The approximation of the quadratic porous medium equation via nonlocal interacting particles subject to repulsive Morse potential

TL;DR

This paper introduces a particle method for a nonlocal equation with repulsive interactions, demonstrating convergence to solutions of the quadratic porous medium equation as particles increase and interactions localize.

Contribution

It develops a particle scheme that converges to weak solutions of a nonlocal equation and captures the transition to the porous medium equation under localization of the potential.

Findings

Particle method converges to weak measure solutions.

Method captures measure-to-$L^inity$ smoothing effect.

Convergence to porous medium equation as potential localizes.

Abstract

We propose a deterministic particle method for a one-dimensional nonlocal equation with interactions through the repulsive Morse potential. We show that the particle method converges as the number of particles goes to infinity towards weak measure solutions to the nonlocal equation. Such a results is proven under the assumption of initial data in the space of probability measures with finite second moment. In particular, our method is able to capture a measure-to-$L^\infty$ smoothing effect of the limit equation. Moreover, as the Morse potential is rescaled to approach a Dirac delta, corresponding to strongly localised repulsive interactions, the scheme becomes a particle approximation for the quadratic porous medium equation. We show that in the joint limit (localised repulsion and increasing number of particles) the reconstructed density converges to a weak solution of the porous medium equation. The strategy relies on various estimates performed at the particle level, including $L^p$ estimates and an entropy dissipation estimate, which benefit from the particular structure of our particle scheme and from the absolutely continuous reconstruction of the density from the particle locations.