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On regularity for the Navier-Stokes equations in Morrey spaces

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  • Let $u$ be a local weak solution of the Navier-Stokes system in a space-time domain $D\subseteq\mathbb R^{n}\times\mathbb R$. We prove that for every $q>3$ there exists $\epsilon>0$ with the following property: If $(x_0,t_0)\in D$ and if there exists $r_0>0$ such that

    sup $|x-x_0|+\sqrt{t-t_0} < r_0$ sup $r\in(0,r_0)$ $ \frac{1}{r^{n+2-q}} \int_{t-r^2}^{t+r^2}\ \ \ \int_{|y-x|\le r} |u(y,s)|^{q}\,dy\,ds \le \epsilon $

    then the solution $u$ is regular in a neighborhood of $(x_0,t_0)$. There is no assumption on the integrability of the pressure or the vorticity.

    Mathematics Subject Classification: Primary: 35Q30, 76D05; Secondary: 35K55, 35K15.

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