Generalized Forchheimer flows of isentropic gases

TL;DR

This paper models generalized Forchheimer flows of isentropic gases in porous media using a doubly nonlinear parabolic equation, providing estimates for solutions based on initial and boundary data.

Contribution

It introduces a new mathematical framework for describing gas flows in porous media with generalized Forchheimer equations, including boundary condition treatment and solution estimates.

Findings

Derived $L^ abla$ and $W^{1,2-a}$ norm estimates for solutions.

Converted volumetric flux boundary condition to a Robin-type boundary condition.

Established solution bounds using trace theorem and Moser's iteration.

Abstract

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.