April 2018, 11(2): 303-336. doi: 10.3934/krm.2018015

A Vlasov-Poisson plasma of infinite mass with a point charge

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author: Xianwen Zhang

Received  March 2017 Published  January 2018

We study the time evolution of the three dimensional Vlasov-Poisson plasma interacting with a positive point charge in the case of infinite mass. We prove the existence and uniqueness of the classical solution to the system by assuming that the initial density slightly decays in space, but not integrable. This result extends a previous theorem for Yukawa potential obtained in [10] to the case of Coulomb interaction.

Citation: Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
References:
[1]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3.

[2]

J. Batt and G. Rein, Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 411-416.

[3]

E. CagliotiS. CaprinoC. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rational Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.

[4]

S. CaprinoG. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164. doi: 10.1137/130916527.

[5]

S. CaprinoG. Cavallaro and C. Marchioro, Remark on a magnetically confined plasma with infinite charge, Rend. Mat. Appl., 35 (2014), 69-98.

[6]

S. CaprinoG. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with infinite charge in $\mathbb{R}^{3}$, Comm. Partial Differential Equations, 40 (2015), 357-385. doi: 10.1080/03605302.2014.944267.

[7]

S. CaprinoG. Cavallaro and C. Marchioro, A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror, Kinet. Relat. Models, 9 (2016), 657-686. doi: 10.3934/krm.2016011.

[8]

S. Caprino, G. Cavallaro and C. Marchioro, The Vlasov-Poisson equation in $\mathbb {R}^{3}$ with infinite charge and velocities, preprint, arXiv: 1608.02336.

[9]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinet. Relat. Models, 3 (2010), 241-254. doi: 10.3934/krm.2010.3.241.

[10]

S. Caprino and C. Marchioro, On a charge interacting with a plasma of unbounded mass, Kinet. Relat. Models, 4 (2011), 215-226. doi: 10.3934/krm.2011.4.215.

[11]

S. CaprinoC. MarchioroE. Miot and M. Pulvirenti, On the attractive plasma-charge system in 2-D, Comm. Partial Differential Equations, 37 (2012), 1237-1272. doi: 10.1080/03605302.2011.653032.

[12]

S. CaprinoC. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass, Comm. Partial Differential Equations, 27 (2002), 791-808. doi: 10.1081/PDE-120002874.

[13]

J. ChenJ. Wei and X. Zhang, Asymptotic growth bounds for the Vlasov-Poisson system with a point charge, Appl. Math. Lett., 46 (2015), 17-24. doi: 10.1016/j.aml.2015.01.020.

[14]

J. ChenX. Zhang and J. Wei, Global weak solutions for the Vlasov-Poisson system with a point charge, Math. Methods Appl. Sci., 38 (2015), 3776-3791. doi: 10.1002/mma.3316.

[15]

Z. Chen and X. Zhang, Sub-linear estimate of large velocities in a collisionless plasma, Comm. Math. Sci., 12 (2014), 279-291. doi: 10.4310/CMS.2014.v12.n2.a4.

[16]

Z. Chen and X. Zhang, Global existence to the Vlasov-Poisson system and propagation of moments without assumption of finite kinetic energy, Comm. Math. Phys., 343 (2016), 851-879. doi: 10.1007/s00220-016-2616-9.

[17]

L. DesvillettesE. Miot and C. Saffirio, Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 373-400. doi: 10.1016/j.anihpc.2014.01.001.

[18] R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477.
[19]

P. E. Jabin, The Vlasov-Poisson system with infinite mass and energy, J. Statist. Phys., 103 (2001), 1107-1123. doi: 10.1023/A:1010321308267.

[20]

R. Kurth, Das Anfangswertproblem der Stellardynamik, Z. Astrophys., 30 (1952), 213-229.

[21]

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.

[22]

C. MarchioroE. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges, Arch. Rational Mech. Anal., 201 (2011), 1-26. doi: 10.1007/s00205-010-0388-5.

[23]

T. Okabe and S. Ukai, On classical solutions in the large in time of two-dimensional Vlasov's equation, Osaka J. Math., 15 (1978), 245-261.

[24]

S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 31 (2006), 349-370. doi: 10.1080/03605300500358004.

[25]

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.

[26]

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

[27]

G. Rein, Collisionless kinetic equation from astrophysics-the Vlasov-Poisson system. in Handbook of Differential Equations: Evolutionary Equations, Amsterdam: Elsevier, (2007), 383-476.

[28]

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

[29]

J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186.

[30]

J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system, Math. Methods Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354.

[31]

J. Schaeffer, Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior, Kinet. Relat. Models, 5 (2012), 129-153. doi: 10.3934/krm.2012.5.129.

[32]

S. Wollman, Global-in-time solutions of the two-dimensional Vlasov-Poisson system, Comm. Pure Appl. Math., 33 (1980), 173-197. doi: 10.1002/cpa.3160330205.

[33]

S. Wollman, Global in time solutions to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl., 176 (1993), 76-91. doi: 10.1006/jmaa.1993.1200.

[34]

X. Zhang and J. Wei, The Vlasov-Poisson system with infinite kinetic energy and initial data in $L^{p}(\mathbb{R}^{6})$, J. Math. Anal. Appl., 341 (2008), 548-558. doi: 10.1016/j.jmaa.2007.10.038.

show all references

References:
[1]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3.

[2]

J. Batt and G. Rein, Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 411-416.

[3]

E. CagliotiS. CaprinoC. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rational Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.

[4]

S. CaprinoG. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164. doi: 10.1137/130916527.

[5]

S. CaprinoG. Cavallaro and C. Marchioro, Remark on a magnetically confined plasma with infinite charge, Rend. Mat. Appl., 35 (2014), 69-98.

[6]

S. CaprinoG. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with infinite charge in $\mathbb{R}^{3}$, Comm. Partial Differential Equations, 40 (2015), 357-385. doi: 10.1080/03605302.2014.944267.

[7]

S. CaprinoG. Cavallaro and C. Marchioro, A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror, Kinet. Relat. Models, 9 (2016), 657-686. doi: 10.3934/krm.2016011.

[8]

S. Caprino, G. Cavallaro and C. Marchioro, The Vlasov-Poisson equation in $\mathbb {R}^{3}$ with infinite charge and velocities, preprint, arXiv: 1608.02336.

[9]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinet. Relat. Models, 3 (2010), 241-254. doi: 10.3934/krm.2010.3.241.

[10]

S. Caprino and C. Marchioro, On a charge interacting with a plasma of unbounded mass, Kinet. Relat. Models, 4 (2011), 215-226. doi: 10.3934/krm.2011.4.215.

[11]

S. CaprinoC. MarchioroE. Miot and M. Pulvirenti, On the attractive plasma-charge system in 2-D, Comm. Partial Differential Equations, 37 (2012), 1237-1272. doi: 10.1080/03605302.2011.653032.

[12]

S. CaprinoC. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass, Comm. Partial Differential Equations, 27 (2002), 791-808. doi: 10.1081/PDE-120002874.

[13]

J. ChenJ. Wei and X. Zhang, Asymptotic growth bounds for the Vlasov-Poisson system with a point charge, Appl. Math. Lett., 46 (2015), 17-24. doi: 10.1016/j.aml.2015.01.020.

[14]

J. ChenX. Zhang and J. Wei, Global weak solutions for the Vlasov-Poisson system with a point charge, Math. Methods Appl. Sci., 38 (2015), 3776-3791. doi: 10.1002/mma.3316.

[15]

Z. Chen and X. Zhang, Sub-linear estimate of large velocities in a collisionless plasma, Comm. Math. Sci., 12 (2014), 279-291. doi: 10.4310/CMS.2014.v12.n2.a4.

[16]

Z. Chen and X. Zhang, Global existence to the Vlasov-Poisson system and propagation of moments without assumption of finite kinetic energy, Comm. Math. Phys., 343 (2016), 851-879. doi: 10.1007/s00220-016-2616-9.

[17]

L. DesvillettesE. Miot and C. Saffirio, Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 373-400. doi: 10.1016/j.anihpc.2014.01.001.

[18] R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477.
[19]

P. E. Jabin, The Vlasov-Poisson system with infinite mass and energy, J. Statist. Phys., 103 (2001), 1107-1123. doi: 10.1023/A:1010321308267.

[20]

R. Kurth, Das Anfangswertproblem der Stellardynamik, Z. Astrophys., 30 (1952), 213-229.

[21]

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.

[22]

C. MarchioroE. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges, Arch. Rational Mech. Anal., 201 (2011), 1-26. doi: 10.1007/s00205-010-0388-5.

[23]

T. Okabe and S. Ukai, On classical solutions in the large in time of two-dimensional Vlasov's equation, Osaka J. Math., 15 (1978), 245-261.

[24]

S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 31 (2006), 349-370. doi: 10.1080/03605300500358004.

[25]

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.

[26]

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

[27]

G. Rein, Collisionless kinetic equation from astrophysics-the Vlasov-Poisson system. in Handbook of Differential Equations: Evolutionary Equations, Amsterdam: Elsevier, (2007), 383-476.

[28]

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

[29]

J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186.

[30]

J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system, Math. Methods Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354.

[31]

J. Schaeffer, Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior, Kinet. Relat. Models, 5 (2012), 129-153. doi: 10.3934/krm.2012.5.129.

[32]

S. Wollman, Global-in-time solutions of the two-dimensional Vlasov-Poisson system, Comm. Pure Appl. Math., 33 (1980), 173-197. doi: 10.1002/cpa.3160330205.

[33]

S. Wollman, Global in time solutions to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl., 176 (1993), 76-91. doi: 10.1006/jmaa.1993.1200.

[34]

X. Zhang and J. Wei, The Vlasov-Poisson system with infinite kinetic energy and initial data in $L^{p}(\mathbb{R}^{6})$, J. Math. Anal. Appl., 341 (2008), 548-558. doi: 10.1016/j.jmaa.2007.10.038.

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