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

In this paper, we prove that the Leray weak solution $u : \mathbb{R}^3\times (0, T)\rightarrow\mathbb{R}^3$ 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.