SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3338-3365, January 2021.
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 nondiagonal parabolic system with a singular potential which trends to infinity logarithmically when the eigenvalues of $Q$ approach 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 an 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 $(0,T_0)$.

中文翻译：

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具有奇异势能的各向异性朗道-德热能产生的梯度流的规律

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3338-3365 页，2021 年 1 月。
在本文中，我们研究了由朗道-德热内斯自由能产生的梯度流，它描述了 $Q$-张量空间中的向列液晶构型。这个自由能密度泛函由三个二次项组成，作为弹性能量密度部分，以及体部的奇异势，被认为是对 Q 的特征值的物理约束的自然强制。该系统是一个具有奇异势的非对角抛物线系统，当 $Q$ 的特征值接近物理边界时，它以对数趋势趋于无穷大。我们给出了一个严格的证明，对于可能具有无限自由能的相当一般的初始数据，系统在任何正时间 $t_0$ 之后都有一个独特的强解。此外，这个独特的强解在足够长的时间 $T_0$ 后脱离物理边界。我们还给出了解接触物理边界的集合的 Hausdorff 测度的估计，从而证明了中间阶段 $(0,T_0)$ 中解的部分正则性结果。