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A geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations. (English) Zbl 1349.35276
Summary: We deal with a suitable weak solution $$(\mathbf v,p)$$ 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 $$(\mathbf fx_0,t_0)$$, which uses the $$L^3$$-norm of $$\mathbf v$$ and the $$L^{3/2}$$-norm of $$p$$ in a shrinking backward parabolic neighbourhood of $$(\mathbf x_0,t_0)$$. The refinement consists in the fact that only the values of $$\mathbf v$$, respectively $$p$$, in the exterior of a space-time paraboloid with vertex at $$(\mathbf x_0,t_0)$$, respectively in a “small” subset of this exterior, are considered. The consequence is that a singularity cannot appear at the point $$(\mathbf x_0,t_0)$$ if $$\mathbf v$$ and $$p$$ are “smooth” outside the paraboloid.

##### MSC:
 35Q30 Navier-Stokes equations 35B65 Smoothness and regularity of solutions to PDEs 35D30 Weak solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
##### Keywords:
Navier-Stokes equation; suitable weak solution; regularity
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