考虑拉伸项的液晶三维等温模型
On a 3D isothermal model for nematic liquid crystals accounting for stretching terms
摘要：
本论文研究了描述不同形状分子运动输运下液晶流动演化的偏微分方程系统的适定性。更具体地说，{ \em速度场} $\ub$的演化由Navier-Stokes不可压缩系统支配，该系统具有应力张量，该应力张量表现出输运项和诱导项之间的特殊耦合。{ \em指向矢场} $\bd$的动力学由抛物型Ginzburg-Landau方程的一个变化来描述，该方程具有适当的物理约束$|\bd|=1$的惩罚。该方程考虑了流场的运动学输运和弹性能引起的内部弛豫。这方面的贡献的主要目的是克服缺乏一个最大值原理的导演方程和证明（没有任何限制的数据和物理常数的问题）的存在性，在物理意义上的边界条件下的$\bd$和$\ub$的全球时间弱解。
英文摘要：
The present contribution investigates the well-posedness of a PDE system describing the evolution of a nematic liquid crystal flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the {\em velocity field} $\ub$ is ruled by the Navier-Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamic of the {\em director field} $\bd$ is described by a variation of a parabolic Ginzburg-Landau equation with a suitable penalization of the physical constraint $|\bd|=1$. Such equation accounts for both the kinematic transport by the flow field and the internal relaxation due to the elastic energy. The main aim of this contribution is to overcome the lack of a maximum principle for the director equation and prove (without any restriction on the data and on the physical constants of the problem) the existence of global in time weak solutions under physically meaningful boundary conditions on $\bd$ and $\ub$.
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