可压缩量子流体模型的适定性和渐近行为研究

The study of the compressible hydrodynamic equations is one of the key fields of partial differential equations. The aim of this project is to study the global existence and decay rates of the smooth solution with small initial perturbation for the compressible quantum hydrodynamic model：When the initial perturbation around a steady state is sufficiently small, .by using energy method, spectrum analysis, Fourier transform method, etc., introducing suitable weighted norm, (1) global existence and decay rates of all order spatial derivatives of the smooth solution are obtained for quantum Hall-MHD equations; (2) global existence and decay rates of the smooth solution are obtained for quantum Navier-Stokes-Poisson equations and full quantum Navier-Stokes-Poisson equations; (3) global existence and decay rates of the smooth solution for bipolar quantum Navier-Stokes-Poisson equations, and global existence of the smooth solution for bipolar full quantum Euler-Poisson equations are investigated. Subjects involved in this project not only have important research value in the field of mathematical theory and physical application, but also are hot research problems with certain difficulty in related fields.

可压缩流体力学方程的研究是目前偏微分方程研究的核心领域之一。本项目主要研究可压缩量子流体模型初始小扰动光滑解的整体存在性和时间衰减率：当稳态附近的初始扰动充分小时，利用能量方法，谱分析方法，傅里叶变换方法等，引入适当的加权范数，(1)得到量子Hall-MHD方程光滑解的整体存在性和各阶空间导数的时间衰减率；(2)得到量子Navier-Stokes-Poisson方程和完全量子Navier-Stokes-Poisson方程光滑解的整体存在性和时间衰减率；(3)得到双极量子Navier-Stokes-Poisson方程光滑解的整体存在性和时间衰减率, 以及双极完全量子Euler-Poisson方程光滑解的整体存在性。本项目所涉及的课题，不仅在数学理论和物理应用方面具有重要的研究价值，而且还是相关领域有相当难度的热点研究问题。

本项目主要研究了量子等离子体可压缩流体方程的适定性和渐近行为。这些偏微分方程的研究可以为材料科学、航空航天、微电子技术等领域的研究提供一定的理论依据。利用能量方法，算子半群理论，傅里叶变换方法等，引入适当的加权范数，本项目证明了以下结果：(1) 三维全空间中量子等离子体可压缩Hall-MHD方程的衰减率；(2) 包括三种物质的等离子体Euler-Poisson系统的长波长极限，以及修正的Korteweg-de-Vries（mKdV）方程的严格推导；(3) 具有给定质量源和一般形式外力项的可压缩量子流体模型的稳态解的存在性和稳定性。其中的一些结果已发表在Journal of Differential Equations, Acta Applicandae Mathematicae 和 Journal of Mathematical Physics上。