论文标题
各向异性Landau-de Gennes能量产生的梯度流量的规律性具有奇异的电位
Regularity of a gradient flow generated by the anisotropic Landau-de Gennes energy with a singular potential
论文作者
论文摘要
在本文中，我们研究了由Landau-DE Gennes自由能产生的梯度流，该梯度在$ q $ tensors的空间中描述了列液晶构型。这种自由能密度的功能由三个二次术语作为弹性能量密度部分组成，并且在整体部分中具有奇异的潜力，这被认为是对$ q $的特征值的物理约束的自然执行。该系统是一个非对角抛物线系统，具有奇异的潜力，当$ q $的特征值接近物理边界时，它会趋向于无限对数。我们提供了一个严格的证据，即对于具有无限自由能的相当通用的初始数据，该系统在任何积极的时间$ t_0 $之后都具有独特的强大解决方案。此外，这种独特的强大解决方案在足够大的时间$ T_0 $之后从物理边界脱离。我们还给出了集合触摸物理边界的集合的Hausdorff度量的估计，因此证明了中间级$（0，T_0）$中解决方案的部分规律性结果。
In this paper we study a gradient flow generated by the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the space of $Q$-tensors. This free energy density functional is composed of three quadratic terms as the elastic energy density part, and a singular potential in the bulk part that is considered as a natural enforcement of a physical constraint on the eigenvalues of $Q$. The system is a non-diagonal parabolic system with a singular potential which trends to infinity logarithmically when the eigenvalues of $Q$ approaches the physical boundary. We give a rigorous proof that for rather general initial data with possibly infinite free energy, the system has a unique strong solution after any positive time $t_0$. Furthermore, this unique strong solution detaches from the physical boundary after a sufficiently large time $T_0$. We also give estimate of the Hausdorff measure of the set where the solution touches the physical boundary and thus prove a partial regularity result of the solution in the intermediate stage $(0,T_0)$.
