大初值有界区域可压缩液晶方程组强解的爆破准则

Blow-Up Criteria for Strong Solutions of Compressible Liquid Crystal Equations in Large Initial Bounded Region

液晶在现实生活中有着非常广泛的应用,它们对光、电、磁和热都具有很高灵敏性,根据这一特点可以设计出各种各样具有热–光、磁–光、电–光等物理效应的仪器。关于它的研究遍及了化学、物理学、生物学、材料科学和电子学,形成了各自专门的学科,例如液晶化学、液晶物理、生物液晶、液晶光学等,也引起了国内外数学家和物理学家们广泛的研究兴趣。液晶工业技术中最受重视的问题就是液晶中观察到的分子分布规律、缺陷、相变现象及其动力学规律,这些问题直接关系到液晶设备的制造,而适当的数学模型就是描述这些现象最有力的工具。针对现实中关注的问题,从数学的角度来看,我们关心的问题是相变的数学机理、动力学方程组解的整体存在性和正则性以及强解的爆破准则。本课题考虑的是一维完全可压缩液晶方程无热传导的Cauchy问题,在大初值有界区域的情况下,利用精细的能量估计方法和非线性泛函分析中的方法,证明这个方程组的强解存在爆破准则。

Liquid crystals are widely used in real life. They are highly sensitive to light, electricity, magnetism and heat. According to this feature, various kinds of instruments with physical effects such as heat light, magnetism light, electricity light can be designed. Its research covers chemistry, physics, biology, materials science and electronics, and forms their own specialized disciplines, such as liquid crystal chemistry, liquid crystal physics, biological liquid crystal, liquid crystal optics, etc., which also arouses extensive interest of mathematicians and physicists at home and abroad. The most important problem in the liquid crystal industry is the molecular distribution law, defect, phase change phenomenon and its dynamics law observed in the liquid crystal. These problems are directly related to the manufacture of liquid crystal equipment, and the appropriate mathematical model is the most powerful tool to describe these phenomena. In view of the problems concerned in reality, from the mathematical point of view, we are concerned with the mathematical mechanism of phase transition, the global existence and regularity of solutions of dynamic equations and the blow-up criteria of strong solutions. In this paper, we consider the Cauchy problem of one-dimensional fully compressible liquid crystal equations without heat conduction. In the case of a large initial value bounded region, we use the method of precise energy estimation and nonlinear functional analysis to prove the existence of blow-up criteria for the strong solution of this system of equations.