量子Landau方程适定性的若干问题研究

两类基本的量子粒子费米子（自旋为半整数）和玻色子（自旋为整数）遵循截然不同的量子力学原理，其描述方程分别为Landau-Fermi-Dirac(LFD)方程和Landau-Bose-Einstein(LBE)方程。本项目将分别研究这两类量子Landau方程在空间齐性和空间非齐性框架下的适定性理论,尤其关心大初值Cauchy 问题解的存在性理论和大时间状态行为，以及LBE方程在某些情形可能出现的爆破及其在Bose-Einstein凝聚中的应用。

Two types of fundamental quantum particles with half-integer spin or integer spin are called fermions or bosons, characterized by different quantum effects thus different evolution equations,which are named Landau-Fermi- Dirac (LFD) equation and Landau-Bose-Einstein (LBE) equation, respectively. This project will focus on the studies of well-posedness of the two types of quantum Landau equations under both spatial homogeneous and spatial non-homogeneous settings，with special attention on the existence theory for the Cauchy problem with large initial data, and large time behavior of the solutions. This project will also consider the blow-up behavior of LBE equation in some cases and its application in explaining Bose-Einstein condensations.