The Aronson-B\'enilan Estimate in Lebesgue Spaces

TL;DR

This paper reviews and clarifies recent extensions of the Aronson-Bénilan estimate in Lebesgue spaces for porous media equations, highlighting modifications, improvements, and limitations of existing strategies.

Contribution

It provides a detailed analysis of recent Lp space extensions of the Aronson-Bénilan estimate, including modifications, improvements, and identified limitations.

Findings

Extended Aronson-Bénilan estimates in Lp spaces.

Identified limitations of current strategies.

Proposed modifications for better estimates.

Abstract

In a celebrated three-pages long paper in 1979, Aronson and B\'enilan obtained a remarkable estimate on second order derivatives for the solution of the porous media equation. Since its publication, the theory of porous medium flow has expanded relentlessly with applications including thermodynamics, gas flow, ground water flow as well as ecological population dynamics. The purpose of this paper is to clarify the use of recent extensions of the Aronson and B\'enilan estimate in Lp spaces, of some modifications and improvements, as well as to show certain limitations of their strategy.