1 Fundamental concepts and equations ........................... 1
1.1 Some mathematical concepts and notation ................. 1
1.2 Governing equations and relations of gas dynamics ...... 15
1.3 Some advanced mathematical concepts and results ........ 32
1.4 Survey of concepts and results from functional
analysis ............................................... 60
2 Theoretical results for the Euler equations ................. 74
2.1 Hyperbolic systems and the Euler equations ............. 74
2.2 Existence of smooth solutions .......................... 79
2.3 Weak solutions ........................................ 106
2.4 Final comments ........................................ 146
3 Some mathematical tools for compressible flows ............. 155
3.1 Renormalized solutions of the steady continuity
equation .............................................. 155
3.2 Vector fields with summable divergence ................ 163
3.3 The equation div υ = ƒ ................................ 165
3.4 Some results for monotone and convex operators ........ 183
4 Weak solutions for steady Navier-Stokes equations
of compressible barotropic flow ............................ 189
4.1 Formulation of problems in bounded and exterior
domains and main results .............................. 189
4.2 Heuristic approach .................................... 194
4.3 Approximations in bounded domains ..................... 200
4.4 Effective viscous flux ................................ 204
4.5 Neumann problem for the Laplacian ..................... 211
4.6 Relaxed continuity equation with dissipation .......... 212
4.7 The Lame system ....................................... 216
4.8 Complete system with dissipation in the relaxed
continuity equation and with artificial pressure ...... 218
4.9 Complete system with relaxed continuity equation and
with artificial pressure .............................. 223
4.10 Complete system with artificial pressure .............. 231
4.11 Complete system of a viscous barotropic gas ........... 239
4.12 Approximations in an exterior domain .................. 245
4.12.1 Relaxation on invading domains ................. 245
4.13 Complete system with relaxed continuity equation on
an exterior domain .................................... 247
4.14 Existence of weak solutions in exterior domains ....... 254
4.15 Existence of weak solutions in bounded and in
exterior Lipschitz domains ............................ 259
4.16 Existence of weak solutions in domains with
noncompact boundaries ................................. 261
4.17 Further results, comments and bibliographic remarks ... 268
5 Strong solutions for steady Navier—Stokes equations
of compressible barotropic flow and small data ............. 279
5.1 Notation and main results ............................. 279
5.2 Heuristic approach .................................... 282
5.3 Auxiliary linear problems ............................. 285
5.4 The linearized system ................................. 290
5.5 The fully nonlinear system ............................ 292
5.6 Bibliographic remarks ................................. 296
6 Some mathematical tools for nonsteady equations ............ 300
6.1 Some auxiliary results from functional analysis ....... 300
6.2 Renormalized solutions of the continuity equation ..... 304
7 Weak solutions for nonsteady Navier-Stokes equations of
compressible barotropic flow ............................... 312
7.1 Formulation of problems and main results .............. 312
7.2 Linear momentum and total energy ...................... 321
7.3 Heuristic approach .................................... 324
7.4 Approximations in bounded domains ..................... 330
7.5 Effective viscous flux ................................ 338
7.6 Continuity equation with dissipation .................. 343
7.7 Galerkin approximation of the system with
dissipation in the continuity equation and with
artificial pressure ................................... 352
7.8 Complete system with dissipation in the continuity
equation and with artificial pressure ................. 361
7.9 Complete system with artificial pressure .............. 368
7.10 Complete system of isentropic Navier-Stokes
equations ............................................. 381
7.11 Existence of solutions in exterior domains ............ 393
7.12 Other problems and bibliographic remarks .............. 404
8 Global behavior of weak solutions .......................... 431
8.1 Formulation of the problem ............................ 431
8.2 Basic assumptions ..................................... 432
8.3 Sequential stabilization of the weak solution ......... 432
8.4 Auxiliary functions ................................... 433
8.5 Existence and estimates of auxiliary functions ........ 435
8.6 Comparison density and a test function ................ 437
8.7 Passing to the limit with the regularization
parameter ............................................. 437
8.8 Comparison density is close to the density as t → ∞ ... 438
8.9 Convergence of the density ............................ 446
8.10 Uniqueness of equilibrium ............................. 452
8.11 Global behavior of weak solutions in time in
bounded domains - arbitrary forces .................... 456
8.12 Bounded absorbing sets ................................ 457
8.13 Asymptotically closed trajectories .................... 458
8.14 Global attractor of short trajectories ................ 459
8.15 Rapidly oscillating external forces ................... 460
8.16 Attractors ............................................ 460
8.17 Time-periodic solutions ............................... 461
8.18 Uniqueness of equilibrium revisited ................... 462
9 Strong solutions of nonsteady compressible Navier-
Stokes equations ........................................... 464
9.1 Problem formulation ................................... 464
9.2 Similarity transformation ............................. 465
9.3 Maximal parabolic regularity .......................... 466
9.4 Resolution of the continuity equation with a given
velocity .............................................. 467
9.5 Further transcription of the problem .................. 469
9.6 Fixed point argument and the existence of a local
solution .............................................. 470
9.7 Uniqueness ............................................ 473
9.8 Global a priori estimate .............................. 474
9.9 Global existence ...................................... 479
9.10 Bibliographical remarks ............................... 480
References .................................................... 485
Index ......................................................... 499
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