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1、Toappearin“D´evelopmentdesMath´ematiquesaucoursdelasecondemoiti´eduXX`emesi`ecle,”J.P.PierManagingEditor,Birkhauser1999.SOMEDEVELOPMENTSONNAVIER-STOKESEQUATIONSINTHESECONDHALFOFTHE20thCENTURYRogerTemamSeptember17,1999DedicatedwithdeeprespecttothememoryofJeanLeray.I
2、NTRODUCTIONThetheoryofNavier-Stokesequations(NSE)constitutesacentralproblemincon-temporarymathematicalphysics.Theseequationsareaphysicallywellacceptedmodelforthedescriptionofverycommonphenomena,andmuchefforthasbeendevotedtothembyfluidmechanicengineers,meteorologists,
3、mathematiciansandothers;never-thelessmanyproblemsarestillatthefrontierofscience.Fromthephysicalviewpointsomeproblemswhichareamazinglysimpleintheirformulationarestillunresolved,suchasthedeterminationofthethermalpropertiesofturbulentflowsorthedescriptionoftheforcesexe
4、rtedbyaturbulentflowonitsboundary(e.g.airplaneorpipe).OnthemathematicalsideNSEareamodelforthestudyofnonlinearphenomenaandnonlinearequationswhichisitselfinitsinfancy:NSEandtherelatedEulerequationsencom-passfourcentralproblemsinnonlinearequations,namely,well-posedness
5、,oscillations,discontinuities,andnonlineardynamics.ThemathematicaltheoryoftheNSEisrathertechnicalandalreadyverylargedespitebeingstillincomplete.Henceitwasnotobviouswhatanarticleaddressingsuchavastsubjectshouldcontain:descriptionofresults(withoutwritingacatalogueofr
6、esults),evolutionoftechniquesandideas,relationswithturbulenceandphysics.Ithasnotbeenpossibletoaddresssatisfactorilyallthesequestionsinthepresentnotes.Therearecertainlymanyimportantomissions,manyimportantsubjectstowhichnotenoughspacehasbeendevoted;choiceshavebeenmad
7、eandundoubtedlyothersinthefieldswouldhavemadedifferentchoices.AlargelistofreferencescompensatesinpartfortheseTypesetbyAMS-TEX12defficiencies.DespitealltheirshortcomingsIstillhopethatthesenotescanbeusefultosomereaders;ImyselfhavelearnednewthingswhilewritingthemandIcorre
8、ctedorconsolidatedthingsIknew.ThenameNavier-Stokesequationswillapplyheretobothincompressibleandcom-pressibleflowequations,althoughsomeauthorsuseth
