Handbook of differential equations: evolutionary equations

677 Pages·2005·3.041 MB·English

by  C.M. Dafermos, Eduard Feireisl

HANDBOOK
OF DIFFERENTIAL EQUATIONS
EVOLUTIONARY EQUATIONS
VOLUME II
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H
ANDBOOK
D E
OF IFFERENTIAL QUATIONS
E E
VOLUTIONARY QUATIONS
Volume II
Editedby
C.M. DAFERMOS
DivisionofAppliedMathematics
BrownUniversity
Providence,USA
E. FEIREISL
MathematicalInstituteASCR
Praha,CzechRepublic
2005
Amsterdam Boston Heidelberg London NewYork Oxford
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Preface
Thisis thesecondvolumein theseries EvolutionaryEquations,partof theHandbookof
Differential Equations project. Whereas Volume I was intended to provide an overview
of diverse abstract approaches, the guiding philosophy of the present volume is to offer
arepresentativesampleofthemostchallengingspecificequationsandsystemsarisingin
scientificapplications.
Threechaptersaredevotedtothemodernmathematicaltheoryoffluiddynamics:Chap-
ter1dealswiththeEulerequations,Chapter5providesageneralintroductiontothetheory
ofincompressibleviscousfluids,andChapter3discussestheasymptoticlimitsofdiscrete
mechanicalsystemsdescribedbytheBoltzmannequation.
In a different direction, Chapter 2 introduces the blow-up phenomena of solutions of
generalparabolicequationsandsystems.
Chapters 4 and 6 are closely related and deal with mathematical problems arising in
materialsscience.
Finally,Chapter7exploresthetopicofnonlinearwaveequations.
Wehavedeliberatelychosendiversetopicsaswellasstylesofpresentationinorderto
exposethereadertotheenormousvarietyofproblems,methodologyandpotentialappli-
cations.
Weshouldliketoexpressourthankstotheauthorswhohavecontributedtothepresent
volume, to the referees who have generously spent time reading the papers, and to the
editorsandstaffofElsevier.
ConstantineDafermos
EduardFeireisl
v
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List of Contributors
Chen,G.-Q.,NorthwesternUniversity,Evanston,IL,USA(Ch.1)
Fila,M.,ComeniusUniversity,Bratislava,Slovakia(Ch.2)
Golse,F.,InstitutUniversitairedeFranceandUniversitéParis7,Paris,France(Ch.3)
Krejcˇí, P., Weierstrass-Institute for Applied Analysis and Stochastics, Berlin, Germany
(Ch.4)
Málek,J.,CharlesUniversity,Praha,CzechRepublic(Ch.5)
Mielke, A., Weierstraß-Institut für Angewandte Analysis und Stochastik and Humboldt-
UniversitätzuBerlin,Berlin,Germany(Ch.6)
Rajagopal,K.R.,TexasA&MUniversity,CollegeStation,TX,USA(Ch.5)
Zhang,P.,AcademyofMathematics&SystemsScience,Beijing,China(Ch.7)
Zheng,Y.,PennStateUniversity,UniversityPark,PA,USA(Ch.7)
vii
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Contents
Preface v
ListofContributors vii
ContentsofVolumeI xi
1. EulerEquationsandRelatedHyperbolicConservationLaws 1
G.-Q.Chen
2. Blow-upofSolutionsofSupercriticalParabolicEquations 105
M.Fila
3. TheBoltzmannEquationandItsHydrodynamicLimits 159
F.Golse
4. Long-TimeBehaviorofSolutionstoHyperbolicEquationswithHysteresis 303
P.Krejcˇí
5. MathematicalIssuesConcerningtheNavier–StokesEquationsandSome
ofItsGeneralizations 371
J.MálekandK.R.Rajagopal
6. EvolutionofRate-IndependentSystems 461
A.Mielke
7. OntheGlobalWeakSolutionstoaVariationalWaveEquation 561
P.ZhangandY.Zheng
AuthorIndex 649
SubjectIndex 659
ix