Singularities and their propagation in optimal transport

In this paper, we investigate the singularities and their propagation properties of poten- tial energy functionals, in the space of all Borel probability measures with compact support Pc(Rm ), (μ) := Rm φ dμ, ∀μ ∈ Pc(Rm ), (we denote (·) by φ(·) for brevity hereafter) associated with semiconcave functions φ defined on Rm . 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(·) (the potential energy associated with u) will propagate globally when u is a weak KAM solution, and the dynamical cost function Ct 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.