本文選題：廣義Kawahara方程 + 正則解　； 參考：《西南交通大學(xué)》2017年碩士論文

【摘要】：本文在有界區(qū)域上研究廣義Kawahara方程的初邊值問題,運用壓縮映射原理得到局部解,結(jié)合能量積分方法、不等式技巧和嵌入定理建立解的先驗估計證明了在有界區(qū)域上整體正則解的存在性和唯一性;并且整體正則解的L2范數(shù)滿足衰減估計.在加強的初值條件下,借助不等式技巧證得在有界區(qū)域上存在與區(qū)間長度無關(guān)的整體正則解,同時得到有界域上存在唯一的弱解.本文共分為四章,主要結(jié)構(gòu)如下:第一章為緒論,首先介紹本文的研究問題,然后簡單介紹了研究背景和研究方法.最后闡述了本文的研究內(nèi)容和方法.第二章首先給出一些基本定義,如工作空間;然后給出了定理1.1證明需要的引理;最后由Banach不動點定理和先驗估計證明了廣義Kawahara方程的與區(qū)間長度有關(guān)的整體正則解.第三章中將定理1.1的初值條件加強,首先建立廣義Kawahara方程的解獨立于區(qū)間長度的先驗估計,然后借助不等式技巧得到廣義Kawahara方程在有界域上與區(qū)間長度無關(guān)的整體正則解,并且證得是唯一存在的.第四章由稠密性證明廣義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.
【學(xué)位授予單位】：西南交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.8

【參考文獻】

相關(guān)期刊論文 前1條

1 ;Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L~2 Initial Data[J];Acta Mathematica Sinica(English Series);2006年05期

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本文編號：1799411