Calculus of Variations and Geometric Measure Theory

G. Bevilacqua - B. Perthame - M. Schmidtchen

The Aronson-Bénilan Estimate in Lebesgue Spaces

created by bevilacqua on 16 Dec 2021
modified on 01 Feb 2022


Accepted Paper

Inserted: 16 dec 2021
Last Updated: 1 feb 2022

Journal: Annales de l’Institut Henri Poincare - Analyse non lineaire
Year: 2022

ArXiv: 2007.15267 PDF


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.