Handbook of differential equations: evolutionary equations

579 Pages·2004·3.347 MB·English

by  C.M. Dafermos, Eduard Feireisl

HANDBOOK
OF DIFFERENTIAL EQUATIONS
EVOLUTIONARY EQUATIONS
VOLUME I
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H
ANDBOOK
D E
OF IFFERENTIAL QUATIONS
E E
VOLUTIONARY QUATIONS
Volume I
Editedby
C.M. DAFERMOS
DivisionofAppliedMathematics
BrownUniversity
Providence,USA
E. FEIREISL
MathematicalInstituteASCR
Praha,CzechRepublic
2004
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Preface
The aim of this Handbookis to acquaintthe reader with the currentstatus of the theory
of evolutionary partial differential equations, and with some of its applications. This is
not an easy task: Unlike other mathematical theories which exhibit a tree-like structure,
withclearlydistinguishabletrunk,mainandsecondarybranches,thetheoryofpartialdif-
ferentialequationshastheappearanceofabushwithverycomplexstructure.Itsrootsas
wellasitsflowersareoftenrelatedtothephysicalworld,anditisfertilizedbyideasand
techniquesborrowedfromvirtuallyeveryotherareaofmathematics.
Evolutionarypartialdifferentialequationsmade their first appearancein the 18thcen-
tury,intheendeavortounderstandthemotionoffluidsandothercontinuousmedia.Itis
remarkable that this research program is still ongoing and many fundamental questions
remainunanswered.Beyondthisarea,however,evolutionarypartialdifferentialequations
havebecomeubiquitous,astheyseem to governthe dynamicsofanyphysical,chemical
biological, ecological or economic system whose state is described by spatially depen-
dentvariables.
Theactiveresearcheffortoverthespanoftwocenturies,combinedwiththewidevari-
etyofphysicalphenomenathathadtobeexplained,hasresultedinanenormousbodyof
literature.Anyattempttoproduceacomprehensivesurveywouldbefutile.Theaimhereis
tocollectreviewarticles,writtenbyleadingexperts,whichwillhighlightthepresentand
expectedfuturedirectionsofdevelopmentofthefield.Theemphasiswillbeonnonlinear
equations,whichposethemostchallengingproblemstoday.Thevariousarticleswilloffer
thereadertheopportunitytocompareandcontrastthebehaviorofhyperbolicandparabolic
equations. Hyperbolic equations are typically associated with media exhibiting “elastic”
response,andencompassthenotoriouslydifficultclassof“hyperbolicconservationlaws”.
Parabolicequationsare associated with variousdiffusivemechanisms.An extremelyim-
portantandchallengingexampleistheNavier–Stokesequation.
VolumeIofthisHandbookwillfocusontheabstracttheoryofevolutionaryequations,
addressing questions of existence, uniquenessand other general qualitative propertiesof
solutions.Futurevolumeswillconsidermoreconcreteproblemsrelatingtospecificappli-
cations.Ourhopeisthatthehandbookwillprovideapanoramaofthisamazinglycomplex
andrapidlydevelopingbranchofmathematics.
ConstantinDafermos
EduardFeireisl
v
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List of Contributors
Arendt,W.,UniversitätUlm,Ulm,Germany(Ch.1)
Bressan,A.,PennStateUniversity,PA,USA(Ch.2)
DiBenedetto,E.,VanderbiltUniversity,Nashville,TN,USA(Ch.3)
Hsiao,L.,AcademyofMathematicsandSystemsScience,AcademiaSinica,Beijing,China
(Ch.4)
Jiang, S., Institute of Applied Physics and Computational Mathematics, Beijing, China
(Ch.4)
Lunardi,A.,UniversitàdiParma,Parma,Italy(Ch.5)
Perthame,B.,ÉcoleNormaleSupérieure,Paris,France(Ch.6)
Serre,D.,ENSLyon,UMPA,Lyon,France(Ch.7)
Urbano,J.M.,UniversidadedeCoimbra,Coimbra,Portugal(Ch.3)
Vespri,V.,UniversitàdiFirenze,Firenze,Italy(Ch.3)
vii
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Contents
Preface v
ListofContributors vii
1. Semigroups and Evolution Equations: Functional Calculus, Regularity and
KernelEstimates 1
W.Arendt
2. FrontTrackingMethodforSystemsofConservationLaws 87
A.Bressan
3. CurrentIssuesonSingularandDegenerateEvolutionEquations 169
E.DiBenedetto,J.M.UrbanoandV.Vespri
4. NonlinearHyperbolic–ParabolicCoupledSystems 287
L.HsiaoandS.Jiang
5. NonlinearParabolicEquationsandSystems 385
A.Lunardi
6. Kinetic Formulations of Parabolic and Hyperbolic PDEs: From Theory to
Numerics 437
B.Perthame
7. L1-stabilityofNonlinearWavesinScalarConservationLaws 473
D.Serre
AuthorIndex 555
SubjectIndex 565
ix