Logarithmically improved regularity criterion for the nematic liquid crystal flows in ${\dot{B}}_{\infty ,\infty }^{-1}$ space

Abstract

In this work, we study the regularity criterion of the three-dimensional nematic liquid crystal flows. It is proved that if the vorticity satisfies

$${\int }_0^T\frac{{\Vert \omega (t,\cdot )\Vert }_{{\stackrel{.}{B}}_{\infty ,\infty }^{-1}}^2}{1+log(e+{\Vert \omega (t,\cdot )\Vert }_{{\stackrel{.}{B}}_{\infty ,\infty }^{-1}})}dt<\infty \text{,}$$

where ${\dot{B}}_{\infty ,\infty }^{-1}$ denotes the critical Besov space, then the solution $(u,d)$ becomes a regular solution on $(0,T]$. This result extends the recent regularity criterion obtained by Fan and Ozawa (2012) [11].