Liquid Crystal Flows with Regularity in One Direction

Abstract

" In this paper, we consider the Cauchy problem for the model of liquid crystal. We show that if the velocity field u satisfies $$\u2202_3u\u2208L^p(0,T;L^q(\\mathbb{R}^3)), \\frac{2}{p}+\\frac{3}{q}+=1+\\frac{1}{q}, 2 \u2039 q \u2264 \u221e,$$ then the solution is in fact smooth."