Blow-up Solutions Of The Generalized Zakharov System With Magnetic Fields In R~3

The nonlocal nonlinear Schr¨odinger equations describe the state of Zakharov system with magnetic fields in plasma at the limit of infinite ion sound velocity,which has a wide range of physics and application background.The thesis mainly studied the equations from the mathematical point of view by using the harmonic analysis and variational method to study the Cauchy problem of 3D generalized Zakharov system with magnetic fields.This thesis is divided into three chapters as follows:In Chapter 1,we introduce the research background,main contents and main work.In Chapter 2,we mainly discuss the sufficient conditions for the existence of local so-lutions and global solutions.Firstly,the existence of local solutions is proved by using the principle of contraction mapping,and two conservation laws are given.Then,using the energy method,we give the sufficient conditions for the existence of the global solution of the system under L~2-critical and L~2-supercritical respectively.In Chapter 3,we study the sharp threshold of the 3D generalized Zakharov system with magnetic fields and the blasting rate.Firstly,by using Virial identity,the sufficient conditions for the existence of blow-up solutions are obtained under L~2-critical and L~2-supercritical condi-tions,respectively.Secondly,by establishing the development flows,the sharp threshold for the existence of blow-up solutions are obtained.Finally,we obtain the blow-up rate by combining the Gagliardo-Nirenberg inequality and Strichartz estimations.