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June 2019, 12(3): 593-636. doi: 10.3934/krm.2019024

Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations

1. 

Inria, Univ. Lille, CNRS, UMR 8524 Laboratoire Paul Painlevé, F-59000 Lille, France

2. 

IRMAR, UMR CNRS 6625, Université de Rennes 1, 263 avenue du General Leclerc, 35042 Rennes Cedex, France

Received  May 2018 Revised  December 2018 Published  February 2019

Fund Project: Research of L. Miguel Rodrigues was partially supported by the ANR project BoND ANR-13-BS01-0009-01

We consider various sets of Vlasov-Fokker-Planck equations modeling the dynamics of charged particles in a plasma under the effect of a strong magnetic field. For each of them in a regime where the strength of the magnetic field is effectively stronger than that of collisions we first formally derive asymptotically reduced models. In this regime, strong anisotropic phenomena occur; while equilibrium along magnetic field lines is asymptotically reached our asymptotic models capture a non trivial dynamics in the perpendicular directions. We do check that in any case the obtained asymptotic model defines a well-posed dynamical system and when self consistent electric fields are neglected we provide a rigorous mathematical justification of the formally derived systems. In this last step we provide a complete control on solutions by developing anisotropic hypocoercive estimates.

Citation: Maxime Herda, Luis Miguel Rodrigues. Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations. Kinetic & Related Models, 2019, 12 (3) : 593-636. doi: 10.3934/krm.2019024
References:
[1]

M. Badsi and M. Herda, Modelling and simulating a multispecies plasma, In CEMRACS 2014-Numerical Modeling of Plasmas, volume 53 of ESAIM Proc. Surveys, EDP Sci., Les Ulis, 53 (2016), 22–37. doi: 10.1051/proc/201653002.

[2]

C. BardosF. GolseT. T. Nguyen and R. Sentis, The Maxwell-Boltzmann approximation for ion kinetic modeling, Phys. D, 376/377 (2018), 94-107. doi: 10.1016/j.physd.2017.10.014.

[3] P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, 2006.
[4]

M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external magnetic field. straight magnetic field lines, HAL preprint, hal-01683869, 2018. doi: 10.1137/070689383.

[5]

F. Bouchut, Global weak solution of the Vlasov-Poisson system for small electrons mass, Comm. Partial Differential Equations, 16 (1991), 1337-1365. doi: 10.1080/03605309108820802.

[6]

F. Bouchut and J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials, Differential Integral Equations, 8 (1995), 487-514.

[7]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, volume 4 of Series in Applied Mathematics (Paris), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000. Edited and with a foreword by Benoît Perthame and Laurent Desvillettes.

[8]

L. CaffarelliJ. DolbeaultP. A. Markowich and C. Schmeiser, On Maxwellian equilibria of insulated semiconductors, Interfaces Free Bound., 2 (2000), 331-339. doi: 10.4171/IFB/23.

[9]

A. de CeccoF. DeluzetC. Negulescu and S. Possanner, Asymptotic transition from kinetic to adiabatic electrons along magnetic field lines, Multiscale Model. Simul., 15 (2017), 309-338. doi: 10.1137/15M1043686.

[10]

J.-Yves Chemin, Perfect Incompressible Fluids, volume 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[11]

C. Cheverry, Can one hear whistler waves?, Comm. Math. Phys., 338 (2015), 641-703. doi: 10.1007/s00220-015-2389-6.

[12]

C. Cheverry, Anomalous transport, J. Differential Equations, 262 (2017), 2987-3033. doi: 10.1016/j.jde.2016.11.012.

[13]

S. De MoorL. Miguel Rodrigues and J. Vovelle, Invariant measures for a stochastic Fokker-Planck equation, Kinet. Relat. Models, 11 (2018), 357-395. doi: 10.3934/krm.2018017.

[14]

P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity, Math. Methods Appl. Sci., 8 (1986), 533-558. doi: 10.1002/mma.1670080135.

[15]

P. Degond, Asymptotic continuum models for plasmas and disparate mass gaseous binary mixtures, In Material Substructures in Complex Bodies, 2007, 1–62. doi: 10.1016/B978-008044535-9/50002-9.

[16]

P. Degond and F. Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field: formal derivation, J. Stat. Phys., 165 (2016), 765-784. doi: 10.1007/s10955-016-1645-2.

[17]

P. Degond and B. Lucquin-Desreux, The asymptotics of collision operators for two species of particles of disparate masses, Math. Models Methods Appl. Sci., 6 (1996), 405-436. doi: 10.1142/S0218202596000158.

[18]

P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory Statist. Phys., 25 (1996), 595-633. doi: 10.1080/00411459608222915.

[19]

J. Dolbeault, Stationary states in plasma physics: Maxwellian solutions of the Vlasov-Poisson system, Math. Models Methods Appl. Sci., 1 (1991), 183-208. doi: 10.1142/S0218202591000113.

[20]

J. Dolbeault, Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (large time behavior and steady states), J. Math. Pures Appl. (9), 78 (1999), 121–157. doi: 10.1016/S0021-7824(01)80006-4.

[21]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828. doi: 10.1090/S0002-9947-2015-06012-7.

[22]

N. El Ghani and N. Masmoudi, Diffusion limit of the Vlasov-Poisson-Fokker-Planck system, Commun. Math. Sci., 8 (2010), 463-479. doi: 10.4310/CMS.2010.v8.n2.a9.

[23]

F. Filbet and L. M. Rodrigues, Asymptotically preserving particle-in-cell methods for inhomogeneous strongly magnetized plasmas, SIAM J. Numer. Anal., 55 (2017), 2416-2443. doi: 10.1137/17M1113229.

[24]

F. Filbet and L. M. Rodrigues, Asymptotics of the three dimensional Vlasov equation in the large magnetic field limit, arXiv preprint, arXiv: 1811.09087, 2018.

[25]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the VlasovPoisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213.

[26]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[27]

R. J Goldston and P. H. Rutherford, Introduction to Plasma Physics, CRC Press, 1995.

[28]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791–817. doi: 10.1016/S0021-7824(99)00021-5.

[29]

L. Grafakos, Classical Fourier Analysis, volume 249 of Graduate Texts in Mathematics, Springer, New York, third edition, 2014. doi: 10.1007/978-1-4939-1194-3.

[30]

B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, volume 1862 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2005. doi: 10.1007/b104762.

[31]

F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 244 (2007), 95-118. doi: 10.1016/j.jfa.2006.11.013.

[32]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218. doi: 10.1007/s00205-003-0276-3.

[33]

M. Herda, On massless electron limit for a multispecies kinetic system with external magnetic field, J. Differential Equations, 260 (2016), 7861-7891. doi: 10.1016/j.jde.2016.02.005.

[34]

M. Herda and L. Miguel Rodrigues, Large-time behavior of solutions to Vlasov-PoissonFokker-Planck equations: From evanescent collisions to diffusive limit, J. Stat. Phys., 170 (2018), 895-931. doi: 10.1007/s10955-018-1963-7.

[35]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, volume 431 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.

[36]

A. J. Majda and A. L. Bertozzi. Vorticity and Incompressible Flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.

[37]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, volume 96 of Applied Mathematical Sciences., Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[38]

É. Miot, On the gyrokinetic limit for the two-dimensional Vlasov-Poisson system, arXiv preprint, arXiv: 1603.04502, 2016.

[39]

K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, volume 38., Springer, 2006.

[40]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.

[41]

C. Negulescu and S. Possanner, Closure of the strongly magnetized electron fluid equations in the adiabatic regime, Multiscale Model. Simul., 14 (2016), 839-873. doi: 10.1137/15M1027309.

[42]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[43]

S. Possanner, Gyrokinetics from variational averaging: Existence and error bounds, J. Math. Phys., 59 (2018), 082702, 34pp. doi: 10.1063/1.5018354.

[44]

F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 10 (2000), 1027-1045. doi: 10.1142/S0218202500000525.

[45]

G. Rein, Collisionless kinetic equations from astrophysics—the Vlasov-Poisson system, In Handbook of Differential Equations: Evolutionary Equations. Vol. III, Handb. Differ. Equ., pages 383–476. Elsevier/North-Holland, Amsterdam, 2007. doi: 10.1016/S1874-5717(07)80008-9.

[46]

L. Saint-Raymond, Control of large velocities in the two-dimensional gyrokinetic approximation, J. Math. Pures Appl. (9), 81 (2002), 379–399. doi: 10.1016/S0021-7824(01)01245-4.

[47]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.

[48]

E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32.

[49]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp. doi: 10.1090/S0065-9266-09-00567-5.

show all references

References:
[1]

M. Badsi and M. Herda, Modelling and simulating a multispecies plasma, In CEMRACS 2014-Numerical Modeling of Plasmas, volume 53 of ESAIM Proc. Surveys, EDP Sci., Les Ulis, 53 (2016), 22–37. doi: 10.1051/proc/201653002.

[2]

C. BardosF. GolseT. T. Nguyen and R. Sentis, The Maxwell-Boltzmann approximation for ion kinetic modeling, Phys. D, 376/377 (2018), 94-107. doi: 10.1016/j.physd.2017.10.014.

[3] P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, 2006.
[4]

M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external magnetic field. straight magnetic field lines, HAL preprint, hal-01683869, 2018. doi: 10.1137/070689383.

[5]

F. Bouchut, Global weak solution of the Vlasov-Poisson system for small electrons mass, Comm. Partial Differential Equations, 16 (1991), 1337-1365. doi: 10.1080/03605309108820802.

[6]

F. Bouchut and J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials, Differential Integral Equations, 8 (1995), 487-514.

[7]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, volume 4 of Series in Applied Mathematics (Paris), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000. Edited and with a foreword by Benoît Perthame and Laurent Desvillettes.

[8]

L. CaffarelliJ. DolbeaultP. A. Markowich and C. Schmeiser, On Maxwellian equilibria of insulated semiconductors, Interfaces Free Bound., 2 (2000), 331-339. doi: 10.4171/IFB/23.

[9]

A. de CeccoF. DeluzetC. Negulescu and S. Possanner, Asymptotic transition from kinetic to adiabatic electrons along magnetic field lines, Multiscale Model. Simul., 15 (2017), 309-338. doi: 10.1137/15M1043686.

[10]

J.-Yves Chemin, Perfect Incompressible Fluids, volume 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[11]

C. Cheverry, Can one hear whistler waves?, Comm. Math. Phys., 338 (2015), 641-703. doi: 10.1007/s00220-015-2389-6.

[12]

C. Cheverry, Anomalous transport, J. Differential Equations, 262 (2017), 2987-3033. doi: 10.1016/j.jde.2016.11.012.

[13]

S. De MoorL. Miguel Rodrigues and J. Vovelle, Invariant measures for a stochastic Fokker-Planck equation, Kinet. Relat. Models, 11 (2018), 357-395. doi: 10.3934/krm.2018017.

[14]

P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity, Math. Methods Appl. Sci., 8 (1986), 533-558. doi: 10.1002/mma.1670080135.

[15]

P. Degond, Asymptotic continuum models for plasmas and disparate mass gaseous binary mixtures, In Material Substructures in Complex Bodies, 2007, 1–62. doi: 10.1016/B978-008044535-9/50002-9.

[16]

P. Degond and F. Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field: formal derivation, J. Stat. Phys., 165 (2016), 765-784. doi: 10.1007/s10955-016-1645-2.

[17]

P. Degond and B. Lucquin-Desreux, The asymptotics of collision operators for two species of particles of disparate masses, Math. Models Methods Appl. Sci., 6 (1996), 405-436. doi: 10.1142/S0218202596000158.

[18]

P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory Statist. Phys., 25 (1996), 595-633. doi: 10.1080/00411459608222915.

[19]

J. Dolbeault, Stationary states in plasma physics: Maxwellian solutions of the Vlasov-Poisson system, Math. Models Methods Appl. Sci., 1 (1991), 183-208. doi: 10.1142/S0218202591000113.

[20]

J. Dolbeault, Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (large time behavior and steady states), J. Math. Pures Appl. (9), 78 (1999), 121–157. doi: 10.1016/S0021-7824(01)80006-4.

[21]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828. doi: 10.1090/S0002-9947-2015-06012-7.

[22]

N. El Ghani and N. Masmoudi, Diffusion limit of the Vlasov-Poisson-Fokker-Planck system, Commun. Math. Sci., 8 (2010), 463-479. doi: 10.4310/CMS.2010.v8.n2.a9.

[23]

F. Filbet and L. M. Rodrigues, Asymptotically preserving particle-in-cell methods for inhomogeneous strongly magnetized plasmas, SIAM J. Numer. Anal., 55 (2017), 2416-2443. doi: 10.1137/17M1113229.

[24]

F. Filbet and L. M. Rodrigues, Asymptotics of the three dimensional Vlasov equation in the large magnetic field limit, arXiv preprint, arXiv: 1811.09087, 2018.

[25]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the VlasovPoisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213.

[26]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[27]

R. J Goldston and P. H. Rutherford, Introduction to Plasma Physics, CRC Press, 1995.

[28]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791–817. doi: 10.1016/S0021-7824(99)00021-5.

[29]

L. Grafakos, Classical Fourier Analysis, volume 249 of Graduate Texts in Mathematics, Springer, New York, third edition, 2014. doi: 10.1007/978-1-4939-1194-3.

[30]

B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, volume 1862 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2005. doi: 10.1007/b104762.

[31]

F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 244 (2007), 95-118. doi: 10.1016/j.jfa.2006.11.013.

[32]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218. doi: 10.1007/s00205-003-0276-3.

[33]

M. Herda, On massless electron limit for a multispecies kinetic system with external magnetic field, J. Differential Equations, 260 (2016), 7861-7891. doi: 10.1016/j.jde.2016.02.005.

[34]

M. Herda and L. Miguel Rodrigues, Large-time behavior of solutions to Vlasov-PoissonFokker-Planck equations: From evanescent collisions to diffusive limit, J. Stat. Phys., 170 (2018), 895-931. doi: 10.1007/s10955-018-1963-7.

[35]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, volume 431 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.

[36]

A. J. Majda and A. L. Bertozzi. Vorticity and Incompressible Flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.

[37]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, volume 96 of Applied Mathematical Sciences., Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[38]

É. Miot, On the gyrokinetic limit for the two-dimensional Vlasov-Poisson system, arXiv preprint, arXiv: 1603.04502, 2016.

[39]

K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion, volume 38., Springer, 2006.

[40]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.

[41]

C. Negulescu and S. Possanner, Closure of the strongly magnetized electron fluid equations in the adiabatic regime, Multiscale Model. Simul., 14 (2016), 839-873. doi: 10.1137/15M1027309.

[42]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[43]

S. Possanner, Gyrokinetics from variational averaging: Existence and error bounds, J. Math. Phys., 59 (2018), 082702, 34pp. doi: 10.1063/1.5018354.

[44]

F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 10 (2000), 1027-1045. doi: 10.1142/S0218202500000525.

[45]

G. Rein, Collisionless kinetic equations from astrophysics—the Vlasov-Poisson system, In Handbook of Differential Equations: Evolutionary Equations. Vol. III, Handb. Differ. Equ., pages 383–476. Elsevier/North-Holland, Amsterdam, 2007. doi: 10.1016/S1874-5717(07)80008-9.

[46]

L. Saint-Raymond, Control of large velocities in the two-dimensional gyrokinetic approximation, J. Math. Pures Appl. (9), 81 (2002), 379–399. doi: 10.1016/S0021-7824(01)01245-4.

[47]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.

[48]

E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32.

[49]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp. doi: 10.1090/S0065-9266-09-00567-5.

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