Singularities of potential energy functionals for weak KAM solutions propagate along a minimizing-movement curve in Wasserstein space, and the curve solves a continuity equation with the minimal-energy gradient selection.
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Singularities and their propagation in optimal transport
Singularities of potential energy functionals for weak KAM solutions propagate along a minimizing-movement curve in Wasserstein space, and the curve solves a continuity equation with the minimal-energy gradient selection.