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A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles
Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data
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 |
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.
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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