Regularity criteria via one directional derivative of the velocity in anisotropic Lebesgue spaces to the 3D Navier-Stokes equations

Abstract

In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative of the velocity in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular solution if the ${\partial }_{3}u$ satisfies $${\int }_0^T\frac{{\Vert {\Vert {\Vert {\partial }_{3}u(t)\Vert }_{L_{x_1}^p}\Vert }_{L_{x_2}^q}\Vert }_{L_{x_3}^r}^{\beta }}{1+\ln \left(\Vert {\partial }_{3}u\left(t\right){\Vert }_{L^2}+e\right)}dt<\infty ,$$ $\text{ where }\frac{2}{\beta }+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1\text{ and }2<p,q,r\le \infty ,1-\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)\ge 0$.