GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D

In this article, we prove the global existence of weak solutions to the nonisothermal nematic liquid crystal system on T2, on the basis of a new approximate system which is different from the classical Ginzburg-Landau approximation. Local in space energy inequalities are employed to recover the estimates on the second order spatial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance.It is shown that these weak solutions satisfy the temperature equation, and also the total energy equation but away from at most finite many singular times, at which the energy concentration occurs and the director field losses its second order derivatives.     

ON GLOBAL BEHAVIOR OF WEAK SOLUTIONS OF COMPRESSIBLE FLOWS OF NEMATIC LIQUID CRYSTALS

In this article,we investigate the global behavior of weak solutions of a simplified Ericksen-Leslie system for compressible flows of nematic liquid crystals in time in a bounded three-dimension domain-arbitrary forces.By adapting the arguments for the compressible Navier-Stokes equations,and carefully analyzing the direction field of liquid crystals in the equations of angular momentum,we show the existence of bounded absorbing sets,global bounded trajectories,and global attractors to weak solutions of compressible flows of nematic liquid crystals with the adiabatic constant γ5/3.     

LOW MACH NUMBER LIMIT OF A COMPRESSIBLE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS MODEL

In this paper, we study the low Mach number limit of a compressible nonisothermal model for nematic liquid crystals in a bounded domain. We establish the uniform estimates with respect to the Mach number, and thus prove the convergence to the solution of the incompressible model for nematic liquid crystals.     

形状记忆合金问题的有限元逼近

１．引言本文讨论非线性微分方程其中　系数　是给定常数,ｆ,ｆ为已知函数．这是形状记忆合金问题的数学模型,未知量ｕ,θ代表位移及Ｋｅｌｖｉｎ温度,其物理背景及数学模型的建立,参见文献［３,４］．最近,文［１,２］讨论了方程组（１．１）－（１．５）的数值求解,提出全离散格式．文［２］用Ｇａｌｅｒｋｉｎ方法,位移ｕ用四阶微分方程的有限个特征向量张成的空间,温度θ用分段线性多项式（折线）空间来近似,给出一个全离散格式,证明了离散近似解的存在唯一性,定性说明收敛于原问题的精确解．文［１］采用［２］中的离散…