Regularity of a Gradient Flow Generated by the Anisotropic Landau--de Gennes Energy with a Singular Potential

Abstract

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 $(0,T_0)$.