Nonlinear Partial Differential Equations with Applications (International Series of Numerical Mathematics)

485 Pages·2013·4.452 MB·English

by  Tomáš Roubíček

ISNM
International Series of Numerical Mathematics
Volume 153
ManagingEditors:
K.-H.Hoffmann,München, Germany
G.Leugering, Erlangen-Nürnberg, Germany
AssociateEditors:
Z.Chen,Beijing,China
R.H.W.Hoppe,Augsburg, Germany/Houston, USA
N.Kenmochi,Chiba,Japan
V.Starovoitov, Novosibirsk,Russia
HonoraryEditor:
J.Todd, Pasadena, USA†
Forfurthervolumes:
www.birkhauser-science.com/series/4819
Tomáš Roubí(cid:254)ek
Nonlinear Partial Differential
Equations with Applications
Second Edition
Tomáš Roubíček
Mathematical Institute
Charles University
Sokolovská 83
186 75 Praha 8
Czech Republic
and
Institute of Thermomechanics of the ASCR
Dolejškova 5
182 00 Praha 8
Czech Republic
and
Institute of Information Theory and Automation of the ASCR
Pod vodárenskou věží 4
182 08 Praha 8
Czech Republic

ISBN 978-3-0348-0512-4 ISBN 978-3-0348-0513-1 (eBook)
DOI 10.1007/978-3-0348-0513-1
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012956219
Mathematics Subject Classification (2010): Primary: 35Jxx, 35Kxx, 35Qxx, 47Hxx, 47Jxx, 49Jxx;
Secondary: 65Nxx, 74Bxx, 74Fxx, 76Dxx, 80Axx

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Contents
Preface xi
Preface to the 2nd edition xv
Notational conventions xvii
1 Preliminary general material 1
1.1 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Normed spaces, Banach spaces, locally convex spaces . . . . 1
1.1.2 Functions and mappings on Banach spaces, dual spaces . . 3
1.1.3 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.5 Fixed-point theorems . . . . . . . . . . . . . . . . . . . . . 8
1.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Continuous and smooth functions. . . . . . . . . . . . . . . 9
1.2.2 Lebesgue integrable functions . . . . . . . . . . . . . . . . . 10
1.2.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Nemytski˘ı mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Green formula and some inequalities . . . . . . . . . . . . . . . . . 20
1.5 Bochner spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Some ordinary differential equations . . . . . . . . . . . . . . . . . 25
I STEADY-STATE PROBLEMS 29
2 Pseudomonotone or weakly continuous mappings 31
2.1 Abstract theory, basic definitions, Galerkin method . . . . . . . . . 31
2.2 Some facts about pseudomonotone mappings . . . . . . . . . . . . 35
2.3 Equations with monotone mappings . . . . . . . . . . . . . . . . . 37
2.4 Quasilinear elliptic equations . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 Boundary-value problems for 2nd-order equations. . . . . . 43
2.4.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.3 Pseudomonotonicity, coercivity, existence of solutions . . . 48
vii
viii Contents
2.4.4 Higher-order equations . . . . . . . . . . . . . . . . . . . . . 56
2.5 Weakly continuous mappings, semilinear equations . . . . . . . . . 61
2.6 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 64
2.6.1 General tools . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.6.2 Semilinear heat equation of type −(cid:2)div(A(x,u)∇(cid:3)u)=g . . . 68
2.6.3 Quasilinear equations of type −div |∇u|p−2∇u +c(u,∇u)=g 75
2.7 Excursion to regularity for semilinear equations . . . . . . . . . . . 85
2.8 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 92
3 Accretive mappings 95
3.1 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 Applications to boundary-value problems . . . . . . . . . . . . . . 99
3.2.1 Duality mappings in Lebesgue and Sobolev spaces . . . . . 99
3.2.2 Accretivity of monotone quasilinear mappings . . . . . . . . 101
3.2.3 Accretivity of heat equation . . . . . . . . . . . . . . . . . . 105
3.2.4 Accretivity of some other boundary-value problems . . . . . 108
3.2.5 Excursion to equations with measures in right-hand sides . 109
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 114
4 Potential problems: smooth case 115
4.1 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Application to boundary-value problems . . . . . . . . . . . . . . . 120
4.3 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 126
4.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Nonsmooth problems; variational inequalities 133
5.1 Abstract inclusions with a potential . . . . . . . . . . . . . . . . . 133
5.2 Application to elliptic variational inequalities . . . . . . . . . . . . 137
5.3 Some abstract non-potential inclusions . . . . . . . . . . . . . . . . 145
5.4 Excursion to quasivariational inequalities . . . . . . . . . . . . . . 154
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.6 Some applications to free-boundary problems . . . . . . . . . . . . 163
5.6.1 Porous media flow: a potential variational inequality . . . . 163
5.6.2 Continuous casting: a non-potential variational inequality . 166
5.7 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 169
6 Systems of equations: particular examples 171
6.1 Minimization-type variational method: polyconvex functionals . . . 171
6.2 Buoyancy-drivenviscous flow . . . . . . . . . . . . . . . . . . . . . 178
6.3 Reaction-diffusion system . . . . . . . . . . . . . . . . . . . . . . . 186
6.4 Thermistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.5 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Contents ix
II EVOLUTION PROBLEMS 199
7 Special auxiliary tools 201
7.1 Sobolev-Bochner space W1,p,q(I;V1,V2) . . . . . . . . . . . . . . . 201
7.2 Gelfand triple, embedding W1,p,p(cid:2)(I;V,V∗)⊂C(I;H) . . . . . . . . 204
7.3 Aubin-Lions lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8 Evolution by pseudomonotone or weakly continuous mappings 213
8.1 Abstract initial-value problems . . . . . . . . . . . . . . . . . . . . 213
8.2 Rothe method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.3 Further estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.4 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.5 Uniqueness and continuous dependence on data . . . . . . . . . . . 247
8.6 Application to quasilinear parabolic equations . . . . . . . . . . . . 251
8.7 Application to semilinear parabolic equations . . . . . . . . . . . . 261
8.8 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 264
8.8.1 General tools . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.8.2 Parabolic equation of type ∂ u−div(|∇u|p−2∇u)+c(u)=g . 266
∂t
8.8.3 Semilinear heat equation c(u)∂ u−div(κ(u)∇u)=g . . . . 277
∂t
8.8.4 Navier-Stokes equation ∂ u+(u·∇)u−Δu+∇π=g, divu=0 . 279
∂t
8.8.5 Some more exercises . . . . . . . . . . . . . . . . . . . . . . 282
8.9 Global monotonicity approach, periodic problems . . . . . . . . . . 288
8.10 Problems with a convex potential: direct method . . . . . . . . . . 294
8.11 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 300
9 Evolution governed by accretive mappings 303
9.1 Strong solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
9.2 Integral solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
9.3 Excursion to nonlinear semigroups . . . . . . . . . . . . . . . . . . 314
9.4 Applications to initial-boundary-value problems . . . . . . . . . . . 319
9.5 Applications to some systems . . . . . . . . . . . . . . . . . . . . . 326
9.6 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 332
10 Evolution governed by certain set-valued mappings 335
10.1 Abstract problems: strong solutions. . . . . . . . . . . . . . . . . . 335
10.2 Abstract problems: weak solutions . . . . . . . . . . . . . . . . . . 339
10.3 Examples of unilateral parabolic problems . . . . . . . . . . . . . . 343
10.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 349
11 Doubly-nonlinear problems 351
11.1 Inclusions of the type ∂Ψ(du)+∂Φ(u)(cid:4)f . . . . . . . . . . . . . 351
dt
11.1.1 Potential Ψ valued in R∪{+∞}. . . . . . . . . . . . . . . . 351
11.1.2 Potential Φ valued in R∪{+∞} . . . . . . . . . . . . . . . 358
11.1.3 Uniqueness and continuous dependence on data . . . . . . . 366
x Contents
11.2 Inclusions of the type dE(u)+∂Φ(u)(cid:4)f . . . . . . . . . . . . . . 367
dt
11.2.1 The case E :=∂Ψ. . . . . . . . . . . . . . . . . . . . . . . . 368
11.2.2 The case E non-potential . . . . . . . . . . . . . . . . . . . 372
11.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
11.3 2nd-order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 377
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
11.5 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 390
12 Systems of equations: particular examples 393
12.1 Thermo-visco-elasticity. . . . . . . . . . . . . . . . . . . . . . . . . 393
12.2 Buoyancy-drivenviscous flow . . . . . . . . . . . . . . . . . . . . . 405
12.3 Predator-prey system . . . . . . . . . . . . . . . . . . . . . . . . . 408
12.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
12.5 Phase-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
12.6 Navier-Stokes-Nernst-Planck-Poisson-typesystem . . . . . . . . . . 420
12.7 Thermistor with eddy currents . . . . . . . . . . . . . . . . . . . . 426
12.8 Thermodynamics of magnetic materials . . . . . . . . . . . . . . . 432
12.9 Thermo-visco-elasticity: fully nonlinear theory . . . . . . . . . . . . 438
Bibliography 449
Index 469