MKdV Limit Of The Euler-poisson System

In this paper,we consider the mKdV limit of the Euler-Poisson system.Under the Gardner-Morikawa transformation,we derive the mKdV equation from the Euler-Poisson system by perturbation method.We employ a careful combination of a series of energy inequalities such as Cauchy inequality,H(?)lder inequality,Gronwall inequality and Sobolev embedding theorem together with analysis of the structure of the remainder system to obtain the uniform bound for the remainders.Thus we establish that the solutions of the Euler-Poisson system converge to the mKdV equation as ?(?)0.This justifies rigorously the mKdV limit of the Euler-Poisson system.This paper consists of the following parts:In part 1,we introduce the physic background of plasma along with the recent progress of studies on the Euler-Poisson system and limit theory,and the main idea of this paper,respectively.In the second part we introduce the concepts,inequalities and equations involved in this paper.In part 3,under the Gardner-Morikawa transformation,we derive the mKdV equation from the Euler-Poisson system formally and we obtain the reminder system.In part 4,we employ a careful combination of energy inequalities in Sobolev space together with analysis of the structure of Poisson equation to obtain energy inequalities about the remainders.In chapter 5,Gronwall inequality is used to prove the main theorem of this paper and the further research works have been forecast.