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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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