Singular Limits in Thermodynamics of Viscous Fluids

Contents

Preface

Notation, Definitions, and Function Spaces

0.1 Notation

0.2 Differential operators

0.3 Function spaces

0.4 Sobolev spaces

0.5 Fourier transform

0.6 Weak convergence of integrable functions

0.7 Non-negative Borel measures

0.8 Parametrized (Young) measures

Fluid Flow Modeling

1.1 Fluids in continuum mechanics

1.2 Balance laws

1.3 Field equations

1.4 Constitutive relations

Weak Solutions, A Priori Estimates

2.1 Weak formulation

2.2 A priori estimates

Existence Theory

3.1 Hypotheses

3.2 Structural properties of constitutive functions

3.3 Main existence result

3.4 Solvability of the approximate system

3.5 Faedo-Galerkin limit

3.6 Artificial diffusion limit

3.7 Vanishing artificial pressure

3.8 Regularity properties of the weak solutions

Asymptotic Analysis – An Introduction

4.1 Scaling and scaled equations

4.2 Low Mach number limits

4.3 Strongly stratified flows

4.4 Acoustic waves

4.5 Acoustic analogies

4.6 Initial data

4.7 A general approach to singular limits for the full Navier- Stokes- Fourier system

Singular Limits – Low Stratification

5.1 Hypotheses and global existence for the primitive system

5.2 Dissipation equation, uniform estimates

5.3 Convergence

5.4 Convergence of the convective term

5.5 Conclusion – main result

Stratified Fluids

6.1 Motivation

6.2 Primitive system

6.3 Asymptotic limit

6.4 Uniform estimates

6.5 Convergence towards the target system

6.6 Analysis of acoustic waves

6.7 Asymptotic limit in entropy balance

Interaction of Acoustic Waves with Boundary

7.1 Problem formulation

7.2 Main result

7.3 Uniform estimates

7.4 Analysis of acoustic waves

7.5 Strong convergence of the velocity field

Problems on Large Domains

8.1 Primitive system

8.2 Uniform estimates

8.3 Acoustic equation

8.4 Regularization and extension to

8.5 Dispersive estimates and time decay of the acoustic waves

8.6 Conclusion – main result

Acoustic Analogies

9.1 Asymptotic analysis and the limit system

9.2 Acoustic equation revisited

9.3 Two-scale convergence

9.4 Lighthill’s acoustic analogy in the low Mach number regime

9.5 Concluding remarks

Appendix

10.1 Mollifiers

10.2 Basic properties of some elliptic operators

10.3 Normal traces

10.4 Singular and weakly singular operators

10.5 The inverse of the div-operator ( Bogovskii’s formula)

10.6 Helmholtz decomposition

10.7 Function spaces of hydrodynamics

10.8 Poincar ´ e type inequalities

10.9 Korn type inequalities

10.10 Estimating

u by means of

and curlxu

10.11 Weak convergence and monotone functions

10.12 Weak convergence and convex functions

10.13 Div-Curl lemma

10.14 Maximal regularity for parabolic equations

10.15 Quasilinear parabolic equations

10.16 Basic properties of the Riesz transform and related operators

10.17 Commutators involving Riesz operators

10.18 Renormalized solutions to the equation of continuity

Bibliographical Remarks

11.1 Fluid flow modeling

11.2 Mathematical theory of weak solutions

11.3 Existence theory

11.4 Analysis of singular limits

11.5 Propagation of acoustic waves

Bibliography

Index