量子流体模型及其诱导系统在弱拓扑中的数学理论

量子效应的出现使得紧性框架难以构建，量子流体动力学模型的弱解特性成为数学界普遍关心的问题。本项目基于量子流体模型和波动系统之间的相互转化关系，拟通过构建和完善奇异非线性波动系统弱解的适定性、爆破和渐近极限等理论，探索研究量子流体模型在弱拓扑中的数学特性，具体内容包括：(1)研究由量子流体力学模型诱导的奇异Schrödinger-Poisson系统弱解的适定性、爆破以及散射问题；(2)研究由量子磁流体模型诱导的非线性Maxwell-Schrödinger(MS)系统弱解的适定性、爆破以及小参数极限，特别考察MS系统在能量空间中的无条件唯一性；(3)建立量子流体模型在弱拓扑中的适定性理论。. 该研究将丰富色散方程的相关结果，拓宽弱拓扑中的分析技巧，为非线性系统弱解的研究提供新思路，同时为等离子体、激光、超导体以及半导体材料等的开发与应用研究提供一定的理论依据。

The emergence of quantum effects makes it difficult to construct compactness, the weak solution characteristic of fluid model is a common concern in mathematics field. Based on the mutual transformation relationship between the dispersive wave system and quantum hydrodynamics model, this project is proposed to study the mathematical characteristics of the quantum hydrodynamic model in the weak topology by constructing and improving the theory of wellposedness, blow up and asymptotic limit of the weak solution of the singular nonlinear dispersive system. Precicely, (1) We will study the wellposedness, blowup and scattering of the solutions of the singular schrodinger-poisson system induced by the quantum fluid dynamics model. (2) The well-posedness, blowup and small parameter limits of solutions of nonlinear Maxwell-Schrodinger (MS) systems induced by quantum magnetohydrodynamic model will be investigated. (3) We will study the characteristics of quantum fluid dynamics model in weak topological space, and establish the posedness theory of weak energy solution.. This project is a frontier topic in the field of the research for PDEs, the study will enrich the related results of dispersion equation, will broaden the analytical skills in the weak topology space, will provide a new train of thoughts for the research of weak solution to the nonlinear coupling system. At the same time, it also will provide a theoretical basis for the development and the application of plasma, laser, superconductor and semiconductor materials.