doi: 10.3934/krm.2022019
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Decay estimates for the $ 3D $ relativistic and non-relativistic Vlasov-Poisson systems

YMSC, Tsinghua University, China

Received  September 2021 Revised  May 2022 Early access May 2022

We study the small data global regularity problem of the $ 3D $ Vlasov-Poisson system for both the relativistic case and the non-relativistic case. The main goal of this paper is twofold. (i) Based on a Fourier method, which works systematically for both the relativistic case and the non-relativistic case, we give a short proof for the global regularity and the sharp decay estimate for the $ 3D $ Vlasov-Poisson system. Moreover, we show that the nonlinear solution scatters to a linear solution in both cases. The result of sharp decay estimates for the non-relativistic case is not new, see Hwang-Rendall-Velázquez [9] and Smulevici [23]. (ii) The Fourier method presented in this paper serves as a good comparison for the study of more complicated $ 3D $ relativistic Vlasov-Nordström system in [24] and $ 3D $ relativistic Vlasov-Maxwell system in [25].

Citation: Xuecheng Wang. Decay estimates for the $ 3D $ relativistic and non-relativistic Vlasov-Poisson systems. Kinetic and Related Models, doi: 10.3934/krm.2022019
References:
[1]

H. Andréasson, The Einstein-Vlasov system/kinetic theory, Living Rev. Relativ., 5 (2022), 33 pp. doi: 10.12942/lrr-2002-7.

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equatino in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. NonLinéaire, 2 (1985), 101-118.  doi: 10.1016/s0294-1449(16)30405-x.

[3]

J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: Paraproducts and Gevrey regularity, Ann. PDE., 2 (2016), Art. 4, 71 pp. doi: 10.1007/s40818-016-0008-2.

[4]

D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, {Princeton University Press}, Princeton, 1993.

[5]

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.

[6]

R. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.

[7]

R. Glassey and J. Schaeffer, On global symmetric solutions to the relativistic Vlasov-Poisson equation in three space dimensions, Math. Meth. Appl. Sci., 24 (2001), 143-157.  doi: 10.1002/1099-1476(200102)24:3<143::AID-MMA202>3.0.CO;2-C.

[8]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Comm. Math. Phys., 271 (2007), 489-509.  doi: 10.1007/s00220-007-0212-8.

[9]

H.-J. 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.

[10]

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.

[11]

S. Klainerman, Remark on the asymptotic behavior of the Klein-Gordon equation in ${\mathbb{R}}^{n+1}$, Comm. Pure Appl. Math., 46 (1993), 137-144.  doi: 10.1002/cpa.3160460202.

[12]

M. LemouF. Méhats and P. Raphaél, On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov Poisson system, Arch. Rational Mech. Anal., 189 (2008), 425-468.  doi: 10.1007/s00205-008-0126-4.

[13]

M. LemouF. Méhats and P. Raphaél, Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system, J. Amer. Math. Soc., 21 (2008), 1019-1063.  doi: 10.1090/S0894-0347-07-00579-6.

[14]

M. LemouF. Méhats and P. Raphaél, Orbital stability of spherical galactic models, Invent. Math., 187 (2012), 145-194.  doi: 10.1007/s00222-011-0332-9.

[15]

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.

[16]

C. Mouhot, Stabilité orbitale pour le systéme de Vlasov-Poisson gravitationel (d'aprés Lemou-Méhats-Raphaél, Guo, Lin, Rein et al.), Astérisque, 352(2013), Exp. No. 1044, vii, 35–82. Séminaire Bourbaki, Exposés, 2011/2012 (2013), 1043–1058.

[17]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.

[18]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[19]

G. Rein, Growth estimates for the Vlasov-Poisson system in the plasma physics case, Math. Nachr., 191 (1998), 269-278.  doi: 10.1002/mana.19981910114.

[20]

G. Rein and A. D. Rendall, Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional cosmological setting, Arch. Rational Mech. Anal., 126 (1994), 183-201.  doi: 10.1007/BF00391558.

[21]

J. Schaeffer, Global existence for the Poisson-Vlasov system with nearly symmetric data, J. Differ. Equ., 69 (1987), 111-148.  doi: 10.1016/0022-0396(87)90105-7.

[22]

J. Schaeffer, Global existence of smooth solutions to the Vlasov Poisson system in three dimensions, Comm. PDE., 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.

[23]

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.

[24]

X. Wang, Propagation of regularity and long time behavior of $3D$ massive relativistic transport equation I: Vlasov-Nordström system, Comm. Math. Phys., 382 (2021), 1843-1934.  doi: 10.1007/s00220-021-03987-2.

[25]

X. Wang, Propagation of regularity and long time behavior of $3D$ massive relativistic transport equation II: Vlasov-Maxwell system, Comm. Math. Phys., 389 (2022), 715-812.  doi: 10.1007/s00220-021-04257-x.

show all references

References:
[1]

H. Andréasson, The Einstein-Vlasov system/kinetic theory, Living Rev. Relativ., 5 (2022), 33 pp. doi: 10.12942/lrr-2002-7.

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equatino in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. NonLinéaire, 2 (1985), 101-118.  doi: 10.1016/s0294-1449(16)30405-x.

[3]

J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: Paraproducts and Gevrey regularity, Ann. PDE., 2 (2016), Art. 4, 71 pp. doi: 10.1007/s40818-016-0008-2.

[4]

D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, {Princeton University Press}, Princeton, 1993.

[5]

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.

[6]

R. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.

[7]

R. Glassey and J. Schaeffer, On global symmetric solutions to the relativistic Vlasov-Poisson equation in three space dimensions, Math. Meth. Appl. Sci., 24 (2001), 143-157.  doi: 10.1002/1099-1476(200102)24:3<143::AID-MMA202>3.0.CO;2-C.

[8]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Comm. Math. Phys., 271 (2007), 489-509.  doi: 10.1007/s00220-007-0212-8.

[9]

H.-J. 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.

[10]

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.

[11]

S. Klainerman, Remark on the asymptotic behavior of the Klein-Gordon equation in ${\mathbb{R}}^{n+1}$, Comm. Pure Appl. Math., 46 (1993), 137-144.  doi: 10.1002/cpa.3160460202.

[12]

M. LemouF. Méhats and P. Raphaél, On the orbital stability of the ground states and the singularity formation for the gravitational Vlasov Poisson system, Arch. Rational Mech. Anal., 189 (2008), 425-468.  doi: 10.1007/s00205-008-0126-4.

[13]

M. LemouF. Méhats and P. Raphaél, Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system, J. Amer. Math. Soc., 21 (2008), 1019-1063.  doi: 10.1090/S0894-0347-07-00579-6.

[14]

M. LemouF. Méhats and P. Raphaél, Orbital stability of spherical galactic models, Invent. Math., 187 (2012), 145-194.  doi: 10.1007/s00222-011-0332-9.

[15]

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.

[16]

C. Mouhot, Stabilité orbitale pour le systéme de Vlasov-Poisson gravitationel (d'aprés Lemou-Méhats-Raphaél, Guo, Lin, Rein et al.), Astérisque, 352(2013), Exp. No. 1044, vii, 35–82. Séminaire Bourbaki, Exposés, 2011/2012 (2013), 1043–1058.

[17]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.

[18]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[19]

G. Rein, Growth estimates for the Vlasov-Poisson system in the plasma physics case, Math. Nachr., 191 (1998), 269-278.  doi: 10.1002/mana.19981910114.

[20]

G. Rein and A. D. Rendall, Global existence of classical solutions to the Vlasov-Poisson system in a three-dimensional cosmological setting, Arch. Rational Mech. Anal., 126 (1994), 183-201.  doi: 10.1007/BF00391558.

[21]

J. Schaeffer, Global existence for the Poisson-Vlasov system with nearly symmetric data, J. Differ. Equ., 69 (1987), 111-148.  doi: 10.1016/0022-0396(87)90105-7.

[22]

J. Schaeffer, Global existence of smooth solutions to the Vlasov Poisson system in three dimensions, Comm. PDE., 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.

[23]

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.

[24]

X. Wang, Propagation of regularity and long time behavior of $3D$ massive relativistic transport equation I: Vlasov-Nordström system, Comm. Math. Phys., 382 (2021), 1843-1934.  doi: 10.1007/s00220-021-03987-2.

[25]

X. Wang, Propagation of regularity and long time behavior of $3D$ massive relativistic transport equation II: Vlasov-Maxwell system, Comm. Math. Phys., 389 (2022), 715-812.  doi: 10.1007/s00220-021-04257-x.

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