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 & 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). Google Scholar

[2]

J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation,, Physica, 74 (1974), 466. doi: 10.1016/0031-8914(74)90355-3. Google Scholar

[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., (). Google Scholar

[4]

C. Cercignani and G. Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications,", Birkh\, (2003). doi: 10.1007/978-3-0348-8165-4_2. Google Scholar

[5]

S. R. De Groot, W. A. van Leuwen and Ch. G. van Weert, "Relativistic Kinetic Theory. Principles and Applications,", North Holland, (1980). Google Scholar

[6]

M. Kunik, S. Qamar and G. Warnecke, Kinetic schemes for the relativistic gas dynamics,, Numer. Math., 97 (2004), 159. doi: 10.1007/s00211-003-0510-9. Google Scholar

[7]

A. Majorana, Relativistic relaxation models for a simple gas,, J. Math. Phys., 31 (1990), 2042. doi: 10.1063/1.528655. Google Scholar

[8]

C. Marle, Sur l'établissement des équations de l'hydrodynamique des fluides relativistes dissipatifs.I.- L'equation de Boltzmann relativiste,, (French), 10 (1969), 67. Google Scholar

[9]

A.D. Rendall, Asymptotics of solutions of the Einstein equations with positive cosmological constant,, Ann. Henri Poincar\'e, 5 (2004), 1041. doi: 10.1007/s00023-004-0189-1. Google Scholar

[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}., (). Google Scholar

[11]

J. Speck and R. M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids,, Comm. Math. Phys., 304 (2011), 229. doi: 10.1007/s00220-011-1207-z. Google Scholar

[12]

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

show all references

References:
[1]

M. Abramovitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", New York: Dover Publications, (1972). Google Scholar

[2]

J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation,, Physica, 74 (1974), 466. doi: 10.1016/0031-8914(74)90355-3. Google Scholar

[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., (). Google Scholar

[4]

C. Cercignani and G. Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications,", Birkh\, (2003). doi: 10.1007/978-3-0348-8165-4_2. Google Scholar

[5]

S. R. De Groot, W. A. van Leuwen and Ch. G. van Weert, "Relativistic Kinetic Theory. Principles and Applications,", North Holland, (1980). Google Scholar

[6]

M. Kunik, S. Qamar and G. Warnecke, Kinetic schemes for the relativistic gas dynamics,, Numer. Math., 97 (2004), 159. doi: 10.1007/s00211-003-0510-9. Google Scholar

[7]

A. Majorana, Relativistic relaxation models for a simple gas,, J. Math. Phys., 31 (1990), 2042. doi: 10.1063/1.528655. Google Scholar

[8]

C. Marle, Sur l'établissement des équations de l'hydrodynamique des fluides relativistes dissipatifs.I.- L'equation de Boltzmann relativiste,, (French), 10 (1969), 67. Google Scholar

[9]

A.D. Rendall, Asymptotics of solutions of the Einstein equations with positive cosmological constant,, Ann. Henri Poincar\'e, 5 (2004), 1041. doi: 10.1007/s00023-004-0189-1. Google Scholar

[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}., (). Google Scholar

[11]

J. Speck and R. M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids,, Comm. Math. Phys., 304 (2011), 229. doi: 10.1007/s00220-011-1207-z. Google Scholar

[12]

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

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