In this paper, we investigate the singularities and their propagation properties of potential energy functionals, in the space of all Borel probability measures with compact support \(\mathscr {P}_c(\mathbb {R}^m)\) , \(\begin{aligned} \Phi (\mu ):=\int _{\mathbb {R}^m}\phi \,d\mu ,\,\,\,\,\forall \mu \in {{\mathscr {P}}}_c(\mathbb {R}^m), \end{aligned}\) (we denote \(\Phi (\cdot )\) by \(\phi (\cdot )\) for brevity hereafter) associated with semiconcave functions \(\phi \) defined on \(\mathbb {R}^m\) . 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 \(u(\cdot )\) (the potential energy associated with \(u\) ) 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 \(u\) .
