# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [5] M. Bostan, A. 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. [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. [7] M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal., 44 (2012), 1415-1447.  doi: 10.1137/100797400. [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. [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. [10] M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external curved magnetic field, in preparation. [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. [12] N. Crouseilles, M. 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. [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. [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. [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. [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. [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. [18] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.  doi: 10.1137/S0036141099364243. [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. [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. [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. [22] E. Miot, On the gyrokinetic limit for the two-dimensional Vlasov-Poisson system, arXiv: 1603.04502. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [5] M. Bostan, A. 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. [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. [7] M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal., 44 (2012), 1415-1447.  doi: 10.1137/100797400. [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. [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. [10] M. Bostan, Asymptotic behavior for the Vlasov-Poisson equations with strong external curved magnetic field, in preparation. [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. [12] N. Crouseilles, M. 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. [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. [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. [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. [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. [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. [18] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.  doi: 10.1137/S0036141099364243. [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. [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. [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. [22] E. Miot, On the gyrokinetic limit for the two-dimensional Vlasov-Poisson system, arXiv: 1603.04502. [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. [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. [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. [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.
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