无界域上一类波动方程的拉回吸引子
发布时间：2019-02-16 19:30
【摘要】：拉回吸引子是描述系统解的长时间渐近行为的紧集,是研究无穷维动力系统的重要工具.本文考虑无界区域上一类波动方程的拉回吸引子,获得了无界区域上非自治Brinkman-Forchheimer方程和非自治Sine-Gordon方程拉回吸引子的存在性.全文共分为五章.第一章,介绍了无穷维动力系统的背景,分自治和非自治两种情形阐述了全局吸引子、一致吸引子和拉回吸引子的研究进展情况,还介绍了本文的主要工作和思想.第二章,介绍了所要研究问题的预备知识,给出了文章需要用到的一些相关符号、不等式、定理以及相关理论,为接下来的讨论做准备.第三章,研究了无界区域L~2(Ω)上非自治Brinkman-Forchheimer方程拉回吸引子的存在性,针对无界区域上Sobolev嵌入不再是紧的,利用一致估计和截断函数技巧证明了无界域上拉回吸引子的存在性,将无界区域分成两部分,有界部分和无界部分,有界部分用Sobolev紧嵌入,无界部分则证明指数一致衰减到零,从而得到紧性.第四章,研究了无界区域H~1(R~n)×L~2(R~n)上非自治Sine-Gordon方程拉回吸引子的存在性,同样的,此时Sobolev紧嵌入不再成立,先对方程的解进行先验估计,通过解的一致估计证明了方程拉回吸收集的存在性,而后构造能量方程证明了解的拉回渐近紧性,从而得到Sine-Gordon方程在无界域上拉回吸引子的存在性.第五章,对本文的总结及对拉回吸引子研究前景的展望。
[Abstract]:The pull back attractor is a compact set which describes the long time asymptotic behavior of the solution of the system and is an important tool for the study of infinite dimensional dynamical systems. In this paper, we consider the pullback attractors of a class of wave equations in unbounded regions, and obtain the existence of pullback attractors for nonautonomous Brinkman-Forchheimer equations and nonautonomous Sine-Gordon equations in unbounded regions. The full text is divided into five chapters. In chapter 1, the background of infinite dimensional dynamical system is introduced. The research progress of global attractor, uniform attractor and pull back attractor is described in two cases of autonomy and nonautonomy. The main work and ideas of this paper are also introduced. In the second chapter, we introduce the preparatory knowledge of the problem, and give some relevant symbols, inequalities, theorems and relevant theories to be used in this paper, so as to prepare for the following discussion. In chapter 3, we study the existence of pullback attractors for the nonautonomous Brinkman-Forchheimer equations on the unbounded domain Ln 2 (惟). For the unbounded region, Sobolev embedding is no longer compact. By using uniform estimation and truncation function technique, the existence of pull back attractor on unbounded domain is proved. The unbounded region is divided into two parts, bounded part and unbounded part, bounded part is Sobolev compactly embedded, and unbounded part is proved that exponentially uniformly attenuates to zero. Thus the compactness is obtained. In chapter 4, we study the existence of the pull back attractor of the nonautonomous Sine-Gordon equation on the unbounded domain H1 (rn) 脳 Ln (rn). Similarly, the Sobolev compactness embedding is no longer true at this time, a priori estimate of the solution of the equation is made. The existence of the pull-back absorption set of the equation is proved by the uniform estimation of the solution, and the tension asymptotically compactness of the solution is proved by constructing the energy equation, and the existence of the trace-back attractor of the Sine-Gordon equation in the unbounded domain is obtained. The fifth chapter, the summary of this paper and the prospect of pull-back attractor research.
【学位授予单位】：广州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2424766
[Abstract]:The pull back attractor is a compact set which describes the long time asymptotic behavior of the solution of the system and is an important tool for the study of infinite dimensional dynamical systems. In this paper, we consider the pullback attractors of a class of wave equations in unbounded regions, and obtain the existence of pullback attractors for nonautonomous Brinkman-Forchheimer equations and nonautonomous Sine-Gordon equations in unbounded regions. The full text is divided into five chapters. In chapter 1, the background of infinite dimensional dynamical system is introduced. The research progress of global attractor, uniform attractor and pull back attractor is described in two cases of autonomy and nonautonomy. The main work and ideas of this paper are also introduced. In the second chapter, we introduce the preparatory knowledge of the problem, and give some relevant symbols, inequalities, theorems and relevant theories to be used in this paper, so as to prepare for the following discussion. In chapter 3, we study the existence of pullback attractors for the nonautonomous Brinkman-Forchheimer equations on the unbounded domain Ln 2 (惟). For the unbounded region, Sobolev embedding is no longer compact. By using uniform estimation and truncation function technique, the existence of pull back attractor on unbounded domain is proved. The unbounded region is divided into two parts, bounded part and unbounded part, bounded part is Sobolev compactly embedded, and unbounded part is proved that exponentially uniformly attenuates to zero. Thus the compactness is obtained. In chapter 4, we study the existence of the pull back attractor of the nonautonomous Sine-Gordon equation on the unbounded domain H1 (rn) 脳 Ln (rn). Similarly, the Sobolev compactness embedding is no longer true at this time, a priori estimate of the solution of the equation is made. The existence of the pull-back absorption set of the equation is proved by the uniform estimation of the solution, and the tension asymptotically compactness of the solution is proved by constructing the energy equation, and the existence of the trace-back attractor of the Sine-Gordon equation in the unbounded domain is obtained. The fifth chapter, the summary of this paper and the prospect of pull-back attractor research.
【学位授予单位】：广州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
【参考文献】
相关期刊论文 前3条
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2 Xueli SONG;;Pullback D-Attractors for A Non-Autonomous Brinkman-Forchheimer System[J];数学研究及应用;2013年01期
3 PARK Jong Yeoul;PARK Sun Hye;;Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains[J];Science China(Mathematics);2011年04期
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