Neumann Problems for the Stokes Equations in Convex Domains

Abstract

This paper studies the Neumann boundary value problems for the Stokes equations in a convex domain in ${\mathbb{R}}^d$. We obtain nontangential-maximal-function estimates in $L^p$ and $W^{1,p}$ estimates for $p$ in certain ranges depending on $d$. These ranges are larger than the known ranges for Lipschitz domains. The proof relies on a $W^{2,2}$ estimate for the Stokes equations in convex domains.