随机长短波方程组的长时间行为

发布时间：2018-05-17 00:20

本文选题：无穷维动力系统 + 随机长短波方程　；参考：《鲁东大学》2017年硕士论文

【摘要】：本文主要对广义(2+1)维非自治长短波方程组,随机(2+1)维长短波方程组的长时间行为等进行了深入的研究,得到了广义(2+1)维长短波方程组的一致吸引子及其近似惯性流形,(2+1)维随机长短波方程组的随机吸引子及其平稳测度.本文共分为四个部分.第一部分,简单介绍了无穷维动力系统,随机微分方程及其长短波方程组的物理背景及其相关理论知识,回顾了已有的部分研究及其取得的成果,最后介绍了本文的主要研究工作.第二部分,考虑了广义(2+1)维非自治长短波方程组的一致吸引子.首先运用一致先验估计以及Gal?rkin方法得到该系统解的存在唯一性.最后使用弱收敛的相关性质证明其一致吸引子的存在性.第三部分,对广义(2+1)维非自治的长短波方程组的近似惯性流形进行研究.考虑该系统通过对相平面进行拓展以及算子投影的方法来完成了对方程组近似惯性流形的构造.第四部分,证明了(2+1)维随机长短波方程组随机吸引子的存在性及其平稳测度.运用It?公式,一致先验估计,得到了解的存在唯一性,又得到了该方程组存在随机吸收集、随机吸引子及其平稳测度.
[Abstract]:In this paper, the long-time behavior of the generalized ~ (21) -dimensional nonautonomous long-wave equations and the random ~ (21) -dimensional long-short wave equations are studied. In this paper, the uniform attractor and its approximate inertial manifold / 21) -dimensional random attractor and its stationary measure are obtained for the system of long and short wave equations with generalized ~ (21) dimension. This paper is divided into four parts. In the first part, the physical background and related theoretical knowledge of infinite dimensional dynamical systems, stochastic differential equations and their short-and short-wave equations are briefly introduced, and some existing studies and their achievements are reviewed. Finally, the main research work of this paper is introduced. In the second part, we consider the uniform attractor of the generalized ~ (21) -dimensional nonautonomous long-short wave equations. First, the existence and uniqueness of the solution of the system are obtained by using the uniform prior estimate and Gal?rkin method. Finally, the existence of uniform attractor is proved by using the correlation property of weak convergence. In the third part, we study the approximate inertial manifold of the nonautonomous long-short wave equations in the generalized ~ (21) dimension. The approximate inertial manifold of the equations is constructed by extending the phase plane and projecting the operator. In the fourth part, we prove the existence and stationary measure of random attractors for random long-short wave equations. Using ITT? The existence and uniqueness of the solution are obtained, and the existence of random absorption set, random attractor and its stationary measure are obtained.
【学位授予单位】：鲁东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63

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