Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

TL;DR

This paper develops a deterministic particle method to approximate nonlocal transport PDEs with nonlinear mobility, proving convergence to entropy solutions and illustrating the importance of entropy conditions through analysis and simulations.

Contribution

It introduces a nonlocal follow-the-leader particle scheme for nonlinear mobility PDEs and proves its convergence to the unique entropy solution, including uniqueness and non-uniqueness results.

Findings

Convergence of particle scheme to entropy solutions in $L^1_{loc}$.

Numerical simulations demonstrating solution behavior and entropy condition importance.

Proof of uniqueness of entropy solutions for the class of PDEs studied.

Abstract

We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in $L^1_{loc}$ towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. As part of the proof, we also prove uniqueness of entropy solutions. We also provide a specific example of non-uniqueness of weak solutions and discuss about the interplay of the entropy condition with the steady states. Finally, we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well-posed semigroup in the continuum limit, and showing the behaviour of solutions for large times.