The Cauchy Problem For The Infinite Dimensional Vector-valued Resonant Nonlinear Schrodinger System

This PhD dissertation is mainly devoted to the study of the Cauchy problem for the infinite dimensional vector-valued resonant nonlinear Schrodinger system.It is one of dispersion equations as a generalization of the classical Schrodinger equation.We mainly study the global well-posedness and scattering theory by the method of"concentration-compactness" or "rigidity" This article is divided into four chapters and the main contents are as follows:In the first chapter,we briefly review the background and progress of the classi-cal nonlinear Schrodinger equation,which leads to the Cauchy problem for the infinite dimensional vector-valued resonant nonlinear Schrodinger system.Finally,the main results and research methods of the problem to be studied in this dissertation are intro-duced.In the second chapter,we briefly recall some necessary theorems and inequalities in harmonic analysis used in this dissertation,and then we introduce some basic concepts and results for the mass-critical,defocusing,infinite dimensional vector-valued resonant nonlinear Schrodinger system,such as mass conservation,energy conservation,local well-posedness and stability of the Cauchy problem for the mass-critical,defocusing,infinite dimensional vector-valued resonant nonlinear Schrodinger system.Then we give the definition of "almost periodic solution",whose quantitative characterization is proven in the Hilbert space L2xh1.Finally,the function spaces U?p(l2),V?p(l2)that we are going to work on are properly defined,and their properties are collected in this chapter.The third chapter deals with long time Strichartz estimate,which is one of the important parts of the dissertation.It is one of the ingredients to exclude the existence of"quasi-sol iton" and "rapid frequency cascade",which guarantee the scattering for mass-critical,defocusing,infinite dimensional vector-valued resonant nonlinear Schrodinger system holds.The establishment of long time Strichartz estimate is based on the U?p(l2)and the V?p(l2)function spaces constructed in the previous chapter,whose proof is quite tedious.In fact,it is based on three bilinear Strichartz estimates,which will be proven in details respectively in this chapter.In Chapter 4,we establish the frequency localized interaction Morawetz estimate,which is another key ingredient in the proof of scattering theory for mass-critical,defo-cusing,infinite dimensional vector-valued resonant nonlinear Schrodinger system.It,together with the long time Strichartz estimate can be used to exclude the existence of the critical element,and then the scattering theory follows,thus we complete the proof of the main result in this dissertation.The derivation of frequency localized interaction Morawetz estimate is somewhat similar to those of the three bilinear Strichartz estimates previously,which will be given in details in this chapter.