A New Prodi–Serrin Type Regularity Criterion in Velocity Directions

Abstract

In this article we generalize (Vasseur in Appl Math 54(1):47–52, 2009) to Lorentz spaces. More specifically, we prove the following. Let u be a Leray–Hopf solution to the Navier–Stokes equation with viscosity $\nu$ and initial value $u_0 \in L^2 ({\mathbb {R}}^3)$. Then there is $c_0 > 0$ such that u is smooth beyond $T > 0$ if

$$\begin{aligned} \left\| {{\mathrm{div}}} \left( \frac{u}{| u |} \right) \right\| _{L^{p, \infty } (0, T ; L^{q, \infty } ({\mathbb {R}}^3))} < c_0 \nu ^{1 - \frac{1}{p}} \Vert u_0 \Vert _{L^2}^{- 1} \end{aligned}$$

(1)

with $\frac{2}{p} + \frac{3}{q} \leqslant \frac{1}{2}$, $q > 6$. We also show that u remains smooth beyond $T > 0$ if

$$\begin{aligned} \left\| {\mathrm{div}} \left( \frac{u}{| u |} \right) \right\| _{L^{p, r} (0, T ; L^{q, \infty } ({\mathbb {R}}^3))} < \infty \end{aligned}$$

(2)

with $\frac{2}{p} + \frac{3}{q} \leqslant \frac{1}{2}, q > 6$ and ${1 \leqslant r < \infty }$.