Existence Of The Global Solutions And Attractors For Fractional Schr(?)dinger Equations

The Schr?dinger equation is the fundamental equation of physics for describing nonrelativistic quantum mechanical behavior, which depicts how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schr?dinger. Using the path integral over Brownian paths derives the standard Schr?dinger equation, while using the path integral over Lévy trajectories deduces the fractional Schr?dinger equation. In this article,we studied the fractional nonlinear Schr?dinger equation and the fractional quasilinear Schr?dinger equations with period boundary value. By using energy method, the existence of the global weak solutions and the existence of the global attractors for these systems are obtained.This article is organized in five chapters.In chapter 1, we introduce the development trend of the partial differential equations and give the physical background for the fractional Schr?dinger equation. Besides, we recall some required definitions, notations, lemmas and present the schedule of this article.In chapter 2, we consider the weakly damped fractional nonlinear Schr?dinger equation with initial condition and periodic boundary condition. With a priori estimates of the solution u in H(7)(8)??, we prove the existence and uniqueness of the global solution of this problem by using the Gelerkin approximations. In the last section of this chapter, by a uniform priori estimates for time, we obtain the existence of the global strong attractor of this initial-boundary value problem.In chapter 3, according to the theorems of approximation, construction of approximate solutions, imbedding in Sobolev spaces and a priori estimates, we deduce the existence of the weak solution for the fractional quasilinear Schr?dinger equation under certain conditions.In chapter 4, we study the weakly damped fractional quasilinear Schr?dinger equation,with initial condition and periodic boundary condition. Using the energy estimates and the Gelerkin approximations derive the existence of the weak solution of this equation, while invoking the theory of semigroup and the theory of absorbing set deduce the existence of the global weak attractor of this problem.In chapter 5, we summarized the work we have done in this article, and also pointed out some unsolved problems for our further study in the future.In this article, the key point is to get a uniformly priori estimates for time about the fractional Schr?dinger equation. During the estimates, we meet some problems which are difficult to be overcome by using the standard method. While we solve these difficulties by using the complicated, meticulous priori estimates.