Singularities and their propagation in optimal transport
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910801459150848 |
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| author | Cannarsa, Piermarco Cheng, Wei Shi, Tianqi Wei, Wenxue |
| author_facet | Cannarsa, Piermarco Cheng, Wei Shi, Tianqi Wei, Wenxue |
| contents | In this paper, we investigate the singularities of potential energy functionals \(ϕ(\cdot)\) 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 \(u(\cdot)\) 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 \(u\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15605 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularities and their propagation in optimal transport Cannarsa, Piermarco Cheng, Wei Shi, Tianqi Wei, Wenxue Analysis of PDEs Dynamical Systems In this paper, we investigate the singularities of potential energy functionals \(ϕ(\cdot)\) 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 \(u(\cdot)\) 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 \(u\). |
| title | Singularities and their propagation in optimal transport |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2501.15605 |