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Hydrodynamics for hypersingular interactions by Γ-convergence of gradient flows

  • Colin McSwiggen
  • Sylvia Serfaty
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Abstract

We consider the dynamics of N particles with Riesz repulsive interaction in one dimension, focusing in particular on the hypersingular case (i.e. inverse power with power larger than or equal to the dimension). We prove convergence of the empirical measure to the solution of a porous medium equation by the method of Γ-convergence of gradient flows introduced by Sandier and the second author. This improves on the prior convergence result by Oelschläger obtained under more stringent assumptions. The method also works for the case where the power is smaller than the dimension, as well as for a class of convex short-range interactions. The proof highlights the crucial role played by the convexity of the interaction, which allows one to show that no excess energy can arise along the flow, implying the needed "slope inequality".

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Cite

BibTeXRIS
BibTeX
@misc{mcswiggen2026hydrodynamics,
  title = {{Hydrodynamics for hypersingular interactions by $Γ$-convergence of gradient flows}},
  author = {Colin McSwiggen and Sylvia Serfaty},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34136},
  eprint = {2610.34136},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34136},
}
RIS
TY  - UNPB
TI  - Hydrodynamics for hypersingular interactions by Γ-convergence of gradient flows
AU  - Colin McSwiggen
AU  - Sylvia Serfaty
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34136
UR  - https://arxiv.org/abs/2610.34136
AB  - We consider the dynamics of N particles with Riesz repulsive interaction in one dimension, focusing in particular on the hypersingular case (i.e. inverse power with power larger than or equal to the dimension). We prove convergence of the empirical measure to the solution of a porous medium equation by the method of Γ-convergence of gradient flows introduced by Sandier and the second author. This improves on the prior convergence result by Oelschläger obtained under more stringent assumptions. The method also works for the case where the power is smaller than the dimension, as well as for a class of convex short-range interactions. The proof highlights the crucial role played by the convexity of the interaction, which allows one to show that no excess energy can arise along the flow, implying the needed "slope inequality".
ER  -