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Small data solutions of the Vlasov-Poisson system and the vector field method. (English) Zbl 1397.35033
Summary: The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension 3 or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velázquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension 4 or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension 3, the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself. The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordström system.

35B40 Asymptotic behavior of solutions to PDEs
35Q83 Vlasov equations
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[1] Bardos, C; Degond, P, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2, 101-118, (1985) · Zbl 0593.35076
[2] Choi, S-H; Ha, S-Y; Lee, H, Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data, J. Differ. Equ., 250, 515-550, (2011) · Zbl 1252.35062
[3] Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space, vol. 41 of Princeton Mathematical Series. Princeton University Press, Princeton (1993) · Zbl 0827.53055
[4] Dafermos, M, A note on the collapse of small data self-gravitating massless collisionless matter, J. Hyperbolic Differ. Equ., 3, 589-598, (2006) · Zbl 1115.35135
[5] Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: XVIth International Congress on Mathematical Physics, pp. 421-432. World Scientific Publishing, Hackensack (2010) · Zbl 1211.83019
[6] Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: Decay for solutions of the wave equation on kerr exterior spacetimes. iii: The full subextremal case. arXiv:1402.7034 (2014) · Zbl 1347.83002
[7] Fajman, D., Joudioux, J., Smulevici, J.: A vector field method for relativistic transport equations with applications. arXiv:1510.04939 (2015) · Zbl 1373.35046
[8] Glassey, RT; Strauss, WA, Absence of shocks in an initially dilute collisionless plasma, Commun. Math. Phys., 113, 191-208, (1987) · Zbl 0646.35072
[9] Guo, Y; Lin, Z, Unstable and stable galaxy models, Commun. Math. Phys., 279, 789-813, (2008) · Zbl 1140.85304
[10] Guo, Y; Rein, G, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys., 271, 489-509, (2007) · Zbl 1130.85002
[11] Hwang, H; Rendall, A; Velázquez, J, Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data, Arch. Ration. Mech. Anal., 200, 313-360, (2011) · Zbl 1228.35252
[12] Klainerman, S, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Commun. Pure Appl. Math., 38, 321-332, (1985) · Zbl 0635.35059
[13] Klainerman, S., Nicolò, F.: The Evolution Problem in General Relativity, vol. 25 of Progress in Mathematical Physics. Birkhäuser Boston Inc., Boston (2003) · Zbl 0157.41502
[14] LeFloch, P., Ma, Y.: The global nonlinear stability of minkowski space for self-gravitating massive fields. arXiv:1511.03324 (2015) · Zbl 1434.83005
[15] Lemou, M; Méhats, F; Raphaël, P, Orbital stability of spherical galactic models, Invent. Math., 187, 145-194, (2012) · Zbl 1232.35170
[16] Lieb, E.H., Loss, M.: Analysis, 2nd edn, vol. 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001) · Zbl 0966.26002
[17] Lindblad, H; Rodnianski, I, The global stability of Minkowski space-time in harmonic gauge, Ann. Math., 171, 1401-1477, (2010) · Zbl 1192.53066
[18] Lions, P-L; Perthame, B, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105, 415-430, (1991) · Zbl 0741.35061
[19] Morawetz, CS, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Commun. Pure Appl. Math., 14, 561-568, (1961) · Zbl 0101.07701
[20] Morawetz, CS, The limiting amplitude principle, Commun. Pure Appl. Math., 15, 349-361, (1962) · Zbl 0196.41202
[21] Morawetz, CS, Time decay for the nonlinear Klein-Gordon equations, Proc. R. Soc. Ser. A, 306, 291-296, (1968) · Zbl 0157.41502
[22] Mouhot, C.: Stabilité orbitale pour le système de Vlasov-Poisson gravitationnel (d’après Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.). Astérisque, 352 (2013), Exp. No. 1044, vii, 35-82. Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043-1058 · Zbl 1287.70008
[23] Mouhot, C; Villani, C, On Landau damping, Acta Math., 207, 29-201, (2011) · Zbl 1239.82017
[24] Pfaffelmoser, K, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95, 281-303, (1992) · Zbl 0810.35089
[25] Rein, G; Rendall, AD, Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting, Arch. Ration. Mech. Anal., 126, 183-201, (1994) · Zbl 0808.35109
[26] Rein, G., Rendall, A.D.: Erratum: Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data” [Commun. Math. Phys. 150 (1992), no. 3, 561-583; MR1204320 (94c:83028)]. Commun. Math. Phys. 176(2), 475-478 (1996) · Zbl 0774.53056
[27] Ringström, H.: On the Topology and Future Stability of the Universe. Oxford Mathematical Monographs. Oxford University Press, Oxford (2013) · Zbl 1270.83005
[28] Sarbach, O., Zannias, T.: Tangent bundle formulation of a charged gas. In: American Institute of Physics Conference Series (January 2014), vol. 1577 of American Institute of Physics Conference Series, pp. 192-207. arXiv:1311.3532 (2014) · Zbl 1295.83035
[29] Schaeffer, J, A small data theorem for collisionless plasma that includes high velocity particles, Indiana Univ. Math. J., 53, 1-34, (2004) · Zbl 1059.35152
[30] Sogge, C.D.: Lectures on Nonlinear Wave Equations. II. Monographs in Analysis. International Press, Boston (1995)
[31] Taylor, M.: The global nonlinear stability of the Minkowski space for the massless Einstein-Vlasov system. arXiv:1602.02611 (2016) · Zbl 1252.35062
[32] Wang, J., Wang, Q.: Global existence for the einstein equations with massive scalar fields. Talk Given at the Conference General Relativity: A Celebration of the 100th Anniversary, Institut Henri Poincaré, Nov 2011. (2015) · Zbl 0593.35076
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