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February  2022, 15(1): 119-146. doi: 10.3934/krm.2021049

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  April 2020 Revised  December 2021 Published  February 2022 Early access  January 2022

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: Small data solution, Vlasov-Poisson system, Vlasov-Yukawa system, vector field method, decay estimate.
Mathematics Subject Classification: Primary: 35Q83; Secondary: 35M31, 35B40.
Citation: Xianglong Duan. Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data. Kinetic and Related Models, 2022, 15 (1) : 119-146. doi: 10.3934/krm.2021049
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. 

show all references

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. 

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