The Well Posed Problems For Bipolar Energy Transport Model

In this paper,the global existence and large time behavior of a bipolar energy transport model including electrons and ions two particles is studied.It's widely used in plasma,gas discharge and semiconductor industry and has strong theoretical significance and practical application value.The model is composed of the mass conservation equation,the energy balance equation and the Poisson equation,which can be derived from the bipolar non-isentropic EulerPoisson equation.We mainly study the global existence and large-time behavior of the smooth solution of the three-dimensional bipolar energy transport model with mixed Dirichlet-Neumann boundary conditions.Firstly,we give the local existence of the smooth solution by the Banach fixed point theorem.Then,with help of the constructing the appropriate energy functional and the techniques of inequalities to deal with the lower order a-prior estimates of space and time,we establish the global existence and exponential decay of the smooth solution to the bipolar energy transport model when the initial perturbation is small enough.The structure of this paper is as follows:The bipolar energy transport model is briefly introduced in the first chapter,which including the research background,some known research results,and the main results of this paper.The second chapter introduces some preparatory knowledge,including some important inequalities,and some basic theorems used in the paper.In chapter 3,the local existence of the smooth solution is proved by the Banach fixed point theorem.The fourth chapter is the main part.Due to the appropriate energy functional,introduction of flux function and entropy function,and appropriate weighted energy estimates,we prove the global existence and large time behavior of smooth solution to the bipolar energy transport model while the initial data is a small perturbation around the steady state.