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Inserted: 9 nov 2026
Last Updated: 9 nov 2026

Pages: 30
Year: 2026

Abstract:

We provide a variational formulation of a finite-dimensional Follow-the-Leader system through a minimizing-movement scheme based on a discrete transport distance with nonlinear mobility. Despite the degeneracy of the kinetic action at maximal density, we prove finiteness of the distance, $C^1$ regularity of geodesics, and suitable Euler--Lagrange relations. For a decreasing external potential, we show that the JKO scheme yields an approximate implicit Euler discretization of the Follow-the-Leader dynamics. Consequently, as $τ\to0^+$, the piecewise affine interpolations converge, up to a subsequence, to a $C^1$ solution of the continuous Follow-the-Leader system. This complements the many-particle convergence theory for the discrete nonlinear-mobility transport distance by establishing the continuous-time limit at fixed number of particles.

Keywords: Optimal transport, minimizing movements, nonlinear mobility, Follow-the-Leader model, JKO scheme, deterministic particle approximation

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