Interpreting systems of continuity equations in spaces of probability measures through PDE duality

TL;DR

This paper introduces a duality-based solution concept for systems of transport equations in probability measure spaces, bridging viscosity and gradient flow solutions, and handling complex nonlinear PDE systems without variational structure.

Contribution

It develops a new duality solution framework for nonlinear transport PDEs in measure spaces, applicable to systems lacking variational structure.

Findings

Duality solutions are equivalent to gradient flow solutions under certain conditions.

The framework applies to nonlinear, nonlocal, and possibly non-diffusive PDE systems.

It extends solution concepts to broader classes of PDEs without variational structure.

Abstract

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear nonlocal, diffusive or not, system of PDEs without any variational structure.