Regularity criterion for the 3D generalized Newtonian fluids

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for $\frac{5}{3}<p<\frac{11}{5}$ provided that the gradient of velocity $\mathrm{\nabla }\mathit{u}$ belongs to the critical space $L^{\frac{2}{2-(3-p)a}}(0,T;{\dot{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-a}({\mathbb{R}}^3))$, where $a\in (\frac{3}{2},\frac{2}{3-p})$ if $p\in (\frac{5}{3},2)$ and $a\in (\frac{1}{p},\frac{2}{3-p})$ if $p\in [2,\frac{11}{5})$.