# American Institute of Mathematical Sciences

May  2013, 12(3): 1341-1347. doi: 10.3934/cpaa.2013.12.1341

## On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state

 1 Universitat Pompeu Fabra, Dept. de Tecnologies de la Informació i les Comunicacions, C/Tànger 122-140, 08018 Barcelona, Spain

Received  April 2012 Revised  May 2012 Published  September 2012

We show that a pair of conjectures raised in [11] concerning the construction of normal solutions to the relativistic Boltzmann equation are valid. This ensures that the results in [11] hold for any range of positive temperatures and that the relativistic Euler system under the kinetic equation of state is hyperbolic and the speed of sound cannot overcome $c/\sqrt{3}$.
Citation: Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341
##### References:
 [1] M. Abramovitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," New York: Dover Publications, 1972. [2] J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica, 74 (1974), 466-488. doi: 10.1016/0031-8914(74)90355-3. [3] A. Bellouquid, J. Calvo, J. Nieto and J. Soler, On the relativistic BGK-Boltzmann model: asymptotics and hydrodynamics, to appear in Journal of Statistical Physics. [4] C. Cercignani and G. Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Birkhäuser, Berlin, 2003. doi: 10.1007/978-3-0348-8165-4_2. [5] S. R. De Groot, W. A. van Leuwen and Ch. G. van Weert, "Relativistic Kinetic Theory. Principles and Applications," North Holland, Amsterdam, 1980. [6] M. Kunik, S. Qamar and G. Warnecke, Kinetic schemes for the relativistic gas dynamics, Numer. Math., 97 (2004), 159-191. doi: 10.1007/s00211-003-0510-9. [7] A. Majorana, Relativistic relaxation models for a simple gas, J. Math. Phys., 31 (1990), 2042-2046. doi: 10.1063/1.528655. [8] C. Marle, Sur l'établissement des équations de l'hydrodynamique des fluides relativistes dissipatifs.I.- L'equation de Boltzmann relativiste, (French), Ann. Inst. H. Poincaré, 10 (1969), 67-127. [9] A.D. Rendall, Asymptotics of solutions of the Einstein equations with positive cosmological constant, Ann. Henri Poincaré, 5 (2004), 1041-1064. doi: 10.1007/s00023-004-0189-1. [10] J. Speck, The stabilizing effect of spacetime expansion on relativistic fluids with sharp results for the radiation equation of state, preprint: arXiv:1201.1963. [11] J. Speck and R. 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. [12] J. L. Synge, "The Relativistic Gas," North Holland, Amsterdam, 1957.

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##### References:
 [1] M. Abramovitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," New York: Dover Publications, 1972. [2] J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica, 74 (1974), 466-488. doi: 10.1016/0031-8914(74)90355-3. [3] A. Bellouquid, J. Calvo, J. Nieto and J. Soler, On the relativistic BGK-Boltzmann model: asymptotics and hydrodynamics, to appear in Journal of Statistical Physics. [4] C. Cercignani and G. Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Birkhäuser, Berlin, 2003. doi: 10.1007/978-3-0348-8165-4_2. [5] S. R. De Groot, W. A. van Leuwen and Ch. G. van Weert, "Relativistic Kinetic Theory. Principles and Applications," North Holland, Amsterdam, 1980. [6] M. Kunik, S. Qamar and G. Warnecke, Kinetic schemes for the relativistic gas dynamics, Numer. Math., 97 (2004), 159-191. doi: 10.1007/s00211-003-0510-9. [7] A. Majorana, Relativistic relaxation models for a simple gas, J. Math. Phys., 31 (1990), 2042-2046. doi: 10.1063/1.528655. [8] C. Marle, Sur l'établissement des équations de l'hydrodynamique des fluides relativistes dissipatifs.I.- L'equation de Boltzmann relativiste, (French), Ann. Inst. H. Poincaré, 10 (1969), 67-127. [9] A.D. Rendall, Asymptotics of solutions of the Einstein equations with positive cosmological constant, Ann. Henri Poincaré, 5 (2004), 1041-1064. doi: 10.1007/s00023-004-0189-1. [10] J. Speck, The stabilizing effect of spacetime expansion on relativistic fluids with sharp results for the radiation equation of state, preprint: arXiv:1201.1963. [11] J. Speck and R. 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. [12] J. L. Synge, "The Relativistic Gas," North Holland, Amsterdam, 1957.
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