June  2020, 13(3): 599-621. doi: 10.3934/krm.2020020

The Vlasov-Maxwell-Boltzmann system near Maxwellians with strong background magnetic field

1. 

School of Mathematics and Statistics and, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

2. 

School of Mathematics and Statistics, and Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China

* Corresponding author: Yan Guo

Received  July 2019 Revised  November 2019 Published  March 2020

In this paper, we are concerned with the construction of global-in-time solutions of the Cauchy problem of the Vlasov-Maxwell-Boltzmann system near Maxwellians with strong uniform background magnetic field. The background magnetic field under our consideration can be any given non-zero constant vector rather than vacuum in the previous results available up to now. Our analysis is motivated by the nonlinear energy method developed recently in [16,24,25] for the Boltzmann equation and the key point in our analysis is to deduce the dissipation estimates of the electronic field and strong background magnetic field.

Citation: Yuanjie Lei, Huijiang Zhao. The Vlasov-Maxwell-Boltzmann system near Maxwellians with strong background magnetic field. Kinetic and Related Models, 2020, 13 (3) : 599-621. doi: 10.3934/krm.2020020
References:
[1]

R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New York-London, 1975.

[2]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Ⅰ, Global existence for soft potential,, J. Funct. Anal., 262 (2012), 915-1010.  doi: 10.1016/j.jfa.2011.10.007.

[3]

K. Asano and S. Ukai, On the Vlasov-Poisson limit of the Vlasov-Maxwell equation. Patterns and waves., Stud. Math. Appl., 18 (1986), North-Holland, Amsterdam, 369–383. doi: 10.1016/S0168-2024(08)70137-1.

[4]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases. An account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Third edition, prepared in co-operation with D. Burnett. Cambridge University Press, London, 1970.

[5]

R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions,, Ann. Inst. H. Poincar Anal. Non Linsaire, 31 (2014), 751-778.  doi: 10.1016/j.anihpc.2013.07.004.

[6]

R.-J. DuanY.-J. LeiT. Yang and H.-J. Zhao, The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials,, Comm. Math. Phys., 351 (2017), 95-153.  doi: 10.1007/s00220-017-2844-7.

[7]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff,, Comm. Math. Phys., 324 (2013), 1-45.  doi: 10.1007/s00220-013-1807-x.

[8]

R.-J. DuanS.-Q. LiuT. Yang and H.-J. Zhao, Stabilty of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials,, Kinetic and Related Models, 6 (2013), 159-204.  doi: 10.3934/krm.2013.6.159.

[9]

R.-J. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Pure. Appl. Math., 64 (2011), 1497-1546.  doi: 10.1002/cpa.20381.

[10]

R.-J. DuanT. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials,, Math. Methods Models Appl. Sci., 23 (2013), 979-1028.  doi: 10.1142/S0218202513500012.

[11]

Y.-Z. FanY.-J. LeiS.-Q. Liu and H.-J. Zhao, The non-cutoff Vlasov- Maxwell-Boltzmann system with weak angular singularity,, Science China Mathematics, 61 (2018), 111-136.  doi: 10.1007/s11425-016-9083-x.

[12]

P. Ghendrih and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions, Preprint, 2018.

[13]

H. Grad, Asymptotic theory of the Boltzmann equation Ⅱ, Rarefied Gas Dynamics (Laurmann, J.A. Ed.), Academic Press, New York, 1 (1963), 26–59.

[14]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off,, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8.

[15]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians,, Invent. Math., 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.

[16]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081-1094.  doi: 10.1512/iumj.2004.53.2574.

[17]

Y. Guo, The Vlasov-Poisson-Laudau system in a periodic box,, J. Amer. Math. Soc., 25 (2012), 759-812.  doi: 10.1090/S0894-0347-2011-00722-4.

[18]

Y. Guo and R. Strain, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system,, Comm. Math. Phys., 310 (2012), 649-673.  doi: 10.1007/s00220-012-1417-z.

[19]

Y. Guo and Y.-J. Wang, Decay of dissipative equation and negative sobolev spaces,, Comm.Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[20]

J. Jang, Vlasov-Maxwell-Boltzmann diffusive limit,, Arch. Ration. Mech. Anal., 194 (2009), 531-584.  doi: 10.1007/s00205-008-0169-6.

[21]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973.

[22]

Y.-J. Lei and H.-J. Zhao, Negative Sobolev spaces and the two-species Vlasov- Maxwell-Landau system in the whole space,, J. Funct. Anal., 267 (2014), 3710-3757.  doi: 10.1016/j.jfa.2014.09.011.

[23]

Y.-J. LeiL.-J. Xiong and H.-J. Zhao, One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space,, Kinet. Relat. Models, 7 (2014), 551-590.  doi: 10.3934/krm.2014.7.551.

[24]

T.-P. LiuT. Yang and S.-H. Yu, Energy method for the Boltzmann equation,, Physica D, 188 (2004), 178-192.  doi: 10.1016/j.physd.2003.07.011.

[25]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles,, Commun. Math. Phys., 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.

[26]

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

[27]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Math. Phys., 268 (2006), 543-567.  doi: 10.1007/s00220-006-0109-y.

[28]

R. M. Strain and Y. Guo, Stability of the relativistic Maxwellian in a collisional plasma,, Comm. Math. Phys., 251 (2004), 263-320.  doi: 10.1007/s00220-004-1151-2.

[29]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287-339.  doi: 10.1007/s00205-007-0067-3.

[30]

R. M. Strain and K.-Y. Zhu, The Vlasov-Poisson-Landau system in ${\mathbb{R}}^3_x$,, Arch. Ration. Mech. Anal., 210 (2013), 615-671.  doi: 10.1007/s00205-013-0658-0.

[31]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305. 

[32]

Y.-J. Wang, Golobal solution and time decay of the Vlasov-Poisson-Landau System in ${\mathbb{R}}^3_x$,, SIAM J. Math. Anal., 44 (2012), 3281-3323.  doi: 10.1137/120879129.

[33]

L.-S. WangQ.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions,, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1049-1097.  doi: 10.1016/S0252-9602(16)30057-1.

[34]

Q.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Posson-Boltzmann system with angular cutoff for soft potential,, J. Differential Equations, 255 (2013), 1196-1232.  doi: 10.1016/j.jde.2013.05.005.

[35]

Q.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials,, Sci. China Math., 57 (2014), 515-540.  doi: 10.1007/s11425-013-4712-z.

[36]

Q.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Posson-Boltzmann system for the whole range of cutoff soft potentials,, J. Funct. Anal., 272 (2017), 166-226.  doi: 10.1016/j.jfa.2016.09.017.

show all references

References:
[1]

R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New York-London, 1975.

[2]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Ⅰ, Global existence for soft potential,, J. Funct. Anal., 262 (2012), 915-1010.  doi: 10.1016/j.jfa.2011.10.007.

[3]

K. Asano and S. Ukai, On the Vlasov-Poisson limit of the Vlasov-Maxwell equation. Patterns and waves., Stud. Math. Appl., 18 (1986), North-Holland, Amsterdam, 369–383. doi: 10.1016/S0168-2024(08)70137-1.

[4]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases. An account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Third edition, prepared in co-operation with D. Burnett. Cambridge University Press, London, 1970.

[5]

R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions,, Ann. Inst. H. Poincar Anal. Non Linsaire, 31 (2014), 751-778.  doi: 10.1016/j.anihpc.2013.07.004.

[6]

R.-J. DuanY.-J. LeiT. Yang and H.-J. Zhao, The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials,, Comm. Math. Phys., 351 (2017), 95-153.  doi: 10.1007/s00220-017-2844-7.

[7]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff,, Comm. Math. Phys., 324 (2013), 1-45.  doi: 10.1007/s00220-013-1807-x.

[8]

R.-J. DuanS.-Q. LiuT. Yang and H.-J. Zhao, Stabilty of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials,, Kinetic and Related Models, 6 (2013), 159-204.  doi: 10.3934/krm.2013.6.159.

[9]

R.-J. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Pure. Appl. Math., 64 (2011), 1497-1546.  doi: 10.1002/cpa.20381.

[10]

R.-J. DuanT. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials,, Math. Methods Models Appl. Sci., 23 (2013), 979-1028.  doi: 10.1142/S0218202513500012.

[11]

Y.-Z. FanY.-J. LeiS.-Q. Liu and H.-J. Zhao, The non-cutoff Vlasov- Maxwell-Boltzmann system with weak angular singularity,, Science China Mathematics, 61 (2018), 111-136.  doi: 10.1007/s11425-016-9083-x.

[12]

P. Ghendrih and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions, Preprint, 2018.

[13]

H. Grad, Asymptotic theory of the Boltzmann equation Ⅱ, Rarefied Gas Dynamics (Laurmann, J.A. Ed.), Academic Press, New York, 1 (1963), 26–59.

[14]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off,, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8.

[15]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians,, Invent. Math., 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.

[16]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081-1094.  doi: 10.1512/iumj.2004.53.2574.

[17]

Y. Guo, The Vlasov-Poisson-Laudau system in a periodic box,, J. Amer. Math. Soc., 25 (2012), 759-812.  doi: 10.1090/S0894-0347-2011-00722-4.

[18]

Y. Guo and R. Strain, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system,, Comm. Math. Phys., 310 (2012), 649-673.  doi: 10.1007/s00220-012-1417-z.

[19]

Y. Guo and Y.-J. Wang, Decay of dissipative equation and negative sobolev spaces,, Comm.Partial Differential Equations, 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[20]

J. Jang, Vlasov-Maxwell-Boltzmann diffusive limit,, Arch. Ration. Mech. Anal., 194 (2009), 531-584.  doi: 10.1007/s00205-008-0169-6.

[21]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973.

[22]

Y.-J. Lei and H.-J. Zhao, Negative Sobolev spaces and the two-species Vlasov- Maxwell-Landau system in the whole space,, J. Funct. Anal., 267 (2014), 3710-3757.  doi: 10.1016/j.jfa.2014.09.011.

[23]

Y.-J. LeiL.-J. Xiong and H.-J. Zhao, One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space,, Kinet. Relat. Models, 7 (2014), 551-590.  doi: 10.3934/krm.2014.7.551.

[24]

T.-P. LiuT. Yang and S.-H. Yu, Energy method for the Boltzmann equation,, Physica D, 188 (2004), 178-192.  doi: 10.1016/j.physd.2003.07.011.

[25]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles,, Commun. Math. Phys., 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.

[26]

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

[27]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Math. Phys., 268 (2006), 543-567.  doi: 10.1007/s00220-006-0109-y.

[28]

R. M. Strain and Y. Guo, Stability of the relativistic Maxwellian in a collisional plasma,, Comm. Math. Phys., 251 (2004), 263-320.  doi: 10.1007/s00220-004-1151-2.

[29]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287-339.  doi: 10.1007/s00205-007-0067-3.

[30]

R. M. Strain and K.-Y. Zhu, The Vlasov-Poisson-Landau system in ${\mathbb{R}}^3_x$,, Arch. Ration. Mech. Anal., 210 (2013), 615-671.  doi: 10.1007/s00205-013-0658-0.

[31]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305. 

[32]

Y.-J. Wang, Golobal solution and time decay of the Vlasov-Poisson-Landau System in ${\mathbb{R}}^3_x$,, SIAM J. Math. Anal., 44 (2012), 3281-3323.  doi: 10.1137/120879129.

[33]

L.-S. WangQ.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions,, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1049-1097.  doi: 10.1016/S0252-9602(16)30057-1.

[34]

Q.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Posson-Boltzmann system with angular cutoff for soft potential,, J. Differential Equations, 255 (2013), 1196-1232.  doi: 10.1016/j.jde.2013.05.005.

[35]

Q.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials,, Sci. China Math., 57 (2014), 515-540.  doi: 10.1007/s11425-013-4712-z.

[36]

Q.-H. XiaoL.-J. Xiong and H.-J. Zhao, The Vlasov-Posson-Boltzmann system for the whole range of cutoff soft potentials,, J. Funct. Anal., 272 (2017), 166-226.  doi: 10.1016/j.jfa.2016.09.017.

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