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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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