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Global magnetic confinement for the 1.5D Vlasov-Maxwell system
Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system
1. | Department of Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois St, Golden, CO 80401, United States |
2. | Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, United States |
References:
[1] |
J. Alcántara and S. Calogero, On a relativistic Fokker-Planck equation in kinetic theory, Kinetic and Related Models, 4 (2011), 401-426.
doi: 10.3934/krm.2011.4.401. |
[2] |
F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Archive for Rational Mechanics and Analysis, 170 (2003), 1-15.
doi: 10.1007/s00205-003-0265-6. |
[3] |
M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1007-1025.
doi: 10.1142/S0218202511005222. |
[4] |
P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 19 (1986), 519-542. |
[5] |
R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Communications on Pure and Applied Mathematics 42 (1989), 729-757.
doi: 10.1002/cpa.3160420603. |
[6] |
R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Mathematical Methods in the Applied Sciences, 13 (1990), 169-179.
doi: 10.1002/mma.1670130207. |
[7] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, $1^{st}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[8] |
R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Archive for Rational Mechanics and Analysis, 92 (1986), 59-90.
doi: 10.1007/BF00250732. |
[9] |
F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, Journal of Functional Analysis, 244 (2007), 95-118.
doi: 10.1016/j.jfa.2006.11.013. |
[10] |
S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Communications on Pure and Applied Analysis, 1 (2002), 103-125. |
[11] |
R. Lai, On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system, Mathematical Methods in the Applied Sciences, 18 (1995), 1013-1040.
doi: 10.1002/mma.1670181302. |
[12] |
R. Lai, On the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Mathematical Methods in the Applied Sciences, 21 (1998), 1287-1296.
doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G. |
[13] |
J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin, 1961. |
[14] |
P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Inventiones Mathematicae, 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
[15] |
S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Communications in Partial Differential Equations, 31 (2006), 349-370.
doi: 10.1080/03605300500358004. |
[16] |
S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity, Submitted, 2014. |
[17] |
S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system, Submitted, 2014. |
[18] |
K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Journal of Differential Equations, 95 (1992), 281-303.
doi: 10.1016/0022-0396(92)90033-J. |
[19] |
J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Communications in Mathematical Physics, 104 (1986), 403-421.
doi: 10.1007/BF01210948. |
[20] |
J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Communications in Partial Differential Equations, 16 (1991), 1312-1335.
doi: 10.1080/03605309108820801. |
[21] |
L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay 78, Vol. 13, Université de Paris-Sud Département de Mathématique, Orsay, 1978. |
[22] |
N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley, New York, NY, 1967. |
[23] |
H. D. Victory, Jr. and B. P. O'Dwyer, On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana University Mathematics Journal, 39 (1990), 105-156.
doi: 10.1512/iumj.1990.39.39009. |
[24] |
C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, Vol. 202, (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[25] |
T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM Journal on Mathematical Analysis, 42 (2010), 459-488.
doi: 10.1137/090755796. |
show all references
References:
[1] |
J. Alcántara and S. Calogero, On a relativistic Fokker-Planck equation in kinetic theory, Kinetic and Related Models, 4 (2011), 401-426.
doi: 10.3934/krm.2011.4.401. |
[2] |
F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Archive for Rational Mechanics and Analysis, 170 (2003), 1-15.
doi: 10.1007/s00205-003-0265-6. |
[3] |
M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1007-1025.
doi: 10.1142/S0218202511005222. |
[4] |
P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 19 (1986), 519-542. |
[5] |
R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Communications on Pure and Applied Mathematics 42 (1989), 729-757.
doi: 10.1002/cpa.3160420603. |
[6] |
R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Mathematical Methods in the Applied Sciences, 13 (1990), 169-179.
doi: 10.1002/mma.1670130207. |
[7] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, $1^{st}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[8] |
R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Archive for Rational Mechanics and Analysis, 92 (1986), 59-90.
doi: 10.1007/BF00250732. |
[9] |
F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, Journal of Functional Analysis, 244 (2007), 95-118.
doi: 10.1016/j.jfa.2006.11.013. |
[10] |
S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Communications on Pure and Applied Analysis, 1 (2002), 103-125. |
[11] |
R. Lai, On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system, Mathematical Methods in the Applied Sciences, 18 (1995), 1013-1040.
doi: 10.1002/mma.1670181302. |
[12] |
R. Lai, On the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Mathematical Methods in the Applied Sciences, 21 (1998), 1287-1296.
doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G. |
[13] |
J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin, 1961. |
[14] |
P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Inventiones Mathematicae, 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
[15] |
S. Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Communications in Partial Differential Equations, 31 (2006), 349-370.
doi: 10.1080/03605300500358004. |
[16] |
S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity, Submitted, 2014. |
[17] |
S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system, Submitted, 2014. |
[18] |
K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Journal of Differential Equations, 95 (1992), 281-303.
doi: 10.1016/0022-0396(92)90033-J. |
[19] |
J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Communications in Mathematical Physics, 104 (1986), 403-421.
doi: 10.1007/BF01210948. |
[20] |
J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Communications in Partial Differential Equations, 16 (1991), 1312-1335.
doi: 10.1080/03605309108820801. |
[21] |
L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay 78, Vol. 13, Université de Paris-Sud Département de Mathématique, Orsay, 1978. |
[22] |
N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics, Wiley, New York, NY, 1967. |
[23] |
H. D. Victory, Jr. and B. P. O'Dwyer, On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana University Mathematics Journal, 39 (1990), 105-156.
doi: 10.1512/iumj.1990.39.39009. |
[24] |
C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, Vol. 202, (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[25] |
T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM Journal on Mathematical Analysis, 42 (2010), 459-488.
doi: 10.1137/090755796. |
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