有界区域上广义Kawahara方程的初边值问题

发布时间：2018-04-25 02:38

本文选题：广义Kawahara方程 + 正则解　；参考：《西南交通大学》2017年硕士论文

【摘要】：本文在有界区域上研究广义Kawahara方程的初边值问题,运用压缩映射原理得到局部解,结合能量积分方法、不等式技巧和嵌入定理建立解的先验估计证明了在有界区域上整体正则解的存在性和唯一性;并且整体正则解的L2范数满足衰减估计.在加强的初值条件下,借助不等式技巧证得在有界区域上存在与区间长度无关的整体正则解,同时得到有界域上存在唯一的弱解.本文共分为四章,主要结构如下:第一章为绪论,首先介绍本文的研究问题,然后简单介绍了研究背景和研究方法.最后阐述了本文的研究内容和方法.第二章首先给出一些基本定义,如工作空间;然后给出了定理1.1证明需要的引理;最后由Banach不动点定理和先验估计证明了广义Kawahara方程的与区间长度有关的整体正则解.第三章中将定理1.1的初值条件加强,首先建立广义Kawahara方程的解独立于区间长度的先验估计,然后借助不等式技巧得到广义Kawahara方程在有界域上与区间长度无关的整体正则解,并且证得是唯一存在的.第四章由稠密性证明广义Kawahara方程在有界域上存在唯一的弱解.
[Abstract]:In this paper, the initial-boundary value problem of the generalized Kawahara equation is studied in the bounded domain. The local solution is obtained by using the contraction mapping principle, and the energy integral method is used. Inequality technique and embedding theorem establish a priori estimate of the solution. It is proved that the existence and uniqueness of the global regular solution in the bounded region, and the L 2 norm of the global regular solution satisfies the decay estimate. Under the condition of strengthened initial value, the existence of a global regular solution independent of interval length on a bounded domain is proved by means of inequality technique. At the same time, the existence of a unique weak solution in a bounded domain is obtained. This paper is divided into four chapters, the main structure is as follows: the first chapter is an introduction, first introduces the research issues of this paper, then briefly introduces the research background and research methods. Finally, the research contents and methods of this paper are described. In chapter 2, we first give some basic definitions, such as workspace; then we give the Lemma needed to prove theorem 1.1; finally, we prove the global regular solution of generalized Kawahara equation by Banach fixed point theorem and a priori estimate. In chapter 3, the initial condition of theorem 1.1 is strengthened. Firstly, the solution of generalized Kawahara equation is established independent of the prior estimate of interval length, and then the global regular solution of generalized Kawahara equation in bounded domain is obtained by means of inequality technique. And the evidence is the only existence. In chapter 4, we prove the existence of unique weak solutions for the generalized Kawahara equation in bounded domain by density.
【学位授予单位】：西南交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

【参考文献】