The Aronson-Bénilan estimate for a Lagrangian particle discretization of the Porous Medium Equation

TL;DR

The paper develops a one-dimensional, nearest-neighbor Lagrangian particle discretization for the porous medium equation $\partial_t\rho=\Delta\rho^m$ with $m>1$ and proves a discrete Aronson-Bénilan estimate that yields uniform-in-$N$ control of the solution's support growth and $L^\infty$ decay. By combining the discrete AB bound with a compactness framework (Aubin-Lions-Savare) and a density reconstruction, the authors prove convergence of the particle scheme to a very weak $L^1$-based solution of the PME for general $L^1$ initial data. The discrete AB estimate also implies a uniform propagation speed bound and a corresponding decay bound, reproducing key qualitative features of the continuum PME at the discrete level. Overall, the work provides a rigorous link between a local, particle-based discretization and the classical diffusion behavior of PME solutions, including convergence and preservation of fundamental properties such as finite speed of propagation and smoothing.

Abstract

We consider a nearest neighbor, Lagrangian particle discretization of the one dimensional porous medium equation. We prove that the particle model satisfies a discrete analog of the celebrated Aronson-Bénilan estimate, which we use to prove a growth estimate for the evolution of the support and an $L^\infty$ decay estimate which are both known to hold in the continuum. These estimates are uniform with respect to the number of particles. We also prove convergence of the scheme towards the solution to the porous medium equation in the full generality of $L^1$ initial data.

Theorems & Definitions (39)

• Proposition 1
• proof
• Lemma 1
• Theorem 1: Discrete Aronson Bénilan estimate
• Theorem 2: Speed of propagation of the support
• Theorem 3: $L^\infty$ decay
• Theorem 4: Convergence of the scheme
• Proposition 2: Global Existence and Discrete Minimum Principle
• proof
• proof : Proof of Theorem \ref{['thm:main_AB']}