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

Marco Di Francesco, Valeria Iorio, Markus Schmidtchen

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.

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

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- 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 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.
Paper Structure (19 sections, 29 theorems, 271 equations)

This paper contains 19 sections, 29 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Morse potential
  4. Weak solutions of the nonlocal equation
  5. One-dimensional Wasserstein distances
  6. One-dimensional continuity equation
  7. Gradient flows and their Lipschitz perturbations on Hilbert spaces
  8. Gradient flow solution and uniqueness of weak solutions of the nonlocal equation
  9. The discrete particle model: well-posedness and estimates
  10. The particle scheme for strictly ordered particles
  11. Particle scheme for initially overlapping particles
  12. Further estimates on the scheme
  13. The nonlocal many-particle limit for fixed $\varepsilon>0$
  14. Definition of the approximation scheme and approximation of the initial data
  15. The case of initial data in $L^p$ for $p > 1$.
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $T\geq 0$ and let $v\in L^\infty([0,T];W^{1,\infty}(\mathbb{R}))$. Assume that $\rho\in L^\infty([0,T];{\mathcal{P}_2(\mathbb{R})})$ is a weak measure solution to the continuity equation eq:continuity on $\mathbb{R}\times [0,T]$ with initial condition $\overline{\rho}\in {\mathcal{P}_2(\mathbb{R

Theorems & Definitions (63)

  • Definition 2.1: Weak solutions to \ref{['eq:QPME']}
  • Definition 2.2: Weak solutions to \ref{['eq:main_intro']}
  • Theorem 2.1
  • Definition 2.3
  • Theorem 2.2
  • Definition 2.4: Gradient flow solutions
  • Theorem 2.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 53 more