Some Mathematical Results Of Unipolar Isentropic Hydrodynamic Model Of Semiconductors

This thesis mainly investigates two types of mathematical problems for the unipolar isentropic hydrodynamic model of semiconductors,which is the EulerPoisson equations with the damping term.It is composed by the following three parts:The first part is the introduction.We mainly introduce the research background of the hydrodynamic model for semiconductors and the research status of the related mathematical problems,and then give an overview of the research contents and main conclusions.The second part is concerned with the Cauchy problem for the unipolar isentropic hydrodynamic model of semiconductors where the damping coefficient relies on time.The specific damping term is(?)for the parameter ??(-1,1).Here,it is called the strong damping when ?< 0 and weak damping when ?>0;while it is called the damping with constant coefficient when ?= 0.Firstly,we consider the one-dimensional case of the above Cauchy problem in the second chapter,where ??(-1,0)?(0,1).For the strong damping case with ??(-1,0),the system is proved to possess a unique global smooth solution timeasymptotically converging to the stationary solution of the unipolar drift-diffusion model for semiconductors in the rate of(1+t)^?for the constant ?> 0.For the weak damping case with ??(0,1),when the doping profile is a positive constant,the system is further proved to admit a unique global smooth solution converging to a constant state in the rate of(1+t)^?for the constant ?> 0,where the index ??[?,?)relies on the initial perturbation.Secondly,we study the multi-dimensional case of the above Cauchy problem in the third chapter,where??(0,1).When the doping profile is a positive constant,the system is proved to admit a unique smooth solution which converges to a constant state in the rate of(1+t)(?)for some number ?> 0,where the index ?? [?,?)still relies on the initial perturbation.In fact,the indices ?and ?in the convergence rates could be large enough once the initial perturbations are sufficiently closed to zero,such that the algebraic part of the corresponding convergence rate can be arbitrarily fast.In addition,compared with exponential convergence rate e^-?tfor ?> 0 corresponding to the constant coefficient damping,both the weak damping with ??(0,1)and the strong damping with ??(-1,0)make the convergence rate of the solution for the system slow down,and the convergence rate for the strong damping case is slower than that for the weak damping case.In view of this,the damping effect with time-dependent coefficient essentially affects the asymptotic behavior of solutions to the Euler-Poisson system.The third part is the fourth chapter.We consider the long time asymptotic behavior of smooth solutions to the initial-boundary value problem for the onedimensional unipolar isentropic hydrodynamic model of semiconductors with the constant coefficient damping on the half line.Here the boundary conditions are proposed physically as the inflow/outflow/impermeable boundary and the insulating boundary,respectively.First of all,the boundary effect causes some essential difficulties in determining the asymptotic profiles for the solutions,so we propose some additional boundary conditions at far field to the steady-state equations such that the steady-state problems are well-posed.Thus,we can determine the steady-state solutions as the asymptotic profiles for the solutions to the original initial-boundary value problems.Secondly,since there are some boundary gaps in L²-sense between the solutions of original initial-boundary value problems and their steady-state solutions,we technically construct some correction functions to delete these gaps.Then,by using the energy estimates,we further prove that the solutions of the initial-boundary value problems time-asymptotically converge to their asymptotic profiles.Finally,we carry out some numerical simulations,which show that,the graphs for the asymptotic profiles in different boundary cases are significantly distinct.