Singularities and their propagation in optimal transport

2025/01/26 by Piermarco Cannarsa, Wei Cheng, Cannarsa, Piermarco +5
Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2501.15605

openalex publication_date 2025/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the singularities of potential energy functionals \(ϕ(⋅)\) associated with semiconcave functions \(ϕ\) in the Borel probability measure space and their propagation properties. Our study covers two cases: when \(ϕ\) is a semiconcave function and when \(u\) is a weak KAM solution of the Hamilton-Jacobi equation \(H(x, Du(x)) = c[0]\) on a smooth closed manifold. By applying previous work on Hamilton-Jacobi equations in the Wasserstein space, we prove that the singularities of \(u(⋅)\) will propagate globally when \(u\) is a weak KAM solution, and the dynamical cost function \(C^t\) is the associated fundamental solution. We also demonstrate the existence of solutions evolving along the cut locus, governed by an irregular Lagrangian semiflow on the cut locus of \(u\).

Related

• Optimal transport in the frame of abstract Lax-Oleinik operator revisited
• Variational construction of singular characteristics and propagation of singularities
• Global Propagation of Singularities for Time Dependent Hamilton-Jacobi\n Equations
• Global propagation of singularities for discounted Hamilton-Jacobi equations
• Singularities of solutions of time dependent Hamilton-Jacobi equations.\n Applications to Riemannian geometry