Regularity and uniqueness for a class of solutions to the hydrodynamic flow of nematic liquid crystals

TL;DR

This paper establishes regularity and uniqueness criteria for weak solutions to the nematic liquid crystal flow, showing smoothness and uniqueness under certain integrability conditions on the velocity and director field gradients.

Contribution

It introduces an $psilon$-regularity criterion and proves interior smoothness and uniqueness for solutions with specific integrability properties.

Findings

Solutions are smooth when $p>2$ and $q>n$.

Uniqueness holds for solutions satisfying certain integrability conditions.

Provides a framework for regularity and uniqueness in nematic liquid crystal flow.

Abstract

In this paper, we establish an $\epsilon$-regularity criterion for any weak solution $(u,d)$ to the nematic liquid crystal flow (1.1) such that $(u,\nabla d)\in L^p_tL^q_x$ for some $p\ge 2$ and $q\ge n$ satisfying the condition (1.2). As consequences, we prove the interior smoothness of any such a solution when $p>2$ and $q>n$. We also show that uniqueness holds for the class of weak solutions $(u,d)$ the Cauchy problem of the nematic liquid crystal flow (1.1) that satisfy $(u,\nabla d)\in L^p_tL^q_x$ for some $p>2$ and $q>n$ satisfying (1.2).