Study On Weak Entropy Solutions To The Euler-Poisson Equations In Bounded Domain

In this paper,we first consider the global existence of L? weak entropy solu-tions for one-dimensional Euler-Poisson equations,for both isentropic and isother-mal case with the initial boundary conditions where x ?[0,1]),t ?[0,T).Secondly,we also study the global existence of L? weak entropy solution for high dimensional Euler-Poisson system,for both isentropic and isothermal case with the initial boundary conditions were x ?[L1,L2](L1,L2 are positive constants),t?[0,T),n is the unit outward normal vector of ?.Finally,the global existence and large time behavior of L? weak entropy solutions for Euler-Poisson equations with damping term are considered(?)with the initial boundary conditions where x ?[L1,L2](L1,L2 are positive constants),t ?[0,T)and H(x,t)<0.This paper is divided into four chapters.In the first chapter,we first in-troduce the development of hydrodynamics,then introduce the research status of Euler-Poisson equations and main contents of the research;In the second chapter,we concerned with the global existence of entropy solutions for one-dimensional Euler-Poisson equations for both isentropic and isothermal case.Firstly,we obtain the uniform bounded of the approximate solutions by adding artificial viscosity,introducing the improved Riemann invariants and the Maxi?mum principle,then the vanishing viscosity method and compensated compactness framework are used to prove the global existence of entropy solutions;In Chapter 3,we study the high dimensional Euler-Poisson equations.We reduced the multi-dimensional compress-ible Euler-Poisson system to one-dimensional system with geometric source term and we study the global existence of weak entropy solutions for one-dimensional systems;In Chapter 4,we first consider the global existence of the weak entropy so-lution for the Euler-Poisson equations with damping,and then we further prove the large time behavior of the weak entropy solution based on the uniformly bounded estimation of density.Moreover,the entropy solutions for both cases are uniformly bounded independent of time.