Time-asymptotic stability for first-order symmetric hyperbolic systems of balance laws in dissipative compressible fluid dynamics
Author:
Heinrich Freistühler
Journal:
Quart. Appl. Math. 80 (2022), 597-606
MSC (2020):
Primary 76N06, 76A02, 76A05, 76E30; Secondary 35L02, 35L03
DOI:
https://doi.org/10.1090/qam/1620
Published electronically:
March 15, 2022
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper identifies a non-(or /iso-)thermal variant of Ruggeri’s 1983 formulation of viscous heat-conductive fluid dynamics as a hyperbolic system of balance laws and shows that both the original model and this variant have (a) time-asymptotically stable equilibria and (b) principal parts deriving from a protopotential: a single scalar function that induces the temporospatial flux as an appropriate part of its Hessian.
References
- Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. First-order systems and applications. MR 2284507
- Stefano Bianchini, Bernard Hanouzet, and Roberto Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (2007), no. 11, 1559–1622. MR 2349349, DOI 10.1002/cpa.20195
- 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
- Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898
- 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
- H. Freistühler, A Galilei invariant version of Yong’s model, arXiv:2012.09059, 2020.
- 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
- K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392. MR 62932, DOI 10.1002/cpa.3160070206
- Robert Geroch and Lee Lindblom, Dissipative relativistic fluid theories of divergence type, Phys. Rev. D (3) 41 (1990), no. 6, 1855–1861. MR 1048881, DOI 10.1103/PhysRevD.41.1855
- S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). MR 0131653
- Yuxi Hu and Na Wang, Global existence versus blow-up results for one dimensional compressible Navier-Stokes equations with Maxwell’s law, Math. Nachr. 292 (2019), no. 4, 826–840. MR 3937620, DOI 10.1002/mana.201700418
- Shuichi Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 169–194. MR 899951, DOI 10.1017/S0308210500018308
- Tong Li and Hailiang Liu, Critical thresholds in hyperbolic relaxation systems, J. Differential Equations 247 (2009), no. 1, 33–48. MR 2510127, DOI 10.1016/j.jde.2009.03.032
- J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. A157 (1867), 49–88.
- I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Zeitschrift für Physik 198 (1967), 329–344.
- Ingo Müller and Tommaso Ruggeri, Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, vol. 37, Springer-Verlag, New York, 1998. With supplementary chapters by H. Struchtrup and Wolf Weiss. MR 1632151, DOI 10.1007/978-1-4612-2210-1
- T. Ruggeri, Entropy principle and relativistic extended thermodynamics: global existence of smooth solutions and stability of equilibrium state, Nuovo Cimento Soc. Ital. Fis. B 119 (2004), no. 7-9, 809–821. MR 2136908
- T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat conducting fluid, Acta Mech. 47 (1983), no. 3-4, 167–183. MR 709990, DOI 10.1007/BF01189206
- Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. MR 798756, DOI 10.14492/hokmj/1381757663
- A. I. Vol’pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sb. 16 (1972), 517–544.
- Wen-An Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266. MR 2058165, DOI 10.1007/s00205-003-0304-3
- Wen-An Yong, Newtonian limit of Maxwell fluid flows, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 913–922. MR 3269638, DOI 10.1007/s00205-014-0769-2
References
- Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. First-order systems and applications. MR 2284507
- Stefano Bianchini, Bernard Hanouzet, and Roberto Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (2007), no. 11, 1559–1622. MR 2349349, DOI 10.1002/cpa.20195
- 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
- Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898
- 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
- H. Freistühler, A Galilei invariant version of Yong’s model, arXiv:2012.09059, 2020.
- 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
- K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392. MR 62932, DOI 10.1002/cpa.3160070206
- Robert Geroch and Lee Lindblom, Dissipative relativistic fluid theories of divergence type, Phys. Rev. D (3) 41 (1990), no. 6, 1855–1861. MR 1048881, DOI 10.1103/PhysRevD.41.1855
- S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). MR 0131653
- Yuxi Hu and Na Wang, Global existence versus blow-up results for one dimensional compressible Navier-Stokes equations with Maxwell’s law, Math. Nachr. 292 (2019), no. 4, 826–840. MR 3937620, DOI 10.1002/mana.201700418
- Shuichi Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 169–194. MR 899951, DOI 10.1017/S0308210500018308
- Tong Li and Hailiang Liu, Critical thresholds in hyperbolic relaxation systems, J. Differential Equations 247 (2009), no. 1, 33–48. MR 2510127, DOI 10.1016/j.jde.2009.03.032
- J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc. A157 (1867), 49–88.
- I. Müller, Zum Paradoxon der Wärmeleitungstheorie, Zeitschrift für Physik 198 (1967), 329–344.
- Ingo Müller and Tommaso Ruggeri, Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, vol. 37, Springer-Verlag, New York, 1998. With supplementary chapters by H. Struchtrup and Wolf Weiss. MR 1632151, DOI 10.1007/978-1-4612-2210-1
- T. Ruggeri, Entropy principle and relativistic extended thermodynamics: global existence of smooth solutions and stability of equilibrium state, Nuovo Cimento Soc. Ital. Fis. B 119 (2004), no. 7-9, 809–821. MR 2136908
- T. Ruggeri, Symmetric-hyperbolic system of conservative equations for a viscous heat conducting fluid, Acta Mech. 47 (1983), no. 3-4, 167–183. MR 709990, DOI 10.1007/BF01189206
- Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. MR 798756, DOI 10.14492/hokmj/1381757663
- A. I. Vol’pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sb. 16 (1972), 517–544.
- Wen-An Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266. MR 2058165, DOI 10.1007/s00205-003-0304-3
- Wen-An Yong, Newtonian limit of Maxwell fluid flows, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 913–922. MR 3269638, DOI 10.1007/s00205-014-0769-2
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
76N06,
76A02,
76A05,
76E30,
35L02,
35L03
Retrieve articles in all journals
with MSC (2020):
76N06,
76A02,
76A05,
76E30,
35L02,
35L03
Additional Information
Heinrich Freistühler
Affiliation:
Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany
Email:
heinrich.freistuehler@uni-konstanz.de
Received by editor(s):
January 30, 2022
Published electronically:
March 15, 2022
Article copyright:
© Copyright 2022
Brown University
