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

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.
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
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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

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