Gradient estimates for the non-stationary Stokes system with the Navier boundary condition

TL;DR

This paper demonstrates that for the non-stationary Stokes system with Navier boundary conditions, spatial regularity can be improved near a flat boundary, contrasting with the no-slip boundary condition where regularity improvement is limited to the interior.

Contribution

It establishes new gradient estimates for the non-stationary Stokes system under Navier boundary conditions, including cases with finite slip length.

Findings

Spatial regularity is improved near the boundary with Navier boundary conditions.

Finite slip length case is more complex than infinite slip length.

Contrasts with no-slip boundary condition where boundary regularity is limited.

Abstract

For the non-stationary Stokes system, it is well-known that one can improve spatial regularity in the interior, but not near the boundary if it is coupled with the no-slip boundary condition. In this note we show that, to the contrary, spatial regularity can be improved near a flat boundary if it is coupled with the Navier boundary condition, with either infinite or finite slip length. The case with finite slip length is more difficult than the case with infinite slip length.