关于几类复杂弹性梁结构方程（组）系统的吸引子研究

发布时间：2018-04-02 09:58

本文选题：结构阻尼　切入点：热弹耦合梁　出处：《太原理工大学》2017年博士论文

【摘要】：本文利用Galerkin方法、Sobolev空间理论、整体吸引子和一致吸引子等理论,研究了四类复杂弹性梁方程(组)系统的初边值问题以及这些系统的长时间整体动力行为最本质的概念吸引子的存在性。首先,利用Galerkin方法结合一些先验估计和不等式技巧和Sobolev空间理论等给出了四类不同系统的整体解的存在性唯一性,从而定义了三类不同的自治无穷维动力系统的半群和一类非自治无穷维动力系统的双参数过程族;其次,通过先验估计结合一些不等式技巧等,给出了复杂弹性梁结构所确定的三类不同的自治无穷维动力系统的有界吸收集的存在性和一类非自治无穷维动力系统的一致有界吸收集的存在性;最后,通过证明了三类自治无穷维动力系统所对应的解半群是渐近光滑的和一类非自治无穷维动力系统过程族是一致渐近紧的,从而根据整体吸引子和一致吸引子的存在性定理,证明了工程上应用广泛的复杂弹性梁结构所确定的具结构阻尼的两类热弹耦合梁方程组系统的整体吸引子和非线性边界条件下一类自治单个弹性梁方程系统的整体吸引子和一类具有局部阻尼的非自治单个弹性梁方程系统的一致吸引子的存在性。具体内容如下:1.第一章介绍了吸引子研究的必要性和研究的现状,以及弹性梁结构所确定的无穷维动力系统的研究现状。2.第二章介绍了本文中用到的一些基本定义、一些常用的基本不等式以及一些基本引理。3.第三章给出了具有结构阻尼的n维热弹耦合梁方程组在齐次边界条件及初始条件下系统的初边值问题和整体吸引子的存在性;4.第四章对于具有热记忆项的热弹耦合梁方程组在齐次边界条件及一定的初始条件下的系统,通过引入一个新的加权空间,把非自治系统自治化,给出了系统的整体解和整体吸引子的存在性;5.第五章给出了非线性边界条件下单个弹性梁方程在一定的初始条件下系统的的整体解和整体吸引子的存在性;6.第六章给出了非线性边界条件下具局部阻尼的非自治单个弹性梁方程在一定的初始条件下系统的一致有界吸收集和一致吸引子的存在性。
[Abstract]:In this paper, the theory of Sobolev space, the global attractor and the uniform attractor are discussed by using the Galerkin method. In this paper, we study the existence of the most essential concept attractors for four classes of complex elastic beam equations (systems) and the most essential concept attractors for the long time global dynamic behavior of these systems. The existence and uniqueness of global solutions for four different systems are obtained by using the Galerkin method and some prior estimators and inequality techniques as well as Sobolev space theory. The semigroup of three kinds of autonomous infinite dimensional dynamical systems and the two-parameter process family of a class of nonautonomous infinite dimensional dynamical systems are defined. The existence of bounded absorption sets for three classes of autonomous infinite dimensional dynamical systems determined by complex elastic beam structures and the existence of uniformly bounded absorption sets for a class of nonautonomous infinite dimensional dynamical systems are given. By proving that the solution Semigroups corresponding to three kinds of autonomous infinite dimensional dynamical systems are asymptotically smooth and that the process families of a class of nonautonomous infinite dimensional dynamical systems are uniformly asymptotically compact, the existence theorems of global attractors and uniform attractors are obtained. It is proved that the global attractor of two classes of thermoelastic coupled beam equations with structural damping and the integral of a class of autonomous single elastic beam equations under nonlinear boundary conditions determined by the widely used complex elastic beam structures in engineering are proved. The existence of uniform attractors for a class of nonautonomous single elastic beam equation systems with local damping. As well as the research status of infinite dimensional dynamic system determined by elastic beam structure. The second chapter introduces some basic definitions used in this paper. Some general inequalities and some basic Lemma .3. in chapter 3, the initial-boundary value problem and global attractor of n-dimensional thermoelastic coupled beam equations with structural damping under homogeneous boundary conditions and initial conditions are given. Chapter 4 for the systems of thermoelastic coupled beam equations with thermal memory under homogeneous boundary conditions and certain initial conditions, By introducing a new weighted space, the nonautonomous system can be autonomous. The global solution and the existence of the global attractor of the system are given. Chapter 5 gives the existence of the global solution and the global attractor of the system under certain initial conditions under the nonlinear boundary conditions. In the chapter, the uniformly bounded suction collection and the existence of uniform attractors for a nonautonomous single elastic beam equation with local damping under certain initial conditions are given under nonlinear boundary conditions.
【学位授予单位】：太原理工大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O19

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