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Large deviations for the solution of a Kac-type kinetic equation
Nonlinear stability of a Vlasov equation for magnetic plasmas
1. | UPMC-Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu F75252 Paris cedex 05, France, France, France |
2. | INRIA Rocquencourt, BANG project-team, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France |
References:
[1] |
M. Acheritogaray, P. Degond, A. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kinetic and Related Models, 4 (2011), 901-918.
doi: 10.3934/krm.2011.4.901. |
[2] |
J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics. With Application to Tokamaks", Wiley/Gauthier-Villars Series in Modern Applied Mathematics, John Wiley & sons, Ltd., Chichester, Gauthier-Villars, Montrouge, 1989. |
[3] |
S.-I. Braginskii, Transport processes in a plasma, in "Reviews of Plasma Physics," Consultants Bureau, 1, New York, (1965), 205-311. |
[4] |
H. Brezis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité dans les plasmas, C. R. Acad. Sciences Paris Série I Math., 321 (1995), 953-959. |
[5] |
F. Bouchut, F. Golse and M. Pulvirenti, "Kinetic Equations and Asymptotic Theory," Series in Appl. Math. (Paris), 4, Gauthiers-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000. |
[6] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Math. Sciences, 106, Springer-Verlag, New York, 1994. |
[7] |
F. Chen, "Introduction To Plasma Physics and Controlled Fusion," Springer, New-York, 1984. |
[8] |
P. Crispel, P. Degond and M.-H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, Journal of Computational Physics, 205 (2005), 408-438.
doi: 10.1016/j.jcp.2004.11.011. |
[9] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Sciences and Technology," Springer, 1990. |
[10] |
B. Després and R. Sart, Reduced resistive MHD in Tokamaks with general density, ESAIM Math. Model. Numer. Anal., 46 (2012), 1081-1106.
doi: 10.1051/m2an/2011078. |
[11] |
L. Desvillettes and S. Mischler, About the splitting algorithm for Boltzmann and B.G.K. equations, Mathematical Models and Methods in Applied Sciences, 6 (1996), 1079-1101.
doi: 10.1142/S0218202596000444. |
[12] |
R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.
doi: 10.1002/cpa.3160420603. |
[13] |
R. J. DiPerna, P.-L. Lions and Y. Meyer, Lp regularity of velocity averages, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 8 (1991), 271-287. |
[14] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. |
[15] |
J. Freidberg, "Plasma Physics and Fusion Energy," Cambridge, 2007. |
[16] |
J.-F. Gerbeau, C. Le Bris and T. Lelièvre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals," Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[17] |
P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution, Kinetic and Related Models, 2 (2009), 707-725.
doi: 10.3934/krm.2009.2.707. |
[18] |
F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.
doi: 10.1016/0022-1236(88)90051-1. |
[19] |
D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425.
doi: 10.1080/03605302.2011.555804. |
[20] |
P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson System, Inventiones Math., 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
[21] |
H. Lütjens and J.-F. Luciani, The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas, Journal of Computational Physics, 227 (2008), 6944-6966.
doi: 10.1016/j.jcp.2008.04.003. |
[22] |
H. Lütjens and J.-F. Luciani, XTOR-2F: A fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks, Journal of Computational Physics, 229 (2010), 8130-8143.
doi: 10.1016/j.jcp.2010.07.013. |
[23] |
C. Mouhot and C. Villani, On Landau damping, Acta Mathematica, 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[24] |
B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205-244.
doi: 10.1090/S0273-0979-04-01004-3. |
[25] |
B. Perthame and P.-E. Souganidis, A limiting case for velocity averaging, Annales Scientifiques de l' École Normale Supérieure, 31 (1998), 591-598.
doi: 10.1016/S0012-9593(98)80108-0. |
[26] |
K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.
doi: 10.1016/0022-0396(92)90033-J. |
[27] |
R. Temam, Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations, 2 (1977), 563-585. |
[28] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," American Mathematical Society, 2000. |
[29] |
J. Simon, Compact sets in the space $L^p(0, T ; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
show all references
References:
[1] |
M. Acheritogaray, P. Degond, A. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kinetic and Related Models, 4 (2011), 901-918.
doi: 10.3934/krm.2011.4.901. |
[2] |
J. Blum, "Numerical Simulation and Optimal Control in Plasma Physics. With Application to Tokamaks", Wiley/Gauthier-Villars Series in Modern Applied Mathematics, John Wiley & sons, Ltd., Chichester, Gauthier-Villars, Montrouge, 1989. |
[3] |
S.-I. Braginskii, Transport processes in a plasma, in "Reviews of Plasma Physics," Consultants Bureau, 1, New York, (1965), 205-311. |
[4] |
H. Brezis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité dans les plasmas, C. R. Acad. Sciences Paris Série I Math., 321 (1995), 953-959. |
[5] |
F. Bouchut, F. Golse and M. Pulvirenti, "Kinetic Equations and Asymptotic Theory," Series in Appl. Math. (Paris), 4, Gauthiers-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000. |
[6] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Math. Sciences, 106, Springer-Verlag, New York, 1994. |
[7] |
F. Chen, "Introduction To Plasma Physics and Controlled Fusion," Springer, New-York, 1984. |
[8] |
P. Crispel, P. Degond and M.-H. Vignal, Quasi-neutral fluid models for current-carrying plasmas, Journal of Computational Physics, 205 (2005), 408-438.
doi: 10.1016/j.jcp.2004.11.011. |
[9] |
R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Sciences and Technology," Springer, 1990. |
[10] |
B. Després and R. Sart, Reduced resistive MHD in Tokamaks with general density, ESAIM Math. Model. Numer. Anal., 46 (2012), 1081-1106.
doi: 10.1051/m2an/2011078. |
[11] |
L. Desvillettes and S. Mischler, About the splitting algorithm for Boltzmann and B.G.K. equations, Mathematical Models and Methods in Applied Sciences, 6 (1996), 1079-1101.
doi: 10.1142/S0218202596000444. |
[12] |
R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.
doi: 10.1002/cpa.3160420603. |
[13] |
R. J. DiPerna, P.-L. Lions and Y. Meyer, Lp regularity of velocity averages, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 8 (1991), 271-287. |
[14] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. |
[15] |
J. Freidberg, "Plasma Physics and Fusion Energy," Cambridge, 2007. |
[16] |
J.-F. Gerbeau, C. Le Bris and T. Lelièvre, "Mathematical Methods for the Magnetohydrodynamics of Liquid Metals," Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780198566656.001.0001. |
[17] |
P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution, Kinetic and Related Models, 2 (2009), 707-725.
doi: 10.3934/krm.2009.2.707. |
[18] |
F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.
doi: 10.1016/0022-1236(88)90051-1. |
[19] |
D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425.
doi: 10.1080/03605302.2011.555804. |
[20] |
P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson System, Inventiones Math., 105 (1991), 415-430.
doi: 10.1007/BF01232273. |
[21] |
H. Lütjens and J.-F. Luciani, The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas, Journal of Computational Physics, 227 (2008), 6944-6966.
doi: 10.1016/j.jcp.2008.04.003. |
[22] |
H. Lütjens and J.-F. Luciani, XTOR-2F: A fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks, Journal of Computational Physics, 229 (2010), 8130-8143.
doi: 10.1016/j.jcp.2010.07.013. |
[23] |
C. Mouhot and C. Villani, On Landau damping, Acta Mathematica, 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[24] |
B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205-244.
doi: 10.1090/S0273-0979-04-01004-3. |
[25] |
B. Perthame and P.-E. Souganidis, A limiting case for velocity averaging, Annales Scientifiques de l' École Normale Supérieure, 31 (1998), 591-598.
doi: 10.1016/S0012-9593(98)80108-0. |
[26] |
K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.
doi: 10.1016/0022-0396(92)90033-J. |
[27] |
R. Temam, Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations, 2 (1977), 563-585. |
[28] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," American Mathematical Society, 2000. |
[29] |
J. Simon, Compact sets in the space $L^p(0, T ; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
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