Scaling-invariant Serrin criterion via one velocity component for the Navier–Stokes equations

Abstract

The classical Ladyzhenskaya–Prodi–Serrin regularity criterion states that if the Leray weak solution $u$ of the Navier–Stokes equations satisfies $u\in L^q(0,T;L^p({\mathbb{R}}^3))$ with $\frac{2}{q}+\frac{3}{p}\le 1$, $p>3$, then it is regular in ${\mathbb{R}}^3\times (0,T)$. In this paper, we prove that the Leray weak solution is also regular in ${\mathbb{R}}^3\times (0,T)$ under the scaling-invariant Serrin condition imposed on one component of the velocity, i.e., $u_3\in L^{q,1}(0,T;L^p({\mathbb{R}}^3))$ with $\frac{2}{q}+\frac{3}{p}\le 1$, $3<p<+\infty$. This result means that if the solution blows up at a time, then all three components of the velocity have to blow up simultaneously.