定态多维量子流体动力学模型解的存在性

Existence of Steady-State Solutions of Quantum Hydrodynamic Model

研究了热平衡态下多维空间的双极量子流体动力学模型.这个模型由二阶椭圆型方程组成,通过研究相应的截断系统,构造不动点算子,利用椭圆型方程的估计,并且由嵌入定理得到不动点算子的紧性,从而得到了解的存在性.在证明过程中,利用最大值原理和二阶椭圆方程的估计,得到解的先验估计,由Leray-Schauder不动点定理,得到了这个模型解的存在性.

A thermal equilibrium of a bipolar quantum hydrodynamic model is analyzed on several dimensional space. This model is composed of elliptic equations. By studying its truncation system, constructing a fixed point operator and using elliptic estimates, the existence of solutions are obtained by applying the compactness of the fixed point operator according to embedding theorem. In the process of the proof, a priori estimate of solutions is acquired by applying the maximum principle and the estimates of elliptic equation. Finally, The existence of solutions of this model is given according to Leray-Schauder＇s fixed point theorem.