Vol. 3, No. 1, 2021

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Nonlinear echoes and Landau damping with insufficient regularity

Jacob Bedrossian

Vol. 3 (2021), No. 1, 121–205

We prove that the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov–Poisson equations on 𝕋x × v cannot, in general, be extended to high Sobolev spaces in the case of gravitational interactions. This is done by showing in every Sobolev space, there exists background distributions such that one can construct arbitrarily small perturbations that exhibit arbitrarily many isolated nonlinear oscillations in the density. These oscillations are known as plasma echoes in the physics community. For the case of electrostatic interactions, we demonstrate a sequence of small background distributions and asymptotically smaller perturbations in Hs which display similar nonlinear echoes. This shows that in the electrostatic case, any extension of Mouhot and Villani’s theorem to Sobolev spaces would have to depend crucially on some additional nonresonance effect coming from the background — unlike the case of Gevrey-ν with ν < 3 regularity, for which results are uniform in the size of small backgrounds. In particular, the uniform dependence on small background distributions obtained in Mouhot and Villani’s theorem in the Gevrey class is false in Sobolev spaces. Our results also prove that the time-scale of linearized approximation obtained by previous work is sharp up to logarithmic corrections.

Vlasov–Poisson, Landau damping, plasma echoes, nonlinear oscillations
Mathematical Subject Classification 2010
Primary: 35B20, 35B40, 76X05
Received: 7 September 2019
Accepted: 20 December 2019
Published: 29 July 2020
Jacob Bedrossian
University of Maryland
College Park, MD
United States