Gevrey regularity for the Vlasov-Poisson system

Velozo Ruiz, Renato

doi:10.1016/j.anihpc.2020.10.006

Velozo Ruiz, Renato ¹

¹ Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1145-1165.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We prove propagation of $\frac{1}{s}$ -Gevrey regularity $(s \in (0, 1])$ for the Vlasov-Poisson system on $T^{d} \times R^{d}$ using a Fourier space method in analogy to the results proved for the 2D-Euler system in [20] and [23]. More precisely, we give quantitative estimates for the growth of the $\frac{1}{s}$ -Gevrey norm and decay of the regularity radius for the solution of the system in terms of the force field and the volume of the support in the velocity variable of the distribution of matter. As an application, we show global existence of $\frac{1}{s}$ -Gevrey solutions ( $s \in (0, 1)$ ) for the Vlasov-Poisson system in $T^{3} \times R^{3}$ . Furthermore, the propagation of Gevrey regularity can be easily modified to obtain the same result in $R^{d} \times R^{d}$ . In particular, this implies global existence of analytic $(s = 1)$ and $\frac{1}{s}$ -Gevrey solutions ( $s \in (0, 1)$ ) for the Vlasov-Poisson system in $R^{3} \times R^{3}$ .

Reçu le : 2020-01-28
Révisé le : 2020-08-14
Accepté le : 2020-10-26

MR Zbl

DOI : 10.1016/j.anihpc.2020.10.006

Mots-clés : Kinetic theory, Vlasov-Poisson system, Gevrey regularity, Propagation of regularity, Global classical solutions

Affiliations des auteurs :

Velozo Ruiz, Renato ¹

¹ Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

@article{AIHPC_2021__38_4_1145_0,
     author = {Velozo Ruiz, Renato},
     title = {Gevrey regularity for the {Vlasov-Poisson} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1145--1165},
     publisher = {Elsevier},
     volume = {38},
     number = {4},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.10.006},
     mrnumber = {4266238},
     zbl = {1472.35082},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.10.006/}
}

TY  - JOUR
AU  - Velozo Ruiz, Renato
TI  - Gevrey regularity for the Vlasov-Poisson system
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 1145
EP  - 1165
VL  - 38
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2020.10.006/
DO  - 10.1016/j.anihpc.2020.10.006
LA  - en
ID  - AIHPC_2021__38_4_1145_0
ER  -

%0 Journal Article
%A Velozo Ruiz, Renato
%T Gevrey regularity for the Vlasov-Poisson system
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 1145-1165
%V 38
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2020.10.006/
%R 10.1016/j.anihpc.2020.10.006
%G en
%F AIHPC_2021__38_4_1145_0

Velozo Ruiz, Renato. Gevrey regularity for the Vlasov-Poisson system. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1145-1165. doi : 10.1016/j.anihpc.2020.10.006. http://www.numdam.org/articles/10.1016/j.anihpc.2020.10.006/

Bibliographie
Cité par

[1] Bardos, C.; Benachour, S. Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de $R^{n}$ , Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 4 (1977) no. 4, pp. 647-687 | Numdam | MR | Zbl

[2] Bardos, C.; Degond, P. 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 | DOI | Numdam | MR | Zbl

[3] Bardos, Claude Analyticité de la solution de l'équation d'Euler dans un ouvert de $R^{n}$ , C. R. Acad. Sci. Paris Sér. A-B, Volume 283 (1976) no. 5, p. A255-A258 (Aii) | MR | Zbl

[4] Bardos, Claude; Benachour, Said; Zerner, Martin Analyticité des solutions périodiques de l'équation d'Euler en deux dimensions, C. R. Acad. Sci. Paris Sér. A-B, Volume 282 (1976) no. 17, p. A995-A998 (Aiii) | MR | Zbl

[5] Batt, Jürgen Global symmetric solutions of the initial value problem of stellar dynamics, J. Differ. Equ., Volume 25 (1977) no. 3, pp. 342-364 | DOI | MR | Zbl

[6] Batt, Jürgen; Rein, Gerhard Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci., Sér. 1 Math., Volume 313 (1991) no. 6, pp. 411-416 | MR | Zbl

[7] Beale, J.T.; Kato, T.; Majda, A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., Volume 94 (1984) no. 1, pp. 61-66 | DOI | MR | Zbl

[8] Bedrossian, Jacob Nonlinear echoes and Landau damping with insufficient regularity, 2020 (preprint) | arXiv | MR | Zbl

[9] Bedrossian, Jacob; Masmoudi, Nader; Mouhot, Clément Landau damping: paraproducts and Gevrey regularity, Ann. PDE, Volume 2 (2016) no. 1 (Art. 4, 71) | MR | Zbl

[10] Benachour, Saïd Analyticité des solutions des équations de Vlassov-Poisson, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 16 (1989) no. 1, pp. 83-104 | Numdam | MR | Zbl

[11] Binney, J.; Tremaine, S. Galactic Dynamics, Princeton Series in Astrophysics, Princeton University Press, 2011 | Zbl

[12] Foias, C.; Temam, R. Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., Volume 87 (1989) no. 2, pp. 359-369 | DOI | MR | Zbl

[13] Gevrey, Maurice Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire, Ann. Sci. Éc. Norm. Supér., Volume 3 (1918) no. 35, pp. 129-190 | DOI | JFM | Numdam | MR

[14] Grenier, Emmanuel; Nguyen, Toan T.; Rodnianski, Igor Landau damping for analytic and Gevrey, 2020 (preprint) | arXiv | MR | Zbl

[15] Horst, E. On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory, Math. Methods Appl. Sci., Volume 3 (1981) no. 2, pp. 229-248 | DOI | MR | Zbl

[16] Horst, E. On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II. Special cases, Math. Methods Appl. Sci., Volume 4 (1982) no. 1, pp. 19-32 | DOI | MR | Zbl

[17] Horst, E. On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci., Volume 16 (1993) no. 2, pp. 75-86 | DOI | MR | Zbl

[18] Kiselev, Alexander; Šverák, Vladimir Small scale creation for solutions of the incompressible two-dimensional Euler equation, Ann. Math. (2), Volume 180 (2014) no. 3, pp. 1205-1220 | DOI | MR | Zbl

[19] Klainerman, Sergiu; Majda, Andrew Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., Volume 34 (1981) no. 4, pp. 481-524 | DOI | MR | Zbl

[20] Kukavica, Igor; Vicol, Vlad On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Am. Math. Soc., Volume 137 (2009) no. 2, pp. 669-677 | DOI | MR | Zbl

[21] Kukavica, Igor; Vicol, Vlad On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, Volume 24 (2011) no. 3, pp. 765-796 | DOI | MR | Zbl

[22] Landau, L. On the vibrations of the electronic plasma, Acad. Sci. USSR. J. Phys., Volume 10 (1946), pp. 25-34 | MR | Zbl

[23] Levermore, C. David; Oliver, Marcel Analyticity of solutions for a generalized Euler equation, J. Differ. Equ., Volume 133 (1997) no. 2, pp. 321-339 | DOI | MR | Zbl

[24] Lifshitz, E.M.; Pitaevskiĭ, L.P. Course of Theoretical Physics [“Landau-Lifshits”], Vol. 10, Pergamon International Library of Science, Technology, Engineering and Social Studies, Pergamon Press, Oxford-Elmsford, N.Y., 1981 (Translated from the Russian by J.B. Sykes and R.N. Franklin) | MR

[25] Lions, P.-L.; Perthame, B. 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

[26] Majda, Andrew J.; Bertozzi, Andrea L. Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[27] Mouhot, Clément; Villani, Cédric On Landau damping, Acta Math., Volume 207 (2011) no. 1, pp. 29-201 | DOI | MR | Zbl

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

[29] Schaeffer, Jack Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Commun. Partial Differ. Equ., Volume 16 (1991) no. 8–9, pp. 1313-1335 | DOI | MR | Zbl

[30] Schaeffer, Jack Asymptotic growth bounds for the Vlasov-Poisson system, Math. Methods Appl. Sci., Volume 34 (2011) no. 3, pp. 262-277 | DOI | MR | Zbl

Cité par Sources :