On the well-posedness of the Darcy–Brinkman–Forchheimer equations for coupled porous media-clear fluid flow

In this paper, we present an existence and uniqueness theory for the Darcy– Brinkman–Forchheimer equations that govern the stationary flow of a porous medium and a clear fluid occupying both bounded and unbounded domains. By extending Ladyzhenskaya’s functional method to the equations at hand, we establish the existence of at least one weak solution. For the case of bounded domains, we additionally show that this solution is unique provided that a suitable smallness assumption, involving the Reynolds number, the porosity of the porous medium and the first eigenvalue of the Laplacian, is satisfied