本项目研究两类重要的发展方程:简化的Ericksen-Leslie系统和修正双分支Camassa-Holm系统。对于简化的Ericksen-Leslie 系统，我们主要研究其整体解的存在性和解的正则性。所用的方法主要有弱收敛方法，Littlewood-Paley 理论，奇异积分及交换子估计。结合局部能量不等式和弱收敛方法对该系统研究是一种新的尝试。 对于修正双分支Camassa-Holm系统，我们主要研究其整体弱解的局部正则性和唯一性。对于这些问题的研究我们所用的方法包括传统的加符号条件分析和构造度量方法以及我们采用的局部正则性提升结合弱强唯一性思想的新方法。

In this project, we consider two important evolution equations : a simplified Ericksen-Leslie system and a modified two-component Camassa-Holm system. For the simplified Ericksen-Leslie system,we mainly study the existence of global solutions and the regularity of solutions.To deal with this problems,we mainly use this methods:weak convergence method,Littlewood-Paley Theory,singular integral and commutators estimates.The combination of local energy inequality and weak convergence method is a novel attempt. For a modified two-component Camassa-Holm system,our main research are the local regularity and uniqueness of weak solutions.Our main methods investigating this problems involve classic analysis of symbol conditions and constructed metrics.Furthermore,we adopt a new method combined the lift of local regularity and weak-strong uniqueness.