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| Preface | xi | |
| Acknowledgement | xv | |
| Notation, Definitions, and Function Spaces | ||
| 0.1 | Notation | xvii |
| 0.2 | Differential operators | xix |
| 0.3 | Function spaces | xx |
| 0.4 | Sobolev spaces | xxv |
| 0.5 | Fourier transform | xxx |
| 0.6 | Weak convergence of integrable functions | xxxiii |
| 0.7 | Non-negative Borel measures | xxxiv |
| 0.8 | Parametrized (Young) measures | xxxv |
| 1 Fluid Flow Modeling | ||
| 1.1 | Fluids in continuum mechanics | 2 |
| 1.2 | Balance laws | 4 |
| 1.3 | Field equations | 8 |
| 1.3.1 | Conservation of mass | 8 |
| 1.3.2 | Balance of linear momentum | 9 |
| 1.3.3 | Total energy | 11 |
| 1.3.4 | Entropy | 12 |
| 1.4 | Constitutive relations | 13 |
| 1.4.1 | Molecular energy and transport terms | 13 |
| 1.4.2 | State equations | 14 |
| 1.4.3 | Effect of thermal radiation | 16 |
| 1.4.4 | Typical values of some physical coefficients | 17 |
| 2 Weak Solutions, A Priori Estimates | ||
| 2.1 | Weak formulation | 21 |
| 2.1.1 | Equation of continuity | 22 |
| 2.1.2 | Balance of linear momentum | 22 |
| 2.1.3 | Balance of total energy | 23 |
| 2.1.4 | Entropy production | 24 |
| 2.1.5 | Constitutive relations | 24 |
| 2.2 | A priori estimates | 25 |
| 2.2.1 | Total mass conservation | 25 |
| 2.2.2 | Energy estimates | 26 |
| 2.2.3 | Estimates based on the Second law of thermodynamics | 27 |
| 2.2.4 | Positivity of the absolute temperature | 32 |
| 2.2.5 | Pressure estimates | 35 |
| 2.2.6 | Pressure estimates, an alternative approach | 39 |
| 3 Existence Theory | ||
| 3.1 | Hypotheses | 44 |
| 3.2 | Structural properties of constitutive functions | 47 |
| 3.3 | Main existence result | 50 |
| 3.3.1 | Approximation scheme | 51 |
| 3.4 | Solvability of the approximate system | 53 |
| 3.4.1 | Approximate continuity equation | 53 |
| 3.4.2 | Approximate internal energy equation | 55 |
| 3.4.3 | Local solvability of the approximate problem | 62 |
| 3.4.4 | Uniform estimates and global existence | 64 |
| 3.5 | Faedo-Galerkin limit | 69 |
| 3.5.1 Estimates independent of the dimension | ||
| of Faedo-Galerkin approximations | 69 | |
| 3.5.2 Limit passage in the approximate continuity | ||
| equation | 72 | |
| 3.5.3 Strong convergence of the approximate temperatures | ||
| and the limit in the entropy equation | 75 | |
| 3.5.4 | Limit in the approximate momentum equation | 82 |
| 3.5.5 The limit system resulting from the | ||
| Faedo-Galerkin approximation | 82 | |
| 3.5.6 The entropy production rate represented | ||
| by a positive measure | 84 | |
| 3.6 | Artificial diffusion limit | 85 |
| 3.6.1 Uniform estimates and limit in the | ||
| approximate continuity equation | 85 | |
| 3.6.2 Entropy balance and strong convergence of the | ||
| approximate temperatures | 87 | |
| 3.6.3 | Uniform pressure estimates | 92 |
| 3.6.4 Limit in the approximate momentum equation | ||
| and in the energy balance | 94 | |
| 3.6.5 | Strong convergence of the densities | 95 |
| 3.6.6 | Artificial diffusion asymptotic limit | 102 |
| 3.7 | Vanishing artificial pressure | 104 |
| 3.7.1 | Uniform estimates | 104 |
| 3.7.2 | Asymptotic limit for vanishing artificial pressure | 106 |
| 3.7.3 Entropy balance and pointwise convergence | ||
| of the temperature | 109 | |
| 3.7.4 | Pointwise convergence of the densities | 113 |
| 3.7.5 | Oscillations defect measure | 118 |
| 3.8 | Regularity properties of the weak solutions | 122 |
| 4 Asymptotic Analysis - An Introduction | ||
| 4.1 | Scaling and scaled equations | 129 |
| 4.2 | Low Mach number limits | 131 |
| 4.3 | Strongly stratified flows | 133 |
| 4.4 | Acoustic waves | 135 |
| 4.4.1 | Low stratification | 135 |
| 4.4.2 | Strong stratification | 137 |
| 4.4.3 | Attenuation of acoustic waves | 138 |
| 4.5 | Acoustic analogies | 139 |
| 4.6 | Initial data | 141 |
| 4.7 A general approach to singular limits for | ||
| the full Navier-Stokes-Fourier system | 142 | |
| 5 Singular Limits - Low Stratification | ||
| 5.1 | Hypotheses and global existence for the primitive system | 150 |
| 5.1.1 | Hypotheses | 150 |
| 5.1.2 | Global-in-time solutions | 152 |
| 5.2 | Dissipation equation, uniform estimates | 153 |
| 5.2.1 | Conservation of total mass | 153 |
| 5.2.2 | Total dissipation balance and related estimates | 154 |
| 5.2.3 | Uniform estimates | 157 |
| 5.3 | Convergence | 160 |
| 5.3.1 | Equation of continuity | 161 |
| 5.3.2 | Entropy balance | 162 |
| 5.3.3 | Momentum equation | 166 |
| 5.4 | Convergence of the convective term | 169 |
| 5.4.1 | Helmholtz decomposition | 170 |
| 5.4.2 | Compactness of the solenoidal part | 171 |
| 5.4.3 | Acoustic equation | 172 |
| 5.4.4 | Formal analysis of the acoustic equation | 175 |
| 5.4.5 | Spectral analysis of the wave operator | 177 |
| 5.4.6 | Reduction to a finite number of modes | 178 |
| 5.4.7 | Weak limit of the convective term - time lifting | 180 |
| 5.5 | Conclusion - main result | 183 |
| 5.5.1 | Weak formulation of the target problem | 183 |
| 5.5.2 | Main result | 185 |
| 5.5.3 | Determining the initial temperature distribution | 186 |
| 5.5.4 | Energy inequality for the limit system | 186 |
| 6 Stratified Fluids | ||
| 6.1 | Motivation | 195 |
| 6.2 | Primitive system | 196 |
| 6.2.1 | Field equations | 196 |
| 6.2.2 | Constitutive relations | 197 |
| 6.2.3 | Scaling | 199 |
| 6.3 | Asymptotic limit | 201 |
| 6.3.1 | Static states | 201 |
| 6.3.2 | Solutions to the primitive system | 201 |
| 6.3.3 | Main result | 203 |
| 6.4 | Uniform estimates | 206 |
| 6.4.1 | Dissipation equation, energy estimates | 206 |
| 6.4.2 | Pressure estimates | 211 |
| 6.5 | Convergence towards the target system | 214 |
| 6.5.1 | Anelastic constraint | 214 |
| 6.5.2 | Determining the pressure | 214 |
| 6.5.3 | Driving force | 217 |
| 6.5.4 | Momentum equation | 219 |
| 6.6 | Analysis of acoustic waves | 220 |
| 6.6.1 | Acoustic equation | 220 |
| 6.6.2 | Spectral analysis of the wave operator | 222 |
| 6.6.3 | Convergence of the convective term | 224 |
| 6.7 | Asymptotic limit in entropy balance | 228 |
| 7 Interaction of Acoustic Waves with Boundary | ||
| 7.1 | Problem formulation | 233 |
| 7.1.1 | Field equations | 233 |
| 7.1.2 | Physical domain and boundary conditions | 235 |
| 7.2 | Main result | 236 |
| 7.2.1 | Preliminaries - global existence | 236 |
| 7.2.2 | Compactness of the family of velocities | 238 |
| 7.3 | Uniform estimates | 239 |
| 7.4 | Analysis of acoustic waves | 241 |
| 7.4.1 | Acoustic equation | 241 |
| 7.4.2 | Spectral analysis of the acoustic operator | 244 |
| 7.5 | Strong convergence of the velocity field | 253 |
| 7.5.1 | Compactness of the solenoidal component | 253 |
| 7.5.2 | Reduction to a finite number of modes | 254 |
| 7.5.3 | Strong convergence | 255 |
| 8 Problems on Large Domains | ||
| 8.1 | Primitive system | 261 |
| 8.2 | Uniform estimates | 264 |
| 8.2.1 Estimates based on the hypothesis of | ||
| thermodynamic stability | 264 | |
| 8.2.2 Estimates based on the specific form of | ||
| constitutive relations | 266 | |
| 8.3 | Acoustic equation | 268 |
| 8.4 | Regularization and extension to M3 | 271 |
| 8.4.1 | Uniform estimates | 271 |
| 8.4.2 | Regularization | 274 |
| 8.4.3 | Extension to the whole space M3 | 276 |
| 8.5 | Dispersive estimates and time decay of acoustic waves | 277 |
| 8.6 | Conclusion - main result | 282 |
| 9 Acoustic Analogies | ||
| 9.1 | Asymptotic analysis and the limit system | 286 |
| 9.2 | Acoustic equation revisited | 287 |
| 9.3 | Two-scale convergence | 291 |
| 9.3.1 | Approximate methods | 295 |
| 9.4 Light hill's acoustic analogy in the low | ||
| Mach number regime | 296 | |
| 9.4.1 | Ill-prepared data | 296 |
| 9.4.2 | Well-prepared data | 297 |
| 9.5 | Concluding remarks | 300 |
| 10 Appendix | ||
| 10.1 | Modifiers | 303 |
| 10.2 | Basic properties of some elliptic operators | 304 |
| 10.2.1 | A priori estimates | 305 |
| 10.2.2 | Fredholm alternative | 307 |
| 10.2.3 | Spectrum of a generalized Laplacian | 309 |
| 10.3 | Normal traces | 311 |
| 10.4 | Singular and weakly singular operators | 314 |
| 10.5 | The inverse of the div-operator (Bogovskii's formula) | 315 |
| 10.6 | Helmholtz decomposition | 323 |
| 10.7 | Function spaces of hydrodynamics | 325 |
| 10.8 | Poincare type inequalities | 327 |
| 10.9 | Korn type inequalities | 329 |
| 10.10 | Estimating Vu by means of div^u and curl^u | 333 |
| 10.11 | Weak convergence and monotone functions | 334 |
| 10.12 | Weak convergence and convex functions | 338 |
| 10.13 | Div-Curl lemma | 341 |
| 10.14 | Maximal regularity for parabolic equations | 343 |
| 10.15 | Quasilinear parabolic equations | 345 |
| 10.16 Basic properties of the Riesz transform and | ||
| related operators | 347 | |
| 10.17 | Commutators involving Riesz operators | 350 |
| 10.18 | Renormalized solutions to the equation of continuity | 352 |
| 11 Bibliographical Remarks | ||
| 11.1 | Fluid flow modeling | 359 |
| 11.2 | Mathematical theory of weak solutions | 360 |
| 11.3 | Existence theory | 361 |
| 11.4 | Analysis of singular limits | 361 |
| 11.5 | Propagation of acoustic waves | 362 |
| Bibliography | 363 | |
| Index | 377 | |
