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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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, arXiv:1411.5150v1.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z.
[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.
[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.
[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.
[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.
[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.
[26]	N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973.
[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.
[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.
[29]	A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.
[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.
[31]	R. M. Strain, Some Applications of an Energy Method in Collisional Kinetic Theory, Phd Thesis, Brown University, Providence RI, 2005.
[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.
[33]	R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.
[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.
[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.
[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.
[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.
[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.
[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.
[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).
[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.
[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.
[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.
[44]	Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, arXiv:1403.2584v1.
[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.

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.
[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.
[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.
[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.
[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, arXiv:1411.5150v1.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z.
[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.
[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.
[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.
[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.
[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.
[26]	N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973.
[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.
[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.
[29]	A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.
[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.
[31]	R. M. Strain, Some Applications of an Energy Method in Collisional Kinetic Theory, Phd Thesis, Brown University, Providence RI, 2005.
[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.
[33]	R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.
[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.
[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.
[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.
[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.
[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.
[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.
[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).
[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.
[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.
[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.
[44]	Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, arXiv:1403.2584v1.
[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.

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