We study here a 2D gyrokinetic model obtained in [5], which naturally appears as the limit of a Vlasov–Poisson system with a very large external uniform magnetic field in the finite Larmor radius regime, when the typical Larmor radius is of order of the Debye length. We show that the Cauchy problem for that system is well-posed in a suitable space, provided that the initial condition satisfies a standard uniform decay assumption in velocity. Our result relies on a stability estimate in Wasserstein distance of order one between two solutions of the system. That stability estimate directly implies the uniqueness (in an appropriate space) of solutions to the Cauchy problem. An extension of the stability estimate to the case of a regularized interaction allows to prove the existence of solutions, as limits of solutions of a similar system with regularized interactions.
Nous étudions ici un modèle gyrocinétique 2D obtenu dans [5], qui apparaît naturellement comme la limite d’un système de Vlasov–Poisson dans un champ magnétique externe très fort, quand le rayon de Larmor typique est de l’ordre de la longueur de Debye. Nous montrons que le problème de Cauchy associé est bien posé dans un espace fonctionnel adapté, à condition que la condition initiale satisfasse une hypothèse assez classique de décroissance polynomiale uniforme en vitesse. Notre résultat est basé sur une estimation de stabilité en distance de Wasserstein d’ordre un entre deux solutions du système. L’estimation de stabilité implique directement l’unicité des solutions au problème de Cauchy (dans un espace fonctionnel adapté). Une extension de l’estimation de stabilité aux cas d’interactions régularisées permet de prouver l’existence de solutions, comme limites de solutions de systèmes similaires avec interactions régularisées.
Accepté le :
Publié le :
Pierre-Antoine Giorgi 1 ; Maxime Hauray 2
CC-BY 4.0
@article{AFST_2025_6_34_2_501_0,
author = {Pierre-Antoine Giorgi and Maxime Hauray},
title = {Well-posedness of a {2D} gyrokinetic model with equal {Debye} length and {Larmor} radius},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {501--537},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {2},
year = {2025},
doi = {10.5802/afst.1818},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1818/}
}
TY - JOUR AU - Pierre-Antoine Giorgi AU - Maxime Hauray TI - Well-posedness of a 2D gyrokinetic model with equal Debye length and Larmor radius JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 501 EP - 537 VL - 34 IS - 2 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1818/ DO - 10.5802/afst.1818 LA - en ID - AFST_2025_6_34_2_501_0 ER -
%0 Journal Article %A Pierre-Antoine Giorgi %A Maxime Hauray %T Well-posedness of a 2D gyrokinetic model with equal Debye length and Larmor radius %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 501-537 %V 34 %N 2 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1818/ %R 10.5802/afst.1818 %G en %F AFST_2025_6_34_2_501_0
Pierre-Antoine Giorgi; Maxime Hauray. Well-posedness of a 2D gyrokinetic model with equal Debye length and Larmor radius. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 2, pp. 501-537. doi : 10.5802/afst.1818. https://afst.centre-mersenne.org/articles/10.5802/afst.1818/
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