A NEW REGULARITY CRITERION FOR THE NAVIER-STOKES EQUATIONS

Abstract

We study the incompressible Navier-Stokes equations in the entire three-dimensional space. We prove that if \(\partial_3u_3 \in L^{s_1}_ t L^{r_1}_x\) and \(u_1u_2 \in L^{s_2}_ t L^{r_2}_x\), then the solution is regular. Here \(\frac{2}{s_1}+\frac{3}{r_1}\leq 1, 3\leq r_1\leq\infty,\frac{2}{s_2}+\frac{3}{r_2}\leq1\) and \(3\leq r_2\leq\infty\).