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

TL;DR

This paper establishes a regularity criterion for a Q-tensor model of nematic liquid crystals with Neumann boundary conditions, proving uniqueness and global weak regularity of solutions based on conditions on the velocity field.

Contribution

It extends previous work by providing a new regularity criterion for Q-tensor models, ensuring solution uniqueness and regularity under specific boundary conditions.

Findings

Proved uniqueness of weak solutions.

Established global in time weak regularity for derivatives.

Extended previous models from orientation vector to Q-tensor.

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.