Logarithmical Blow-up Criteria for the Nematic Liquid Crystal Flows

We investigate the blow-up criterion for the local in time classical solution of the nematic liquid crystal flows in dimension two and three. More precisely, $0<T_{\ast }<+\mathrm{\infty }$ is the maximal time interval if and only if (i) for $n=3$, {align*} \int_{0}^{T_{}}\frac{|\omega|{\dot{B}^{0}{\infty,\infty}}+|\nabla d|{\dot{B}^{0}{\infty,\infty}}^{2}}{\sqrt{1+$\text{ln}$(e+|\omega|{\dot{B}^{0}{\infty,\infty}} +|\nabla d|{\dot{B}^{0}{\infty,\infty}})}}$\text{d}$t=\infty, {align} or {align*} \int_{0}^{T_{}}\frac{|\nabla u|{\dot{B}^{-1}{\infty,\infty}}^{2}+|\nabla d|{\dot{B}^{0}{\infty,\infty}}^{2}}{\sqrt{1+$\text{ln}$(e+|\nabla u|{\dot{B}^{-1}{\infty,\infty}} +|\nabla d|{\dot{B}^{0}{\infty,\infty}})}}$\text{d}$t=\infty; {align} and (ii) for $n=2$, {align*} \int_{0}^{T_{}}\frac{|\nabla d|{\dot{B}^{0}{\infty,\infty}}^{2}}{\sqrt{1+$\text{ln}$(e +|\nabla d|{\dot{B}^{0}{\infty,\infty}})}}$\text{d}$t=\infty. {align}