Global Smooth Solutions And Blow-up Solutions For The Generalized Zakharov Equations

Abstract:Zakarov system is a very important kind of nonlinear partial dif-ferential equations which describes the interaction of plasma and laser. It is of importance and interests from both theories and applications. In this thesis, we mainly study two important types of Zakharov systems:the Zakharov system with quantum effects describing the quantum corrections to the Zakharov system for langmuir waves in plasma, and the Zakharov system with magnetic field de-scribing the spontaneous generation of a magnetic field in a cold plasma.In the first chapter, we mainly discuss the Zakharov system with quantum effects in the framework of the scalar mode in two space dimensions:By making a priori integral estimates and utilizing the so-called continuity method, we prove the existence and uniqueness of global smooth solutions to the Cauchy problem for the above system in the sobolev space. In the second chapter, we study the blow-up solution of the magnetic Za-kharov system in the three space dimensions: If T is the finite blow-up time, we derive the integral estimate below: implies thatEspecially, for a radially symmetric solution to the Cauchy problem of thr Za-kharov(ZSM), there holds...