向列型液晶中变分波方程(组)的初值问题研究

项目名称： 向列型液晶中变分波方程(组)的初值问题研究

项目编号： No.11301128

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 胡燕波

作者单位： 杭州师范大学

项目金额： 22万元

中文摘要： 各向异性介质中非线性波的研究是具有重要理论和现实意义的前沿热点课题之一。本项目研究产生于向列型液晶理论的非线性变分波方程(组)的初值问题。研究带源项的变分波方程及其渐近方程的初值问题，探讨源项对两类弱解的存在性和正则性的影响。运用Young测度理论和依赖能量坐标方法，研究变分波方程组及其渐近方程组初值问题的两类弱解的整体适定性。建立相应的耗散格式和守恒格式，对相关问题数值模拟。

中文关键词： 变分波方程；依赖能量坐标；守恒解；特征分解；Young测度

英文摘要： The study of nonlinear waves arising in anisotropic media is one of the forefront focus subjects, which has important theoretical and practical significance. This project studies the initial value problems of the variational wave equation (equations) arising in nematic liquid crystals. The initial value problems of the variational wave equation with a source term and its asymptotic equation are investigated and the effect of the source term for the existence and regularity to two class of weak solutions are discussed. The global well-posedness of two class of weak solutions for the initial value problems to the variational wave equations and its asymptotic equations are considered by using the Young measure theory and the method of energy-dependent coordinates. Moreover, the corresponding dissipative numerical schemes and conservative numerical schemes used to compute the relevant problems are established.

英文关键词： variational wave equation；energy-dependent coordinates；conservative solutions；characteristic decompositions；Young measure