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

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: v · u = 0 and au + β(T(u, p)v -(T(u, p)u, v)v) = 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 v 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.