On the Robin-Transmission Boundary Value Problems for the Nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes Systems

Abstract

The purpose of this paper is to study a boundary value problem of Robin-transmission type for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes systems in two adjacent bounded Lipschitz domains in \({{\mathbb{R}}^{n} (n\in \{2,3\})}\), with linear transmission conditions on the internal Lipschitz interface and a linear Robin condition on the remaining part of the Lipschitz boundary. We also consider a Robin-transmission problem for the same nonlinear systems subject to nonlinear transmission conditions on the internal Lipschitz interface and a nonlinear Robin condition on the remaining part of the boundary. For each of these problems we exploit layer potential theoretic methods combined with fixed point theorems in order to show existence results in Sobolev spaces, when the given data are suitably small in \({L^2}\)-based Sobolev spaces or in some Besov spaces. For the first mentioned problem, which corresponds to linear Robin and transmission conditions, we also show a uniqueness result. Note that the Brinkman–Forchheimer-extended Darcy equation is a nonlinear equation that describes saturated porous media fluid flows.