Ill-posedness for nonlinear Schrodinger and wave equations

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1、ILL-POSEDNESSFORNONLINEARSCHRODINGERANDWAVEEQUATIONS¨MICHAELCHRIST,JAMESCOLLIANDER,ANDTERENCETAOdAbstract.ThenonlinearwaveandSchr¨odingerequationsonR,withgeneralpowernon-linearityandwithboththefocusinganddefocusingsigns,areprovedtobeill-posedinthesSobolevspaceHwhenevertheexponentsislo

2、werthanthatpredictedbyscalingorGalileaninvariances,orwhentheregularityistoolowtosupportdistributionalsolutions.Thisextendspreviouswork[7]oftheauthors,whichtreatedtheone-dimensionalcubicnonlinearSchr¨odingerequation.Inthedefocusingcasesolitonorblowupexamplesareunavailable,andaproofofil

3、l-posednessrequirestheconstructionofothersolutions.In[7]thiswasachievedusingcertainlong-timeasymptoticbehaviorwhichoccursonlyforlowpowernonlinearities.Hereweanalyzeinsteadaclassofsolutionsforwhichthezero-dispersionlimitprovidesagoodapproximation.Themethodisrathergeneralandshouldbeappl

4、icabletowiderclassesofnonlinearequations.1.IntroductionThispaperisconcernedwiththelowregularitybehavior(andinparticularill-posedness)oftheCauchyproblemforthegeneralizednonlinearSchr¨odingerequation(−iu(t,x)+∆u(t,x)=ω

5、u

6、p−1u(t,x)tx(gNLS)u(0,x)=u(x)∈Hs(Rd)0andthe(complex)nonlinearwaveeq

7、uationu(t,x)=ω

8、u

9、p−1u(t,x)(gNLW)u(0,x)=u(x)∈Hs(Rd)0s−1d∂tu(0,x)=u1(x)∈H(R)inRd,whereω=±1andp>1,andu:R×Rd→Cisacomplex-valuedfield.Here∆xdenotesarXiv:math/0311048v1[math.AP]4Nov2003P∂22theLaplacian∆x:=j∂x2,while2:=−∂t+∆xisthed’Alembertian.Thesignω=+1isjreferredtoasthedefocusingca

10、se,whilethesignω=−1isfocusing.WesaythattheNLSequation(gNLS)islocallywell-posedinHsifforeveryu0∈HstheredexistatimeT=T(ku0kHs)>0anda(distributional)solutionu:[−T,T]×R→Cto(gNLS)whichisinthespaceC0([−T,T];Hxs),andsuchthatthesolutionmapu07→uisuniformlycontinuous1fromHstoC0([−T,T];Hs).Furth

11、ermore,thereisanadditionalspaceXinwhichxDate:August13,2003.1991MathematicsSubjectClassification.35Q55,35L15.Keywordsandphrases.zero-dispersionlimit,ill-posedness,NLW-typeequations,NLS-typeequations.M.C.issupportedinpartbyN.S.F.grantDMS9970660.J.C.issupportedinpartbyN.S.F.grantDMS010059

12、5,N.S