On Serrin Interior Regularity Criterion for Navier-Stokes Equations

Abstract

We revisit Serrin's interior spatial regularity criterion for distributional solutions to the Navier-Stokes equations in ${\mathbb{R}}^3$ and considerably relax the hypotheses in two main directions. More precisely, we show that if $u\in L_t^{s^{\prime }}{L}_x^s$ locally is a distributional solution to the Navier-Stokes equations with $\frac{2}{s^{\prime }}+\frac{3}{s}=1$ for $s^{\prime }\in [4,\infty )$, then $u\in L_t^q(C_x^{\infty })$ locally for all $q\in (2,s^{\prime })$. If $s^{\prime }\in (2,4)$, the same conclusion holds provided that in addition $u\in L_t^4(L_x^p)$ locally, for some $p>1$. In particular, we remove any integrability hypothesis on the vorticity, and we reduce the requirement of integrability in time all the way to $L^4$ from $L^{\infty }$. To achieve this, we employ a new bootstrap argument, distinct from Serrin's, and we argue that a reduction of the exponent in time integrability does not follow from Serrin's original argument.

15 pages