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

Shuangqian  Liu; Qinghua  Xiao

doi:10.3934/krm.2016005

Article Contents

2016, Volume 9, Issue 3: 515-550. Doi: 10.3934/krm.2016005

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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
2.
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071

Received: September 30, 2015

Revised: November 30, 2015

Published: August 2016

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Abstract

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:

Relativistic Vlasov-Maxwell-Boltzmann system,
large time stability,
coercivity,
time-velocity weighted energy method,
optimal temporal decay.

Mathematics Subject Classification: Primary: 35Q20; Secondary: 83A05, 35B40.

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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.