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1、Degasperis-ProcesiExistenceofRarefactionWaveSolutionsoftheDegasperis-ProcesiEquationCandidateZhouHuangSupervisorCollegeProgramSpecializationDegreeUniversityDateAssociateProfessor.TANGTai-ManMathematicsandComputationalSciencePureMathematicsPartialDifferent
2、ialEquationMasterofScienceXiangtanUniversityMay27th,2010Degasperis-ProcesiDegasperis-ProcesiDegasperis-ProcesiDegasperis-Procesi,:Degasperis-ProcesiIAbstractWeconsidertheglobalexistenceofrarefactionwavesolutionsoftheDegasperis-Procesiequation.Byrarefacti
3、onwavesolutions,wemeansolutionswithgivenendstates,withtheleftendstatelessthantherightstate.Weprovetheglobalex-istenceofthiskindofsolutionstotheinitialvalueproblemofDegasperis-Procesiequation.Thisworkprovidesthebasisforthestudyofthenonlinearstabilityofrar
4、-efactionwaves.ItreducestheproblemtothestudyoftheasymptoticbehavioroftheweakssolutionofaperturbedDegasperis-Procesiequation.Thisarticleisdividedinto4chapters.Chapter1isintroduction.Prelimi-nariesneededintheproofsoftheconclusionsaregiveninChapter2.InChapt
5、er3,weprovetheexistenceoftheglobalrarefactionwavesolutions.Finally,weconcludeinChapter4andgivesomecommentsonpossiblefuturework.KeyWords:Degasperis-Procesiequation;Globalweaksolution;rar-efactionwaveII......................................................
6、.11.1.....................................................11.2Degasperis-Procesi.......................11.32.12.2.....................................................2..................................................4....................................
7、.......4.....................................................5..............................................7................................................21.............................................................22................................
8、................................26..........................27III§1.1Degasperis-ProcesiCauchy(1.1)(1.2)2P−Pxx=23u2,0u(0,x)=u0(x)→u±,(t,x)∈R+×R,(t,x)∈R+×R,x∈R,x→±∞(1.1)(1.2)ut−utxx+4uux=3uxuxx+uuxxx,u(0,x)=u0(x)→u±,x→±∞(1.1
