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

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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

Abstract
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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. Batt, Global symmetric solutions of the initial value problem of stellar dynamics,, J. Diff. Eqns., 25 (1977), 342. 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. Academy of Sci. Paris Sér. I Math., 313 (1991), 411.
[3]	E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass,, Arch. Rational Mech. Anal., 159 (2001), 85. doi: 10.1007/s002050100150.
[4]	S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass,, Commun. PDE, 27 (2002), 791. doi: 10.1081/PDE-120002874.
[5]	R. Glassey, "The Cauchy Problem in Kinetic Theory,'', SIAM, (1996). doi: 10.1137/1.9781611971477.
[6]	R. Glassey and J. Schaeffer, Time decay for solutions to the linearized Vlasov equation,, Trans. Th. Stat. Phys., 23 (1994), 411. doi: 10.1080/00411459408203873.
[7]	R. Glassey and J. Schaeffer, On time decay rates in Landau damping,, Commun. PDE, 20 (1995), 647. doi: 10.1080/03605309508821107.
[8]	R. Glassey and W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities,, Arch. Rat. Mech. Anal., 92 (1986), 59. doi: 10.1007/BF00250732.
[9]	E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 16 (1993), 75. doi: 10.1002/mma.1670160202.
[10]	E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov-Equation, Parts I and II,, Math. Meth. Appl. Sci., 3 (1981), 229.
[11]	P.-E. Jabin, The Vlasov-Poisson system with infinite mass and energy,, J. Statist. Phys., 103 (2001), 1107. doi: 10.1023/A:1010321308267.
[12]	R. Kurth, Das Anfangswertproblem der stellardynamik,, Z. Astrophys., 30 (1952), 213.
[13]	L. D. Landau, On the vibrations of the electronic plasma,, Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574.
[14]	P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1991), 415. doi: 10.1007/BF01232273.
[15]	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.
[16]	S. Pankavich, Explicit solutions of the one-dimensional Vlasov-Poisson system with infinite mass,, Math. Methods Appl. Sci., 31 (2008), 375. doi: 10.1002/mma.915.
[17]	S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass,, Math. Methods Appl. Sci., 30 (2007), 529. doi: 10.1002/mma.796.
[18]	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. doi: 10.1080/00411450701703480.
[19]	S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics,, Comm. Partial Differential Equations, 31 (2006), 349.
[20]	K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Diff. Eqns., 95 (1992), 281.
[21]	J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Commun. Part. Diff. Eqns., 16 (1991), 1313.
[22]	J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system,, Mathematical Methods in the Applied Sciences., 34 (2011), 262. doi: 10.1002/mma.1354.
[23]	J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics,, Comm. PDE, 28 (2003), 1057. doi: 10.1081/PDE-120021186.
[24]	J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 26 (2003), 273. doi: 10.1002/mma.354.
[25]	N. G. VanKampen and B. U. Felderhof, "Theoretical Methods in Plasma Physics,'', North-Holland, (1967).
[26]	S. Wollman, Global-in-time solutions of the two-dimensional Vlasov-Poisson system,, Comm. Pure Appl. Math., 33 (1980), 173. doi: 10.1002/cpa.3160330205.

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References:

[1]	J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics,, J. Diff. Eqns., 25 (1977), 342. 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. Academy of Sci. Paris Sér. I Math., 313 (1991), 411.
[3]	E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass,, Arch. Rational Mech. Anal., 159 (2001), 85. doi: 10.1007/s002050100150.
[4]	S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass,, Commun. PDE, 27 (2002), 791. doi: 10.1081/PDE-120002874.
[5]	R. Glassey, "The Cauchy Problem in Kinetic Theory,'', SIAM, (1996). doi: 10.1137/1.9781611971477.
[6]	R. Glassey and J. Schaeffer, Time decay for solutions to the linearized Vlasov equation,, Trans. Th. Stat. Phys., 23 (1994), 411. doi: 10.1080/00411459408203873.
[7]	R. Glassey and J. Schaeffer, On time decay rates in Landau damping,, Commun. PDE, 20 (1995), 647. doi: 10.1080/03605309508821107.
[8]	R. Glassey and W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities,, Arch. Rat. Mech. Anal., 92 (1986), 59. doi: 10.1007/BF00250732.
[9]	E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 16 (1993), 75. doi: 10.1002/mma.1670160202.
[10]	E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov-Equation, Parts I and II,, Math. Meth. Appl. Sci., 3 (1981), 229.
[11]	P.-E. Jabin, The Vlasov-Poisson system with infinite mass and energy,, J. Statist. Phys., 103 (2001), 1107. doi: 10.1023/A:1010321308267.
[12]	R. Kurth, Das Anfangswertproblem der stellardynamik,, Z. Astrophys., 30 (1952), 213.
[13]	L. D. Landau, On the vibrations of the electronic plasma,, Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574.
[14]	P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1991), 415. doi: 10.1007/BF01232273.
[15]	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.
[16]	S. Pankavich, Explicit solutions of the one-dimensional Vlasov-Poisson system with infinite mass,, Math. Methods Appl. Sci., 31 (2008), 375. doi: 10.1002/mma.915.
[17]	S. Pankavich, Local existence for the one-dimensional Vlasov-Poisson system with infinite mass,, Math. Methods Appl. Sci., 30 (2007), 529. doi: 10.1002/mma.796.
[18]	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. doi: 10.1080/00411450701703480.
[19]	S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics,, Comm. Partial Differential Equations, 31 (2006), 349.
[20]	K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Diff. Eqns., 95 (1992), 281.
[21]	J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Commun. Part. Diff. Eqns., 16 (1991), 1313.
[22]	J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system,, Mathematical Methods in the Applied Sciences., 34 (2011), 262. doi: 10.1002/mma.1354.
[23]	J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics,, Comm. PDE, 28 (2003), 1057. doi: 10.1081/PDE-120021186.
[24]	J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 26 (2003), 273. doi: 10.1002/mma.354.
[25]	N. G. VanKampen and B. U. Felderhof, "Theoretical Methods in Plasma Physics,'', North-Holland, (1967).
[26]	S. Wollman, Global-in-time solutions of the two-dimensional Vlasov-Poisson system,, Comm. Pure Appl. Math., 33 (1980), 173. doi: 10.1002/cpa.3160330205.

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