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

In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative velocity anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular solution if ? 3 satisfies ? 0 T ? ( t ) L x 1 p 2 q r ? + ln ? e d < ? , where = and ? ? ? .