一类不可压弹性复杂流体方程的数学理论研究

The focus of this project is to study analytic issues in the theory of elastic complex fluids, such as variational wave equation and incompressible nematic liquid crystal flows. We are particularly interested in the global existence, uniqueness, partial regularity and singularities. . Because of the complex microstructures of the hydrodynamic systems, the mathematical models are usually very difficult and challenge on mathematical analysis. And on the other hand, all these models have widely applications in physics and daily lives. Therefore they has become the most concerned and basic problems and attracted lots of physicist, materials scientist and mathematicians. . There are many important mathematical problems in the modeling and analysis of elastic complex fluids. The complexity of the systems requires new mathematical methods and in many cases, new ideas. The proposed research in this project will incorporate theories of physics, partial differential equations, harmonic analysis, geometric measure theory and geometric analysis. New tools from analysis will play an increasingly important role in the mathematical modeling and analysis of important physical problems. Due to the inherent nonlinearities, many mathematical questions such as existence, uniqueness, partial regularity and singularity of weak solutions remain open. The research of the PI is designed to enhance and contribute to this important link between analysis and the understanding of challenging physical problems.

本课题主要研究液晶流体等一类不可压弹性复杂流体的偏微分方程问题，特别是方程弱解的存在性、唯一性、正则性和奇性解的存在性。由于液晶材料复杂的微观结构和丰富的物理特性，数学家和物理学家所建立的偏微分方程模型都具有非常大的数学难度和挑战性，同时也具有鲜明的物理意义、广泛的应用价值和非常重要的数学理论价值。因此，一直以来都是数学界、物理界和材料学界最为关心的基本问题之一。. 液晶变分波动方程、不可压液晶流体方程弱解的存在唯一性、正则性和奇性解的存在性等问题特色鲜明，综合性强，与其他领域有很强的交叉互补性，深入的数学研究需要综合运用物理学、偏微分方程、调和分析和几何分析等学科的知识和工具。其中有些问题具有非常强的探索性，原有的理论已经无法完全解决，需要探究新的研究方法和技巧。这既可以推动数学理论特别是偏微分方程理论的发展，又可以诠释和发现物理现象、推动材料科学的发展。

本课题主要研究液晶流体等一类不可压弹性复杂流体的偏微分方程问题，特别是方程弱解的存在性、唯一性、正则性和奇性解的存在性。由于液晶材料复杂的微观结构和丰富的物理特性，数学家和物理学家所建立的偏微分方程模型都具有非常大的数学难度和挑战性，同时也具有鲜明的物理意义、广泛的应用价值和非常重要的数学理论价值。因此，一直以来都是数学界、物理界和材料学界最为关心的基本问题之一。.液晶变分波动方程、不可压液晶流体方程弱解的存在唯一性、正则性和奇性解的存在性等问题特色鲜明，综合性强，与其他领域有很强的交叉互补性，深入的数学研究需要综合运用物理学、偏微分方程、调和分析和几何分析等学科的知识和工具。其中有些问题具有非常强的探索性，原有的理论已经无法完全解决，需要探究新的研究方法和技巧。这既可以推动数学理论特别是偏微分方程理论的发展，又可以诠释和发现物理现象、推动材料科学的发展