-
Previous Article
A review of the mean field limits for Vlasov equations
- KRM Home
- This Issue
-
Next Article
Time decay of solutions to the compressible Euler equations with damping
On the Geometrical Gyro-Kinetic theory
1. | LMBA (UMR 6205) Université de Bretagne-Sud, F-56017 Vannes, France |
2. | IRMA (UMR 7501) Université de Strasbourg, F-67094 Strasbourg Cedex, France |
References:
[1] |
J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks, Physics of Plasmas, 2 (1995), 459-471.
doi: 10.1063/1.871465. |
[2] |
L Brouwer, Über abbildung von mannigfaltigkeiten, Mathematische Annalen, 71 (1912), 97-115. |
[3] |
D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535.
doi: 10.1063/1.864113. |
[4] |
E. Frénod, P. A. Raviart and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field, J. Math. Pures et Appl., 80 (2001), 815-843.
doi: 10.1016/S0021-7824(01)01215-6. |
[5] |
E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymp. Anal., 18 (1998), 193-214. |
[6] |
E. Frénod and E. Sonnendrücker, Long time behavior of the two dimensionnal Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.
doi: 10.1142/S021820250000029X. |
[7] |
E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.
doi: 10.1137/S0036141099364243. |
[8] |
E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508.
doi: 10.1063/1.863762. |
[9] |
X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion, 50 (2010), 043002.
doi: 10.1088/0029-5515/50/4/043002. |
[10] |
C. S. Gardner, Adiabatic invariants of periodic classical systems, Physical Rieview, 115 (1959), 791-794.
doi: 10.1103/PhysRev.115.791. |
[11] |
P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions, Kinet. Relat. Models, 2 (2009), 707-725.
doi: 10.3934/krm.2009.2.707. |
[12] |
F. Golse and L. Saint Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures. Appl., 78 (1999), 791-817.
doi: 10.1016/S0021-7824(99)00021-5. |
[13] |
V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation, Journal of Computational Physics, 217 (2006), 395-423.
doi: 10.1016/j.jcp.2006.01.023. |
[14] |
V. Grandgirard, Y. Sarazin, P Angelino, A. Bottino, N. Crouseilles, G. Darmet, G. Dif-Pradalier, X. Garbet, Ph. Ghendrih, S. Jolliet, G. Latu, E. Sonnendrücker and L. Villard, Global full-$f$ gyrokinetic simulations of plasma turbulence, Plasma Physics and Controlled Fusion, 49 (2007), B173. |
[15] |
T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673.
doi: 10.1063/1.866544. |
[16] |
T. S. Hahm, Nonlinear gyrokinetic equations for turbulence in core transport barriers, Physics of Plasmas, 3 (1996), 4658-4664.
doi: 10.1063/1.872034. |
[17] |
T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948.
doi: 10.1063/1.866641. |
[18] |
T. S. Hahm, Lu Wang and J. Madsen, Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence, Physics of Plasmas, 16 (2009), 022305.
doi: 10.2172/938981. |
[19] |
V. I. Istratescu, Fixed Point Theory an Introduction, Dordrecht-Boston, Mass., 1981. |
[20] |
G. Kawamura and A. Fukuyama, Refinement of the gyrokinetic equations for edge plasmas with large flow shears, Physics of Plasmas, 15 (2008), 042304.
doi: 10.1063/1.2902016. |
[21] |
M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory, International Atomic Energy Agency, Vienna, 1965. |
[22] |
R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458.
doi: 10.1063/1.524053. |
[23] |
R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749.
doi: 10.1063/1.863594. |
[24] |
R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747.
doi: 10.1063/1.525429. |
[25] |
T. G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics, 15 (1961), 79-101.
doi: 10.1016/0003-4916(61)90167-1. |
[26] |
T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order, Physics of Fluids, 21 (1978), 384-389.
doi: 10.1063/1.862226. |
[27] |
P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4350-2. |
[28] |
F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014.
doi: 10.1088/0741-3335/50/6/065014. |
[29] |
F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002.
doi: 10.1088/0741-3335/51/6/065002. |
[30] |
F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics, Plasma Physics and Controlled Fusion, 52 (2010), 045004.
doi: 10.1088/0741-3335/52/4/045004. |
[31] |
H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, General gyrokinetic equations for edge plasmas, Contributions to Plasma Physics, 46 (2006), 477-489.
doi: 10.1002/ctpp.200610034. |
[32] |
H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas, Physics of Plasmas, 14 (2007), 056110.
doi: 10.1063/1.2472596. |
show all references
References:
[1] |
J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks, Physics of Plasmas, 2 (1995), 459-471.
doi: 10.1063/1.871465. |
[2] |
L Brouwer, Über abbildung von mannigfaltigkeiten, Mathematische Annalen, 71 (1912), 97-115. |
[3] |
D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535.
doi: 10.1063/1.864113. |
[4] |
E. Frénod, P. A. Raviart and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field, J. Math. Pures et Appl., 80 (2001), 815-843.
doi: 10.1016/S0021-7824(01)01215-6. |
[5] |
E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymp. Anal., 18 (1998), 193-214. |
[6] |
E. Frénod and E. Sonnendrücker, Long time behavior of the two dimensionnal Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.
doi: 10.1142/S021820250000029X. |
[7] |
E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.
doi: 10.1137/S0036141099364243. |
[8] |
E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508.
doi: 10.1063/1.863762. |
[9] |
X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion, 50 (2010), 043002.
doi: 10.1088/0029-5515/50/4/043002. |
[10] |
C. S. Gardner, Adiabatic invariants of periodic classical systems, Physical Rieview, 115 (1959), 791-794.
doi: 10.1103/PhysRev.115.791. |
[11] |
P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions, Kinet. Relat. Models, 2 (2009), 707-725.
doi: 10.3934/krm.2009.2.707. |
[12] |
F. Golse and L. Saint Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures. Appl., 78 (1999), 791-817.
doi: 10.1016/S0021-7824(99)00021-5. |
[13] |
V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation, Journal of Computational Physics, 217 (2006), 395-423.
doi: 10.1016/j.jcp.2006.01.023. |
[14] |
V. Grandgirard, Y. Sarazin, P Angelino, A. Bottino, N. Crouseilles, G. Darmet, G. Dif-Pradalier, X. Garbet, Ph. Ghendrih, S. Jolliet, G. Latu, E. Sonnendrücker and L. Villard, Global full-$f$ gyrokinetic simulations of plasma turbulence, Plasma Physics and Controlled Fusion, 49 (2007), B173. |
[15] |
T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673.
doi: 10.1063/1.866544. |
[16] |
T. S. Hahm, Nonlinear gyrokinetic equations for turbulence in core transport barriers, Physics of Plasmas, 3 (1996), 4658-4664.
doi: 10.1063/1.872034. |
[17] |
T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948.
doi: 10.1063/1.866641. |
[18] |
T. S. Hahm, Lu Wang and J. Madsen, Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence, Physics of Plasmas, 16 (2009), 022305.
doi: 10.2172/938981. |
[19] |
V. I. Istratescu, Fixed Point Theory an Introduction, Dordrecht-Boston, Mass., 1981. |
[20] |
G. Kawamura and A. Fukuyama, Refinement of the gyrokinetic equations for edge plasmas with large flow shears, Physics of Plasmas, 15 (2008), 042304.
doi: 10.1063/1.2902016. |
[21] |
M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory, International Atomic Energy Agency, Vienna, 1965. |
[22] |
R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458.
doi: 10.1063/1.524053. |
[23] |
R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749.
doi: 10.1063/1.863594. |
[24] |
R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747.
doi: 10.1063/1.525429. |
[25] |
T. G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics, 15 (1961), 79-101.
doi: 10.1016/0003-4916(61)90167-1. |
[26] |
T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order, Physics of Fluids, 21 (1978), 384-389.
doi: 10.1063/1.862226. |
[27] |
P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4350-2. |
[28] |
F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014.
doi: 10.1088/0741-3335/50/6/065014. |
[29] |
F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002.
doi: 10.1088/0741-3335/51/6/065002. |
[30] |
F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics, Plasma Physics and Controlled Fusion, 52 (2010), 045004.
doi: 10.1088/0741-3335/52/4/045004. |
[31] |
H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, General gyrokinetic equations for edge plasmas, Contributions to Plasma Physics, 46 (2006), 477-489.
doi: 10.1002/ctpp.200610034. |
[32] |
H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas, Physics of Plasmas, 14 (2007), 056110.
doi: 10.1063/1.2472596. |
[1] |
Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239 |
[2] |
Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 |
[3] |
Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227 |
[4] |
Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021020 |
[5] |
Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065 |
[6] |
Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014 |
[7] |
Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 |
[8] |
Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879 |
[9] |
David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353 |
[10] |
Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789 |
[11] |
Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481 |
[12] |
Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 |
[13] |
Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 |
[14] |
Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 |
[15] |
Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 |
[16] |
Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 |
[17] |
Runzhang Xu. Preface: Special issue on advances in partial differential equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : i-i. doi: 10.3934/dcdss.2021137 |
[18] |
Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure and Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211 |
[19] |
Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097 |
[20] |
Chaudry Masood Khalique, Muhammad Usman, Maria Luz Gandarais. Nonlinear differential equations: Lie symmetries, conservation laws and other approaches of solving. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : i-ii. doi: 10.3934/dcdss.2020415 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]