9780198569039

The Porous Medium Equation Mathematical Theory

by
  • ISBN13:

    9780198569039

  • ISBN10:

    0198569033

  • Format: Hardcover
  • Copyright: 2006-12-28
  • Publisher: Clarendon Pr

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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

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