Nonlinear stability of a Vlasov equation for magnetic plasmas

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June 2013, 6(2): 269-290. doi: 10.3934/krm.2013.6.269

Nonlinear stability of a Vlasov equation for magnetic plasmas

Frédérique Charles ^1, , Bruno Després ^1, , Benoît Perthame ^2, and Rémis Sentis ^1,

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

Received April 2012 Revised November 2012 Published February 2013

Abstract
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The mathematical description of laboratory fusion plasmas produced in Tokamaks is still challenging. Complete models for electrons and ions, as Vlasov-Maxwell systems, are computationally too expensive because they take into account all details and scales of magneto-hydrodynamics. In particular, for most of the relevant studies, the mass electron is negligible and the velocity of material waves is much smaller than the speed of light. Therefore it is useful to understand simplified models. Here we propose and study one of those which keeps both the complexity of the Vlasov equation for ions and the Hall effect in Maxwell's equation. Based on energy dissipation, a fundamental physical property, we show that the model is nonlinear stable and consequently prove existence.

Keywords: Maxwell's equations., plasma physics, kinetic averaging lemma, Vlasov equations.

Mathematics Subject Classification: 35B35, 35L60, 82D1.

Citation: 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

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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