Mean‑field particle derivation for m ≠ 1

Determine whether the nonlocal conservation law ∂_t u − div(u^m ∇g ∗ u) = 0 on the d‑dimensional torus admits a rigorous derivation as the mean‑field limit of an interacting particle system when m ≠ 1, analogous to the known particle system derivation in the m = 1 case. Construct an appropriate particle dynamics, specify the interaction scaling, and prove convergence of empirical measures to the PDE.

Background

For m = 1, the equation has a well‑established mean‑field limit interpretation from repulsive Coulomb‑type particle systems, with propagation of convergence to the limiting PDE at positive times. Extending this derivation to m ≠ 1 would connect microscopic dynamics with the macroscopic nonlinear mobility law and justify particle‑based numerical schemes.

The challenge is to formulate a consistent particle model whose continuum limit yields the nonlinearity u^m in the flux, especially given the potential for shocks and lack of gradient‑flow structure for m > 1.