American Institute of Mathematical Sciences

Advanced
March  2012, 5(1): 129-153. doi: 10.3934/krm.2012.5.129

Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior

Jack Schaeffer 1,

1. 

Department of Mathematics Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States

Received  August 2011 Revised  August 2011 Published  January 2012

A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as $|x|\to\infty$ is considered. Hence, the total positive charge and the total negative charge are both infinite. It is shown, in three spatial dimensions, that smooth solutions may be continued as long as the velocity support remains finite. Also, in the case of spherical symmetry, a bound on velocity support is obtained and hence solutions exist globally in time.
Keywords: velocity support, Vlasov equation., Collisionless plasma.
Mathematics Subject Classification: 35L60, 35Q99, 82C21, 82C22, 82D1.
Citation: Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
References:
[1]

J. Diff. Eqns., 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[2]

C. R. Academy of Sci. Paris Sér. I Math., 313 (1991), 411-416.  Google Scholar

[3]

Arch. Rational Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.  Google Scholar

[4]

Commun. PDE, 27 (2002), 791-808. doi: 10.1081/PDE-120002874.  Google Scholar

[5]

SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[6]

Trans. Th. Stat. Phys., 23 (1994), 411-453. doi: 10.1080/00411459408203873.  Google Scholar

[7]

Commun. PDE, 20 (1995), 647-676. doi: 10.1080/03605309508821107.  Google Scholar

[8]

Arch. Rat. Mech. Anal., 92 (1986), 59-90. doi: 10.1007/BF00250732.  Google Scholar

[9]

Math. Meth. Appl. Sci., 16 (1993), 75-86. doi: 10.1002/mma.1670160202.  Google Scholar

[10]

Math. Meth. Appl. Sci., 3 (1981), 229-248 and 4 (1982), 19-32. Google Scholar

[11]

J. Statist. Phys., 103 (2001), 1107-1123. doi: 10.1023/A:1010321308267.  Google Scholar

[12]

Z. Astrophys., 30 (1952), 213-229.  Google Scholar

[13]

Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574-586.  Google Scholar

[14]

Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.  Google Scholar

[15]

Osaka J. Math., 15 (1978), 245-261.  Google Scholar

[16]

Math. Methods Appl. Sci., 31 (2008), 375-389. doi: 10.1002/mma.915.  Google Scholar

[17]

Math. Methods Appl. Sci., 30 (2007), 529-548. doi: 10.1002/mma.796.  Google Scholar

[18]

Transport Theory Statist. Phys., 36 (2007), 531-562. doi: 10.1080/00411450701703480.  Google Scholar

[19]

Comm. Partial Differential Equations, 31 (2006), 349-370.  Google Scholar

[20]

J. Diff. Eqns., 95 (1992), 281-303.  Google Scholar

[21]

Commun. Part. Diff. Eqns., 16 (1991), 1313-1335.  Google Scholar

[22]

Mathematical Methods in the Applied Sciences., 34 (2011), 262-277. doi: 10.1002/mma.1354.  Google Scholar

[23]

Comm. PDE, 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186.  Google Scholar

[24]

Math. Meth. Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354.  Google Scholar

[25]

North-Holland, Amsterdam, 1967. Google Scholar

[26]

Comm. Pure Appl. Math., 33 (1980), 173-197. doi: 10.1002/cpa.3160330205.  Google Scholar

show all references

References:
[1]

J. Diff. Eqns., 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[2]

C. R. Academy of Sci. Paris Sér. I Math., 313 (1991), 411-416.  Google Scholar

[3]

Arch. Rational Mech. Anal., 159 (2001), 85-108. doi: 10.1007/s002050100150.  Google Scholar

[4]

Commun. PDE, 27 (2002), 791-808. doi: 10.1081/PDE-120002874.  Google Scholar

[5]

SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[6]

Trans. Th. Stat. Phys., 23 (1994), 411-453. doi: 10.1080/00411459408203873.  Google Scholar

[7]

Commun. PDE, 20 (1995), 647-676. doi: 10.1080/03605309508821107.  Google Scholar

[8]

Arch. Rat. Mech. Anal., 92 (1986), 59-90. doi: 10.1007/BF00250732.  Google Scholar

[9]

Math. Meth. Appl. Sci., 16 (1993), 75-86. doi: 10.1002/mma.1670160202.  Google Scholar

[10]

Math. Meth. Appl. Sci., 3 (1981), 229-248 and 4 (1982), 19-32. Google Scholar

[11]

J. Statist. Phys., 103 (2001), 1107-1123. doi: 10.1023/A:1010321308267.  Google Scholar

[12]

Z. Astrophys., 30 (1952), 213-229.  Google Scholar

[13]

Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574-586.  Google Scholar

[14]

Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.  Google Scholar

[15]

Osaka J. Math., 15 (1978), 245-261.  Google Scholar

[16]

Math. Methods Appl. Sci., 31 (2008), 375-389. doi: 10.1002/mma.915.  Google Scholar

[17]

Math. Methods Appl. Sci., 30 (2007), 529-548. doi: 10.1002/mma.796.  Google Scholar

[18]

Transport Theory Statist. Phys., 36 (2007), 531-562. doi: 10.1080/00411450701703480.  Google Scholar

[19]

Comm. Partial Differential Equations, 31 (2006), 349-370.  Google Scholar

[20]

J. Diff. Eqns., 95 (1992), 281-303.  Google Scholar

[21]

Commun. Part. Diff. Eqns., 16 (1991), 1313-1335.  Google Scholar

[22]

Mathematical Methods in the Applied Sciences., 34 (2011), 262-277. doi: 10.1002/mma.1354.  Google Scholar

[23]

Comm. PDE, 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186.  Google Scholar

[24]

Math. Meth. Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354.  Google Scholar

[25]

North-Holland, Amsterdam, 1967. Google Scholar

[26]

Comm. Pure Appl. Math., 33 (1980), 173-197. doi: 10.1002/cpa.3160330205.  Google Scholar

[1]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004
[2]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003
[3]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[4]

Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021046
[5]

Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021, 16 (2) : 187-219. doi: 10.3934/nhm.2021004
[6]

Lifen Jia, Wei Dai. Uncertain spring vibration equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021073
[7]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[8]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[9]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029
[10]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015
[11]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[13]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[14]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[15]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[16]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384
[17]

Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021065
[18]

José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025
[19]

Samira Shahsavari, Saeed Ketabchi. The proximal methods for solving absolute value equation. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 449-460. doi: 10.3934/naco.2020037
[20]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences