A generalization of the Darcy–Forchheimer equation involving an implicit, pressure-dependent relation between the drag force and the velocity

We study mathematical properties of steady flows described by the system of equations generalizing the classical porous media models of Darcy and Forchheimer. The considered generalizations are outlined by implicit relations between the drag force and the velocity, that are in addition parametrized by the pressure. We analyze such drag force–velocity relations which are described through a maximal monotone graph varying continuously with the pressure. Large-data existence of a solution to this system is established, whereupon we show that under certain assumptions on data, the pressure satisfies a maximum or minimum principle, even if the drag coefficient depends on the pressure exponentially.