The Porous Medium Equation Mathematical Theory

by Vazquez, Juan Luis

• ISBN13: 9780198569039
• ISBN10: 0198569033
• Format: Hardcover
• Copyright: 2006-12-28
• Publisher: Clarendon Press

Summary

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academicsin mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physicalapplications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes,providing comments, historical notes or recommended reading, and exercises for the reader.

Table of Contents

1 Introduction	1
1.1 The subject	1
1.1.1 The porous medium equation	1
1.1.2 The PME as a nonlinear parabolic equation	2
1.2 Peculiar features of the PME	4
1.2.1 Finite propagation and free boundaries	4
1.2.2 The role of special solutions	6
1.3 Nonlinear diffusion. Related equations	7
1.4 Contents	9
1.4.1 The main problems and the classes of solutions	9
1.4.2 Chapter overview	10
1.4.3 What is not covered	12
1.5 Reading the book	13
Notes	14
PART ONE
2 Main applications	19
2.1 Gas flow through a porous medium	19
2.1.1 Extensions	21
2.2 Nonlinear heat transfer	21
2.3 Groundwater flow. Boussinesq's equation	23
2.4 Population dynamics	25
2.5 Other applications and equations	26
2.6 Images, concepts and names taken from the applications	26
Notes	27
Problems	28
3 Preliminaries and basic estimates	30
3.1 Quasilinear equations and the PME	30
3.1.1 Existence of classical solutions	30
3.1.2 Weak theories and the PME	31
3.2 The GPME with good Φ Main estimates	33
3.2.1 Maximum principle and comparison	34
3.2.2 Other boundedness estimates	35
3.2.3 The stability estimate. L¹ contraction	36
3.2.4 The energy identity	38
3.2.5 Estimate of a time derivative	40
3.2.6 The BV estimates	42
3.3 Properties of the PME	43
3.3.1 Elementary invariance	43
3.3.2 Scaling	44
3.3.3 Conservation and dissipation	45
3.4 Alternative formulations of the PME and associated equations	46
3.4.1 Formulations	46
3.4.2 Dual equation	47
3.4.3 The p-Laplacian equation in d = 1	48
Notes	49
Problems	49
4 Basic examples	52
4.1 Some very simple solutions	52
4.2 Separation of variables	53
4.2.1 Positive λ. Nonlinear eigenvalue problem	53
4.2.2 Negative λ = —1 less than 0. Blow-up	54
4.3 Planar travelling waves	55
4.3.1 Limit solutions	57
4.3.2 Finite propagation and Darcy's law	58
4.4 Source-type solutions. Self-similarity	59
4.4.1 Comparison of ZKB profiles with Gaussian profiles.	61
4.4.2 Self-similarity. Derivation of the ZKB solution	62
4.4.3 Extension to m less than 1	65
4.5 Blow-up. Limits for the existence theory	66
4.5.1 Optimal existence versus blow-up	67
4.5.2 Non-contractivity in uniform norm	67
4.6 Two solutions in groundwater infiltration	68
4.6.1 The Polubarinova-Kochina solution	68
4.6.2 The dipole solution	69
4.6.3 Signed self-similar solutions	71
4.7 General planar front solutions	72
4.7.1 Solutions with a blow-up interface	73
Notes	75
Problems	77
5 The Dirichlet problem I. Weak solutions	81
5.1 Introducing generalized solutions	81
5.2 Weak solutions for the complete GPME	84
5.2.1 Concepts of weak and very weak solution	85
5.2.2 Definition of weak solutions for the HDP	86
5.2.3 About the initial data	87
5.2.4 Examples of weak solutions for the PME	89
5.3 Uniqueness of weak solutions	90
5.3.1 Non-existence of classical solutions	91
5.3.2 The subclass of energy solutions	92
5.4 Existence of weak energy solutions for general Φ. Case of non-negative data	92
5.4.1 Improvement of the assumption on ƒ	96
5.4.2 Non-positive solutions	96
5.5 Existence of weak signed solutions	97
5.5.1 Constant boundary data	101
5.6 Some properties of weak solutions	101
5.7 Weak solutions with non-zero boundary data	103
5.7.1 Properties of radial solutions	107
5.8 Universal bound in sup norm	108
5.9 Construction of the Friendly Giant	111
5.10 Properties of fast diffusion	114
5.10.1 Extinction in finite time	114
5.10.2 Singular fast diffusion	116
5.11 Equations of inhomogeneous media. A short review	116
Notes	119
Problems	122
6 The Dirichlet problem II. Limit solutions, very weak solutions and some other variants	126
6.1 L¹ theory. Stability. Limit solutions	127
6.1.1 Stability of weak solutions	127
6.1.2 Limit solutions in the L¹ setting	128
6.2 Theory of very weak solutions	130
6.2.1 Uniqueness of very weak solutions	132
6.2.2 Traces of very weak solutions	135
6.3 Problems in different domains	137
6.4 Limit solutions build a semigroup	138
6.5 Weak solutions with bounded forcing	140
6.5.1 Relating the concepts of solution	142
6.6 More general initial data. The case L¹δ	143
6.7 More general initial data. The case H-¹	145
6.7.1 Review of-functional analysis	145
6.7.2 Basic identities	146
6.7.3 General setting. Existence of H-¹ solutions	147
Notes	148
Problems	149
7 Continuity of local solutions	152
7.1 Continuity in several space dimensions	152
7.2 Problem, assumptions and result	155
7.3 Lemmas controlling the size of upsilon	157
7.4 Proof of the continuity theorem	164
7.4.1 Behaviour near a vanishing point	164
7.4.2 Behaviour near a non-vanishing point	165
7.4.3 End of proof	166
7.5 Continuity of weak solutions of the Dirichlet problem	167
7.5.1 Initial regularity	167
7.5.2 Boundary regularity	169
7.6 Holder continuity for porous media equations	170
7.7 Continuity of weak solutions in 1D	175
7.8 Existence of classical solutions	177
7.9 Extensions	178
7.9.1 Fast diffusions	178
7.9.2 When continuity fails	178
7.9.3 Equations with measurable coefficients	178
7.9.4 Other	179
Notes	179
Problems	180
8 The Dirichlet problem III. Strong solutions	181
8.1 Regularity for the PME. Bounds for ut	181
8.1.1 Bounds for ut if u > or equal to 0	182
8.1.2 Bound for ut for signed solutions	184
8.2 Strong solutions	185
8.2.1 The energy identity. Dissipation	187
8.2.2 Super- and subsolutions. Barriers	188
Notes	191
Problems	191
9 The Cauchy problem. L¹-theory	194
9.1 Definition of strong solution. Uniqueness	195
9.2 Existence of non-negative solutions	197
9.3 The fundamental estimate for the CP	199
9.4 Boundedness of the solutions	202
9.5 Existence with general L¹ data	204
9.5.1 Mass conservation	206
9.5.2 More properties of L¹ solutions	206
9.5.3 Sub- and supersolutions. More on comparison	207
9.6 Solutions with special properties	208
9.6.1 Invariance and symmetry	208
9.6.2 Aleksandrov's reflection principle	209
9.6.3 Solutions with compactly supported data	210
9.6.4 Solutions with finite moments	211
9.6.5 Centre of mass and mean deviation	215
9.7 The Cauchy-Dirichlet problem in unbounded domains	216
9.8 The Cauchy problem for the GPME	217
9.8.1 Weak theory	217
9.8.2 Limit L¹ theory	220
9.8.3 Relating the Cauchy-Dirichlet and Cauchy problems	221
Notes	221
Problems	222
10 The PME as an abstract evolution equation. Semigroup approach	229
10.1 Maximal monotone operators and semigroups	230
10.1.1 Generalities on maximal monotone operators	230
10.1.2 Evolution problem associated to an m.m.o. Semigroup	233
10.1.3 Complete evolution equation	234
10.1.4 Application to the GPME	235
10.2 Discretizations, mild solutions and accretive operators	236
10.2.1 The ITD method	237
10.2.2 Problem of convergence. Mild solutions	238
10.2.3 Accretive operators	240
10.2.4 The Crandall—Liggett theorem	242
10.3 Mild solutions of the filtration equation	244
10.3.1 Problems in bounded domains	245
10.3.2 Problem in the whole space	246
10.3.3 Cauchy problem with a peculiar nonlinearity	248
10.4 Time discretization and mass transfer problems	250
10.5 Other concepts of solution	252
Notes	254
Problems	255
11 The Neumann problem and problems on manifolds	257
11.1 Problem and weak solutions	257
11.1.1 Concept of weak solution	258
11.1.2 Examples of solutions of the HNP	260
11.2 Existence and uniqueness for the HNP	260
11.2.1 Uniqueness and energy solutions	260
11.2.2 Existence and properties for good data	261
11.2.3 Existence for L¹ data	262
11.2.4 Neumann problem and abstract ODE theory	262
11.2.5 Convergence to the Cauchy problem	263
11.3 Results for the HNP with a power equation	263
11.4 Other boundary value problems	266
11.4.1 Exterior problems	266
11.4.2 Mixed problems	267
11.4.3 Nonlinear boundary conditions	267
11.4.4 Dynamic boundary conditions	268
11.4.5 Boundary conditions of combustion type	268
11.5 The porous medium flow on a Riemannian manifold	268
11.5.1 Initial value problem	269
11.5.2 Initial value problem for the PME	271
11.5.3 Homogeneous Dirichlet, Neumann and other problem	272
Notes	273
Problems	273
PART TWO 12 The Cauchy problem with growing initial data	279
12.1 The Cauchy problem with large initial data	280
12.2 The Aronson–Caffarelli estimate	281
12.2.1 Precise a priori control on the initial data	283
12.3 Existence under optimal growth conditions	284
12.3.1 Functional preliminaries	284
12.3.2 Growth estimates for good solutions	284
12.3.3 Estimates in the spaces L¹(ρα)	289
12.3.4 Existence results	290
12.4 Uniqueness of growing solutions	293
12.5 Further properties of the solutions	297
12.6 Special solutions	299
12.6.1 Bounded solutions	299
12.6.2 Periodic solutions	300
12.6.3 Problems in a half space	300
12.6.4 Problems in intervals	301
12.7 Boundedness of local solutions	301
12.8 The PME in cones and tubes. Higher growth rates	302
12.8.1 Solutions in conical domains	302
12.8.2 Solutions in tubes	303
Notes	305
Problems	306
13 Optimal existence theory for non-negative solutions	309
13.1 Measures as initial data. Initial trace	310
13.2 Existence of initial traces in the CP	312
13.3 Pierre's uniqueness theorem	315
13.4 Uniqueness without growth restrictions	321
13.5 Dirichlet problem with optimal data	325
13.5.1 The special solution	326
13.5.2 The double trace results	326
13.6 Weak implies continuous	328
13.7 Complements	328
13.7.1 Signed solutions	328
Notes	330
Problems	331
14 Propagation properties	332
14.1 Basic definitions. The free boundary	333
14.2 Evolution properties of the positivity set	334
14.2.1 Persistence	334
14.2.2 Expansion and penetration of the support	335
14.2.3 Finite propagation	337
14.3 Initial behaviour. Waiting times	340
14.3.1 Waiting times for general solutions of the Cauchy problem	342
14.3.2 Addendum for comparison. Positivity for the heat equation	343
14.3.3 Examples of infinite waiting time near a corner	344
14.4 Hölder continuity and vertical lines	345
14.5 Describing the free boundary by the time function	347
14.6 Properties of solutions in the whole space	348
14.6.1 Finite propagation for L¹ data	348
14.6.2 Monotonicity properties for solutions with compact support	349
14.6.3 Free boundary behaviour	351
14.7 Propagation of signed solutions	352
Notes	353
Problems	354
15 One-dimensional theory. Regularity and interfaces	357
15.1 Cauchy problem. Regularity of the pressure	358
15.2 New comparison theorems	363
15.2.1 Shifting comparison	363
15.2.2 Counting intersections and lap number	365
15.3 The interface	368
15.3.1 Generalities	368
15.3.2 Left-hand interface and inner interfaces	371
15.3.3 Waiting time	372
15.4 Equation of the interface and Lipschitz continuity	376
15.4.1 Semiconvexity	378
15.5 C¹ regularity	379
15.5.1 Local linear behaviour and C¹ regularity near moving points	379
15.5.2 Limited regularity. Interfaces with a corner point *	383
15.5.3 Initial behaviour	385
15.6 Local solutions. Basic estimates	385
15.6.1 The local estimate for vx	385
15.6.2 The local lower estimate for vxx	386
15.6.3 Boundary behaviour	387
15.7 Interfaces of local solutions	387
15.7.1 Review of the regularity in the local case	388
15.8 Higher regularity	389
15.8.1 Second derivative estimate	389
15.8.2 Cinfinity regularity of v and s(t)*	391
15.8.3 Higher interface equations and convexity properties	392
15.8.4 Concavity results	393
15.8.5 Analyticity*	393
15.9 Solutions and interfaces for changing-sign solutions*	394
Notes	394
Problems	396
16 Full analysis of self-similarity	401
16.1 Scale invariance and self-similarity	401
16.1.1 Subfamilies	403
16.1.2 Invariance implies self-similarity	404
16.2 Three types of time self-similarity	405
16.3 Self-similarity and existence theory	407
16.4 Phase-plane analysis	408
16.4.1 The autonomous ODE system	409
16.4.2 Analysis of system (S1)	410
16.4.3 Some special solutions. Straight lines in phase plane	413
16.4.4 The special dimensions	414
16.5 An alternative phase plane	414
16.6 Sign-change trajectories. Complete inversion	418
16.6.1 Global analysis. Applications	420
16.7 Beyond blow-up growth. The oscillating signed solution	421
16.8 Phase plane for Type II	423
16.9 Other types of exact solutions	424
16.9.1 Ellipsoidal solutions of ZKB type	425
16.10 Self-similarity for GPME	426
Notes	427
Problems	428
17 Techniques of symmetrization and concentration	431
17.1 Functional preliminaries	431
17.1.1 Rearrangement	431
17.1.2 Schwarz symmetrization	432
17.1.3 Mass concentration	433
17.2 Concentration theory for elliptic equations	434
17.2.1 Solutions are less concentrated than their data	435
17.2.2 Integral super- and subsolutions	437
17.2.3 Comparison of solutions	438
17.3 Symmetrization and comparison. Elliptic case	439
17.3.1 Standard symmetrization result revisited	440
17.3.2 General symmetrization-concentration comparison	442
17.3.3 Problem in the whole space	444
17.4 Comparison theorems for the evolution	444
17.5 Smoothing effect and decay for the PME with L¹ functions or measures as data	446
17.5.1 The calculation of the best constant	447
17.5.2 Cases m less than or equal to 1	448
17.6 Smoothing exponents and scaling properties	449
17.7 Smoothing effect and time decay from Lp	450
Notes	452
Problems	453
18 Asymptotic behaviour I. The Cauchy problem	454
18.1 ZKB asymptotics for the PME	456
18.2 Proof of convergence for non-negative solutions	458
18.2.1 Completing the general case	464
18.3 Convergence of supports and interfaces	466
18.4 Continuous scaling version. Fokker—Planck equation	468
18.5 A Lyapunov method	469
18.6 The entropy approach. Convergence rates	473
18.6.1 Rates of convergence	475
18.7 Asymptotic behaviour in one space dimension	477
18.7.1 Adjusting the centre of mass. Improved convergence	477
18.7.2 Closer analysis of the velocity. N-waves	481
18.7.3 The quest for optimal rates	482
18.8 Asymptotic behaviour for signed solutions	483
18.8.1 Actual rates for M = 0	486
18.8.2 Asymptotics for the PME with forcing	487
18.8.3 Asymptotic expansions	488
18.9 Introduction to the fast diffusion case	488
18.9.1 Stabilization with convergence in relative pointwise error	489
18.9.2 Solutions of the FDE that remain two-signed	489
18.10 Various topics	491
18.10.1 Asymptotics of non-integrable solutions	491
18.10.2 Asymptotics of filtration equations	491
18.10.3 Asymptotics of superslow diffusion	492
18.10.4 Asymptotics of the PME in inhomogeneous media	493
18.10.5 Asymptotics for systems	494
18.10.6 Other	494
Notes	494
Problems	495
19 Regularity and finer asymptotics in several dimensions	498
19.1 Lipschitz and C¹ regularity for large times	499
19.1.1 Lipschitz continuity for the pressure	499
19.1.2 Lipschitz continuity of the free boundary	502
19.1.3 C¹α regularity	505
19.2 Focusing solutions and limited regularity	505
19.2.1 Propagation and hole filling. Unbounded speed	508
19.2.2 Asymptotic convergence	509
19.2.3 Continuation after the singularity	510
19.2.4 Multiple holes	510
19.3 Lipschitz continuity from space to time	510
19.4 Cinfinity regularity	513
19.4.1 Eliminating the admissibility condition	514
19.5 Further regularity results	514
19.5.1 Conservation of initial regularity	515
19.5.2 Concavity results	515
19.5.3 Eventual concavity	515
19.6 Various	516
19.6.1 Precise Hölder regularity	516
19.6.2 Fast diffusion flows	518
Notes	518
Problems	519
20 Asymptotic behaviour II. Dirichlet and Neumann problems	521
20.1 Large-time behaviour of the HDP. Non-negative solutions	521
20.1.1 Rate of convergence	526
20.1.2 Linear versus nonlinear	527
20.1.3 On general initial data	529
20.2 Asymptotic behaviour for signed solutions	529
20.2.1 Description of the ω-limit in d = 1	533
20.3 Asymptotics of the PME in a tubular domain	535
20.3.1 Basic asymptotic result	536
20.3.2 Lateral propagation. Logarithmic speed	537
20.4 Other Dirichlet problems	540
20.5 Asymptotics of the Neumann problem	543
20.5.1 Case of zero mass	544
20.5.2 Case of non-zero mass	544
20.6 Asymptotics on compact manifolds	546
Notes	546
Problems	547
COMPLEMENTS
21 Further applications	551
21.1 Thin liquid film spreading under gravity	551
21.1.1 Higher order models for thin films	552
21.1.2 Related application	553
21.2 The equations of unsaturated filtration	553
21.3 Immiscible fluids. Oil equations	554
21.4 Boundary layer theory	555
21.5 Spread of magma in volcanos	556
21.6 Signed solutions in groundwater flow	557
21.7 Limits of kinetic and radiation models	557
21.7.1 Carleman's model	557
21.7.2 Rosseland model	558
21.7.3 Marshak waves	559
21.8 The PME as the limit of particle models	559
21.9 Diffusive coagulation-fragmentation models	560
21.10 Diffusion in semiconductors	561
21.11 Contrast enhancement in image processing	561
21.12 Stochastic models. PME with noise	563
21.13 General filtration equations	563
21.14 Other	564
A Basic facts	565
A.1 Notations and basic facts	565
A.1.1 Points and sets	565
A.1.2 Functions	566
A.1.3 Integrals and derivatives	567
A.1.4 Functional spaces	567
A.1.5 Some integrals and constants	568
A.1.6 Various	569
A.2 Nonlinear operators	569
A.3 Maximal monotone graphs	570
A.3.1 Comparison of maximal monotone graphs	571
A.4 Measures	572
A.5 Marcinkiewicz spaces	573
A.6 Some ideas of potential theory	574
A.7 A lemma from measure theory	574
A.8 Results for semiharmonic functions	575
A.9 Three notes on the Giant and elliptic problems	577
A.9.1 Nonlinear elliptic approach. Calculus of variations	578
A.9.2 Another dynamical proof of existence	579
A.9.3 Another construction of the Giant	580
A.10 Optimality of the asymptotic convergence for the PME	581
A.11 Non-contractivity of the PME flow in Lp spaces	583
A.11.1 Other contractivity properties	586
Bibliography	588
Index	621