A Logarithmically Improved Blow-up Criterion for a Simplified Ericksen-Leslie System Modeling the Liquid Crystal Flows

Abstract

" In this paper, we prove a logarithmically improved blow-up criterion in terms of the homogeneous Besov spaces for a simplified 3D Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystal. The result shows that if a local smooth solution (u,d) satisfies $$\u222b^T_0\\frac{||u||^{\\frac{2}{1-r}}_{\\dot{B}^{-r}{\u221e,\u221e}}+||\u2207 d||\u00b2_{L^\u221e}}{1+1n(e+||u||_H^S+||\u2207 d||_H^S)}dt\u2039\u221e$$ with 0 \u2264 r \u2039 1 and s \u2265 3, then the solution (u,d) can be smoothly extended beyond the time T."