The Almost-periodic Solutions And Long Time Stability Of Some Partial Differential Equations

In this thesis,we mainly give a quantitative version of the infinite-dimensional KAM theorem,apply the infinite dimensional KAM theory to prove that there exists an almost-periodic solution for a class of almost-periodically forced nonlinear beam equation,and by the partial Birkhoff normal form to show the long time stability for KdV equation with a quasi-periodical forcing.This thesis consists of four chapters and an appendix.In Chapter 1,we mainly introduce the Hamiltonian invariant torus and the historical background,significance and the present research status of KAM theory.Meanwhile,the main content of this thesis is introduced.In Chapter 2,we discuss the relation between the size of perturbations and the dimension of tori based on the KAM theorem which is in Ann.Sc.Norm.Sup.Pisa,23(1996)and can be applied to Hamiltonian PDEs.In other words,we give a quantitative version of the KAM theorem mentioned above.Moreover,in order to obtain non-degenerate normal form,after infinite KAM iterations,we make action and normal variables of KAM tori of the reduced Hamiltonian system in a domain instead of shrinking them to the origin.In Chapter 3,we consider the almost-periodically forced nonlinear beam equation,and prove that under appropriate assumptions,by repeatedly using the infinite-dimensional KAM theorem,there exist almost-periodic solutions.In Chapter 4,we discuss the long time stability of the quasi-periodically forced KdV equation.Since the perturbation of the KdV equation is unbounded,the method of studying the long time stability of solution of the Hamiltonian PDEs with bound perturbation is not applicable here.By transforming the Hamiltonian system into an autonomous one and utilizing the weak-tame property,we can obtain the long time stability of solutions of the KdV equation.In the appendix,we list some lemmas which are used in this thesis.