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 (1 < q < ∞) in a domain of Rⁿ (n ≥ 2) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: ν·u = 0 and αu + β(T(u,p)ν - <T(u,p)ν,ν>ν) = h on the boundary of the domain with α, β ≥ 0 and α + β = 1, where u denotes a velocity vector, p a pressure, T(u, p) the stress tensor for the Stokes flow, and ν the unit outer normal to the boundary of the domain. It presents the slip condition when β = 1 and the non-slip one when α = 1, respectively.