June  2020, 13(3): 531-548. doi: 10.3934/krm.2020018

Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions

Aix Marseille Université, CNRS, Centrale Marseille, I2M, Centre de Mathématiques et Informatique, UMR 7373, 39 rue Frédéric Joliot Curie, 13453 Marseille Cedex 13, France

Received  June 2019 Published  March 2020

We investigate the Vlasov-Poisson equations perturbed by a strong external uniform magnetic field. We study the asymptotic behavior of the solutions, based on averaging techniques. We analyze the case of general initial conditions. By filtering out the oscillations, we are led to a profile. We prove strong convergence results and establish second order estimates.

Citation: Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic & Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018
References:
[1]

A. A. Arsen'ev, Global existence of weak solution of Vlasov's system of equations, Z. Vychisl. Mat. Fiz., 15 (1975), 136-147.   Google Scholar

[2]

J. Batt, Global symmetric solutions of the initial value problem in stellar dynamics, J. Differential Equations, 25 (1977), 342-364.  doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[3]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[4]

M. Bostan, Gyro-kinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957.  doi: 10.1137/090777621.  Google Scholar

[5]

M. BostanA. Finot and M. Hauray, The effective Vlasov-Poisson system for strongly magnetized plasmas, C. R. Acad. Sci. Paris, Ser. I, 354 (2016), 771-777.  doi: 10.1016/j.crma.2016.04.014.  Google Scholar

[6]

M. Bostan and A. Finot, The effective Vlasov-Poisson system for the finite Larmor radius regime, SIAM J. Multiscale Model. Simul., 14 (2016), 1238-1275.  doi: 10.1137/16M1060479.  Google Scholar

[7]

M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal., 44 (2012), 1415-1447.  doi: 10.1137/100797400.  Google Scholar

[8]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188.  doi: 10.1137/15M1033034.  Google Scholar

[9]

M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external magnetic field. Straight magnetic field lines, SIAM J. Math. Anal., 51 (2019), 2713–2747. doi: 10.1137/18M122813X.  Google Scholar

[10]

M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external curved magnetic field, in preparation. Google Scholar

[11]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.  doi: 10.1080/03605300008821529.  Google Scholar

[12]

N. CrouseillesM. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations, J. Comput. Phys., 248 (2013), 287-308.  doi: 10.1016/j.jcp.2013.04.022.  Google Scholar

[13]

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.  Google Scholar

[14]

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

[15]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[16]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213.   Google Scholar

[17]

E. Frénod and E. Sonnendrücker, Long time behavior of the two-dimensional Vlasov-equation with strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.  doi: 10.1142/S021820250000029X.  Google Scholar

[18]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.  doi: 10.1137/S0036141099364243.  Google Scholar

[19]

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.  Google Scholar

[20]

D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma, ath. Model. Numer. Anal., 46 (2012), 929-947.  doi: 10.1051/m2an/2011068.  Google Scholar

[21]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar

[22]

E. Miot, On the gyrokinetic limit for the two-dimensional Vlasov-Poisson system, arXiv: 1603.04502. Google Scholar

[23]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in 3 dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[24]

L. Saint-Raymond, The gyro-kinetic approximation for the Vlasov-Poisson system, Math. Models Methods Appl. Sci., 10 (2000), 1305-1332.  doi: 10.1142/S0218202500000641.  Google Scholar

[25]

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

[26]

T. Ukai and S. Okabe, On the classical solution in the large time of the two dimensional Vlasov equations, Osaka J. Math., 15 (1978), 245-261.   Google Scholar

show all references

References:
[1]

A. A. Arsen'ev, Global existence of weak solution of Vlasov's system of equations, Z. Vychisl. Mat. Fiz., 15 (1975), 136-147.   Google Scholar

[2]

J. Batt, Global symmetric solutions of the initial value problem in stellar dynamics, J. Differential Equations, 25 (1977), 342-364.  doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[3]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[4]

M. Bostan, Gyro-kinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957.  doi: 10.1137/090777621.  Google Scholar

[5]

M. BostanA. Finot and M. Hauray, The effective Vlasov-Poisson system for strongly magnetized plasmas, C. R. Acad. Sci. Paris, Ser. I, 354 (2016), 771-777.  doi: 10.1016/j.crma.2016.04.014.  Google Scholar

[6]

M. Bostan and A. Finot, The effective Vlasov-Poisson system for the finite Larmor radius regime, SIAM J. Multiscale Model. Simul., 14 (2016), 1238-1275.  doi: 10.1137/16M1060479.  Google Scholar

[7]

M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal., 44 (2012), 1415-1447.  doi: 10.1137/100797400.  Google Scholar

[8]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188.  doi: 10.1137/15M1033034.  Google Scholar

[9]

M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external magnetic field. Straight magnetic field lines, SIAM J. Math. Anal., 51 (2019), 2713–2747. doi: 10.1137/18M122813X.  Google Scholar

[10]

M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external curved magnetic field, in preparation. Google Scholar

[11]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.  doi: 10.1080/03605300008821529.  Google Scholar

[12]

N. CrouseillesM. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations, J. Comput. Phys., 248 (2013), 287-308.  doi: 10.1016/j.jcp.2013.04.022.  Google Scholar

[13]

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.  Google Scholar

[14]

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

[15]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[16]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213.   Google Scholar

[17]

E. Frénod and E. Sonnendrücker, Long time behavior of the two-dimensional Vlasov-equation with strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.  doi: 10.1142/S021820250000029X.  Google Scholar

[18]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.  doi: 10.1137/S0036141099364243.  Google Scholar

[19]

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.  Google Scholar

[20]

D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma, ath. Model. Numer. Anal., 46 (2012), 929-947.  doi: 10.1051/m2an/2011068.  Google Scholar

[21]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar

[22]

E. Miot, On the gyrokinetic limit for the two-dimensional Vlasov-Poisson system, arXiv: 1603.04502. Google Scholar

[23]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in 3 dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[24]

L. Saint-Raymond, The gyro-kinetic approximation for the Vlasov-Poisson system, Math. Models Methods Appl. Sci., 10 (2000), 1305-1332.  doi: 10.1142/S0218202500000641.  Google Scholar

[25]

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

[26]

T. Ukai and S. Okabe, On the classical solution in the large time of the two dimensional Vlasov equations, Osaka J. Math., 15 (1978), 245-261.   Google Scholar

[1]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283

[2]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039

[3]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016

[4]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051

[5]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223

[6]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955

[7]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129

[8]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050

[9]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[10]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046

[11]

Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032

[12]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353

[13]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485

[14]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723

[15]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027

[16]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729

[17]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015

[18]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757

[19]

Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058

[20]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (76)
  • HTML views (85)
  • Cited by (0)

Other articles
by authors

[Back to Top]