Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. BigorgneD. FajmanJ. JoudiouxJ. 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. ChoiS.-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. FajmanJ. 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. FajmanJ. 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. HwangA. 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. WatsonA 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. 
  • 加载中

Article Metrics

HTML views(320) PDF downloads(171) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint