【作者】 李本图；

【导师】 李栋龙；

【作者基本信息】 广西大学 ， 应用数学， 2007， 硕士

【摘要】 Ginzburg-Landau方程具有十分丰富的物理背景和内涵，它出现在流体力学系统及等离子传播和超导体理论已有很长的历史。许多物理学家和数学家对其物理性质和数学理论进行了深入的研究，取得了丰硕的成果。本文主要研究以下三维Ginzburg-Landau方程的整体吸引子及其维数估计：ut=ρu+（1+iγ）△u-（1+iμ）|u|^2σu，u（0，x）=u₀（x），x∈Ω，u（x，t）是Ω-周期的，Ω=（0，L₁）×（0，L₁）×（0，L₁）∈Ω。其中u（t）是定义在三维空间R³⁺¹的未知复值函数，△是R³的拉普拉斯算子ρ＞0，γ，μ是实参数。本文共分为两章。第一章是本文的概述，叙述了无穷维动力系统，Ginzburg-Landau方程的历史，物理背景及研究状况，介绍所取得的主要结果。第二章是本文的主体部分，共分为四节。第2．1节是引言，给出了本文所考虑问题的假设条件，为下文做好必要的准备。第2．2节证明三维Ginzburg-Landau方程的局部解的存在唯一性。第2．3节利用先验估计的方法证明三维Ginzburg-Landau方程的整体解的存在性。第2．4节证明三维Ginzburg-Landau方程的整体吸引子的存在性并对其维数进行估计。更多还原

【Abstract】 Ginzburg-Landau equation has very abundant physical background andconnotation, which appears at hydromechanics system, plasma propation andsuperconductor theory for a long history. Many physical and mathematicianshave studied it’s physical property and mathematical theory deeply and ob-tained plenteous result.In this dissertation, we aim to study global attractor for three dimensionalGinzburg-Landau equation and the estimate of the dimensionality of ti’s at-tractor,the equation as following:u_t=ρu+（1+iγ）△u-（1+iμ）|u|^2σu,u（0,x）=u₀（x）, x∈Ωu（x,t） isΩ-periodic,Ω=（0, L₁）×（0, L₁）×（0, L₁）∈Ω。where u（t） is an unknown complex function which is defined in three dimen-sional space R³⁺¹,△is a Laplacian operator in R³ ,ρ＞0,γ,μare realparameters.This thesis is composed of two chapters. The first chapter devotes to intro-duce the summarize of the dissertation. we recount about the history, physicalbackground and the present circustance of unlimited dimensional dynamic sys-tem and Ginzburg-Landau equation, introduce some main obtained results.The second chapter is the primary proportion of this paper which composedof four sections.The 2.1 section is the introduction which gives the conditions we will needin order to discuss the problem directly in the following section. In the 2.2 section, we proof that the local solution for this three dimensionalGinzburg-Landau equation is existent and unique.In the 2.3 section, we obtain the existence of global solution for this threedimensional Ginzburg-Landau equation by using the method of priori estimate.In the 2.4 section, we obtain the existence of global attractor for this threedimensional Ginzburg-Landau equation and estimate the dimensionality of thisattractor.更多还原