兰州大学机构库 >数学与统计学院
非柱形区域上两类发展方程解的长时间行为研究
Alternative TitleThe long-time behavior of solutions for two types of nonlinear evolutionary equations on non-cylindrical domains
周峰
Thesis Advisor孙春友
2016-06-07
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword非柱形区域 拉回D–吸引子 复Ginzburg-Landau (CGL) 方程 非线性弱耗散波方程 变分解
Abstract随着社会的发展与科技的进步,在物理学、生物学、控制论等学科领域涌现出越来越多的非柱形区域问题.由于此类问题非自治的固有性以及非自治动力系统发展的滞后性, 关于其动力学行为的研究至今还比较少.本文针对两类具体的非柱形区域上的复Ginzburg-Landau (CGL) 方程与非线性弱耗散波方程解的动力学行为进行了研究.首先,本文对同胚区域上的CGL方程与非线性弱耗散波方程解的适定性及其动力学行为进行了研究.主要包括:(1)借助于同胚变换以及新建立的一些重要的不等式,证明此两类方程强解的存在唯一性.(2)选择合适的函数空间给出弱解的定义,并证明弱解的存在唯一性.(3)分别建立两类系统的拉回吸引子.由于区域变化以及方程本身的特性带来的困难,我们分别运用有限ε–网与收缩函数的思想方法来证明CGL方程与弱耗散波方程的渐近紧性.以上问题的研究构成本文第三章与第五章的主要内容.同时,我们引入了一些新的方法与技巧建立系统的拉回吸引子,降低了现有方法(如[67, 102]) 对区域变化的要求.在第五章中,我们还建立了有效的判定准则来研究同胚区域波方程能否保持双曲性的题.其次,本文研究了单调区域上的CGL方程解的适定性及其动力学行为.研究单调区域上发展方程的经典方法是“惩罚法”.然而, 由于CGL方程本身的复值性及非线性性等特征,现有的惩罚函数无法应用CGL 方程的研究. 结合已有的惩罚函数及CGL方程的特点,本文给出了一种新型的惩罚函数.特别地,此类惩罚函数对于研究其他非柱形区域问题,如波方程的单调区域问题将起到积极促进作用.同时,沿用S. Bonaccorsi [14] 等引入的“选择”的思想,本文证明了变分解的唯一性.然后, 我们利用无穷维动力系统的一些思想方法证明系统存在紧的拉回吸收集,进而建立此类非自治动力系统的拉回吸引子.此亦为我们第四章的主要内容.最后,我们基于所取得的研究成果及当前的研究现状,列出将要研究的部分问题.
Other AbstractWith the development of society and technology, there are more and more partial differential equations on non-cylindrical domains coming from physics, biological mathematics and control theory, etc. Due to their inherent characteristics and the lag of development for non-autonomous dynamical systems, there are a few research on the dynamics of evolution equations on non-cylindrical domains. In this paper, we considered the dynamics of the complex Ginzburg-Landau (CGL) equations and the nonlinear weakly dissipative wave equations on two kinds of concrete non-cylindrical domains. Firstly we studied the dynamics for the CGL equations and nonlinear weakly dissipative wave equations on non-cylindrical domains that the spatial domains Ot are obtained from a bounded base domain O by a diffeomorphism respectively, mainly includes: (1) Using a suitable change of variables and some established important inequalities, we established the existence and uniqueness of strong solutions for these two types of equations. (2) We gave a definition of the weak solutions by selecting a proper function space, and proved the results of the existence and uniqueness of weak solutions. (3) Due to the domain is varying and the inherent characteristics for the two types of the equations, we applied finite ε-net and contraction function methods to obtain the asymptotic compactness of the two dissipative systems respectively. These are the main contents of chapter 3 and chapter 5. At the same time, we introduced some new methods and techniques to establish the existence of pullback attractor for the dynamical systems, which can improve the restriction on varying domains in the aforementioned works on the topic, such as [67, 102]. In addition, We also constructed a sufficient condition to ensure the transformation is hyperbolic in Appendix of chapter 5, which may be of independent interest. Secondly we investigated the dynamics of the CGL equations on non-cylindrical domains that the spatial domains Ot satisfying a monotonicity condition. It is well known that the classical method to study this type region is the“penalty method”. However, the existing penalty function cannot be applied directly to the study of CGL equations due to their inherent characteristics. Combining with the existing penalty function and the characteristics of CGL equations, we presented a new type of penalty function. In particular, this method will play an important role in studying the dynamics of other d...
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Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225444
Collection数学与统计学院
Recommended Citation
GB/T 7714
周峰. 非柱形区域上两类发展方程解的长时间行为研究[D]. 兰州. 兰州大学,2016.
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