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

Neustupa, Jiří

Mathematica Bohemica, Tome 139 (2014) no. 4, pp. 685-698.

Résumé

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.

Détail

DOI : 10.21136/MB.2014.144145

Classification : 35B65, 35Q30, 76D03, 76D05
Keywords: Navier-Stokes equation; suitable weak solution; regularity