A GEOMETRIC IMPROVEMENT OF THE VELOCITY-PRESSURE LOCAL REGULARITY CRITERION FOR A SUITABLE WEAK SOLUTION TO THE NAVIER-STOKES EQUATIONS

We deal with a suitable weak solution (v, p) to the Navier-Stokes equations in a domain Omega subset of R-3. We refine the criterion for the local regularity of this solution at the point ( fx(0), t(0)), which uses the L-3 -norm of v and the L-3/2 -norm of p in a shrinking backward parabolic neighbourhood of (x(0), t(0)). The refinement consists in the fact that only the values of v, respectively p, in the exterior of a space-time paraboloid with vertex at (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 (x(0), t(0)) if v and p are "smooth" outside the paraboloid.