On Serrin Interior Regularity Criterion for Navier-Stokes Equations
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 $u\in{L_t^{s'}L_x^s}$ locally is a distributional solution to the Navier-Stokes equations with $\frac2{s'}+\frac3s=1$ for $s'\in[4,\infty)$, then $u\in L^q_t(C_x^\infty)$ locally for all $q\in(2,s')$. If $s'\in(2,4)$, 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.