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

[1]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[2]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[3]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[4]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[5]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[6]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[7]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[8]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[9]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[10]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[11]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[12]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[13]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[14]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[15]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[16]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[17]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[18]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[19]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[20]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]