Well-Posedness Of Sonic Boundary Value Problem To Unipolar Hydrodynamic Model Of Semiconductors

This thesis focuses on the well-posedness of sonic boundary value problem for three types of unipolar hydrodynamic model of semiconductors.In chapter 1,we first introduce the research background and status of the hydrodynamic model of semiconductors,and briefly summarize the main results of this thesis.In chapter 2,we consider sonic boundary value problem to one dimensional unipolar isothermal hydrodynamic model of semiconductors with transonic dop-ing profile,and categorize the transonic doping profile into two types:subsonic-dominated and supersonic-dominated.Firstly,when the doping profile is subsonic-dominated,the system is proved to possess a unique interior subsonic solution and an interior supersonic solution by the compactness analysis combining the energy method and the Green function method,and we apply the phase-plane analysis to obtain the non-existence of the above solutions with a small relaxation time.Based on the idea of construction,we prove the existence of transonic shock solutions with a large relaxation time.Secondly,when the doping profile is supersonic-dominated,we discuss the existence and non-existence of all types of stationary solutions of this problem.In chapter 3,we concern sonic boundary value problem to one dimensional unipolar full hydrodynamic model of semiconductors with weak semiconductor ef-fect.Particularly,the weak semiconductor effect means that both momentum and energy relaxation time are sufficiently large,which makes the system nearly isen-tropic.Under the framework of compactness,we utilize the Schauder fixed point theorem to prove the existence and uniqueness of interior subsonic solution,and show the existence of interior supersonic solutions and transonic shock solutions by the continuous perturbation method.In addition,when the doping profile is a subsonic constant,the existence of C~1-smooth transonic solutions is obtained by the phase-plane analysisIn chapter 4,we study the well-posedness of radial solutions to sonic bound-ary value problem for unipolar isothermal multi-dimensional hydrodynamic model of semiconductors model in an annular domain.We first obtain the existence and uniqueness of radial subsonic solution and the existence of radial supersonic solu-tions by the Schauder fixed point theorem,where radial supersonic solutions are proved by a two-step iteration.The difficulty of non-autonomous system can be overcome owing to the special structure of transonic solutions.Therefore,it can be proved that there exist infinitely many radial transonic shock solutions to the system with a large relaxation time,and radial C~1-smooth transonic solutions are obtained with a small relaxation time.Here adopting the idea of local analysis,we first prove the existence of smooth transonic solutions by the local continu-ation method,and then acquire the C~1 regularity of the solutions by the local approximation method.