On a generalized resolvent estimate for the Stokes system with Robin boundary condition

Abstract

We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the $L_q$ framework in a domain of ${\mathbf{R}}^n$ ($n\ge 2$) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: $\nu \cdot u=0$ and on the boundary of the domain with $\alpha ,\beta \ge 0$ and $\alpha +\beta =1$, where $u$ denotes a velocity vector, $p$ a pressure, $T(u,p)$ the stress tensor for the Stokes flow, and $\nu$ the unit outer normal to the boundary of the domain. It presents the slip condition when $\beta =1$ and the non-slip one when $\alpha =1$, respectively.