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

Abstract

In this paper, we prove that the Leray weak solution $u:{\mathbb{R}}^3\times (0,T)\to {\mathbb{R}}^3$ of the Navier-Stokes equations is regular in ${\mathbb{R}}^3\times (0,T)$ under the scaling invariant Serrin condition imposed on one component of the velocity $u_3\in L^{q,1}(0,T;L^p({\mathbb{R}}^3))$ with \[ \frac{2}{q}+\frac{3}{p}\leq 1,\quad 3<p<+\infty. \] This result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

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