# American Institute of Mathematical Sciences

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
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] R. Glassey, "The Cauchy Problem in Kinetic Theory,'', SIAM, (1996). doi: 10.1137/1.9781611971477. Google Scholar [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. Google Scholar [7] R. Glassey and J. Schaeffer, On time decay rates in Landau damping,, Commun. PDE, 20 (1995), 647. doi: 10.1080/03605309508821107. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] P.-E. Jabin, The Vlasov-Poisson system with infinite mass and energy,, J. Statist. Phys., 103 (2001), 1107. doi: 10.1023/A:1010321308267. Google Scholar [12] R. Kurth, Das Anfangswertproblem der stellardynamik,, Z. Astrophys., 30 (1952), 213. Google Scholar [13] L. D. Landau, On the vibrations of the electronic plasma,, Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics,, Comm. Partial Differential Equations, 31 (2006), 349. Google Scholar [20] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Diff. Eqns., 95 (1992), 281. Google Scholar [21] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Commun. Part. Diff. Eqns., 16 (1991), 1313. Google Scholar [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. Google Scholar [23] J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics,, Comm. PDE, 28 (2003), 1057. doi: 10.1081/PDE-120021186. Google Scholar [24] J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 26 (2003), 273. doi: 10.1002/mma.354. Google Scholar [25] N. G. VanKampen and B. U. Felderhof, "Theoretical Methods in Plasma Physics,'', North-Holland, (1967). Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] R. Glassey, "The Cauchy Problem in Kinetic Theory,'', SIAM, (1996). doi: 10.1137/1.9781611971477. Google Scholar [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. Google Scholar [7] R. Glassey and J. Schaeffer, On time decay rates in Landau damping,, Commun. PDE, 20 (1995), 647. doi: 10.1080/03605309508821107. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] P.-E. Jabin, The Vlasov-Poisson system with infinite mass and energy,, J. Statist. Phys., 103 (2001), 1107. doi: 10.1023/A:1010321308267. Google Scholar [12] R. Kurth, Das Anfangswertproblem der stellardynamik,, Z. Astrophys., 30 (1952), 213. Google Scholar [13] L. D. Landau, On the vibrations of the electronic plasma,, Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics,, Comm. Partial Differential Equations, 31 (2006), 349. Google Scholar [20] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Diff. Eqns., 95 (1992), 281. Google Scholar [21] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Commun. Part. Diff. Eqns., 16 (1991), 1313. Google Scholar [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. Google Scholar [23] J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics,, Comm. PDE, 28 (2003), 1057. doi: 10.1081/PDE-120021186. Google Scholar [24] J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system,, Math. Meth. Appl. Sci., 26 (2003), 273. doi: 10.1002/mma.354. Google Scholar [25] N. G. VanKampen and B. U. Felderhof, "Theoretical Methods in Plasma Physics,'', North-Holland, (1967). Google Scholar [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. Google Scholar
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