【作者】 张峰；

【导师】 杨晗；

【作者基本信息】 西南交通大学 ， 基础数学， 2014， 硕士

【摘要】 本文研究了一类修正二元Camassa-Holm方程组的柯西问题在低正则初值意义下整体守恒解的存在性。全文分为五章：在第一章,回顾了从一般的浅水波方程到扩展的二元Camassa-Holm方程组的研究过程,介绍了这项问题的物理背景以及研究意义,综述了前人的研究成果,并给出了本文所要研究的问题与研究的意义所在。在第二章,介绍了文章所需的基础知识,利用积分变换,将原方程改写成卷积的形式,并推导了相应的守恒律。最后,给出了本文的主要结论。在第三章,通过引入新的变量,将原方程组转化为一个等价的半线性系统。第四章主要分为两部分,前半部分应用压缩映像原理证明了半线性系统在相关完备度量空间上局部解的存在性。后半部分,结合第二章得到的先验估计,将局部解拓展到整体解。在第五章,证明了关于半线性系统的整体解是原问题的整体守恒解,同时得到了原方程组柯西问题的整体守恒解连续依赖于初值,从而主要结论得证。更多还原

【Abstract】 This paper is designed to consider the global conservative solutions for the Cauchy problem of a modified two-component Camassa-Holm system. The paper consists of five chapters.The first chapter is concerned with the background of shallow water equation. Moreover, the author introduces some known results of two-component Camassa-Holm system.By using the basic theory of ordinary differential equations, two-component Camassa-Holm system can be rewritten into convolution style, and the conservation law is obtained. The main results and the preliminaries of this dissertation are given in the second chapter.By introducing some independent and dependent variables, the original system is rewritten as a semilinear system in the third chapter.The forth chapter is mainly divided into two parts. The local well-posedness for the semilinear system is obtained as fixed points of a contractive transformation in the first half part. Then local solutions can be extended to global solutions by using the estimates which established in the second half part.In the fifth chapter, returning to the original variables, the author obtains the global conservative solutions of original Cauchy problem. The solutions depends continuously on the initial data.更多还原