A priori estimates for gaseous flows of Forchheimer-type in heterogeneous porous media

TL;DR

This paper derives a priori estimates for isentropic gas flows modeled by Forchheimer-type equations in heterogeneous porous media, focusing on short-time behavior and employing advanced inequalities and iterative methods.

Contribution

It introduces new weighted Sobolev inequalities and applies Moser iteration to obtain solution estimates for nonlinear parabolic equations with spatially varying coefficients.

Findings

Established short-time solution estimates based on initial and boundary data.

Developed multi-weight Sobolev and trace inequalities for heterogeneous media.

Applied Moser iteration to nonlinear Forchheimer-type flow equations.

Abstract

We study isentropic fluid flows of gases of the Forchheimer-type in heterogeneous porous media. The governing equation is a doubly nonlinear parabolic equation with coefficients depending on the spatial variables. Its solutions are subject to a nonlinear Robin boundary condition. We establish the estimates of the solutions for short time in terms of the initial and boundary data. For the proof, the multi-weight versions of the Sobolev inequality, parabolic Sobolev inequality and trace theorem are derived. They are then used to implement the Moser iteration for suitable weighted norms.