# 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 and 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), 342-364. 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), 411-416. [3] E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rational Mech. Anal., 159 (2001), 85-108. 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), 791-808. doi: 10.1081/PDE-120002874. [5] R. Glassey, "The Cauchy Problem in Kinetic Theory,'' SIAM, Philadelphia, PA, 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), 411-453. doi: 10.1080/00411459408203873. [7] R. Glassey and J. Schaeffer, On time decay rates in Landau damping, Commun. PDE, 20 (1995), 647-676. 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), 59-90. 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), 75-86. 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), 229-248 and 4 (1982), 19-32. [11] P.-E. Jabin, The Vlasov-Poisson system with infinite mass and energy, J. Statist. Phys., 103 (2001), 1107-1123. doi: 10.1023/A:1010321308267. [12] R. Kurth, Das Anfangswertproblem der stellardynamik, Z. Astrophys., 30 (1952), 213-229. [13] L. D. Landau, On the vibrations of the electronic plasma, Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574-586. [14] 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. [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), 245-261. [16] 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. [17] 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. [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), 531-562. doi: 10.1080/00411450701703480. [19] S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 31 (2006), 349-370. [20] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eqns., 95 (1992), 281-303. [21] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Commun. Part. Diff. Eqns., 16 (1991), 1313-1335. [22] J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Mathematical Methods in the Applied Sciences., 34 (2011), 262-277. doi: 10.1002/mma.1354. [23] J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. PDE, 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186. [24] J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354. [25] N. G. VanKampen and B. U. Felderhof, "Theoretical Methods in Plasma Physics,'' North-Holland, Amsterdam, 1967. [26] 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.

show all references

##### References:
 [1] J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Diff. Eqns., 25 (1977), 342-364. 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), 411-416. [3] E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass, Arch. Rational Mech. Anal., 159 (2001), 85-108. 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), 791-808. doi: 10.1081/PDE-120002874. [5] R. Glassey, "The Cauchy Problem in Kinetic Theory,'' SIAM, Philadelphia, PA, 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), 411-453. doi: 10.1080/00411459408203873. [7] R. Glassey and J. Schaeffer, On time decay rates in Landau damping, Commun. PDE, 20 (1995), 647-676. 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), 59-90. 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), 75-86. 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), 229-248 and 4 (1982), 19-32. [11] P.-E. Jabin, The Vlasov-Poisson system with infinite mass and energy, J. Statist. Phys., 103 (2001), 1107-1123. doi: 10.1023/A:1010321308267. [12] R. Kurth, Das Anfangswertproblem der stellardynamik, Z. Astrophys., 30 (1952), 213-229. [13] L. D. Landau, On the vibrations of the electronic plasma, Akad. Nauk SSSR. Shurnal Eksper. Fiz., 16 (1946), 574-586. [14] 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. [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), 245-261. [16] 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. [17] 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. [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), 531-562. doi: 10.1080/00411450701703480. [19] S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Comm. Partial Differential Equations, 31 (2006), 349-370. [20] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eqns., 95 (1992), 281-303. [21] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Commun. Part. Diff. Eqns., 16 (1991), 1313-1335. [22] J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Mathematical Methods in the Applied Sciences., 34 (2011), 262-277. doi: 10.1002/mma.1354. [23] J. Schaeffer, The Vlasov-Poisson system with steady spatial asymptotics, Comm. PDE, 28 (2003), 1057-1084. doi: 10.1081/PDE-120021186. [24] J. Schaeffer, Steady spatial asymptotics for the Vlasov-Poisson system, Math. Meth. Appl. Sci., 26 (2003), 273-296. doi: 10.1002/mma.354. [25] N. G. VanKampen and B. U. Felderhof, "Theoretical Methods in Plasma Physics,'' North-Holland, Amsterdam, 1967. [26] 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.
 [1] Christophe Pallard. Growth estimates and uniform decay for a collisionless plasma. Kinetic and Related Models, 2011, 4 (2) : 549-567. doi: 10.3934/krm.2011.4.549 [2] Baptiste Fedele, Claudia Negulescu. Numerical study of an anisotropic Vlasov equation arising in plasma physics. Kinetic and Related Models, 2018, 11 (6) : 1395-1426. doi: 10.3934/krm.2018055 [3] Oǧul Esen, Serkan Sütlü. Matched pair analysis of the Vlasov plasma. Journal of Geometric Mechanics, 2021, 13 (2) : 209-246. doi: 10.3934/jgm.2021011 [4] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [5] Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic and Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729 [6] Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic and Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015 [7] Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic and Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004 [8] Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 [9] Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 [10] Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159 [11] Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control and Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73 [12] Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic and Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011 [13] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345 [14] Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic and Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35 [15] Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173 [16] Pedro Isaza, Jorge Mejía. On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1239-1255. doi: 10.3934/cpaa.2011.10.1239 [17] Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379 [18] Frédérique Charles, Bruno Després, Benoît Perthame, Rémis Sentis. Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinetic and Related Models, 2013, 6 (2) : 269-290. doi: 10.3934/krm.2013.6.269 [19] Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169 [20] Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic and Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429

2020 Impact Factor: 1.432