Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov–Poisson system

Stephan de Bièvre (1, 2) , Arthur Vavasseur (3, 4) , Thierry Goudon (5, 3, 4)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 MEPHYSTO-POST - Méthodes quantitatives pour les modèles aléatoires de la physique
3 COFFEE - COmplex Flows For Energy and Environment
4 LJAD - Laboratoire Jean Alexandre Dieudonné
5 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)

We are interested in a kinetic equation intended to describe the interaction of particles with their environment. The environment is modeled by a collection of local vibrational degrees of freedom. We establish the existence of weak solutions for a wide class of initial data and external forces. We also identify a relevant regime which allows us to derive, quite surprisingly, the attractive Vlasov–Poisson system from the coupled Vlasov-Wave equations.

Mots clés: Inelastic Lorentz gas.; Interacting particles; Vlasov–like equations

Domaines: Equations aux dérivées partielles [math.AP]

Stephan de Bièvre, Arthur Vavasseur, Thierry Goudon. Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov–Poisson system. SIAM Journal on Mathematical Analysis, 2016, 48 (6), pp.3984--4020. ⟨hal-01286519⟩