A regularity criterion for the solution of the nematic liquid crystal flows in terms of ${\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1}$-norm

In this paper, we investigate regularity criterion for the solution of the nematic liquid crystal flows in dimension three and two. We prove the solution $(u,d)$ is smooth up to time $T$ provided that there exists a positive constant ${\epsilon }_0>0$ such that (i) for n=3, |(u,\nabla d)|{L^{\infty}(0,T;\dot{B}^{-1}{\infty,\infty})}\leq \varepsilon_{0}, and (ii) for $n=2$, |\nabla d|{L^{\infty}(0,T;\dot{B}^{-1}{\infty,\infty})}\leq \varepsilon_{0}.