A note on regularity criterion for 3D shear thickening fluids in terms of velocity

Abstract

We show that a weak solution for unsteady 3D shear thickening flows becomes a strong solution, for ${2\le p<\frac{11}{5}}$, provided that the velocity field u belongs to the critical space

$$\begin{aligned} L^\beta (0, T; L^\alpha (\Omega )), \quad \frac{\frac{2}{3-p}}{\beta }+\frac{3}{\alpha }\le \frac{p-1}{3-p}, \quad \frac{3(3-p)}{p-1}<\alpha \le \infty . \end{aligned}$$

Here $\Omega$ is $\mathbb {R}^3$ or the periodic domain $[0, 1]^3$. The main ingredient is to prove and use a variant of Gagliardo–Nirenberg’s inequality including the terms $\Vert u\Vert _{\alpha }$ and $\Vert |\mathcal {D}u|^{\frac{p-2}{2}} \nabla ^2 u\Vert _{2}$, where $\mathcal {D}u$ is the deformation rate tensor.