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: neustupa@math.cas.cz

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


Full text available as PDF.

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.


[Previous Article] [Next Article] [Contents of This Number] [Contents of Mathematica Bohemica]
[Full text of the older issues of Mathematica Bohemica at DML-CZ]