Existence and stability of stationary solutions for the general nonisentropic hydrodynamic semiconductor models.

摘要

In this thesis, we discuss a nonisentropic hydrodynamic model for semiconductor with heat sources. The model is derived from the Boltzmann equation by taking the first three moments with proper closures, and consists of a set of Euler equations with certain source terms for hydrodynamical quantities such as the density, the velocity and the temperature, and a Poisson's equation for the electric potential, which makes the model self-consistent. That is, the nonisentropic (full) hydrodynamic semiconductor model with the heat source is a hyperbolic-parabolic-elliptic coupled system. We first investigate the thermal equilibrium and non-thermal equilibrium stationary solutions for one- and multi-dimensional full hydrodynamic semiconductor model supplemented with the proper boundary conditions and the more general semiconductor devices in terms of the Schauder's fixed point principle, the Stampacchia truncation methods, the iteration method, and the basic energy estimates. For the non-thermal equilibrium stationary solutions, we deal mainly with the subsonic case. Next, we present the asymptotic analysis such as the zero-electron-mass limit, the zero-relaxation-time limit and the quasi-neutral limit for the stationary hydrodynamic models with general boundary data. Finally, using a delicate energy method, we study the large time asymptotic stability of these stationary solutions. Furthermore, we can establish the global existences and asymptotic behavior of smooth solutions for one- and multi-dimensional Cauchy problems with initial data which are close to the stationary states for the nonisentropic hydrodynamic model with the heat source, and we find that the solutions tend to the stationary solutions exponentially fast as t → +infinity. At the same time, we also obtain the global existences and asymptotic behaviors of smooth solutions for multi-dimensional initial boundary value problems with small perturbed initial data and the proper boundary value conditions for the nonisentropic hydrodynamic model with the heat source, and prove the similar convergence results as for the Cauchy problem.