Gradient estimates for Stokes systems in domains

TL;DR

This paper establishes boundary regularity and weak type estimates for solutions to the stationary Stokes system with Dini mean oscillation coefficients in domains with Dini boundary smoothness.

Contribution

It proves boundary continuity of solutions and weak type-$(1,1)$ estimates for the gradient and pressure in Dini smooth domains, extending regularity results for Stokes systems.

Findings

Solutions are continuous up to the boundary.

Weak type-$(1,1)$ estimates hold for the gradient and pressure.

Results apply to domains with $C^{1, m{Dini}}$ boundary.

Abstract

We study the stationary Stokes system with Dini mean oscillation coefficients in a domain having $C^{1,\rm{Dini}}$ boundary. We prove that if $(u, p)$ is a weak solution of the system with zero Dirichlet boundary condition, then $(Du,p)$ is continuous up to the boundary. We also prove a weak type-$(1,1)$ estimate for $(Du, p)$.