December  2017, 10(4): 977-1009. doi: 10.3934/krm.2017039

The two dimensional Vlasov-Poisson system with steady spatial asymptotics

a. 

Department of Mathematics, School of Science, Nanchang University, Nanchang, Jiangxi 330031, China

b. 

School of Automation, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

c. 

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

* Corresponding author: Zili Chen

Received  April 2015 Revised  November 2016 Published  March 2017

We consider a two dimensional collisionless plasma interacting with a fixed background of positive charge, the density of which depends only upon velocity variable $v$ and decays as $|v| \to \infty $. Suppose that mobile negative ions balance the positive charge as spatial variable $|x|\to \infty $, then on the mesoscopic level the system is characterized by the two dimensional Vlasov-Poisson system with steady spatial asymptotics, whose total positive charge and total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior are shown to exist locally in time, and an "almost optimal" criterion for the continuation of these solutions is established.

Citation: Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
References:
[1]

A. A. Arsen'ev, Global existence of a weak solution of Vlasov's system of equations, Dokl. Akad. Nauk SSSR, 220 (1975), 1249-1250.

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson system in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.

[3]

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

[4]

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.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

E. Horst and R. Hunze, Weak solutions of the inintial valume problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci., 6 (1984), 262-279. doi: 10.1002/mma.1670060118.

[11]

R. Illner and H. Neunzert, An existence theorem for the unmodified Vlasov equation, Math. Methods Appl. Sci., 1 (1979), 530-554. doi: 10.1002/mma.1670010410.

[12]

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

[13]

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.

[14]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005.

[15]

E. Miot, A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system, Commun. Math. Phys., 346 (2016), 469-482. doi: 10.1007/s00220-016-2707-7.

[16]

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.

[17]

C. Pallard, Large velocities in a collisionless plasma, J. Differ. Equ., 252 (2012), 2864-2876. doi: 10.1016/j.jde.2011.09.020.

[18]

C. Pallard, Moment propagation for weak solutions to the Vlasov-Poisson system, Commun. Partial Differ. Equ., 37 (2012), 1273-1285. doi: 10.1080/03605302.2011.606863.

[19]

C. Pallard, Space moments of the Vlasov-Poisson system: Propagation and regularity, SIAM J. Math. Anal., 46 (2014), 1754-1770. doi: 10.1137/120881178.

[20]

S. Pankavich, Global existence and increased spatial decay for the radial Vlasov-Poisson system with steady spatial asymptotics, Transport Theory Statist. Phys., 36 (2007), 531-562. doi: 10.1080/00411450701703480.

[21]

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

[22]

S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass, Math. Methods Appl. Sci., 30 (2007), 529-548. doi: 10.1002/mma.796.

[23]

S. Pankavich, Explicit solutions of the one-dimensional Vlasov-Poisson system with infinite mass, Math. Methods Appl. Sci., 31 (2008), 375-389. doi: 10.1002/mma.915.

[24]

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.

[25]

G. Rein, Collisionless kinetic equation from astrophysics-the Vlasov-Poisson system, in Handbook of Differential Equations: Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), vol. 3, Elsevier, (2007), 383–476. doi: 10.1016/S1874-5717(07)80008-9.

[26]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19 (2009), 199-228. doi: 10.1142/S0218202509003401.

[27]

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

[28]

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

[29]

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

[30]

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.

[31]

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.

[32]

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]

A. A. Arsen'ev, Global existence of a weak solution of Vlasov's system of equations, Dokl. Akad. Nauk SSSR, 220 (1975), 1249-1250.

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson system in 3 space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.

[3]

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

[4]

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.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

E. Horst and R. Hunze, Weak solutions of the inintial valume problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci., 6 (1984), 262-279. doi: 10.1002/mma.1670060118.

[11]

R. Illner and H. Neunzert, An existence theorem for the unmodified Vlasov equation, Math. Methods Appl. Sci., 1 (1979), 530-554. doi: 10.1002/mma.1670010410.

[12]

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

[13]

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.

[14]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005.

[15]

E. Miot, A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system, Commun. Math. Phys., 346 (2016), 469-482. doi: 10.1007/s00220-016-2707-7.

[16]

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.

[17]

C. Pallard, Large velocities in a collisionless plasma, J. Differ. Equ., 252 (2012), 2864-2876. doi: 10.1016/j.jde.2011.09.020.

[18]

C. Pallard, Moment propagation for weak solutions to the Vlasov-Poisson system, Commun. Partial Differ. Equ., 37 (2012), 1273-1285. doi: 10.1080/03605302.2011.606863.

[19]

C. Pallard, Space moments of the Vlasov-Poisson system: Propagation and regularity, SIAM J. Math. Anal., 46 (2014), 1754-1770. doi: 10.1137/120881178.

[20]

S. Pankavich, Global existence and increased spatial decay for the radial Vlasov-Poisson system with steady spatial asymptotics, Transport Theory Statist. Phys., 36 (2007), 531-562. doi: 10.1080/00411450701703480.

[21]

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

[22]

S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass, Math. Methods Appl. Sci., 30 (2007), 529-548. doi: 10.1002/mma.796.

[23]

S. Pankavich, Explicit solutions of the one-dimensional Vlasov-Poisson system with infinite mass, Math. Methods Appl. Sci., 31 (2008), 375-389. doi: 10.1002/mma.915.

[24]

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.

[25]

G. Rein, Collisionless kinetic equation from astrophysics-the Vlasov-Poisson system, in Handbook of Differential Equations: Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), vol. 3, Elsevier, (2007), 383–476. doi: 10.1016/S1874-5717(07)80008-9.

[26]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19 (2009), 199-228. doi: 10.1142/S0218202509003401.

[27]

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

[28]

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

[29]

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

[30]

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.

[31]

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.

[32]

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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