MATHEMATICA BOHEMICA, Vol. 139, No. 4, pp. 685-698, 2014

A geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations

Jiri Neustupa

Jiří Neustupa, Academy of Sciences of the Czech Republic, Institute of Mathematics, Žitná 25, 115 67 Prague, Czech Republic, e-mail:

Abstract: 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.

Keywords: Navier-Stokes equation; suitable weak solution; regularity

Classification (MSC 2010): 35Q30, 76D03, 76D05

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