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Small Data Solutions for the Vlasov–Poisson System with a Repulsive Potential

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Abstract

In this paper, we study small data solutions for the Vlasov–Poisson system with the simplest external potential, for which unstable trapping holds for the associated Hamiltonian flow. We prove sharp decay estimates in space and time for small data solutions to the Vlasov–Poisson system with the repulsive potential \(\frac{-|x|^2}{2}\) in dimension two or higher. The proofs are obtained through a commuting vector field approach. We exploit the uniform hyperbolicity of the Hamiltonian flow, by making use of the commuting vector fields contained in the stable and unstable invariant distributions of phase space for the linearized system. In dimension two, we make use of modified vector field techniques due to the slow decay estimates in time. Moreover, we show an explicit teleological construction of the trapped set in terms of the non-linear evolution of the force field.

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Notes

  1. We stress the trapped set in the exterior of black hole backgrounds is eventually absolutely r-normally hyperbolic for every r according to [HPS77, Chapter 1, Definition 4].

  2. We refer to a distribution in phase space \({{\mathbb {R}}}^n_x\times {{\mathbb {R}}}^n_v\) as a map \((x,v)\mapsto \Delta _{(x,v)}\subseteq T_{(x,v)}({{\mathbb {R}}}^n_x\times {{\mathbb {R}}}^n_v)\), where \(\Delta _{(x,v)}\) are vector subspaces satisfying suitable conditions (in the standard sense used in differential geometry).

  3. In general relativity, many-particle systems can be composed by particles moving at the speed of light for which their mass vanishes. Nonetheless, we only comment on stability results for relativistic collisionless many-particle systems for which the mass of their particles is one.

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Acknowledgements

RVR would like to express his gratitude to his advisors Mihalis Dafermos and Clément Mouhot for their continued guidance and encouragements. RVR also would like to thank Léo Bigorgne and Jacques Smulevici for many helpful discussions. RVR received funding from the ANID grant 72190188, the Cambridge Trust grant 10469706, and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant 101034255. AVR received funding from the grant FONDECYT Iniciación 11220409.

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  1. Facultad de matemática, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna, 4860, Santiago, Chile

    Anibal Velozo Ruiz

  2. Laboratory Jacques-Louis Lions (LJLL), University Pierre and Marie Curie (Paris 6), 4 place Jussieu, 75252, Paris, France

    Renato Velozo Ruiz

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Velozo Ruiz, A., Velozo Ruiz, R. Small Data Solutions for the Vlasov–Poisson System with a Repulsive Potential. Commun. Math. Phys. 405, 80 (2024). https://doi.org/10.1007/s00220-024-04970-3

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