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

Deterministic particle approximation for nonlocal transport equations with nonlinear mobility