Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative

Abstract

In this paper, we study the regularity of the weak solutions for the incompressible 3D Navier–Stokes equations with the partial derivative of the velocity. By the embedded technology, we prove that the weak solution $u$ is regular on (0, T] if ${\partial }_{3}u\in L^p\left(0,T;L^q\left(R^3\right)\right)$ with $\left(2/p\right)+\left(3/q\right)=\left(70/37\right)+\left(15/37q\right),\left(15/4\right)\le q\le \infty$ , or $\left(2/p\right)+\left(3/q\right)=\left(34/19\right)+\left(9/19q\right),\left(9/4\right)\le q\le \infty$ .