两类随机非局部偏微分方程在有界区域上鞅解的存在性

发布时间：2018-01-19 14:00

本文关键词： 非局部有界区域广义Burgers方程 Ginzburg-Landau方程白噪声鞅解　出处：《四川师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：Burgers方程和Ginzburg-Landau方程由于分别模拟了冲击波的传播和超导现象而受到物理学家和数学家的关注.本文主要对有界区域上的随机非局部广义Burgers方程和Ginzburg-Landau方程感兴趣,证明了其鞅解的存在性.第一章介绍了 Burgers方程和Ginzburg-Landau方程的物理背景和已有研究,列出了文章所需的定义、不等式及引理,并给出了论文的主要结论.第二章研究了有界区域上的随机非局部广义Burgers方程.首先通过在适当的分数阶加权Sobolev空间上考虑,克服了有界区域上非局部Laplace算子带来的困难,然后通过Galerkin近似、Prokhorov定理、Skorokhod定理以及鞅表示定理获得了系统鞅解的存在性.第三章利用类似的方法证明了有界区域上的随机非局部Ginzburg-Landau方程鞅解的存在性.最后一章对以后的研究工作进行思考和展望.
[Abstract]:The Burgers and Ginzburg-Landau equations by physicists and mathematicians concerned because of the shock wave propagation and superconductivity were simulated. This paper focuses on the random non local generalized Burgers equation and the Ginzburg-Landau equation in bounded domain of interest, prove the existence of solutions of the martingale. The first chapter introduces the physical background and the existing research of Burgers equation and the Ginzburg-Landau equation, a list of the definitions required, inequality and lemma, and gives the main conclusions of the thesis. The second chapter studies the random non local generalized Burgers equation in bounded domain. Firstly, by considering the fractional order weighted Sobolev space properly, overcomes the difficulties of non local operators bring Laplace bounded domain, then by Galerkin approximation, Prokhorov theorem, Skorokhod theorem and martingale representation theorem to obtain the existence of solutions for the system of martingale. In the three chapter, we prove the existence of martingale solutions for stochastic nonlocal Ginzburg-Landau equations on bounded domains by using similar methods. In the last chapter, we consider and prospect future research.

【学位授予单位】：四川师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2

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