The Global Attractors For Some Nonlinear Evolution Equations

In this paper, we consider the dynamical behavior for some nonlinear evolution equations, such as Long-Short wave equations, Hasegawa-Mima equation and Hirota equation. The existence of globally smooth solutions, the existence of global attractors, its fractal dimensions and the approximate inertial manifolds for this systems are obtained.This paper is organized in five chapters.In chapter 1, we give the physical background for the nonlinear evolution equations, such as Long-Short wave equations, Hasegawa-Mima equation and Hirota equation. We recall some important known results and briefly describe our research results of the present paper.In chapter 2, we consider a class of generalized Long-Short wave equations in one dimension. In section 2.2, by uniformly a priori estimates and Galerkin method, we prove the existence of the global attractors for the periodic boundary value problem, and we get the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors. In section 2.3, we construct the two approximate inertial manifolds for the periodic boundary value problem. We obtain the order of approximation of the manifolds to the global attractors.In chapter 3, we consider a class of generalized Long-Short wave equations in one dimension. In section 3.2, by Kato's method for abstract quasi-linear evolution equations, we consider the initial value problem. In section 3.3, by introducing weighted space and using the method of priori estimates, by using the weighted function space and the interpolation inequality, we prove the existence of the global attractors for the Long-Short wave equations on an unbounded domain.In chapter 4, we consider the periodic boundary value problem of the Hasegawa-Mima equation. In section 4.2, by uniformly a priori estimates and Galerkin method, we prove the existence of the globally smooth solution for the two-dimensional Hasegawa-Mima equation. In section 4.3, we also prove the existence of the globally smooth solution for a class of generalized Hasegawa-Mima equa-tion(GHM equation) in two dimension. In some special case, we prove that the...