Geometric regularity criteria for incompressible Navier–Stokes equations with Navier boundary conditions

Abstract

We study the regularity criteria for weak solutions to the 3D incompressible Navier–Stokes equations in terms of the direction of vorticity, taking into account the boundary conditions. A boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions, provided that the directions of vorticity are coherently aligned up to the boundary. As an application, we establish the boundary regularity for weak solutions to Navier–Stokes equations in round balls, half-spaces and right circular cylindrical ducts, subject to the classical Navier and kinematic boundary conditions.