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

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 $$∫^T_0frac{||u||^{frac{2}{1-r}}_{dot{B}^{-r}{∞,∞}}+||∇ d||²_{L^∞}}{1+1n(e+||u||_H^S+||∇ d||_H^S)}dt‹∞$$ with 0 ≤ r ‹ 1 and s ≥ 3, then the solution (u,d) can be smoothly extended beyond the time T.