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September  2016, 9(3): 515-550. doi: 10.3934/krm.2016005

The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction

Shuangqian Liu 1, and  Qinghua Xiao 2,

1. 

Department of Mathematics, Jinan University, Guangzhou 510632, China
2. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071

Received  October 2015 Revised  December 2015 Published  May 2016

We are concerned with the Cauchy problem of the relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. For perturbative initial data with suitable regularity and integrability, we prove the large time stability of solutions to the relativistic Vlasov-Maxwell-Boltzmann system, and also obtain as a byproduct the convergence rates of solutions. Our proof is based on a new time-velocity weighted energy method and some optimal temporal decay estimates on the solution itself. The results also extend the case of ``hard ball" model considered by Guo and Strain [Comm. Math. Phys. 310: 49--673 (2012)] to the short range interactions.
Keywords: large time stability, optimal temporal decay., time-velocity weighted energy method, Relativistic Vlasov-Maxwell-Boltzmann system, coercivity.
Mathematics Subject Classification: Primary: 35Q20; Secondary: 83A05, 35B4.
Citation: Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic & Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005
References:
[1]

S. Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052. doi: 10.1063/1.1793328.  Google Scholar

[2]

C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8165-4.  Google Scholar

[3]

S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, Relativistic Kinetic Theory. Principles and Applications, North-Holland Publishing Co., Amsterdam-New York, 1980.  Google Scholar

[4]

R. J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 751-778. doi: 10.1016/j.anihpc.2013.07.004.  Google Scholar

[5]

R. J. Duan, Y. J. Lei, T. Yang and H. J. Zhao, The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials,, , ().   Google Scholar

[6]

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

[7]

R. J. Duan, S. Q. Liu, T. Yang and H. J. Zhao, Stability of the nonrelativistic Vlasov- Maxwell-Boltzmann system for angular non-cutoff potentials, Kinetic and Related Models, 6 (2013), 159-204.  Google Scholar

[8]

R. J. Duan, T. Yang and H. J. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386. doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[9]

R. J. Duan, T. Yang and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Mathematical Models and Methods in Applied Sciences, 23 (2013), 979-1028. doi: 10.1142/S0218202513500012.  Google Scholar

[10]

R. J. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb R^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6.  Google Scholar

[11]

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

[12]

M. Dudyński and M. L. Ekiel-J.ezewska, Global existence proof for relativistic Boltzmann equation, J. Statist.Phys., 66 (1992), 991-1001. doi: 10.1007/BF01055712.  Google Scholar

[13]

M. Escobedo, S. Mischler and M. A. Valle, Homogeneous Boltzmann Equation in Quantum Relativistic Kinetic Theory, Electronic Journal of Differential Equations, Monograph, vol. 4, Southwest Texas State University, San Marcos, TX, 2003.  Google Scholar

[14]

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

[15]

R. T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Comm. Math. Phys., 264 (2006), 705-724. doi: 10.1007/s00220-006-1522-y.  Google Scholar

[16]

R. T. Glassey and W. A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68. doi: 10.1080/00411459108204708.  Google Scholar

[17]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347. doi: 10.2977/prims/1195167275.  Google Scholar

[18]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys., 24 (1995), 657-678. doi: 10.1080/00411459508206020.  Google Scholar

[19]

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

[20]

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

[21]

Y. Guo and R. M. 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

[22]

S. Y. Ha, Y. D. Kim, H. Lee and S. E. Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262. doi: 10.4310/MAA.2007.v14.n3.a3.  Google Scholar

[23]

L. Hsiao and H. J. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499. doi: 10.1002/mma.736.  Google Scholar

[24]

L. Hsiao and H. J. Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. doi: 10.1016/j.jde.2005.10.022.  Google Scholar

[25]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16 (2006), 1839-1859. doi: 10.1142/S021820250600173X.  Google Scholar

[26]

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

[27]

Y. J. Lei and H.-J. Zhao, The Vlasov-Maxwell-Boltzmann system with a uniform ionic background near Maxwellians, J. Differential Equations, 260 (2016), 2830-2897. doi: 10.1016/j.jde.2015.10.021.  Google Scholar

[28]

S. Q. Liu and H. J. Zhao, Optimal large-time decay of the relativistic Landau-Maxwell system, J. Differential Equations, 256 (2014), 832-857. doi: 10.1016/j.jde.2013.10.004.  Google Scholar

[29]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.  Google Scholar

[30]

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

[31]

R. M. Strain, Some Applications of an Energy Method in Collisional Kinetic Theory, Phd Thesis, Brown University, Providence RI, 2005. Google Scholar

[32]

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

[33]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.  Google Scholar

[34]

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

[35]

R. M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601. doi: 10.1137/090762695.  Google Scholar

[36]

R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft-potentials, Comm. Math. Phys., 300 (2010), 529-597. doi: 10.1007/s00220-010-1129-1.  Google Scholar

[37]

R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359. doi: 10.3934/krm.2011.4.345.  Google Scholar

[38]

R. M. Strain and K. Y. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb R^3$, Kinetic and Related Models, 5 (2012), 383-415. doi: 10.3934/krm.2012.5.383.  Google Scholar

[39]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

[40]

L. S. Wang, Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions, Acta Mathematica Scientia, 2016 (in press). Google Scholar

[41]

Q. H. Xiao, Large-time behavior of the two-species relativistic Landau-Maxwell system in $\mathbb R^3_x$, J. Differential Equations, 259 (2015), 3520-3558. doi: 10.1016/j.jde.2015.04.031.  Google Scholar

[42]

Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system with angular cutoff for soft potentials, J. Differential Equations, 255 (2013), 1196-1232. doi: 10.1016/j.jde.2013.05.005.  Google Scholar

[43]

Q. H. Xiao, L. 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

[44]

Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials,, , ().   Google Scholar

[45]

T. Yang and H. J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equa- tions in the whole space, J. Differential Equations, 248 (2010), 1518-1560. doi: 10.1016/j.jde.2009.11.027.  Google Scholar

show all references

References:
[1]

S. Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052. doi: 10.1063/1.1793328.  Google Scholar

[2]

C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8165-4.  Google Scholar

[3]

S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, Relativistic Kinetic Theory. Principles and Applications, North-Holland Publishing Co., Amsterdam-New York, 1980.  Google Scholar

[4]

R. J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 751-778. doi: 10.1016/j.anihpc.2013.07.004.  Google Scholar

[5]

R. J. Duan, Y. J. Lei, T. Yang and H. J. Zhao, The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials,, , ().   Google Scholar

[6]

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

[7]

R. J. Duan, S. Q. Liu, T. Yang and H. J. Zhao, Stability of the nonrelativistic Vlasov- Maxwell-Boltzmann system for angular non-cutoff potentials, Kinetic and Related Models, 6 (2013), 159-204.  Google Scholar

[8]

R. J. Duan, T. Yang and H. J. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386. doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[9]

R. J. Duan, T. Yang and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Mathematical Models and Methods in Applied Sciences, 23 (2013), 979-1028. doi: 10.1142/S0218202513500012.  Google Scholar

[10]

R. J. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb R^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6.  Google Scholar

[11]

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

[12]

M. Dudyński and M. L. Ekiel-J.ezewska, Global existence proof for relativistic Boltzmann equation, J. Statist.Phys., 66 (1992), 991-1001. doi: 10.1007/BF01055712.  Google Scholar

[13]

M. Escobedo, S. Mischler and M. A. Valle, Homogeneous Boltzmann Equation in Quantum Relativistic Kinetic Theory, Electronic Journal of Differential Equations, Monograph, vol. 4, Southwest Texas State University, San Marcos, TX, 2003.  Google Scholar

[14]

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

[15]

R. T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Comm. Math. Phys., 264 (2006), 705-724. doi: 10.1007/s00220-006-1522-y.  Google Scholar

[16]

R. T. Glassey and W. A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68. doi: 10.1080/00411459108204708.  Google Scholar

[17]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347. doi: 10.2977/prims/1195167275.  Google Scholar

[18]

R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys., 24 (1995), 657-678. doi: 10.1080/00411459508206020.  Google Scholar

[19]

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

[20]

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

[21]

Y. Guo and R. M. 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

[22]

S. Y. Ha, Y. D. Kim, H. Lee and S. E. Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262. doi: 10.4310/MAA.2007.v14.n3.a3.  Google Scholar

[23]

L. Hsiao and H. J. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499. doi: 10.1002/mma.736.  Google Scholar

[24]

L. Hsiao and H. J. Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. doi: 10.1016/j.jde.2005.10.022.  Google Scholar

[25]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16 (2006), 1839-1859. doi: 10.1142/S021820250600173X.  Google Scholar

[26]

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

[27]

Y. J. Lei and H.-J. Zhao, The Vlasov-Maxwell-Boltzmann system with a uniform ionic background near Maxwellians, J. Differential Equations, 260 (2016), 2830-2897. doi: 10.1016/j.jde.2015.10.021.  Google Scholar

[28]

S. Q. Liu and H. J. Zhao, Optimal large-time decay of the relativistic Landau-Maxwell system, J. Differential Equations, 256 (2014), 832-857. doi: 10.1016/j.jde.2013.10.004.  Google Scholar

[29]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.  Google Scholar

[30]

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

[31]

R. M. Strain, Some Applications of an Energy Method in Collisional Kinetic Theory, Phd Thesis, Brown University, Providence RI, 2005. Google Scholar

[32]

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

[33]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.  Google Scholar

[34]

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

[35]

R. M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601. doi: 10.1137/090762695.  Google Scholar

[36]

R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft-potentials, Comm. Math. Phys., 300 (2010), 529-597. doi: 10.1007/s00220-010-1129-1.  Google Scholar

[37]

R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359. doi: 10.3934/krm.2011.4.345.  Google Scholar

[38]

R. M. Strain and K. Y. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb R^3$, Kinetic and Related Models, 5 (2012), 383-415. doi: 10.3934/krm.2012.5.383.  Google Scholar

[39]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

[40]

L. S. Wang, Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions, Acta Mathematica Scientia, 2016 (in press). Google Scholar

[41]

Q. H. Xiao, Large-time behavior of the two-species relativistic Landau-Maxwell system in $\mathbb R^3_x$, J. Differential Equations, 259 (2015), 3520-3558. doi: 10.1016/j.jde.2015.04.031.  Google Scholar

[42]

Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system with angular cutoff for soft potentials, J. Differential Equations, 255 (2013), 1196-1232. doi: 10.1016/j.jde.2013.05.005.  Google Scholar

[43]

Q. H. Xiao, L. 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

[44]

Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials,, , ().   Google Scholar

[45]

T. Yang and H. J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equa- tions in the whole space, J. Differential Equations, 248 (2010), 1518-1560. doi: 10.1016/j.jde.2009.11.027.  Google Scholar

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