Generalized Forchheimer flows of isentropic gases

We consider generalized Forchheimer flows of either isentropic gases or slightly compressible fluids in porous media. By using Muskat's and Ward's general form of the Forchheimer equations, we describe the fluid dynamics by a doubly nonlinear parabolic equation for the appropriately defined pseudo-pressure. The volumetric flux boundary condition is converted to a time-dependent Robin-type boundary condition for this pseudo-pressure. We study the corresponding initial boundary value problem, and estimate the $L^\infty$ and $W^{1,2-a}$ (with $0<a<1$) norms for the solution on the entire domain in terms of the initial and boundary data. It is carried out by using a suitable trace theorem and an appropriate modification of Moser's iteration.