Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces

Study of the scalar oseen equation

This paper is devoted to the scalar Oseen equation, a linearized form of the Navier- Stokes equations. Because of the various decay properties in various directions of RN, the problem is set in Sobolev spaces with anisotropic weights. In a first step, some weighted Hardy-type inequalities are obtained, which yield some norm equivalences. In a second step, we establish existence results

The Oseen equations in Rn and weighted Sobolev spaces

In this paper, we study the nonhomogeneous Oseen equations in Rn. We prove an existence and uniqueness result in weighted Sobolev spaces. As the main tool, we prove an existence and uniqueness theorem of a scalar model of those equations