The local regularity theory for the Stokes and Navier--Stokes equations near the curved boundary

Abstract

In this paper, we study local regularity of the solutions to the Stokes equations near a curved boundary under no-slip or Navier boundary conditions. We extend previous boundary estimates near a flat boundary to that near a curved boundary, under very low starting regularity assumptions. Compared with the flat case, the proof for the curved case is more complicated and we adapt new techniques such as the ``normal form" after the mollification, recovering vertical derivative estimates from horizontal derivative estimates, and transferring temporal derivatives to spatial derivatives, to deal with the higher order perturbation terms generated by boundary straightening. As an application, we propose a new definition of boundary regular points for the incompressible Navier--Stokes equations that guarantees higher spatial regularity.