Reduced and. generalized stokes resolvent equations in asymptotically flat layers, part I: Unique solvability

We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer Omega(0) = Rn-1 x (-1, 1). Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In the first part we prove the unique solvability in L-q-Sobolev spaces, 1 < q < infinity, by extending the known results in the case of an infinite layer Omega(0) via a perturbation argument to asymptotically flat layers which are sufficiently close to Omega(0). Combining this result with standard cut-off techniques and the parametrix constructed in the second part, we prove the unique solvability for an arbitrary asymptotically flat layer. Moreover, we show equivalence of unique solvability of the generalized and the reduced Stokes resolvent equations, which is essential for the second part of this contribution.