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Morrey space techniques applied to the interior regularity problem of the Navier-Stokes equations. (English) Zbl 1022.35036

A new criterion of the interior regularity to the weak solutions of the evolutionary Navier-Stokes equations is formulated in the framework of Morrey spaces. Namely, the following theorem is proved: Let $\frac{7}{2}, $\left({t}_{0},{x}_{0}\right)\in \left(0,T\right)×{\Omega }$, $0<{r}_{0}<\text{dist}\left(\partial {\Omega },{x}_{0}\right)$ and $0<{t}_{0}-{r}_{0}^{2}$ with $\partial {\Omega }$ the boundary of ${\Omega }$. Suppose that $u$ is a weak solution of equation

$\partial u/\partial t-{\Delta }u+\nabla ·\left(u\otimes u\right)+\nabla \pi =0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{1.em}{0ex}}\left(0,T\right)×{\Omega }$
$\text{div}u=0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{1.em}{0ex}}\left(0,T\right)×{\Omega }·$

Then $u$ is regular in $\left({t}_{0}-{r}_{1}^{2},{t}_{0}\right]×{B}_{{r}_{1}}\left({x}_{0}\right)$ for ${r}_{1}={r}_{0}/2$ provided that

$sup{r}^{1-5/q}{\left({\int }_{t-{r}^{2}}^{t}{\int }_{|x-y|

where the constant $ϵ$ is sufficiently small and the supremum is taken over all $\left(t-{r}^{2},t\right]×{B}_{r}\left(x\right)\subset \left({t}_{0}-{r}_{0}^{2},{t}_{0}\right]×{B}_{{r}_{0}}\left({x}_{0}\right)·$

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory