一维双极量子漂移-扩散方程稳态解的存在性和经典极限

研究了来自于半导体器件和等离子体中的一维双极量子漂移-扩散模型的稳态解.在有合适边界条件的有界区域里,先利用Schauder不动点定理和能量估计的技巧,证明一维双极量子漂移-扩散模型的稳态解的存在性和唯一性;其次,研究双极量子漂移-扩散模型的稳态解的经典极限,即当普朗克常数ε趋于零时,量子漂移-扩散模型的稳态解趋向于经典漂移-扩散模型的稳态解.

Existence and classical limit of stationary solutions to a one-dimensional bipolar quantum drift-diffusion equation

We study the stationary solutions of a one-dimensional bipolar quantum drift-diffusion model from semiconductor devices and plasmas.In a bounded interval supplemented by the proper boundary conditions,we first show the existence and uniqueness of the stationary solutions to the one-dimensional bipolar quantum drift-diffusion model.The proof can be completed by the Schauder fixed-point principle and the careful energy estimates.Then,we study the classical limit of the stationary solutions to the bipolar quantum drift-diffusion model.Namely,we show that the stationary solution to the quantum drift-diffusion model approaches that to the drift-diffusion model as the scaled Planck constant ε tends to zero.