# A NEW REGULARITY CRITERION FOR THE NAVIER-STOKES EQUATIONS

Volume 4, Issue 2, pp 126-129
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### Authors

HU YUE - School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China. WU-MING LI - School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China.

### 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 $3\leq r_2\leq\infty$.

### Keywords

• Navier-Stokes equations
• Leray-Hopf weak solution
• Regularity.

•  35Q30
•  76D03

### References

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