Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems

Abstract

Systems of Vlasov-Poisson type are kinetic models describing dilute plasma. The structure of the model differs according to whether it describes the electrons or positively charged ions in the plasma. In contrast to the electron case, where the well-posedness theory for Vlasov-Poisson systems is well established, the well-posedness theory for ion models has been investigated more recently. In this article, we prove global well-posedness for two Vlasov-Poisson systems for ions, posed on the whole three-dimensional Euclidean space R 3 , under minimal assumptions on the initial data and the confining potential.

Backward problem for the 1D ionic Vlasov-Poisson equation

“…In the one-dimensional setting, weak solutions were constructed globally in time by D. Han-Kwan and M. Iacobelli [19]. The global well-posedness has then been proved by M. Griffin-Pickering and M. Iacobelli in the case of the whole space in dimension three [16] and of the torus in dimension two and three [17]. They showed existence of strong solutions for measure initial data with bounded moments and uniqueness with bounded density.…”

“…We use the same techniques for our proof but with the added difficulty of dealing with a nonlinear coupling for the Poisson equation. To tackle this difficulty, we use a decomposition of the electric field introduced by D. Han-Kwan and M. Iacobelli in [19] (see also [16,17], where they used the same decomposition). The paper is thus organised as follows.…”

Recent Developments on Quasineutral Limits for Vlasov-Type Equations

“…Since then, these results have been extended to the three-dimensional torus T 3 [9], and many works have refined the assumptions and techniques, such as [65,51,58,20]-this list is non-exhaustive, see [34] for a more detailed discussion about the well-posedness of Vlasov type systems. In contrast, the solution theory for the ion model was developed only more recently: weak global solutions in R 3 were obtained in the 90s by Bouchut [13], while global well-posedness theory for classical solutions in two and three dimensions was the subject of a series of recent works by the authors [32,31] (x ∈ T 3 , R 3 ) and Cesbron and the second author [18] (x in bounded domains).…”