On a Vlasov–Poisson system in a bounded set with direct reflection boundary conditions
Annales Henri Lebesgue, Volume 5 (2022), pp. 703-727.

Metadata

KeywordsVlasov–Poisson system, reflection boundary conditions, generalized characteristics

Abstract

The Vlasov–Poisson system models a collisionless plasma. In a bounded domain it is known that singularities can occur. Existence of global in time continuous solutions to the Vlasov–Poisson system is proven in a one-dimensional bounded domain, with direct reflection boundary conditions and initial data even with respect to the v-variable. Local in time uniqueness is proven. Generalized characteristics are used. Electroneutrality is obtained in the limit.


References

[Ale93] Alexandre, Radjesvarane Weak solutions of the Vlasov–Poisson initial boundary value problem, Math. Methods Appl. Sci., Volume 16 (1993) no. 8, pp. 533-607 | MR | Zbl

[Ars75] Arsenev, Alekseĭ Existence in the large of a weak solution of Vlasov’s system of equations, Zh. Vychisl. Mat. Mat. Fiz., Volume 15 (1975), pp. 136-147 | MR | Zbl

[BA94] Ben Abdallah, Naoufel Weak solutions of the initial-boundary value problem for the Vlasov–Poisson system, Math. Methods Appl. Sci., Volume 17 (1994) no. 6, pp. 451-476 | MR | Zbl

[BD85] Bardos, Claude W.; Degond, Pierre Global existence for the Vlasov–Poisson equation in 3 space variables with small initial data, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 2 (1985) no. 2, pp. 101-118 | Numdam | MR | Zbl

[DL88] DiPerna, Ronald J.; Lions, Pierre-Louis Solutions globales du type Vlasov–Poisson, C. R. Acad. Sci., Paris, Sér. I, Volume 307 (1988) no. 12, pp. 655-658 | MR | Zbl

[Gio19] Giorgi, Pierre-Antoine Analyse mathématique de modèles cinétiques en physique des plasmas, Ph. D. Thesis, Aix-Marseille University, Marseille, France (2019)

[Guo94] Guo, Yan Regularity for the Vlasov Equation with boundary in a Half Space, Indiana Univ. Math. J., Volume 43 (1994) no. 1, pp. 255-319 | Zbl

[Guo95] Guo, Yan Singular solutions of the Vlasov–Maxwell system on a half line, Arch. Ration. Mech. Anal., Volume 131 (1995) no. 3, pp. 241-304 | MR | Zbl

[HKH15] Han-Kwan, Daniel; Hauray, Maxime Stability issues in the quasineutral limit of the one-dimensional Vlasov–Poisson equation, Commun. Math. Phys., Volume 334 (2015) no. 2, pp. 1101-1152 | DOI | MR | Zbl

[HKR16] Han-Kwan, Daniel; Rousset, Frédéric Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 6, pp. 1445-1495 | DOI | MR | Zbl

[HM18] Holding, Thomas; Miot, Evelyne Uniqueness and stability for the Vlasov–Poisson system with spatial density in Orlicz spaces, Mathematical analysis in fluid mechanics: selected recent results. International conference on vorticity, rotation and symmetry (IV) – complex fluids and the issue of regularity, CIRM, Luminy, Marseille, France, May 8–12, 2017. Proceedings (Danchin, Raphaël, ed.) (Contemporary Mathematics), Volume 710, American Mathematical Society, 2018, pp. 145-162 | DOI | MR | Zbl

[HS08] Hwang, Hyung J.; Schaeffer, Jack Uniqueness for weak solutions of a one-dimensional boundary value problem for the Vlasov–Poisson system, J. Differ. Equations, Volume 244 (2008) no. 10, pp. 2665-2691 | DOI | MR | Zbl

[Loe06] Loeper, Grégoire Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., Volume 86 (2006) no. 1, pp. 68-79 | DOI | MR | Zbl

[LP91] Lions, Pierre-Louis; Perthame, Benoît Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system, Invent. Math., Volume 105 (1991) no. 2, pp. 415-430 | DOI | MR | Zbl

[Mio16] Miot, Evelyne A uniqueness criterion for unbounded solutions to the Vlasov–Poisson system, Commun. Math. Phys., Volume 346 (2016) no. 2, pp. 469-482 | DOI | MR | Zbl

[Mis99] Mischler, Stéphane On the trace problem for solutions of the Vlasov equation, Commun. Partial Differ. Equations, Volume 25 (1999) no. 7-8, pp. 1415-1443 | DOI | MR | Zbl

[Pal12] Pallard, Christophe Moment propagation for weak solutions to the Vlasov–Poisson system, Commun. Partial Differ. Equations, Volume 37 (2012) no. 7, pp. 1273-1285 | DOI | MR | Zbl

[Pfa92] Pfaffelmoser, Klaus Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data, J. Differ. Equations, Volume 95 (1992) no. 2, pp. 281-303 | DOI | MR | Zbl

[Pou90] Poupaud, Frédéric Stationary solutions of Vlasov–Poisson equations, C. R. Acad. Sci., Paris, Sér. I, Volume 311 (1990) no. 6, pp. 307-312 | MR | Zbl

[Rei97] Rein, Gerhard Self-gravitating systems in Newtonian theory – The Vlasov–Poisson system, Mathematics of gravitation, Part I (Warsaw, 1996) (Banach Center Publications), Volume 41, Polish Academy of Sciences, 1997, pp. 179-194 | MR | Zbl

[Rob97] Robert, Raoul Unicité de la solution faible à support compact de l’équation de Vlasov–Poisson, C. R. Acad. Sci., Paris, Sér. I, Volume 324 (1997) no. 8, pp. 833-877 | Zbl

[Sch91] Schaeffer, Jack Global existence of smooth solutions to the Vlasov–Poisson system in three dimensions, Commun. Partial Differ. Equations, Volume 16 (1991), pp. 1313-1335 | DOI | MR | Zbl

[Wec95] Weckler, Jürge On the initial-boundary-value problem for the Vlasov–Poisson system: existence of weak solutions and stability, Arch. Ration. Mech. Anal., Volume 130 (1995) no. 2, pp. 145-161 | DOI | MR | Zbl