American Institute of Mathematical Sciences

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Differential operators . . . . . . . . . . . . . . . . . . . . . .3
1.2 Measures and distributions . . . . . . . . . . . . . . . . . . . . .4
1.2.1 Continuously differentiable functions . . . . . . . . . . . . . . 4
1.2.2 Compactness of sets of continuous functions . . . . . . . . . . . 5
1.2.3 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
1.2.4 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Bochner and Sobolev spaces . . . . . . . . . . . . . . . . . . . . .7
1.3.1 Lebesgue (Bochner) spaces . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . .10
1.3.3 Dual Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Embeddings for dual Sobolev spaces . . . . . . . . . . . . . . . .12
1.3.5 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
1.3.6 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . .14
1.4 Fourier analysis . . . . . . . . . . . . . . . . . . . . . . . . . .14
1.4.1 Fourier transform of tempered distributions . . . . . . . . . . . 15
1.4.2 Boundedness of Fourier multipliers . . . . . . . . . . . . . . . .16
1.5 Compactness of families of functions . . . . . . . . . . . . . . . .17
1.5.1 Weak compactness of integrable functions . . . . . . . . . . . . .17
1.5.2 Young measures . . . . . . . . . . . . . . . . . . . . . . . . . .18
1.6 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . .19
2 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Dynamical systems and attractors . . . . . . . . . . . . . . . . . .22
2.2 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Exponential attractor . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Method of trajectories . . . . . . . . . . . . . . . . . . . . . . .36
X Contents
2.5 Method of Lyapunov exponents . . . . . . . . . . . . . . . . . . . .43
2.5.1 Coverings of ellipsoids . . . . . . . . . . . . . . . . . . . . . 45
2.5.2 Lyapunov exponents and solution operators . . . . . . . . . . . . 48
2.6 Extension to the case without uniqueness . . . . . . . . . . . . . .53
2.7 Bibliographical remarks. . . . . . . . . . . . . . . . . . . . . . .56
3 Viscous fluid systems . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 Fields and balance laws . . . . . . . . . . . . . . . . . . . . . . 59
3.1.1 Balance law - integral formulation . . . . . . . . . . . . . . . .61
3.1.2 Balance law - formal interpretation . . . . . . . . . . . . . . . 64
3.1.3 Weak sequential stability . . . . . . . . . . . . . . . . . . . . 65
3.2 Kinematics of mass transport . . . . . . . . . . . . . . . . . . . .65
3.2.1 Renormalization. . . . . . . . . . . . . . . . . . . . . . . . . .66
3.3 Momentum equation . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Total energy balance . . . . . . . . . . . . . . . . . . . . . . . .71
3.4.1 Kinetic and internal energy balance . . . . . . . . . . . . . . . 72
3.4.2 Conservative boundary conditions and the total
energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
3.5 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5.1 Entropy production as a non-negative Radon measure . . . . . . . .74
3.6 Fluid description . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6.1 General constitutive relations . . . . . . . . . . . . . . . . . .76
3.6.2 Navier-Stokes-Fourier system . . . . . . . . . . . . . . . . . . .77
3.7 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . .81
3.7.1 Hypothesis of thermodynamic stability . . . . . . . . . . . . . . 82
3.7.2 Monoatomic gas state equation. . . . . . . . . . . . . . . . . . .82
3.7.3 Third law of thermodynamics, entropy . . . . . . . . . . . . . . .83
3.7.4 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.7.5 Transport coefficients . . . . . . . . . . . . . . . . . . . . . .85
4 Complete fluid systems, analysis . . . . . . . . . . . . . . . . . . .87
4.1 Equilibrium states . . . . . . . . . . . . . . . . . . . . . . . . .88
4.1.1 Total mass conservation . . . . . . . . . . . . . . . . . . . . . 88
4.1.2 Total dissipation balance . . . . . . . . . . . . . . . . . . . . 88
4.1.3 Static states . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 Weak sequential stability . . . . . . . . . . . . . . . . . . . . . 94
4.2.1 Stability of solutions under a uniform energy bound . . . . . . . 95
4.3 Global-in-time solutions . . . . . . . . . . . . . . . . . . . . . .117
4.3.1 Existence theory . . . . . . . . . . . . . . . . . . . . . . . . .117
4.4 Long-time behavior of conservative systems . . . . . . . . . . . . .120
5 Large time behavior . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1 Large time behavior revisited . . . . . . . . . . . . . . . . . . . 127
5.1.1 Uniform decay of density oscillations . . . . . . . . . . . . . . 129
5.2 Energy “blow up” . . . . . . . . . . . . . . . . . . . . . . . . . .138
Contents XI
5.2.1 A general result . . . . . . . . . . . . . . . . . . . . . . . . .138
5.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . .143
5.3 Rapidly oscillating driving forces . . . . . . . . . . . . . . . . .146
5.3.1 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . .149
5.3.2 Uniform bounds via iteration . . . . . . . . . . . . . . . . . . .151
5.3.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .153
6 Complete and simplified fluid systems . . . . . . . . . . . . . . . . 155
6.1 Dynamical systems approach . . . . . . . . . . . . . . . . . . . . .156
6.1.1 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.1.2 Global trajectories and attractors . . . . . . . . . . . . . . . .158
6.2 Dynamics of reduced systems . . . . . . . . . . . . . . . . . . . . 159
6.2.1 Barotropic compressible fluids . . . . . . . . . . . . . . . . . .161
6.2.2 Incompressible fluids . . . . . . . . . . . . . . . . . . . . . . 162
6.3 Bibliographical comments . . . . . . . . . . . . . . . . . . . . . .165
7 Incompressible model of Ladyzhenskaya . . . . . . . . . . . . . . . . 169
7.1 Statement of the problem. . . . . . . . . . . . . . . . . . . . . . 170
7.1.1 Assumptions on the stress tensor . . . . . . . . . . . . . . . . .171
7.1.2 Mathematical formulation of the problem. . . . . . . . . . . . . .173
7.1.3 Existence and uniqueness results . . . . . . . . . . . . . . . . .175
7.2 2D Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . .180
7.2.1 Dissipativity and absorbing set . . . . . . . . . . . . . . . . . 181
7.2.2 Differentiability of the solution operator . . . . . . . . . . . .183
7.2.3 Estimates of the trace. Attractor dimension . . . . . . . . . . . 184
7.3 2D Ladyzhenskaya model; case r = 2 . . . . . . . . . . . . . . . . .187
7.3.1 Dissipativity and absorbing set . . . . . . . . . . . . . . . . . 188
7.3.2 Differentiability of the solution semigroup . . . . . . . . . . . 190
7.3.3 Estimates of the trace. Attractor dimension. . . . . . . . . . . .192
7.4 3D Ladyzhenskaya model; case r > 11/5 . . . . . . . . . . . . . . . 194
7.4.1 Existence of exponential attractor . . . . . . . . . . . . . . . .195
7.4.2 Explicit dimension estimates for r > 12/5 . . . . . . . . . . . . 203
8 2D Model of Ladyzhenskaya for r < 2 . . . . . . . . . . . . . . . . . 217
8.1 2D Ladyzhenskaya model; case r ? (1, 2) . . . . . . . . . . . . . . 218
8.1.1 Basic mathematical theory . . . . . . . . . . . . . . . . . . . . 218
8.1.2 Dissipativity and the dynamics of trajectories . . . . . . . . . .224
8.1.3 Exponential attractor . . . . . . . . . . . . . . . . . . . . . . 226
8.1.4 Improved estimates for r > 4/3 . . . . . . . . . . . . . . . . . .234
8.2 Fluids with pressure-dependent viscosities . . . . . . . . . . . . .238
8.2.1 Basic mathematical theory . . . . . . . . . . . . . . . . . . . . 241
8.2.2 Dissipativity and absorbing set . . . . . . . . . . . . . . . . . 246
8.2.3 Existence of exponential attractor . . . . . . . . . . . . . . . .251
8.3 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . 257
XII Contents
9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .259
9.1 Embeddings and interpolations . . . . . . . . . . . . . . . . . . . 259
9.2 Korn and Poincar´e type inequalities . . . . . . . . . . . . . . . .262
9.3 Div-Curl lemma and related topics . . . . . . . . . . . . . . . . . 264
9.4 A result of P.-L. Lions on weak continuity . . . . . . . . . . . . .267
9.5 Coverings of compacts in Bochner spaces. . . . . . . . . . . . . . .272
9.5.1 Bochner spaces . . . . . . . . . . . . . . . . . . . . . . . . . .272
9.5.2 Asymptotics of the sequences . . . . . . . . . . . . . . . . . . .275
9.5.3 Aubin-Lions lemma . . . . . . . . . . . . . . . . . . . . . . . . 277
9.6 Lieb-Thirring inequality . . . . . . . . . . . . . . . . . . . . . .280
9.7 Properties of the stress tensor . . . . . . . . . . . . . . . . . . 283
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .285
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
                             

Eduard Feireisl received the RNDr degree from the Charles University in Prague, and the Ph.D. and DrSc (doctor of science) degree form the Academy of Sciences of the Czech Republic. Currently, he is Research Director at the Institute of Mathematics of the Academy of Sciences in Prague. He published 2 monographs and over 150 research papers in journals, books, and refereed conference proceedings. He is in the editorial board of SIAM Journal on Mathematical Analalysis, Journal of Evolution Equations, Czechoslovak Mathematical Journal, Discrete and Continuous Dynamical Systems Ser. S, Central European Journal of Mathematics, Mathematica Bohemica. His main research interest include partial differential equations, dynamical systems, with applications to fluid mechanics.

Dalibor Pražák received his Ph.D. at Charles University, Prague, under the guidance of J. Málek. He is currently associate professor at Department of Mathematical Analysis, Charles University, Prague. His research interests include mathematical fluid mechanics and dynamical systems in general.