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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.
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.
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.
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.
|
[5] |
R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Communications on Pure and Applied Mathematics 42 (1989), 42 (1989), 729.
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.
doi: 10.1002/mma.1670130207. |
[7] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory,, $1^{st}$ edition, (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.
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.
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.
|
[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.
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.
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, (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.
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.
doi: 10.1080/03605300500358004. |
[16] |
S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity,, Submitted, (2014). Google Scholar |
[17] |
S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system,, Submitted, (2014). Google Scholar |
[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.
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.
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.
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, (1978).
|
[22] |
N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics,, Wiley, (1967). Google Scholar |
[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.
doi: 10.1512/iumj.1990.39.39009. |
[24] |
C. Villani, Hypocoercivity,, Memoirs of the American Mathematical Society, (2009).
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.
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.
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.
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.
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.
|
[5] |
R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Communications on Pure and Applied Mathematics 42 (1989), 42 (1989), 729.
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.
doi: 10.1002/mma.1670130207. |
[7] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory,, $1^{st}$ edition, (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.
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.
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.
|
[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.
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.
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, (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.
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.
doi: 10.1080/03605300500358004. |
[16] |
S. Pankavich and N. Michalowski, A Short Proof Of Increased Parabolic Regularity,, Submitted, (2014). Google Scholar |
[17] |
S. Pankavich and J. Schaeffer, Global classical solutions of the "one and one-half'' dimensional Vlasov-Maxwell-Fokker-Planck system,, Submitted, (2014). Google Scholar |
[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.
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.
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.
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, (1978).
|
[22] |
N. G. van Kampen and B. U. Felderhof, Theoretical Methods in Plasma Physics,, Wiley, (1967). Google Scholar |
[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.
doi: 10.1512/iumj.1990.39.39009. |
[24] |
C. Villani, Hypocoercivity,, Memoirs of the American Mathematical Society, (2009).
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.
doi: 10.1137/090755796. |
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