一维热扩散系统及相关模型解的整体适定性研究
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摘要
热扩散系统描述的是固体介质中热扩散的过程,是根据固体介质中热扩散的基本理论所建立的一类数学模型。在本文中,我们主要研究一维热扩散系统及相关模型初边值问题解的整体适定性,包括第Ⅰ型的热扩散系统,带有第二声的热扩散系统和第Ⅲ型的热扩散系统等,得到了它们解的整体存在性、渐近稳定性和整体(一致)吸引子的存在性。
本文共分为五章：
第一章是绪论,主要介绍了所研究问题的相关研究背景、研究进展、本文的主要工作、本文研究中经常用到的半群方法基本理论、吸引子存在性基本理论和一些常用的不等式。
第二章研究了第Ⅰ型热扩散系统解的整体适定性,首先利用半群方法得到了自治系统解的整体存在性和指数稳定性,其次利用半群方法得到了非自治系统和半线性系统解的整体存在性,利用多乘子方法得到了非自治系统解的渐近稳定性,并讨论了其它几种边界条件的情况,给出了相关结论,最后利用压缩函数的方法得到过程族的一致渐近紧性,从而得到了非自治系统一致吸引子的存在性。该方法的主要优点在于我们可以利用在建立吸收集时所需要的能量估计,来直接验证紧性条件。
第三章我们讨论了带有第二声的热扩散系统解的整体适定性,分别考察了两端固定且无热扩散流的边界条件情形和两端固定且温度和化学势恒定的边界条件情形,首先利用半群方法得到了系统解的整体存在性,其次利用能量方法、多乘子方法和边界控制技巧得到了解的指数稳定性,最后利用压缩函数的方法得到半群的渐近紧性,从而得到了系统整体吸引子的存在性。
第四章我们讨论了第Ⅲ型热扩散系统解的整体适定性,首先利用半群方法得到了系统解的整体存在性,其次利用能量方法和多乘子方法得到了解的指数稳定性,最后利用压缩函数的方法得到半群的渐近紧性,从而得到了系统整体吸引子的存在性。
第五章总结了本文的主要工作,并对未来的研究方向作了展望。
Thermodiffusion system describes the process of thermodiffusion in a solid body, such a system is established based on the fundamental theories of thermodiffusion. This dissertation is concerned with the initial-boundary value problems for the global well-posedness of the one-dimensional thermodiffusion system and related models. We obtain the existence of global solutions, asymptotic stability and the global(uniform) attractors for the thermodiffusion system of type I, the thermodiffusion system with second sound, and the thermodiffusion system of type Ⅲ.
This dissertation is divided into five chapters.
Chapter1is preface, in which we mainly introduce the background and the pre-vious results, our main work, the fundamental theories for semigroup approaches, the fundamental concepts and existence theorems for global(uniform) attractors, and some classic inequalities.
In Chapter2, we consider the initial-boundary value problems for the one-dimensional thermodiffusion system in a bounded region. Using the semigroup approaches and the multiplier methods, we obtain the global existence and asymptotic behavior of so-lutions for homogeneous, nonhomogeneous and semilinear thermodiffusion system subject to various boundary conditions, respectively. At the end, we get the existence of uniform attractors for the one-dimensional non-autonomous thermodiffusion sys-tem in a bounded region by establishing the uniformly asymptotic compactness of the family of processes generated by the global solutions. The main advantage of this method is that we only need to verify compactness condition with the type of energy estimates same as those for establishing the absorbing set.
In Chapter3, we consider the initial-boundary value problem for a one-dimensional linear model of thermodiffusion with second sound in a bounded region. Using the semigroup approached and the energy methods we obtain the global existence and exponential stability, subject to two boundary conditions (i.e., the rigidly clamped medium with zero heat and diffusion flux; the rigidly clamped medium with tem-perature and chemical potential held on the boundary), respectively. Moreover, the existence of global attractors is established by using the method of contractive func-tions.
In Chapter4, we first establish the global existence and exponential stability for the solutions to the initial-boundary value problem for the one-dimensional model of thermodiffusion of type III by the semigroup approach and the energy method. Fur-thermore, we prove the existence of global attractors by the method of contractive functions.
In Chapter5, we summarize of the results of the dissertation, and predict the work in the future.
本文共分为五章：
第一章是绪论,主要介绍了所研究问题的相关研究背景、研究进展、本文的主要工作、本文研究中经常用到的半群方法基本理论、吸引子存在性基本理论和一些常用的不等式。
第二章研究了第Ⅰ型热扩散系统解的整体适定性,首先利用半群方法得到了自治系统解的整体存在性和指数稳定性,其次利用半群方法得到了非自治系统和半线性系统解的整体存在性,利用多乘子方法得到了非自治系统解的渐近稳定性,并讨论了其它几种边界条件的情况,给出了相关结论,最后利用压缩函数的方法得到过程族的一致渐近紧性,从而得到了非自治系统一致吸引子的存在性。该方法的主要优点在于我们可以利用在建立吸收集时所需要的能量估计,来直接验证紧性条件。
第三章我们讨论了带有第二声的热扩散系统解的整体适定性,分别考察了两端固定且无热扩散流的边界条件情形和两端固定且温度和化学势恒定的边界条件情形,首先利用半群方法得到了系统解的整体存在性,其次利用能量方法、多乘子方法和边界控制技巧得到了解的指数稳定性,最后利用压缩函数的方法得到半群的渐近紧性,从而得到了系统整体吸引子的存在性。
第四章我们讨论了第Ⅲ型热扩散系统解的整体适定性,首先利用半群方法得到了系统解的整体存在性,其次利用能量方法和多乘子方法得到了解的指数稳定性,最后利用压缩函数的方法得到半群的渐近紧性,从而得到了系统整体吸引子的存在性。
第五章总结了本文的主要工作,并对未来的研究方向作了展望。
Thermodiffusion system describes the process of thermodiffusion in a solid body, such a system is established based on the fundamental theories of thermodiffusion. This dissertation is concerned with the initial-boundary value problems for the global well-posedness of the one-dimensional thermodiffusion system and related models. We obtain the existence of global solutions, asymptotic stability and the global(uniform) attractors for the thermodiffusion system of type I, the thermodiffusion system with second sound, and the thermodiffusion system of type Ⅲ.
This dissertation is divided into five chapters.
Chapter1is preface, in which we mainly introduce the background and the pre-vious results, our main work, the fundamental theories for semigroup approaches, the fundamental concepts and existence theorems for global(uniform) attractors, and some classic inequalities.
In Chapter2, we consider the initial-boundary value problems for the one-dimensional thermodiffusion system in a bounded region. Using the semigroup approaches and the multiplier methods, we obtain the global existence and asymptotic behavior of so-lutions for homogeneous, nonhomogeneous and semilinear thermodiffusion system subject to various boundary conditions, respectively. At the end, we get the existence of uniform attractors for the one-dimensional non-autonomous thermodiffusion sys-tem in a bounded region by establishing the uniformly asymptotic compactness of the family of processes generated by the global solutions. The main advantage of this method is that we only need to verify compactness condition with the type of energy estimates same as those for establishing the absorbing set.
In Chapter3, we consider the initial-boundary value problem for a one-dimensional linear model of thermodiffusion with second sound in a bounded region. Using the semigroup approached and the energy methods we obtain the global existence and exponential stability, subject to two boundary conditions (i.e., the rigidly clamped medium with zero heat and diffusion flux; the rigidly clamped medium with tem-perature and chemical potential held on the boundary), respectively. Moreover, the existence of global attractors is established by using the method of contractive func-tions.
In Chapter4, we first establish the global existence and exponential stability for the solutions to the initial-boundary value problem for the one-dimensional model of thermodiffusion of type III by the semigroup approach and the energy method. Fur-thermore, we prove the existence of global attractors by the method of contractive functions.
In Chapter5, we summarize of the results of the dissertation, and predict the work in the future.
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