∞, p > 3. Then the following inequality is true: ̂ (ξ , t)| ≤ C|ξ |−1 , |u

ξ ∈ S(t),

where S(t) denotes n-dimensional sphere centered at the origin with a time-dependent radius, and H = {φ ∈ L2 (R3 )3 , div φ = 0}. Proof. Applying the Fourier transform to (6.5.1), we get ̂ = G(ξ , t), ̂ t + |ξ |2 u u

(6.5.7)

where (τ̃ij = |∇u|p−1 ),

G(ξ , t) = −ℱ (u ⋅ ∇u) − ξ ℱ (π) − ξ ℱ (τ̃ij ),

(6.5.8)

and ℱ is the Fourier transform, the pressure term ℱ (π) is a function. 2 Multiplying (6.5.7) by e|ξ | t , we obtain 2 d |ξ |2 t ̂ ] = e|ξ | t G(ξ , t). [e u dt

(6.5.9)

Then we have t

2

2

̂ (ξ , t)| ≤ e−|ξ | t |u ̂ 0 (ξ )| + ∫ e−|ξ | (t−s) |G(ξ , s)|ds. |u

(6.5.10)

0

Furthermore, assume |G(ξ , t)| ≤ C|ξ |,

(6.5.11)

we get ̂ (ξ , t)| ≤ e |u

−|ξ |2 t

t

2

̂ 0 (ξ )| + C ∫ e−|ξ | (t−τ) |ξ |ds. |u 0

(6.5.12)

6.5 Other decay estimates of incompressible non-Newtonian fluids | 217

Due to u0 ∈ L1 , after Fourier transform, it belongs to L∞ , that is, ̂ 0 (ξ )| ≤ C, |u

∀ξ , C > 0.

(6.5.13)

From (6.5.12) and (6.5.13), we know 2

̂ (ξ , t)| ≤ Ce−|ξ | t + |u

2 C (1 − e−|ξ | t ). |ξ |

The inequality of the lemma yields, since the radius of S(t) is bounded uniformly with respect to t. Lemma 6.5.2. Under the assumptions of Lemma 6.5.1, we have |G(ξ , t)| ≤ C|ξ |.

(6.5.14)

Proof. |ℱ (u ⋅ ∇u)| = ∫ div(u ⊗ u)eixξ dx| ≤ ∫ ∑ |ui uj ||ξj |dx, n n ij i j

R

i

j

2

‖u u ‖L1 ≤ ‖u ‖‖u ‖ ≤ ‖u0 ‖

R

⇒ |ℱ (u ⋅ ∇u)| ≤ C|ξ |,

|ℱ (∇ ⋅ τ̃ij )| ≤ ∑ ∫ |ξj ||∇u|p−1 dx, ij Rn

2

(6.5.15)

2(p−3)

∫ |Duij |p−1 dx ≤ ‖Duij ‖ p−2 ‖Duij ‖Lpp−2 ≤ k.

Rn

Now we need to estimate the pressure term, applying div in (6.5.1), we have Δπ = ∑ ij

𝜕2 (−ui uj + τ̃ij ), 𝜕xi 𝜕xj

(6.5.16)

|ξ |2 ℱ (π) = ∑ ξi ξj [−ℱ (ui uj ) + ℱ (τ̃ij )]. ij

Since ui uj , τ̃ij ∈ L1 , we have |ξ ||ℱ (π)| ≤ C|ξ |, and (6.5.14) is satisfied. Lemma 6.5.3. Under the assumptions of Lemma 6.5.1, then π ∈ L5/3 (R3 ), and ℱ (π) ∈ L5/2 (R3 ), for the fixed t. Proof. From (6.5.16), (∇π, ∇φ) = (u ⋅ ∇u, ∇φ) + (∇τ̃ij , ∇φ).

(6.5.17) 15

15

Now we want to obtain that π ∈ L5/3 (I, QT ), if u ⋅ ∇u, ∇τ̃ij ∈ L 14 , then ∇π ∈ L 14 , by Sobolev imbedding we have ‖π‖L5/3 ≤ ‖∇π‖L15/14 ≤ C.

218 | 6 The regularity of solutions and related problems 15

Next, we prove u ⋅ ∇u and ∇τ̃ij ∈ L 14 . 1−α1

‖u ⋅ ∇u‖L15/14 ≤ ‖∇u‖‖u‖L28/13 ≤ ‖∇u‖‖u‖α1 ‖u‖Lp 2p

≤ ϵ1 ‖∇u‖2 + ϵ2 ‖u‖ p−2 + ϵ3 ‖∇u‖pLp ,

(6.5.18)

1 where α1 = p(p−2) . Finally,

‖∇2 u(|∇u|p−2 )‖L15/14 ≤ ‖∇2 u‖‖∇u‖L(28/13)(p−2)

≤ ‖∇2 u‖‖∇u‖1−α ‖∇u‖αL3p

≤ ϵ̃1 ‖∇2 u‖2 + ϵ̃2 ‖∇u‖q(1−α) + ϵ̃3 ‖∇u‖αp , L3p

(6.5.19)

3p(41−14p) 1 , 21 + q1 + 3p = 1. where α = 14(p−2)(2−3p) Applying the Hausdorff–Young inequality,

‖ℱ (π)‖L5/2 ≤ C‖π‖L5/3 ,

(6.5.20)

which implies the integrability of the Fourier transform of the pressure. Theorem 6.5.1. Assume u ∈ Lp (I, W 1,p (R3 )3 )∩C(I, H)∩L2 (I, W 1,2 (R3 )3 ), u → 0, |x| → ∞, p > 3. Then u and π satisfy u + (u ⋅ ∇)u − div τν + ∇π = 0, { { t ∇ ⋅ u = 0, { { 1,2 3 3 1 3 3 {u(x, 0) = u0 (x), u0 ∈ W (R ) ∩ H ∩ L (R ) ,

(6.5.21)

then we have 1

‖u(⋅, t)‖2 ≤ C(t + 1)− 2 , where C is dependent of the L1 and L2 norm of u0 , and have the energy inequality d ∫ |u|2 dx + C1 ∫ |∇u|2 dx + C2 ∫ |∇u|p dx ≤ 0. dt R3

R3

(6.5.22)

R3

We introduce notation τ̃ij = |∇u|p−1 , H = {φ ∈ L2 (R3 )3 , div φ = 0}. Proof. First, we have the following energy inequality: d ∫ |u|2 dx + C1 ∫ |∇u|2 dx ≤ 0. dt R3

(6.5.23)

R3

Applying Fourier transform and the Plancherel theorem to (6.5.23) yield d ̂ |2 dξ + C1 ∫ |u ̂ ||ξ |2 dξ ≤ 0. ∫ |u dt R3

R3

(6.5.24)

6.5 Other decay estimates of incompressible non-Newtonian fluids | 219

Rewrite the above inequality, and we get d ̂ |2 dξ − 2C1 ∫ |ξ |2 |u ̂ |2 dξ , ̂ |2 dξ ≤ −2C1 ∫ |ξ |2 |u ∫ |u dt S(t)c

R3

(6.5.25)

S(t)

where S(t) is the ball centered at the origin with radius r(t) = [

1/2

n ] . 2C1 (t + 1)

(6.5.26)

Then d n ̂ |2 dξ ≤ − ̂ |2 dξ − 2C1 ∫ |u ̂ |2 |ξ |2 dξ , ∫ |u ∫ |u dt t+1 R3

S(t)c

R3

R3

d n ̂ |2 dξ + ̂ |2 dξ ∫ |u ∫ |u dt (t + 1)

S(t)

n ̂ |2 dξ − 2C1 ∫ |u ̂ |2 |ξ |2 dξ . ≤ ∫ |u t+1 S(t)

(6.5.27)

(6.5.28)

S(t)

It is also that n n d ̂ |2 dξ . ̂ |2 dξ + ̂ |2 dξ ≤ ∫ |u ∫ |u ∫ |u dt t+1 (t + 1) R3

(6.5.29)

S(t)

R3

From Lemma 6.5.1, we get ̂ (ξ , t)| ≤ C|ξ |−1 , |u

ξ ∈ S(t).

(6.5.30)

Then d n C ̂ |2 dξ + ̂ |2 dξ ≤ ∫ |u ∫ |u ∫ |ξ |−2 dξ . dt t+1 (t + 1) R3

(6.5.31)

S(t)

R3

Multiply (6.5.31) by (t + 1)n , we have d ̂ |2 dξ ] ≤ C(t + 1)n−1 ∫ |ξ |−2 dξ , [(t + 1)n ∫ |u dt

(6.5.32)

S(t)

R3

the right-hand side of the above inequality can be expressed n−1

Cω0 (t + 1)

r(t)

∫ r n−1 r −2 dr, 0

(6.5.33)

220 | 6 The regularity of solutions and related problems where ω0 is the volume of the n dimensional ball, r(t) is defined by (6.5.26), n

2 d 3 ̂ |2 dξ ] ≤ Cω0 (t + 1)n−1 [ [(t + 1)n ∫ |u ] dt 2C1 (t + 1)

−1

≤ C(t + 1)n/2 .

(6.5.34)

R3

Integrating with respect to time gives us ̂ |2 dx ≤ ∫ |u ̂ (ξ , 0)|2 dξ + C[(t + 1)(n/2+1) − 1]. (t + 1)n ∫ |u R3

(6.5.35)

R3

Since u0 ∈ L2 (R3 ), we can deduce 1

∫ |u|2 dx ≤ C(t + 1)− 2 . R3

Next, we give the other decay estimate of non-Newtonian fluid [88]. Consider the following initial boundary value problem: ut − Δu + (u ⋅ ∇)u − ∇ ⋅ (|e(u)|p−2 e(u)) + ∇π = 0 (x, t) ∈ Rn × (0, ∞),

∇ ⋅ u = 0,

n

(x, t) ∈ R × (0, ∞),

u(x, 0) = u0 ,

n

x∈R .

(6.5.36) (6.5.37) (6.5.38)

Lemma 6.5.4. Assume u0 ∈ W 1,2 (Rn ) ∩ L2σ ∩ L1 (Rn ) (n = 2, 3), u is a weak solution of (6.5.36)–(6.5.38), then we have sup ‖u(t)‖ ≤ ‖u0 ‖,

(6.5.39)

0≤t≤∞

for n = 2, 2 < p < 3, t

t

2

̂ (ξ , t)| ≤ ‖u0 ‖L1 + C|ξ | ∫ ‖u(s)‖ ds + C|ξ |(∫ ‖u(s)‖ |u 0

≤ C + C|ξ |t + C|ξ |t for n = 3,

11 5

4−p 2

2 4p−2

ds)

4−p 2

0

(6.5.40)

,

≤ p < 3, t

̂ (ξ , t)| ≤ ‖u0 ‖L1 + C|ξ | ∫ ‖u(s)‖2 ds |u 0 t

+ C|ξ |(∫ ‖u(s)‖

14−2p 19−5p

ds)

19−5p 8

,

(6.5.41)

0

where C is a constant dependent on v0 . The space Lqσ (Rn )n denotes the closure of ∞ C0,σ (Rn )n in Lq , which is the set of smooth divergence-free with compact supports in Rn .

6.5 Other decay estimates of incompressible non-Newtonian fluids | 221

Proof. It is easy to obtain (6.5.39) by energy inequality. Now prove (6.5.40) and (6.5.41), and applying the Fourier transform to (6.5.36), we get ̂ t + |ξ |2 u ̂ = ℱ [∇ ⋅ (|∇u|p−2 e(u)) − (u ⋅ ∇)u − ∇π] = G(ξ , t). u

(6.5.42)

Similar to the above discussion, the following estimates yield: |ℱ [(u ⋅ ∇)u]| ≤ |ℱ [div(u ⊗ u)]| ≤ |ξ |‖u‖2 ,

(6.5.43)

|ℱ [∇π]| ≤ |ℱ [∇ ⋅ (|∇u|p−2 Du)]| + |ℱ [(u ⋅ ∇)u]|.

(6.5.44)

In the case of n = 2, p−2 p−2 p−1 ℱ [∇ ⋅ (|e(u)| e(u))] ≤ |ξ ||ℱ [|e(u)| e(u)]| ≤ |ξ | ∫ |∇u| dx 2

p−2

≤ C|ξ |‖u‖‖D u‖

Rn

(6.5.45)

.

In the case of n = 3, 7−p 5p−11 p−2 2 ℱ [∇ ⋅ (|∇u| Du)] ≤ C|ξ |‖u‖ 4 ‖D u‖ 4 .

(6.5.46)

Combining the above inequalities, we have |G(ξ , t)| ≤ C|ξ |‖u‖2 + C|ξ |‖u‖‖D2 u‖p−2 , 2

|G(ξ , t)| ≤ C|ξ |‖u‖ + C|ξ |‖u‖

7−p 4

2

‖D u‖

n = 2,

5p−11 4

,

n = 3.

(6.5.47) (6.5.48)

From (6.5.42), we have 2 2 d ̂ e|ξ | t ) ≤ G(ξ , t)e|ξ | t (u dt

Integrating with respect to time yields 2

t

2

̂ (ξ , t) ≤ e−|ξ | t u ̂ 0 (ξ ) + ∫ G(ξ , t)e−|ξ | (t−s) ds. u 0

According to the theorem for existence of the solution, we also assume that T ∫0 ‖D2 u(t)‖2 dt ≤ C. Then for n = 2, 2 < p < 3, t

t

̂ | ≤ |u ̂ 0 (ξ )| + C ∫ |ξ |‖u(s)‖2 ds + C ∫ |ξ |‖u(s)‖‖D2 u(s)‖p−2 ds |u 0

t

≤ ‖u0 ‖L1 + C|ξ | ∫ ‖u(s)‖2 ds 0

0

222 | 6 The regularity of solutions and related problems t

+ C|ξ |(∫ ‖u(s)‖

2 4−p

ds)

4−p 2

t

× (∫ ‖D2 u(s)‖2 ds) 0

0 t

t

2

≤ ‖u0 ‖L1 + C|ξ | ∫ ‖u(s)‖ ds + C|ξ |(∫ ‖u(s)‖ 0

≤ C + C|ξ |t + C|ξ |t For n = 3,

11 3

p−2 2

4−p 2

2 4−p

4−p 2

ds)

0

.

≤ p < 3, t

t

̂ (ξ , t)| ≤ |u ̂ 0 (ξ )| + C ∫ |ξ |‖u(s)‖2 ds + C ∫ |ξ |‖u(s)‖‖u(s)‖ |u 0

7−p 2

0 t

t

2

2

2

≤ ‖u0 ‖L1 + C|ξ | ∫ ‖u(s)‖ ds + C|ξ |(∫ ‖D u(s)‖ ds) 0

0

t

t

‖D2 u(s)‖ 5p−11 8

14−2p

≤ ‖u0 ‖L1 + C|ξ | ∫ ‖u(s)‖2 ds + C|ξ |(∫ ‖u(s)‖ 19−5p ds) 0

5p−11 4

ds

t

× (∫ ‖u(s)‖

(4−2p) 19−5p

ds)

0

19−5p 8

.

0 1,2

n

Lemma 6.5.5. Assume u0 ∈ W (R ) ∩ (6.5.36)–(6.5.38), then we have (1) n = 2, 2 < p < 3

L2σ

∩ L1 (Rn ), n = 2, 3, u is a weak solution of

1

‖u‖ ≤ C(1 + t)− 2 . (2) n = 3,

11 5

19−5p 8

(6.5.49)

≤p 2, and let f (t) = (1 + t)2 , from (6.5.39), we have 1

t

1

t

28−4p

̂ (ξ , t)|2 dξ ≤ C(1 + t) 2 + C(1 + t) 2 ∫ ‖u(s)‖4 ds + C(1 + t) 2 ∫ ‖u(s)‖ 19−5p ds (1 + t)2 ∫ |u 0

R3 1

1

0

t

≤ C(1 + t) 2 + C(1 + t) 2 ∫ ‖u(s)‖2 ds, 0

by the generalized Gronwall inequality, it follows that 3

‖u(t)‖ ≤ C(1 + t)− 4 .

7 Global attractors and time-spatial chaos 7.1 Global attractor of low regularity With the application of harmonic analysis, the existence, uniqueness, continuous dependence, namely, well-posedness of lower regular global solution for a kind of nonlinear dispersive equation can be obtained by the Bourgain method. Under the weak initial value condition, the existence and asymptotic smoothness of global attractors are proved. In this section, we illustrate the KdV equation with an example. Using Bourgain space, we can split the solution into two parts. One is regular (belongs to H 3 (R)), the other decays to zero in L2 (R) as t → ∞. Finally, the regularity of attractors is obtained by applying the energy equation method. Consider the following weakly damped, forced KdV equation [31]: ut + uux + uxxx + γu = f ,

(x, t) ∈ R × R,

(7.1.1)

with the initial value u|t=0 = u0 (x) ∈ L2 (R).

(7.1.2)

An important role is played by the Bourgain space-time function space X s,b , for s, b ∈ R, which is defined as the completion of the Schwartz space φ(R2 ) with respect to the norm ‖f ‖2X s,b = ∫ ∫⟨τ − ξ 3 ⟩2b ⟨ξ ⟩2s |f ̂(ξ , τ)|2 dξdτ,

(7.1.3)

R R

where f ̂ = f ̂(ξ , τ) is the Fourier transform of f = f (x, t), and ⟨λ⟩ = √1 + |λ|2 .

(7.1.4)

Sometimes, the Fourier transform of Rn is denoted by ℱ (f )(ξ ) = f ̂(ξ ) =

1 (2π)

n 2

∫ f (z)e−iξz dz.

(7.1.5)

Rn

The inverse Fourier transform is denoted by ℱ −1 . ∀s ∈ R, b > 21 , the continuous embedding X s,b ⊆ Cb (R, Hxs (R)) holds. To obtain the local time estimates in the space X s,b , we introduce the function ψ ∈ Cc∞ (R), which is equal to 1 in the interval [−1, 1), and to 0 outside the interval (−2, 2). We set t ψδ (t) = ψ( ). δ https://doi.org/10.1515/9783110549614-007

(7.1.6)

228 | 7 Global attractors and time-spatial chaos For any given interval I = [a, b], let ψI (t) = ψ( (2t−a−b) ). Considering the seminorms (b−a) in X s,b ,

‖u‖X s,b

[−δ,δ]

= ‖ψδ u‖X s,b ,

‖u‖X s,b = ‖ψI u‖X s,b . I

3

Consider the Airy group {W(t)}t∈R , W(t) = e−t𝜕x . The Bourgain spaces are such that u = u(x, t) ∈ X s,b , W(−t)u ∈ Htb Hxs (R2 ), and we can write ‖u‖X s,b = ‖W(−⋅)u‖H b H s (R2 ) = ‖⟨τ − ξ 3 ⟩b ⟨ξ ⟩s u‖̂ L2 t

x

τ,ξ

(7.1.7)

(R2 ) .

It is easy to obtain the following estimate: ‖ψδ W(t)u0 ‖X s,b = ‖ψδ ‖H b (R) ‖u0 ‖H s (R) .

(7.1.8)

Lemma 7.1.1. Assume that s ∈ R, 0 < δ ≤ 1, − 21 < b − 1 ≤ 0 ≤ b ≤ b , g ∈ X s,b −1 , and define

t

w(t) = ψδ (t) ∫ W(t − t )g(t )dt .

(7.1.9)

0

Then there exists a constant C1 > 0 independent of s, b, b and δ, such that the following inequality holds: ‖w‖X s,b ≤ C1 δb −b ‖g‖X s,b −1 .

Lemma 7.1.2. There exists a constant C2 > 0, such that for s ∈ R, we have ‖ψδ u‖X s,b ≤ C2 δ

(1−2b) 2

‖u‖X s,b ,

(7.1.10) 1 2

< b ≤ 1, and u ∈ Xbs , (7.1.11)

Recall now the well-known Strichartz-type inequalities establishing dispersivetype regularizations of the free Airy group, for any u0 ∈ L2 (R), ≤ C3 ‖u0 ‖L2x (R) ,

(7.1.12)

‖𝜕x W(t)u0 ‖L∞ 2 2 ≤ C3 ‖u0 ‖L2 (R) , x x Lt (R )

(7.1.13)

‖W(t)u0 ‖L8

x,t (R

1 6

‖(𝜕x ) W(t)u0 ‖L6 for some constant C3 > 0.

x,t (R

2)

2)

≤ C3 ‖u0 ‖L2x (R) ,

(7.1.14)

7.1 Global attractor of low regularity | 229

Lemma 7.1.3. Let ‖ ⋅ ‖Y be a seminorm in Lx,t (R2 ) which is stable under multiplication by functions in L∞ t (R) in the sense that 2 ∀ψ ∈ L∞ t (R), ∀f ∈ Lx,t (R ).

‖ψf ‖Y ≤ C‖ψ‖L∞ ‖f ‖Y , t (R)

(7.1.15)

Assume that the following (Strichartz-type) inequality holds: ∀u0 ∈ L2x (R).

‖W(⋅)u0 ‖Y ≤ C‖u0 ‖L2x (R) ,

(7.1.16)

Then, for all b > 21 , the following inequality holds: ‖f ‖Y ≤ Cb ‖f ‖X 0,b ,

∀f ∈ X 0,b ,

(7.1.17)

1

where Cb = Cb1/2 (2b − 1)− 2 . Hence, due to Lemma 7.1.3, we follow from (7.1.12)–(7.1.14) that for any b > f ∈ X 0,b , ‖f ‖L8

2)

x,t (R

≤ Cb ‖f ‖X 0,b ,

(7.1.18)

Cb ‖f ‖X 0,b ,

(7.1.19)

≤ Cb ‖f ‖X 0,b ,

(7.1.20)

‖𝜕x f ‖L∞ 2 2 ≤ x Lt (R ) 1 6

‖(𝜕x ) f ‖L6

2)

x,t (R

1 , 2

where Cb is dependent of b > 21 . By interpolating (7.1.18) with L2x,t = X 0,0 , we obtain ‖f ‖L4

2)

x,t (R

≤ Cb ‖f ‖

b

X 0, 2

(7.1.21)

.

Similarly, interpolating (7.1.20), we find 1

‖(𝜕x ) 8 f ‖L4

2)

x,t (R

≤ Cb ‖f ‖

X 0,

3b 4

.

(7.1.22)

Bilinear estimates Proposition 7.1.1. Given s ∈ (− 43 , 0]. Then there exists a constant Cs and a constant b ∈ ( 21 , 1) such that for any function u ∈ X s,b , 2 2 D(u )X s,b−1 ≤ Cs ‖u‖X s,b .

(7.1.23)

Proposition 7.1.2. Given s ∈ (− 21 , 0]. Then there exists a constant Cs such that for b ∈ 9 ], b ∈ [ 21 , min{b , 1 + s}], and for any function u ∈ X s,b , ( 21 , 16 2 D(u )X s,b −1 ≤ Cs ‖u‖X s,b ‖u‖ − 21 ,b . X As a corollary of this result, we have the following.

(7.1.24)

230 | 7 Global attractors and time-spatial chaos ̂ , τ) of u is supported in the Corollary 7.1.1. Assume that the Fourier transform û = u(ξ 9 set {(ξ , τ); |ξ | ≥ N}, N > 0. Then, for 21 < b ≤ 16 , we get C 2 2 D(u )X 0,b−1 ≤ 0 ‖u‖X 0,b . √N

(7.1.25)

Proof of Proposition 7.1.2. Let v̂ = ⟨ξ ⟩s ⟨τ − ξ 3 ⟩b û where û is the Fourier transform of u. By the duality argument, and by setting ρ = −s, (7.1.24) stems from the following result: there exists a constant C = Cs such that for any G and v, with ‖G‖L2 (R2 ) = ‖v‖L2 (R2 ) = 1, Q=∫ D

̂ 1 , τ1 )v(ξ ̂ 2 , τ2 )G(ξ , τ) ⟨ξ1 ⟩ρ ⟨ξ2 ⟩ρ ξ v(ξ dτ1 dξ1 dτdξ ⟨τ − ξ 3 ⟩1−b ⟨τ1 − ξ13 ⟩b ⟨τ2 − ξ23 ⟩b ⟨ξ ⟩ρ

≤ C‖v‖

1

X ρ− 2 ,0

where

(7.1.26)

, D = {σ = {ξ , τ, ξ1 , τ1 } ∈ R2 × R2 },

and ξ2 = ξ − ξ1 ,

τ2 = τ − τ1 .

We discuss Q according to the following three cases: First case: |ξ1 | ≤ |ξ2 | ≤ 21 . Set D1 ⊂ D. These inequalities are valid in D1 . In that case, the function ⟨ξ1 ⟩ρ ⟨ξ2 ⟩ρ |ξ |/⟨ξ ⟩ρ is bounded, and we only need to estimate Q1 = ∫ D1

̂ 1 , τ1 )v(ξ ̂ 2 , τ2 )G(ξ , τ) v(ξ dτ1 dξ1 dτdξ . 3 1−b ⟨τ − ξ ⟩ ⟨τ1 − ξ13 ⟩b ⟨τ2 − ξ23 ⟩b

Let ̂ , τ) = w(ξ

(7.1.27)

̂ , τ) v(ξ χ(ξ ), ⟨τ − ξ 3 ⟩b

where χ is the characteristic function of the interval [− 21 , 21 ]. Then, due to b − 1 ≤ 0 and ‖G‖L2 (R2 ) = 1, Q1 ≤ ‖G(ξ , τ)⟨τ − ξ 3 ⟩b −1 ‖L2

ξ ,τ

(R) ‖ŵ

̂ L2 ∗ w‖

ξ ,τ

(R)

1 ‖G(ξ , τ)‖L2 (R) ‖W 2 ‖L2x,t (R2 ) ξ ,τ 2π 1 −1 ̂ , τ)χ(ξ , τ)|)⟨τ − ξ 3 ⟩−b ‖2L4 (R) , ‖ℱ (|v(ξ = x,t 2π ≤

(7.1.28)

where ∗ denotes the convolution operator. Now, we apply (7.1.21), the embeddings X s,a ⊂ X s,0 for all a ≥ 0, and the following inverse inequality: ̂ ‖F −1 (|v|χ)‖ X 0,0 ≤ ‖v‖X ρ−1/2,0 to obtain Q1 ≤ C‖v‖X ρ−1/2,0 .

(7.1.29)

7.1 Global attractor of low regularity | 231

Second case: |ξ1 | ≤ 21 ≤ |ξ2 |. Set D2 be the corresponding region. In that case, we use |ξ | ≤ 2|ξ2 | and the function ⟨ξ1 ⟩ρ ⟨ξ2 ⟩ρ /⟨ξ ⟩ρ is bounded to get ̂ 2 , τ2 )|)⟨τ2 − ξ23 ⟩−b ‖L∞ L2 Q2 ≤ C‖G(ξ , τ)⟨τ − ξ 3 ⟩b −1 ‖L2 ‖ℱ −1 (|ξ2 | |v(ξ x t

ξ ,τ

̂ 1 , τ1 )χ(ξ1 )|)⟨τ1 − ξ13 ⟩−b ‖L2 L∞ . × ‖ℱ (|v(ξ x t

(7.1.30)

−1

Using (7.1.19), we have ̂ 2 , τ2 )|)⟨τ2 − ξ23 ⟩−b ‖L∞ L2 ≤ C‖v‖L2 (R2 ) = C. ‖ℱ −1 (|ξ2 | |v(ξ x t x,t

(7.1.31)

2 From the embedding X 0,b ⊂ L2x L∞ t (R ) and (7.1.24), we have

̂ 1 , τ1 )χ(ξ1 )|)⟨τ1 − ξ13 ⟩−b ‖L2 L∞ ‖ℱ −1 (‖v(ξ x t

̂ 1 , τ1 )|χ(ξ1 ))‖X 0,0 ≤ C‖v‖ ≤ C‖ℱ −1 (|v(ξ

We obtain Q2 ≤ C‖v‖

Third case: tant:

1 2

1

X ρ− 2 ,0

1

X ρ− 2 ,0

(7.1.32)

.

.

≤ |ξ1 | ≤ |ξ2 |. In this case, the following algebraic inequality is impor-

3|ξξ1 ξ2 | = |ξ 3 − ξ13 − ξ23 | = |(τ − ξ 3 ) − (τ1 − ξ13 ) − (τ2 − ξ23 )| ≤ |τ − ξ 3 | − |τ1 − ξ13 | + |τ2 − ξ23 |.

(7.1.33)

First subcase: |ξξ1 ξ2 | ≤ |τ1 − ξ13 |. Set D31 be the corresponding region. For b ≤ 1 − ρ, we use |ξ | ≤ 2|ξ2 | to obtain Q31 ≤ C ∫ D31

̂ 1 , τ1 )v(ξ ̂ 2 , τ2 )G(ξ , τ)| ρ−b 1−2b |v(ξ |ξ1 | |ξ2 | dτ1 dξ1 dτdξ . 3 1−b ⟨τ − ξ ⟩ ⟨τ2 − ξ23 ⟩b

It follows from (7.1.21) and 1 − 2b ≤ 0, b − 1 ≤

b 2

(7.1.34)

that

Q31 ≤ C ℱ −1 (G(ξ , τ)⟨τ − ξ 3 ⟩b −1 )L4 x,t −1 ̂ 1−2b ⟨τ2 − ξ23 ⟩−b )‖L4 ‖vX ρ−b,0 ℱ (|v(ξ2 , τ2 )||ξ2 | x,t

≤ C‖v‖X ρ−b,0 .

(7.1.35)

Second subcase: |ξξ1 ξ2 | ≤ |τ2 − ξ23 |. Set D32 be the corresponding region. Then Q32 ≤ C ∫ D32

̂ 1 , τ1 )v(ξ ̂ 2 , τ2 )G(ξ , τ)| ρ−b 1−2b |v(ξ |ξ1 | |ξ2 | dτ1 dξ1 dτdξ . 3 1−b ⟨τ − ξ ⟩ ⟨τ1 − ξ13 ⟩b

We get ̂ 1 , τ1 )||ξ1 |ρ−b ⟨τ1 − ξ13 ⟩−b )L4 Q32 ≤ C‖v‖X 1−2b,0 ℱ −1 (|v(ξ

x,t

232 | 7 Global attractors and time-spatial chaos ‖ℱ −1 (G)⟨τ − ξ 3 ⟩b −1 ‖L4

x,t

≤ C‖v‖

1

X ρ− 2 ,0

,

where we used (7.1.21) and the relations b ≥ 21 , ρ − b ≤ 1 − 2b ≤ 0, b − 1 ≤ b2 . Third subcase: |ξξ1 ξ2 | ≤ |τ − ξ 3 |. In this case, using |ξ | ≤ 2|ξ2 | again. Set D33 be the corresponding region. Then ̂ 1 , τ1 )| |ξ1 |ρ+b −1 |v(ξ ̂ 2 , τ2 )| |ξ2 |2b −1 |v(ξ |G(τ, ξ )|dτ1 dξ1 dτdξ . 3 b ⟨τ1 − ξ1 ⟩ ⟨τ2 − ξ23 ⟩b

Q33 ≤ C ∫ D33

Thus ̂ 1 , τ1 )|ξ1 |ρ+b −1 |)⟨τ1 − ξ13 ⟩−b L4 Q33 ≤ C‖G‖L2 ℱ −1 (|v(ξ τ,ξ x,t 2b −1 3 −b ℱ −1 (|v(ξ ̂ 2 , τ2 )|ξ2 | |)⟨τ2 − ξ2 ⟩ L4 . x,t

Owing to (7.1.22), we obtain Q33 ≤ C‖v‖

1

−1,0

X − 8 +ρ+b

provided b ≤

1 2

+

1 16

=

9 . 16

‖v‖

1

−1,0

X − 8 +2b

≤ C‖v‖

1

X ρ− 2 ,0

,

(7.1.36)

The proof of the proposition is completed.

Next, we prove the L2 energy equation of the solutions and deduce the global wellposedness and the existence of bounded absorbing sets in L2 (R). 9 We consider the case − 21 < s ≤ 0, 21 < b < b ≤ 16 Assume γ ∈ R, f = f (x, t) ∈ s,b−1 s X[−T ,T ] , for some Tf > 0. For each u0 (x) ∈ H (R), we seek a local solution of (7.1.1) in f

f

the mild sense on an interval [−δ, δ] (0 < δ ≤ 1) sufficiently small, as the fixed point in X s,b of the map Σ(u) given by t

1 Σ(u) = ψ1 (t)W(t)u0 + ψδ (t) ∫ W(t − s)[2ψTf (s)f − 2γψ1 (s)u(s) 2 0

2

(7.1.37)

− ((ψδ (s)u(s)) )x ]ds.

By using the previous estimates, we can choose b, b , such that ϵ = b − 3b + 1 > 0, ‖Σ(u)‖X s,b ≤ ‖ψ1 ‖H b (R) ‖u0 ‖Hxs (R) + C1 ‖f ‖X s,b−1 t

[−Tf ,Tf ]

+ γC1 δ‖ψ1 ‖H b (R) ‖u‖X s,b t

1 + C1 Cs C22 δϵ ‖u‖2X s,b , 2

and ‖Σ(u) − Σ(v)‖X s,b ≤ γC1 δ‖ψ1 ‖H b (R) ‖u − v‖X s,b t

7.1 Global attractor of low regularity | 233

1 + C1 Cs C22 δϵ ‖u + v‖X s,b ‖u − v‖X s,b . 2 For δ > 0 sufficiently small, depending on ‖u0 ‖H s (R) , ‖f ‖X s,b−1

[−Tf ,Tf ]

s, b and b , the map ∑ is

a strict contraction on a closed ball of Xbs centered at the origin. Hence, there exists a unique fixed-point u which is a mild solution of (7.1.1) on the interval [−δ, δ]. Moreover, we have the following estimates: ‖u‖X s,b

[−δ,δ]

(7.1.38)

≤ C(‖u0 ‖H s (R) + ‖f ‖X s,b−1 ), [−Tf ,Tf ]

where constant c is independent of the data. 25 9 Noting that if 21 < b < 48 , it is possible to choose b , such that b ≤ b ≤ 16 and b − 3b + 1 > 0 as needed in the above computations. Then we obtain the following theorem. s,b−1 Theorem 7.1.1. Let γ ∈ R and f ∈ X[−T , with − 21 < s ≤ 0, ,T ] f

f

1 2

0. Let

δ = δ(‖u0 ‖H s (R) ) be as described above. Then, for each u0 ∈ H (R), there exists a unique s,b solution u ∈ X[−δ,δ] of equation (7.1.1). Moreover, t → u(t) belongs to Cb ([−δ, δ], L2 (R)) and the map which associates (γ, f , u0 ) to the corresponding unique solution is continus,b−1 s,b 2 ous from R × X[−T × H s (R) → X[−T ,T ] ∩ C([−T , T ]; L (R)), for ∀0 < T < δ(‖u0 ‖H s (R) ). ,T ] f

f

Theorem 7.1.1 applies, in particular, to the case where f is independent of time and belongs to H s (R). To prove the global existence of the solutions, this is obtained with the help of one of the invariants of the KdV. For a smooth initial condition ũ 0 ∈ Cc∞ (R) and a 0,b given by Theorem 7.1.1, for a smooth forcing term f ̃ ∈ Cc∞ (R), the local solution X[−δ,δ] small δ > 0 coincides with the classical solution, which exists globally and belongs to C ∞ (R × R). Multiplying equation (7.1.1) by 2u,̃ we obtain the L2 energy-type equation d 2 2 ̃ ̃ ̃ ̃ ‖u(t)‖ L2 (R) + 2γ‖u(t)‖L2 (R) = 2(f , u(t))L2 (R) , dt

∀t ∈ R.

(7.1.39)

̃ + 2 ∫(f ̃, u(s)) L2 (R) ds,

(7.1.40)

Integrating (7.1.39) to find 2 ̃ ‖u(t)‖ L2 (R)

t

+

2 ̃ 2γ ∫ ‖u(s)‖ L2 (R) ds 0

=

‖ũ 0 ‖2L2 (R)

t

0

for t ∈ [−δ, δ]. Now, we consider approximations of u0 ∈ L2 (R) and f ∈ L2 (R) by smooth functions ũ 0 ∈ L2 (R) and f ̃ ∈ L2 (R) converging to u0 and f in L2 (R), respectively. By the continuity with respect to the data of the local solution given by Theorem 7.1.1, we have that the 0,b ̃ solutions ũ with initial condition u(0) = ũ 0 and forcing term f ̃ converge in X[−δ,δ] for all

0,b δ > 0 to the solution u ∈ X[−δ,δ] with the initial condition u(0) = u0 and forcing term f .

234 | 7 Global attractors and time-spatial chaos By taking the limit in (7.1.40) and using the continuity of the solution with respect to the data, in particular using that 2 ̃ ‖u(t)‖2L2 (R) = lim ‖u(t)‖ L2 (R) ,

̃ L2 (R) , (f , u(t))L2 (R) = lim(f ̃, u(t))

∀t ∈ [−δ, δ].

(7.1.41)

One can extend the solution u indefinitely and obtain a global solution u = u(t), t ∈ R 0,b ∩ Cb ([−δ, δ]; L2 (R)) for all δ > 0. with u ∈ X[−δ,δ] Theorem 7.1.2. Let γ ∈ R, f ∈ L2 (R) and u0 ∈ L2 (R). Then there exists a solution u ∈ 0,b C(R; L2 (R)) of equation (7.1.1) which is the unique solution belonging to X[−T,T] for all T > 0, and all

1 2

0. In particular, ‖u‖X 0,b

[−T,T]

≤ C(γ, ‖f ‖L2 (R) , ‖u0 ‖L2 (R) , T)

(7.1.43)

for some constant C depending monotonically on the data. According to Theorem 7.1.2, we can define a group associated with equation (7.1.1). Definition 7.1.1. For γ ∈ R, f ∈ L2 (R), we denote by {S(t)}t∈R the group in L2 (R) defined 0,b by S(t)u0 = u(t) where u = u(t) is the unique solution of (7.1.1) which belongs to X[−T,T] for all T > 0. We want to find the existence of bounded absorbing sets for the solution operator {S(t)}t∈R . By applying the Cauchy–Schwarz and Young inequalities, we can obtain 1 d ‖u(t)‖2L2 (R) + γ‖u(t)‖2L2 (R) ≤ ‖f ‖2L2 (R) . dt γ

(7.1.44)

Integrating in time, we get ‖u(t)‖2L2 (R) ≤ ‖u(t0 )‖2L2 (R) e−γ(t−t0 ) +

1 ‖f ‖2 2 (1 − e−γ(t−t0 ) ), γ 2 L (R)

whence we deduce that lim sup ‖u(t)‖L2 (R) ≤ t→∞

1 ‖f ‖ 2 . γ L (R)

(7.1.45)

7.1 Global attractor of low regularity | 235

Proposition 7.1.3. Let γ > 0, f ∈ L2 (R). Then the solution operator associated with equation (7.1.1) possesses a bounded absorbing set in L2 (R) with the radius of absorbing ball given according to (7.1.45). Now, we consider the decomposition of solution. The splitting method used is obtaining by writing u = v + ω and splitting the nonlinear term as uux = uvx + PN ((vω)x + ωωx ) + QN ((vω)x + ωωx ), where PN denotes the Fourier spectral projector associated with a cut off of the higher modes which retains only the modes with spatial frequency |ξ | ≤ N, for N > 0. The operator QN is the complement QN = I − PN . We now consider the equations vt + vvx + vxxx + γv = f − PN ((vω)x + ωωx ),

ωt + QN (ωωx ) + ωxxx + γω = −QN (vω)x ,

(7.1.46) (7.1.47)

with the initial values v|t=0 = PN u0 ,

ω|t=0 = QN u0 .

(7.1.48)

First, we use v = u−ω to write the equation for ω without including of v, and deduce the global existence of ω and the decay in time of ω(t) in L2 (R). Then the global existence of v follows, and we prove the regularity of v in H 3 (R) and the L2 energy equation for v. Owing to v = u − ω, we can obtain ωt + QN (ωωx ) + ωxxx + γω = −QN (uω)x ,

(7.1.49)

ω|t=0 = QN u0 .

(7.1.50)

The local well-posedness of the equation for ω follows as that for equation (7.1.1). For this, we use the fact that the solution u ∈ X 0,b locally in time. We consider more general initial conditions of the form ω|t=0 = ω0 ∈ L2 (R),

(7.1.51)

where t0 ∈ R is arbitrary, and QN ω0 = ω0 . Then, applying the fixed-point argument, we find a solution ω̃ of the equation t

1 ̃ ω̃ = ψ1 (t)W(t)ω0 + ψδ (t) ∫ W(t − s)[−2γψ1 (s)ω(s) 2 0

2

̃ ̃ − (QN (ψδ (s)u(s)ψδ (s)ω(s))) x − (QN (ψδ (s)ω(s)) )x ]ds.

(7.1.52)

236 | 7 Global attractors and time-spatial chaos Combining the above estimates, we can choose ϵ = b − 3b + 1 > 0, such that ̃ X 0,b ̃ X 0,b ≤ ‖ψ1 ‖H b (R) ‖ω0 ‖L2 (R) + γC1 δ‖ψ1 ‖H b (R) ‖ω‖ ‖ω‖ x t

t

1 ̃ 2X 0,b + C1 Cs C22 δϵ ‖u‖X 0,b ‖ω‖ ̃ X 0,b . + C1 Cs C22 δϵ ‖ω‖ 2

Hence, for δ sufficiently small, we obtain ̃ X 0,b ≤ C‖ω0 ‖L2 (R) , ‖ω‖

25 1 0 (and starting with n sufficiently large such that tn − T ≥ 0). It follows from the Arzela–Ascoli theorem that {vn (tn +⋅)}n is precompact in C([−T, T]; L2loc (R)) for every T > 0. Then, using again (7.1.72), we know by interpolation that {vn (tn + ⋅)}n is

240 | 7 Global attractors and time-spatial chaos s actually precompact in C([−T, T]; Hloc (R)) for all 0 ≤ s < 3. Thus, by a diagonalization process, we obtain a subsequence such that s strongly in C([−T, T]; Hloc (R)), ∀s ∈ [0, 3), ̄ { vnj (tnj + ⋅) → u(⋅) ∞ weakly star in L ([−T, T]; H 3 (R)), ∀T > 0.

(7.1.74)

Because ū t ∈ L∞ ([−T, T]; L2 (R)), for any s < 3 ū ∈ C(R; H s (R)),

̄ ‖u(t)‖ H 3 (R) ≤ C(ρ, N),

∀t ∈ R,

(7.1.75)

the uniform bound follows from the boundedness of v in H 3 (R) (see (7.1.69)). The fact that ū is weakly continuous with values H 3 (R) (by the Strauss theorem). We also find that weakly in H 3 (R), ∀t ∈ R.

̄ vnj (tnj + t) → u(t)

(7.1.76)

From (7.1.60), we get that ‖ωn (tn + t)‖L2 (R) → 0

uniformly for t ≥ −T, ∀T > 0.

(7.1.77)

From (7.1.75) and (7.1.77), one can pass to the limit in the weak formulation of the equation for vnj to find that ū is a solution of the weakly damped, forced KdV equation (7.1.1). We now write the integral form (7.1.71) of the L2 energy equation for vn with t = tn and t0 = tn − T. ‖vn (tn )‖2L2 (R)

= ‖vn (tn −

T)‖2L2 (R) e−2γT

− (PN (2vn ωn −

T

+ ∫ e−2γ(T−s) [2(f , vn )L2 (R)

0 2 ωn ), vnx )L2 (R) ]ds,

(7.1.78)

where, for simplicity of notation, we omitted the argument tn − T + s of the functions inside the time integral. By using the uniform bound for v in H 3 (R), decay (7.1.77) of ωn and the weak-star limit of vnj in (7.1.74), we get lim sup ‖vnj (tnj )‖2L2 (R) j→∞

T

≤ C(R, N)e

−2γT

̄ + 2 ∫ e−2γ(T−s) (f , u(−T + s))L2 (R) ds.

(7.1.79)

0

By substituting the L2 energy equation for ū into (7.1.79), we obtain 2 ̄ lim sup ‖vnj (tnj )‖2L2 (R) ≤ 2C(R, N)e−2γT + ‖u(0)‖ L2 (R) . j→∞

(7.1.80)

7.1 Global attractor of low regularity | 241

Let T → ∞ to deduce that 2 ̄ lim sup ‖vnj (tnj )‖2L2 (R) ≤ ‖u(0)‖ L2 (R) . j→∞

(7.1.81)

This together with the weak convergence (7.1.76) and the decay (7.1.77), implies that strongly in L2 (R),

̄ { unj (tnj ) = vnj (tnj ) + ωnj (tnj ) → u(0)

weakly in H 3 (R).

(7.1.82)

This yields that the solution operator is asymptotically compact in L2 (R); there exists a global attractor A in L2 (R). Moreover, it also follows that A is a bounded set in H 3 (R). By interpolation, A is compact in any H s (R), for 0 ≤ s < 3. It remains to show that A is compact in H 3 (R). We assume that the sequence of initial conditions {u0n }n ∈ A . Because the global attractor is invariant and is bounded in H 3 (R), the corresponding trajectories un (t) = S(t)u0n are uniformly bounded in H 3 (R) for all t ∈ R. We want to show that unj (tnj ) condu ̄ verges to u(0) in H 3 (R). For this purpose, we use the notation un = dtn in the previous equation to obtain unt + (un un )x + unxxx + γun = 0.

(7.1.83)

̄ From the equation for un , we see that proving that unj (tnj ) converges to u(0) in H 3 (R) 2 ̄ amounts to proving that unj (tnj ) converges to u (0) strongly in L (R). Since the trajectories {un (tn + ⋅)}n are uniformly bounded in H 3 (R) and converge strongly in C([−T, T]; L2 (R)) to u,̄ we get strongly in C([−T, T]; H −s (R)), ∀s ∈ [0, 3)

unj (tnj + ⋅) → ū (⋅) {

weakly star in L∞ ([−T, T]; L2 (R)), ∀T > 0.

(7.1.84)

Now, we consider the L2 energy equation for un ‖un (tn )‖2L2 (R)

=

‖un (tn

−

T)‖2L2 (R) e−2γT

T

− ∫ e−2γ(T−s) (unx , u2 n )ds

(7.1.85)

0

where, for simplicity, we omitted the argument tn − T + s of the functions inside the time integral. We want to pass to the limit in (7.1.85). On the one hand, due to (7.1.84), as j → ∞, T

∫ e−2γ(T−s) ((unj )x − ū x , ū 2 )L2 (R) ds → 0. 0

(7.1.86)

242 | 7 Global attractors and time-spatial chaos 23 25 On the other hand, for 0 < s ≤ 48 , and 21 < b < 48 , we take b such that 1 − b < b < 21 and proceed as in (7.1.56)–(7.1.58) (but with −s, and using (7.1.24) with b = b instead of (7.1.25)) to find T 2 ∫ e−2γ(T−s) ((unj )x , u2 nj − ū )L2 (R) ds ≤ C‖unj ‖X s,b ‖unj − ū ‖X −s,b . [0,T]

0

[0,T]

(7.1.87)

−s,b (see Theorem 7.1.1), we can pass From (7.1.84) and the well-posedness of (7.1.83) in X[0,T] to the limit as j → ∞ and have T

lim ∫ e−2γ(T−s) ((unj )x (tnj − T + s), unj (tnj − T + s)2 )L2 (R) ds

j→∞

0

T

= ∫ e−2γ(T−s) (ū x (−T + s), ū (−T + s)2 )L2 (R) ds.

(7.1.88)

0

Then, it follows from (7.1.85) that T

lim sup ∫ e−2γ(T−s) ‖unj (tnj )‖ ≤ C(ρ)e−2γT j→∞

0

T

− ∫ e−2γ(T−s) (ū x (−T + s), ū (−T + s)2 )L2 (R) ds.

(7.1.89)

0

By substituting the corresponding L2 energy equation for ū into the above equation, we have that lim sup ‖unj (tnj )‖2 ≤ 2C(ρ)e−2γT + ‖ū (0)‖2L2 (R) .

(7.1.90)

lim sup ‖unj (tnj )‖2 ≤ ‖ū (0)‖L2 (R) .

(7.1.91)

unj (tnj ) → ū (0) strongly in L2 (R).

(7.1.92)

j→∞

Let T → ∞, j→∞

Therefore,

̄ As mentioned above, it follows that unj (tnj ) converges to u(0) in H 3 (R), which gives the

desired asymptotic compactness in H 3 (R), the compactness of A in H 3 (R) as well.

Theorem 7.1.3. Let γ > 0 and f ∈ L2 (R). Then, the solution operator {S(t)} in L2 (R) associated with equation (7.1.1) possesses a connected global attractor A in L2 (R) which

7.1 Global attractor of low regularity | 243

is compact in H 3 (R). More precisely, A is a connected and compact set in H 3 (R), it is invariant for the system; it attracts (in the L2 (R)-metric) all the orbits of the system uniformly with respect to bounded sets (in L2 (R)) of initial conditions; and (with respect to the inclusion relation) A is maximal among the bounded invariant sets and minimal among the globally attracting sets. Similarly, one can yield the following result. Theorem 7.1.4. Let γ > 0 and f ∈ H k (R), k ∈ N. Then, for each m = 0, 1, . . . , k, the solution operator {Sm (t)}t∈R associated with equation (7.1.1) in the phase space H m (R) is well-defined and possesses a connected global attractor A in H m (R) which is the same for all m = 0, 1, . . . , k. Moreover, the global attractor A is compact in H k+3 (R). Now, we prove the L2 weak continuity of the solution operator. Since we are interested in the weak continuity locally in time, we can take γ = 0 for simplicity. The proof for γ ∈ R is similar. Moreover, since the solutions are bounded in L2 (R) on a finite interval of time (globally in time for γ = 0), we can also consider a suitably small interval of time depending on the L2 (R)-norm of the initial condition. This result can then be iterated to yield the weak continuity on arbitrarily large intervals of time. Consider that uϵ0 converges weakly to u0 in L2 (R), and u(t) is solution to the following equation: ut + uxxx + uux = 0,

(7.1.93)

with u(0) = u0 . This solution is understood in the mild sense of the Bourgain–KPV solution to KdV. Consider the solution uϵ (t) of (7.1.93) with initial data uϵ0 . For t small enough, it remains to prove that uϵ (t) converges weakly to u(t) in L2x (R). ̂ )χ( Nξ ), First step: Fix N and consider the projector PN defined by ℱ (PN u)(ξ ) = u(ξ where χ is the characteristic function of the interval [−1, 1]. Consider vϵ,N that solves vt + vxxx + vvx = 0

(7.1.94)

with initial condition v(0) = PN uϵ0 . For fixed N, we know that vϵ,N (t) is bounded in the space C([−T, T], Hx2 (R)) ∩ C 1 ([−T, T], Hx−1 (R)). Because PN uϵ0 is smooth, and ‖PN uϵ0 ‖Hx2 (R) ≤ KN 2 ,

(7.1.95)

where K is independent of ϵ. If L2x,loc (R) denotes the Fréchet space endowed with its natural topology (L2x convergence on bounded sets), vϵ,N strongly converges to vN in C([−T, T]; L2x,loc (R)). We can

244 | 7 Global attractors and time-spatial chaos let ϵ → 0 in (7.1.95) (convergence in D ), and vN is solution to (7.1.95) with initial data PN u0 . On the other hand, the limit vN ∈ L∞ ([−T, T]; Hx2 (R)) (weak-star convergence) and vtN ∈ L∞ ([−T, T]; Hx−1 (R)). Therefore, vN ∈ C([−T, T]; Hxs (R)) for each s < 2 and is even weakly continuous in t with values in Hx2 (R). By uniqueness of solution of (7.1.94) in C([−T, T]; Hxs (R)) for s > 32 , vN is the solution of (7.1.94) with initial data PN u0 . Second step: Let ωϵ,N (t) = uϵ (t) − vϵ,N (t) is solution to ωt + ωxxx + ωωx + (vω)x = 0,

(7.1.96)

with initial data ω(0) = (Id − PN )uϵ0 (ϵ = 0). We can prove that on bounded intervals of time, for any s ∈ [0, 43 ), ‖ω(t)‖H −s (R) ≤ K‖ω(0)‖H −s (R) .

(7.1.97)

Here, K depends on T but is independent of N and ϵ. In fact, (7.1.97) comes from the well-posedness of the KdV equation in Bourgain −s, 1

spaces Xloc 2 . Since uϵ0 is bounded in L2 (R), we have ‖ω(0)‖H −s (R) = ‖(Id − PN )uϵ0 ‖H −s (R) ≤

‖uϵ0 ‖L2 (R) Ns

≤ KN −s .

(7.1.98)

Let N be fixed, for a smooth test function ψ with compactly supported and satisfies ‖ψ‖L2 (R) = 1, |(u(t) − uϵ (t), ψ)L2 (R) | = |(u(t) − vN (t) + vN (t) − vϵ,N (t) + vϵ,N (t) − uϵ (t), ψ)L2 (R) | = |(ω0,N (t) + vN (t) − vϵ,N (t) + ωϵ,N (t), ψ)L2 (R) | ≤ 2KN −s + |(vN (t) − vϵ,N (t), ψ)L2 (R) |,

where we used (7.1.98). Let ϵ → 0 with N fixed, from the first step, it follows that lim sup |(vN (t) − uϵ (t), ψ)L2 (R) | ≤ 2KN −s . ϵ→0

(7.1.99)

Let N → ∞ to conclude that uϵ (t) converges to u(t) in the distribution sense. We get rid of the condition “ψ is smooth and compactly supported” by a density argument, since ωϵ,N is a bounded sequence in L∞ ([−T, T]; L2 (R)). This concludes the proof.

7.2 Attractors and their spatial complexity of reaction-diffusion equations on bounded domain We consider the initial boundary value problem of the following quasi-linear parabolic equation: ut = aΔx u − λ0 u − f (u) + g, { u|𝜕Ω = 0, u|t=0 = u0 ,

x ∈ Ω,

(7.2.1)

7.2 Attractors and their spatial complexity | 245

in an unbounded domain Ω ⊂ ℝn , and u(t) = (u1 (t, x), u2 (t, x), . . . , uk (t, x)) is an unknown vector-valued function, f and g are given functions, λ0 > 0 is a positive constant and a is a k × k matrix satisfying the condition a + a∗ > 0.

(7.2.2)

The long-time behavior of solutions is of great interest. It is well-known that, under the appropriate assumptions on the nonlinear term f (u), this behavior can be described in terms of an attractor of the corresponding dynamical system see, e. g., [83, 91]. The nonlinear term f (u) satisfies the assumptions f (u) ∈ C 2 (ℝk , ℝk ), { { { f (u) ⋅ u ≥ −C, { { { {f (u) ≥ −k.

(7.2.3)

We first introduce some functional spaces, for a detailed study of these spaces, see [90]: Wbl,p (Ω) = {u ∈ D (Ω) : ‖u‖W l,p = sup ‖u‖W l,p (Ω∩B1 x0 ∈Ω

b

x0 )

< ∞}

with the appropriate choice of exponents l and p (here and below, BRx0 denotes the

R-ball in ℝn centered at x0 , and W l,p (V) is a Sobolev space of functions whose derivatives up to order l belong to Lp (V)). A function ϕ ∈ Cloc (ℝn ) is a weight function with the growth rate μ ≥ 0 if the condition ϕ(x + y) ≤ Cϕ eμ|x| ϕ(y),

ϕ(x) > 0,

(7.2.4)

is satisfied for every x, y ∈ ℝn . It is not difficult to deduce from (7.2.4) that ϕ(x + y) ≥ Cϕ−1 e−μ|x| ϕ(y)

(7.2.5)

is also satisfied for every x, y ∈ ℝn . The following example of a weight function is of fundamental importance: ϕϵ,x0 (x) = e−ϵ|x−x0 | ,

s ∈ ℝ, x0 ∈ ℝn .

Definition 7.2.1. Let Ω ⊂ ℝn be some unbounded domain, and let ϕ be a weight function with the growth rate μ. We set Lpϕ (Ω) = {u ∈ D (Ω) : ‖u, Ω‖pϕ,0,p ≡ ∫ ϕ(x)|u(x)|p dx < ∞}. Ω

246 | 7 Global attractors and time-spatial chaos Analogously, the weighted Sobolev space Wϕl,p (Ω), l ∈ ℕ, is defined as the space of

distributions whose derivatives up to the order l, inclusive, belong to Lpϕ (Ω). To sims,p instead of Wes,p plify the notation, we will write W{ϵ} −ϵ|x| . We also define another class of weighted Sobolev spaces, l,p Wb,ϕ (Ω) = {u ∈ D (Ω) : ‖u, Ω‖pb,ϕ,l,p = sup ϕ(x0 )‖u, Ω ∩ B1x0 ‖pl,p < ∞}. x0 ∈Ω

Here and below, we denote by BRx0 the ball in ℝn of radius R centered at x0 , and ‖u, l,p V‖l,p denotes ‖u‖W l,p (V) . Below we write Wbl,p instead of Wb,1 .

Proposition 7.2.1. Let u ∈ Lpϕ (Ω), where ϕ is a weight function with the growth rate μ. Then for any 1 ≤ q ≤ ∞, the following estimate is valid: q

(∫ ϕ (x0 )(∫ e Ω

−ϵ|x−x0 |

p

q

1 q

|u(x)| dx) dx0 ) ≤ C ∫ ϕ(x)|u(x)|p dx

Ω

(7.2.6)

Ω

for every ϵ > μ, where the constant C depends only on ϵ, μ, Cϕ and is independent of Ω. Let u ∈ L∞ ϕ (Ω). Then the following analogue of estimate (7.2.6) is valid: sup {ϕ(x0 ) sup e−ϵ|x−x0 | |u(x)|} ≤ C sup{ϕ(x)|u(x)|}.

x0 ∈Ω

x∈Ω

x∈Ω

Proof. Assume that q = 1, from (7.2.4), we get ∫ ∫ ϕ(x0 )e−ϵ|x−x0 | |u(x)|p dxdx0 Ω Ω

≤ Cϕ ∫ ∫ e−μ|x−x0 | e−ϵ|x−x0 | ϕ(x)|u(x)|p dxdx0 Ω Ω

≤ Cϕ (∫ e−(ϵ−μ)|y| dy)(∫ ϕ(x)|u(x)|p dx) Rn

Ω p

≤ C1 ∫ ϕ(x)|u(x)| dx. Ω

For the case q = ∞, applying the (7.2.4), we have sup {ϕ(x0 ) ∫ e−ϵ|x−x0 | |u(x)|p dx}

x0 ∈Ω

Ω

≤ Cϕ ∫ sup {e−μ|x−x0 | e−ϵ|x−x0 | }ϕ(x)|u(x)|p dx x∈Ω

x0 ∈Ω

≤ Cϕ ∫ ϕ(x)|u(x)|p dx. Ω

(7.2.7)

7.2 Attractors and their spatial complexity | 247

Therefore, inequality (7.2.7) holds, for both q = 1 and q = ∞. For the case 1 < q < ∞, the result can be obtained by the interpolation inequality ‖ ⋅ ‖Lq ≤ ‖ ⋅ ‖αL1 ‖ ⋅ ‖1−α L∞ ,

α=

1 . q

We need some regularity assumptions on the domain Ω ⊂ ℝn , and assume that there exists a positive number R0 > 0 such that, for every point x0 ∈ Ω, there exists a smooth domain Vx0 ⊂ Ω such that BRx00 ∩ Ω ⊂ Vx0 ⊂ BRx00 +1 ∩ Ω.

(7.2.8) R +2

Moreover, we also assume that there exists a diffeomorphism θx0 : B20 → Bx00 that θx0 (x) = x0 + px0 (x), θx0 (B10 ) = Vx0 , and ‖px0 ‖CN + ‖p−1 x0 ‖C N ≤ K,

such

(7.2.9)

where the constant K is independent of x0 ∈ Ω and N is large enough. For simplicity, we assume from now on that (7.2.8) and (7.2.9) hold for R0 = 2. We note that, in the case where Ω is bounded, conditions (7.2.8) and (7.2.9) are equivalent to the boundary 𝜕Ω being a smooth manifold. Now, for unbounded domains, the sole smoothness of the boundary is not sufficient to obtain the regular structure of Ω as |x| → ∞, since some uniform smoothness conditions with respect to x0 ∈ Ω are required. It is, however, more convenient to formulate these conditions in the forms (7.2.8) and (7.2.9). Proposition 7.2.2. Let the domain Ω satisfy (7.2.8) and (7.2.9). The weight function satisfies (7.2.4), and R must be a positive number. Then the following estimates yield: C2 ∫ ϕ(x)|u(x)|p dx ≤ ∫ ϕ(x0 ) ∫ |u(x)|p dxdx0 Ω

Ω

Ω∩BRx

0

≤ C1 ∫ ϕ(x)|u(x)|p dx.

(7.2.10)

Ω

Proof. We only outline the proof. Let us change the order of integration in the middle part of (7.2.10): ∫ϕ(x0 ) ∫ |u(x)|p dxdx0 Ω

Ω∩BRx

0

= ∫ |u(x)|p (∫ χΩ∩BRx (x0 )ϕ(x0 )dx0 )dx. Ω

Ω

(7.2.11)

248 | 7 Global attractors and time-spatial chaos Here, χΩ∩BRx is the characteristic function of the set Ω ∩ BRx . It follows from inequalities (7.2.4) and (7.2.5) that C1 ϕ(x) ≤ inf ϕ(x0 ) ≤ sup ϕ(x0 ) ≤ C2 ϕ(x), x0 ∈BRx

x0 ∈BRx

(7.2.12)

and assumptions (7.2.8) and (7.2.9) imply that 0 < C1 ≤ Vol(Ω ∩ BRx ) ≤ C2

(7.2.13)

uniformly with respect to x ∈ Ω. Estimate (7.2.10) follows from estimates (7.2.11)– (7.2.13). Proposition 7.2.2 is proved. Corollary 7.2.1. Let (7.2.8) and (7.2.9) be valid. Then the equivalent norm in a weighted Sobolev space Wϕl,p is given by ‖u, Ω‖ϕ,l,p = (∫ ϕ(x0 )‖u, Ω ∩ Ω

BRx0 ‖pl,p dx0 )

1 p

.

(7.2.14)

In particular, norms (7.2.14) are equivalent for different R ∈ ℝ+ . To study equation (7.2.1), we also need weighted Sobolev spaces with fractional derivatives s ∈ ℝ+ (and not for s ∈ ℤ only). If V is a bounded domain, the norm in the space W s,p (V), s = [s] + l, 0 < l < 1, [s] ∈ ℤ+ , is given by ‖u, V‖ps,p = ‖u, V‖p[s],p + ∑

∫ ∫

|α|=[s] x∈V y∈V

|Dα u(x) − Dα u(y)|p dxdy. |x − y|n+lp

(7.2.15)

It is not difficult to prove, by arguing as in Proposition 7.2.2 and using (7.2.15), that for any bounded domain V with a sufficiently smooth boundary, C1 ‖u, V‖ps,p ≤ ∫ ‖u, V ∩ BRx0 ‖ps,p dx0 ≤ C2 ‖u, V‖ps,p .

(7.2.16)

x0 ∈V

Definition 7.2.2. We define the space Wϕs,p (Ω) for any s ∈ ℝ+ as the space of distributions whose norm (7.2.14) is finite. It is not difficult to check that these norms are also equivalent for different R > 0. We now note that the weight functions ϕϵ,x0 = e−ϵ|x−x0 |

(7.2.17)

satisfy conditions (7.2.4) uniformly with respect to x0 ∈ ℝn . Consequently, all estimates obtained above for the arbitrary weights remain valid for family (7.2.17) with constants independent of x0 ∈ ℝn . These estimates are written explicitly in the form of corollaries as follows.

7.2 Attractors and their spatial complexity | 249

Corollary 7.2.2. Let u ∈ Lp{δ} (Ω), 0 < δ < ϵ. Then the following estimate holds uniformly with respect to y ∈ ℝn : q

( ∫e−qδ|x0 −y| (∫ e−ϵ|x−x0 | |u(x)|p dx) dx0 ) Ω

1 q

Ω

≤ Cϵ,q ∫ e

−δ|x−y|

|u(x)|p dx.

(7.2.18)

Ω

L∞ {δ} (Ω),

Moreover, if u ∈

δ < ϵ, we have

sup {e−δ|x0 −y| sup{e−ϵ|x−x0 | |u(x)|}} ≤ Cϵ,δ sup{e−δ|x−y| |u(x)|}.

x0 ∈Ω

x∈Ω

x∈Ω

(7.2.19)

l,p Corollary 7.2.3. Let u ∈ Wb,ϕ (Ω), and ϕ be a weight function with the growth rate μ < ϵ. Then

C1 ‖u, Ω‖pb,ϕ,l,p ≤ sup {ϕ(x0 ) ∫ e−ϵ|x−x0 | ‖u, Ω ∩ B1x ‖pl,p dx} x0 ∈Ω

≤

x∈Ω p C2 ‖u, Ω‖b,ϕ,l,p .

(7.2.20)

Proof. First, sup {ϕ(x0 ) ∫ e−ϵ|x−x0 | ‖u, Ω ∩ B1x ‖pl,p dx}

x0 ∈Ω

x∈Ω

≤ C sup{ϕ(x)‖u, Ω ∩ B1x ‖pl,p } sup { ∫ ϕ(x0 )ϕ−1 (x)e−ϵ|x−x0 | dx} x∈Ω

x0 ∈Ω

x∈Rn

≤ C1 sup{ϕ(x)‖u, Ω ∩ B1x ‖pl,p } sup { ∫ eμ|x−x0 | e−ϵ|x−x0 | dx} x∈Ω

≤

x0 ∈Ω

C2 ‖u, Ω‖pb,ϕ,l,p .

x∈Rn

On the other hand, by the inequality ‖u, Ω ∩ B1x0 ‖pl,p ≤ C ∫ e−ϵ|x−x0 | ‖u, Ω ∩ B1x ‖pl,p dx,

(7.2.21)

Ω

we get ‖u, Ω‖pb,ϕ,l,p = sup {ϕ(x0 )‖u, Ω ∩ B1x0 ‖pl,p } x0 ∈Ω

≤ C sup {ϕ(x0 ) ∫ e−ϵ|x−x0 | ‖u, Ω ∩ B1x ‖pl,p dx}. x0 ∈Ω

x∈Ω

We also need the following subclass of weight functions with an exponential growth rate.

250 | 7 Global attractors and time-spatial chaos Definition 7.2.3. A function ϕ ∈ Cloc (ℝn ), ϕ > 0, is a weight function with polynomial growth rate μ if the following inequality is valid for every x, y ∈ ℝn : μ

ϕ(x + y) ≤ Cϕ ((1 + |y1 |2 )(1 + |y2 |2 ) ⋅ ⋅ ⋅ (1 + |yn |2 )) 2 ϕ(x).

(7.2.22)

The next analogue of Corollary 7.2.3 is valid for such weight functions. Corollary 7.2.4. Let ϕ be a weight function with a polynomial growth rate μ < N. Then the following estimate yields: C1 sup ϕ(x0 )u(x0 ) x0 ∈Ω

≤ sup{ϕ(x) sup((1 + |x1 − y1 |2 ) ⋅ ⋅ ⋅ (1 + |xn − yn |2 )) x∈Ω

y∈Ω

−N 2

u(y)}

≤ C2 sup ϕ(x0 )u(x0 ). x0 ∈Ω

Next, we derive some a priori estimates for the solutions of the reaction-diffusion system ut = aΔx u − λ0 u − f (u) + g,

x ∈ Ω, u|𝜕Ω = 0,

u|t=0 = u0 ,

(7.2.23)

in the unbounded domain Ω ⊂ ℝn . Recalling that a is a constant k ×k matrix satisfying the condition a + a∗ > 0, λ0 > 0, and the nonlinear term f (u) satisfies the assumptions f (u) ∈ C 2 (ℝk , ℝk ), { { { f (u) ⋅ u ≥ −C, { { { {f (u) ≥ −k.

(7.2.24)

We also impose the additional growth restriction for the nonlinearity f (u), |f (u)| ≤ C(1 + |u|p ),

(7.2.25)

where the exponent p is arbitrary for n ≤ 4 and p < 1 + 4/(n − 4), n ≥ 5. The external force g ∈ Lqb (Ω) for certain q ≥ 2 and q > n/2 (if n ≤ 3, then the exponent q = 2 is allowed), and the initial data u0 ∈ Φb (Ω) = Wb2,q (Ω) ∩ {u0 |𝜕Ω = 0}. The solution of (7.2.23) is a function u ∈ L∞ (ℝ+ ; Wb2,q (Ω)) ∩ C([0, ∞); Lqb (Ω))

(7.2.26)

that satisfies the equation in the sense of distributions. It follows from the Sobolev embedding theorem and from choice of the exponent q (q > n/2) that the solution u ∈ L∞ (ℝ+ × Ω); consequently, the nonlinear term in (7.2.23) is well-defined and belongs to L∞ . Therefore, it follows from (7.2.26) and from equation (7.2.23) that ut ∈ L∞ (ℝ+ ; Lqb (Ω)).

(7.2.27)

7.2 Attractors and their spatial complexity | 251

Moreover, it can be shown by using standard arguments that q 1 u ∈ C([0, T]; We2,q −ϵ|x| (Ω)) ∩ C ([0, T]; Le−ϵ|x| (Ω))

for every T > 0 and every ϵ > 0. We note, in contrast to the case of bounded domains, for generic u0 ∈ Φ, the corresponding solution u(t) is not continuous at t = 0 as a function with values in Φb (Ω). Theorem 7.2.1. Let the above assumptions hold and let u(t) be a solution of (7.2.23). Then the following estimate is valid: ‖u(t)‖Φb (Ω) ≤ Q(‖u(0)‖Φb (Ω) )e−αt + Q(‖g‖Lq (Ω) ), b

(7.2.28)

where α > 0 is a positive constant depending only on (7.2.23), and Q is a monotonic function that also depends only on (7.2.23) (and is independent of u and u0 ). Proof. We divide the proof of this theorem into several lemmas. Lemma 7.2.1. Let the above assumptions hold; then the following estimate holds for every x0 ∈ Ω: ‖u(T), Ω ∩

B1x0 ‖20,2

T+1

+ ∫ ‖u(t), Ω ∩ B1x0 ‖21,2 dt T

≤ Ce−αT (e−ϵ|x−x0 | , |u(0)|2 ) + C(|g|2 , e−ϵ|x−x0 | ), where the positive constants C, α, ϵ are independent of x0 , and (u, v) denotes the inner product in L2 (Ω). This estimate can be obtained in a standard way by multiplying equation (7.2.23) by u(t)e−ϵ|x−x0 | (ϵ > 0 small enough), integration by parts, and using the assumption f (u) ⋅ u ≥ −C, the positivity of a, it is easy to see that ‖∇x (e−ϵ|x−x0 | )‖ ≤ ϵe−ϵ|x−x0 | .

(7.2.29)

Lemma 7.2.2. Let the above assumptions hold; then the following estimate yields: T+1

‖u(T), Ω ∩ B1x0 ‖21,2 + ∫ ‖u(t), Ω ∩ B1x0 ‖22,2 dt T

≤ Ce−αT (e−ϵ|x−x0 | , |u(0)|2 + |∇x u(0)|2 ) + C(|g|2 , e−ϵ|x−x0 | ),

(7.2.30)

where the positive constants C, α, ϵ are independent of x0 . Proof. Multiplying equation (7.2.23) by the expression n

∑ 𝜕xi (ϕϵ,x0 (x)𝜕xi u(t)) = ϕϵ,x0 Δx u(t) + ∇x ϕϵ,x0 ⋅ ∇x u(t), i=1

(7.2.31)

252 | 7 Global attractors and time-spatial chaos where ϕϵ,x0 = e−ϵ|x−x0 | and ϵ > 0 is small enough. Then, after the standard integration by parts and using the monotonicity assumption f (u) ≥ −k and inequality (7.2.29), we have 1 𝜕 (ϕ , |∇ u(t)|2 ) + λ0 (ϕϵ,x0 , |∇x u(t)|2 ) + μ(ϕϵ,x0 , |Δx u(t)|2 ) 2 t ϵ,x0 x ≤ k(ϕϵ,x0 , |∇x u(t)|2 ) + C|a|ϵ(ϕϵ,x0 |Δx u(t)|, |∇x u(t)|) + (ϕϵ,x0 , |g||Δx u(t)| + ϵ|g||∇x u(t)|).

(7.2.32)

Estimating the last two terms in the right-hand side of (7.2.32) by the Hölder inequality, we derive that 𝜕t (ϕϵ,x0 , |∇x u(t)|2 ) + λ0 (ϕϵ,x0 , |∇x u(t)|2 ) + μ(ϕϵ,x0 , |Δx u(t)|2 ) ≤ 2k(ϕϵ,x0 , |∇x u(t)|2 ) + C(ϕϵ,x0 , |g|2 ).

(7.2.33)

Applying the Gronwall inequality to (7.2.33) and using inequality in Lemma 7.2.1 for estimating the t-integral over the right-hand side of (7.2.33), we obtain (ϕϵ,x0 , |∇x u(T)|2 ) ≤ Ce−αT (ϕϵ,x0 , |∇x u(0)|2 + |u(0)|2 ) + C(ϕϵ,x0 , |g|2 ).

(7.2.34)

Estimates (7.2.33) and (7.2.34) imply that T+1

∫ (ϕϵ,x0 , |Δx u(t)|2 )dt T

≤ C1 e−αT (ϕϵ,x0 , |∇x u(0)|2 + |u(0)|2 ) + C1 (ϕϵ,x0 , |g|2 ).

(7.2.35)

Moreover, according to our regularity assumptions on the boundary 𝜕Ω, we have elliptic regularity for Δv, ‖v‖W 2,2

ϕϵ,x

0

≤ C(‖Δx v‖L2

ϕϵ,x

0

+ ‖v‖L2

ϕϵ,x 0

(7.2.36)

).

Estimates (7.2.34), (7.2.35) and (7.2.36) imply Lemma 7.2.2. The next thing is to obtain the estimate for the Wb2,2 -norm, analogous to (7.2.28). To this end, we introduce the following norm, depending on ϵ > 0 and x0 ∈ Ω: ‖v‖2Dϵ,x = ‖v‖2W 2,2 0

ϕϵ,x

0

+ ‖f (v)‖2L2

ϕϵ,x

0

(7.2.37)

.

Lemma 7.2.3. Let the above assumptions hold and ϵ > 0 be small enough. Then the following estimate holds for the solutions of equation (7.2.23): ‖u(t)‖2Dϵ,x ≤ Ce2kt (1 + ‖u(0)‖2Dϵ,x + ‖g‖2L2 0

0

ϕϵ ,x0

),

(7.2.38)

where the constant k is the same as in (7.2.24) and the constant C is independent of x0 and ϵ.

7.2 Attractors and their spatial complexity | 253

Proof. We give only the formal derivation of estimate (7.2.38), which can easily be justified using, for example, the standard difference approximations for the derivative 𝜕t u and regularity. Let us differentiate equation (7.2.23) with respect to t and denote θ(t) = ut . Then this function satisfies the equation θt = aΔx θ − λ0 θ − f (u)θ, { θ(0) = aΔx u0 − λ0 u0 − f (u0 ) + g,

θ|𝜕Ω = 0.

(7.2.39)

Multiplying this equation by θ(t)ϕϵ,x0 and integrating over x ∈ Ω. Then, integration by parts and using the monotonicity assumption f (u) ≥ −k and inequality (7.2.29) (where ϵ is small enough), we derive the estimate 𝜕t (ϕϵ,x0 , |θ(t)|2 ) ≤ 2k(ϕϵ,x0 , |θ(t)|2 ).

(7.2.40)

Applying the Gronwall inequality, we obtain that ‖𝜕t u(t)‖2L2

ϕϵ ,x0

≤ Ce2kt (1 + ‖u0 ‖2Dϵ,x + ‖g‖2L2

ϕϵ ,x0

0

(7.2.41)

).

We can consider parabolic equation (7.2.23) as an elliptic boundary value problem at a fixed-point T, aΔx u(T) − f (u(T)) = hu := 𝜕t u(T) − g,

u(T)|𝜕Ω = 0,

(7.2.42)

with the right-hand side hu ∈ L2ϕϵ,x (Ω). Multiplying the equation by uϕϵ,x0 and by the 0

expression (7.2.31), we have

‖u(T)‖2W 2,2

ϕϵ ,x0

≤ C(1 + ‖hu ‖2L2

ϕϵ ,x0

(7.2.43)

).

Estimates (7.2.41) and (7.2.43) immediately show that ‖u(T)‖2W 2,2

ϕϵ ,x0

≤ Ce2kt (1 + ‖u0 ‖2Dϵ,x + ‖g‖2L2

ϕϵ ,x0

0

(7.2.44)

).

The W 2,2 part of estimate (7.2.38) is proved, and L2ϕϵ,x (Ω)-norm of f (u) is an immedi0

ate corollary of inequalities (7.2.41) and (7.2.44) and equation (7.2.23). Lemma 7.2.3 is proved. Applying the supx0 ∈Ω to the both sides of inequality (7.2.38) and using Corollary 7.2.3, we have ‖u(t)‖2W 2,2 ≤ Ce2kt (1 + ‖u0 ‖2W 2,2 + ‖f (u0 )‖2L2 + ‖g‖2L2 ). b

b

b

b

(7.2.45)

Moreover, according to our growth restrictions on f and the Sobolev embedding theorem, ‖f (u0 )‖2L2 ≤ Q(‖u0 ‖2W 2,2 ) b

b

(7.2.46)

254 | 7 Global attractors and time-spatial chaos for the appropriate monotonic function (Q(z) = C(1 + |z|p )). Inequalities (7.2.45) and (7.2.46) show ‖u(t)‖W 2,2 ≤ Cekt (Q(‖u0 ‖W 2,2 ) + ‖g‖L2 ).

(7.2.47)

b

b

b

We note, however, that the obtained estimate of the Wb2,2 -norm diverges exponentially with respect to t → ∞, which is not good for the study of the attractor. In order to remove this divergence, we need the following smoothing property. Lemma 7.2.4. Let the above assumptions hold; then the following estimate is valid for any solution of problem (7.2.23): ‖u(1)‖W 2,2 (Ω) ≤ Q(‖u(0)‖W 1,2 (Ω) ) + C‖g‖L2 (Ω) , b

(7.2.48)

b

b

for a certain monotonic function Q. Proof. Fix an arbitrary x0 ∈ Ω and a sufficiently small ϵ > 0. It follows from estimate (7.2.30) and Proposition 7.2.1 that 1

∫ ‖u(t)‖2W 2,2 dt ≤ C(1 + ‖u(0)‖2W 2,2 0

ϕϵ ,x0

ϕϵ ,x0

+ ‖g‖2L2

ϕϵ ,x0

(7.2.49)

).

It follows from (7.2.49) that there exists a point T = T(x0 ) ∈ [0, 1] such that ‖u(T)‖2W 2,2

ϕϵ ,x0

≤ C(1 + ‖u(0)‖2W 2,2

ϕϵ ,x0

+ ‖g‖2L2

ϕϵ ,x0

(7.2.50)

).

According to our growth restrictions on the nonlinearity f (u), the Sobolev embedding theorem, and Propositions 7.2.1 and 7.2.2, we derive that 2 −pϵ|x−x0 | |u(T, x)|2p dx) f (u(T))L2 (Ω) ≤ C(1 + ∫ e ϕpϵ ,x0 x∈Ω

≤ C1 (1 + ∫ e−pϵ|x−x0 | ‖u(T), Vx ‖2p 0,2p dx) x∈Ω

≤ C2 (1 + ∫ e−pϵ|x−x0 | ‖u(T), Vx ‖2p 2,2 dx) x∈Ω

p

≤ C3 (1 + ∫ e−pϵ|x−x0 | ( ∫ e−δ|y−x| ‖u(T), Vy ‖22,2 dy) dx) x∈Ω

y∈Ω

≤ C4 (1 + ∫ e−ϵ|x−x0 | ‖u(T), Vx ‖22,2 dx) x∈Ω

p

≤ C5 (1 + ‖u(T)‖2W 2,2 ) ϕϵ ,x0

p

(7.2.51)

7.2 Attractors and their spatial complexity | 255

where δ > ϵ and Vx is the same as in conditions (7.2.8) and (7.2.9). Here, we have also used the obvious formula ‖v, Vx ‖l,p ≤ Cδ ∫ e−δ|x−y| ‖v, Vy ‖l,p dy,

(7.2.52)

y∈Ω

which holds for every δ > 0. Estimates (7.2.50) and (7.2.51) imply that ‖u(T)‖2Dpϵ,x ≤ C(1 + ‖u(0)‖2W 1,2 0

ϕϵ ,x0

p

+ ‖g‖2L2

ϕϵ ,x0

) .

(7.2.53)

Applying now estimate (7.2.38) (with ϵ replaced by pϵ) at the initial time moment t = T instead of t = 0, we derive from (7.2.53) that ‖u(1)‖2W 2,2

ϕpϵ ,x0

≤ C(1 + ‖u(0)‖2W 1,2

ϕϵ ,x0

p

+ ‖g‖2L2

ϕϵ ,x0

) .

(7.2.54)

We note that all constants C in the previous estimates were, in fact, independent of the choice of x0 ∈ Ω; consequently, applying the supx0 ∈Ω to both sides of (7.2.54) and using Corollary 7.2.3, we derive estimate (7.2.48) and finish the proof of Lemma 7.2.4. Thus, we have proved the analogue of estimate (7.2.28) for q = 2. Lemma 7.2.5. Let the above assumptions hold; then ‖u(t)‖W 2,2 (Ω) ≤ Q(‖u0 ‖W 2,2 (Ω) )e−αt + Q(‖g‖L2 (Ω) ) b

b

b

(7.2.55)

for some positive α > 0 and a certain monotonic function Q. This lemma is a simple corollary of estimates (7.2.30), (7.2.47) and (7.2.48). Our task now is, starting from the Wb2,2 -estimate (7.2.55) and using the parabolic regularity theorems, to improve step by step this estimate to the Wb2,q -estimate (7.2.28). We first derive 2−μ,q

the Wb

-estimate for a sufficiently small positive μ.

Lemma 7.2.6. Let the above assumptions hold; then for every μ > 0, the following estimate is valid: ‖u(t)‖W 2−μ,q (Ω) ≤ Qμ (‖u0 ‖Φb (Ω) )e−αt + Qμ (‖g‖Lq (Ω) ) b

b

(7.2.56)

for some positive α > 0 and a certain monotonic function Qμ depending on μ. Proof. We recall that the domain Ω is assumed to satisfy conditions (7.2.8) and (7.2.9) with R0 = 2. Let us consider the cutoff function ψ(x) ∈ C0∞ (ℝn ) such that ψ(x) = 1 if x ∈ B10 , and ψ(x) = 0 if x ∉ B20 . We set ψx0 (x) = ψ(x − x0 ) and vx0 (t) = ψx0 u(t). Then it follows from equation (7.2.1) and from condition (7.2.8) that vx0 (t) is a solution of the problem 𝜕t vx0 − aΔx vx0 + λ0 vx0 = hx0 (t) = ψx0 g − 2∇x ψx0 ⋅ a∇x u,

{

− Δx ψx0 ⋅ avx0 − ψx0 f (u(t)),

vx0 |Vx = 0, 0

The next standard regularity result is important.

vx0 |t=0 = ψx0 u(0).

(7.2.57)

256 | 7 Global attractors and time-spatial chaos Proposition 7.2.3. Let the domains Vx0 satisfy assumptions (7.2.8) and (7.2.9). Then for every 1 ≥ μ > 0, 1 < r < ∞, and t ∈ [0, 1], the following estimate is valid for the solution vx0 of problem (7.2.57): ‖vx0 (t), Vx0 ‖2−μ,r ≤ C(‖vx0 (0), Vx0 ‖2−μ,r + sup ‖hx0 , Vx0 ‖0,r ),

(7.2.58)

s∈[0,t]

where the constant C = C(r, μ) is independent of x0 . Moreover, the following version of the smoothing property is valid for every t ∈ ℝ+ : ‖vx0 (t + 1), Vx0 ‖2−μ,r ≤ C (‖vx0 (t), Vx0 ‖1,2 + sup ‖hx0 , Vx0 ‖0,r ),

(7.2.59)

s∈[t,t+1]

where the constant C is independent of x0 . It is easy to obtain the estimates (7.2.58) and (7.2.59) from the analytic semigroups theory. Moreover, assumptions (7.2.8) and (7.2.9) imply that the constants C and C are independent of x0 . We continue to prove estimate (7.2.56) by induction with respect to the exponent q. Let (7.2.56) be valid for some exponent l ≥ 2; then, applying the operator supx0 ∈Ω to both sides of (7.2.58) (with the appropriate r = r(l), which will be specified below), we have ‖u(t)‖W 2−μ,r (Ω) ≤ C(‖u0 ‖Φb (Ω) + ‖g‖Lq (Ω) ) b

b

+ C( sup ‖u(s)‖W 1,r (Ω) + ‖f (u(s))‖Lr (Ω) ). s∈[0,1]

(7.2.60)

b

b

2−μ,l

Let us estimate the right-hand side of (7.2.60) by the Wb -norms of u(s), which are assumed to be known. The third term on the right-hand side of (7.2.60) can be estimated in this way if r ≤ r1 (l) =

nl , n − l(1 − μ) 2−μ,l

where r1 = r1 (l) is the Sobolev maximal exponent of the embedding Wb usual, r1 = ∞, if n < l(1 − μ)). We note that

⊂ W 1,r1 (as

r1 (l) r1 (2) n > = > δ1 > 1. l 2 n − 2(1 − μ) Analogously, using the growth restriction (7.2.3) and Sobolev’s embedding theorem nl W 2−μ,l ⊂ Lpμ , pμ (l) = n−l(2−μ) , we deduce the estimate p f (u(s))Lr (Ω) ≤ C(1 + ‖u(s)‖W 2−μ,l (Ω) ) , b b

r ≤ r2 (l) =

pμ (l) p

.

(7.2.61)

7.2 Attractors and their spatial complexity | 257

Moreover, according to our growth restrictions, p < n/(n − 4) ( n ≤ 4), we have the embedding W 2,2 ⊂ Lr , and Lemma 7.2.5 implies the estimate of the Lr -norm of f (u) for every r < ∞, we have r2 (l) r2 (2) n n−4 > = ⋅ > δ2 > 1, l 2 p(n − 4) n − 4 + 2μ

(7.2.62)

if μ > 0 is small enough. Let r(l) := min{q, r1 (l), r2 (l)}. Then r(l) ≥ min{q, δl},

δ = min{δ1 , δ2 } > 1,

(7.2.63)

if μ is small enough, and (7.2.60) and (7.2.61) imply that ‖u(t)‖W 2−μ,r(l) (Ω) ≤ C(1 + ‖g‖Lq (Ω) ) b

b

+ C sup ‖u(s)‖p s∈[t−1,t]

2−μ,l

Wb

(Ω)

,

t ≤ 1.

(7.2.64)

We now assume that t ≥ 1. Then, using estimate (7.2.59) instead of (7.2.58) and arguing as in the proof of (7.2.64), we have ‖u(t)‖W 2−μ,r(l) (Ω) ≤ C(1 + ‖g‖Lq (Ω) ) + C sup ‖u(s)‖p b

b

2−μ,l

Wb

s∈[t−1,t]

(Ω)

.

(7.2.65)

Thus, if the estimate (7.2.56) was proved for some q = l, then estimates (7.2.64) and (7.2.65) would imply this estimate for q = r(l) > l (if μ > 0 was small enough). We also recall that estimate (7.2.56) for q = 2 is proved in Lemma 7.2.5. Therefore, starting with l0 = 2 and iterating estimates (7.2.64) and (7.2.65) with lk+1 = r(lk ), we finally obtain estimate (7.2.56) with l = q (the finiteness of the number of iterations is guaranteed by estimate (7.2.63)). Lemma 7.2.6 is proved. According to our assumptions on the exponent q (q > n/2), the embedding ⊂ Cb holds if μ > 0 is small enough. Therefore, estimate (7.2.56) implies the following estimate:

2−μ,q Wb

‖u(t)‖Cb (Ω) ≤ Q(‖u0 ‖Φb (Ω) )e−αt + Q(‖g‖Lq (Ω) ), b

(7.2.66)

with a positive constant α > 0 and some monotonic function Q. To prove that (7.2.56) is valid for μ = 0, we introduce a function v̄x0 = v̄x0 (x) as solution of the equation aΔx v̄x0 − λ0 v̄x0 + ψx0 g = 0,

v̄x0 |𝜕Vx = 0, 0

(7.2.67)

where ψx0 and Vx0 are the same as in the proof of Lemma 7.2.6. Then, due to the Lq -regularity theorem for the Laplacian, we get ‖v̄x0 , Vx0 ‖2,q ≤ C‖g, Vx0 ‖0,q .

(7.2.68)

258 | 7 Global attractors and time-spatial chaos Moreover, owing to assumptions (7.2.8) and (7.2.9), the constant C is independent of x0 ∈ Ω. Let Wx0 (t) = vx0 (t) − v̄x0 , where vx0 is the same as in the proof of the previous lemma. Then this function satisfies the equation 𝜕t Wx0 − aΔx Wx0 + λ0 Wx0 = h̄ x0 (t) = −2∇x ψx0 ⋅ a∇x u(t),

{

− Δx ψx0 ⋅ au(t) − ψx0 f (u(t)),

Wx0 |𝜕Vx = 0,

Wx0 |t=0 = ψx0 u0 − v̄x0 .

0

(7.2.69)

The proof of Theorem 7.2.1 is based on (7.2.56) and on the following standard regularity result for auxiliary problem (7.2.69). Proposition 7.2.4. Let the above assumptions hold, and let β > 0 be a positive number. Then the solutions of equation (7.2.69) satisfy the estimate ‖Wx0 (t), Vx0 ‖2,q ≤ C(‖Wx0 (0), Vx0 ‖2,q + sup ‖h̄ x0 , Vx0 ‖β,q ), s∈[0,1]

(7.2.70)

which is valid for t ≤ 1 and a constant C that is independent of x0 . Moreover, the following result of the smoothing property holds for every t ≥ 0 and μ > 0: ‖Wx0 (t + 1), Vx0 ‖2+β−μ,q ≤ C(‖Wx0 (t), Vx0 ‖1,2 + sup ‖h̄ x0 (s), Vx0 ‖β,q ), s∈[t,t+1]

(7.2.71)

where the constant C = C(β, μ) is also independent of x0 . Estimates (7.2.70) and (7.2.71) can be obtained from the analytic semigroups theory. The fact that the constant C is independent of x0 is guaranteed by regularity assumptions on the domains Vx0 . Due to the assumption f ∈ C 1 and the embedding W 2−μ,q ⊂ C for sufficiently small μ > 0, we have ‖f (u(s))‖W 1,q (Ω) ≤ Q(‖u(s)‖W 2−μ,q (Ω) ) b

(7.2.72)

b

for a certain monotonic function Q (depending only on f ). Consequently, discussing as in the proof of Lemma 7.2.6 and from the estimates (7.2.70), (7.2.71), and (7.2.72), we derive that for t ≤ 1, ‖u(t)‖W 2,q (Ω) ≤ C(‖u0 ‖Φb (Ω) + ‖g‖Lq (Ω) ) + sup Q1 (‖u(s)‖W 2−μ,q (Ω) ) b

b

s∈[0,1]

b

(7.2.73)

and for the appropriate function Q1 , and the smoothing property ‖u(t + 1)‖W 2,q (Ω) ≤ sup Q1 (‖u(s)‖W 2−μ,q (Ω) ) + C‖g‖Lq (Ω) b

s∈[t,t+1]

b

b

(7.2.74)

is also valid for every t ≥ 0. Inserting estimate (7.2.56) into the right-hand side of (7.2.73) and (7.2.74), Theorem 7.2.1 follows after simple transformations.

7.2 Attractors and their spatial complexity | 259

Remark 7.2.1. Arguing as in the proof of Theorem 7.2.1, we can deduce the following smoothing property for the solutions of (7.2.23): ‖u(1)‖Φb (Ω) ≤ Q(‖u(0)‖L2 (Ω) ).

(7.2.75)

b

The smoothing property from Wb1,2 (Ω) to Wb2,q (Ω) is proved in Lemma 7.2.3 and Lemma 7.2.6. The smoothing property from L2b to Wb1,2 can be proved in a standard way (see the proof of Lemma 7.2.2), multiplying by t in (7.2.31). From a priori estimate (7.2.28), we can easily verify the existence of a solution for problem (7.2.23). Theorem 7.2.2. Let the above assumptions hold. Then for every u0 ∈ Φb (Ω), equation (7.2.23) possesses a unique solution u(t), and the following estimate holds two solutions u1 (t) and u2 (t) of equation (7.2.23): ‖u1 (t) − u2 (t)‖L2 (Ω) ≤ Cekt ‖u1 (0) − u2 (0)‖L2 (Ω) , b

(7.2.76)

b

where the constant k is the same as in (7.2.24) and the constant C depends only on the equation. Proof. The existence of a solution of (7.2.23) for the case where the domain Ω is bounded can be deduced from a priori estimate (7.2.28) by using the Leray–Schauder fixed-point principle. The existence of a solution in the unbounded domain Ω can be proved after approximating the unbounded domain Ω by the bounded ones ΩN and passing to the limit N → ∞. Now, we prove estimate (7.2.76), which immediately implies the uniqueness. Let u1 (t) and u2 (t) be two solutions of (7.2.23) and let v(t) = u1 (t) − u2 (t). Then this function satisfies the equation 𝜕t v = aΔx v − λ0 v − l(t)v,

v|𝜕Ω = 0,

v|t=0 = u1 (0) − u2 (0),

(7.2.77)

where 1

l(t) = ∫ f (su1 (t) + (1 − s)u2 (t))ds,

l(t) ∈ ℒ(ℝk , ℝk ).

0

Moreover, according to our assumptions on f , we have l(t) ≥ −k; consequently, multiplying equation (7.2.77) by v(t)ϕϵ,x0 integrating over x ∈ Ω, and arguing as in the proof of Lemmata 7.2.1 and 7.2.2, we have t+1

‖v(t)‖2L2 ϕ

ϵ,x0

(Ω)

+ ∫ ‖v(s)‖W 1,2 t

ϕϵ,x

0

(Ω) ds

≤ Ce2kt ‖v(0)‖2L2

ϕϵ,x

0

(Ω) .

(7.2.78)

Applying the operator supx0 ∈Ω to both sides of (7.2.78) and using Corollary 7.2.3, we obtain inequality (7.2.76), and thus prove Theorem 7.2.2.

260 | 7 Global attractors and time-spatial chaos Corollary 7.2.5. Let the above assumptions hold; then problem (7.2.23) defines a semigroup St in the phase space Φb (Ω), St : Φb (Ω) → Φb (Ω),

u(t) = St u0 ,

(7.2.79)

where u(t) is a solution of (7.2.23) with u(0) = u0 . Estimate (7.2.76) allows us to extend by continuity the semigroup St from Φb (Ω) to Moreover, due to smoothing property (7.2.75), the semigroup Ŝt thus obtained will act from L2b (Ω) to Φb (Ω) if t > 0. Thus, it is possible to define a solution of problem (7.2.23) for every initial data from L2b (Ω). We conclude this section by formulating some results on the smoothing property for the difference of the solutions of (7.2.23), which are of fundamental importance to our study of the attractor of semigroup (7.2.79). L2b (Ω).

Theorem 7.2.3. Let the above assumptions hold, then for every two solutions u1 (t), u2 (t) ∈ Φb and every ϵ > 0, the following estimate holds: ‖u1 (1) − u2 (1), Ω ∩ B1x0 ‖21,2 ≤ C‖u1 (0) − u2 (0)‖2L2

ϕϵ,x

0

(Ω) ,

(7.2.80)

where the constant C = C(‖u1 ‖Φb , ‖u2 ‖Φb , ϵ) is independent of x0 ∈ Ω. Analogously, ‖u1 (1) − u2 (1), Ω ∩ B1x0 ‖q2,q ≤ C1 ‖u1 (0) − u2 (0)‖qL2

ϕϵ,x

0

(Ω)

,

(7.2.81)

where C1 is also independent of x0 ∈ Ω. Obviously, the first estimate is an immediate corollary of the second one, but it is more convenient for us to formulate them separately. They are very important to study the entropy of the attractor. Proof. The proof of these estimates is based on a standard analysis of linear equation (7.2.77) and can be obtained in the spirit of the proof of Theorem 7.2.1: ‖l(t)‖W 1,q ∩C b

b (Ω)

≤ Q(‖u1 (0)‖Φb , ‖u2 (0)‖Φb ),

due to (7.2.28) and the facts that f ∈ C 2 , Wb2,q ⊂ C. In order to prove the first estimate of the theorem, it is sufficient to multiply equation (7.2.77) by t ∑ni=1 𝜕xi (ϕϵ,x0 𝜕xi v(t)), integrate over x ∈ Ω and apply the Gronwall inequality using estimates (7.2.81) and (7.2.78). The second estimate can be deduced from the first one, by using, for example, the iteration procedure described in the proof of Lemma 7.2.6. Theorem 7.2.3 is proved. In the following, we prove the existence of a locally compact attractor A for the semigroup St generated by equation (7.2.23).

7.2 Attractors and their spatial complexity | 261

According to Theorem 7.2.1, the semigroup St : Φb (Ω) → Φb (Ω) generated by equation (7.2.23) possesses a bounded absorbing set ℬ in the phase space Φb (Ω), that is, for any other bounded subset of B ⊂ Φb (Ω) there exists T = T(B) such that St B ⊂ ℬ,

t ≥ T.

Nevertheless, in contrast to the case of bounded domains, the compact attractor in Φb (Ω) usually does not exist in the case of unbounded domains; for example, the Chafee–Infante equation in ℝn (k = 1, f (u) = u3 − λu, λ > λ0 ) does not possess a compact attractor in the topology of Φb (Ω). The existence of the global attractor for problem (7.2.1) in the case Ω = Rn was first established in [1]. Consequently, we construct below the attractor A of semigroup (7.2.79), which attracts bounded subsets of Φb (Ω) 2,q only in a local topology of the space Φloc = Wloc (Ω); i. e., A is the (Φb , Φloc )-attractor of (7.2.79). Noting that the space Φloc (Ω) is a reflexive, metrizable F-space that is generated by seminorms ‖⋅, Ω ∩ B1x0 ‖2,q , x0 ∈ Ω. A set A ⊂ Φb (Ω) is defined to be the attractor of the semigroup St if the following assumptions hold: (1) The set A is compact in Φloc (Ω) and is bounded in Φb (Ω); (2) The set A is strictly invariant with respect to St ; St A = A ,

t ≥ 0.

(3) The set A is an attracting set for St in a local topology; for every neighborhood O(A ) of A in the topology of the space Φloc (Ω) and for every bounded subset B in the uniform topology of Φb (Ω), there exists T = T(O, B) such that St B ⊂ O(A ),

t ≥ T.

We recall that the first condition means that the restriction A |Ω1 is compact in the space W 2,q (Ω1 ) for every bounded Ω1 ⊂ Ω. Analogously, the third condition means that for every bounded Ω1 ⊂ Ω, every bounded B in Φb (Ω) and every W 2,q (Ω1 )-neighborhood O(A |Ω1 ) of the restriction A |Ω1 , there exists T = T(Ω1 , O, B) such that (St B)|Ω1 ⊂ O(A |Ω1 ),

t ≥ T.

Theorem 7.2.4. Let the above assumptions hold, then the semigroup St defined by (7.2.79) possesses an attractor A that has the structure A = 𝒦|t=0

(7.2.82)

where we denote by 𝒦 the set of all solutions of (7.2.23) that are defined and bounded for all t ∈ R, sup ‖u(t)‖Φb (Ω) < ∞. t∈R

262 | 7 Global attractors and time-spatial chaos Proof. According to the existence theorem of attractor for abstract semigroups, it is sufficient to verify the following conditions: (1) The semigroup St possesses a compact absorbing set K in Φloc -topology. (2) The operators St have closed graphs on K in the Φloc -topology for every fixed t ≥ 0. We need the following lemmas to verify the first condition. Lemma 7.2.7. Let the domain Ω satisfy assumptions (7.2.8) and (7.2.9). Then for every g ∈ Lqb (Ω), the problem aΔx v − λ0 v + g = 0,

v|𝜕Ω = 0,

(7.2.83)

possesses a unique solution v = v(g) ∈ Wb2,q (Ω), and the following estimate is valid: ‖v‖W 2,q (Ω) ≤ C‖g‖Lq (Ω) .

(7.2.84)

b

b

Maximal regularity (7.2.84) follows from estimate (7.2.28). The existence of a solution and its uniqueness can be verified as in Theorem 7.2.3. Lemma 7.2.8. Let u(t) be a solution of equation (7.2.23), v = v(g) be a solution constructed in Lemma 7.2.7, and W(t) = u(t)−v. Then there exists a positive μ > 0 depending only on the equation such that ‖W(1)‖W 2+μ,q (Ω) ≤ Q(‖u(0)‖Φb (Ω) ) + Q(‖g‖W q (Ω) ) b

b

(7.2.85)

for a certain monotonic function Q. Based on the smoothing property (7.2.71) and arguing as at the end of the proof of Theorem 7.2.1, we can derive the estimate ‖W(1)‖W 2+β,q (Ω) ≤ sup Q1 (‖u(s)‖Φb (Ω) ) s∈[0,1]

b

(7.2.86)

for a certain monotonic function Q1 and positive β. Inserting now estimate (7.2.28) into the right-hand side of (7.2.86), we obtain (7.2.85). Estimates (7.2.28) and (7.2.85) imply that the set 2+β,q

K = v(g) + B(0, R, Wb

2+β,q B(0, R, Wb )

= {W ∈

2+β,q Wb (Ω)

),

: ‖W‖W 2+β,q ≤ R} b

is an absorbing set for semigroup (7.2.79) generated by equation (7.2.23) if R is large enough. Note that the absorbing set K obtained is compact in Φloc (Ω). Thus, the first assumption of the abstract theorem on the attractor’s existence is verified. Let us verify the second one.

7.2 Attractors and their spatial complexity | 263

Lemma 7.2.9. Let ℬ be a bounded set in Φb (Ω) and ϕ be a positive weight function with the exponential growth rate such that ∫ℝn ϕ(x)dx < ∞. Then the topologies induced on 2,q

ℬ by the embeddings ℬ ⊂ Φloc (Ω) and ℬ ⊂ Φϕ (Ω) = Wϕ (Ω) coincide.

This lemma is more or less evident; a rigorous proof is omitted here. Let us fix ϕ(x) = e−ϵ|x| (ϵ > 0 is small enough). Then, due to Lemma 7.2.9, in order to prove that St has a closed graph on K in the Φloc -topology, it is sufficient to prove that the convergences u0 = Φϕ − lim un0 , n→∞

v = Φϕ − lim St un0 , n→∞

(7.2.87)

with un0 , u0 ∈ K imply that v = St u0 . But, according to estimate (7.2.78), the semigroup St is globally Lipschitz-continuous in the L2ϕ -topology, consequently, St u0 = L2ϕ − lim St un0 .

(7.2.88)

n→∞

Convergences (7.2.87) and (7.2.88) imply that v = St u0 . Thus, all assumptions of the abstract theorem on the attractor’s existence are verified. The theorem is proved. Next, we recall the definition of ϵ-entropy and give upper and lower estimates of it as ϵ → 0 for the typical sets in functional spaces. Definition 7.2.4. Let 𝕄 be a metric space and let K be a precompact subset of it. For a given ϵ > 0, let Nϵ (K) = Nϵ (K, 𝕄) be the minimal number of ϵ-balls in 𝕄 that cover the set K (this number is finite by the Hausdorff criteria). Kolmogorov’s ϵ-entropy of K in 𝕄 is the number ℍϵ (K) = ℍϵ (K, 𝕄) = ln Nϵ (K).

(7.2.89)

Example 7.2.1. Let K be a compact, n-dimensional Lipschitz manifold in 𝕄. Then obvious estimates imply that n

n

1 1 C1 ( ) ≤ Nϵ (K) ≤ C2 ( ) , ϵ ϵ

(7.2.90)

and 1 ℍϵ (K) = (n + o(1)) ln , ϵ

ϵ → 0+ .

(7.2.91)

Definition 7.2.5. The fractal (box-counting) dimension of the set K ⊂ 𝕄 is defined to be dimF (K) = dimF (K, 𝕄) = lim sup ϵ→0

ℍϵ (K) ln ϵ1

.

(7.2.92)

We note that the fractal dimension dimF (K) ∈ [0, ∞] is defined for any compact set in 𝕄, but it may not be an integer if K is not a manifold.

264 | 7 Global attractors and time-spatial chaos Example 7.2.2. Let 𝕄 = [0, 1], and let K be the ternary Cantor set in 𝕄, then, as is known, d

d

1 1 C1 ( ) ≤ Nϵ (K) ≤ C2 ( ) , ϵ ϵ

d=

ln 2 , ln 3

and dimF (K) = d =

ln 2 . ln 3

(7.2.93)

Example 7.2.3. Let K be a set of all analytic functions f in a ball B(R) of radius R > 1 in ℂn satisfying ‖f ‖L∞ (B(R)) ≤ 1, and let 𝕄 be the space C(B), where BRe = {z ∈ ℂn : Im zi = 0, |z| ≤ 1}. Thus, K consists of all functions from C(BRe ) that can be extended holomorphically to the ball B(R) ⊂ ℂn , and the C-norm of this extension is not greater than 1. Then n+1

1 C1 (ln ) ϵ

n+1

1 ≤ ℍϵ (K, 𝕄) ≤ C2 (ln ) ϵ

(7.2.94)

.

Example 7.2.4. Let 𝕄 be the same as in the previous example, and let K be the set of all functions f in 𝕄 that can be extended to the entire function f ̂ in ℂn and that satisfy the estimate |f ̂(z)| ≤ k1 ek2 |z| ,

z ∈ ℂn .

(7.2.95)

Then C1

(ln ϵ1 )n+1

(ln ln ϵ1 )n

≤ ℍϵ (K) ≤ C2

(ln ϵ1 )n+1

(ln ln ϵ1 )n

(7.2.96)

.

Example 7.2.5. Let Ω be a smooth, bounded domain in ℝn and W l1 ,p1 (Ω) ⊂ W l2 ,p2 (Ω),

0 ≤ li < ∞,

1 < pi < ∞,

l1 > l2 ,

that is, according to the embedding theorem, l1 l 1 1 − > 2 − . n p1 n p2 Let 𝕄 = W l2 ,p2 (Ω) and K be the unitary ball in W l1 ,p1 (Ω), then n

n

1 l1 −l2 1 l1 −l2 C1 ( ) ≤ ℍϵ (K) ≤ C2 ( ) . ϵ ϵ

(7.2.97)

The following class of functions will be used in order to obtain the lower bounds of the ϵ-entropy of attractors.

7.2 Attractors and their spatial complexity | 265

Definition 7.2.6. Let 𝔹σ (ℝn ) = 𝔹σ (ℝn , ℂ) denote the subspace of L∞ (ℝn , ℂ) that consists of all functions φ whose Fourier transforms (in the sense of distributions) satisfy the condition supp φ̂ ⊂ [−σ, σ]n .

(7.2.98)

It is well known that every function φ ∈ 𝔹σ can be extended to the entire function ̃ φ(z) ∈ A(ℂn ) that satisfies the estimate: n

̃ + iy)| ≤ C‖φ, ℝn ‖0,∞ eσ ∑i=1 |yi | . sup |φ(x

x∈ℝn

(7.2.99)

Moreover, every function φ ∈ L∞ that possesses the entire extension φ̃ satisfying (7.2.99) belongs to the space 𝔹σ . Example 7.2.6. Let k = B(0, 1, 𝔹σ ) be a unit ball in 𝔹σ and 𝕄 = Cb (BR0 ); then n

1 1 ℍϵ (B(0, 1, 𝔹σ ), Cb (BR0 )) ≤ C(R + k ln ) ln , ϵ ϵ

ϵ ≤ ϵ0 < 1.

(7.2.100)

Moreover, C, k are independent of R. We also formulate the lower bounds for the entropy. Proposition 7.2.5. The following estimate holds for R ≥ R0 , ϵ < ϵ0 : 1 ℍϵ (B(0, 1, 𝔹σ ), Cb (BR0 )) ≥ CRn ln , ϵ

(7.2.101)

where the constant C is independent of R, ϵ. Proof. For each k = (k1 , k2 , . . . , kn ) ∈ ℤn , we define a function n

φk (z) = ∏

4 sin2 (

i=1

σz σ2 ( 2 i

σzi 2

− πki )

− πki )2

.

Direct calculations reveal that φk ∈ 𝔹σ . Let LN be the subspace of 𝔹σ , spanned by φk (z), k ∈ [−N, N]n ≡ KN . LN consists of all functions u(z) = ∑ ak φk (z), K∈KN

ak ∈ ℝ.

(7.2.102)

The proof of this proposition is based on the following two lemmas. Lemma 7.2.10. The following estimate holds: sup |ak | ≤ ‖u, ℝn ‖0,∞ ≤ C sup |ak |,

k∈KN

where the constant C is independent of N.

k∈KN

u ∈ LN ,

(7.2.103)

266 | 7 Global attractors and time-spatial chaos Proof. The left inequality is an immediate corollary of the equality u( 2πk ) = ak , and σ the right one can be deduced easily from the assertion ∑

1

n k∈ℤn ∏i=1 (πki

−

σzi 2 ) 2

< ∞,

∀

σz ∉ ℤn , 2π

|φk (z)| ≤ 1.

Lemma 7.2.11. Let N ≥ N0 , R ≥ bN (b ≥ 1), then 1 ‖u, BR0 ‖0,∞ ≥ ‖u, ℝn ‖0,∞ , 2

∀u ∈ LN .

(7.2.104)

Proof. The estimate in the general case is similar to the case n = 1. Let u ∈ LN . Then, according to Lemma 7.2.10, ‖u, |z| > ℝ‖0,∞ ≤ sup {|ak |} ∑ k∈KN

k∈KN

≤ C‖u, ℝ1 ‖0,∞

( σR 2

1

− πN)2

2N (R − lN)2

1 ≤ ‖u, ℝ1 ‖0,∞ , 2 if R ≥ 2lN, l =

2π σ

and N is large enough.

The last thing to prove is Proposition 7.2.5. Fix R large enough and ϵ small enough, define N = [ Rb ] + 1, where b is the same as in Lemma 7.2.11, and consider the subspace LN . We divide the segment [− C1 , C1 ], where C is defined in Lemma 7.2.10, by points aj = 1 1 4ϵj j, j = [− 4Cϵ ], . . . , [ 4Cϵ ], and define a function φJ (z) = ∑ aJ(k) φk (z),

(7.2.105)

k∈KN

1 1 where J : [−N, N]n → [− 4cϵ ], . . . , [ 4cϵ ] an arbitrary integer map. It follows from Lemma 7.2.10 that φJ (z) ∈ B(0, 1, 𝔹σ ), and Lemma 7.2.11 implies that

‖φJ1 − φJ2 , BR0 ‖0,∞ ≥ 2ϵ,

J1 ≠ J2 .

Hence (2N+1)n

1 ] + 1) Nϵ (B(0, 1, 𝔹σ )) ≥ (2[ 4Cϵ

.

Consequently, ℍϵ (B(0, 1, 𝔹σ ), Cb (BR0 )) ≥ (2N + 1)n ln The proof of Proposition 7.2.5 is completed.

C 1 ≥ C1 Rn ln . 2ϵ ϵ

(7.2.106)

7.2 Attractors and their spatial complexity | 267

Corollary 7.2.6. Let A be the attractor, R ≥ ln ϵ1 , ϵ ≤ ϵ0 then C1 Rn ln

1 1 ≤ ℍϵ (A , Cb (BR0 )) ≤ C2 Rn ln . ϵ ϵ

(7.2.107)

For R ≪ ln ϵ1 , we formulate the following result. Proposition 7.2.6. For every δ > 0, there exists Cδ > 0 such that n+1−δ

1 ℍϵ (B(0, 1, 𝔹σ ), C(B10 )) ≥ Cδ (ln ) ϵ

.

(7.2.108)

Consequently, estimate (7.2.100) is also sharp for the case R ≪ ln ϵ1 . Remark 7.2.2. Together with the spaces 𝔹σ , we consider a slightly more general class 𝔹σ,ξ , ξ ∈ ℝk that consists of functions φ whose Fourier transforms φ̂ satisfy the assumption supp φ̂ ⊂ ξ + [−σ, σ]n .

(7.2.109)

We note that the space 𝔹σ,ξ is isomorphic to 𝔹σ , and this isomorphism is given

by multiplication on the function eiξx . Consequently, estimates (7.2.100) and (7.2.108) remain valid for the class 𝔹σ,ξ as well. We will also need the space of real parts of functions from 𝔹σ,ξ (ℝn , ℂ). Definition 7.2.7. We define the space 𝔹Re σ,ξ by the expression n ∞ n n 𝔹Re σ,ξ (ℝ , ℝ) = {φ ∈ L (ℝ ) : ∃u ∈ 𝔹σ,ξ (ℝ , ℂ), φ = Re u}.

(7.2.110)

Remark 7.2.3. Obviously, 𝔹Re σ,ξ ⊂ 𝔹σ,ξ + 𝔹σ,−ξ . Moreover, the analogues of estimates (7.2.107) and (7.2.108) are valid for this space as well. Next, we obtain the upper estimates for ϵ-entropy of the attractor A of equation. In other words, we obtain the upper estimates for ϵ-entropy of the restrictions A |Ω∩BRx 0 of the attractor A . Theorem 7.2.5. Let the assumptions of above be valid, and let VolΩ,x0 (R) = Vol(Ω ∩ BRx0 ).

(7.2.111)

Then for every R ∈ ℝ+ , x0 ∈ Ω, ϵ ≤ ϵ0 < 1, 1 1 ℍϵ (A |Ω∩BRx , Wb2,q (Ω ∩ BRx0 )) ≤ C VolΩ,x0 (R + k ln ) ln , 0 ϵ ϵ where the constants C, k, ϵ0 are independent of R and x0 ∈ R.

(7.2.112)

268 | 7 Global attractors and time-spatial chaos Proof. We give a sketch of this proof. We define a family of weight functions with the growth rate 1 by the formula eR−|x−x0 | , ΨR,x0 (x) { 1,

|x − x0 | > R, |x − x0 | ≤ R.

(7.2.113)

Then obviously, 2,q ℍϵ (A |Ω∩BRx , Wb2,q (Ω ∩ BRx0 )) ≤ ℍϵ (A , Wb,Ψ

R,x0

0

(7.2.114)

(Ω)).

Hence, instead of estimating the entropy of restrictions A |Ω∩BRx , it is sufficient to 0

2,q estimate the entropy of the attractor in the weighted Sobolev spaces Wb,Ψ

R,x0

(Ω).

Now let u1 (t) and u2 (t) be two solutions of equation (7.2.23) that belong to the attractor A . Then, according to estimates (7.2.81), ‖u1 (1) − u2 (1)‖W 2,q

q/2 R,x0

b,Ψ

(Ω)

≤ C‖u1 (0) − u2 (0)‖L2

b,ΨR,x 0

(7.2.115)

(Ω) .

Here, the constant C is independent of u1 , u2 ∈ A . Moreover, since ΨR,x0 (x + y) ≤ e|x| ΨR,x0 (y), then CΨR,x ≡ 1 and, consequently, C is also independent of R and x0 . By applying the 0

operator supz∈Ω ΨR,x0 (z)q/2 to both sides of (7.2.81) (in which x0 is replaced by z), we have ‖u1 (1) − u2 (1)‖W 2,q

q/2 R,x0

b,Ψ

(Ω)

≤ C(sup ΨR,x0 (z) ∫ e z∈Ω

−ϵ|x−z|

‖u1 (0) − u2 (0), Ω ∩

x∈Ω

B1x ‖20,2 dx)

q 2

.

(7.2.116)

Applying estimate (7.2.17) to the right-hand side of the previous formula, we derive (7.2.115). Estimate (7.2.115), together with description (7.2.82) of the attractor A , immediately implies that ℍϵ (A , W 2,q q/2 (Ω)) ≤ ℍ ϵ (A , L2b,ΨR,x (Ω)). b,ΨR,x

2C

0

(7.2.117)

0

Estimate (7.2.117) reduces our problem to estimate the entropy of the attractor in the space L2b,ΨR,x (Ω). 0

Let u1 and u2 be two arbitrary solutions of equation (7.2.23) that belong to the attractor. Then the following estimate is valid: ‖u1 (1) − u2 (1)‖W 1,2

b,ΨR,x

0

(Ω)

≤ C‖u1 (0) − u2 (0)‖L2

b,ΨR,x

0

(Ω) ,

(7.2.118)

where the constant C depends only on the equation. Equation (7.2.118) implies the following recurrent estimate:

7.2 Attractors and their spatial complexity | 269

Lemma 7.2.12. Let estimate (7.2.118) be valid, then ℍ ϵ (A , L2b,ΨR,x ) ≤ ℍϵ (A , L2b,ΨR,x ) + k ln Mk (ϵ), 2k

0

0

(7.2.119)

where ln Mk (ϵ) ≤ C VolΩ,x0 (R + L ln

2k ). ϵ

(7.2.120)

Moreover, the constants C and L are independent of k, R, ϵ ≤ ϵ0 and x0 . Estimate (7.2.112) is an immediate corollary of (7.2.119). Since A is bounded in Φb , there exists R0 > 0 such that ℍR0 (A , L2b,ΨR,x ) = 0 for every R and x0 . Estimate (7.2.119) now implies 0 that ℍ R0 (A , L2b,ΨR,x ) ≤ Ck VolΩ,x0 (R + L ln 0

2k

2k ). R0

(7.2.121)

Fixing k ∼ ln(R0 /ϵ) and using (7.2.114) and (7.2.117), we obtain (7.2.112) and finish the proof of Theorem 7.2.5. Corollary 7.2.7. Since Cb (Ω) ⊂ Wb2,q (Ω), 1 1 ℍϵ (A , C(Ω ∩ Bkx0 )) ≤ C VolΩ,x0 (R + k ln ) ln . ϵ ϵ

(7.2.122)

Corollary 7.2.8. Let Ω = ℝn , then VolΩ,x0 (r) = Cr n and, consequently, n

1 1 ̃ ℍϵ (A , Wb2,q (BRx0 )) ≤ C(R + k ln ) ln . ϵ ϵ

(7.2.123)

Taking R = ln(1/ϵ), we have ln( ϵ1 )

ℍϵ (A , Wb2,q (Bx0

n+1

1 )) ≤ C1 (ln ) ϵ

.

(7.2.124)

Note that estimate (7.2.123) gives the same type of upper bounds for R = 1, R = ln(1/ϵ). Corollary 7.2.9. Let Ω be a bounded domain, then Theorem 7.2.5 implies the estimate 1 ℍϵ (A , Wb2,q (Ω)) ≤ C Vol(Ω) ln , ϵ

(7.2.125)

which reflects the well-known fact that in this case the attractor A has the finite fractal dimension. Corollary 7.2.10. Let Ω = ℝk × ωn−k be a cylindrical domain (ω in a bounded domain). Then estimate (7.2.112) gives the following bound of the ϵ-entropy of the attractor A : k

1 1 ℍϵ (A , Wb2,q (Ω ∩ BRx0 )) ≤ C(R + k ln ) ln . ϵ ϵ

(7.2.126)

270 | 7 Global attractors and time-spatial chaos We now recall the concept of the topological entropy per unit volume. Definition 7.2.8. Let A ⊂ Φb (Ω) be a compact set in the space Φloc (Ω). Then the ϵ-entropy per unit volume is defined to be ℍ̄ ϵ (A ) = lim sup R→∞

ℍϵ (A , Wb2,q (Ω ∩ BRx0 )) VolΩ,0 (R)

.

(7.2.127)

Corollary 7.2.11. The following estimate is valid: 1 ℍ̄ ϵ (A ) ≤ C ln . ϵ

(7.2.128)

Estimate (7.2.128) is an immediate corollary of estimate (7.2.112) and the trivial assertion lim

R→∞

VolΩ,x0 (R + C1 ) VolΩ,x0 (R)

= 1.

Next, we use the technique of infinite-dimensional manifolds to develop the lower bounds for the entropy of the attractor A . We restrict ourselves to considering only the spatially homogeneous case Ω = ℝn , g = 0. In this case, the equation f (z) + λ0 z = 0 always has at least one solution z0 = (z01 , z02 , . . . , z0k ) ∈ ℝk (due to assumptions (7.2.24)) and, consequently, equation (7.2.23) has at least one spatially homogeneous equilibrium u(t) ≡ z0 . We obtain the lower bounds for the attractor’s entropy under the additional assumption that equation (7.2.23) possesses at least one exponentially unstable, spatially homogeneous equilibrium z0 ∈ ℝk . (without loss of generality, we assume below that z0 = 0). To be more precise, it is assumed that equation (7.2.23) has the form ut = aΔx u + Bu − f ̃(u),

(7.2.129)

where f ̃ ∈ C 2 (ℝk , ℝk ) such that f ̃(0) = f ̃ (0) = 0, the matrix B ∈ L (ℝk , ℝk ),(B = −f (z0 ) − λ0 ), and the spectrum σ(L ) in the space L2 (ℝn ) of the linearization L := aΔx + B satisfy the assumption σ(L ) ∩ {Re z > 0} ≠ 0.

(7.2.130)

The main aim of this section is to show that assumptions (7.2.129) and (7.2.130) are sufficient for obtaining the lower bounds of the entropy of the attractor of the same type as the upper ones obtained previously. We begin with studying the linear nonhomogeneous problem vt − L v = h(t), which corresponds to the linearization of (7.2.129) at u = 0.

(7.2.131)

7.2 Attractors and their spatial complexity | 271

Definition 7.2.9. Let γ ∈ ℝ be arbitrary, then the space 𝕃γ (E), where E is a certain Banach subspace of distributions D (ℝn ), is defined by the expression −γt 𝕃γ (E) = {u ∈ L∞ ‖u(t)‖E < ∞}. loc (ℝ− , E) : ‖u‖𝕃γ (E) = sup e

(7.2.132)

t≤0

Lemma 7.2.13. Let γ > sup Re σ(L ), then for every h ∈ 𝕃γ (Lqb (ℝn )), equation (7.2.131) 2−μ,q

possesses a backward solution u(t), t ≤ 0, which is unique in the class u ∈ 𝕃γ (Wb Thus, a linear operator 2−μ,q

𝕋γ : 𝕃γ (Lqb ) → 𝕃γ (Wb

),

u(t) = (𝕋γ h)(t),

(ℝn )).

(7.2.133)

is well-defined for every μ > 0. Moreover, there exists a positive exponent ϵ > 0, such that ‖(𝕋γ h)(t), B1x0 ‖2−μ,q ≤ Cμ sup e(γ−ϵ)(t−s) (sup e−ϵ|x−x0 | ‖h(s), B1x ‖0,q ), x∈ℝn

s∈(−∞,t]

(7.2.134)

where the constant Cμ is independent of x0 and t. Proof. According to the smoothing property for solutions of the linear equation (7.2.131), it is sufficient to deduce estimate (7.2.134) only for the W 1,2 -norm on the lefthand side (instead of the W 2−μ,q -norm). We also note that, without loss of generality, we may assume that γ = 0. Let us first consider the case where h ∈ 𝕃0 (L2 (ℝn )), the general case will be reduced to it. It is well known that the operator L generates an analytic semigroup in L2 (ℝn ), and, consequently, due to the spectral mapping theorem, σ(eL )\{0} = eσ(L ) . Moreover, according to our assumptions, sup Re σ(L ) < 0(γ = 0). Therefore, there exists a positive ν > 0 such that sup Re σ(L ) < −2ν. Thus, the spectral radius of the exponent eL satisfies the inequality r(eL ) ≤ e−2ν < 1.

(7.2.135)

Consequently, the Duhamel formula t

v(t) = ∫ eL (t−s) h(s)ds,

(7.2.136)

−∞

defines a solution v ∈ 𝕃0 (L2 (ℝn )) that satisfies the estimate ‖v(t)‖L2 (ℝn ) ≤ C sup e−ν(t−s) ‖h(s)‖L2 (ℝn ) , s∈(−∞,t]

Moreover, this solution is unique in the class 𝕃0 (L2 ).

t ≤ 0.

(7.2.137)

272 | 7 Global attractors and time-spatial chaos Estimate (7.2.137), together with a standard (L2 , W 1,2 )-smoothing property for the solutions of (7.2.131), yield ‖v(t)‖1,2 ≤ C1 sup e−ν(t−s) ‖h(s)‖0,2 . s∈(−∞,t]

(7.2.138)

It is convenient for us to write the last estimate in the equivalent form sup e−ν(t−s) ‖v(s)‖1,2 ≤ C2 sup e−ν(t−s) ‖h(s)‖0,2 .

s∈(−∞,t]

s∈(−∞,t]

(7.2.139)

In order to reduce the general case h ∈ 𝕃0 (L2b ) to the one considered above, we fix an arbitrary x0 ∈ ℝn and introduce a new unknown function Wx0 (t) = v(t)ϕ̃ ϵ,x0 , where 1

2 2 ϕ̃ ϵ,x0 (x) = e−ϵ(1+|x−x0 | ) , and ϵ > 0 is a small parameter that will be specified below. We note that weight functions ϕ̃ ϵ,x0 are equivalent to ϕϵ,x0 but are smooth and satisfy the conditions

|D2 ϕ̃ ϵ,x0 | ≤ Cϵϕ̃ ϵ,x0 .

|∇x ϕ̃ ϵ,x0 | ≤ Cϵϕ̃ ϵ,x0 ,

(7.2.140)

It is not difficult to verify that the function Wx0 satisfies the equation 𝜕t Wx0 − L Wx0 = ϕ̃ ϵ,x0 h + K1 (x)Wx0 + K2 (x)∇x Wx0 = hx0 (t);

(7.2.141)

estimates (7.2.140) imply that |Ki (x)| ≤ C2 ϵ, i = 1, 2. Obviously, hx0 ∈ 𝕃0 (L2 ), consequently, estimate (7.2.139) yields sup e−ν(t−s) ‖Wx0 (s)‖1,2 ≤ sup e−ν(t−s) ‖hx0 (s)‖0,2

s∈(−∞,t]

s∈(−∞,t]

≤ C3 sup e−ν(t−s) ‖ϕϵ,x0 h(s)‖0,2 s∈(−∞,t]

+ C3 ϵ sup e−ν(t−s) ‖Wx0 (s)‖1,2 . s∈(−∞,t]

(7.2.142)

Fixing ϵ > 0 small enough in (7.2.142), we derive that ‖v(t), B1x0 ‖ ≤ C sup e−ν(t−s) ‖ϕϵ,x0 v(s)‖1,2 s∈(−∞,t]

≤ C1 sup e−ν(t−s) sup {ϕ ϵ ,x0 (x)‖h(s), B1x ‖0,2 }. x∈ℝn

s∈(−∞,t]

2

(7.2.143)

Estimate (7.2.134) is proved. Applying the operator supt∈ℝ− e−γt supx0 ∈ℝn to both sides of inequality (7.2.134), we derive that ‖v‖𝕃

2−μ,q ) γ (Wb

Lemma 7.2.13 is proved.

≤ C5 ‖h‖𝕃γ (Lq ) . b

(7.2.144)

7.2 Attractors and their spatial complexity | 273

Corollary 7.2.12. Let the assumptions of Lemma 7.2.13 hold, and let ϕ be a weight function that satisfies (7.2.1) with a sufficiently small growth rate. Then the operator 𝕋γ con2−μ,q

structed in Lemma 7.2.13 is bounded as the operator from Lγ (Lqb,ϕ ) to Lγ (Wb,ϕ ).

This assertion is an immediate corollary of (7.2.134) and (7.2.4). Let us now study the homogeneous problem (7.2.131) (h ≡ 0). Lemma 7.2.14. Let the spectrum of L satisfy assumption (7.2.130). Then there exist γ > 0, σ > 0, ξ0 ∈ ℝk , e ∈ ℝk , and the operator 𝒫γ : 𝔹σ,ξ0 → 𝕃γ (Wb2,q (ℝn , ℂk )), where the space 𝔹σ,ξ0 = 𝔹σ,ξ0 (ℝn , ℂ) is defined by (7.2.109) such that the following are true: (1) For every u0 ∈ 𝔹σ,ξ0 (ℝn ), the function v ∈ 𝕃γ (Wb2,q (ℝn )) defined by v(t) = 𝒫γ (u0 )(t), t ≤ 0, is a solution of (7.2.131) with h ≡ 0. (2) 2γ > sup Re σ(L ). (3) Let 𝒮γ (u0 ) = 𝒫γ (u0 )(0) and Πe z = (z ⋅ e)/|e|2 be the orthogonal projection onto the vector e. Then Πe 𝒮γ (u0 ) = u0 , ∀u0 ∈ 𝔹σ,ξ0 . (4) ∀N ∈ ℝ+ , u ∈ 𝔹σ,ξ0 , the following estimate holds: ‖𝒫γ (u0 )(t), B1x0 ‖2,q ≤ CN eγt sup

x∈ℝn

‖u0 , B1x ‖0,∞

1

(1 + |x − x0 |2N ) 2

(7.2.145)

where the constant CN is independent of x0 ∈ ℝn . Proof. Applying the x-Fourier transform to homogeneous equation (7.2.131), we have ̂ − L ̂(ξ )U(t) ̂ = 0, 𝜕t U(t)

2

L ̂(ξ ) = a|ξ | + B.

(7.2.146)

Moreover, assumption (7.2.130) implies that there exist two points ξ0 ∈ ℝk and λ̂0 ∈ σ(L ̂(ξ0 )) such that Re λ̂0 > 0. Without loss of generality, we may also assume that Re σ(L ̂(ξ )) < λ̂0 + ϵ, ∀ξ ∈ ℝk where ϵ > 0 is small enough to satisfy ϵ < Re λ̂0 /3. ̂ ) the spectrum of σ(L ̂(ξ )). Then (since the matrix-valued funcLet us denote by λ(ξ ̂ ) is an analytic function with respect ̂ tion L (ξ ) is a polynomial with respect to ξ ) λ(ξ to ξ on the corresponding (2k)-sheeted Riemann surface. Moreover, without loss of generality, we may assume that ξ0 ≠ 0, and it is not a branch point of this function. ̂ ) in the neighborhood of ξ such that We denote by λ̂0 (ξ ) the analytic branch of λ(ξ 0 ̂λ (ξ ) = λ̂ . 0 0 r Thus, we have proved that there exists a neighborhood Bξ0 of ξ0 and smooth func0 r r tion λ̂ : B 0 → ℂ, and e : B 0 → ℂk such that 0

ξ0

0

ξ0

L ̂(ξ )e0 (ξ ) = λ̂0 (ξ )e0 (ξ ),

e0 (ξ ) ≠ 0.

r Obviously, we may fix r0 > 0 in such way that Re λ̂0 (ξ ) > Re λ0 − ϵ for every ξ ∈ Bξ0 , 0 and r0 < |ξ0 |. Moreover, since e0 (ξ0 ) ≠ 0, then either Re e0 (ξ0 ) ≠ 0 or Im e0 (ξ0 ) ≠ 0.

274 | 7 Global attractors and time-spatial chaos We set e = Re e0 (ξ0 ) if Re e0 (ξ0 ) ≠ 0 and e = Im e 0 (ξ0 ) otherwise. Then it is possible to normalize the eigenvector e0 (ξ0 ) in such way that Πe e0 (ξ ) ≡ 1,

r

∀ξ ∈ Bξ0 ,

(7.2.147)

0

where we decrease the radius r0 if necessary. Let us now fix the exponent σ > 0 and the corresponding space 𝔹σ,ξ0 in such a r0

way that supp ϕ̂ ⊂ Bξ2 , for every ϕ ∈ 𝔹σ,ξ0 and define the solution of (7.2.131) via 0

̂ )e (ξ ). ̂ ξ ) = eλ̂0 (ξ )t ϕ(ξ U(t, 0

(7.2.148)

We claim that the operator 𝒫γ : ϕ → U, where γ = Re λ̂0 − ϵ, defined by (7.2.148), satisfies all the assumptions of the lemma. r0

Let us define a cutoff function ψ ∈ C0∞ (ℝn ) such that ψ(ξ ) ≡ 1, ξ ∈ Bξ2 , ψ(ξ ) = 0, r

ξ ∉ Bξ0 . Then formula (7.2.148) can be rewritten in the equivalent form

0

0

̂ ), ̂ ξ ) = eγt Ψ(t, ξ )ϕ(ξ U(t,

x ∈ ℝn ,

(7.2.149)

where Ψ(t, ξ ) = e(λ0 (ξ )−λ0 +ϵ)t ψ(ξ )e0 (ξ ). Moreover, it is not difficult to verify that, due to our construction of functions ψ, λ̂0 , e0 , ̂

̂

∫ |DN Ψ(t, ξ )|2 dξ ≤ CN , ℝn

t ∈ ℝ− .

(7.2.150)

Thus, the operator 𝒫γ can be represented as a convolution operator γt

𝒫γ (u0 )(t) = e (Fξ Ψ(t, ξ )) ∗ u0 , −1

u0 ∈ 𝔹σ,ξ0 .

(7.2.151)

Moreover, it follows that the convolution’s kernel K(t, x) in (7.2.151) satisfies the estimate 1 , ∀N ∈ ℝ+ , (7.2.152) |K(t, x)| = |Fξ−1 Ψ(t, ξ )| ≤ CN 1 (1 + |x|2N ) 2 and, consequently, |U(t, x0 )| ≤ C̃ N eγt sup

x∈ℝn

‖u0 , B1x ‖0,∞

1

(1 + |x − x0 |2N ) 2

.

(7.2.153)

Estimate (7.2.145) is an immediate corollary of (7.2.153) and the smoothing property for linear equation (7.2.131). This estimate implies, in particular, that the operator 𝒫γ is indeed a bounded operator from Bσ,ξ0 to 𝕃γ (Wb2,q ). The remaining properties of 𝒫γ , noted in Lemma 7.2.14 are evident. The fact that for every u0 ∈ 𝔹σ,ξ0 , U = 𝒫γ u0 is a solution of (7.2.131) follows from representation (7.2.148). The second assertion is a corollary of our choice of the exponent ϵ, 2γ = 2(λ̂0 − ϵ) > λ̂0 + ϵ > sup Re σ(L ), since ϵ < proved.

λ̂0 . 3

The third one is a corollary of normalization (7.2.147). Lemma 7.2.14 is

7.2 Attractors and their spatial complexity | 275

Corollary 7.2.13. Let the assumptions of Lemma 7.2.14 hold; then, for every weight function ϕ with a polynomial growth rate (7.2.22), the following estimate is valid: ‖𝒫γ (u0 )(t)‖W 2,q (ℝn ) ≤ Ceγt ‖u0 ‖L∞

b,ϕ1/q

b,ϕ

(ℝn ) ,

u0 ∈ 𝔹σ,ξ0

(7.2.154)

where the constant C is independent of the choice of the weight ϕ. Thus, we have constructed the complex-valued solution 𝒫γ (u0 ) of equation (7.2.131), but we need in the following only the real-valued solutions of this equation. Since the operator L has real coefficients, then Re 𝒫γ (u0 ) is the appropriate real-valued solution. Moreover, the assertions of Lemma 7.2.14 remain valid for this operator, with the exception of (3), which should be replaced by Πe 𝒮γ (u0 ) = Re u0 ,

(7.2.155)

∀u0 ∈ 𝔹σ,ξ0 .

However, that Re u0 , u0 ∈ 𝔹σ,ξ0 if and only if u0 = 0 (due to the fact that, by definition, r supp û 0 ⊂ 𝔹ξ0 , r0 < |ξ0 |). 0

Proposition 7.2.7. Let √nσ < |ξ0 |, then every entire function u0 ∈ 𝔹σ,ξ0 (ℝ, ℂ) is uniquely determined by its real part. Moreover, for every N ∈ ℝ+ , the following estimate is valid: |u0 (x0 )| ≤ CN sup

x∈ℝn

‖ Re u0 , B1x ‖0,∞

1

(1 + |x − x0 |2N ) 2

,

(7.2.156)

where the constant CN is independent of x0 ∈ ℝn and, consequently, the spaces 𝔹Re σ,ξ0 and 𝔹σ,ξ0 are isomorphic. We denote this isomorphism by ℛ. Proof. Since ū 0 ∈ 𝔹σ,−ξ0 and √nσ < |ξ0 |, then supp û 0 ∩ supp û̄ 0 = ⊬. Let ψ(ξ ) ∈ C0∞ (ℝn ) be a cutoff function such that ψ(ξ ) ≡ 1, ξ ∈ ξ0 + [−σ, σ]n , ψ(ξ ) = 0, ξ ∈ −ξ0 + [−σ, σ]n . Let K(x) = Fξ−1→x ψ. Then u0 = 2K ∗ Re u0 ,

(7.2.157)

1

and |K(x)| ≤ CN (1+|x|2N )− 2 . Estimate (7.2.156) is an immediate corollary of (7.2.157). We write below 𝒫γ instead of Re 𝒫γ and 𝒮γ instead of Re 𝒮γ . Corollary 7.2.14. Let ϕ be a weight function with the polynomial growth rate (7.2.22), and let the assumptions of Lemma 7.2.14 hold. Then the following estimate is valid: ‖u0 ‖L∞ ≤ C‖𝒮γ u0 ‖L∞ , b,ϕ

b,ϕ

u0 ∈ 𝔹σ,ξ0 ,

𝒮γ u0 = (Re 𝒫γ u0 )(0).

This corollary follows from (7.2.155), (7.2.156) and Corollary 7.2.4.

(7.2.158)

276 | 7 Global attractors and time-spatial chaos Theorem 7.2.6. Let the assumptions of Theorem 7.2.4 be valid, and let equation (7.2.23) be represented in the form (7.2.129) with the exponentially unstable linear part (7.2.130) assumed to be satisfied. Then there exists r > 0 and a C 1 -map n

U0 : B(0, r, 𝔹σ,ξ0 (ℝ , ℂ)) → A ,

(7.2.159)

where B(0, r, 𝔹σ,ξ0 ) is a r-ball in the space 𝔹σ,ξ0 centered at 0, the constants σ and ξ0 are the same as in Lemma 7.2.14, and for every u0 ∈ B(0, r, 𝔹σ,ξ0 ), the following estimate is valid: ‖U0 (u0 ) − 𝒮γ (u0 )‖Φb (ℝn ) ≤ C‖u0 ‖2L∞ (ℝn ) .

(7.2.160)

b

Moreover, this map is Lipschitz-continuous in the sense that for every N ∈ ℝ+ and every x0 ∈ ℝn , we have the estimates ‖u1 − u2 , B1x ‖0,∞ { 1 { ‖ U (u ) − U (u ), B ‖ ≤ C sup , { 0 1 0 2 N 1 x0 2,q { { x∈Ω (1 + |x − x |2N ) 2 { 0 { { ‖U0 (u1 ) − U0 (u2 ), B1x ‖2,q { 1 { { ‖u − u , B ‖ ≤ C sup , { 1 2 x0 0,∞ N 1 x∈Ω (1 + |x − x0 |2N ) 2 {

(7.2.161)

which are valid for every u1 , u2 ∈ B(0, r, 𝔹σ,ξ0 ). Proof. The proof of this theorem is based on the implicit function theorem and on the following lemma. Lemma 7.2.15. Let f ∈ C 2 satisfy f (0) = f (0) = 0 and the exponent μ > 0 be fixed in such way that the embedding W 2−μ,q ⊂ C holds. Then the Nemitsky operator Fu = f (u) 2−μ,q ), 𝕃2γ (Lqb )). belongs to the space C 1 (𝕃γ (Wb This lemma can be directly verified. We are now going to find the backward solutions of problem (7.2.129) near the equilibrium using the implicit function theorem. To this end, we rewrite this equation in the form ut − L u = −f ̃(u),

t ≤ 0.

Fix γ as in Lemma 7.2.14 and μ as in Lemma 7.2.15, and consider the equation u + 𝕋2γ f ̃(u) = 𝒫γ u0 ,

2−μ,q

u ∈ 𝕃γ (Wb

(7.2.162)

),

where u0 ∈ 𝔹σ,ξ0 and σ satisfies the conditions of Lemma 7.2.14. We note that every solution of (7.2.162) is simultaneously a solution of equation (7.2.129). Hence, it is suf2−μ,q ficient to prove u ∈ 𝕃γ (Wb ). 2−μ,q

We introduce the function F : 𝕃γ (Wb

2−μ,q

) × 𝔹σ,ξ0 → 𝕃γ (Wb

F (u, u0 ) = u + 𝕋2γ f ̃(u) − 𝒫γ u0 .

) via

7.2 Attractors and their spatial complexity | 277

2−μ,q

It follows from Lemmata 7.2.13, 7.2.14 and 7.2.15 that the function F ⊂ C 1 (𝕃γ (Wb 2−μ,q 𝔹σ,ξ0 , 𝕃γ (Wb ))

)×

and Du F (0, 0) = Id. Hence, due to the implicit function theorem, there exists a neighborhood B(0, r, 𝔹σ,ξ0 ) and a C 1 -function 2−μ,q

U : B(0, r, 𝔹σ,ξ0 ) → 𝕃γ (Wb

),

such that F (U (u0 ), u0 ) ≡ 0 and, consequently, U (u0 )(t) is a backward solution of problem (7.2.129). Equation (7.2.162) and Lemmata 7.2.13, 7.2.14 and 7.2.15 now imply that ‖U (u0 ) − 𝒫γ (u0 )‖𝕃

2−μ,q ) 2γ (Wb

≤ C f ̃(U (u0 ))𝕃

q 2γ (Lb )

≤ C1 ‖U (u0 )‖2𝕃

≤ C2 ‖u0 ‖2𝔹σ,ξ .

2−μ,q ) γ (Wb

(7.2.163)

0

Recall that the function u(t) := U (u0 )(t) satisfies equation (7.2.129). Consequently, due to the smoothing property for nonlinear equation (7.2.129) (see Proposition 7.2.3 and the end of the proof of Theorem 7.2.1) and the fact that f ̃(0) = 0, ‖u(t + 1)‖Φb ≤ Q(‖u(t)‖W 2−μ,q )‖u(t)‖W 2−μ,q . b

b

Therefore, ‖U (u0 )‖𝕃γ (Φb ) ≤ Q(‖U (u0 )‖𝕃 ≤ C‖u0 ‖𝔹σ,ξ , 0

2−μ,q (Ω)) 0 (Wb

)‖U (u0 )‖𝕃

∀u0 ∈ B(0, r, 𝔹σ,ξ0 ).

2−μ,q (Ω)) γ (Wb

(7.2.164)

Analogously, the function ω(t) = U (u0 )(t) − 𝒫γ (u0 ) satisfies the equation 𝜕t ω(t) − aΔx ω(t) − Bω(t) = −f ̃(u(t)). Applying the smoothing property to this equation and using (7.2.164) and the fact that f ̃(0) = f ̃ (0) = 0 we deduce from (7.2.163) that ‖U (u0 ) − 𝒫2γ u0 ‖𝕃γ (Φb ) ≤ C‖U (u0 ) − 𝒫γ (u0 )‖𝕃 ≤

C1 ‖u0 ‖2𝔹σ,ξ . 0

2−μ,q ) 2γ (Wb

+ C‖U (u0 )‖2𝕃

2−μ,q ) γ (Wb

(7.2.165)

We now set U0 (u0 ) = U (u0 )|t=0 . Then (7.2.165) together with the definition of 𝒮γ imply estimate (7.2.160). The assertion U0 (B(0, μ0 , 𝔹σ,ξ0 )) ⊂ A follows immediately from description (7.2.82) of the attractor A and from the fact that the solution u(t) = U (u0 )(t), t ≤ 0 of problem (7.2.129) can be extended, due to Theorems 7.2.1 and 7.2.2, to a complete solution u(t), t ∈ R, u(0) = U0 (u0 )(0). Thus, it remains to verify estimates (7.2.161). Let u10 , u20 ∈ B(0, r, 𝔹σ,ξ0 ) and ui (t) = U (ui0 )(t) be the corresponding backward

278 | 7 Global attractors and time-spatial chaos solutions of (7.2.129), U0 = u10 − u20 and U(t) = u1 (t) − u2 (t). Then the last function satisfies U + 𝕋2γ (f ̃(u1 ) − f ̃(u2 )) − 𝒫γ U0 = 0.

(7.2.166)

Let us x0 ∈ ℝn , ϵ > 0, and the corresponding weight function ϕϵ,x0 (x) = e−ϵ|x−x0 | . Then 2−μ,q

the operator 𝕋2γ : 𝕃2γ (Lqϕ

ϵ,x0

) → 𝕃2γ (Wϕ

ϵ,x0

), due to Lemma 7.2.13, Corollary 7.2.12 and

the fact that 2γ > sup Re σ(L ) and 𝕃2γ (V) ⊂ 𝕃γ (V), γ > 0. Consequently, equation (7.2.166) implies that ≤ C1 ‖f ̃(u1 ) − f ̃(u2 )‖𝕃2γ (Lq

‖U − 𝒫γ U0 ‖𝕃

2−μ,q γ (Wb,ϕϵ ,x ) 0

b,ϕϵ ,x0

),

(7.2.167)

where C1 is independent of x0 . We recall that f ̃ ∈ C 2 and f ̃(0) = f ̃ (0) = 0, therefore, |f ̃(u1 ) − f ̃(u2 )| ≤ Q(|u1 | + |u2 |)(|u1 | + |u2 |)|u1 − u2 |,

(7.2.168)

for some monotonic function Q. Estimates (7.2.168) and (7.2.164) imply that ̃ 1 2 f (u ) − f ̃(u )𝕃

q 2γ (Lb,ϕϵ ,x

̂ 1‖ ≤ Q(‖u 𝕃 1

)

0

+ ‖u2 ‖𝕃

2−μ,q ) 0 (Wb

× (‖u ‖𝕃

2−μ,q ) γ (Wb

2−μ,q ) 0 (Wb

2

)

+ ‖u ‖𝕃

̂ ≤ Q(2Cr)2Cr‖U‖ 𝕃

2−μ,q ) γ (Wb

2−μ,q γ (Wb,ϕϵ ,x

0

)

1

)‖U‖𝕃 2

2−μ,q γ (Wb,ϕϵ ,x

0

)

∀u , u ∈ B(0, r, 𝔹σ,ξ0 ).

,

(7.2.169)

Decreasing r if necessary, we may assume that δ ̃ 1 2 ‖U‖𝕃 (W 2−μ,q ) . f (u ) − f ̃(u )𝕃2γ (Lq ) ≤ γ b,ϕϵ ,x0 b,ϕϵ ,x0 2C1

(7.2.170)

Estimates (7.2.167) and (7.2.170) yield ‖U − 𝒫γ U0 ‖𝕃

2−μ,q γ (Wb,ϕϵ ,x

0

)

1 ≤ ‖U‖𝕃 (W 2−μ,q ) . γ b,ϕϵ ,x0 2

(7.2.171)

Consequently, 1 3 ‖U‖𝕃 (W 2−μ,q ) ≤ ‖𝒫γ U0 ‖𝕃 (W 2−μ,q ) ≤ ‖U‖𝕃 (W 2−μ,q ) . γ γ γ b,ϕϵ ,x0 b,ϕϵ ,x0 b,ϕϵ ,x0 2 2

(7.2.172)

We now note that, according to (7.2.154) and (7.2.158), we may estimate 𝒫γ U0 in terms of U0 in spaces with polynomial weight functions only, but estimate (7.2.172) contains exponential weights ϕϵ,x0 . In order to overcome this difficulty, we multiply (7.2.172) by N

QN,y0 (x0 ) = (1 + |x0 − y0 |2 )− 2 , where N ∈ ℕ and y0 ∈ ℝn are arbitrary and take a sup x0 ∈ ℝn . Then, due to (7.2.4), we have CN−1 ‖U‖𝕃

2−μ,q ) γ (Wb,Q N,y0

≤ ‖𝒫γ U0 ‖𝕃

2−μ,q ) γ (Wb,Q N,y0

≤ CN ‖U‖𝕃

2−μ,q ) γ (Wb,Q N,y0

,

(7.2.173)

7.2 Attractors and their spatial complexity | 279

where the constant CN depends on N but is independent of y0 ∈ ℝn . Moreover, since U(t) is a solution of (7.2.77), then due to smoothing property (7.2.81), we have ‖U(0)‖Φb,Q

N,y0

≤ CN ‖U‖𝕃

2−μ,q ) γ (Wb,Q N,y0

(7.2.174)

,

where CN is also independent of y0 . Applying estimate (7.2.154) and (7.2.158) to inequality (7.2.173) and using (7.2.174), we finally derive that C1 ‖U(0)‖Φb,Q

N,y0

≤ ‖U0 ‖L∞

QN/q,y

0

≤ C2 ‖U(0)‖Φb,Q

N,y0

,

(7.2.175)

where the constants C1 , C2 depend on N but are independent of y0 . Since y0 ∈ ℝn is arbitrary, then estimates (7.2.161) are immediate corollaries of (7.2.175), and Theorem 7.2.6 is proved. Corollary 7.2.15. Let the assumptions of Theorem 7.2.6 be valid, and let ϕ be a weight function with polynomial growth rate (7.2.22). Then the map U0 realizes the Lipschitzcontinuous homeomorphism between B(0, r, 𝔹σ,ξ0 ) and its image U0 (B(0, r, 𝔹σ,ξ0 )) in the sense that C1 ‖u10 − u20 ‖L∞ ≤ ‖U0 (u0 )1 − U0 (u20 )‖Φb,ϕq ≤ C2 ‖u1 − u2 ‖L∞ . b,ϕ

b,ϕ

(7.2.176)

Estimate (7.2.176) is an immediate corollary of (7.2.161) and Corollary 7.2.4. Corollary 7.2.16. Let {Th : h ∈ ℝn } be a group of spatial translations (Th u0 )(x) = u0 (x + h), and let 𝕂 := B(0, r, 𝔹σ,ξ0 (ℝk , ℂ), C) where r and ξ0 are the same as in Theorem 7.2.6. Then, obviously, Th A = A and Th 𝕂 = 𝕂. Moreover, the map U0 : 𝕂 → A commutes with this group, Th U0 (u0 ) = U0 (Th (u0 )), ∀h ∈ ℝn .

(7.2.177)

Assertion (7.2.177) is an immediate corollary of our construction of the map U0 and of the uniqueness part of the implicit function theorem. Corollary 7.2.17. Let u10 , u20 ∈ B(0, μ, 𝔹σ,ξ0 ) and μ ≤ r, where r, σ and ξ0 are the same as in Theorem 7.2.6. Then for every R > R0 , 1 2 1 2 2 U0 (u0 ) − U0 (u0 )W 2,p (BR ) ≥ L Re(u0 − u0 )L∞ (BR ) − Cμ , 0 0 b where C and L are independent of R. In fact, when we use the fact that ∏e 𝒮r u0 = Re u0 , we obtain 1 2 U0 (u0 ) − U0 (u0 )Φb (BR ) 0

≥ ‖𝒮γ u10 − 𝒮γ u20 ‖Φb (BR ) − U0 (u10 ) − Sγ u10 Φ (ℝn ) b 0

(7.2.178)

280 | 7 Global attractors and time-spatial chaos + U0 (u20 ) − Sγ u20 Φ (ℝn ) b

≥ L‖𝒮γ u10 − 𝒮γ u20 ‖L∞ (BR ) − C1 (‖u10 ‖2𝔹σ,ξ + ‖u20 ‖2𝔹σ,ξ ) 0

0

≥ L Re(u10 − u20 )L∞ (BR ) − 2C1 μ2 .

0

(7.2.179)

0

We are now ready to obtain the lower bounds for the ϵ-entropy of the attractor A . Theorem 7.2.7. Let the assumptions of Theorem 7.2.6 hold. Then the attractor A of problem (7.2.129) possesses the entropy estimates C2 Rn ln

n

1 1 1 ≤ ℍϵ (A , Wb2,q (BR0 )) ≤ C1 (R + K ln ) ln , ϵ ϵ ϵ

ϵ ≤ ϵ0 < 1.

(7.2.180)

Moreover, for every δ > 0, there exists Cδ > 0 such that n+1−δ

n+1

1 ≤ ℍϵ (A , Wb2,q (B10 )) ≤ C(ln ) ϵ

1 Cδ (ln ) ϵ

.

(7.2.181)

1

ϵ 2 ) ≤ r, and functions u10 , u20 ∈ B(0, μ, 𝔹Re Proof. Let ϵ > 0 be small enough, μ = ( 2CL σ,ξ0 ) such that

‖u10 − u20 ‖L∞ (BR ) ≥ 0

ϵ . L

(7.2.182)

Then it follows from (7.2.178) that ϵ 1 2 U0 (ℛu0 ) − U0 (ℛu0 )W 2,q (BR ) ≥ , 0 b 2

(7.2.183)

where ℛ is the isomorphism constructed in Proposition 7.2.7. Estimates (7.2.182) and (7.2.183), together with the fact that U0 (ℛui0 ) ∈ A , imply that 1

ℍ ϵ (A , Wb2,q (BR0 )) ≥ℍ ϵ (B(0, ( 4

L

=ℍ

1

( 2Cϵ )2 L

ϵ 2 Re ) , 𝔹σ,ξ0 ), Cb (BR0 )) 2CL

R (B(0, 1, 𝔹Re σ,ξ0 ), Cb (B0 ))

(7.2.184)

Estimates (7.2.180) and (7.2.181) are immediate corollaries of (7.2.184), (7.2.101) and (7.2.102) and Theorem 7.2.5. Theorem 7.2.7 is proved. Corollary 7.2.18. Let the assumptions of Theorem 7.2.7 hold, then 0 < C1 ln

1 1 ≤ ℍ̄ ϵ (A ) ≤ C2 ln . ϵ ϵ

(7.2.185)

We still need to study the spatial complexity of the attractor and spatial chaos. We continue to study the attractor of a spatially homogeneous system (7.2.129) in Ω = ℝn under the assumptions of Theorem 7.2.6. We recall that, in this case, the group {Th : h ∈ ℝn } of spatial shifts acts on the attractor of (7.2.129) Th A = A ,

(Th u)(x) = u(x + h),

h ∈ ℝn ,

(7.2.186)

7.2 Attractors and their spatial complexity | 281

group (7.2.186) is treated as a dynamical system with multidimensional “time” h ∈ ℝn . As a simple corollary of the estimates obtained previously (Theorem 7.2.6), we verify that the topological entropy hsp (A ) of semigroup is infinite, and define new quantitative characteristics ĥ sp (A ) of the complexity of dynamics that are finite and positive. We recall that the usual way to indicate the chaotic behavior of a dynamical system Th : A → A is to find a closed invariant subset M ⊂ A in the corresponding phase space and construct a homeomorphism τ : M → M such that τ : (Th |M , M) → (T̂ h , M ),

T̂ h = τ ∘ Th ∘ τ−1 ,

(7.2.187)

where (T̂ h , M ) is a model example of a dynamical system whose chaotic behavior is evident. Moreover, homeomorphism (7.2.187) is usually constructed only for the appropriate discrete subgroups of Th , and the model examples (T̂ h , M ) are the appropriate Bernoulli shifts. It is worth emphasizing, however, that the (multidimensional) symbolic dynamics with a finite number of symbols (Bernoulli shifts) are not adequate for understanding spatial dynamics (7.2.186), since the topological entropy of such shifts is finite, whereas we have the dynamics with the infinite topological entropy. Consequently, we introduce below another model example (T̂ h , M ) of chaos that is close to the standard Bernoulli shifts but is adopted to the case of infinite topological entropy; we also construct the Lipschitz-continuous embedding of this model into the dynamical system (7.2.186). Definition 7.2.10. Let ϕ(x) > 0, ϕ ∈ Cb (ℝn ), be a weight function that satisfies lim|x|→∞ ϕ(x) = 0, and let A be a compact set in Φb,ϕ that is invariant with respect to the Th action. Then, for every R ∈ ℝ+ , we define a new metric on A via dR,ϕ (x, y) =

sup ‖Th x − Th y‖Φb,ϕ ,

h∈[−R,R]n

Set

1 ℍ (A , dR,ϕ ), (2R)n ϵ

(7.2.188)

1 1 ĥ sp (A , ϕ) = lim sup 1 lim sup ℍϵ (A , dR,ϕ ). ϵ→0 ln ϵ R→∞ (2R)n

(7.2.189)

hsp (A , ϕ) = hsp (A , ϕ, Th ) = lim lim sup ϵ→0

and

x, y ∈ A .

R→∞

Remark 7.2.4. Quantity (7.2.188) coincides with the usual definition of the topological entropy for the group Th : A → A , and (7.2.189) is one possible generalization of this concept for the case where the topological entropy is infinite. Lemma 7.2.16. Let the above assumptions hold, then for every ϕ such as in Definition 7.2.10, hsp (A , ϕ) = hsp (A ) = lim lim sup ϵ→0

R→∞

1 ℍ (A , Wb2,q ([−R, R]n )), (2R)n ϵ

(7.2.190)

282 | 7 Global attractors and time-spatial chaos and similarly, 1 1 hsp (A ) = ĥ sp (A ) = lim lim sup ℍϵ (A , Wb2,q ([−R, R]n )). ϵ→0 ln 1 R→∞ (2R)n ϵ

(7.2.191)

Proof. Since ϕ(x) → 0, |x| → ∞, then for every ϵ > 0 there exists L = L(ϵ) such that ϕ(x) < ϵ for |x| > L(ϵ). Consequently, n

ℍϵ (A , dR,ϕ ) = ℍϵ/C (A , Wb2,q ([−R − L(ϵ), R + L(ϵ)] ))

(7.2.192)

for the appropriate C, which is independent of R. Therefore, due to the fact that L(ϵ) is independent of R, we have ĥ sp (A , ϕ) ≤ ĥ sp (A ).

hsp (A , ϕ) ≤ hsp (A ),

(7.2.193)

The opposite inequalities follow from the evident estimate sup ϕ(x + h) ≥ ϕ(0) > 0,

h∈[−R,R]n

|xi | ≤ R.

It is well known that the topological entropy hsp (A ) depends only on the topology on A and is independent of the choice of the metric preserving the topology. However, we note that the modified topological entropy ĥ sp (A ) does not possess this property and, rigorously speaking, is not a topological invariant. It is worth mentioning that it is possible to construct a topological invariant using the quantity ĥ sp (A ). To this end, for every metric d that generates the local topology of Φloc on A , one should consider the quantity ĥ sp (A , d) computed with respect to this metric, and take an infimum with respect to all such metrics, ĥ top (A ) = inf ĥ sp (A , d). d

(7.2.194)

We also mention that, due to embedding constructed below, the quantity (7.2.194) is also usually strictly positive. Remark 7.2.5. Nevertheless, ĥ sp is Lipschitz-continuous invariant; it is preserved under Lipschitz-continuous homeomorphisms. Moreover, if τ is Hölder-continuous with the Hölder constant 0 < α < 1, then 1 ĥ sp (τ(M)) ≤ ĥ sp (M). α

(7.2.195)

Theorem 7.2.8. Let the assumptions of Theorem 7.2.7 hold, and let A be the attractor of equation (7.2.129). Then the group {Th : h ∈ ℝn } of spatial shifts on the attractor has the infinite topological entropy hsp (A ) = ∞.

(7.2.196)

Moreover, its modified topological entropy is finite and strictly positive, 0 < C1 ≤ ĥ sp (A ) ≤ C2 < ∞.

(7.2.197)

7.2 Attractors and their spatial complexity | 283

This theorem is an immediate corollary of estimate (7.2.180) and Lemma 7.2.16. Let us now study the spatial chaos generated by the action of {Th : h ∈ ℝn } on the attractor A . We first obtain embedding (7.2.187) for the case of continuous dynamics h ∈ ℝn , and after that we simplify the obtained model considering the case of discrete dynamics h ∈ ℝn . Theorem 7.2.9. Let the assumptions of Theorem 7.2.6 be valid, and let r, σ, ξ0 be the same as in Theorem 7.2.6. Also, let 𝕂 be the ball B(0, r, 𝔹σ,ξ0 ) endowed with the local n topology of L∞ loc (ℝ ). Then the map U0 : 𝕂 → A defined in Theorem 7.2.6 realizes the homeomorphism U0 : (Th , 𝕂) → (Th , U0 (𝕂)),

U0 ∘ Th = Th ∘ U0 .

(7.2.198)

Moreover, this homeomorphism is Lipschitz-continuous if we endow the spaces 𝕂 and A with the topology L∞ b,ϕ and Φb,ϕq , respectively (where ϕ is an arbitrary weight function with the polynomial growth rate). Consequently, it preserves the modified topological entropy 0 < C1 ≤ ĥ sp (𝕂) = ĥ sp U0 (𝕂) ≤ C2 < ∞.

(7.2.199)

This theorem is an immediate corollary of Theorem 7.2.6 and Corollaries 7.2.15 and 7.2.16. Thus, the r-ball 𝕂 of the space 𝔹σ,ξ0 , together with the group of spatial shifts {Th : h ∈ ℝn } acting on it, can be considered as a model example for the topological description of the spatial chaos in the reaction-diffusion systems in unbounded domains. However, this model is rather complicated by itself, and it seems reasonable to simplify it. Definition 7.2.11. Let 𝔻 := {z ∈ ℂ : |z| ≤ 1} be a unitary disc on the complex plane n and M := 𝔻ℤ be the space of all functions U : ℤn → 𝔻. We endow this space by a Fréchet topology generated by the system of seminorms ‖U, BR0 ‖0,∞ =

sup |U(l)|,

l∈ℤn ,|l|≤R

(7.2.200)

and denote the space thus obtained by Mloc . It is evident that Mloc is a compact metric space and its topology coincides with Tikhonov’s topology on the Descartes product n 𝔻ℤ . The spaces Mb and Mb,ϕ , where ϕ is a weight function, can be defined analogously. We define a group {Tl : l ∈ ℤn } on M by the standard expression (Tl )(m) = U(l + m),

l, m ∈ ℤn ,

U ∈ Mb .

(7.2.201)

We only consider the action of a discrete subgroup {Tδl , l ∈ ℤn } of the group of spatial shifts on the attractor A , where δ > 0 is an appropriate positive number, and use the dynamical system {Tl , M } as the model for embedding (7.2.187). In order to construct such an embedding, we use the Whittaker–Shannon–Kotel’nikov interpolation formula to represent the functions from 𝔹σ,ξ0 .

284 | 7 Global attractors and time-spatial chaos Proposition 7.2.8. Every function u(x) from the class 𝔹σ ,ξ0 can be represented in the form u(x) = ∑ u(δk)gρ,k (x), l∈ℤn

n

gρ,k (x) = eiξ0 (x−δk) ∏ j=1

π , σ + ρ

δ=

(7.2.202)

sin ρ(xj − δk j ) sin(σ + ρ)(x j − δk j ) . ρ(σ + ρ)(x j − δk j )2

(7.2.203)

Moreover, gρ,k ∈ 𝔹σ +2ρ,ξ0 . We now fix σ , ρ > 0, such that σ + 2ρ < σ, and define a map κ : M → 𝔹σ,ξ0 via κ(U) = ∑ U(l)gρ,l (x),

(7.2.204)

l∈ℤn

where the functions gρ,l are defined in (7.2.203). The next result is then valid. Lemma 7.2.17. Let the above assumptions hold and let 0 < ν < 1, then n

j

j 2

− ν2

|κ(U)(x)| ≤ sup |U(l)|(∏(1 + |x − l | )) l∈ℤn

j=1

.

(7.2.205)

Moreover, for every R > √n R ‖κ(U), BδR 0 ‖0,∞ ≥ ‖U, B0 ‖0,∞ .

(7.2.206)

Proof. Estimate (7.2.206) follows immediately from the fact that κ(U)(δl) = U(l) for every l ∈ ℤ, (see (7.2.202) and (7.2.203)). The estimate (7.2.205) is based on the evident estimate |gρ,l (x)| ≤

C , ∏nj=1 (1 + |x j − lj |2 )

l ∈ ℤn , x ∈ ℝn

(7.2.207)

and can also be verified in a direct way. Corollary 7.2.19. Let the above assumptions hold; then there exists a constant C = C(σ , ρ) such that ‖κ(M )‖L∞ (ℝn ) ≤ C. b

(7.2.208)

Moreover, for every weight function with a polynomial growth rate ν < 1, the following estimate is valid: C1 ‖U‖Mb,ϕ ≤ ‖κ(U)‖L∞ ≤ C2 ‖U‖Mb,ϕ . b,ϕ

(7.2.209)

7.2 Attractors and their spatial complexity | 285

This corollary follows from estimates (7.2.205), (7.2.206). We now assume that σ, r > 0 are the same as in Theorems 7.2.6 and 7.2.9. Then estimate (7.2.208) implies that the map ̃ κ(U) =

r κ(U), C

U ∈ M,

(7.2.210)

is an embedding M in 𝕂, where C is as defined in (7.2.208). Moreover, we show that this embedding is Lipschitz-continuous in the appropriate metrics. Lemma 7.2.18. Let the above assumptions hold, then the set κ(̃ M ) is invariant with respect to the discrete group Tδl , and this group commutes with the map K̃ defined by (7.2.210); i. e., κ̃ ∘ Tl = Tδl ∘ κ,̃

l ∈ ℤn .

This lemma is an immediate corollary of the fact that κ(U)(δl) = U(l). We now note that the topological entropy hsp and the modified topological entropy ̂h can be defined for discrete groups as well. Moreover, the assertions of Lemma 7.2.16 sp and Remark 7.2.5 also remain valid for this case. Consequently, due to (7.2.209), the map κ̃ : (Tl , M ) → (Tδl , K(̃ M )) ⊂ (Th , 𝕂),

(7.2.211)

preserves the modified topological entropy 0 < ĥ sp (Tl , M ) = δ−n ĥ sp (Tδl , κ(̃ M )).

(7.2.212)

The multiplier δ−n appears due to the “time” rescaling l → δl. Thus, for the case of a discrete group of shifts Tδl , we have constructed the Lipschitz-continuous embedding of the dynamical system (Tl , M ) into the dynamical system (Th , 𝕂). Combining this embedding with the embedding constructed in Theorem 7.2.9, we obtain the next result. Theorem 7.2.10. Let the assumptions of Theorem 7.2.9 hold and δ > 0 be the same as in Lemma 7.2.18. Then the map τ = U0 ∘ κ̃ is a Lipschitz-continuous (in weighted metrics described in Corollary 7.2.19)isomorphism between M and τ(M ) ⊂ A that commutes with the spatial shifts, τ : (Tl , M ) → (Tδl , τ(M )),

τ ∘ Tl = Tδl ∘ τ, ∀l ∈ ℤn ,

(7.2.213)

and, consequently, this homeomorphism preserves the modified topological entropy 0 < ĥ sp (Tl , M ) = δ−n ĥ sp (Tδl , τ(M )).

(7.2.214)

286 | 7 Global attractors and time-spatial chaos Thus, we have constructed the Lipschitz-continuous embedding of the model dynamical system (Tl , M ) into the dynamical system (Tδl , A ) generated by the discrete spatial shifts on the attractor A of equation (7.2.129). Moreover, if we consider only the subset MN ⊂ M of functions U : ℤn → {a1 , a2 , . . . , aN } where a1 , a2 , . . . , aN ∈ 𝔻 are arbitrary different complex numbers in the unitary ball, we obtain the standard symbolic dynamics with N symbols (multidimensional Bernoulli shifts). Consequently, Theorem 7.2.10 allows us to embed the symbolic dynamics with N symbols into the discrete spatial shifts of the attractor A for every N ∈ ℕ. Moreover, the following corollary shows that an arbitrary finite-dimensional (discrete) dynamics can be realized as a restriction of the discrete spatial shifts to the appropriate invariant subset of the attractor. Corollary 7.2.20. Let the assumptions of the previous theorem hold, let K ⊂ ℂN be an arbitrary compact set in ℂN and let ϕ : K → K be a homeomorphism. Define a dynamical system {Gn : n ∈ ℤ} on K by iteration of this homeomorphism Gn z = (ϕ)n z,

z ∈ K.

(7.2.215)

Then there exists a homeomorphism τ : K → τ(K) ⊂ A such that τ ∘ Gn = Tnp⃗ ∘ τ,

n ∈ ℤ,

(7.2.216)

where p⃗ = Nδe1 = Nδ(1, 0, . . . , 0) and δ is the same as in Theorem 7.2.10. Proof. Due to Theorem 7.2.10, it is sufficient to construct the embedding of this system into the model system (Tl , M ). Without loss of generality, we may also assume that K is a subset of the N-dimensional polydisc K ⊂ 𝔻n . Let us define an embedding θ : K → M: θ(z)(l1 , l2 , . . . ln ) = [Gn (z)]k

(7.2.217)

where l ∈ ℤn , l1 = nN + K, n ∈ ℤ, K ∈ {0, 1, . . . , N − 1}, z ∈ K ⊂ 𝔻N , and [z]k denotes the K th coordinate of z ∈ ℤN . It is not difficult to verify that θ : K → θ(K) ⊂ M is indeed a homeomorphism (since Gn : K → K is a homeomorphism). Moreover, it follows from the definition of θ that θ(Gn (z)) = TnNe1 θ(z),

z ∈ K, n ∈ ℤ.

(7.2.218)

The corollary is now an immediate consequence of (7.2.218) and Theorem 7.2.10. Remark 7.2.6. For simplicity, we have formulated and proved Corollary 7.2.20 only for the dynamical system (Gn , K) with one-dimensional “time,” but its generalization for the multidimensional case is straightforward.

7.2 Attractors and their spatial complexity | 287

We previously constructed several invariant (with respect to spatial shifts) subsets B ⊂ A of the attractor; when {Th : h ∈ ℝn } is restricted to these subsets, it demonstrates chaotic behavior, has infinite topological entropy hsp (B) = ∞, positive modified entropy ĥ sp (B) > 0, and so on. We note, however, that all sets thus constructed are not invariant with respect to the temporal dynamics {St : t ≥ 0} generated by equation (7.2.129) (in fact, the image Ũ 0 (𝕂) constructed in Theorem 7.2.9 belongs to an exponentially unstable manifold of the equilibrium u0 ≡ 0). Thus, it seems reasonable to study the spatial complexity of sets St B, t ≥ 0, where B is a spatially invariant subset of the attractor A . Lemma 7.2.19. Let the assumptions of Theorem 7.2.7 hold, and let B be a subset of the phase space Φb of equation (7.2.129) that is compact in Φloc and invariant with respect to the spatial shifts {Th : h ∈ ℝn }. Then ĥ sp (St B) ≤ ĥ sp (B),

hsp (St B) ≤ hsp (B),

∀t ≥ 0,

(7.2.219)

where St : Φb → Φb is a semigroup generated by equation (7.2.129). Proof. The set B is obviously bounded in Φb , and, consequently, due to estimates (7.2.81) and (7.2.3), the semigroup St is Lipschitz-continuous in the space Φb,ϕ for every weight function that satisfies assumption (7.2.1). But the (modified) topological entropy does not increase under the Lipschitz-continuous mappings (see Remark 7.2.5). The lemma is proved. Theorem 7.2.11. Let the assumptions of Theorem 7.2.9 hold, and let the matrix a in equation (7.2.129) be normal, i. e., aa∗ = a∗ a.

(7.2.220)

Let B be a subset of the attractor A that is compact in Φloc and invariant with respect to {Th : h ∈ ℝn }. Then the quantities hsp (B) and ĥ sp (B) under the temporal dynamics preserve hsp (St B) = hsp (B),

ĥ sp (St B) = ĥ sp (B),

∀t ≥ 0.

(7.2.221)

This theorem is a corollary of the next lemma, which claims that the semigroup St is backward Hölder-continuous on the attractor with the Hölder exponent arbitrarily close to 1. Lemma 7.2.20. Let the above assertions hold, and let u1 (t), u2 (t) ∈ A , t ∈ R, be two arbitrary solutions of (7.2.129) belonging to the attractor. Then, for every 0 < α < 1 and every fixed T > 0, there exist ϵ > 0 and a constant C = C(α, T, ϵ) such that ‖u1 (0) − u2 (0), B1x0 ‖2,q ≤ C sup e−ϵ|x−x0 | ‖u1 (T) − u2 (T), B1x ‖α0,2 . x∈Ω

(7.2.222)

288 | 7 Global attractors and time-spatial chaos The proof of this lemma is based on the following convexity result. Proposition 7.2.9. Let H be a Hilbert space and B : D(B) → H be a linear unbounded operator in it. Also, let v ∈ C 1 ([t0 , t1 ], H) ∩ C([t0 , t1 ], D(B)) be a solution of the equation 𝜕t v − Bv = P(t)v,

‖P(t)‖H→H ≤ P0 .

(7.2.223)

Also assume that B = B+ + B− + B − where B+ is a symmetric operator and B− and B− are skew-symmetric operators such that for every ω ∈ H:

(B+ ω, B− ω)H ≥ −γ‖B+ ω‖H ‖ω‖H − β‖ω‖2H , 2 ‖B − ω‖H

(7.2.224)

β‖ω‖2H .

(7.2.225)

(P(t)u(t), u(t)) . ‖u(t)‖2H

(7.2.226)

≤ γ‖B+ ω‖H ‖ω‖H +

Set t

l(t) = 2 ln ‖u(t)‖H − ∫ ψ(s)ds,

ψ(t) = 2

t0

Then, for every t0 ≤ t ≤ t1 we have l(t) ≤ α± l(t0 ) + (1 − α± )l(t1 ) + e4γ(t1 −t0 ) (t1 − t0 )2 (8γ 2 + 4β + 2P02 ),

(7.2.227)

where α± =

e±4γt1 − e±4γt . e±4γt1 − e±4γt0

(7.2.228)

In (7.2.228), one takes the negative sign if l(t0 ) ≤ l(t1 ) and the positive sign if l(t0 ) ≥ l(t1 ). Corollary 7.2.21. Let the assumptions of Lemma 7.2.19 hold, and let the solution U(t) be defined on (−∞, t1 ] and remain bounded, ‖U(t)‖H ≤ K. Then for every μ > 0 and t ∈ (−∞, t1 ), there exists a constant C = C(t, t1 , μ, K) such that ‖U(t)‖H ≤ C‖U(t1 )‖αH ,

α = e4γ(t−t1 ) − μ.

(7.2.229)

Proof. Taking the exponent from both sides of inequality (7.2.227) and using the fact t that −2P0 (t − t0 ) ≤ ∫t ψ(s)ds ≤ 2P0 (t − t0 ), we have 0

1−α±

‖U(t)‖H ≤ C(t, t1 , t0 )‖U(t1 )‖H

α

‖U(t2 )‖H± .

(7.2.230)

Since ‖U(t2 )‖H ≤ K, (7.2.230) implies ‖U(t)‖H ≤ C (K, t, t1 , t0 )‖U(t1 )‖αH ,

(7.2.231)

where α = min{1 − α+ , 1 − α− }. Let t2 = −N, N > 0 is large enough, then α± = 1 − α+ =

eγt − e4γN → e−4γ(t1 −t) , N → ∞. e4γt1 − e4γN

(7.2.232)

Therefore, for every μ > 0, we may find N = N(μ) such that α ≥ e−4γ(t1 −t) − μ. Corollary 7.2.21 is proved.

7.2 Attractors and their spatial complexity | 289

Proof of Lemma 7.2.20. Let U(t) = u1 (t) − u2 (t), then U(t) satisfies 𝜕t U = aΔx U − λ0 U − l(t)U,

(7.2.233)

1

where l(t) = ∫0 f (su1 (t) + (1 − s)u2 (t))ds. We recall that each ui (t), i = 1, 2 is a complete bounded solution belonging to the A . Consequently, due to Theorems 7.2.1 and 7.2.4, ‖ui (t)‖Cb (ℝN ) ≤ ‖ui (t)‖Φb ≤ C. Therefore, the function l(t) are uniformly bounded; ‖l(t)‖Cb (ℝN ) ≤ C1 and C1 are independent of ui . We now fix an arbitrary x0 ∈ ℝn and consider a function Wx0 (t) = U(t)ϕ̃ ϵ,x0 , where the weight function ϕ̃ is the same as in the proof of Lemma 7.2.13 and ϵ is a small ϵ,x0

parameter. Then it is not difficult to verify Wx0 satisfies

𝜕t Wx0 (t) − aΔx Wx0 (t) + K1 (x)Wx0 (t) + K2 (x)∇x Wx0 (t) + l(t)Wx0 (t) = 0,

(7.2.234)

where K1 (x)Wx0 (t) = (

Δx ϕ̃ ϵ,x0 |∇x ϕ̃ ϵ,x0 |2 −2 )aWx0 (t), ϕ̃ ϕ̃ 2 ϵ,x0

ϵ,x0

(7.2.235)

̃ K2 (x)∇x Wx0 (t) = 2ϕ̃ −1 ϵ,x0 ∇x ϕϵ,x0 ⋅ a∇x Wx0 (t) n

̃ = 2ϕ̃ −1 ϵ,x0 ∑ 𝜕xi ϕϵ,x0 a𝜕xi Wx0 (t). i

(7.2.236)

Moreover, it follows from (7.2.140) that |Ki (x)| + |∇x Ki (x)| ≤ Cϵ for the appropriate constant C. Let us now verify that equation (7.2.234) satisfies all the assumptions of Lemma 7.2.19. Let H = [L2 (ℝn )]k , RW = K2 (x)∇x W, then we have 1 1 B+ = (a + a∗ )Δx − λ0 − (R + R∗ ), 2 2 1 1 ∗ B− = (a − a )Δx , B− = − (R − R∗ ), 2 2

(7.2.237)

and P(t)W = −K1 (x)W − l(t)W. Then, obviously, B+ is symmetric and B− , B − are skewsymmetric. In order to verify assumptions (7.2.224) and (7.2.225), we first compute the operator R∗ : ∗ ̃ ̃ ̃ −1 ∗ R∗ W = −2ϕ̃ −1 ϵ,x0 ∇x ϕϵ,x0 ⋅ a ∇x W − 2∇x ⋅ (∇x ϕϵ,x0 ϕϵ,x0 )a W.

Then (B+ W, B− W) =

1 ((a + a∗ )Δx W, (a − a∗ )Δx W) 4

(7.2.238)

290 | 7 Global attractors and time-spatial chaos ∗ ∗ ̃ − (ϕ̃ −1 ϵ,x0 ∇x ϕϵ,x0 ⋅ (a − a )∇x W, (a − a )Δx W) ∗ ∗ + (∇x (∇x ϕ̃ ϵ,x0 ϕ̃ −1 ϵ,x0 )a W, (a − a )Δx W).

(7.2.239)

Since a is normal, the first term in the right-hand side of (7.2.239) is equal to zero identically. Integration by parts in the second term, we derive that ∗ ∗ ̃ (ϕ̃ −1 ϵ,x0 ∇x ϕϵ,x0 ⋅ (a − a )∇x W, (a − a )Δx W)

1 ∗ ∗ ̃ = − ({∇x (ϕ̃ −1 ϵ,x0 ∇x ϕϵ,x0 )}(a − a )∇x W, (a − a )∇x W) 2 ≤ Cϵ‖∇x W‖2H .

(7.2.240)

It follows from the interpolation inequality, the regularity theorem for the Laplace operator in ℝn , and the fact that ϵ > 0 is small enough that ‖∇x W‖2H ≤ C‖W‖W 2,2 (Rn ) ‖W‖H ≤ C1 ‖B+ W‖H ‖W‖H .

(7.2.241)

Finally, due to the Hölder inequality, ∗ ∗ ̃ ({∇x (ϕ̃ −1 ϵ,x0 ∇x ϕϵ,x0 )}a W, (a − a )Δx W)

≥ −Cϵ‖W‖H ‖Δx W‖H

≥ −C2 ϵ‖B+ W‖H ‖W‖H .

(7.2.242)

Combining estimates (7.2.239)–(7.2.242), we have (B+ W, B− W) ≥ −γ‖B+ W‖H ‖W‖H ,

γ = Cϵ,

thus assumption (7.2.224) is verified. Let us verify assumption (7.2.225). Since B − is a first-order differential operator, due to (7.2.241), we have 2 2 2 ‖B − W‖H ≤ Cϵ(‖∇x W‖H + ‖W‖H )

≤ C1 ϵ(‖B+ W‖H ‖W‖H + ‖W‖2H ).

(7.2.243)

Thus, assumption (7.2.225) is also verified. We now note that ui (t) ∈ A implies that ‖Wx0 (t)‖L2 (Rn ) ≤ K where K is independent of x0 . Thus, all assumptions of Lemma 7.2.20 and Corollary 7.2.21 are verified, according to (7.2.229) (with t1 = T and t = −1), ‖Wx0 (−1)‖0,2 ≤ C(ϵ, μ, T)‖Wx0 (T)‖α0,2 .

(7.2.244)

Here, α = e−Cϵ(T+1) − μ, μ > 0 can be chosen arbitrarily small and the constant C is independent of x0 . Since ϵ, μ can be chosen arbitrarily small, the Hölder exponent α < 1 in (7.2.244) is arbitrarily close to 1.

7.2 Attractors and their spatial complexity | 291

Estimate (7.2.244) immediately implies that ‖U(−1), B1x0 ‖0,2 ≤ C (α, ϵ, T) sup ‖U(T), B1x ‖α0,2 , x∈Rn

(7.2.245)

where α is arbitrarily close to 1 and ϵ = ϵ(α) > 0. Estimate (7.2.222) is an immediate corollary of (7.2.244). Lemma 7.2.20 is proved. We are now ready to complete the proof of Theorem 7.2.11. We note that estimate (7.2.222) implies that the restriction of ST |A to the attractor A is invertible, for every weight function ϕ with a polynomial growth rate and for every 0 < α < 1, the operator ST−1 |A is uniformly Hölder-continuous with the exponent α, ST−1 : A ∩ Φb,ϕ → A ∩ Φb,ϕα ,

(7.2.246)

that is, for every u1 , u2 ∈ A , ‖u1 − u2 ‖Φb,ϕα ≤ C(T, α)‖ST u1 − ST u2 ‖Φb,ϕα , due to Lemma 7.2.16 and estimate (7.2.195), we have ĥ sp (B) ≤ αĥ sp (ST B),

hsp (ST B) = hsp (B).

(7.2.247)

Passing to the limit α → 1 in (7.2.247). Theorem 7.2.11 is proved. Remark 7.2.7. We have constructed the set B = U0 (𝕂) ⊂ A , spatial shifts restricted to this set are isomorphic to the model dynamics (Th , 𝕂) (or to (Tl , M ) for discrete spatial shifts). Estimate (7.2.246) now implies that the set ST B ⊂ A is also homeomorphic to (Th , 𝕂) (or (Tl , M ), respectively). Thus, the spatial chaos constructed in above section is preserved under the time evolution {St : t ∈ ℝ+ }. Now study the spatial complexity of individual solutions u(t) ∈ A of equation (7.2.129). Definition 7.2.12. Let u0 ∈ A . We denote by Hsp (u0 ) the hull of this point with respect to the spatial shifts n

Hsp (u0 ) = [Th u0 , h ∈ ℝ ]Φ , loc

(7.2.248)

where [⋅]Φloc denotes a closure in the space Φloc and define the quantities hsp (u0 ) and ĥ (u ) as sp

0

hsp (u0 ) = hsp (Hsp (u0 )),

ĥ sp (u0 ) = ĥ sp (Hsp (u0 )).

(7.2.249)

The next corollary shows that quantities (7.2.249) are constants along the trajectories of (7.2.129).

292 | 7 Global attractors and time-spatial chaos Corollary 7.2.22. Let the assumptions of Theorem 7.2.11 be valid. Then, for every u0 ∈ A , we have hsp (St u0 ) = hsp (u0 ),

ĥ sp (St u0 ) = ĥ sp (u0 ),

∀t ≥ 0.

(7.2.250)

Moreover, the quantity ĥ sp (u0 ) is finite for every u0 ∈ A , and there exists a point u0 ∈ A such that hsp (u0 ) = ∞,

ĥ sp (u0 ) > C > 0.

(7.2.251)

Proof. Assertions (7.2.250) are immediate corollaries of Theorem 7.2.11. Thus, we only need to verify the existence of a point u0 that satisfies (7.2.251). We recall that, due to Theorem 7.2.10, it is sufficient to find a point U0 ∈ M such that its hull (with respect to the discrete shift group {Tl : l ∈ ℤn }) has a positive modified topological entropy. But the space M , obviously, possesses a topologically transitive orbit, that is, there exists U0 ∈ M such that n

M = [Tl U0 : l ∈ ℤ ]M . loc

Fixing now u0 = τ(U0 ) ⊂ A , where τ : M → A is defined in Theorem 7.2.10, we obtain a point of A that satisfies (7.2.251), and Corollary 7.2.22 is proved. Remark 7.2.8. It follows from the proof of Corollary 7.2.22 that there is a point u0 ∈ A with an extremely complicated spatial structure. In particular, (Tl , M ) ⊂ (Tδl , Hsp (u0 )). Consequently, due to Corollary 7.2.20, any finite-dimensional dynamics can be realized by restricting the discrete spatial shift group to the appropriate subset of the hull Hsp (u0 ) of this point.

8 Non-Newtonian generalized fluid and their applications Non-Newtonian characteristics are exhibited by numerous fluids including physiological liquids, geological suspensions, industrial tribological liquids and biotechnological liquids. It is difficult to propose a single model which can exhibit all the properties of non-Newtonian fluids. To describe the viscoelastic properties of such fluids, recently, constitutive equations with ordinary and fractional time derivatives have been introduced. Fractional calculus has proved to be very successful in the description of constitutive relations of viscoelastic fluids. The starting point of the fractional derivative model of viscoelastic fluids is usually a classical differential equation which is modified by replacing the time derivative of an integer order with fractional order and may be formulated both in the Riemann–Liouville or Caputo sense. With the development of fractional calculus, fractional derivatives and fractional partial differential equations have been applied to the numerical solution of the complex problems in fluids and continuum mechanics.

8.1 An inverse problem of a heated generalized second grade fluid Fractional material models were first considered in the mid-twentieth century. As long ago as in the 1970s, Caputo and Mainardi had considered the linear models of dissipation in inelastic solids. Nowadays, it has turned out that many phenomena in engineering, physics, chemistry, biology, and many other sciences can be described very successfully by models using mathematical tools from fractional calculus. In recent years, it has been found that fractional calculus is a powerful tool for modeling the viscoelastic behaviors and non-Newtonian fluid characteristics and particularly suitable for building the time-dependent constitutive model.

8.1.1 Formulation 8.1.1.1 Direct problem governing equations The fractional Stokes’ first problem for a heated generalized second grade fluid (FSFPHGSGF) is employed in this work 2 𝜕u(x, t) 1−γ 𝜕u (x, t) = (ν + η0 Dt ) + f (x, t), 𝜕t 𝜕x2 0 ≤ x ≤ L, 0 ≤ t ≤ T, 0 < γ < 1 https://doi.org/10.1515/9783110549614-008

(8.1.1)

294 | 8 Non-Newtonian generalized fluid and their applications with initial and boundary conditions u(x, 0) = ϕ(x),

0≤x≤L

u(0, t) = φ(x),

u(L, t) = θ(t),

(8.1.2) 0≤t≤T

(8.1.3)

In the above equations, u is the velocity in the y coordinate direction, ν and η are the positive constants, and ν and η are defined by ν = μ/ρ, η = α1 /ρ, where ρ is the density of fluid, μ is coefficient of viscosity and α1 is the stress moduli. The symbol 1−γ 0 Dt u(x, t) is the Riemann–Liouville time fractional derivative of order 1 − γ defined by [29, 76] 1−γ

0 Dt u(x, t) =

1

t

Γ(γ) 𝜕t𝜕

∫ 0

u(x, ξ ) dξ (t − ξ )1−γ

(8.1.4)

where 0 < γ < 1. Note that in the limit γ → 0, equation (8.1.1) reduces to the classic Stokes’ first problem for a heated generalized second grade fluid. 8.1.1.2 Inverse problem In order to identify the unknown order of the Riemann–Liouville fractional derivative, we present an inverse fractional Stokes’ first problem for a heated generalized second grade fluid (IFSFP-HGSGF). Hence, the new IFSFP-HGSGF must be formulated according to the FSFP-HGSGF equations (8.1.1)–(8.1.3) introduced in the previous section [89, 92]. We consider a one-dimensional IFSFP-HGSGF on a finite slab in which the Riemann–Liouville fractional derivative γ at the domain (0, 1) is desired and unknown, while everything else in equations (8.1.1)–(8.1.3) is known. The IFSFP-HGSGF is described mathematically by the system 2 𝜕u(x, t) 1−γ 𝜕u (x, t) = (ν + η0 Dt ) + f (x, t), 𝜕t 𝜕x2 0 ≤ x ≤ L, 0 ≤ t ≤ T

(8.1.5)

u(x, 0) = ϕ(x),

0 ≤ x ≤ L,

known,

(8.1.6)

u(0, t) = φ(x),

0 ≤ t ≤ T,

known,

(8.1.7)

u(L, t) = θ(t),

0 ≤ t ≤ T,

known,

(8.1.8)

0 < γ < 1,

unknown,

(8.1.9)

8.1.2 Outline of the optimization method In this section, our goal is to identify the unknown Riemann–Liouville fractional derivative γ ∈ (0, 1), which minimizes a given cost function S depending on the fluid state u(γ) satisfying the FSFP-HGSGF equations (8.1.1)–(8.1.3).

8.1 An inverse problem of a heated generalized second grade fluid | 295

8.1.2.1 Implicit numerical method for the direct problem In order to solve the direct problem numerically, we make discrete the FSFP-HGSGF equations (8.1.1)–(8.1.3) in a two-dimensional domain Ω = [0, L] × [0, T], and let xj = jh,

j = 0, 1, 2, . . . , M,

tk = kτ,

k = 0, 1, 2, . . . , N,

L M T h= N

h=

where M and N are positive integers, h and τ are the space and time steps, respectively. The numerical solution u at the mesh point (xj , tk ) is denoted by ukj . Using the relationship between the Grünwald–Letnikov fractional derivative and the Riemann–Liouville fractional derivative, we can approximate the fractional derivative by k

1−γ

γ−1 ∑ λm f (t − mτ) 0 Dt f (t) ≈ τ

(8.1.10)

m=0

1−γ

where λm = (−1)m (m ), m = 0, 1, 2, . . . , k. Then the FSFP-HGSGF equations (8.1.1)– (8.1.3) can be approximated by the following implicit numerical approximation scheme [68, 69]: 1−γ

ukj − ut τ

m=0

u0j

uk0 ukM where

δx2 ulj

=

ulj−1

−

2ulj

+

k

= ητγ−1 ∑ λm

δx2 uk−m j h2

+ν

δx2 ukj h2

+ fjk ,

k = 1, 2, . . . , N, j = 1, 2, . . . , M − 1,

(8.1.11)

= ϕ(xj ),

j = 0, 1, 2, . . . , M,

(8.1.12)

= φ(tk ),

k = 0, 1, 2, . . . , N,

(8.1.13)

k = 0, 1, 2, . . . , N,

(8.1.14)

= θ(tk ),

ulj+1 , fjk

= f (xj , tk ).

8.1.2.2 Optimization process Let us define the coast function as the squared 2-norm m

S(γ) := ‖Y − U‖2 = ∑(yi − ui )2 i=1

(8.1.15)

where U = (u1 , u2 , . . . , um )T is a vector of estimated or computed velocity and Y = (y1 , y2 , . . . , ym )T is a vector of measured velocity at some measured time t = tk . Hence, we are interested in solving the following problem: min S(γ)

(8.1.16)

0 < γ < 1.

(8.1.17)

γ

subject to

296 | 8 Non-Newtonian generalized fluid and their applications 8.1.2.3 Levenberg–Marquardt algorithm The LMA is an iterative technique that locates the minimum of a multivariate function that is expressed as the sum of the squares of nonlinear real-valued functions. LMA has been applied to the solution of a variety of inverse problems involving the estimation of unknown parameters. The iterative algorithm for finding the unknown parameter is T

γ k+1 = γ k + (H + μk Dk ) (Jk ) (Y − U) −1

(8.1.18)

where H = (Jk )T Jk is Hessian matrix, D is a diagonal matrix, uk is a positive scalar (damping parameter), γ k is the estimated parameter, k is the number of iterations, and the Jacobian matrix J is the sensitivity matrix defined by J=

𝜕U . 𝜕γ

The elements of the sensitivity matrix are called the sensitivity coefficients. The stopping criterion of the iterative algorithm is |γ k+1 − γ k | < ϵ

(8.1.19)

where ϵ is a chosen tolerance. Remark 8.1.1. The goal of the term uk Dk is to damp the oscillations and instabilities due to the ill-conditioned character of the problem by making its components large as compared to those of H if necessary. Here, Dk is taken as Dk = diag(H). 8.1.2.4 Fractional sensitivity equation based on the Riemann–Liouville fractional derivative The sensitivity matrix plays a fundamental role in the parameter estimation. Differentiating equations (8.1.1)–(8.1.3) with respect to γ, and noting that Jγ = 𝜕u , we obtain 𝜕γ the fractional sensitivity equation: 𝜕Jγ 𝜕t

𝜕2 Jγ

2 𝜕 𝜕f 1−γ 𝜕 u(x, t) [ D ( )] + t 2 2 𝜕γ 0 𝜕γ 𝜕x 𝜕x 0 ≤ x ≤ L, 0 ≤ t ≤ T

=ν

+η

(8.1.20)

with initial and boundary conditions 𝜕ϕ , 𝜕γ 𝜕φ Jγ (0, t) = , 𝜕γ

Jγ (x, 0) =

0≤x≤L Jγ (L, t) =

(8.1.21) 𝜕θ , 𝜕γ

0 ≤ t ≤ T.

(8.1.22)

Here, for calculating the second term of the right-hand side in equation (8.1.20), we need to differentiate the fractional derivative with respect to γ. For this purpose, using

8.1 An inverse problem of a heated generalized second grade fluid | 297

the relationship equation (8.1.10) between the Grünwald–Letnikov fractional derivative and the Riemann–Liouville fractional derivative, we can obtain 2 𝜕 1−γ 𝜕 u(x, t) [ Dt ( )] 𝜕γ 0 𝜕x2

=

2 k−m

δx uj 𝜕 γ−1 k [τ ∑ λm ( 𝜕γ h2 m=0 k

= τγ−1 ln τ ∑ λm m=0 k

= τγ−1 ln τ ∑ λm

)]

δx2 uk−m j h2

δx2 uk−m j

m=0

h2

k

+ τγ−1 ∑ dm m=0 k

+ τγ−1 ∑ dm

δx2 uk−m j h2

δx2 uk−m j h2

m=0

k

+ τγ−1 ln τ ∑ λm m=0

2 1−γ 𝜕 Jγ 𝜕x 2

+ 0 Dt

k−m δx2 Jγ,j

h2

(8.1.23)

where dm = 𝜕γm . Concerning dm , for m = 0, we have d0 = 0. For m ≥ 1, we have 𝜕(λ )

𝜕(λm ) 𝜕γ 𝜕 1−γ = [(−1)m (m )] 𝜕γ

dm =

=

(−1)m Γ(2 − γ) 𝜕 [ ] 𝜕γ Γ(m + 1)Γ(2 − γ − m)

= λm [

𝜕 (Γ(2 𝜕γ

− γ))

Γ(2 − γ)

−

𝜕 (Γ(2 𝜕γ

− γ − m))

Γ(2 − γ − m)

]

= λm [−ψ(2 − γ) + ψ(2 − γ − m)] = λm [−ψ(2 − γ) + ψ(γ + m − 1) − π cot π(2 − γ − m)]

(8.1.24)

where ψ(x) =

d ln Γ(x) Γ (x) = dx Γ(x)

is the digamma function, and the reflection rule ψ(1 − x) = ψ(x) + π cot πx has been used in the last step of the above expression. Substituting equations (8.1.23) and equations (8.1.24) into equation (8.1.20), we obtain the following governing equation: 𝜕Jγ 𝜕t

1−γ

= η0 Dt (

𝜕2 Jγ 𝜕x

)+ν 2

𝜕2 Jγ 𝜕x2

+ Source

(8.1.25)

298 | 8 Non-Newtonian generalized fluid and their applications where Source = ητγ−1 ln τ(

δx2 ukj h2

k

) + ητγ−1 ∑ λm m=1

× [ln τ − ψ(2 − γ) + ψ(γ + m − 1) − π cot π(2 − γ − m)](

δx2 uk−m j h2

)+

𝜕f . 𝜕γ

(8.1.26)

Using the implicit numerical method, we can easily obtain Jγ . Remark 8.1.2. The digamma function is introduced in order to better express the fractional sensitivity equation, which is absolutely different from the classical sensitivity equation. Furthermore, the fractional sensitivity equation is highly dependent on the direction problem. 8.1.2.5 Computational algorithms Suppose that the measured velocity Y is given, the initial guess γ 0 is available for the unknown parameter γ, and choose a value for μ0 , for example, μ0 = 0.1, and the iteration number is initialized (k = 0). Then the computational algorithm is listed as following: Step 1. Solve the direct problem given by equations (8.1.1)–(8.1.3) with the available estimation γ k ; Step 2. Compute S(γ k ) from equation (8.1.15); Step 3. Compute the sensitivity matrix Jk and the matrix Dk = diag(H); Step 4. Calculate the new estimation γ k+1 from equation (8.1.18); Step 5. Solve the direct problem with the new estimation γ k+1 , and compute S(γ k+1 ) as defined in step 2; Step 6. If S(γ k+1 ) ≥ S(γ k ), replace μk by μk+1 = 10μk and return to step 4; Step 7. If S(γ k+1 ) ≤ S(γ k ), accept the new estimation γ k+1 , and replace μk by μk+1 = k 0.1μ ; Step 8. Check the stopping criteria given by equation (8.1.19). Stop the iterative procedure if it is satisfied; otherwise, replace k by k + 1, and return to step 3. 8.1.3 Illustrative examples Example. Consider the following FSFP-HGSGF: 2 𝜕u(x, t) 𝜕u2 (x, t) 1−γ 𝜕u (x, t) = 0 Dt [ ]+ 2 𝜕t 𝜕x 𝜕x 2 Γ(2 + γ) 2γ 1+γ + ex [(1 + γ)t γ − t −t ] Γ(1 + 2γ) 0 < x < 1, 0 < t ≤ 1, 0 < γ < 1

(8.1.27)

8.1 An inverse problem of a heated generalized second grade fluid | 299

with initial and boundary conditions u(0, t) = t 1+γ , u(1, t) = et

0≤t≤1

1+γ

u(x, 0) = 0,

,

0≤t≤1

0 ≤ x ≤ 1.

(8.1.28) (8.1.29) (8.1.30)

The exact solution of this problem is u(x, t) = ex t 1+γ .

(8.1.31)

In the following discussions, the exact velocity value is taken as u(x, t = 1/2) at the grid point, the exact value of the parameter γ is taken as 0.75, and the tolerance ϵ is 1 1 , τ = 100 . We will discuss two cases in which the measurement chosen as 10−4 , h = 20 values contain or does not contain the random measurement error. The first case is that the exact velocity value is taken as the measurement value, while in the second case, the random measurement error is added into the velocity value. 8.1.3.1 Case without random measurement error In this section, we take the exact velocity value as the measurement value. Having chosen a initial guess γ 0 , LMA based on a fractional derivative is implemented. (1) The optimal estimation of the Riemann–Liouville fractional derivative γ Having chosen an initial guess γ 0 = 0.6, the optimal estimation of the Riemann– Liouville fractional derivative γ is attained as listed in Table 8.1.1. From Table 8.1.1, we can observe that, after 3 iterations, the estimated value of the Riemann–Liouville fractional derivative γ is 0.7512. Table 8.1.1: Optimal estimation of the fractional derivative γ. Exact value

γ0

Number of iterations

γ

0.75

0.6

3

0.7512

Figure 8.1.1 demonstrates the comparison between the estimated value (when γ = 0.7512) and the exact solution (when γ = 0.75). From Figure 8.1.1, we can observe that the estimated values are consistently excellent compared to the exact solution. (2) Effect of the initial guess γ 0 In this subsection, we choose a different initial guess γ 0 in the iterative procedure. The computational results including the number of iterations and the final estimated results are listed in Table 8.1.2. From Table 8.1.2, we can obviously find out that the optimal estimation of the Riemann–Liouville fractional derivative γ varies little during the iterative procedure, that

300 | 8 Non-Newtonian generalized fluid and their applications

Figure 8.1.1: Comparison between the estimated value (γ = 0.7512) and the exact solution (γ = 0.75). Table 8.1.2: Optimal estimation of the fractional derivative γ for different initial guess γ 0 . Exact value

γ0

Number of iterations

γ

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

0.2 0.3 0.5 0.6 0.75 0.8 0.9 0.99

3 3 3 3 2 2 2 2

0.75116 0.75117 0.75118 0.75118 0.75118 0.75122 0.75123 0.75121

is, a different initial guess γ 0 has little impact on the final estimation. This demonstrates that LMA based on the fractional derivative is effective and efficient for solving the inverse problem of estimating the unknown order of the Riemann–Liouville fractional derivative. 8.1.3.2 Case with random measurement error In this section, the random measurement error σ is added into the exact velocity value as the measurement value, that is, umeasurement = uexact (1 + σ).

(8.1.32)

Having chosen an initial guess γ 0 , LMA based on the fractional derivative is implemented. (1) The optimal estimation of Riemann–Liouville fractional derivative γ

8.1 An inverse problem of a heated generalized second grade fluid | 301 Table 8.1.3: Optimal estimation of the fractional derivative γ for different random measurement errors σ. γ0

σ (%)

Number of iterations

γ

0.6 0.6 0.6 0.6 0.6

1 2 3 5 10

3 3 3 3 2

0.7568 0.7225 0.7084 0.6806 0.6133

Table 8.1.4: Optimal estimation of the fractional derivative γ for different initial guesses γ 0 and different random measurement errors σ. γ0

σ (%)

Number of iterations

γ

0.2 0.4 0.6 0.9 0.95 0.2 0.4 0.6 0.9 0.95 0.2 0.4 0.6 0.9 0.95 0.2 0.4 0.6 0.9 0.95 0.2 0.4 0.6 0.9 0.95

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 5 5 5 5 5 10 10 10 10 10

3 3 3 2 2 3 3 3 2 2 3 3 3 2 2 3 3 3 2 1 3 3 2 2 3

0.736764 0.736784 0.736787 0.736835 0.736824 0.722512 0.722530 0.722532 0.722578 0.722563 0.708400 0.708415 0.708418 0.708460 0.708440 0.680582 0.680595 0.680596 0.680630 0.680647 0.613302 0.613308 0.613296 0.613291 0.613309

Having chosen an initial guess γ 0 = 0.6 and a different random measurement error σ = 1 %, 2 %, 3 %, 5 %, 10 %, the optimal estimation of the Riemann–Liouville fractional derivative γ is obtained. Table 8.1.3 lists the number of iterations and the final estimated value of the Riemann–Liouville fractional derivative γ for different random measurement errors σ.

302 | 8 Non-Newtonian generalized fluid and their applications From Table 8.1.3, we can obviously observe that the deviation grows larger between the estimated value and the exact value of the Riemann–Liouville fractional derivative γ as the random measurement error σ becomes larger. We can also find out that the magnitudes of the discrepancies are directly proportional to the size of the random measurement error. (2) Effect of the initial guess γ 0 In this subsection, we choose a different initial guess γ 0 for different random measured errors σ in the iterative procedure. The computational results including the number of iterations and the final estimated results are listed in Table 8.1.4. From Table 8.1.4, we can obviously find out that the optimal estimation of the Riemann–Liouville fractional derivative γ varies little during the iterative procedure for a fixed random measurement error σ, that is, different initial guesses γ 0 have little impact on the final estimation. This demonstrates that LMA based on the fractional derivative is also effective and efficient for solving the inverse problem of estimating the unknown order of the Riemann–Liouville fractional derivative. Figure 8.1.2 demonstrates the comparison between the estimated value (γ = 0.7368), measurement value (σ = 1 %), and the exact solution (γ = 0.75). Figure 8.1.3 demonstrates the comparison between the estimated value (γ = 0.7225), measurement value (σ = 2 %), and the exact solution (γ = 0.75). Figure 8.1.4 demonstrates the comparison between the estimated value (γ = 0.7084), measurement value (σ = 3 %), and the exact solution (γ = 0.75). Figure 8.1.5 demonstrates the comparison between the estimated value (γ = 0.6806), measurement value (σ = 5 %) and the exact solution (γ = 0.75). Figure 8.1.6 demonstrates the comparison between the estimated value (γ = 0.6133), measurement value (σ = 10 %), and the exact solution (γ = 0.75). From Figures 8.1.2, 8.1.3, 8.1.4, 8.1.5, and 8.1.6, we can obviously observe that the estimated values are in consistently excellent with respect to the measurement values, which

Figure 8.1.2: Comparison between the estimated value (γ = 0.7368), measurement value (σ = 1 %) and the exact solution (γ = 0.75).

8.1 An inverse problem of a heated generalized second grade fluid | 303

Figure 8.1.3: Comparison between the estimated value (γ = 0.7225), measurement value (σ = 2 %) and the exact solution (γ = 0.75).

Figure 8.1.4: Comparison between the estimated value (γ = 0.7084), measurement value (σ = 3 %) and the exact solution (γ = 0.75).

contain the random measurement error. This also demonstrates that the proposed numerical algorithm is effective for solving the inverse problem of estimating the unknown order of the Riemann–Liouville fractional derivative. Remark 8.1.3. In the computational process, we choose μ0 = 0.1. Of course, we can choose some other values such as μ0 = 1, 10, and so on. In fact, the purpose of the matrix μk Dk is to dampen the oscillations and instabilities by making their components large as compared to the components of (J k )T J k if necessary. The damping parameter μ0 can be chosen large in the beginning of the iterations for the problem is generally ill-conditioned in the region around the initial guess which may be far from the exact parameter. The damping parameter μk is then gradually reduced in the iteration pro-

304 | 8 Non-Newtonian generalized fluid and their applications

Figure 8.1.5: Comparison between the estimated value (γ = 0.6806), measurement value (σ = 5 %) and the exact solution (γ = 0.75).

Figure 8.1.6: Comparison between the estimated value (γ = 0.6133), measurement value (σ = 10 %) and the exact solution (γ = 0.75).

cess; hence, the number of iterations may vary at most if we choose some other μ0 . However, there is little impact on the final estimation of the unknown parameter.

8.2 A numerical study of a generalized Maxwell fluid through a porous medium Oscillating (or transient) flow of non-Newtonian fluids through a channel or tube is a fundamental flow regime encountered in many biological and industrial transport processes. The quasi-periodic blood flow in the cardiovascular system, movement of food bolus in the gastrointestinal tract and urodynamic transport in the human

8.2 A numerical study of a generalized Maxwell fluid through a porous medium

| 305

ureter are just several examples of oscillating flow in biological systems. Industrial applications of oscillating flows include slurry and waste conveyance systems employing roller pumps and finger pumps. The low Reynolds’ numbers characterizing such flows, and the fact that the dimensions of the channels and macromolecules in the fluid can be of the same order of magnitude, and can lead to effects unseen in macroscopic systems [60].

8.2.1 Mathematical model The constitutive equation for the shear stress-strain relationship of viscoelastic fluids obeying the fractional Maxwell model is given by [87] 𝜕α (1 + λ̃1α α )τ̃ = μγ̇ 𝜕t ̃

(8.2.1)

where λ̃1 , t,̃ τ,̃ μ, γ̇ are the relaxation time, time, shear stress, viscosity, and rate of shear strain, respectively, and α is a fractional parameter such that 0 ≤ α ≤ 1. If α = 0, this model reduces to the classical Newtonian model and when α = 1, the model reduces to the classical Maxwell model. The fractional parameter α characterizes the rheological behavior of material that is intermediate between the Newtonian and Maxwell viscoelastic fluids. This model is composed of a Hooke element connected in series with a Scott–Blair element. The well-known Darcy law states that, for the flow of a Newtonian fluid through a porous medium, the pressure gradient caused by the friction drag is directly proportional to the velocity [79, 82, 84, 85]. Darcy resistance quantifies the impedance to the flow in the bulk of the porous space. For generalized Maxwell fluid flows in porous media, the Darcy resistance can be expressed as follows: (1 + λ̃1α

μφ 𝜕α )R = − ũ 𝜕t α̃ K̃

(8.2.2)

where R, φ, K,̃ and ũ designate the Darcy resistance, porosity of the porous medium, permeability, and axial velocity, respectively. Figure 8.2.1 shows the geometry of oscillating flow through a porous medium, for the present problem.

Figure 8.2.1: Geometry of oscillating peristaltic flow through a uniform porous medium.

306 | 8 Non-Newtonian generalized fluid and their applications The constitutive equation for the geometry under consideration (Figure 8.2.1), i. e., oscillating peristaltic flow through a uniform porous medium takes the form: π h(̃ ξ ̃ , t)̃ = a − ϕ̃ cos2 (ξ ̃ − ct)̃ λ

(8.2.3)

where h,̃ λ, a, c, ϕ̃ are the transverse oscillating displacement, wavelength, half-width of the channel, wave velocity, and amplitude, respectively. The governing equations of motion for generalized Maxwell fluid flow through a porous medium using the above formulations can then be shown to take the form: 𝜕 𝜕p̃ 𝜕τ̃ξ ̃ ξ ̃ 𝜕τ̃ξ ̃ η̃ 𝜕 𝜕 + ũ + Rξ ̃ + ṽ )ũ = − + + 𝜕η̃ 𝜕η̃ 𝜕t ̃ 𝜕ξ ̃ 𝜕ξ ̃ 𝜕ξ ̃ 𝜕 𝜕 𝜕 𝜕p̃ 𝜕τ̃η̃ ξ ̃ 𝜕τ̃η̃ η̃ ρ( + ũ + ṽ )ṽ = − + + + Rη̃ 𝜕η̃ 𝜕η̃ 𝜕η̃ 𝜕t ̃ 𝜕ξ ̃ 𝜕ξ ̃ ρ(

(8.2.4)

where ρ, ξ ̃ , u,̃ η,̃ p,̃ and Rξ ̃ , Rη̃ are the fluid density, axial coordinate, transverse velocity, transverse coordinate, pressure, and components of Darcy resistance, respectively. It is pertinent to introduce the following nondimensional parameters: cλ̃ ξ̃ η̃ ϕ̃ ct ̃ ũ ṽ a , η = , t = , λ1 = 1 , u = , v = , ϕ = , δ = λ a λ λ c cδ a λ ̃ 2 φK̃ ρcaδ pa aτ̃ h 2 , τ= , Re = , K = 2 (8.2.5) h = = 1 − ϕ cos π(ξ − t), p = a μcλ μc μ a ξ =

where δ, Re and K are the wave number, Reynolds’ number, and permeability parameter, respectively. Substituting the values of shear stress and Darcy resistance from equations (8.2.1) and (8.2.2) into equation (8.2.4), using the non-dimensional parameters from equation (8.2.5) and thereafter applying the long wavelength and low Reynolds’ number approximations, equation (8.2.4) reduces to (1 + λ1α

𝜕α 𝜕p 𝜕2 u u = 2 − ) 𝜕t α 𝜕ξ K 𝜕η

𝜕p =0 𝜕η

(8.2.6)

The following boundary and initial conditions are prescribed: 𝜕u(ξ , η, t) = 0, η=0 𝜕η

u(ξ , η, t)η=h = 0,

𝜕p =0 𝜕ξ t=0

(8.2.7)

Integrating equation (8.2.6) with respect to η and using the first and second condition of equation (8.2.7), the axial velocity is obtained as follows: u= where k 2 =

1 . K

𝜕α 𝜕p cosh(kη) 1 (1 + λ1α α ) ( − 1) 2 𝜕t 𝜕ξ cosh(kh) k

(8.2.8)

8.2 A numerical study of a generalized Maxwell fluid through a porous medium

| 307

h

The volumetric flow rate is defined as Q = ∫0 udη, which, by virtue of equation (8.2.8), reduces to Q=

1 𝜕α 𝜕p (1 + λ1α α ) (tanh(kh) − kh) 3 𝜕t 𝜕ξ k

(8.2.9)

The transformations between the wave and the laboratory frames, in dimensionless form, are given by x = ξ − t,

y = η,

U = u − 1,

V =v

(8.2.10)

where the left-hand side parameters are in the wave frame and the right-hand side parameters are in the laboratory frame. Using the transformations defined in equation (8.2.10), it follows that equation (8.2.3) can be reduced to h = 1 − ϕ cos2 (πx).

(8.2.11)

The volumetric flow rate in the wave frame is given by h

h

q = ∫ Udy = ∫(u − 1)dη 0

(8.2.12)

0

which, on integration, yields q = Q − h.

(8.2.13)

Averaging the volumetric flow rate along one time period gives 1

1

0

0

Q̄ = ∫ Qdt = ∫(q + h)dt.

(8.2.14)

Subsequent integration, yields ϕ Q̄ = q + 1 − . 2

(8.2.15)

Equation (8.2.9), in view of equation (8.2.15), gives ϕ 3 𝜕α 𝜕p 1 𝜕p 1 k (Q̄ + h − 1 + 2 ) = . ( )+ α 𝜕t α 𝜕x λ1 𝜕x λ1α tanh(kh) − kh

(8.2.16)

Using equations (8.2.8) and equations (8.2.16), the stream function (ψ) in the wave 𝜕ψ frame (U = 𝜕y ) is obtained as ψ = [(

Q̄ + h − 1 +

ϕ 2

tanh(kh) − kh

)(

sinh(ky) − ky) − y]. cosh(ky)

(8.2.17)

It is evident from equation (8.2.17) that the stream function is independent of fractional parameter and relaxation time.

308 | 8 Non-Newtonian generalized fluid and their applications 8.2.2 HPM solutions Equation (8.2.16) can be simplified to yield 1 A f =− α λ1α λ1

Dαt f + where f (x, t) =

𝜕p 𝜕x

and A = −

ϕ

̄ k 3 (Q+h−1+ ) 2 , tanh(kh)−kh

(8.2.18)

with the initial condition

f (x, 0) = 0.

(8.2.19)

Using the HPM, we construct the following homotopy: ̂ Dαt f = −q[

1 A f + α ]. λ1α λ1

(8.2.20)

̂ to expand the solution: Furthermore following, we use the homotopy parameter “q” f = f0 + qf̂ 1 + q̂ 2 f2 + q̂ 3 f3 + q̂ 4 f4 + ⋅ ⋅ ⋅ .

(8.2.21)

When q̂ → 1, equation (8.2.21) becomes the approximate solution of equation (8.2.18). Substituting equation (8.2.21) in equation (8.2.20) and comparing the like powers of q,̂ we obtain the following set of fractional partial differential equations (FPDE): q̂ 0 : Dαt f0 = 0

(8.2.22)

q̂ 1 : Dαt f1 = −

(8.2.23)

1 A f − λ1α 0 λ1α 1 q̂ 2 : Dαt f2 = − α f1 λ1 1 3 α q̂ : Dt f3 = − α f2 λ1 1 q̂ 4 : Dαt f4 = − α f3 λ1

(8.2.24) (8.2.25) (8.2.26)

and so on. The method is based on applying the operator Jtα (the inverse operator of the Caputo derivative Dαt ) on both sides of equations (8.2.22)–(8.2.26), which leads to f0 = 0 f1 = − f2 = −

α

A t λ1α Γ(α + 1)

A t 2α λ12α Γ(2α + 1)

(8.2.27) (8.2.28) (8.2.29)

8.2 A numerical study of a generalized Maxwell fluid through a porous medium

f3 = − f4 =

A t 3α 3α Γ(3α + 1) λ1

A t 4α . λ14α Γ(4α + 1)

| 309

(8.2.30) (8.2.31)

Thus, the exact solution may be obtained as ∞

∞

r=0

r=0 α

r

f (x, t) = ∑ fr = ∑ [ψ(α)]

t rα Γ(rα + 1)

= Eα (ψ(α)t )

(8.2.32)

where (−1)γ λArα , r 1 [ψ(α)] = { 0,

r≥1 r=0

γ

t and Eα = ∑∞ r=0 Γ(rα+1) , (α > 0) is the Mittag-Leffler function in one parameter. The pressure difference across one wavelength (△p) and the friction force across one wavelength (F) are defined by the following integrals: 1

△p = ∫ 0

𝜕p dx 𝜕x

1

F = ∫ h(− 0

𝜕p )dx. 𝜕x

(8.2.33) (8.2.34)

8.2.3 Numerical results and discussion Numerical results have been presented in this section to study the effects of fractional viscoelastic behavior on oscillating peristaltic flow through a uniform porous medium. Mathematica software is used to plot all the figures and 100 terms of the Mittag-Leffler function have been employed in the computations. All figures have been plotted based on equations (8.2.33)–(8.2.34) and for the integrations, Simpson’s one-third rule has been implemented. The graphical plots are presented for the effects of relevant values of the control parameters, i. e., the relaxation time (λ1 ), fractional parameter (α), and permeability parameter (K) in Figures 8.2.2–8.2.7. Figures 8.2.2, 8.2.3, and 8.2.4 illustrate the variation of the volumetric flow rate with pressure gradient for different values of the relaxation time (λ1 ), fractional parameter (α), and permeability parameter (K) based on equation (8.2.33). Different regions on the basis of the values of pressure gradient have been examined in this study. The region for p > 0 is the entire pumping region, the region for △p = 0 is the free pumping region, and the region for △p < 0 is the co-pumping region.

310 | 8 Non-Newtonian generalized fluid and their applications

Figure 8.2.2: Pressure versus averaged flow rate for various values of λ1 at t = 1, α = 1/4, K = 0.1, φ = 0.5.

Figure 8.2.3: Pressure versus averaged flow rate for various values of α at t = 1, λ1 = 1, K = 0.1, φ = 0.5.

Figure 8.2.2 shows that in the entire pumping region, the volumetric flow rate as well as pressure decrease with an increase in magnitude of relaxation time; a similar response is computed for the free pumping region. However, in the co-pumping region, both volumetric flow rate and pressure increase with a rise in magnitude of relaxation time. Inspection of Figure 8.2.2 further reveals that for a Newtonian fluid (λ1 → 0), the magnitude of the pressure is greater than that for the Maxwell fluid in the pumping region whereas it is markedly less in the co-pumping region. Figure 8.2.3 indicates that both volumetric flow rate and pressure are enhanced, both in the pumping region and also in the free pumping region by increasing the magnitude of the fractional parameter. However, the opposite effect is computed for the co-pumping region where

8.2 A numerical study of a generalized Maxwell fluid through a porous medium

| 311

Figure 8.2.4: Pressure versus averaged flow rate for various values of K at t = 1, λ1 = 1, α = 1/4, φ = 0.5.

Figure 8.2.5: Frictional force versus averaged flow rate for various values of λ1 at t = 1, α = 1/4, K = 0.1, φ = 0.5.

both volumetric flow rate and pressure are depressed with increasing magnitude of the fractional parameter. It is further observed that for a viscoelastic fluid based on the classical Maxwell model (α= 1), the pressure is greater than that for viscoelastic fluids with the fractional Maxwell model in the pumping region; the reverse trend is apparent in the co-pumping region. Figure 8.2.4 shows that the pressure diminishes in the pumping region at a critical value of volumetric flow rate and thereafter it increases in the pumping, free pumping, and co-pumping regions with increasing magnitude of permeability parameter. Increasing permeability which corresponds to progressively lesser solid fibers in the porous medium serves to reduce the Darcy resistance. This significantly influences pressures.

312 | 8 Non-Newtonian generalized fluid and their applications

Figure 8.2.6: Frictional force versus averaged flow rate for various values of α at t = 1, λ1 = 0.1, K = 0.1, φ = 0.5.

Figure 8.2.7: Frictional force versus averaged flow rate for various values of K at t = 1, α = 1/4, λ1 = 1, φ = 0.5.

Frictional force (F) in the case of the oscillating flow of generalized Maxwell fluids is calculated over one wave period in terms of the averaged volumetric flow rate. Figures 8.2.5, 8.2.6, and 8.2.7 show the variation of frictional force with averaged flow rate for different values of the pertinent parameters based on equation (8.2.34). Figure 8.2.5 illustrates that the frictional force is enhanced with increasing relaxation time at a certain value of volumetric flow rate; following this the opposite behavior is observed, i. e., there is a subsequent decrease in frictional force with increasing relaxation time. From Figure 8.2.6, it is observed that the impact of fractional parameter on frictional force is similar in a quantitative sense to that for pressure but opposite in a qualitative sense. The influence of the permeability parameter on frictional force is shown in Figure 8.2.7. This figure indicates that the magnitude of frictional force in-

8.3 Viscoelastic fluid with fractional derivative models | 313

creases in a very small interval of averaged flow rate and then reduces with increasing magnitude of the permeability parameter. Permeability therefore significantly influences not only the pressure distributions (as computed in Figure 8.2.4) but also the frictional force in the conduit.

8.2.4 Conclusions The influence of viscoelastic behavior, fractional characteristics, and permeability on the flow patterns in two-dimensional oscillating peristaltic non-Newtonian flow in a porous medium, have been studied analytically and numerically. The effects of the key physical parameters on volumetric flow rate, pressure, frictional force, and trapping have been examined. An important observation of the present simulations is that the pressure in the conduit is reduced by increasing the magnitude of relaxation time in the pumping region, whereas it is constant in the free pumping region and enhanced in the co-pumping region. The effect of the fractional parameter on the pressure is opposite to that for relaxation time, in all pumping regions. The present results also reveal that pressure reduces in the pumping region at a critical value of volumetric flow rate and subsequently increases in all three regions with a rise in the permeability parameter. The study further shows that the effects of all pertinent parameters on frictional force are quantitatively similar but qualitatively opposite to that of pressure. Furthermore, the present computations show that trapping can be reduced by increasing the amplitude ratio whereas it may be enhanced with increasing the magnitude of the permeability parameter.

8.3 Viscoelastic fluid with fractional derivative models Linear viscoelasticity is certainly the field of the most extensive applications of fractional calculus. This is due to the nonlocal character of fractional derivatives, leading to their ability to model more adequately phenomena with memory. On the other hand, the same nonlocality property makes it difficult to design fast and accurate numerical techniques for fractional order differential equations [50, 52]. Since many industrial and natural processes can be modeled as viscoelastic flows: from polymer extrusion to processes in geophysics, such numerical algorithms are essential [4]. Viscoelastic flows are intensively modeled in literature under different constitutive equations and in various media. In this section, we consider an initial-boundary value problem for the velocity distribution of a viscoelastic flow with generalized fractional Oldroyd-B constitutive model. Let Ω be a bounded rectangular domain in ℝd , d = 1, 2, with boundary 𝜕Ω. Let T > 0 be a fixed time, and 0 < α < β < 1, a, b ≥ 0, μ > 0, be given constant parameters. Applying the generalized fractional Oldroyd-B constitutive model leads to the

314 | 8 Non-Newtonian generalized fluid and their applications following initial-boundary value problem for the flow velocity u(x, t): β

(1 + aDαt )ut = μ(1 + bDt )Δu + F(x, t), u(x, t) = 0,

x ∈ Ω, t ∈ (0, T],

x ∈ 𝜕Ω, t ∈ [0, T],

u(x, 0) = f (x),

x ∈ Ω,

(8.3.1) γ

where Δ is the Laplacian acting on spatial variables, ut = 𝜕u/𝜕t and Dt is the Riemann– Liouville fractional derivative of order γ: γ

t

Dt f (t) :=

d f (τ) 1 dτ, ∫ Γ(1 − γ) dt (t − τ)γ

0 < γ < 1.

0

If a ≠ 0, an additional initial condition ut (x, 0) < ∞ is assumed. The generalized Oldroyd-B model equation (8.3.1) encompasses a large class of fluids [93]: Newtonian fluid(a = b = 0), generalized second grade fluid(a = 0, b > 0), fractional Maxwell model(b = 0, a > 0).

8.3.1 Preliminaries t

Denote by ∗ by the classical convolution: t

t

(f ∗ g)(t) = ∫ f (t − τ)g(τ)dτ, 0

which we use with respect to time variable (with respect to spatial variables we use nonclassical convolutions, defined in Section 8.3.4). Let ωα (t) :=

t α−1 , Γ(α)

α > 0.

(8.3.2)

Denote the Laplace transform of a function f (t) by ̂f to ℒ{f }. Then ℒ{ωα }(s) = s , −α

α > 0.

The Laplace transform for the Riemann–Liouville fractional differential operator Dαt with 0 < α < 1 is given by t α α + ℒ{Dt f }(s) = s ̂f (s) − (ω1−α ∗ f )(0 ),

and thus α α ℒ{Dt f }(s) = s ̂f (s),

if f (0) < ∞.

(8.3.3)

8.3 Viscoelastic fluid with fractional derivative models | 315

Denote by Eα,β (z) the Mittag-Leffler function: zk , Γ(αk + β) k=0 ∞

Eα,β (z) = ∑

α > 0, β ∈ ℝ, z ∈ ℂ.

The following identity is satisfied: ℒ{t

β−1

Eα,β (−λt α )}(s) =

sα−β , sα + λ

t > 0, λ ∈ ℝ.

(8.3.4)

Lemma 8.3.1. Let a ≠ 0, α ∈ (0, 1), t ∈ (0, T]. The ordinary fractional differential equation (1 + aDαt )y(t) = f (t),

y(0) < ∞,

has a unique solution given by t

y(t) = hα (a, t) ∗ f (t), where hα (a, t) := a−1 t α−1 Eα,α (−a−1 t α ).

(8.3.5)

Proof. The assertion follows applying Laplace transform to the equation and using properties (8.3.3) and (8.3.4). Note that, by applying Lemma 8.3.1 to the governing equation in (8.3.1) and then integrating both sides (or applying only the second step if a = 0), we can recast problem (8.3.1) into a Volterra integral equation: t

u(x, t) = f (x) + ∫ κ(t − τ)Δu(x, τ)dτ + F1 (x, t), 0

where the kernel κ(t) is given by μ(1 + bω1−β (t)), κ(t) = { t μhα (a, t) ∗ (1 + bω1−β (t)),

if a = 0, if a ≠ 0,

and the function F1 (x, t) is t

∫0 F(x, τ)dτ,

F1 (x, t) = {

t

τ

∫0 hα (a, τ) ∗ F(x, τ)dτ,

if a = 0, if a ≠ 0.

316 | 8 Non-Newtonian generalized fluid and their applications 8.3.2 Eigenfunction expansion of the solution and properties of the time-dependent components Let {λn }n∈N and {φn (x)}n∈N be the Dirichlet eigenvalues and eigenfunctions of −Δ on the domain Ω, and let 0 < λ1 < λ2 < ⋅ ⋅ ⋅. Denote by (⋅, ⋅) the inner product in L2 (Ω). Applying eigenfunction decomposition, the solution has the form ∞

u(x, t) = ∑ un (t)φn (x), n=1

where the functions un (t) satisfy the following ordinary differential equation: β

(1 + aDαt )un (t) = −λn μ(1 + bDt )un (t) + Fn (t), un (0) = fn , un (0) = fn ,

if a = 0,

un (0) < ∞,

if a ≠ 0,

(8.3.6)

where fn = (f , φn ), Fn (t) = (F(⋅, t), φn ). We solve this problem by applying Laplace transform and obtain the formal eigenfunction expansion of the solution: ∞

∞

t

n=1

n=1

0

u(x, t) = ∑ fn Gn (t)φn (x) + ∑ (∫ Hn (t − τ)Fn (τ)dτ)φn (x),

(8.3.7)

where the function Gn (t) and Hn (t) are defined by their Laplace transforms as follows: 1 + asα , s(1 + asα ) + μλn (+bsβ ) 1 ̂n (s) = H . α s(1 + as ) + μλn (+bsβ ) ̂ (s) = G n

(8.3.8) (8.3.9)

To prove that the series (8.3.7) is convergent we need estimates for the time-dependent components Gn (t) and Hn (t). Since in some aspects there are essential differences between the two cases a = 0 (parabolic equation) and a ≠ 0 (nonparabolic), we consider them separately. Denote by Σθ the sector Σθ := {s ∈ ℂ; s ≠ 0, |arg s| < θ}. For ρ > 0 and θ ∈ (0, π) denote by Γρ,θ the contour Γρ,θ := {re−iθ : r ≥ ρ} ∪ {ρeiψ : |ψ| ≤ θ} ∪ {reiθ : r ≥ ρ}, which is oriented counterclockwise.

8.3 Viscoelastic fluid with fractional derivative models | 317

Lemma 8.3.2. Let a ≠ 0, ρ > 0, φ0 = π/(α + 1) and fix φ ∈ (π/2, φ0 ). Then for any n ∈ ℕ ̂ (s) has no poles in the sector Σ and there hold the estimates the function G φ0 n ̂ (s)| ≤ C|s|−1 , |G n

̂ (s)| ≤ C(|s|−β + a|s|α−β ), |λn G n

s ∈ Σφ .

(8.3.10)

̂ (s) by d (s). Then Proof. Let s0 = reiθ , r > 0, 0 < θ ≤ φ0 . Denote the denominator of G n n ℑ(dn (s0 )) = r sin θ + ar α+1 sin(α + 1)θ + μλn br β sin βθ > 0, ̂ (s) lie outside the and thus s0 cannot be a zero of dn (s). This means that all poles of G n sector Σφ0 . For s ∈ Σφ define the function g(s) :=

s(1 + asα ) . μ(1 + bsβ )

(8.3.11)

Form (8.3.8) and (8.3.11), we get the representations: ̂ (s) = G n

g(s) , s(g(s) + λn )

̂ (s) = g(s)( 1 − G ̂ (s)). λn G n n s

(8.3.12)

We prove that if s ∈ Σφ then g(s) ∈ Σπ−δ for some δ ∈ (0, π/2). Let s = reiψ , |ψ| < φ, r > 0. Then g(s) = =

reiψ (1 + ar α eiαψ ) reiψ (1 + ar α eiαψ )(1 + br β e−iβψ ) = μ(1 + br β eiβψ ) μ(1 + br β eiβψ )(1 + br β e−iβψ )

1 reiψ + ar α+1 ei(α+1)ψ + br β+1 ei(1−β)ψ + abr α+β+1 ei(α+1−β)ψ , μ (1 + br β cos(βψ))2 + (br β sin(βψ))2

and by noting 0 < α < β < 1 we obtain arg(g(s)) ≤ (α + 1)ψ. Hence g(s) ∈ Σπ−δ with δ = π − (α + 1)ψ, i. e., δ ∈ (0, π/2). Based on this, we prove that for any real constant c > 0, g(s) ≤ C, g(s) + c

s ∈ Σφ ,

(8.3.13)

which together with the first identity in (8.3.12) gives the first estimate in (8.3.10). Indeed, the elementary inequality x2 + 2ax + 1 ≥ 1 − a2 for any x, a ∈ ℝ implies 1 + 2x cos θ + x2 ≥ sin2 θ,

x ∈ ℝ.

Since g(s) ∈ Σπ−δ , i. e., g(s) = reiθ , r > 0, |θ| < π − δ, using (8.3.14) it follows g(s) 2 r2 1 ≤ , = 2 r + 2cr cos θ + c2 sin2 θ g(s) + c which implies (8.3.13) with C = (sin δ)−1 .

(8.3.14)

318 | 8 Non-Newtonian generalized fluid and their applications Further, the first estimate in (8.3.10) together with (8.3.12) gives α ̂ (s)| ≤ C g(s) = C 1 + as . |λn G n s μ(1 + bsβ )

(8.3.15)

Since for s = reiψ , ψ ∈ (π/2, π) 2

2 1/2

|1 + bsβ | = ((1 + br β cos(βψ)) + (br β sin(βψ)) )

≥ br β sin(βπ),

inserting this estimate in (8.3.15) gives the second estimate in (8.3.10). This completes the proof of the lemma. Theorem 8.3.1. Let a ≠ 0. The functions Gn (t) and Hn (t), n ∈ ℕ, are continuous for t ≥ 0 and have the following properties: Gn (0) = 1,

Hn (0) = 0,

|Gn (t)| ≤ C,

(8.3.16)

t ≥ 0,

|λn Gn (t)| ≤ C(t

β−1

+ at

β−α−1

(8.3.17) ),

t

Hn (t) = hα (a, t) ∗ Gn (t), t

∫ |λn Hn (τ)|dτ ≤ C,

t > 0, t ≥ 0,

t ≥ 0,

(8.3.18) (8.3.19) (8.3.20)

0

where the function hα (a, t) is defined in (8.3.5) and the constants C do not depend on n and t. Proof. Applying the property of the Laplace transform f (0) = lim ŝf (s) s→+∞

to (8.3.8) and (8.3.9), we obtain (8.3.16). Further, taking the inverse Laplace transform of (8.3.8), we get Gn (t) =

1 + asα 1 ds, ∫ est 2πi s(1 + asα ) + μλn (1 + bsβ )

(8.3.21)

Br

with Br the Bromwich path: Br = {s; ℜs = σ}, where σ > 0 and σ > ℜsj , sj being the singularities of the Laplace transform. According to Lemma 8.3.2, the function under the integral has no poles in the sector Σφ0 , where φ0 = π/(α + 1). Therefore, we can bend the Bromwich path in equation (8.3.21) into the contour Γ := Γ1/t,φ ,

t > 0, φ ∈ (π/2, φ0 ),

8.3 Viscoelastic fluid with fractional derivative models | 319

and prove (8.3.17) by applying the first estimate in (8.3.10) (note that cos φ < 0): |Gn (t)| ≤ C ∫ eℜ(s)t |s|−1 |ds| Γ ∞

φ

1/t

0

≤ C( ∫ ert cos φ r −1 dr + ∫ ecos ψ dψ) ≤ C. Further, applying the second estimate in (8.3.10) we get |λn Gn (t)| ≤ C ∫ eℜ(s)t (|s|−β + a|s|α−β )|ds|, Γ

and noting that for γ ∈ (0, 1) ∞

∫e

ℜ(s)t

|s| |ds| ≤ C( ∫ e −γ

Γ

rt cos φ −γ

φ

r dr + ∫ ecos ψ t γ−1 dψ) ≤ Ct γ−1 , 0

1/t

we obtain (8.3.18). Based on the Laplace transforms of Gn (t) and Hn (t), (8.3.8) and (8.3.9), and the property (8.3.3) it follows that (1 + aDαt )Hn (t) = Gn (t). Then, since Hn (0) = 0, Lemma 8.3.1 implies (8.3.19). Now, from (8.3.19) and the Young inequality for the classical convolution we get t

t

t

∫ |λn Hn (τ)|dτ ≤ ∫ |hα (a, τ)|dτ ∫ |λn Gn (τ)|dτ 0

0

0

and, inserting (8.3.18) and the inequality |hα (a, t)| ≤ Ct α−1 , which is implied by the boundedness of the Mittag-Leffler function, we establish the last estimate (8.3.20). For the case a = 0, properties of the time-dependent components Gn (t) and Hn (t) (in this case Gn (t) = Hn (t)), implies that estimates (8.3.17), (8.3.18), and (8.3.20) hold for a = 0. In addition, the following result is proven. Theorem 8.3.2. Assume a = 0. Then the functions Gn (t) and Hn (t) in the eigenfunction expansion (8.3.7) have the representation ∞

Gn (t) = Hn (t) = ∫ e−rt Kn (r)dr,

n ∈ ℕ,

(8.3.22)

0

where Kn (r) =

μλn r β sin βπ b . π (−r + μλn br β cos βπ + μλn )2 + (μλn br β sin βπ)2

(8.3.23)

320 | 8 Non-Newtonian generalized fluid and their applications Proof. The function under the Laplace inverse integral (8.3.21) has a branch point 0, so we cut off the negative part of the real axis. When a = 0, this function has no poles in the main sheet of the Riemann surface including its boundaries on the cut, since ℑ(s + μλn (1 + bsβ )) ≠ 0. So, we can bend the Bromwich path into the Hankel path, which starts from −∞ along the lower side of the negative real axis, encircles the origin counterclockwise and ends at −∞ along the upper side of the negative real axis, and obtain from (8.3.21) the representation (8.3.22). Based on the properties of the time-dependent components we prove that the obtained formal solution (8.3.7) of problem (8.3.1) is a continuous function under some conditions. For the sake of brevity, we consider here only the 1D case. The proof in 2D case is analogous. Consider the one-dimensional problem (8.3.1) on Δ = (0, 1) × (0, T): β

(1 + aDαt )ut = μ(1 + bDt )uxx + F(x, t), u(0, t) = u(1, t) = 0, u(x, 0) = f (x),

ut (x, 0) < ∞.

(8.3.24)

Then the corresponding eigenvalues and eigenfunctions are respectively λn = n2 π 2 and φn (x) = √2 sin(nπx), n ∈ ℕ. Theorem 8.3.3. Let f (x) ∈ C 2 ([0, 1]), f (0) = f (1) = 0, F(x, t) ∈ C(Δ). Then the function u(x, t) defined in (8.3.7) satisfies u ∈ C(Δ). Proof. First, note that φn (x) are bounded functions on [0, 1]. Since f (x) ∈ C 2 ([0, 1]) and f (0) = f (1) = 0, then after integration by parts in the identity fn = (f , φn ) we get |fn | ≤ Cn−2 . This together with the estimate (8.3.17) implies that the first series in (8.3.7) is uniformly convergent on Δ. Analogously, the uniform convergence of the second series in (8.3.7) is implied by the estimate (8.3.20) and the boundedness of |F(x, t)| on Δ. Since all terms in the series are continuous functions on Δ, their sum is a continuous function on Δ. Remark 8.3.1. In the homogeneous case F ≡ 0, with f satisfying assumptions of Theorem 8.3.3, we can prove better regularity of the solution. Indeed, by differentiating termwise the series (8.3.7) and using similar argument as in the proof of Theorem 8.3.3, together with estimates (8.3.18), it follows uxx ∈ C(Δ). To prove this in the inhomogeneous case, we would need additional assumptions on F. Remark 8.3.2. The obtained estimates for the functions Gn (t) and Hn (t) are useful for further study of solution regularity in different settings, e. g., in Sobolev spaces for the fractional diffusion equation.

8.3 Viscoelastic fluid with fractional derivative models | 321

8.3.3 Finite difference approximation In this section, we construct difference schemes [58] for the one-dimensional problem (8.3.24). Assume x ∈ [0, 1], t ∈ [0, T]. Let M and N be the number of time and space nodes, and τ = T/M, h = 1/N be the time and space steps, respectively. Let xj := jh, tk := kτ, j = 0, 1, . . . , N, k = 0, 1, . . . , M. We approximate problem (8.3.24) by an implicit and an explicit finite difference schemes, based on the Grünwald–Letnikov approximation of the Riemann–Liouville fractional derivative [20]: k α k (Dαt u)j = τ−α ∑ (−1)m ( )uk−m + O(τ), j m m=0

(8.3.25)

and the usual approximations for the integer order derivatives uxx and ut : δx2 ulj =

ulj−1 − 2ulj + ulj+1 h2

,

δt+ ulj =

l ul+1 j − uj

τ

,

δt− ulj =

ulj − ul−1 j τ

.

In this way, we obtain δt ukj +

a k α b k β 2 k−m δx uj ), = μ(δx2 ukj + β ∑ wm ∑ wm δt uk−m j α τ m=0 τ m=0

(8.3.26)

α where wm = (−1)m (mα ) and δt ulj = δt+ ulj for the explicit scheme and δt ulj = δt− ulj for the implicit scheme. The initial and boundary conditions are discretized in a standard way (nonhomogeneous boundary conditions can also be considered). In the case a = 0, the explicit and implicit schemes obtained from equation (8.3.26). It should be possible to prove convergence and stability results for the schemes equation (8.3.26) in the general case a ≠ 0. However, since this is out of the scope of the paper, we leave it as an open problem for future research. It is proven that for the explicit scheme the following stability condition is required:

4μτ(1 + bτ−β ) ≤ h2 .

(8.3.27)

In order to satisfy this condition, for reasonably fine mesh in space, we should take extremely small time steps if β increases. We considered some test problems and found out that difficulties in numerical implementation of the explicit scheme appear for β ≥ 0.3. In addition, we encountered similar problems also for a ≠ 0. The implicit scheme is unconditionally stable. However, its numerical implementation is based on solving a system of algebraic equation for each time step, which again leads to problems when using fine meshes, especially for large times T or 2D problems. The above mentioned difficulties motivates us to look also for nonstandard methods for numerical solution of problem (8.3.1).

322 | 8 Non-Newtonian generalized fluid and their applications 8.3.4 Duhamel-type representation of the solution Applying the convolutional calculus of Dimovski [5], in this section we obtain a Duhamel-type representation of the solution of the one-dimensional problem (8.3.24). Basic in a convolutional calculus is the notion of convolution. Definition 8.3.1. Let L : X → X be a linear operator defined on a linear space X. A bilinear, commutative, and associative operation ∗ : X × X → X is said to be a convolution of the operator L if L(f ∗ g) = (Lf ) ∗ g

for any f , g ∈ X.

We define in the space C([0, 1]) of continuous functions on [0, 1] the operator L, which is right inverse of the operator D = d2 /dx2 and satisfies (Lf )(0) = (Lf )(1) = 0. It is given explicitly by x

1

Lf (x) = ∫(x − ξ )f (ξ )dξ − x ∫(1 − ξ )f (ξ )dξ . 0

0

The following operation is a convolution of the operator L: 1

ξ

0

x

1 (f ∗ g)(x) = − ∫(∫ f (ξ + x − η)g(η)dη 2 x

ξ

− ∫ f (|ξ − x − η|)g(|η|)sgn((ξ − x − η)η)dη)dξ .

(8.3.28)

−x

Moreover, x

Lf = {x} ∗ f .

(8.3.29)

The following properties hold true: x

x

x

D(f ∗ g) = (Df ) ∗ g + D((Ff ) ∗ g),

x

x

F(f ∗ g) = F((Ff ) ∗ g),

(8.3.30)

where F is the defining projector, F = I − LD. It incorporates the boundary conditions and has the explicit form Ff (x) = f (0)(1 − x) + f (1)x. Denote x

x

̃ g := D(f ∗ g). f∗

(8.3.31)

Theorem 8.3.4. The operation 1

1 d ̃ g)(x) = − (f ∗ (∫ f (1 + x − η)g(η)dη 2 dx x

1

x

+ ∫ f (|1 − x − η|)g(|η|)sgn((1 − x − η)η)dη) −x

(8.3.32)

8.3 Viscoelastic fluid with fractional derivative models | 323 x

̃f is a convolution of the operator L in C 1 ([0, 1]) such that the representation Lf = {Lx}∗ holds. Moreover, for m, n ∈ ℕ x 0, ̃ sin(mπx) = { sin(nπx)∗ sin(nπx), (−1)n−1 nπ 2

m ≠ n, m = n.

(8.3.33)

Proof. Expression (8.3.32) is obtained by differentiation of (8.3.28). Property (8.3.33) can be proven directly. The rest follows the properties of the original convolution (8.3.28). Detailed proof is a matter of direct but tedious check, so it will be omitted. x

t

̃ and on the classical convolution ∗, we define a bivariBased on the convolution ∗ x,t

̃ of two functions f (x, t) and g(x, t): ate convolution ∗ t

x,t

x

̃ g)(x, t) = ∫ f (x, t − τ)∗ ̃ g(x, τ)dτ. (f ∗

(8.3.34)

0

The Duhamel-type representation of the solution is given in the following. Theorem 8.3.5. If f (x) ∈ C 2 ([0, 1]), f (0) = f (1) = 0 and Ft (x, t) ∈ C(Δ), then the solution of problem (8.3.24) has the following representation: x

̃ f + u(x, t) = U ∗

𝜕 x,t ̃ F), (V ∗ 𝜕t

(8.3.35)

where U(x, t) is a particular solutions of (8.3.24) with f (x) = (x 3 − x)/6 and F ≡ 0; V(x, t) is a particular solutions of (8.3.24) with f ≡ 0, F(x, t) = x. The functions U(x, t) and V(x, t) have the following series expansions: U(x, t) =

2 ∞ (−1)n G (t) sin(nπx), ∑ π 3 n=1 n3 n t

2 ∞ (−1)n−1 V(x, t) = ∑ (∫ Hn (τ)dτ) sin(nπx). π n=1 n

(8.3.36)

0

Proof. Inserting the Fourier coefficients of the functions (x 3 − x)/6 and x in (8.3.7), we obtain the eigenfunction expansions (8.3.36). By the estimates (8.3.17) and (8.3.20), it is clear that the series in (8.3.36) are uniformly convergent, as well as the series after termwise differentiation with respect to x. Therefore, Ux and Vx are continuous functions on [0, 1] × [0, T] and the expansion in (8.3.35) is well-defined. We prove that formula (8.3.35) is equivalent to (8.3.7). Indeed, inserting the eigenfunction expansions of arbitrary initial function f (x) and source function F(x, t), and using the properties t

t

(f ∗ g) = f ∗ g + f (0)g(t),

(8.3.37)

324 | 8 Non-Newtonian generalized fluid and their applications (8.3.33) and the separability property of the bivariate convolution x,t

x

t

̃ (g1 (x)g2 (t)) = (f1 (x)∗ ̃ g1 (x))(f2 (t) ∗ g2 (t)), (f1 (x)f2 (t)) ∗ we get (8.3.7). Remark 8.3.3. Note that the functions {(x 3 −x)/6} and {x}, appearing in Theorem 8.3.5, play special role in our convolutional approach. Due to the representation (8.3.29) of the operator L as a convolution operator, the function {x} can be identified with the operator L. Since L{x} = {(x 3 − x)/6}, the function {(x 3 − x)/6} can be identified with L2 . Applying properties (8.3.34) and (8.3.37), we can differentiate the convolution product in (8.3.35) and obtain the following. Corollary 8.3.1. Under the assumptions of Theorem 8.3.5, we have x

x,t

̃ f (x) + V ∗ ̃( u(x, t) = U ∗

x 𝜕F ̃ F(x, 0). ) + V∗ 𝜕t

(8.3.38)

8.3.5 Numerical experiments For the numerical implementation of the Duhamel-type representations of the solution equation (8.3.35)/(8.3.38), the particular solutions U(x, t) and V(x, t) can be calculated in advance by an appropriate method. Then the solution u(x, t) of (8.3.24) with arbitrary initial function f (x) and source function F(x, t) is computed according to (8.3.35)/(8.3.38) applying only numerical integration and differentiation. As usual, the right-hand side F(x, t) can incorporate nonhomogeneous boundary conditions. Note that such conditions are often considered with this problem, e. g., when flow between two moving parallel plates is modeled. Indeed, the solution of the problem with nonhomogeneous boundary conditions β

(1 + aDαt )ut = μ(1 + bDt )uxx , u(0, t) = ϕ(t), u(x, 0) = 0,

u(1, t) = ψ(t),

ut (x, 0) < ∞

is given by u = v + ϕ(t)(1 − x) + ψ(t)x, where v is a solution of (8.3.24) with f ≡ 0 and F(x, t) = −(1 + aDαt )(ϕ (t)(1 − x) + ψ (t)x).

8.3 Viscoelastic fluid with fractional derivative models | 325

For numerical implementations, the function F(x, t) can be computed using the Grünwald–Letnikov approximation of the Riemann–Liouville fractional derivative (8.3.25). In order to compare the proposed method for numerical computation of the solution to a standard finite difference method, and to visualize the solution in some practically interesting cases, we present in this section some numerical examples. We carried out numerical experiments only for the case a = 0 because of two reasons. First, the existing numerical studies for (8.3.1) are mostly concerned with this case. Second, for a = 0 it is easy to compute the kernels U and V, using their eigenfunction expansions, which makes our method self-contained. Expansions (8.3.36) and Theorem 8.3.2 imply that the particular solutions of 1D problem (8.3.24) with a = 0 are given by U(x, t) = V(x, t) =

2 ∞ (−1)n G (t) sin(nπx), ∑ π 3 n=1 n3 n

2 ∞ (−1)n−1 Sn (t) sin(nπx), ∑ π n=1 n

(8.3.39) (8.3.40)

where ∞

∞

Gn (t) = ∫ e

−rt

Kn (r)dr,

Sn (t) = ∫

0

0

1 − e−rt Kn (r)dr, r

2 2

with function Kn (r) given in (8.3.23), where λn = n π . In the numerical tests in this section we use representations equation (8.3.39) and equation (8.3.40) for the numerical computation of U(x, t) and V(x, t). To get an impression of the behavior of the time-dependent components in the series equation (8.3.39) and equation (8.3.40), on Figures 8.3.1–8.3.2, we give some plots of Gn (t) and Sn (t). First, consider a test problem with known exact solution in closed form and compare our method, based on formula (8.3.35), with the explicit finite difference method from. We choose the explicit scheme because it is easy for numerical implementation; a feature which is also characteristic for our method. Example 8.3.1. Consider the following problem: β

ut = (1 + Dt )uxx + F(x, t),

u(0, t) = u(1, t) = u(x, 0) = 0.

(8.3.41)

To compute its solution based on representation equation (8.3.38), we compute first the kernel V(x, t) using equation (8.3.40); see Figure 8.3.3. Recall that V is a solution of equation (8.3.41) with F(x, t) = x. In Figure 8.3.4, the solution of problem equation (8.3.41) is given for F(x, t) = (2t + π 2 t 2 + 2π 2 ω3−β (t)) sin(πx),

(8.3.42)

326 | 8 Non-Newtonian generalized fluid and their applications

Figure 8.3.1: Plot of the function Gn (t) for β = 0.5, μ = b = 1, n = 1, 2, 3.

Figure 8.3.2: Plot of the function Sn (t) for β = 0.5, μ = b = 1, n = 1, 2, 3.

where ω3−β (t) is defined by (8.3.2). For this special choice of the F problem equation (8.3.41) has an exact solution: uexact (x, t) = t 2 sin(πx). We make numerical tests to compare two methods for solving equation (8.3.41): the convolutional calculus method (CCM) (representation (8.3.38)) and the explicit finite difference method (FDM). We choose β = 0.2 because of the stability requirement (8.3.27) on the explicit scheme. Note that our method is applicable in the whole range β ∈ (0, 1). In Figure 8.3.5, results of the comparisons are presented. The errors calculated by the formula max |u(xk , t) − uexact (xk , t)|

0≤k≤N

8.3 Viscoelastic fluid with fractional derivative models | 327

Figure 8.3.3: The kernel V (x, t) of problem (8.3.41) with β = 0.2.

Figure 8.3.4: The solution of problem (8.3.41) with F (x, t) given by (8.3.42).

are given and the times necessary to achieve the corresponding accuracy. It is seen that in order to achieve similar accuracy for this test problem, we need 9 × 103 more time if we use FDM, compared to CCM. (The time necessary for computation of the kernel V(x, t) is not included, since this is usually done in advance. This time is less than 10 min.) Example 8.3.2. Consider the problem β

ut = (1 + Dt )uxx ,

u(0, t) = u(1, t) = 0,

u(x, 0) = f (x).

(8.3.43)

It models velocity distribution of a flow with nonzero initial velocity f (x) and situated between two parallel plates at rest (with a no-split condition). In

328 | 8 Non-Newtonian generalized fluid and their applications

Figure 8.3.5: Comparisons of CCM with FDM for the test problem from Example 8.3.1 (N-number of space nodes).

Figure 8.3.6: The kernel U(x, t) of problem equation (8.3.43) with β = 0.25.

Figures 8.3.6–8.3.7, we give the graphs of two solutions of equation (8.3.43) with β = 0.25: the left is of the kernel U(x, t) (corresponds to initial function f (x) = (x 3 −x)/6, computed using equation (8.3.39)), the right is with initial function f (x) = sin(2πx) and is computed based on representation (8.3.38). A comparison with the exact solution for this case uexact (x, t) = G2 (t) sin(2πx), gives error of the order of 10−4 for M = N = 100. The relatively short time intervals in the figures are chosen because of the fast decay of the solution. Example 8.3.3. The solution of equation (8.3.43) with f (x) = x(1 − x) are computed for different values of β : β = 0.25, 0.5, and 0.75. In Figures 8.3.8–8.3.9, the graph of the solution is given for β = 0.25 and the decays of the solutions at x = 0.5 are compared for different values of β.

8.3 Viscoelastic fluid with fractional derivative models | 329

Figure 8.3.7: The solution of problem equation (8.3.43) with f (x) = sin(2πx).

Figure 8.3.8: Plot of the solution from Example 8.3.3.

Example 8.3.4. Consider the problem β

ut = (1 + Dt )uxx , u(0, t) = ϕ(t),

u(1, t) = 0,

u(x, 0) = 0.

(8.3.44)

It models the velocity distribution of a flow between two parallel plates, one of which is moving. The flow is initially at rest. In Figures 8.3.10–8.3.11, the graphs of the solution for β = 0.5 are given in two cases: the flow is induced by a linear acceleration (ϕ(t) = t 2 ) or oscillation (ϕ(t) = sin(4πt)) of the moving plate, together with a no-slip

330 | 8 Non-Newtonian generalized fluid and their applications

Figure 8.3.9: Plot of the solution from Example 8.3.3 with x = 0.5.

condition. The solution has the form u = v + ϕ(t)(1 − x), where v is the solution of problem equation (8.3.41) with F(x, t) = −ϕ (t)(1 − x). We calculate v using formula equation (8.3.38). For the computation of the convolution products in the numerical tests of this section, we have used Simpson’s rule for numerical integration (O(h4 )), and central differences for numerical differentiation (O(h2 )). So, the numerical approximation of the convolutions is not optimal and further improvement of the accuracy is possible. We take T ≤ 1 (T ≪ 1 when the solution has a fast decay). However, there are no obstacles in numerical implementation to work with large times, since the solution is computed in each point independently. Also, there are no limitations on the order β ∈ (0, 1) of fractional differentiation.

8.3.6 Two-dimensional problem In this section, we briefly present the application of the convolutional approach to the 2D problem on the unit square Ω = [0, 1] × [0, 1]: β

(1 + aDαt )ut = μ(1 + bDt )(uxx + uyy ) + F(x, y, t),

u(0, y, t) = u(1, y, t) = u(x, 0, t) = u(x, 1, t) = 0, u(x, y, 0) = f (x, y),

ut (x, y, 0) < ∞.

(8.3.45)

To find Duhamel-type representation of the solution, we apply the general scheme for constructing multidimensional convolutional calculi. Define the bivariate convolution

8.3 Viscoelastic fluid with fractional derivative models | 331

Figure 8.3.10: Flow between two parallel plate, one of which is oscillating.

Figure 8.3.11: Flow between two parallel plates, one of which is moving with linear acceleration. x,y

̃ of functions f , g ∈ C(Ω) as a composition of two convolutions (8.3.32): ∗ x,y

̃ g)(x, y) = − (f ∗

1 𝜕 𝜕 K(x, y), 4 𝜕x 𝜕y

where 1 1

K(x, y) = ∫ ∫ f (1 + x − ξ , 1 + y − η)g(ξ , η)dξdη x y

332 | 8 Non-Newtonian generalized fluid and their applications 1 1

+ ∫ ∫ f (1 + x − ξ , |1 − y − η|)g(ξ , |η|)sgn((1 − y − η)η)dξdη x −y 1 1

+ ∫ ∫ f (|1 − x − ξ |, 1 + y − η)g(|ξ |, η)sgn((1 − x − ξ )ξ )dξdη −x y 1 1

+ ∫ ∫ f (|1 − x − ξ |, |1 − y − η|)g(|ξ |, |η|)sgn((1 − x − ξ )(1 − y − η)ξη)dξdη. −x −y

Define also the convolution t

x,y,t

x,y

̃ G(x, y, τ)dτ. ̃ G)(x, y, t) = ∫ F(x, y, t − τ) ∗ (F ∗ 0 4

Theorem 8.3.6. Let 𝜕𝜕xf2(x,y) ∈ C(Ω), f (0, y) = f (1, y) = f (x, 0) = f (x, 1) = 0 and 𝜕y2 Ft (x, y, t) ∈ C(Ω × [0, T]). Then the solution of problem equation (8.3.45) has the representation x,y

̃( u(x, y, t) = U ∗

𝜕4 f 𝜕 x,y,t ̃ F), ) + (V ∗ 2 2 𝜕t 𝜕x 𝜕y

where U(x, y, t) is a particular solution of (8.3.45) with F ≡ 0 and f (x, y) = (x 3 − x)(y3 − y)/36; V(x, y, t) is a particular solution of equation (8.3.45) with f ≡ 0 and F(x, y, t) = xy. Corollary 8.3.2. Under the assumptions of Theorem 8.3.6, we have x,y

̃( u(x, y, t) = U ∗

x,y,t 𝜕F x,y 𝜕4 f ̃ ( )+V∗ ̃ F(x, y, 0). )+V ∗ 2 2 𝜕t 𝜕x 𝜕y

8.3.7 Conclusion In this work, an initial-boundary value problem for the velocity distribution of a viscoelastic flow with generalized fractional Oldroyd-B constitutive model is studied and a Duhamel-type representation of its solution is obtained. The representation is used for the numerical computation of the solution in the particular case of a generalized second grade fluid. Numerical results show that this representation can serve as a basis for an efficient, accurate, and fast numerical technique for solution computation, with the advantage that the solution is calculated in each point independently. The presented technique has the potential for further accuracy improvement and application to multidimensional problems.

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Index ϵ-entropy 263 absorbing set 136 Agmon inequality 143 analytic semigroups theory 258 approximated inertial manifolds 179 attracting set 261 backward solution 271 Bernoulli shifts 286 bipolar fluids 15 bounded domain 93 Bourgain space-time function space 227 Calderon–Zygmond estimate 207 Caratheodory theorem 73 convergence of solution and attractors 211

inertial manifolds 159 infinite topological entropy 281 interpolation inequality 247 inverse problem 293 isomorphism 275 isothermal compressible 8 Kolmogorov’s ϵ-entropy 263 L2 decay rates 195 Leray–Schauder fixed-point principle 259 Lipschitz-continuous homeomorphism 279 low regularity 227 modified Boussinesq approximation 55, 133

decay estimates 191

non-Newtonian fluids 1 non-Newtonian generalized fluid 293 numerical study 304

exponential attractors 119

other related models 9

Fourier transform 197 fractal dimension 136, 263 fractional derivative models 313 Fréchet derivative 152

partial regularity of solutions 191 Plancherel theorem 196

Gagliardo–Nirenberg inequality 213 Galerkin method 44 global attractors 89, 227

singular points 202 skew-symmetric operators 288 spatial chaos 280 spatial complexity of the attractor 280 stationary solution 191 Strichartz-type inequalities 228 symmetric operator 288

Hausdorff dimension 136 Hausdorff–Young inequality 218 Hölder exponent 290 incompressible and isothermal 3 incompressible bipolar non-Newtonian fluids 191 incompressible monopolar non-Newtonian fluids 191 incompressible non-Newtonian fluids 159

https://doi.org/10.1515/9783110549614-010

quasi-linear parabolic equation 244

topological entropy 285 unbounded domain 107 uniformly differentiable 145 weighted Sobolev space 246