Existence Of Weak Solution And Relaxation Limit Of The Bipolar Isothermal Euler-Poisson Equations

With the development of science and technology, and the extensive application of computers,investigation of more and more mathematical model from science, technology, and even variousfields of social science, have been becoming the focus of science study, especially in the last years,having the technology of submicronic devices, many people have more motivations in semicon-ductor physics fields. In past thirty nears, people raise many microcosmic and macroscopic modelsto describe the motion of semiconductor material and devices with electrified particles(electronsor holes). They consist of quantum mechanical models, kinetic models, ?uid dynamical models,while for the mini semiconductor devices, the effect of quantum is not ignored, then correspondingmodels also include some quantum mechanics. Roughly speaking, these hydrodynamic modelsconsisting of the effect of quantum are: drift-diffusion equation with quantum and hydrodynami-cal model for semiconductor with quantum.The analysis and computational mathematics of macroscopic models attracted more and morepure mathematician and applied mathematician, and they obtained many important results. Forinstance, for one-dimensional hydrodynamical semiconductor model, the existence of subsonicand transonic steady-state solutions, the blasting mechanism of smooth solutions for the time-dependent hydrodynamical models, the existence of weak solutions, the existence, uniqueness,large time behavior of smooth"small"solutions and the relaxation limit of weak solutions andsmooth solutions, i.e., the relationship of the weak solutions of normal hydrodynamical modeland classical hydrodynamical model–drift-diffusion equation, all of these have been attained. Weknow that classical(or quantum) full hydrodynamical model in our life is bipolar semiconductormodel. But since the interaction and the special structure of two carriers, the study of the bipolarsemiconductor model is far from being well, the discussion of all kinds basic problems is activeand not perfect.In this thesis, We mainly concentrate on isothermal bipolar semiconductor model(Euler-Poisson system), the system can be derived from semiconductor device or plasma objects. Us-ing the modified Glimm scheme and the estimate of compactness, we firstly establish the Cauchyproblem with large initial data. Next, we use the entropy inequality and dela Valle′e-Poussin cri-terion of weak compactness, we show that the entropy-admissible weak solutions converges tothe solutions to the bipolar isothermal drift-diffusion equation whileτ?→0. This implies theinherent relationship of the bipolar isothermal Euler-Poisson system and the corresponding drift- diffusion equation.