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The non-relativistic limit of the Vlasov–Maxwell system with uniform macroscopic bounds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 545-594.

We study in this paper the non-relativistic limit from Vlasov–Maxwell to Vlasov–Poisson, which corresponds to the regime where the speed of light is large compared to the typical velocities of particles. In contrast with [4, 14, 43] which handle the case of classical solutions, we consider measure-valued solutions, whose moments and electromagnetic field are assumed to satisfy some uniform bounds. To this end, we use a functional inspired by the one introduced by Loeper in his proof of uniqueness for the Vlasov–Poisson system [33]. We also build a special class of measure-valued solutions, that enjoy no higher regularity with respect to the momentum variable, but whose moments and electromagnetic fields satisfy all required conditions to enter our framework.

Nous étudions dans cet article la limite non-relativiste de Vlasov–Maxwell vers Vlasov–Poisson, ce qui correspond au régime où la vitesse de la lumière est grande par rapport à la vitesse typique des particles. Contrairement à [4, 14, 43] qui traitent le cas de solutions classiques, nous considérons des solutions à valeurs dans les mesures, dont les moments et champ électro-magnétique sont supposés satisfaire certaines bornes uniformes. À cette fin, nous utilisons une fonctionnelle inspirée par celle introduite par Loeper dans sa preuve d’unicité pour le système de Vlasov–Poisson [33]. Nous construisons également une classe particulière de solutions à valeurs dans les mesures, qui ne sont pas lisses par rapport à la variable de moment, mais dont les moments et champ électro-magnétique satisfont toutes les conditions requises pour entrer dans notre cadre de travail.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1702
Classification: 35Q83,  35B40,  35Q61
Keywords: Vlasov–Maxwell, Vlasov–Poisson, Non-relativistic limit
Nicolas Brigouleix 1; Daniel Han-Kwan 1

1 CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France
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     title = {The non-relativistic limit of the {Vlasov{\textendash}Maxwell} system with uniform macroscopic bounds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {545--594},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Nicolas Brigouleix; Daniel Han-Kwan. The non-relativistic limit of the Vlasov–Maxwell system with uniform macroscopic bounds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 545-594. doi : 10.5802/afst.1702. https://afst.centre-mersenne.org/articles/10.5802/afst.1702/

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