Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data

Xianglong Duan

doi:10.3934/krm.2021049

Article Contents

2022, Volume 15, Issue 1: 119-146. Doi: 10.3934/krm.2021049

This issue Previous Article Next Article Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system

Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data

Xianglong Duan^1,2,

1.
Beijing Advanced Innovation Center for Imaging Theory and Technology, Capital Normal University, Beijing 100048, China
2.
Laboratoire de Mathématiques d'Orsay, Univ Paris-Sud 11, CNRS, Université Paris-Saclay, F-91405, Orsay, France

Received: March 31, 2020

Revised: November 30, 2021

Published: February 2022

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we present sharp decay estimates for small data solutions to the following two systems: the Vlasov-Poisson (V-P) system in dimension 3 or higher and the Vlasov-Yukawa (V-Y) system in dimension 2 or higher. We rely on a modification of the vector field method for transport equation as developed by Smulevici in 2016 for the Vlasov-Poisson system. Using the Green's function in particular to estimate the bilinear terms, we improve Smulevici's result by removing the requirement of some $ v $-weighted $ L^p $ integrability for the initial data and extend the result to the Vlasov-Yukawa system.

Keywords:

Mathematics Subject Classification: Primary: 35Q83; Secondary: 35M31, 35B40.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	C. Bardos and 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 (1985), 101-118. doi: 10.1016/S0294-1449(16)30405-X.
[2]	L. Bigorgne, A vector field method for massless relativistic transport equations and applications, J. Funct. Anal., 278 (2020), 108365, 44 pp. doi: 10.1016/j.jfa.2019.108365.
[3]	L. Bigorgne, Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes, preprint, arXiv: 2006.03579.
[4]	L. Bigorgne, Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions, preprint, arXiv: 1712.09698.
[5]	L. Bigorgne, Sharp asymptotic behavior of solutions of the 3d vlasov-maxwell system with small data, Comm. Math. Phys., 376 (2020), 893-992. doi: 10.1007/s00220-019-03604-3.
[6]	L. Bigorgne, Sharp asymptotics for the solutions of the three-dimensional massless Vlasov-Maxwell system with small data, Ann. Henri Poincaré, 22 (2021), 219-273. doi: 10.1007/s00023-020-00978-2.
[7]	L. Bigorgne, Asymptotic properties of the solutions to the Vlasov-Maxwell system in the exterior of a light cone, Int. Math. Res. Not. IMRN, (2021), 3729–3793. doi: 10.1093/imrn/rnaa062.
[8]	L. Bigorgne, D. Fajman, J. Joudioux, J. Smulevici and M. Thaller, Asymptotic stability of minkowski space-time with non-compactly supported massless Vlasov matter, Arch. Ration. Mech. Anal., 242 (2021), 1-147. doi: 10.1007/s00205-021-01639-2.
[9]	S.-H. Choi, S.-Y. Ha and H. Lee, Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data, J. Differential Equations, 250 (2011), 515-550. doi: 10.1016/j.jde.2010.10.005.
[10]	D. Fajman, J. Joudioux and J. Smulevici, A vector field method for relativistic transport equations with applications, Anal. PDE, 10 (2017), 1539-1612. doi: 10.2140/apde.2017.10.1539.
[11]	D. Fajman, J. Joudioux and J. Smulevici, Sharp asymptotics for small data solutions of the Vlasov-Nordström system in three dimensions, preprint, arXiv: 1704.05353.
[12]	D. Fajman, J. Joudioux and J. Smulevici, The stability of the Minkowski space for the Einstein-Vlasov system, Anal. PDE, 14 (2021), 425-531. doi: 10.2140/apde.2021.14.425.
[13]	R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. xii+241 pp. doi: 10.1137/1.9781611971477.
[14]	H. Hwang, A. Rendall and J. Velázquez, Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data, Arch. Ration. Mech. Anal., 200 (2011), 313-360. doi: 10.1007/s00205-011-0405-3.
[15]	S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332. doi: 10.1002/cpa.3160380305.
[16]	H. Lindblad and M. Taylor, Global stability of Minkowski space for the Einstein-Vlasov system in the harmonic gauge, Arch. Ration. Mech. Anal., 235 (2020), 517-633. doi: 10.1007/s00205-019-01425-1.
[17]	P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.
[18]	K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.
[19]	J. Smulevici, Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE, 2 (2016), Art. 11, 55 pp. doi: 10.1007/s40818-016-0016-2.
[20]	C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis, II. International Press, Boston, MA, 1995. vi+159 pp.
[21]	X. Wang, Decay estimates for the 3D relativistic and non-relativistic Vlasov-Poisson systems, preprint, arXiv: 1805.10837.
[22]	G. N. Watson, A Treatise on the Theory of Bessel FFunctions, 2$^{nd}$ eddtion, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.
[23]	H. Yukawa, On the interaction of elementary particles, Proc. Phys. Math. Soc. Japan, 17 (1935), 48-57.