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 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 t0. Furthermore, this unique strong solution detaches from the physical boundary after a sufficiently large time T0. 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,T0).