Calculus of Variations and Geometric Measure Theory

M. Di Francesco - S. Fagioli - E. Radici

Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

created by radici on 30 Jan 2018
modified on 28 Oct 2022


Published Paper

Inserted: 30 jan 2018
Last Updated: 28 oct 2022

Journal: Journal of Differential Equations
Year: 2018

ArXiv: 1801.08770 PDF


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 L1loc 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 behavior of solutions for large times.