Regularity of weak solutions and the number of singular points to the 3D simplified nematic liquid crystal system

Abstract

In this paper, we investigate the regularity of weak solutions to the Cauchy problem of the 3D simplified nematic liquid crystal flows with lower bound on pressure, and show that if the negative part of the pressure Π is controlled, or if the positive part of quantity |u|2+|∇d|2+2Π is controlled, then the weak solution (u,d) is global-in-time smooth. We also study the singular points of weak solutions, and prove that if a weak solution (u,d) on R3×(0,T) satisfies‖u(⋅,t)‖Lp(R3)+‖∇d(⋅,t)‖Lp(R3)≤c(T−t)p−32p for some 3<p<∞, where c is a positive constant, then the number of singular points is finite.