# American Institute of Mathematical Sciences

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 and 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), 1386-1405. doi: 10.1137/0527076. [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), 2231-2240. doi: 10.1002/mma.555. [3] Klaus Bichteler, On the Cauchy problem of the relativistic Boltzmann equation, Comm. Math. Phys., 4 (1967), 352-364. doi: 10.1007/BF01653649. [4] B. Boisseau and W. A. van Leeuwen, Relativistic Boltzmann theory in $D+1$ spacetime dimensions, Ann. Physics, 195 (1989), 376-419. doi: 10.1016/0003-4916(89)90249-2. [5] Simone Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052. doi: 10.1063/1.1793328. [6] C. Cercignani and G. M. Kremer, On relativistic collisional invariants, J. Statist. Phys., 96 (1999), 439-445. doi: 10.1023/A:1004545104959. [7] C. Cercignani and G. M.Kremer, Trend to equilibrium of a degenerate relativistic gas, J. Statist. Phys., 98 (2000), 441-456. doi: 10.1023/A:1018695426728. [8] C. Cercignani and G. M. Kremer, Dispersion and absorption of plane harmonic waves in a relativistic gas, Contin. Mech. Thermodyn., 13 (2001), 171-182. [9] C. Cercignani and G. M.Kremer, Moment closure of the relativistic Anderson and Witting model equation, Phys. A, 290 (2001), 192-202. doi: 10.1016/S0378-4371(00)00403-9. [10] Carlo Cercignani, Speed of propagation of infinitesimal disturbances in a relativistic gas, Phys. Rev. Lett., 50 (1983), 1122-1124. doi: 10.1103/PhysRevLett.50.1122. [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, HI, 1998), 29 (2000), 607-614. [12] Carlo Cercignani, Reinhard Illner and Mario Pulvirenti, "The mathematical Theory of Dilute Gases," Applied Mathematical Sciences, Springer-Verlag, New York, volume 106, 1994. [13] Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Progress in Mathematical Physics, Birkhäuser Verlag, Basel, 22, 2002. [14] Carlo Cercignani and Armando Majorana, Propagation of infinitesimal disturbances in a gas according to a relativistic kinetic model, Meccanica, 19 (1984), 175-181. doi: 10.1007/BF01743729. [15] S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, "Relativistic Kinetic Theory," Principles and applications, North-Holland Publishing Co., Amsterdam, 1980. [16] J. J. Dijkstra and W. A. van Leeuwen, Mathematical aspects of relativistic kinetic theory, Phys. A, 90 (1978), 450-486. doi: 10.1016/0378-4371(78)90004-3. [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), 321-366. doi: 10.2307/1971423. [18] Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics, J. Statist. Phys., 57 (1989), 199-245. doi: 10.1007/BF01023641. [19] 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. [20] Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation'', Investigación Oper., 6 (1985), 2228. [21] Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Comm. Math. Phys., 115 (1988), 607-629. [22] 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. [23] Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation - mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. [24] Robert T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996, [25] Robert T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Comm. Math. Phys., 264 (2006), 705-724. doi: 10.1007/s00220-006-1522-y. [26] Robert T. Glassey and Walter A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68. doi: 10.1080/00411459108204708. [27] Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347. [28] 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. [29] Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z. [30] Yan Guo and Robert M. Strain, Momentum Regularity and Stability of the Relativistic Vlasov-Maxwell-Boltzmann System, (2010), preprint, arXiv:1012.1158v1. [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), 251-262. [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), 295-312. doi: 10.1142/S0219891609001848. [33] Ling Hsiao and Hongjun Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499. doi: 10.1002/mma.736. [34] Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. doi: 10.1016/j.jde.2005.10.022. [35] Zhenglu Jiang, On the relativistic Boltzmann equation, Acta Math. Sci. (English Ed.), 18 (1998), 348-360. [36] Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: global existence, Transport Theory Statist. Phys., 28 (1999), 617-628. doi: 10.1080/00411459908214520. [37] André Lichnerowicz and Raymond Marrot, Propriétés statistiques des ensembles de particules en relativité restreinte, C. R. Acad. Sci. Paris, 210 (1940), 759-761. [38] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461. [39] Jared Speck and Robert M. Strain, Hilbert Expansion from the Boltzmann equation to relativistic Fluids, in press 2010, Comm. Math. Phys., 49pp, arXiv:1009.5033v1. [40] J. M. Stewart, "Non-equilibrium Relativistic Kinetic Theory," volume 10 of Lectures Notes in Physics, Springer-Verlag, Berlin, 1971. [41] Robert Strain, Asymptotic stability of the relativistic Boltzmann equation for the Soft Potentials, Comm. Math. Phys., 300 (2010), 529-597. [42] Robert M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601, arXiv:1004.5407v1. [43] Robert M. Strain, "An Energy Method In Collisional Kinetic Theory," Ph.D. thesis, Division of Applied Mathematics at Brown University in Providence, RI, May 2005. [44] 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. [45] J. L. Synge, "The Relativistic Gas," North-Holland Publishing Company, Amsterdam, 1957, [46] Stephen Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity," Wiley, New York, 1972. [47] Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302. doi: 10.1007/s002050050054. [48] Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. doi: 10.1016/j.jde.2009.11.027. [49] Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations, 246 (2009), 3776-3817. doi: 10.1016/j.jde.2009.02.021.

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##### 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. [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), 2231-2240. doi: 10.1002/mma.555. [3] Klaus Bichteler, On the Cauchy problem of the relativistic Boltzmann equation, Comm. Math. Phys., 4 (1967), 352-364. doi: 10.1007/BF01653649. [4] B. Boisseau and W. A. van Leeuwen, Relativistic Boltzmann theory in $D+1$ spacetime dimensions, Ann. Physics, 195 (1989), 376-419. doi: 10.1016/0003-4916(89)90249-2. [5] Simone Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052. doi: 10.1063/1.1793328. [6] C. Cercignani and G. M. Kremer, On relativistic collisional invariants, J. Statist. Phys., 96 (1999), 439-445. doi: 10.1023/A:1004545104959. [7] C. Cercignani and G. M.Kremer, Trend to equilibrium of a degenerate relativistic gas, J. Statist. Phys., 98 (2000), 441-456. doi: 10.1023/A:1018695426728. [8] C. Cercignani and G. M. Kremer, Dispersion and absorption of plane harmonic waves in a relativistic gas, Contin. Mech. Thermodyn., 13 (2001), 171-182. [9] C. Cercignani and G. M.Kremer, Moment closure of the relativistic Anderson and Witting model equation, Phys. A, 290 (2001), 192-202. doi: 10.1016/S0378-4371(00)00403-9. [10] Carlo Cercignani, Speed of propagation of infinitesimal disturbances in a relativistic gas, Phys. Rev. Lett., 50 (1983), 1122-1124. doi: 10.1103/PhysRevLett.50.1122. [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, HI, 1998), 29 (2000), 607-614. [12] Carlo Cercignani, Reinhard Illner and Mario Pulvirenti, "The mathematical Theory of Dilute Gases," Applied Mathematical Sciences, Springer-Verlag, New York, volume 106, 1994. [13] Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Progress in Mathematical Physics, Birkhäuser Verlag, Basel, 22, 2002. [14] Carlo Cercignani and Armando Majorana, Propagation of infinitesimal disturbances in a gas according to a relativistic kinetic model, Meccanica, 19 (1984), 175-181. doi: 10.1007/BF01743729. [15] S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, "Relativistic Kinetic Theory," Principles and applications, North-Holland Publishing Co., Amsterdam, 1980. [16] J. J. Dijkstra and W. A. van Leeuwen, Mathematical aspects of relativistic kinetic theory, Phys. A, 90 (1978), 450-486. doi: 10.1016/0378-4371(78)90004-3. [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), 321-366. doi: 10.2307/1971423. [18] Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics, J. Statist. Phys., 57 (1989), 199-245. doi: 10.1007/BF01023641. [19] 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. [20] Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation'', Investigación Oper., 6 (1985), 2228. [21] Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Comm. Math. Phys., 115 (1988), 607-629. [22] 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. [23] Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation - mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. [24] Robert T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996, [25] Robert T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Comm. Math. Phys., 264 (2006), 705-724. doi: 10.1007/s00220-006-1522-y. [26] Robert T. Glassey and Walter A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68. doi: 10.1080/00411459108204708. [27] Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347. [28] 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. [29] Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z. [30] Yan Guo and Robert M. Strain, Momentum Regularity and Stability of the Relativistic Vlasov-Maxwell-Boltzmann System, (2010), preprint, arXiv:1012.1158v1. [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), 251-262. [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), 295-312. doi: 10.1142/S0219891609001848. [33] Ling Hsiao and Hongjun Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499. doi: 10.1002/mma.736. [34] Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. doi: 10.1016/j.jde.2005.10.022. [35] Zhenglu Jiang, On the relativistic Boltzmann equation, Acta Math. Sci. (English Ed.), 18 (1998), 348-360. [36] Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: global existence, Transport Theory Statist. Phys., 28 (1999), 617-628. doi: 10.1080/00411459908214520. [37] André Lichnerowicz and Raymond Marrot, Propriétés statistiques des ensembles de particules en relativité restreinte, C. R. Acad. Sci. Paris, 210 (1940), 759-761. [38] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461. [39] Jared Speck and Robert M. Strain, Hilbert Expansion from the Boltzmann equation to relativistic Fluids, in press 2010, Comm. Math. Phys., 49pp, arXiv:1009.5033v1. [40] J. M. Stewart, "Non-equilibrium Relativistic Kinetic Theory," volume 10 of Lectures Notes in Physics, Springer-Verlag, Berlin, 1971. [41] Robert Strain, Asymptotic stability of the relativistic Boltzmann equation for the Soft Potentials, Comm. Math. Phys., 300 (2010), 529-597. [42] Robert M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601, arXiv:1004.5407v1. [43] Robert M. Strain, "An Energy Method In Collisional Kinetic Theory," Ph.D. thesis, Division of Applied Mathematics at Brown University in Providence, RI, May 2005. [44] 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. [45] J. L. Synge, "The Relativistic Gas," North-Holland Publishing Company, Amsterdam, 1957, [46] Stephen Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity," Wiley, New York, 1972. [47] Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302. doi: 10.1007/s002050050054. [48] Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. doi: 10.1016/j.jde.2009.11.027. [49] Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations, 246 (2009), 3776-3817. doi: 10.1016/j.jde.2009.02.021.
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