HANDBOOK OF MATHEMATICAL FLUID DYNAMICS VOLUME IV H ANDBOOK M OF ATHEMATICAL F D LUID YNAMICS Volume IV Editedby S. FRIEDLANDER UniversityofIllinois-Chicago Chicago,Illinois,USA D. SERRE EcoleNormaleSupérieuredeLyon Lyon,France Amsterdam•Boston•Heidelberg•London•NewYork•Oxford Paris•SanDiego•SanFrancisco•Singapore•Sydney•Tokyo North-HollandisanimprintofElsevier Radarweg29,POBox211,1000AEAmsterdam,TheNetherlands TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Firstedition2007 Copyright©2007ElsevierB.V.Allrightsreserved Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyformor byanymeanselectronic,mechanical,photocopying,recordingorotherwisewithoutthepriorwritten permissionofthepublisher PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRightsDepartmentinOx- ford,UK:phone(+44)(0)1865843830;fax(+44)(0)1865853333;email:[email protected]. AlternativelyyoucansubmityourrequestonlinebyvisitingtheElsevierwebsiteathttp://elsevier.com/ locate/permissions,andselectingObtainingpermissiontouseElseviermaterial Notice Noresponsibilityisassumedbythepublisherforanyinjuryand/ordamagetopersonsorpropertyasa matterofproductsliability,negligenceorotherwise,orfromanyuseoroperationofanymethods,prod- ucts,instructionsorideascontainedinthematerialherein.Becauseofrapidadvancesinthemedical sciences,inparticular,independentverificationofdiagnosesanddrugdosagesshouldbemade LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN-13: 9780444528346 ForinformationonallNorth-Hollandpublications visitourwebsiteatbooks.elsevier.com PrintedandboundinTheNetherlands 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1 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 vi ContentsoftheHandbooks VolumeIII 1. Fromparticlestofluids,R.EspositoandM.Pulvirenti 1 2. Two-dimensionalEulersystemandthevortexpatchesproblem,J.-Y.Chemin 83 3. Harmonic analysis tools for solving the incompressible Navier–Stokes equa- tions,M.Cannone 161 4. Boundarylayers,E.Grenier 245 5. Stabilityoflarge-amplitudeshockwavesofcompressibleNavier–Stokesequa- tions,K.Zumbrun(withAppendixbyH.K.JenssenandG.Lyng) 311 6. Some mathematical problems in geophysical fluid dynamics, R. Temam and M.Ziane 535 Preface This is the fourth volume in a series of survey articles covering many aspects of mathe- matical fluid dynamics. Our original intention was to stop with three volumes. However, even over 2000 pages in the first three volumes do not cover a number of lively areas in thesubject.Wethereforedecidedtoeditanothervolume,notwiththeunrealisticgoalof beingcompletelycomprehensive,butinordertoaddcertaintopicswherethereiscurrently considerableactivityandprogress.Wehopethatthetotalcollectionofarticlesillustrates thebreathanddepthofmathematicalfluiddynamics,whichremainsavitalsourceofopen mathematicalproblemsandexcitingphysics. Volume 4 starts with an article illustrating that topology and geometry (branches of mathematicsnotconventionallyassociatedwiththestudyoffluids)haveanimportantrole toplayinexaminingthebehaviourofthemotionofanidealfluid.Thisisfollowedbyan articleaboutinviscidcompressiblefluids,afieldhavingnotreceivedattentioninthesecond andthirdvolumes.Itisdevotedtotheinteractionofthemostelementarysolutions,namely multi-dimensional shock waves. The next paper makes the important bridge between the compressibleandincompressiblefluidmodels.Itisalsothefirstofthreeinthisvolumeto dealwithasymptoticanalysisforfluids. Theremainingsixarticlesaddressvarioustopicsconcernedwithincompressibleviscous fluidsandtheNavier–Stokesequations.Animportantmathematicalissue(relatedtoaone- milliondollarprize!)is,ofcourse,theregularityofthesolutionsoftheseequations:known results and open questions are presented in the 4th article. The next two articles discuss geophysical fluid dynamics. Mathematical models for the ocean and the atmosphere are presentedanditisshownhowcertainmodelscanbeusedtoanalyzemathematicallysome precise geophysical phenomena. The 5th has an emphasis on asymptotic analysis, with applicationstoeithermid-latitudesorthetropics.The6thismoreconcernedwithperiodic forcingdue,forexample,tocircadianorseasonalrhythms.Closeconnectionswithphysics continuein the nextarticles.The mathematicalproperties of the equationsgoverningthe flow of fluids with pressure and shear rate dependent viscosities is the topic of the 7th article. The 8th paper is a very comprehensive review of results concerning flows in domains withoutletsatinfinity. The final article explains how homogenization theory provides models for flows in porous media and other non-homogeneous situations. For instance, it gives a rigorous derivationofDarcy’slaw. Againwerepeatourheartfeltthankstoauthorsofthearticlesinvolume4andalsothe authors of all the articles in the previous volumes. We are deeply indebted to them for the immense amountof work that writing such excellentsurvey articles requires. We are vii viii Preface also very grateful to the referees who generously spent much timeand thought to ensure the high quality, scholarship and enduring value of the Handbook of Mathematical Fluid Dynamics. We thank the Editors and staff at Elsevier who were unfailingly helpful and professionalandwhoproducedtheexcellentprintingofthepublishedbooks. ChicagoandLyon,September,2006 SusanFriedlanderandDenisSerre [email protected]@umpa.ens-lyon.fr List of Contributors Conca, C., Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas,UniversidaddeChile,andCentrodeModelamientoMatemático,UMI2807 CNRS-UChile,Casilla170/3-Correo3,Santiago,Chile(Ch.9) Gallagher,I.,InstitutdeMathématiquesdeJussieuUMR7586,UniversitéParisVII,175, rueduChevaleret,75013Paris,France(Ch.5) Ghrist, R., Department of Mathematics and Coordinated Science Laboratory, University ofIllinois,Urbana,IL61801,USA(Ch.1) Málek,J.,CharlesUniversity,FacultyofMathematicsandPhysics,MathematicalInstitute, Sokolovská83,18675Prague8,CzechRepublic(Ch.7) Pileckas,K.,VilniusUniversity,FacultyofMathematicsandInformatics,NaugardukoStr., 24,Vilnius,LithuaniaandInstituteofMathematicsandInformatics,Akademijos4,Vilnius, Lithuania(Ch.8) Rajagopal,K.R.,DepartmentofMechanicalEngineering,TexasA&MUniversity,College Station,TX77843,USA(Ch.7) Saint-Raymond,L., LaboratoireJ.-L.LionsUMR 7598,UniversitéParisVI, 175,ruedu Chevaleret,75013Paris,France(Ch.5) Schochet, S., Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, 69978 Tel-Aviv,Israel(Ch.3) Sell,G.R.,SchoolofMathematics,UniversityofMinnesota,Minneapolis,MN55455,USA (Ch.6) Seregin, G., Steklov Institute of Mathematics at St. Petersburg, St. Petersburg, Russia (Ch.4) Serre,D.,ÉcoleNormaleSupérieuredeLyon,69364LyonCedex07,France(Ch.2) Vanninathan, M., TIFR Center, IISc. Campus, P.O. Box 1234, Bangalore-560012, India (Ch.9) ix Contents ContentsoftheHandbooks v Preface vii ListofContributors ix 1. OntheContactTopologyandGeometryofIdealFluids 1 R.Ghrist 2. ShockReflectioninGasDynamics 39 D.Serre 3. TheMathematicalTheoryoftheIncompressibleLimitinFluidDynamics 123 S.Schochet 4. LocalRegularityTheoryoftheNavier–StokesEquations 159 G.Seregin 5. OntheInfluenceoftheEarth’sRotationonGeophysicalFlows 201 I.GallagherandL.Saint-Raymond 6. TheFoundationsofOceanicDynamicsandClimateModeling 331 G.R.Sell 7. MathematicalPropertiesoftheSolutionstotheEquationsGoverningtheFlow ofFluidswithPressureandShearRateDependentViscosities 407 J.MálekandK.R.Rajagopal 8. TheNavier–StokesSysteminDomainswithCylindricalOutletstoInfinity. Leray’sProblem 445 K.Pileckas 9. PeriodicHomogenizationProblemsinIncompressibleFluidEquations 649 C.ConcaandM.Vanninathan AuthorIndex 699 SubjectIndex 707 xi