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 ∂3u satisfies∫0T‖‖‖∂3u(t)‖Lx1p‖Lx2q‖Lx3rβ1+ln⁡(‖∂3u(t)‖L2+e)dt<∞, where 2β+1p+1q+1r=1 and 2<p,q,r≤∞,1−(1p+1q+1r)≥0.