Large-time decay of the soft potential relativistic
Boltzmann equation in $\mathbb{R}^3_x$

Robert M. Strain; Keya Zhu

doi:10.3934/krm.2012.5.383

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June 2012, 5(2): 383-415. doi: 10.3934/krm.2012.5.383

Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$

Robert M. Strain ^1, and Keya Zhu ^2,

1.	University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104, United States
2.	University of Pennsylvania, Department of Mathematics, David RittenhouseLab, 209 South 33rd Street, Philadelphia, PA 19104-6395, United States

Received December 2011 Revised February 2012 Published April 2012

Abstract
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For the relativistic Boltzmann equation in $\mathbb{R}^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988 [13].

Keywords: relativistic Maxwellian, Relativity, Boltzmann, stability, kinetic theory., whole space, collisional kinetic theory.

Mathematics Subject Classification: Primary: 76P05; Secondary: 83A0.

Citation: Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic & Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383

References:

[1]	Håkan Andréasson, Regularity of the gain term and strong $L^1$ convergence to equilibrium for the relativistic Boltzmann equation, SIAM J. Math. Anal., 27 (1996), 1386-1405. doi: 10.1137/0527076. Google Scholar
[2]	Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. Google Scholar
[3]	Russel E. Caflisch, The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic, Comm. Math. Phys., 74 (1980), 97-109. Google Scholar
[4]	Simone Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052. doi: 10.1063/1.1793328. Google Scholar
[5]	Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002. Google Scholar
[6]	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
[7]	L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9. Google Scholar
[8]	R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), 130 (1989), 321-366. doi: 10.2307/1971423. Google Scholar
[9]	Renjun Duan and Robert M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6. Google Scholar
[10]	Renjun Duan and Robert M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Commun. Pure Appl. Math., 64 (2011), 1497-1546. Google Scholar
[11]	Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics, J. Statist. Phys., 57 (1989), 199-245. doi: 10.1007/BF01023641. Google Scholar
[12]	Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. Google Scholar
[13]	Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Comm. Math. Phys., 115 (1988), 607-629. doi: 10.1007/BF01224130. Google Scholar
[14]	Marek Dudyński and Maria L. Ekiel-Jeżewska, Global existence proof for relativistic Boltzmann equation, J. Statist. Phys., 66 (1992), 991-1001. doi: 10.1007/BF01055712. Google Scholar
[15]	Marek Dudyński and Maria L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation, Phys. Rev. Lett., 55 (1985), 2831-2834. doi: 10.1103/PhysRevLett.55.2831. Google Scholar
[16]	Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation,'' Investigación Oper., 6 (1985), 2228. Google Scholar
[17]	Seung-Yeal Ha, Yong Duck Kim, Ho Lee and Se Eun Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262. Google Scholar
[18]	Seung-Yeal Ha, Ho Lee, Xiongfeng Yang and Seok-Bae Yun, Uniform $L^2$-stability estimates for the relativistic Boltzmann equation, J. Hyperbolic Differ. Equ., 6 (2009), 295-312. Google Scholar
[19]	Ling Hsiao and Hongjun Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499. doi: 10.1002/mma.736. Google Scholar
[20]	Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. Google Scholar
[21]	Robert T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. Google Scholar
[22]	Robert 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
[23]	Robert T. Glassey and Walter 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
[24]	Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347. doi: 10.2977/prims/1195167275. Google Scholar
[25]	Robert T. Glassey and Walter 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
[26]	Philip T. Gressman and Robert M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci. USA, 107 (2010), 5744-5749. doi: 10.1073/pnas.1001185107. Google Scholar
[27]	Philip T. Gressman and Robert 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
[28]	Philip T. Gressman and Robert M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384. doi: 10.1016/j.aim.2011.05.005. Google Scholar
[29]	Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z. Google Scholar
[30]	Yan Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353. doi: 10.1007/s00205-003-0262-9. Google Scholar
[31]	Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y. Google Scholar
[32]	Yan Guo and Robert 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
[33]	Yan Guo and Walter A. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894. doi: 10.1002/cpa.3160480803. Google Scholar
[34]	Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: Global existence, Transport Theory Statist. Phys., 28 (1999), 617-628. doi: 10.1080/00411459908214520. Google Scholar
[35]	Zhenglu Jiang, On the relativistic Boltzmann equation, Acta Math. Sci. (English Ed.), 18 (1998), 348-360. Google Scholar
[36]	Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320. doi: 10.1007/BF03167846. Google Scholar
[37]	P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584. Google Scholar
[38]	Tai-Ping Liu and Shih-Hsien Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math., 60 (2007), 295-356. doi: 10.1002/cpa.20172. Google Scholar
[39]	Tai-Ping Liu and Shih-Hsien Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011. Google Scholar
[40]	Clément Mouhot and Cédric Villani, On Landau damping, Acta Math., 207 (2012), 29-201. Google Scholar
[41]	Jared Speck and Robert M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Comm. Math. Phys., 304 (2011), 229-280. doi: 10.1007/s00220-011-1207-z. Google Scholar
[42]	Robert M. Strain and Yan 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
[43]	Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. Google Scholar
[44]	Robert M. Strain and Yan 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
[45]	Robert 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
[46]	Robert 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
[47]	Robert 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
[48]	Robert M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, preprint, arXiv:1011.5561v2, 2010. Google Scholar
[49]	Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 57-99. doi: 10.2977/prims/1195183569. Google Scholar
[50]	Ivan Vidav, Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., 30 (1970), 264-279. doi: 10.1016/0022-247X(70)90160-5. Google Scholar
[51]	Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302. doi: 10.1007/s002050050054. Google Scholar
[52]	Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. Google Scholar
[53]	Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations, 246 (2009), 3776-3817. Google Scholar

show all references

References:

[1]	Håkan Andréasson, Regularity of the gain term and strong $L^1$ convergence to equilibrium for the relativistic Boltzmann equation, SIAM J. Math. Anal., 27 (1996), 1386-1405. doi: 10.1137/0527076. Google Scholar
[2]	Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. Google Scholar
[3]	Russel E. Caflisch, The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic, Comm. Math. Phys., 74 (1980), 97-109. Google Scholar
[4]	Simone Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052. doi: 10.1063/1.1793328. Google Scholar
[5]	Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002. Google Scholar
[6]	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
[7]	L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9. Google Scholar
[8]	R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), 130 (1989), 321-366. doi: 10.2307/1971423. Google Scholar
[9]	Renjun Duan and Robert M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6. Google Scholar
[10]	Renjun Duan and Robert M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Commun. Pure Appl. Math., 64 (2011), 1497-1546. Google Scholar
[11]	Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics, J. Statist. Phys., 57 (1989), 199-245. doi: 10.1007/BF01023641. Google Scholar
[12]	Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. Google Scholar
[13]	Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Comm. Math. Phys., 115 (1988), 607-629. doi: 10.1007/BF01224130. Google Scholar
[14]	Marek Dudyński and Maria L. Ekiel-Jeżewska, Global existence proof for relativistic Boltzmann equation, J. Statist. Phys., 66 (1992), 991-1001. doi: 10.1007/BF01055712. Google Scholar
[15]	Marek Dudyński and Maria L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation, Phys. Rev. Lett., 55 (1985), 2831-2834. doi: 10.1103/PhysRevLett.55.2831. Google Scholar
[16]	Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation,'' Investigación Oper., 6 (1985), 2228. Google Scholar
[17]	Seung-Yeal Ha, Yong Duck Kim, Ho Lee and Se Eun Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262. Google Scholar
[18]	Seung-Yeal Ha, Ho Lee, Xiongfeng Yang and Seok-Bae Yun, Uniform $L^2$-stability estimates for the relativistic Boltzmann equation, J. Hyperbolic Differ. Equ., 6 (2009), 295-312. Google Scholar
[19]	Ling Hsiao and Hongjun Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499. doi: 10.1002/mma.736. Google Scholar
[20]	Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. Google Scholar
[21]	Robert T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. Google Scholar
[22]	Robert 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
[23]	Robert T. Glassey and Walter 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
[24]	Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347. doi: 10.2977/prims/1195167275. Google Scholar
[25]	Robert T. Glassey and Walter 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
[26]	Philip T. Gressman and Robert M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci. USA, 107 (2010), 5744-5749. doi: 10.1073/pnas.1001185107. Google Scholar
[27]	Philip T. Gressman and Robert 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
[28]	Philip T. Gressman and Robert M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384. doi: 10.1016/j.aim.2011.05.005. Google Scholar
[29]	Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z. Google Scholar
[30]	Yan Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353. doi: 10.1007/s00205-003-0262-9. Google Scholar
[31]	Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y. Google Scholar
[32]	Yan Guo and Robert 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
[33]	Yan Guo and Walter A. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894. doi: 10.1002/cpa.3160480803. Google Scholar
[34]	Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: Global existence, Transport Theory Statist. Phys., 28 (1999), 617-628. doi: 10.1080/00411459908214520. Google Scholar
[35]	Zhenglu Jiang, On the relativistic Boltzmann equation, Acta Math. Sci. (English Ed.), 18 (1998), 348-360. Google Scholar
[36]	Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320. doi: 10.1007/BF03167846. Google Scholar
[37]	P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584. Google Scholar
[38]	Tai-Ping Liu and Shih-Hsien Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math., 60 (2007), 295-356. doi: 10.1002/cpa.20172. Google Scholar
[39]	Tai-Ping Liu and Shih-Hsien Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011. Google Scholar
[40]	Clément Mouhot and Cédric Villani, On Landau damping, Acta Math., 207 (2012), 29-201. Google Scholar
[41]	Jared Speck and Robert M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Comm. Math. Phys., 304 (2011), 229-280. doi: 10.1007/s00220-011-1207-z. Google Scholar
[42]	Robert M. Strain and Yan 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
[43]	Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. Google Scholar
[44]	Robert M. Strain and Yan 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
[45]	Robert 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
[46]	Robert 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
[47]	Robert 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
[48]	Robert M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, preprint, arXiv:1011.5561v2, 2010. Google Scholar
[49]	Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 57-99. doi: 10.2977/prims/1195183569. Google Scholar
[50]	Ivan Vidav, Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., 30 (1970), 264-279. doi: 10.1016/0022-247X(70)90160-5. Google Scholar
[51]	Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302. doi: 10.1007/s002050050054. Google Scholar
[52]	Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. Google Scholar
[53]	Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations, 246 (2009), 3776-3817. Google Scholar

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