[Sergiu Klainerman]PDE

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1、PARTIALDIFFERENTIALEQUATIONSSERGIUKLAINERMAN1.BasicdefinitionsandexamplesTostartwithpartialdi?erentialequations,justlikeordinarydi?erentialorintegralequations,arefunctionalequations.Thatmeansthattheunknown,orunknowns,wearetryingtodeterminearefunctions.Inthecaseofpartialdi?erentialequa-t

2、ions(PDE)thesefunctionsaretobedeterminedfromequationswhichinvolve,inadditiontotheusualoperationsofadditionandmultiplication,partialderivativesofthefunctions.Thesimplestexample,whichhasalreadybeendescribedinsection1ofthiscompendium,istheLaplaceequationinR3,?u=0(1)222where?u=?u+?u+?u.Theo

3、thertwoexamplesdescribedinthesection?x2?y2?z2offundamentalmathematicalde?nitionsaretheheatequation,withk=1,??tu+?u=0,(2)andthewaveequationwithk=1,??2u+?u=0.(3)tIntheselasttwocasesoneisaskedto?ndafunctionu,dependingonthevariablest,x,y,z,whichveri?esthecorrespondingequations.Observethatbo

4、th(2)and(3)involvethesymbol?whichhasthesamemeaningasinthe?rstequation,thatis222222?u=(?+?+?)u=?u+?u+?u.Bothequationsarecalledevolution?x2?y2?z2?x2?y2?z2equations,simplybecausetheyaresupposedtodescribethechangerelativetothetimeparametertofaparticularphysicalobject.Observethat(1)canbeinte

5、rpretedasaparticularcaseofboth(3)and(2).Indeedsolutionsu=u(t,x,y,z)ofeither(3)or(2)whichareindependentoft,i.e.?tu=0,verify(1).Avariationof(3),importantinmodernparticlephysics,istheKlein-Gordonequa-tion,describingthefreeevolution,i.e.intheabsenceinteractions,ofamassiveparticle.??2u+?u?m2

6、u=0.(4)tAnotherbasicequationofmathematicalphysics,whichdescribesthetimeevolutionofaquantumparticle,istheSchr¨odingerequation,i?tu+k?u=0(5)withuafunctionofthesamevariables(t,x,y,z)withvaluesinthecomplexspaceCandk=h>0,whereh>0correspondstothePlanckconstantandm>02m12SERGIUKLAINERMANthemass

7、oftheparticle.Aswithourothertwoevolutionequations,(2)and(3).abovewesimplifyourdiscussionbytakingk=1.ObservethatallthreePDEmentionedabovesatisfythefollowingsimplepropertycalledtheprincipleofsuperposition:Ifu1,u2aresolutionsofanequationsoisanylinearcombinationofthemλ1u1+λ2u2where