Scaling-invariant Serrin criterion via one velocity component for the Navier-Stokes equations (Q6200391)

Summary: 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 \(\mathbb{R}^3\times(0,T)\) in 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.