Vector field methods for kinetic equations with applications to classical and relativistic systems
Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 3, 10 p.

The aim of this talk is to present an extension of the vector field method of Klainerman, which is typically applied in the context of non-linear wave equations, to the case of kinetic equations of Vlasov type. We first describe how our method yields sharp decay estimates for velocity averages for the linear classical and relativistic transport equations and then explain how it can be applied to various models of mathematical physics, such as the Vlasov-Poisson, Vlasov-Nordström and Vlasov-Einstein systems.

Publié le :
DOI : https://doi.org/10.5802/slsedp.103
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author = {Smulevici, Jacques},
title = {Vector field methods for kinetic equations with applications to classical and relativistic systems},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:3},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2016-2017},
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Smulevici, Jacques. Vector field methods for kinetic equations with applications to classical and relativistic systems. Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 3, 10 p. doi : 10.5802/slsedp.103. http://www.numdam.org/articles/10.5802/slsedp.103/

[1] S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38(3):321–332 (1985). | Article | MR 784477 | Zbl 0635.35059

[2] C. Bardos, P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2(2):101–118(1985). | Article | Numdam | MR 794002 | Zbl 0593.35076

[3] D. Fajman, J. Joudioux and J. Smulevici, A vector field method for relativistic kinetic transport equations with applications, to appear in Analysis and PDE. | Article | MR 3683922 | Zbl 1373.35046

[4] D. Fajman, J. Joudioux, and J. Smulevici. Sharp asymptotics for small data solutions of the Vlasov-Nordström system in three dimensions. arXiv:1704.05353.

[5] M. Dafermos, A note on the collapse of small data self-gravitating massless collisionless matter, J. Hyperbolic Differ. Equ., 3(4):589–598 (2006). | Article | MR 2289606 | Zbl 1115.35135

[6] M. Taylor, Stability of the Minkowski space for the massless Einstein-Vlasov system, Ann. PDE, 3(1) Art. 9, 177 pp, 2017. | Article | MR 3629140 | Zbl 06919598

[7] J. Smulevici. Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method. Ann. PDE, 2(2): Art. 2, 11pp, 2016. | Article | MR 3595457 | Zbl 1397.35033

[8] W. W. Y. Wong. A commuting-vector-field approach to some dispersive estimates. arXiv:1701.01460, January 2017. | Article | MR 3761138

[9] S. Calogero. Spherically symmetric steady states of galactic dynamics in scalar gravity. Classical Quantum Gravity, 20(9):1729–1741, 2003. | Article | MR 1981446 | Zbl 1030.83018

[10] S. Calogero. Global classical solutions to the 3D Nordström-Vlasov system. Comm. Math. Phys., 266(2):343–353, 2006. | Article | Zbl 1123.35080

[11] S. Calogero and G. Rein. Global weak solutions to the Nordström-Vlasov system. J. Differential Equations, 204(2):323–338, 2004. | Article | Zbl 1060.35027

[12] C. Pallard. On global smooth solutions to the 3D Vlasov-Nordström system. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23(1):85–96, 2006. | Article | Numdam | Zbl 1092.85001

[13] S. Friedrich. Global Small Solutions of the Vlasov-Norstrom System. arXiv:math-ph/0407023, July 2004.

[14] R. T. Glassey and W. A. Strauss. Absence of shocks in an initially dilute collisionless plasma. Comm. Math. Phys., 113(2):191–208, 1987. | Article | MR 919231 | Zbl 0646.35072

[15] H. Hwang, A. D. Rendall, and J. J. L. Velázquez. Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data. Archive for Rational Mechanics and Analysis, 200(1):313–360, 2011. | Article | MR 2781595 | Zbl 1228.35252

[16] H. Ringström. On the topology and future stability of the universe. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2013. | Article | Zbl 1270.83005

[17] G. Rein and A. D. Rendall. Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Comm. Math. Phys., 150(3):561–583, 1992. | Article | MR 1204320 | Zbl 0774.53056

[18] L. Andersson, P. Blue, and J. Joudioux. Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime. arXiv:1612.09304, December 2016. | Article | MR 3772194

[19] Robert T. Glassey. The Cauchy problem in kinetic theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. | Article | Zbl 0858.76001

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