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1、MasterDissertationSubmittedtoShanghaiJiaoTongUniversityTHEERRORESTIMATEFORASPECIALNONLINEARDEGENERATEPARABOLIC-HYPERBOLICEQUATIONAuthor:JiatongNiuAdvisor:Prof.YachunLiSpecialty:AppliedMathematicsDepartmentofMathematicsShanghaiJiaoTongUniversityNovember,2009上海交通大学硕士学位论文一一一个个个特特特殊殊殊的的的非非非线线
2、线性性性退退退化化化抛抛抛物物物双双双曲曲曲方方方程程程的的的误误误差差差估估估计计计摘摘摘要要要本篇论文主要讨论了一个特殊的非线性退化抛物双曲方程的误差估计.文章首先介绍了退化抛物双曲方程的物理模型,以两相流边界层为例详细地推导了Prandtl边界层方程,它是典型的各向异性退化抛物双曲方程.在第二章中我们介绍了关于二阶非线性退化抛物双曲方程的研究现状,对各向同性及各向异性,Cauchy问题及Dirichlet边值问题等各方面的研究结果做了归纳比较.接下来我们介绍了守恒律方程的误差估计,主要用的是Kuznetsov方法及Cockburn等人的改进方法,这也是我们对退化抛物
3、双曲方程做误差估计的方法来源.第三章我们介绍了退化抛物双曲方程的熵解的定义,提出主要定理并给出详细的证明.关键词:Prandtl边界层方程;非线性退化抛物双曲;误差估计;Kuznetsov方法;熵解IABSTRACTTHEERRORESTIMATEFORASPECIALNONLINERADEGENERATEPARABOLIC-HYPERBOLICEQUATIONABSTRACTThispaperfocusesontheerrorestimatesaclassofnonlineardegenerateparabolichyperbolicequation.Thepaper¯
4、rstdescribesthedeteriorationparabolicphysicalmodelofhy-perbolicequationstotwo-phaseboundarylayer,wedetailedthederivationofthePrandtlboundarylayerequations,whichisatypicalofanisotropicdegenerateparabolichyperbolicequations.Inthesecondchapterweintroducedthepresentsituationofthesecond-ordern
5、onlineardegenerateparabolichyperbolicequation,generalizedandcomparedtheresearchresultsfortheisotropicandanisotropic,CauchyproblemandDirichletboundaryvalueproblems.Next,weintroducederrorestimatesoftheconserva-tionlawequation,primarilyusingtheKuznetsovapproximationtheoryandtheimprovedmethod
6、ofCockburnandothers,whichisalsooursourcetodoerrorestimationfordegenerateparabolichyperbolicequations.InChapterThree,wein-troducedthethede¯nitionofentropysolutionsofdegenerateparabolichyperbolicequation,proposedthemaintheoremandprovedit.KEYWORDS:::Prandtlboundarylayerequations,Nonilinearde
7、generateII上海交通大学硕士学位论文parabolic;errorestimate;entropysolutions;Kuznetsov'sapproximationtheoryIII目目目录录录中文摘要.....................................I英文摘要.....................................II目录........................................IV第第第一一一章章章物物物理理理模模模型型型1第第第二二二章章章预预预备