Maximum estimates for generalized Forchheimer flows in heterogeneous porous media

TL;DR

This paper develops methods to estimate the maximum pressure and its rate of change in complex, heterogeneous porous media flows governed by generalized Forchheimer equations, accounting for degeneracies and singularities.

Contribution

It introduces a novel approach using weighted De Giorgi iteration and inequalities to derive long-time maximum estimates for pressure in heterogeneous Forchheimer flows.

Findings

Established $L^ abla$-norm bounds for pressure and its time derivative.

Proved weighted parabolic Poincaré-Sobolev inequalities.

Derived long-time $L^ abla$-bounds using energy inequalities.

Abstract

This article continues our previous study of generalized Forchheimer flows in heterogeneous porous media. Such flows are used to account for deviations from Darcy's law. In heterogeneous media, the derived nonlinear partial differential equation for the pressure can be singular and degenerate in the spatial variables, in addition to being degenerate for large pressure gradient. Here we obtain the estimates for the $L^\infty$-norms of the pressure and its time derivative in terms of the initial and the time-dependent boundary data. They are established by implementing De Giorgi's iteration in the context of weighted norms with the weights specifically defined by the Forchheimer equation's coefficient functions. With these weights, we prove suitable weighted parabolic Poincar\'e-Sobolev inequalities and use them to facilitate the iteration. Moreover, local in time $L^\infty$-bounds are combined with uniform Gronwall-type energy inequalities to obtain long-time $L^\infty$-estimates.