Handbook of Mathematical Fluid Dynamics

688 Pages·2004·3.074 MB·English

by  S. Friedlander, D. Serre

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HANDBOOK
OF MATHEMATICAL
FLUID DYNAMICS
VOLUME III
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H
ANDBOOK
M
OF ATHEMATICAL
F D
LUID YNAMICS
Volume III
Editedby
S. FRIEDLANDER
UniversityofIllinois-Chicago,Chicago,Illinois,USA
D. SERRE
EcoleNormaleSupérieuredeLyon,Lyon,France
2004
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Contents of the Handbooks
VolumeI
1. TheBoltzmannequationandfluiddynamics,C.Cercignani 1
2. Areviewofmathematicaltopicsincollisionalkinetictheory,C.Villani 71
3. Viscousand/orheatconductingcompressiblefluids,E.Feireisl 307
4. Dynamicflowswithliquid/vaporphasetransitions,H.FanandM.Slemrod 373
5. TheCauchyproblemfortheEulerequationsforcompressiblefluids,
G.-Q.ChenandD.Wang 421
6. StabilityofstrongdiscontinuitiesinfluidsandMHD,A.Blokhinand
Y.Trakhinin 545
7. Onthemotionofarigidbodyinaviscousliquid:Amathematicalanalysiswith
applications,G.P.Galdi 653
VolumeII
1. Statisticalhydrodynamics,R.Robert 1
2. Topicsonhydrodynamicsandvolumepreservingmaps,Y.Brenier 55
3. WeaksolutionsofincompressibleEulerequations,A.Shnirelman 87
4. NearidentitytransformationsfortheNavier–Stokesequations,
P.Constantin 117
5. PlanarNavier–Stokesequations:Vorticityapproach,M.Ben-Artzi 143
6. AttractorsofNavier–Stokesequations,A.V.Babin 169
7. Stabilityandinstabilityinviscousfluids,M.RenardyandY.Renardy 223
8. Localizedinstabilitiesinfluids,S.FriedlanderandA.Lipton-Lifschitz 289
9. Dynamotheory,A.D.Gilbert 355
10. Water-wavesasaspatialdynamicalsystem,F.DiasandG.Iooss 443
11. SolvingtheEinsteinequationsbyLipschitzcontinuousmetrics:
Shockwavesingeneralrelativity,J.Groah,B.TempleandJ.Smoller 501
v
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Preface
Thisvolumecomprisesthethirdvolumeinthecurrent3volumeseriesofarticlescovering
manyaspects of mathematicalfluid dynamics.In the three volumeswe have endeavored
to illustrate the breadth and depth of a subject that has its roots stretching back several
centuries. Yet it remains a vital source of open questions that are challenging from both
mathematical and physical points of view. We have concentrated on mathematical issue
arisingfromfluidmodels,partlytokeepthelengthundercontrolbutmainlybecausephys-
ical and numericalaspects of fluid dynamicsare well developedin other comprehensive
collections.
Volume1ismoreorlessspecializedtocompressiblefluidsandbeginswithkineticmod-
elsthataretheverysourceofalmostallotherfluidmodelsundervariouslimitingregimes.
Volume2containsawiderangeofmaterialwiththemajorityofthearticlesaddressingis-
suesrelatedtoincompressiblefluidsandspecificphysicalproblems.Thetablesofcontents
ofVolumes1and2canbefoundonp.vofVolume3.
Volume3beginswiththefundamentalandchallengingissuesofhowandwhyadiscrete
systemofahugenumberofparticlescanbereplacedbyafluidcontinuum.TheEulerequa-
tions for an inviscid fluid are in some sense the most basic analytical description of this
continuum.EvenintwodimensionsthereareimportantcurrentdevelopmentsfortheEuler
equations;onesuchconcernstheshapeandmotionofvortexpatcheswhicharediscussed
inthesecondarticle.TheremainingarticlesinVolume3concentrateontheNavier–Stokes
equationsforviscousfluids.Anareaofmathematicsthatisbecomingincreasinglyimpor-
tantinfluiddynamicsisharmonicanalysis.TechniquesandresultsobtainedfortheNavier–
Stokesequationsusingharmonicanalysisaredescribedinthethirdarticle.Thefourthand
sixtharticlesdiscussmathematicalissuesthataredirectlymotivatedbyphysicalandgeo-
physical phenomena, namely boundary layers and fluid dynamics relevant to the oceans
and atmosphere. The monographlength fifth article returns to the topic of compressible
fluidsandgivesacomprehensivetreatmentofthestabilityoflargeamplitudeshockwaves.
We repeatourheartfeltthankstoalltheauthorswhohaveworkedso hardtowrite the
excellentarticlesinthethreevolumesoftheHandbookofMathematicalFluidDynamics.
Wealsothankthemanyrefereeswhohavegenerouslycontributedmuchtimetohelpen-
surethehighqualityofthearticles.OnceagainwethanktheeditorsandstaffofElsevier
forthequalityoftheproductionofthesesubstantialvolumes.Wehopethatthefascinating
varietyofmathematicsthatarisesfromthestudyoffluidmotionwillcontinuetoinspire,
motivateandchallengescientistsinthefutureasithasinthepast.
ChicagoandLyon,January,2004
SusanFriedlanderandDenisSerre
[email protected]@umpa.ens-lyon.fr
vii
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