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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&#39;s and Ward&#39;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&#39;s iteration.