量子力学是现代物理学的一个重要分支，主要研究微观粒子的运动. 量子流体是统计物理学和凝聚态物理学的一个重要研究领域. 它可以用来描述很多物理现象, 例如超导性, 超流性, 波色-爱因斯坦凝聚等. 目前关于量子流体力学方程的弱解存在性的研究以及半导体中的量子Euler-Poisson方程的研究已经取得了较多的进展, 但是关于解的爆破机制, 渐近极限, 以及耦合量子流体力学方程的研究相对较少. 本项目首先会研究量子流体力学方程组解的爆破机制. 其次本项目将会对带粘性量子流体力学方程组的小参数极限进行一些研究, 如半经典极限和粘性消失极限等. 最后我们将会研究量子Navier-Stokes-Maxwell方程和量子Navier-Stokes-Landau-Lifshitz方程的弱解存在性.

Quantum mechanics is an important branch of modern physics, which mainly studies the motion of microscopic particles. Quantum fluid is an important research field in statistical physics and condensed matter physics. It can be used to describe many physical phenomena, such as superconductivity, superfluidity, the Bose-Einstein condensation. Recently, there are amount of the researches about the existence of the weak solution to the quantum hydrodynamic model and the researches about the quantum Euler-Poisson equation, but there are relatively few studies about the blow-up mechanism, asymptotic limit and the coupled quantum hydrodynamic model. In this program, we will investigate the blow-up mechanism of quantum hydrodynamic equations, and establish the blow-up criteria for several kinds of quantum hydrodynamic models. At this time, we will focus on the small parameter limit about the several kinds of viscous quantum hydrodynamic equations, such as the semiclassical limit and the vanishing viscosity limit. At last, we will study the global existence of the weak solution to the quantum Navier-Stokes-Maxwell equation and the quantum Navier-Stokes-Landau-Lifshitz equation.