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Nonlinear Partial Differential Equations with Applications

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Nonlinear Partial Differential Equations with Applications ISNM 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: Tomásˇ 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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P.O.Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN-10: 3-7643-7293-1 e-ISBN: 3-7643-7397-0 ISBN-13: 978-3-7643-7293-4 9 8 7 6 5 4 3 2 1 www.birkhauser.ch To the memory of professor Jindˇrich Neˇcas Contents Preface xi Notational conventions xv 1 Preliminary general material 1 1.1Functionalanalysis........................... 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 Convexsets........................... 6 1.1.4 Compactness.......................... 7 1.1.5 Fixed-pointtheorems..................... 8 1.2Functionspaces............................. 8 1.2.1 Continuousandsmoothfunctions............... 9 1.2.2 Lebesgueintegrablefunctions................. 10 1.2.3 Sobolevspaces......................... 14 1.3 Nemytski˘ımappings.......................... 19 1.4Greenformulaandsomeinequalities................. 20 1.5Bochnerspaces............................. 22 1.6Someordinarydifferentialequations................. 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 2.3Equationswithmonotonemappings................. 35 2.4 Quasilinear elliptic equations . 40 2.4.1 Boundary-value problems for 2nd-order equations . 41 2.4.2 Weakformulation....................... 42 2.4.3 Pseudomonotonicity, coercivity, existence of solutions . 46 2.4.4 Higher-orderequations..................... 53 2.5 Weakly continuous mappings, semilinear equations . 56 viii Contents 2.6Examplesandexercises........................ 58 2.6.1 Generaltools.......................... 59 − A ∇ 2.6.2 Semilinear heat equation of type div( (x, u) u)=g ... 62 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 2.8Bibliographicalremarks........................ 86 3 Accretive mappings 89 3.1Abstracttheory............................. 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 Accretivityofheatequation.................. 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 3.3Exercises................................ 106 3.4Bibliographicalremarks........................ 107 4 Potential problems: smooth case 109 4.1Abstracttheory............................. 109 4.2 Application to boundary-value problems . 114 4.3Examplesandexercises........................ 120 4.4Bibliographicalremarks........................ 124 5 Nonsmooth problems; variational inequalities 125 5.1Abstractinclusionswithapotential................. 125 5.2 Application to elliptic variational inequalities . 129 5.3Someabstractnonpotentialinclusions................ 135 5.4 Excursion to quasivariational inequalities . 144 5.5Exercises................................ 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 5.7Bibliographicalremarks........................ 159 6 Systems of equations: particular examples 161 6.1 Minimization-type variational method: polyconvex functionals . 161 6.2Buoyancy-drivenviscousflow..................... 168 6.3Reaction-diffusionsystem....................... 172 6.4Thermistor............................... 175 6.5 Semiconductors . 178 Contents ix II EVOLUTION PROBLEMS 185 7 Special auxiliary tools 187 1,p,q 7.1 Sobolev-Bochner space W (I; V1,V2) ............... 187 7.2 Gelfand triple, embedding W 1,p,p (I;V,V∗)⊂C(I;H)........ 190 7.3Aubin-Lionslemma........................... 193 8 Evolution by pseudomonotone or weakly continuous mappings 199 8.1Abstractinitial-valueproblems.................... 199 8.2Rothemethod.............................. 201 8.3Furtherestimates............................ 215 8.4Galerkinmethod............................ 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 8.8Examplesandexercises........................ 242 8.8.1 Generaltools.......................... 242 ∂ p−2 8.8.2 Parabolic equation of type ∂tu−div(|∇u| ∇u)+c(u)=g . 244 ∂ 8.8.3 Semilinear heat equation c(u) ∂tu − div(κ(u)∇u)=g .... 252 ∂ 8.8.4 Navier-Stokes equation ∂tu+(u·∇)u−∆u+∇π=g, div u=0 . 255 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 8.11Bibliographicalremarks........................ 272 9 Evolution governed by accretive mappings 275 9.1Strongsolutions............................. 275 9.2Integralsolutions............................ 280 9.3Excursiontononlinearsemigroups.................. 286 9.4 Applications to initial-boundary-value problems . 291 9.5Applicationstosomesystems..................... 295 9.6Bibliographicalremarks........................ 302 10 Evolution governed by certain set-valued mappings 305 10.1 Abstract problems: strong solutions . 305 10.2Abstractproblems:weaksolutions.................. 309 10.3 Examples of unilateral parabolic problems . 313 10.4Bibliographicalremarks........................ 318 11 Doubly-nonlinear problems 321 d 11.1 Inclusions of the type ∂Ψ( dt u)+∂Φ(u) f ............. 321 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 d 11.2 Inclusions of the type dt E(u)+∂Φ(u) f .............. 334 11.2.1 The case E := ∂Ψ........................ 335 11.2.2 The case E nonpotential.................... 339 11.2.3Uniqueness........................... 341 11.32nd-orderequations.......................... 342 11.4Exercises................................ 351 11.5Bibliographicalremarks........................ 355 12 Systems of equations: particular examples 357 12.1Thermo-visco-elasticity......................... 357 12.2Buoyancy-drivenviscousflow..................... 361 12.3Predator-preysystem......................... 365 12.4 Semiconductors . 368 12.5Phase-fieldmodel............................ 372 12.6Navier-Stokes-Nernst-Planck-Poisson-typesystem.......... 376 References 383 Index 399 Preface The theoretical foundations of differential equations have been significantly devel- oped, especially during the 20th century. This growth can be attributed to fast and successful development of supporting mathematical disciplines (such as functional 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. A great number of applications involve distributed-parameter systems (which can be, in particular, described by partial differential equations1) often involving various nonlinearities. This book focuses on the theory of such equations with the aim of bringing it as fast as possible to a stage applicable to real-world tasks. This competition between rigor and applicability naturally needs many compromises to keep the scope reasonable. As a result (or, conversely, the reason for it) the book is primarily meant for graduate or PhD students in programs such as mathematical modelling or applied mathematics. Although some preliminary knowledge of mod- 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 prototype tasks addressed in this book are boundary-value problems for semilinear2 equations of the type −∆u + c(u)=g,
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