A Dirichlet-Transmission Problem for Darcy–Forchheimer–Brinkman and Navier–Stokes Equations in Bounded Lipschitz Domain

A boundary value problem of Dirichlet-transmission type for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes equations in two adjacent bounded Lipschitz domains from \(\mathbb {R}^{3}\) ${\mathbb{R}}^3$ has been studied. The existence and uniqueness of a weak solution in some Sobolev spaces is obtained using a layer potential approach and a fixed point theorem, when the boundary data are chosen in some \(L^{2}\) $L^2$ -based Sobolev spaces and are suitable small.