The Well - Posedness And Scattering Theory Of Nonlinear Schrdinger Equations

In this dissertation, we study the system of the coupled nonlinear Schrodinger equation in dimension three. By establishing the a priori interaction Morawetz estimate in the spirit of the work of Tao-Visan-Zhang, we prove the glob-al well-posedness and scattering of the solution (u, v) to the defocusing coupled equation. Furthermore, we adapt the standard method from to obtain the blowup result for the focusing case under the restriction:energy is negative and the initial data belongs to S:=H1∩L2(|x|2dx) or the initial data is radial.In chapter 1, we introduce the development process of nonlinear Schrodinger equation, and state two main results in this paper.In chapter 2, we give notation and recall some known results, which will be used in the following writing.In chapter 3, firstly,we apply the Banach fixed point argument to obtain the local well-posedness theory for (1.0.1). Then combining with mass and energy conservation, we complete the proof of the global well-posedness for (1.0.1) with λ=1. Finally, we prove the interaction Morawetz estimate. Then we obtain the finite global Strichartz norms based on the interaction Morawetz estimate. As a result, we utilize the global Strichartz norms to obtain scattering.In chapter 4, we will prove finite time blowup under the assumptions in Theorem 1.2. To do this, we will follow the convexity method of Glassey [10]. More precisely, we derive the proof into two cases. Firstly, we will consider the variance V(t):=||xu(t)||L22+||xu(t)||L22 for (uo,vo)∈H1(R3)∩L2(·|x|2dx). Then we redefine the Virial function VR(t):=∫R3Ψ(x)(|u(t, x)|2+|v(t,x)|2)dx for (u0, v0) being radial function. Thereafter, we show that as a function of t> 0, they are decreasing and concave, which suggests the existence of a blowup time T* at which V(T*)= 0 and a blow timeT*R at which VR(T*R)= 0.