单轴和双轴向列型液晶动力学方程组的数学理论研究

The dynamical equations of nematic liquid crystals can be viewed as a mixture of the Navier-Stokes equation and the heat flow of harmonic maps. Due to the strong nonlinearity and strong coupling terms, there are many difficulties to overcome. So far, the results of the dynamical equations of nematic liquid crystals have mainly focused on the case of constant viscosity coefficient, isothermal and uniaxial liquid crystal molecules. However, the studies of complex cases have just started (or have not even started), which included viscosity coefficient dependent on density (or temperature), non-isothermal and biaxial liquid crystal molecules. The purpose of this project is to study these models, and the main contents are organized as follows: 1. Well-posedness, asymptotic behavior and blowup criterion of the solutions to the dynamical equations of nonisothermal uniaxial nematic liquid crystals; 2. Well-posedness and convergent rate of the strong solutions to the dynamical equations of uniaxial nematic liquid crystals with viscosity coefficient dependent on density (or temperature); 3. Vanishing viscosity limit for the dynamical equations of uniaxial nematic liquid crystals ; 4. Low Mach number limit for the dynamical equations of nonisothermal uniaxial nematic liquid crystals; 5. Well-posedness of the solutions to the dynamical equations of biaxial nematic liquid crystals. The research of this project can enrich and develop the theory and method of hydrodynamic flow.

向列型液晶动力学方程组是由经典的Navier-Stokes方程和调和映照热流方程耦合而成。由于它具有强非线性性和强耦合性的特点，给问题的研究带来了很大的困难。目前，关于向列型液晶动力学方程组的研究结果主要集中在常粘性系数、等温和单轴液晶分子情形。然而，对于更加复杂的情形，包括粘性系数依赖于密度（或者温度）、非等温以及双轴液晶分子情形的研究还很少，有些研究刚开始起步, 有些还未开始。本项目拟研究对象就是这几类方程组，主要研究内容为：1. 非等温单轴向列型液晶动力学方程组解的适定性、渐近行为和爆破准则问题；2. 单轴向列型液晶动力学方程组在变粘性系数下强解的适定性和衰减率问题；3. 单轴向列型液晶动力学方程组的粘性消失极限问题；4. 非等温单轴向列型液晶动力学方程组的小马赫数极限问题；5.双轴向列型液晶动力学方程组解的适定性问题。本项目的研究能进一步丰富和发展流体力学方程的理论和方法。

向列型液晶动力学方程组是液晶动力学中的基本偏微分方程组，它是由经典的Navier-Stokes方程和调和映照热流方程耦合而成的。本项目主要研究了单轴和双轴向列型液晶动力学方程组的数学理论。我们的研究成果包括以下四个方面：a) 研究了非均匀不可压缩的单轴向列型液晶动力学方程组的初值具有真空的情形强解的整体存在性；b) 研究了具有大初值的非等温不可压缩单轴液晶动力学方程组的二维Cauchy问题强解的整体存在性；c) 研究了有界区域上粘性系数依赖于密度的不可压缩单轴向列型液晶动力学方程组解的适定性和衰减估计；d) 研究了三维非等温可压缩单轴液晶动力学方程组的初值具有真空的情形强解的整体存在性。我们的研究成果具有及其重要的数学和物理意义，有着十分广泛和发展前景。