Para-differential calculus and application to the Cauthy problem for nonlinear systems

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1、Para-differentialCalculusandApplicationstotheCauchyProblemforNonlinearSystemsGuyMetivierUniversiteBordeaux1,IMBUMRCNRS525133405TalenceCedex,Franceguy.metivier@math.u-bordeaux1.frMay9,2008ContentsIIntroductiontoSystems81NotationsandExamples.91.1Firstordersystems........................91.1.1Nota

2、tions.........................91.1.2Planewaves........................101.1.3Thesymbol........................121.2Examples.............................131.2.1Gasdynamics.......................131.2.2Maxwell'sequations...................151.2.3Magneto-hydrodynamics.................201.2.4Elastici

3、ty.........................252ConstantCoecientSystems.FourierSynthesis262.1Themethod............................262.1.1TheFouriertransform.................262.1.2Solvingtheevolutionequation(2.1.1).........282.2Examples.............................302.2.1Theheatequation....................302.2.2

4、Schr•odingerequation..................302.2.3Thewaveequation....................312.3Firstordersystems:hyperbolicity...............322.3.1Thegeneralformalism..................322.3.2Stronglyhyperbolicsystems..............322.3.3Symmetrichyperbolicsystems.............342.3.4Smoothlydiagonalizables

5、ystems,hyperbolicsystemswithconstantmultiplicities...............352.3.5Existenceanduniquenessforstronglyhyperbolicsys-tems............................362.4Higherordersystems.......................3612.4.1SystemsofSchr•odingerequations............362.4.2Elasticity.........................373TheMe

6、thodofSymmetrizers393.1Themethod............................393.2Theconstantcoecientscase..................413.2.1Fouriermultipliers....................413.2.2Therstordercase....................423.3Hyperbolicsymmetricsystems.................433.3.1Assumptions.......................443.3.2Exist

7、enceanduniqueness................443.3.3Energyestimates.....................45IIThePara-DierentialCalculus494Pseudo-dierentialoperators504.1Fourieranalysisoffunctionalspaces..............504.1.1Smoothingandapproximat