Locally Space-Time Anisotropic Regularity Criteria for the Navier–Stokes Equations in Terms of Two Vorticity Components

Abstract

In this paper we prove the regularity of Leray weak solutions of the Navier–Stokes equations as long as the vorticity projection to a plane is bounded in the scale critical space $$L^p(0,T;L^q)$$ , $$2/p+3/q=2$$ , $$q \in (3/2,\infty )$$ . The plane may vary in space and time while the unit vector $$v=v(x,t)$$ orthogonal to the plane is locally a Holder function in space with the coefficient 1/2. This extends previous works by Chae and Choe and by Miller. We further show that a generalized form of this criterion improves several other regularity criteria in terms of the vorticity direction known from the literature.