The non-relativistic limit of the Vlasov–Maxwell system with uniform macroscopic bounds

Brigouleix, Nicolas; Han-Kwan, Daniel

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 2, pp. 545-594.

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.

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.

Classification : 35Q83, 35B40, 35Q61
Keywords: Vlasov–Maxwell, Vlasov–Poisson, Non-relativistic limit
Mots clés : Vlasov–Maxwell, Vlasov–Poisson, Limite non-relativiste

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