几类非线性反应扩散方程整体动力行为研究

发布时间：2017-12-31 11:11

本文关键词：几类非线性反应扩散方程整体动力行为研究　出处：《太原理工大学》2016年博士论文　论文类型：学位论文

【摘要】：本文研究了几类自治耗散动力系统的长时间动力行为,证明了全局吸引子的存在性.第一章介绍了无穷维动力系统的理论和应用背景,全局吸引子的发展及研究进展,总结了全局吸引子存在的理论和方法及混合有限元两重网格方法.第二章给出了本文用到的空间及不等式.第三章考虑了经典的反应扩散方程的解在无界域上的长时间动力行为.首先证明了系统存在有界吸收集;其次通过证明单位算子在某个有界吸收集上是连续的,得到了解半群的渐近紧性,从而得到了反应扩散方程在无界区域上存在全局吸引子.第四章考虑了非经典的反应扩散方程的解在无界域上的长时间动力行为.首先证明了系统存在有界吸收集;其次利用解的分解技巧,得到解半群的w-极限紧性,从而得到了非经典的反应扩散方程在无界区域上存在全局吸引子.第五章考虑了带记忆项的非经典反应扩散方程的解在有界域上的长时间动力行为.首先证明了系统存在有界吸收集,其次利用解的分解技巧和紧性的传递性,得到解半群的渐近紧性,从而得到反应扩散方程在弱拓扑空间及强拓扑空间上存在全局吸引子.第六章利用稳定的混合有限元两重网格方法求解定常非线性反应扩散方程.首先证明了离散系统解的存在唯一性;其次采用了最低等阶非协调(NCP_1-P_1)元来逼近,给出了误差估计;最后通过数值算例验证了理论分析的结果.第七章考虑了在混合变分形式下的非定常反应扩散系统的二重网格方法,首先通过引入新的变量使得原方程通过混合变分形式来表示,不但降低了解的正则性要求,而且由于采用混合变分形式,可以同时求出两个变量的数值解.然后使用最低等阶协调P_1-P_1元对方程进行离散.同时为了使格式满足LBB稳定性条件,通过定义单元上的压力投影引入稳定化项.这种新方法没有稳定化参数的限制,与其它方法相比,数值结果显示新的方法有更好的稳定性,并且可以减少计算量,节约计算时间.
[Abstract]:In this paper, the long-time dynamic behavior of some autonomous dissipative dynamical systems is studied, and the existence of global attractors is proved. In chapter 1, the theory and application background of infinite dimensional dynamical systems are introduced. The development and research progress of global attractor. The existence theory and method of global attractor and the mixed finite element double grid method are summarized. In chapter 2, the space and inequality used in this paper are given. In chapter 3, the solutions of classical reaction-diffusion equation in unbounded domain are considered. First, it is proved that the system has bounded absorption set. Secondly, it is proved that the unit operator is continuous on a bounded absorption set, and the asymptotic compactness of the solution semigroup is obtained. In chapter 4th, the long-time dynamic behavior of the solution of the non-classical reaction-diffusion equation in the unbounded domain is considered. Firstly, the existence of bounded absorption of the system is proved. Set; Secondly, we obtain the w-limit compactness of solution Semigroups by using the decomposition technique of solutions. Thus, the existence of global attractors for nonclassical reaction-diffusion equations in unbounded domain is obtained. In chapter 5th, the long-time dynamic behavior of solutions of nonclassical reaction-diffusion equations with memory term in bounded domain is considered. There are bounded absorption sets in the system. Secondly, the asymptotic compactness of solution Semigroups is obtained by using the decomposition technique of solution and the transitivity of compactness. Thus, the existence of global attractors in weak topological space and strong topological space for reaction-diffusion equation is obtained. In Chapter 6th, stable mixed finite element double grid method is used to solve steady nonlinear reaction-diffusion equation. Existence and uniqueness of solution of scattered system; Secondly, the lowest equal-order nonconforming NCP1-PSP _ 1 element is used to approximate and the error estimate is given. Finally, numerical examples are given to verify the results of the theoretical analysis. Chapter 7th considers the double grid method for unsteady reaction-diffusion systems in mixed variational form. Firstly, by introducing new variables, the original equation is represented by mixed variational form, which not only reduces the regularity requirement of the solution, but also adopts the mixed variational form. The numerical solution of two variables can be obtained at the same time, then the equations are discretized by using the lowest equal-order concordant PSTs 1-P _ 1 element, and in order to make the scheme satisfy the LBB stability condition. The stability term is introduced by defining the pressure projection on the element. The new method has no limit of stabilization parameters. Compared with other methods, the numerical results show that the new method has better stability. And can reduce the amount of calculation, save calculation time.
【学位授予单位】：太原理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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