A uniqueness and regularity criterion for Q-tensor models with Neumann boundary conditions

Abstract

We give a regularity criterion for a $Q$-tensor system modeling a nematic Liquid Crystal, under homogeneous Neumann boundary conditions for the tensor $Q$. Starting of a criterion only imposed on the velocity field ${\bf u}$ two results are proved; the uniqueness of weak solutions and the global in time weak regularity for the time derivative $(\partial_t {\bf u},\partial_t Q)$. This paper extends the work done in [F. Guill\'en-Gonz\'alez, M.A. Rodr\'iguez-Bellido \& M.A. Rojas-Medar, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr. 282 (2009), no. 6, 846-867] for a nematic Liquid Crystal model formulated in $({\bf u},{\bf d})$, where ${\bf d}$ denotes the orientation vector of the liquid crystal molecules.