Singularities and their propagation in optimal transport

In this paper, we investigate the singularities of potential energy functionals $\phi(\cdot)$ associated with semiconcave functions $\phi$ in the Borel probability measure space and their propagation properties. Our study covers two cases: when $\phi$ 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 $u(\cdot)$ 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 $u$.