Wardrop equilibria : long-term variant, degenerate anisotropic PDEs and numerical approximations

TL;DR

This paper establishes a connection between long-term Wardrop equilibria in congested networks and anisotropic PDEs, providing regularity results and numerical methods for their approximation.

Contribution

It demonstrates the equivalence of measure-based minimization problems with anisotropic PDE formulations and extends regularity analysis to degenerate cases.

Findings

Proves Sobolev regularity for minimizers despite degeneracy.

Establishes equivalence between measure-based and PDE formulations.

Develops numerical approximation methods for the PDEs.

Abstract

As shown in [15], under some structural assumptions, working on congested traffic problems in general and increasingly dense networks leads, at the limit by {\Gamma}-convergence, to continuous minimization problems posed on measures on generalized curves. Here, we show the equivalence with another problem that is the variational formulation of an anisotropic, degenerate and elliptic PDE. For particular cases, we prove a Sobolev regularity result for the minimizers of the minimization problem despite the strong degeneracy and anisotropy of the Euler-Lagrange equation of the dual. We extend the analysis of [6] to the general case. Finally, we use the method presented in [5] to make numerical simulations.