# Article

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Keywords:
Navier-Stokes equation; suitable weak solution; regularity
Summary:
We deal with a suitable weak solution \$(\bold v,p)\$ to the Navier-Stokes equations in a domain \$\Omega \subset \mathbb R^3\$. We refine the criterion for the local regularity of this solution at the point \$(\bold fx_0,t_0)\$, which uses the \$L^3\$-norm of \$\bold v\$ and the \$L^{3/2}\$-norm of \$p\$ in a shrinking backward parabolic neighbourhood of \$(\bold x_0,t_0)\$. The refinement consists in the fact that only the values of \$\bold v\$, respectively \$p\$, in the exterior of a space-time paraboloid with vertex at \$(\bold x_0,t_0)\$, respectively in a ”small” subset of this exterior, are considered. The consequence is that a singularity cannot appear at the point \$(\bold x_0,t_0)\$ if \$\bold v\$ and \$p\$ are “smooth” outside the paraboloid.
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