Dynamical Behavior For Three Dimensional Ginzburg-Landau Equation

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.