Optimal transport with nonlinear mobilities: A deterministic particle approximation result

We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ > 0. This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.