Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem PDF
Preview Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem
Lecture Notes in Mathematics 2097 Tatsuo Nishitani Hyperbolic Systems with Analytic Coefficients Well-posedness of the Cauchy Problem Lecture Notes in Mathematics 2097 Editors-in-Chief: J.-M.Morel,Cachan B.Teissier,Paris AdvisoryBoard: CamilloDeLellis(Zurich) MarioDiBernardo(Bristol) AlessioFigalli(Pisa/Austin) DavarKhoshnevisan(SaltLakeCity) IoannisKontoyiannis(Athens) GaborLugosi(Barcelona) MarkPodolskij(Heidelberg) SylviaSerfaty(ParisandNY) CatharinaStroppel(Bonn) AnnaWienhard(Heidelberg) Forfurthervolumes: http://www.springer.com/series/304 Tatsuo Nishitani Hyperbolic Systems with Analytic Coefficients Well-posedness of the Cauchy Problem 123 TatsuoNishitani DepartmentofMathematics GraduateSchoolofScience OsakaUniversity Toyonaka,Osaka,Japan ISBN978-3-319-02272-7 ISBN978-3-319-02273-4(eBook) DOI10.1007/978-3-319-02273-4 SpringerChamHeidelbergNewYorkDordrechtLondon LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2013955050 MathematicsSubjectClassification(2010):35L45,35L40,35L55 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface In this monographwe discuss the C1 well-posednessof the Cauchy problemfor hyperbolicsystems.Wearemainlyconcernedwiththefollowingtwoquestionsfor differentialoperatorsoforderqwithsmoothm(cid:2)mmatrixcoefficients: (A) UnderwhichconditionsonlowerordertermsistheCauchyproblemC1well posed? (B) WhenistheCauchyproblemC1wellposedforanylowerorderterm? For scalar case, that is m D 1, the question (B) has been answered. As for the question (A), in particular for second order scalar equations, that is m D 1 and q D 2,so manyworksare devotedto thisquestionandthe situationisfairly well understood.Contrary to the scalar case, for systems that is if m (cid:3) 2 we have no satisfactoryresult. Evenfordifferentialoperatorswithcharacteristicsofconstantmultiplicitywith realanalyticmatrixcoefficients,thequestion(A)hasbeensolvedveryrecently. So in this monograph, assuming that the coefficients are real analytic in a neighborhoodoftheorigin,westudythesetwoquestions.Ofcoursethisanalyticity assumption is rather restrictive but which allows us to make detailed studies on the Cauchy problem. We hope that this study can throw light on the studies of the Cauchy problem for hyperbolic systems with less regular, in particular C1 coefficients. The contents are organizedas follows. In Chap.1 after giving the definition of C1 well-posednessoftheCauchyproblemwe showthattheCauchyproblemfor symmetrichyperbolicsystemsisC1wellposedforanylowerorderterm.Thenwe giveanexampleoffirstorder2(cid:2)2systemwhichisnotsymmetrizablebutforwhich theCauchyproblemis C1 wellposedforanylowerorderterm.Actuallythereis a class of non-symmetrizablesystems for which the Cauchy problem is C1 well posedforanylowerorderterm.Thisisamainobjectionwhenwetrytoanswerto theproblem(B).WeprovetheLax-Mizohatatheoremexhibitingnaiveideaswhich are usedin Chaps.2 and3. For firstordersystemswith characteristicsof constant multiplicities, the necessity of the Levi condition for the C1 well-posedness is provedwhichisusedinChap.2.InChap.2,westudynecessaryconditionsaboutthe v vi Preface problem(B)form(cid:2)mfirstordersystemswithrealanalyticcoefficients.Weprove rather general necessary conditions in terms of minors of the principal symbols. Contrary to the scalar case the multiplicity of characteristics is irrelevant for the problem (B) since for symmetric or symmetrizable hyperbolic systems (of first order)theCauchyproblemisalwaysC1wellposedforanylowerorderterm.Here the maximalsize of the Jordanblocks,whichis supposedto measurethe distance fromdiagonalmatrices,playsanimportantroleintheproblem(B). InChap.3,westudytwoquestions(A)and(B)forfirstorder2(cid:2)2systemswith two independentvariables with real analytic coefficients. For this special case we cangiveanecessaryandsufficientconditionforthequestions(A)and(B),thatis, in this case we have complete answer for (A) and (B). The results provide many instructive examples. For instance, we can exhibit a first order 2(cid:2)2 system with analyticcoefficientswhichisstrictlyhyperbolicoutsidetheinitiallineforwhichno lowerordertermcouldbetakensothattheCauchyproblemisC1wellposed.This cannothappenfor second orderhyperbolicscalar operatorswith two independent variableswithanalyticcoefficients. In Chap.4, we introduce a new class of hyperbolicsystems, that is hyperbolic systems with nondegenerate characteristics which generalizes strictly hyperbolic systems. Strictly hyperbolic systems are hyperbolic systems with nondegenerate characteristicsoforderone.Thetheoryofstrictlyhyperbolicsystemsisrich,butfirst orderstrictlyhyperbolicsystemhardlyexists.WeprovethattheCauchyproblemfor hyperbolic systems with nondegenerate characteristics is C1 well posed for any lowerorderterm.Wealsoshowthatnondegeneratecharacteristicsarestable,thatis anyhyperbolicsystemwhichisclosetoahyperbolicsystemwithanondegenerate characteristicoforderrhasanondegeneratecharacteristicofthesameordernearby. This shows, in particular, that near any hyperbolic system with a nondegenerate characteristicof orderr (cid:3) 2 there is no strictly hyperbolicsystem, which givesa greatdifferencefromthescalarcaseandshowsacomplexityofhyperbolicsystems. We also discuss hyperbolic systems which are perturbations of symmetric systems andprovethatif the dimensionof the linear space that the symbolof the symmetricsystemspansislargeenough,thengenericallysuchhyperbolicsystemis similartoasymmetricsystem. Osaka,Japan TatsuoNishitani March2013 Contents 1 Introduction .................................................................. 1 1.1 Well-PosednessoftheCauchyProblem.............................. 1 1.2 SymmetricHyperbolicSystems....................................... 5 1.3 SystemsWhichAreNotSymmetrizable ............................. 12 1.4 Lax-MizohataTheorem ............................................... 18 1.5 LeviCondition......................................................... 22 1.6 ALemmaonHyperbolicPolynomials ............................... 26 2 NecessaryConditionsforStrongHyperbolicity.......................... 31 2.1 NecessaryConditionsforStrongHyperbolicity ..................... 31 2.2 KeyPropositions....................................................... 33 2.3 ProofofTheorem2.1(SimplestCase) ............................... 37 2.4 ProofofTheorem2.1(GeneralCase) ................................ 44 2.5 ProofsofPropositions2.4and2.5.................................... 50 2.6 ProofofKeyProposition.............................................. 55 2.7 ProofofKeyProposition,AsymptoticDiagonalization............. 63 2.8 InvolutiveCharacteristics ............................................. 71 2.9 LocalizationatInvolutiveCharacteristics ............................ 75 2.10 ConcludingRemarks .................................................. 82 3 TwobyTwoSystemswithTwoIndependentVariables.................. 85 3.1 ReductiontoAlmostDiagonalSystems.............................. 85 3.2 NonnegativeRealAnalyticFunctions................................ 90 3.3 Well-PosednessandPseudo-CharacteristicCurves.................. 92 3.4 StronglyHyperbolic2(cid:2)2Systems................................... 96 3.5 NonnegativeFunctionsandNewtonPolygons....................... 98 3.6 BehaviorAroundPseudo-CharacteristicCurves..................... 104 3.7 ProofofProposition3.2............................................... 108 3.8 EnergyEstimatesNearPseudo-CharacteristicCurves............... 112 3.9 EnergyEstimatesofHigherOrderDerivatives ...................... 121 3.10 WeightedEnergyEstimates........................................... 125 3.11 ConditionsforWell-Posedness ....................................... 136 vii viii Contents 3.12 ConstructionofAsymptoticSolutions................................ 142 3.13 ProofofNecessity..................................................... 150 3.14 EquivalenceofConditions ............................................ 153 3.15 ConcludingRemarks .................................................. 160 4 SystemswithNondegenerateCharacteristics............................. 161 4.1 NondegenerateCharacteristics........................................ 161 4.2 NondegenerateDoubleCharacteristics............................... 169 4.3 Symmetrizability(SpecialCase)...................................... 174 4.4 StabilityandSmoothnessofNondegenerateCharacteristics........ 179 4.5 Symmetrizability(GeneralCase)..................................... 194 4.6 WellPosedCauchyProblem.......................................... 209 4.7 NondegenerateCharacteristicsofSymmetricSystems.............. 213 4.8 HyperbolicPerturbationsofSymmetricSystems.................... 216 4.9 Stabilityof SymmetricSystemsUnderHyperbolic Perturbations........................................................... 221 4.10 SomeSpecialCases ................................................... 225 4.11 ConcludingRemarks .................................................. 229 References......................................................................... 231 Index............................................................................... 235 Chapter 1 Introduction Abstract In thischapterwe show thatthe Cauchyproblemfor symmetrichyper- bolic systems is C1 well posed for any lower order term proving the existence of solutions and the bound of the supports at the same time, by steering a course somewhat close to the boundary value problems rather than the initial value problems. We give an example, which would be the simplest one, is not symmetrizablebuttheCauchyproblemisC1wellposedforanylowerorderterm. We show the well-posedness by the classical method of characteristic curves. We also give a proofof the Lax-Mizohatatheoremexhibitingnaive ideasto construct an asymptoticsolutionto systems whichwill be usedin Chaps.1 and 2 in a more involved way. The Levi condition for first order systems with characteristics of constant multiplicities is discussed in a somewhat intermediate form. We use this forprovingmoregeneralresultinChap.2. 1.1 Well-Posedness of theCauchyProblem Letusstudyadifferentialoperatoroforderd withm(cid:2)mmatrixcoefficients X 1 @ P.x;D/D A .x/D’; D D ’ j i @x j j’j(cid:2)d whereA .x/arem(cid:2)mmatrixvaluedsmoothfunctiondefinedinaneighborhood(cid:2) ’ oftheoriginofRnC1withasystemofcoordinatesx D.x ;x ;:::;x /D.x ;x0/. 0 1 n 0 Weassumethatx Dconst:arenoncharacteristicandthenwithoutrestrictionswe 0 canassumethat A .x/DI .d;0;:::;0/ T.Nishitani,HyperbolicSystemswithAnalyticCoefficients,LectureNotes 1 inMathematics2097,DOI10.1007/978-3-319-02273-4__1, ©SpringerInternationalPublishingSwitzerland2014
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