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}\leq 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}\leq 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.