On the transmission problems for the Oseen and Brinkman systems on Lipschitz domains in compact Riemannian manifolds

Abstract

The purpose of this work is to show the well‐posedness in L2‐Sobolev spaces of the Poisson‐transmission problem for the Oseen and Brinkman systems on complementary Lipschitz domains in a compact Riemannian manifold. The Oseen system appears as a perturbation of order one of the Stokes system, given in terms of the Levi‐Civita connection, while the Brinkman system is a zero order perturbation of the Stokes system. The technical details of this paper rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by a first order differential operator given in terms of the Levi‐Civita connection. The compactness of this differential operator requires to restrict ourselves to low dimensional compact Riemannian manifolds.