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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 115–151 | Cite as

Generalized Symplectization of Vlasov Dynamics and Application to the Vlasov–Poisson System

  • Robert Axel NeissEmail author
Article

Abstract

In this paper, we study a Hamiltonian structure of the Vlasov–Poisson system, first mentioned by Fröhlich et al. (Commun Math Phys 288:1023–1058, 2009). To begin with, we give a formal guideline to derive a Hamiltonian on a subspace of complex-valued \({\mathcal{L}^{2}}\) integrable functions α on the one particle phase space \({\mathbb{R}^{2d}_{{\bf Z}}}\); s.t. \({f={\left|{\alpha}\right|}^2}\) is a solution of a collisionless Boltzmann equation. The only requirement is a sufficiently regular energy functional on a subspace of distribution functions \({f \in \mathcal{L}^{1}}\). Secondly, we give a full well-posedness theory for the obtained system corresponding to Vlasov–Poisson in \({d \geqq 3}\) dimensions. Finally, we adapt the classical globality results (Lions and Perthame in Invent Math 105:415–430, 1991; Pfaffelmoser in J Differ Equ 95:281–303, 1992; Schaeffer in Commun Partial Differ Equ 16(8–9):1313–1335, 1991) for d = 3 to the generalized system.

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Notes

Acknowledgements

I would like to thank Markus Kunze for his support and many useful discussions and hints, and also Antti Knowles, for a short discussion during his visit to our department. I would also like to thank the referees for their detailed and insightful comments on the paper.

Compliance with ethical standards

Conflict of interest

I declare that there are no conflicts of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnCologneGermany

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