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On Linear Damping Around Inhomogeneous Stationary States of the Vlasov-HMF Model

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Abstract

We study the dynamics of perturbations around an inhomogeneous stationary state of the Vlasov-HMF (Hamiltonian Mean-Field) model, satisfying a linearized stability criterion (Penrose criterion). We consider solutions of the linearized equation around the steady state, and prove the algebraic decay in time of the Fourier modes of their density. We prove moreover that these solutions exhibit a scattering behavior to a modified state, implying a linear damping effect with an algebraic rate of damping.

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Notes

  1. The Lebesgue measure on \([-\pi ,\pi ]\) divided by the length of the torus \(2\pi \).

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Authors and Affiliations

  1. INRIA-Rennes Bretagne Atlantique and IRMAR (UMR 6625), Université de Rennes I, Rennes, France

    Erwan Faou & Romain Horsin

  2. Laboratoire de Mathématiques d’Orsay (UMR 8628), Université Paris-Saclay, Gif-sur-Yvette, France

    Frédéric Rousset

Authors
  1. Erwan Faou
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  2. Romain Horsin
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  3. Frédéric Rousset
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Correspondence to Erwan Faou.

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Dedicated to the memory of Walter Craig.

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This work was partially supported by the ERC starting Grant GEOPARDI No. 279389.

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Faou, E., Horsin, R. & Rousset, F. On Linear Damping Around Inhomogeneous Stationary States of the Vlasov-HMF Model. J Dyn Diff Equat 33, 1531–1577 (2021). https://doi.org/10.1007/s10884-021-10044-y

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  • DOI: https://doi.org/10.1007/s10884-021-10044-y

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