A causal formulation of dissipative relativistic fluid dynamics with or without diffusion
Author:
Heinrich Freistühler
Journal:
Quart. Appl. Math. 81 (2023), 507-515
MSC (2020):
Primary 35L65, 35Q75, 76N30, 76N17
DOI:
https://doi.org/10.1090/qam/1656
Published electronically:
February 10, 2023
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Additional Information
Abstract: The article proposes a causal five-field formulation of dissipative relativistic fluid dynamics as a quasilinear symmetric hyperbolic system of second order. The system is determined by four dissipation coefficients $\eta ,\zeta ,\kappa ,\mu$, free functions of the fields, which quantify shear viscosity, bulk viscosity, heat conductivity, and diffusion.
References
- Fábio S. Bemfica, Marcelo M. Disconzi, and Jorge Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Phys. Rev. D 98 (2018), no. 10, 104064, 26. MR 3954719, DOI 10.1103/physrevd.98.104064
- Guy Boillat, Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques, C. R. Acad. Sci. Paris Sér. A 278 (1974), 909–912 (French). MR 342870
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 4th ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2016. MR 3468916, DOI 10.1007/978-3-662-49451-6
- C. Eckart, The thermodynamics of irreversible processes. 3: Relativistic theory of the simple fluid, Phys. Rev. 58 (1940), 919–924.
- H. Freistühler and M. Sroczinski, Global existence and deacy of small solutions for quasi-linear uniformly dissipative mixed-order hyperbolic-hyperbolic systems, preprint, 2022.
- Heinrich Freistühler and Blake Temple, Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2166, 20140055, 17. MR 3190223, DOI 10.1098/rspa.2014.0055
- Heinrich Freistühler and Blake Temple, Causal dissipation for the relativistic dynamics of ideal gases, Proc. A. 473 (2017), no. 2201, 20160729, 20. MR 3668121, DOI 10.1098/rspa.2016.0729
- Heinrich Freistühler and Blake Temple, Causal dissipation in the relativistic dynamics of barotropic fluids, J. Math. Phys. 59 (2018), no. 6, 063101, 17. MR 3816451, DOI 10.1063/1.5007831
- S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). MR 131653
- Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273–294 (1977). MR 420024, DOI 10.1007/BF00251584
- G. A. Kluitenberg, S. R. de Groot, and P. Mazur, Relativistic thermodynamics of irreversible processes. II. Heat conduction and diffusion; physical part, Physica 19 (1953), 1079–1094. MR 63825, DOI 10.1016/S0031-8914(53)80122-9
- L. D. Landau and E. M. Lifshitz, Fluid mechanics, Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, MA, 1959. Translated from the Russian by J. B. Sykes and W. H. Reid. MR 108121
- Tommaso Ruggeri and Alberto Strumia, Main field and convex covariant density for quasilinear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincaré Sect. A (N.S.) 34 (1981), no. 1, 65–84. MR 605357
- Tommaso Ruggeri and Masaru Sugiyama, Classical and relativistic rational extended thermodynamics of gases, Springer, Cham, [2021] ©2021. MR 4260900, DOI 10.1007/978-3-030-59144-1
- Matthias Sroczinski, Asymptotic stability of homogeneous states in the relativistic dynamics of viscous, heat-conductive fluids, Arch. Ration. Mech. Anal. 231 (2019), no. 1, 91–113. MR 3894547, DOI 10.1007/s00205-018-1274-9
- Matthias Sroczinski, Asymptotic stability in a second-order symmetric hyperbolic system modeling the relativistic dynamics of viscous heat-conductive fluids with diffusion, J. Differential Equations 268 (2020), no. 2, 825–851. MR 4021905, DOI 10.1016/j.jde.2019.08.028
- M. Sroczinski, Global existence and decay of small solutions for quasi-linear uniformly dissipative second-order hyperbolic-hyperbolic systems, preprint, 2022.
- S. Weinberg, Gravitation and cosmology: Principles and applications of the general theory of relativity, John Wiley & Sons, New York, 1972.
References
- Fábio S. Bemfica, Marcelo M. Disconzi, and Jorge Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Phys. Rev. D 98 (2018), no. 10, 104064, 26. MR 3954719, DOI 10.1103/physrevd.98.104064
- Guy Boillat, Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques, C. R. Acad. Sci. Paris Sér. A 278 (1974), 909–912 (French). MR 342870
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 4th ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2016. MR 3468916, DOI 10.1007/978-3-662-49451-6
- C. Eckart, The thermodynamics of irreversible processes. 3: Relativistic theory of the simple fluid, Phys. Rev. 58 (1940), 919–924.
- H. Freistühler and M. Sroczinski, Global existence and deacy of small solutions for quasi-linear uniformly dissipative mixed-order hyperbolic-hyperbolic systems, preprint, 2022.
- Heinrich Freistühler and Blake Temple, Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2166, 20140055, 17. MR 3190223, DOI 10.1098/rspa.2014.0055
- Heinrich Freistühler and Blake Temple, Causal dissipation for the relativistic dynamics of ideal gases, Proc. A. 473 (2017), no. 2201, 20160729, 20. MR 3668121, DOI 10.1098/rspa.2016.0729
- Heinrich Freistühler and Blake Temple, Causal dissipation in the relativistic dynamics of barotropic fluids, J. Math. Phys. 59 (2018), no. 6, 063101, 17. MR 3816451, DOI 10.1063/1.5007831
- S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). MR 0131653
- Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273–294 (1977). MR 420024, DOI 10.1007/BF00251584
- G. A. Kluitenberg, S. R. de Groot, and P. Mazur, Relativistic thermodynamics of irreversible processes. II. Heat conduction and diffusion; physical part, Physica 19 (1953), 1079–1094. MR 63825
- L. D. Landau and E. M. Lifshitz, Fluid mechanics, Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1959. Translated from the Russian by J. B. Sykes and W. H. Reid. MR 0108121
- Tommaso Ruggeri and Alberto Strumia, Main field and convex covariant density for quasilinear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincaré Sect. A (N.S.) 34 (1981), no. 1, 65–84. MR 605357
- Tommaso Ruggeri and Masaru Sugiyama, Classical and relativistic rational extended thermodynamics of gases, Springer, Cham, [2021] ©2021. MR 4260900, DOI 10.1007/978-3-030-59144-1
- Matthias Sroczinski, Asymptotic stability of homogeneous states in the relativistic dynamics of viscous, heat-conductive fluids, Arch. Ration. Mech. Anal. 231 (2019), no. 1, 91–113. MR 3894547, DOI 10.1007/s00205-018-1274-9
- Matthias Sroczinski, Asymptotic stability in a second-order symmetric hyperbolic system modeling the relativistic dynamics of viscous heat-conductive fluids with diffusion, J. Differential Equations 268 (2020), no. 2, 825–851. MR 4021905, DOI 10.1016/j.jde.2019.08.028
- M. Sroczinski, Global existence and decay of small solutions for quasi-linear uniformly dissipative second-order hyperbolic-hyperbolic systems, preprint, 2022.
- S. Weinberg, Gravitation and cosmology: Principles and applications of the general theory of relativity, John Wiley & Sons, New York, 1972.
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Additional Information
Heinrich Freistühler
Affiliation:
Department of Mathematics, University of Konstanz, 78457 Konstanz, Germany
ORCID:
0000-0002-0741-886X
Received by editor(s):
November 2, 2022
Received by editor(s) in revised form:
January 16, 2023
Published electronically:
February 10, 2023
Dedicated:
This paper is dedicated to Constantine Dafermos on the occasion of his eigthieth birthday.
Article copyright:
© Copyright 2023
Brown University
