March  2011, 4(1): 345-359. doi: 10.3934/krm.2011.4.345

Coordinates in the relativistic Boltzmann theory

1. 

University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104, United States

Received  September 2010 Revised  October 2010 Published  January 2011

It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates which may illumniate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator. We show the equivalence between some known representations [27, 15], and others which seem to be new. One of these representations has been used recently to solve several open problems in [42, 41, 30, 39].
Citation: Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345
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. doi: 10.1137/0527076. Google Scholar

[2]

Håkan Andréasson, Simone Calogero and Reinhard Illner, On blowup for gain-term-only classical and relativistic Boltzmann equations,, Math. Methods Appl. Sci., 27 (2004), 2231. doi: 10.1002/mma.555. Google Scholar

[3]

Klaus Bichteler, On the Cauchy problem of the relativistic Boltzmann equation,, Comm. Math. Phys., 4 (1967), 352. doi: 10.1007/BF01653649. Google Scholar

[4]

B. Boisseau and W. A. van Leeuwen, Relativistic Boltzmann theory in $D+1$ spacetime dimensions,, Ann. Physics, 195 (1989), 376. doi: 10.1016/0003-4916(89)90249-2. Google Scholar

[5]

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

[6]

C. Cercignani and G. M. Kremer, On relativistic collisional invariants,, J. Statist. Phys., 96 (1999), 439. doi: 10.1023/A:1004545104959. Google Scholar

[7]

C. Cercignani and G. M.Kremer, Trend to equilibrium of a degenerate relativistic gas,, J. Statist. Phys., 98 (2000), 441. doi: 10.1023/A:1018695426728. Google Scholar

[8]

C. Cercignani and G. M. Kremer, Dispersion and absorption of plane harmonic waves in a relativistic gas,, Contin. Mech. Thermodyn., 13 (2001), 171. Google Scholar

[9]

C. Cercignani and G. M.Kremer, Moment closure of the relativistic Anderson and Witting model equation,, Phys. A, 290 (2001), 192. doi: 10.1016/S0378-4371(00)00403-9. Google Scholar

[10]

Carlo Cercignani, Speed of propagation of infinitesimal disturbances in a relativistic gas,, Phys. Rev. Lett., 50 (1983), 1122. doi: 10.1103/PhysRevLett.50.1122. Google Scholar

[11]

Carlo Cercignani, Propagation phenomena in classical and relativistic rarefied gases,, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, 29 (2000), 607. Google Scholar

[12]

Carlo Cercignani, Reinhard Illner and Mario Pulvirenti, "The mathematical Theory of Dilute Gases,", Applied Mathematical Sciences, 106 (1994). Google Scholar

[13]

Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications,", Progress in Mathematical Physics, 22 (2002). Google Scholar

[14]

Carlo Cercignani and Armando Majorana, Propagation of infinitesimal disturbances in a gas according to a relativistic kinetic model,, Meccanica, 19 (1984), 175. doi: 10.1007/BF01743729. Google Scholar

[15]

S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, "Relativistic Kinetic Theory," Principles and applications,, North-Holland Publishing Co., (1980). Google Scholar

[16]

J. J. Dijkstra and W. A. van Leeuwen, Mathematical aspects of relativistic kinetic theory,, Phys. A, 90 (1978), 450. doi: 10.1016/0378-4371(78)90004-3. Google Scholar

[17]

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. doi: 10.2307/1971423. Google Scholar

[18]

Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics,, J. Statist. Phys., 57 (1989), 199. doi: 10.1007/BF01023641. Google Scholar

[19]

Marek Dudyński and Maria L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation,, Phys. Rev. Lett., 55 (1985), 2831. doi: 10.1103/PhysRevLett.55.2831. Google Scholar

[20]

Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation'',, Investigación Oper., 6 (1985). Google Scholar

[21]

Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions,, Comm. Math. Phys., 115 (1988), 607. Google Scholar

[22]

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

[23]

Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation - mathematical and physical aspects,, J. Tech. Phys., 48 (2007), 39. Google Scholar

[24]

Robert T. Glassey, "The Cauchy Problem in Kinetic Theory,", Society for Industrial and Applied Mathematics (SIAM), (1996). Google Scholar

[25]

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

[26]

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

[27]

Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian,, Publ. Res. Inst. Math. Sci., 29 (1993), 301. Google Scholar

[28]

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

[29]

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

[30]

Yan Guo and Robert M. Strain, Momentum Regularity and Stability of the Relativistic Vlasov-Maxwell-Boltzmann System,, (2010), (2010). Google Scholar

[31]

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. Google Scholar

[32]

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. doi: 10.1142/S0219891609001848. Google Scholar

[33]

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

[34]

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

[35]

Zhenglu Jiang, On the relativistic Boltzmann equation,, Acta Math. Sci. (English Ed.), 18 (1998), 348. Google Scholar

[36]

Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: global existence,, Transport Theory Statist. Phys., 28 (1999), 617. doi: 10.1080/00411459908214520. Google Scholar

[37]

André Lichnerowicz and Raymond Marrot, Propriétés statistiques des ensembles de particules en relativité restreinte,, C. R. Acad. Sci. Paris, 210 (1940), 759. Google Scholar

[38]

P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II,, J. Math. Kyoto Univ., 34 (1994), 391. Google Scholar

[39]

Jared Speck and Robert M. Strain, Hilbert Expansion from the Boltzmann equation to relativistic Fluids,, in press 2010, (2010). Google Scholar

[40]

J. M. Stewart, "Non-equilibrium Relativistic Kinetic Theory,", volume \textbf{10} of Lectures Notes in Physics, 10 (1971). Google Scholar

[41]

Robert Strain, Asymptotic stability of the relativistic Boltzmann equation for the Soft Potentials,, Comm. Math. Phys., 300 (2010), 529. Google Scholar

[42]

Robert M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum,, SIAM J. Math. Anal., 42 (2010), 1568. Google Scholar

[43]

Robert M. Strain, "An Energy Method In Collisional Kinetic Theory,", Ph.D. thesis, (2005). Google Scholar

[44]

Robert M. Strain and Yan Guo, Stability of the relativistic Maxwellian in a collisional plasma,, Comm. Math. Phys., 251 (2004), 263. doi: 10.1007/s00220-004-1151-2. Google Scholar

[45]

J. L. Synge, "The Relativistic Gas,", North-Holland Publishing Company, (1957). Google Scholar

[46]

Stephen Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,", Wiley, (1972). Google Scholar

[47]

Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms,, Arch. Rational Mech. Anal., 139 (1997), 291. doi: 10.1007/s002050050054. Google Scholar

[48]

Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space,, J. Differential Equations, 248 (2010), 1518. doi: 10.1016/j.jde.2009.11.027. Google Scholar

[49]

Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system,, J. Differential Equations, 246 (2009), 3776. doi: 10.1016/j.jde.2009.02.021. 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. doi: 10.1137/0527076. Google Scholar

[2]

Håkan Andréasson, Simone Calogero and Reinhard Illner, On blowup for gain-term-only classical and relativistic Boltzmann equations,, Math. Methods Appl. Sci., 27 (2004), 2231. doi: 10.1002/mma.555. Google Scholar

[3]

Klaus Bichteler, On the Cauchy problem of the relativistic Boltzmann equation,, Comm. Math. Phys., 4 (1967), 352. doi: 10.1007/BF01653649. Google Scholar

[4]

B. Boisseau and W. A. van Leeuwen, Relativistic Boltzmann theory in $D+1$ spacetime dimensions,, Ann. Physics, 195 (1989), 376. doi: 10.1016/0003-4916(89)90249-2. Google Scholar

[5]

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

[6]

C. Cercignani and G. M. Kremer, On relativistic collisional invariants,, J. Statist. Phys., 96 (1999), 439. doi: 10.1023/A:1004545104959. Google Scholar

[7]

C. Cercignani and G. M.Kremer, Trend to equilibrium of a degenerate relativistic gas,, J. Statist. Phys., 98 (2000), 441. doi: 10.1023/A:1018695426728. Google Scholar

[8]

C. Cercignani and G. M. Kremer, Dispersion and absorption of plane harmonic waves in a relativistic gas,, Contin. Mech. Thermodyn., 13 (2001), 171. Google Scholar

[9]

C. Cercignani and G. M.Kremer, Moment closure of the relativistic Anderson and Witting model equation,, Phys. A, 290 (2001), 192. doi: 10.1016/S0378-4371(00)00403-9. Google Scholar

[10]

Carlo Cercignani, Speed of propagation of infinitesimal disturbances in a relativistic gas,, Phys. Rev. Lett., 50 (1983), 1122. doi: 10.1103/PhysRevLett.50.1122. Google Scholar

[11]

Carlo Cercignani, Propagation phenomena in classical and relativistic rarefied gases,, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, 29 (2000), 607. Google Scholar

[12]

Carlo Cercignani, Reinhard Illner and Mario Pulvirenti, "The mathematical Theory of Dilute Gases,", Applied Mathematical Sciences, 106 (1994). Google Scholar

[13]

Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications,", Progress in Mathematical Physics, 22 (2002). Google Scholar

[14]

Carlo Cercignani and Armando Majorana, Propagation of infinitesimal disturbances in a gas according to a relativistic kinetic model,, Meccanica, 19 (1984), 175. doi: 10.1007/BF01743729. Google Scholar

[15]

S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, "Relativistic Kinetic Theory," Principles and applications,, North-Holland Publishing Co., (1980). Google Scholar

[16]

J. J. Dijkstra and W. A. van Leeuwen, Mathematical aspects of relativistic kinetic theory,, Phys. A, 90 (1978), 450. doi: 10.1016/0378-4371(78)90004-3. Google Scholar

[17]

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. doi: 10.2307/1971423. Google Scholar

[18]

Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics,, J. Statist. Phys., 57 (1989), 199. doi: 10.1007/BF01023641. Google Scholar

[19]

Marek Dudyński and Maria L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation,, Phys. Rev. Lett., 55 (1985), 2831. doi: 10.1103/PhysRevLett.55.2831. Google Scholar

[20]

Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation'',, Investigación Oper., 6 (1985). Google Scholar

[21]

Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions,, Comm. Math. Phys., 115 (1988), 607. Google Scholar

[22]

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

[23]

Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation - mathematical and physical aspects,, J. Tech. Phys., 48 (2007), 39. Google Scholar

[24]

Robert T. Glassey, "The Cauchy Problem in Kinetic Theory,", Society for Industrial and Applied Mathematics (SIAM), (1996). Google Scholar

[25]

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

[26]

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

[27]

Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian,, Publ. Res. Inst. Math. Sci., 29 (1993), 301. Google Scholar

[28]

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

[29]

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

[30]

Yan Guo and Robert M. Strain, Momentum Regularity and Stability of the Relativistic Vlasov-Maxwell-Boltzmann System,, (2010), (2010). Google Scholar

[31]

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. Google Scholar

[32]

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. doi: 10.1142/S0219891609001848. Google Scholar

[33]

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

[34]

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

[35]

Zhenglu Jiang, On the relativistic Boltzmann equation,, Acta Math. Sci. (English Ed.), 18 (1998), 348. Google Scholar

[36]

Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: global existence,, Transport Theory Statist. Phys., 28 (1999), 617. doi: 10.1080/00411459908214520. Google Scholar

[37]

André Lichnerowicz and Raymond Marrot, Propriétés statistiques des ensembles de particules en relativité restreinte,, C. R. Acad. Sci. Paris, 210 (1940), 759. Google Scholar

[38]

P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II,, J. Math. Kyoto Univ., 34 (1994), 391. Google Scholar

[39]

Jared Speck and Robert M. Strain, Hilbert Expansion from the Boltzmann equation to relativistic Fluids,, in press 2010, (2010). Google Scholar

[40]

J. M. Stewart, "Non-equilibrium Relativistic Kinetic Theory,", volume \textbf{10} of Lectures Notes in Physics, 10 (1971). Google Scholar

[41]

Robert Strain, Asymptotic stability of the relativistic Boltzmann equation for the Soft Potentials,, Comm. Math. Phys., 300 (2010), 529. Google Scholar

[42]

Robert M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum,, SIAM J. Math. Anal., 42 (2010), 1568. Google Scholar

[43]

Robert M. Strain, "An Energy Method In Collisional Kinetic Theory,", Ph.D. thesis, (2005). Google Scholar

[44]

Robert M. Strain and Yan Guo, Stability of the relativistic Maxwellian in a collisional plasma,, Comm. Math. Phys., 251 (2004), 263. doi: 10.1007/s00220-004-1151-2. Google Scholar

[45]

J. L. Synge, "The Relativistic Gas,", North-Holland Publishing Company, (1957). Google Scholar

[46]

Stephen Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,", Wiley, (1972). Google Scholar

[47]

Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms,, Arch. Rational Mech. Anal., 139 (1997), 291. doi: 10.1007/s002050050054. Google Scholar

[48]

Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space,, J. Differential Equations, 248 (2010), 1518. doi: 10.1016/j.jde.2009.11.027. Google Scholar

[49]

Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system,, J. Differential Equations, 246 (2009), 3776. doi: 10.1016/j.jde.2009.02.021. Google Scholar

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