Handbook of differential equations: Evolutionary equations

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Preface......Page 4
List of Contributors......Page 5
Contents......Page 6
Contents of Volume I......Page 7
Contents of Volume II......Page 8
Contents of Volume III......Page 9
Incompressible Euler Equations: The Blow-up Problem and Related Results......Page 10
Introduction......Page 12
Basic properties......Page 13
Preliminaries......Page 16
Kato's local existence and the BKM criterion......Page 19
Refinements of the BKM criterion......Page 22
Constantin-Fefferman-Majda's and other related results......Page 25
Vortex tube collapse......Page 29
Squirt singularity......Page 30
Self-similar blow-up......Page 32
Asymptotically self-similar blow-up......Page 34
Distortions of the Euler equations......Page 35
The Constantin-Lax-Majda equation......Page 36
The 2D quasi-geostrophic equation and its 1D model......Page 37
The 2D Boussinesq system and Moffatt's problem......Page 42
Deformations of the Euler equations......Page 44
Dichotomy: singularity or global regular dynamics?......Page 48
Spectral dynamics approach......Page 52
Conservation laws for singular solutions......Page 56
References......Page 59
Mathematical Methods in the Theory of Viscous Fluids......Page 65
Balance laws......Page 67
Conservation of mass......Page 68
Balance of momentum......Page 69
Second law of thermodynamics, entropy......Page 70
Kinetic energy dissipation......Page 71
Navier-Stokes-Fourier system......Page 72
Constitutive theory......Page 73
Thermodynamics stability......Page 74
Effect of thermal radiation......Page 75
Real gas state equation......Page 76
Total mass conservation......Page 77
A priori estimates based on the energy dissipation......Page 78
Renormalized equation of continuity and refined density estimates......Page 80
A priori estimates, summary......Page 82
Weak sequential stability......Page 83
Preliminaries, Div-Curl lemma, and related results......Page 84
Strong convergence of the temperature......Page 85
Strong convergence of the density......Page 87
Existence theory......Page 92
Bibliographical comments......Page 93
Long-time behavior......Page 94
Stationary driving force......Page 95
Bibliographical comments......Page 97
Singular limits......Page 98
Uniform estimates......Page 99
Asymptotic limit......Page 102
Acoustic equation......Page 103
Bibliographical comments......Page 104
References......Page 105
Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains......Page 110
Introduction......Page 112
Main definitions......Page 119
Existence of the global attractor......Page 124
Attractors for semigroups having a global Lyapunov function......Page 126
Dimension of the global attractor......Page 129
Robustness of the global attractor......Page 132
Inertial manifolds......Page 135
Construction of exponential attractors......Page 136
Robust families of exponential attractors......Page 141
Uniform attractors......Page 145
Pullback attractors......Page 149
Finite dimensional reduction of nonautonomous systems......Page 152
Weighted and uniformly local phase spaces: basic dissipative estimates......Page 156
Attractors and locally compact attractors......Page 161
The finite dimensional case......Page 164
The infinite dimensional case: entropy estimates......Page 169
Infinite dimensional exponential attractors......Page 174
Complexity of space-time dynamics: entropy theory......Page 176
Lower bounds on the entropy, the Kotelnikov formula, and spatial chaos......Page 182
Sinai-Bunimovich space-time chaos in PDEs......Page 186
Ill-posed dissipative systems and trajectory attractors......Page 189
References......Page 198
The Cahn-Hilliard Equation......Page 208
Introduction......Page 210
Backwards diffusion and regularization......Page 211
The Cahn-Hilliard equation and phase separation......Page 213
The constant mobility - quartic polynomial case......Page 215
The degenerate mobility - logarithmic free energy case......Page 216
Existence, uniqueness, and regularity......Page 218
Linear stability and spinodal decomposition......Page 221
Long time behavior and limiting motions......Page 223
The Mullins-Sekerka problem......Page 224
Upper bounds for coarsening......Page 226
References......Page 232
Mathematical Analysis of Viscoelastic Fluids......Page 236
The equations describing viscoelastic flows......Page 238
General considerations......Page 239
Differential models......Page 240
Integral models......Page 241
Molecular models......Page 242
Local existence......Page 244
Global existence......Page 245
Hyperbolic shocks......Page 246
Breakup of liquid jets......Page 247
Existence theory......Page 249
Inflow boundaries......Page 250
The high Weissenberg number limit......Page 252
Stress boundary layers......Page 253
The reentrant corner singularity......Page 255
Flow instabilities......Page 257
Constitutive instabilities......Page 259
Characteristics and change of type......Page 260
Controllability of viscoelastic flows......Page 262
Concluding remarks......Page 263
References......Page 264
Application of Monotone Type Operators to Parabolic and Functional Parabolic PDE's......Page 273
Introduction......Page 275
Basic definitions......Page 276
Evolution equations with monotone operators......Page 278
Evolution equations with pseudomonotone operators......Page 281
Second order and higher order nonlinear parabolic differential equations......Page 284
Definition of the weak solution......Page 285
Application of monotone operators......Page 287
Application of pseudomonotone operators......Page 290
Strongly nonlinear equations......Page 292
Existence theorems......Page 293
Examples......Page 295
Strongly nonlinear equations......Page 299
Parabolic equations containing functional dependence in the main part......Page 300
Existence theorems......Page 301
Examples......Page 303
Non-uniformly parabolic equations......Page 306
Modified conditions on nonlocal terms......Page 307
Existence of solutions in (0,)......Page 308
Boundedness of solutions......Page 310
Attractivity......Page 312
Stabilization of solutions......Page 313
Systems of parabolic equations and functional parabolic equations......Page 317
Contact problems......Page 319
References......Page 325
Recent Results on Hydrodynamic Limits......Page 328
Introduction......Page 330
Conservation laws and balance laws......Page 331
Convex entropy......Page 332
Relative entropy......Page 334
Admissible balance laws......Page 335
Weak entropy solutions......Page 336
Dissipative solutions......Page 337
A structure lemma......Page 338
Weak/strong principle......Page 340
Stability......Page 341
Approximated solutions......Page 342
Isentropic gas dynamics......Page 343
Isothermal gas dynamic......Page 345
A bi-fluid model......Page 346
A counterexample: The Euler system with temperature......Page 350
Rarefaction waves......Page 351
Axisymmetric solution with vacuum at the origin......Page 352
Kinetic equations......Page 353
Nonlinear Fokker-Planck equation......Page 355
Kinetic formulation of balance laws......Page 357
Coupled system of Navier-Stokes Fokker-Planck equations......Page 359
Scaling......Page 361
Asymptotic limit......Page 362
Hydrodynamic limit from Fokker-Planck equation to isothermal system of gas......Page 365
Hydrodynamic limits to isentropic gas dynamics......Page 366
Hydrodynamic limits of the Fokker-Planck Navier-Stokes system......Page 374
Conclusion and open problems......Page 377
References......Page 378
Introduction to Stefan-Type Problems......Page 382
Introduction......Page 386
Weak formulation......Page 388
The energy balance.......Page 389
The temperature-phase rule.......Page 390
Classical formulation......Page 391
Metastability.......Page 392
The one-dimensional Stefan problem.......Page 393
An ideal experiment.......Page 394
Equivalence between the CSP and the WSP.......Page 395
A Stefan-type problem arising in ferromagnetism......Page 396
The quasi-steady Stefan problem and the Hele-Shaw problem.......Page 399
The hyperbolic Stefan problem.......Page 400
Historical note......Page 402
Free boundary problems.......Page 403
Undercooling and superheating.......Page 404
Contact angle condition.......Page 405
Limit as sigma->0.......Page 406
First mode: Directional solidification......Page 407
Glass formation.......Page 408
The entropy balance.......Page 409
Linearization.......Page 411
Phase transitions in heterogeneous systems......Page 412
Mass diffusion.......Page 413
Phase separation.......Page 414
A transformation of variable.......Page 415
A nonparabolic system of equations.......Page 417
Approach via nonequilibrium thermodynamics......Page 418
Balance laws and Gibbs formula.......Page 419
Entropy balance.......Page 420
Phenomenological laws.......Page 421
Diffuse-interface models and length-scales......Page 422
Two relaxation dynamics.......Page 423
The Penrose-Fife and phase-field models.......Page 424
Macroscopic-mesoscopic, and microscopic length-scales.......Page 425
Analysis of the weak formulation of the Stefan model......Page 426
L2-techniques......Page 427
Interpretation.......Page 428
Regularity.......Page 432
An L1-result.......Page 434
L-results.......Page 436
Time-integral transformation.......Page 438
Inversion of the Laplace operator.......Page 440
Change of pivot space.......Page 441
L1-semigroups.......Page 442
Weak formulation......Page 445
Existence of a weak solution......Page 447
Modelling remarks.......Page 453
Convexity and other analytical tools......Page 454
Convex and lower semicontinuous functions......Page 455
Legendre-Fenchel transformation and subdifferential......Page 456
Saddle points......Page 459
Compactness by strict convexity......Page 462
Maximal monotone operators......Page 464
Cauchy problem.......Page 466
T-accretiveness.......Page 467
Perimeter and curvature......Page 468
Gamma-convergence......Page 469
Acknowledgments......Page 471
Bibliography......Page 472
References......Page 473
The KdV Equation......Page 490
Historical background......Page 492
Preliminary profile solution......Page 494
Third-order KdV equations......Page 496
Higher-order KdV equation......Page 497
The tanh-coth method......Page 498
The sine-cosine method......Page 499
Hirota's bilinear method......Page 500
Conservation laws......Page 501
The KdV equation......Page 502
Using the tanh-coth method......Page 503
Using the sine-cosine method......Page 504
Multiple-soliton solutions of the KdV equation......Page 505
The modified KdV equation......Page 512
Using the tanh-coth method......Page 513
Using the sine-cosine method......Page 514
Multiple-solitons of the mKdV equation......Page 515
Using the tanh-coth method......Page 522
The coth-csch ansatz......Page 524
Using the coth-csch ansatz......Page 525
Multiple-solitons of the potential KdV equation......Page 526
The generalized KdV equation......Page 528
Using the tanh-coth method......Page 529
The Gardner equation......Page 531
Using the tanh method......Page 532
A sinh ansatz......Page 533
A csch ansatz......Page 534
A sech-tanh ansatz......Page 535
Using the tanh method......Page 536
Using cosh ansatz......Page 538
Using sinh ansatz......Page 539
Using sech and csch ansatze......Page 540
Fifth-order KdV equation......Page 541
Using the tanh-coth method......Page 542
The first criterion......Page 543
The Lax equation......Page 544
The Kaup-Kupershmidt (KK) equation......Page 545
The second criterion......Page 546
Multiple-solitons of the fifth-order KdV equation......Page 547
The Lax equation......Page 548
The Sawada-Kotera equation......Page 551
The Kaup-Kupershmidt equation......Page 553
The sech method......Page 554
The sech method......Page 556
The coupled KdV or the Hirota-Satsuma equations......Page 558
Using the tanh-coth method......Page 559
Multiple-soliton solutions of the Hirota-Satsuma system......Page 561
Multiple-soliton solutions by another method......Page 562
Compactons and the K(n,n) equation......Page 564
The K(n,n) equation......Page 566
Variant of the K(n,n) equation......Page 568
References......Page 570
Author Index......Page 574
Subject Index......Page 590