液晶方程组的一些数学理论研究

Liquid crystal flows is a hyperbolic-parabolic system with strongly coupling and nonlinearity. Because of its rich physical background and practical applications, and many unsolved mathematical problems, the research on this system has been always the central issues in applied mathematics and computational mechanics. In this project, we try our best to adopt the classical theory and some new techniques, which were developed in the studies of nonlinear partial differential equations as well as harmonic and geometric analysis, to study some mathematically important problems of liquid crystal flows. We think that the results obtained in future will develop and enrich the theory of the liquid crystal flows..

液晶方程组是具有强耦合性、强非线性的双曲-抛物耦合组。由于它丰富的物理背景和实际应用，以及该领域存在众多有待解决的数学问题，所以它一直是应用数学和计算流体力学研究的前沿热点问题。本项目拟从数学理论研究的角度出发，利用近年来逐步完善的非线性偏微分方程理论以及调和分析、几何分析中的新的思想、新方法去研究液晶方程组相关数学问题，从而进一步发展和完善液晶方程组的研究成果。

本项目主要研究对象是液晶方程组解的适定性问题。通过3年的研究工作，项目组成员基本完成了预期成果，目前我们取得了三项研究成果：1、证明了3维齐次不可压液晶方程组Cauchy问题古典解的衰减估计；2、证明了仅依赖密度上界的2维可压缩液晶方程组Cauchy问题强解的blow-up准则；3、证明了2维非齐次不可压液晶方程组Cauchy问题大初值强解的全局存在性。