The Well-posedness And Infinite-dimensional Dynamical Systems For Two Kinds Of Nonlinear Evolution Equations

In this paper, we study the well-posedness and infinite-dimensional dynamical systems for the (2+1)-dimensional long wave-short wave equations, (1+1)-dimensional stochastic long wave-short wave equations, and the nonlinear elastodynamic system. We prove the existence of global smooth solutions, the existence of global attractor, and the approximate inertial manifolds for the (2+1)-dimensional long wave-short wave equations. We prove the existence of the random attractors for the (1+1)-dimensional stochastic long wave-short wave equations of the periodic boundary value problem and the initial value problem in an infinite lattice. At last, we prove the local existence of the classical solution to the mixed initial-boundary value problem of Neumann type for the nonlinear elastodynamic system. This paper closely contact of physics, mechanics and stochastic analysis, so it is very important in theoretical significance and application value.This paper is organized in seven chapters.In Chapter 1, we give the physical background for the nonlinear evolution equations, such as long wave-short wave equations and the nonlinear elastodynamic system. The development and application of the stochastic partial differential equations are introduced. We recall some known results and briefly describe our work in the present paper.In Chapter 2, we consider the initial value problem of a class of generialized (2+1)-dimensional long wave-short wave equations. By using a priori estimates and the Galerkin method, we obtain the existence and uniqueness of the global smooth solutions for the periodic boundary value problem and the initial value problem.In Chapter 3, we prove the existence of global attractor for the initial value problem of a class of (2+1)-dimensional long wave-short wave equations. In section 1, by using a priori estimates and the Galerkin method, we get the existence of the global smooth solutions. In section 2,3,4, by the theory of the seimgroup, we get the existence of global attractor.In Chapter 4, we constract the approximate inertial manifolds for the initial value problem of (2+1)-dimensional long wave-short wave equations. In section 1, we get the a priori estimates by using the Brezis-Gallouet inequality. In section 2, we get the existence of the global smooth solutions by the Galerkin method. In section 3, the approximate inertial manifolds are constracted by using the abstract differential equations.In Chapter 5,we consider the long time behavior of the solution for (1+1)-dimensional stochas-tic long wave-short wave equations. We prove the existence of the random attractor.In Chapter 6, we discuss the initial value problem of (1+1)-dimensional stochastic long wave-short wave equations in a infinite lattice. By stochastic process z(t)= eiW1(t), we change stochastic equations into deterministic equations. Then using the general theory of attractor, we get the existence of the random attractor.In Chapter 7, we consider the local existence of the classic solution for the mixed initial-boundary value problem of Neumann type for the nonlinear elastodynamic system. In order to obtain this result, we prove the existence of solutions for the Neumann type of the second order linear hyperbolic system with variable coefficients (in Sobolev spaces) outside of a domain by using linear evolution operators and integro-differential equations.