Table Of Content

This page intentionally left blank HANDBOOK OF MATHEMATICAL FLUID DYNAMICS VOLUME III This page intentionally left blank 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 Amsterdam • Boston • Heidelberg • London • NewYork •Oxford•Paris • San Diego •San Francisco • Singapore •Sydney • Tokyo ELSEVIERB.V. ELSEVIERInc. ELSEVIERLtd ELSEVIERLtd SaraBurgerhartstraat25 525BStreet,Suite1900 TheBoulevard,LangfordLane 84TheobaldsRoad P.O.Box211,1000AEAmsterdam SanDiego,CA92101-4495 Kidlington,OxfordOX51GB LondonWC1X8RR TheNetherlands USA UK UK ©2004ElsevierB.V.Allrightsreserved. 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PrintedinTheNetherlands 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 This page intentionally left blank 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 This page intentionally left blank