Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media

This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source terms and boundary fluxes are integrated into the model. It covers isentropic gas flows, ideal gases and slightly compressible fluids. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressure, and study the corresponding initial boundary value problem. The maximum estimates of the solution are established by using suitable trace theorem and adapting appropriately the Moser's iteration. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications.