A locally anisotropic regularity criterion for the Navier--Stokes equation in terms of vorticity

Abstract

In this paper, we will prove a regularity criterion that guarantees solutions of the Navier--Stokes equation must remain smooth so long as the the vorticity restricted to a plane remains bounded in the scale critical space $L^4_t L^2_x$, where the plane may vary in space and time as long as the gradient of the vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier--Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in family of scale critical mixed Lebesgue spaces. This regularity criterion also can be seen as interpolating between Chae and Choe's regularity criterion in terms of two vorticity components and Beir\~{a}o da Veiga and Berselli's regularity criterion in terms of the gradient of vorticity direction.