Global Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids

TL;DR

This paper develops global estimates for the pressure and its derivatives in generalized Forchheimer flows of slightly compressible fluids, providing insights into their behavior over time in porous media.

Contribution

It introduces a reformulation of Forchheimer flows as a degenerate parabolic equation and derives boundary and large-time estimates without complex calculations.

Findings

Global $L^\infty$ estimates for pressure and derivatives

Large-time estimates independent of initial data

Simplified approach due to pressure's nonlinear structure

Abstract

This paper is focused on the generalized Forchheimer flows of slightly compressible fluids in porous media. They are reformulated as a degenerate parabolic equation for the pressure. The initial boundary value problem is studied with time-dependent Dirichlet boundary data. The estimates up to the boundary and for all time are derived for the $L^\infty$-norm of the pressure, its gradient and time derivative. Large-time estimates are established to be independent of the initial data. Particularly, thanks to the special structure of the pressure's nonlinear equation, the global gradient estimates are obtained in a relatively simple way, avoiding complicated calculations and a prior requirement of H\"older estimates.