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Keywords:
Navier-Stokes equation; suitable weak solution; regularity
Navier-Stokes equation; suitable weak solution; regularity
Summary:
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.
References:
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