Table Of ContentISNM
International Series of Numerical Mathematics
Volume 153
Managing Editors:
Karl-Heinz Hoffmann, Bonn
D. Mittelmann, Tempe
Associate Editors:
R.E. Bank, La Jolla
H. Kawarada, Chiba
R.J. LeVeque, Seattle
C. Verdi, Milano
Honorary Editor:
J. Todd, Pasadena
Nonlinear Partial
Differential Equations
with Applications
Tomásˇ Roubícˇek
Birkhäuser Verlag
Basel . Boston . Berlin
Author:
Tomáˇs Roubícˇek
School of Mathematics
Charles University
Faculty of Mathematics and Physics
Sokolovská 83
186 75 Praha 8 - Karlin
Czech Republic
2000 Mathematics Subject Classification: Primary 35Jxx, 35Kxx, 35Qxx, 47Hxx, 47Jxx, 49Jxx; Secondary 65Mxx,
65Nxx, 74Bxx, 74Fxx, 76Dxx, 76Rxx, 80Axx
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available
in the Internet at <http://dnb.ddb.de>.
ISBN 3-7643-7293-1 Birkhäuser Verlag, Basel – Boston – Berlin
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© 2005 Birkhäuser Verlag, P.O.Box 133, CH-4010 Basel, Switzerland
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ISBN-10: 3-7643-7293-1 e-ISBN: 3-7643-7397-0
ISBN-13: 978-3-7643-7293-4
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To the memory of professor Jindˇrich Neˇcas
Contents
Preface xi
Notational conventions xv
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 27
2 Pseudomonotone or weakly continuous mappings 29
2.1 Abstract theory, basic definitions, Galerkin method . . . . . . . . . 29
2.2 Some facts about pseudomonotone mappings . . . . . . . . . . . . 33
2.3 Equations with monotone mappings . . . . . . . . . . . . . . . . . 35
2.4 Quasilinear elliptic equations . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Boundary-value problems for 2nd-order equations. . . . . . 41
2.4.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.3 Pseudomonotonicity,coercivity, existence of solutions . . . 46
2.4.4 Higher-order equations . . . . . . . . . . . . . . . . . . . . . 53
2.5 Weakly continuous mappings, semilinear equations . . . . . . . . . 56
viii Contents
2.6 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 58
2.6.1 General tools . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6.2 Semilinear heat equation of type −div(A(x,u)∇u)=g . . . 62
(cid:1) (cid:2)
2.6.3 Quasilinear equations of type −div |∇u|p−2∇u +c(u,∇u)=g 69
2.7 Excursion to regularity for semilinear equations . . . . . . . . . . . 79
2.8 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 86
3 Accretive mappings 89
3.1 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2 Applications to boundary-value problems . . . . . . . . . . . . . . 93
3.2.1 Duality mappings in Lebesgue and Sobolev spaces . . . . . 93
3.2.2 Accretivity of monotone quasilinear mappings . . . . . . . . 95
3.2.3 Accretivity of heat equation . . . . . . . . . . . . . . . . . . 99
3.2.4 Accretivity of some other boundary-value problems . . . . . 102
3.2.5 Excursion to equations with measures in right-hand sides . 103
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 107
4 Potential problems: smooth case 109
4.1 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2 Application to boundary-value problems . . . . . . . . . . . . . . . 114
4.3 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 124
5 Nonsmooth problems; variational inequalities 125
5.1 Abstract inclusions with a potential . . . . . . . . . . . . . . . . . 125
5.2 Application to elliptic variational inequalities . . . . . . . . . . . . 129
5.3 Some abstract nonpotential inclusions . . . . . . . . . . . . . . . . 135
5.4 Excursion to quasivariational inequalities . . . . . . . . . . . . . . 144
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6 Some applications to free-boundary problems . . . . . . . . . . . . 152
5.6.1 Porous media flow: a potential variational inequality . . . . 152
5.6.2 Continuous casting: a nonpotential variational inequality . 156
5.7 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 159
6 Systems of equations: particular examples 161
6.1 Minimization-type variational method: polyconvex functionals . . . 161
6.2 Buoyancy-drivenviscous flow . . . . . . . . . . . . . . . . . . . . . 168
6.3 Reaction-diffusion system . . . . . . . . . . . . . . . . . . . . . . . 172
6.4 Thermistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.5 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Contents ix
II EVOLUTION PROBLEMS 185
7 Special auxiliary tools 187
7.1 Sobolev-Bochner space W1,p,q(I;V1,V2) . . . . . . . . . . . . . . . 187
7.2 Gelfand triple, embedding W1,p,p(cid:1)(I;V,V∗)⊂C(I;H) . . . . . . . . 190
7.3 Aubin-Lions lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8 Evolution by pseudomonotone or weakly continuous mappings 199
8.1 Abstract initial-value problems . . . . . . . . . . . . . . . . . . . . 199
8.2 Rothe method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.3 Further estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.4 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.5 Uniqueness and continuous dependence on data . . . . . . . . . . . 228
8.6 Application to quasilinear parabolic equations . . . . . . . . . . . . 232
8.7 Application to semilinear parabolic equations . . . . . . . . . . . . 239
8.8 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 242
8.8.1 General tools . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.8.2 Parabolic equation of type ∂ u−div(|∇u|p−2∇u)+c(u)=g . 244
∂t
8.8.3 Semilinear heat equation c(u)∂ u−div(κ(u)∇u)=g . . . . 252
∂t
8.8.4 Navier-Stokes equation ∂ u+(u·∇)u−∆u+∇π=g, divu=0 . 255
∂t
8.8.5 Some more exercises . . . . . . . . . . . . . . . . . . . . . . 257
8.9 Global monotonicity approach, periodic problems . . . . . . . . . . 262
8.10 Problems with a convex potential: direct method . . . . . . . . . . 267
8.11 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 272
9 Evolution governed by accretive mappings 275
9.1 Strong solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
9.2 Integral solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.3 Excursion to nonlinear semigroups . . . . . . . . . . . . . . . . . . 286
9.4 Applications to initial-boundary-value problems . . . . . . . . . . . 291
9.5 Applications to some systems . . . . . . . . . . . . . . . . . . . . . 295
9.6 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 302
10 Evolution governed by certain set-valued mappings 305
10.1 Abstract problems: strong solutions. . . . . . . . . . . . . . . . . . 305
10.2 Abstract problems: weak solutions . . . . . . . . . . . . . . . . . . 309
10.3 Examples of unilateral parabolic problems . . . . . . . . . . . . . . 313
10.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 318
11 Doubly-nonlinear problems 321
11.1 Inclusions of the type ∂Ψ(du)+∂Φ(u)(cid:3)f . . . . . . . . . . . . . 321
dt
11.1.1 Potential Ψ valued in R∪{+∞}. . . . . . . . . . . . . . . . 321
11.1.2 Potential Φ valued in R∪{+∞} . . . . . . . . . . . . . . . 327
11.1.3 Uniqueness and continuous dependence on data . . . . . . . 332
x Contents
11.2 Inclusions of the type dE(u)+∂Φ(u)(cid:3)f . . . . . . . . . . . . . . 334
dt
11.2.1 The case E :=∂Ψ. . . . . . . . . . . . . . . . . . . . . . . . 335
11.2.2 The case E nonpotential . . . . . . . . . . . . . . . . . . . . 339
11.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
11.3 2nd-order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 342
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
11.5 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 355
12 Systems of equations: particular examples 357
12.1 Thermo-visco-elasticity. . . . . . . . . . . . . . . . . . . . . . . . . 357
12.2 Buoyancy-drivenviscous flow . . . . . . . . . . . . . . . . . . . . . 361
12.3 Predator-prey system . . . . . . . . . . . . . . . . . . . . . . . . . 365
12.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
12.5 Phase-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
12.6 Navier-Stokes-Nernst-Planck-Poisson-typesystem . . . . . . . . . . 376
References 383
Index 399
Preface
Thetheoreticalfoundationsofdifferentialequationshavebeensignificantlydevel-
oped,especiallyduringthe20thcentury.Thisgrowthcanbeattributedtofastand
successfuldevelopmentofsupportingmathematicaldisciplines(suchasfunctional
analysis, measure theory, and function spaces) as well as to an ever-growing call
for applications especially in engineering, science, and medicine, and ever better
possibility to solve more and more complicated problems on computers due to
constantly growing hardware efficiency as well as development of more efficient
numerical algorithms.
Agreatnumberofapplicationsinvolvedistributed-parametersystems(which
can be, in particular, described by partial differential equations1) often involving
variousnonlinearities.This book focuses onthe theory ofsuchequations withthe
aimofbringingitasfastaspossibletoastageapplicabletoreal-worldtasks.This
competitionbetweenrigorandapplicabilitynaturallyneedsmanycompromisesto
keepthescopereasonable.Asaresult(or,conversely,thereasonforit)thebookis
primarily meant for graduate or PhD students in programssuch as mathematical
modellingorappliedmathematics.Althoughsomepreliminaryknowledgeofmod-
ern methods in linear partial differential equations is useful, the book is basically
self-contained if the reader consults Chapter 1 where auxiliary material is briefly
presented without proofs.
The prototypetasksaddressedinthis bookareboundary-valueproblemsfor
semilinear2 equations of the type
(cid:1) (cid:2)
−∆u+c(u)=g, or more general −div κ(u)∇u +c(u)=g, (0.1)
or, still more general, for quasilinear3 equations of the type
(cid:1) (cid:2)
−div a(u,∇u) +c(u,∇u)=g, (0.2)
1Theadjective“partial”referstooccurrenceofpartialderivatives.
2Inthisbooktheadjective“semilinear”willrefertoequations wherethehighestderivatives
standlinearlyandtheinducedmappingsonfunctionspacesareweaklycontinuous.
3The adjective “quasilinear” refers to equations where the highest derivatives occur lin-
earlybutmultipliedbyfunctionscontaininglower-orderderivatives,whichmeansheretheform
− ni,j=1aij(x,u,∇u)∂2u/∂xi∂xj+c(x,u,∇u)=g.Afterapplyingthechainrule,onecansee
tha(cid:0)t(0.2)isonlyaspecialcase,namelyanequation intheso-calleddivergenceform.
