【作者】 李栋龙；

【导师】 郭柏灵；

【作者基本信息】 中国工程物理研究院北京研究生部 ， 应用数学， 2003， 博士

【摘要】 该文由两部分构成。首先，在第一部分，考虑Ginzburg-Landau方程在三维空间的整体吸引子的存在性，整体吸引子的正则性，指数吸引子和三无界区域R³上的整体吸引子。第二部分，考虑广义Ginzburg-Landau方程在三维空间的整体吸引子的存在性和时间周期解的存在性。 该文由七章构成。在第一章，介绍Ginzburg-Landau方程的物理背景，研究状况及本文的工作内容。在第二章，考虑Ginzburg-Landau方程在三维空间的整体吸引子的存在性，首先考虑Ginzburg-Landau方程的局部解的存在性，对于一给定的扰动项N（u），证明N（u）是收缩的且是局部Lipschitz连续的。因此，可获得局部解的存在性。然后，使用先验估计的方法获得整体解的存在性。同时也获得了紧的有界吸收集的存在性，从而得到整体吸引子的存在性，最后讨论了它的Hausdorff维数和分形维数。第三章，在三维空间中研究Ginzburg-Landau方程整体光滑解的存在性和整体吸引子的正则性，首先利用系列精细的估计证明整体光滑解的存在性。进而研究在H^m中的吸引子的存在性。考虑H¹中的解半群{S^（1）（t）}t≥0，采用分解其解算子S¹（t）=S₁¹（t）+S₂¹（t）的方法，证明了S₁¹（t）u₀比S¹（t）u₀更加正则，而当t→∞时，‖S₂¹（t）u₀‖₁趋于零，其中u₀∈H²。进而证明H¹中的吸引子A₁等于H^m（2σ+1≥m≥2）中的吸引子A_m。第四章，研究Ginzburg-Landau方程在三维空间的指数吸引子的存在性。首先证明解算子S（t）是Lipschitz连续的，然后证明离散解算子S_*=S（t_*）满足挤压性，从而得到指数吸引子M的存在性。它的分形维数是有限的。第五章，讨论Ginzburg-Landau方程在三维无界区域R³上的整体吸引子。通过引入加权空间，以克服古典Sobolev空间在无界区域嵌入的非紧性。在加权空间进行先验估计，获得解算子S（t）在加权空间紧的有界吸收集，从而在加权空间得到整体吸引子的存在性。第六章，它是论文的第二部分，研究带导数项的广义Ginzburg-Landau方程在三维空间的整体吸引子的存在性。主要困难在于解在空间H¹，H²的先验估计。我们充分考虑方程的特点，将高阶非线性项化为一非负定二次型使得解在H¹中的吸收集的存在性得以证明。通过能量估计的巧妙组合以及多种形式的不等式进行精细的估计以获得H“的吸收集的存在性.从而得到整体吸引子的存在性，讨论了它的Hausdorff维数和分形维数.第七章，讨论广义Ginzburg一Landau方程在三维空间的时间周期解的存在性。首先应用Galerkin方法和Larey一schaude:不动点定理证明近似解的存在性，然后进行近似解的高阶导数的先验估计（关于空间变量和时间变量）。最后，使用标准的紧性讨论的方法得到这个系统的时间周期解的存在性。更多还原

【Abstract】 This dissertation consists of two part. In first part we consider the existence of the global attractor the complex Ginzburg-Landau equation in three dimen-sions space, the regularity of the global attractor, the exponential attractors and the existence of the global attractor in whole R³. Then, in secondary part, we consider the existence of the global attractor and the time periodic solution for generlized complex Ginzburg-Landau equation in three dimensions space.This dissertation consists of seven chapters. In chapter 1, we briefly in-troduce the background in physics and developments in the Ginzburg-Landau equation. In which the main work of the dissertation is described. In chapter 2, we study the existence of the global attractor the complex Ginzburg-Landau equation in three dimensions space. First, we consider existence of local solu-tion. For a given perturbation N（u）, we prove N（u） is contractive and locally Lipschitz continuous. Therefore, we obtain existence of local solution. Then we obtain the existence of global solution by the using the method of priori estimate. At the same time, we give the existence of bounded absorbing set. Moreover, we obtain the existence of global attractor. Finally, we study the dimensionality of Hausdorff and fractal of attractor. In chapter 3, we study the existence of global smooth solution and regularity of the global attractor for the complex Ginzburg-Landau equation in three dimensions space. First, applying a, series fine estimate, we prove existence of global smooth solution. Moreover, we obtain the existence of the attractor in H^m. Considering solution semigroup in H¹, we decomposing solution operator We prove that is more regular than the solution tends to zero as t goes to infinity, uniformly for . Furthermore, attractor A₁ in H¹ is equal to attractor A_m in H^m（2σ + 1 ≥ m ≥ 2）. In chapter 4, we study the existence of the exponential attractor of the complex Ginzburg-Landau equation in three dimen-sions space. We first show that the solution operator S（t） is Lipschitz continuous, then prove the discrete solution operator S_* = 5(t_*) satisfy the squeezing property, finally, we get the existence of the exponential attractor M. whose fractal dimensionality is finite. In chapter 5, we study the existence of the global attrac-tor for the complex Ginzburg-Landau equation in 3-D unbounded domain. By introducing weighted space and using the method of priori estimatehe, uniformly compactness are achieved for S（t） in weighted space to overcome the noncom-pactness of the classical Sobolev embedding in unbounded domain. In chapter 6, it’s the secondary part of this dissertation, we study the existence of the global attractor for the generalized complex Ginzburg-Landau equation with derivative term in three dimensions space. Considering enough the property of the equa-tion, we prove existence of absorbing set of the solution in H¹ by the method of changeing the higher order nonlinear term as nonnegative guadratic form and prove existence of absorbing set of the solution in H² by the method of linear combining of energy inequality and multiform classical inequality. Moreover, we obtain the existence of global attractor. Finally, we study the dimensionality of Hausdorff and fractal of attractor. In chapter 7, we discuss the existence of time periodic solution of the generalized complex Ginzburg-Landau equation in three dimensions space. First, we apply the method of Galerkin and the fixed theorem of the Larey-Schauder to prove the existence of the approximate solution. Next, we give the priori estimates of the higher order derivatives （with respect to spa-tial variable and time variable） of the approximate solution. Finally, we use the method of standard compactness arguments to get the existence of time periodic solution of this system.更多还原