Abstract
We are concerned in this paper with the non-relativistic global limits of the entropy solutions to the Cauchy problem of 3 × 3 system of relativistic Euler equations modeling the conservation of baryon numbers, momentum, and energy respectively. Based on the detailed geometric properties of nonlinear wave curves in the phase space and the Glimm’s method, we obtain, for the isothermal flow, the convergence of the entropy solutions to the solutions of the corresponding classical non-relativistic Euler equations as the speed of light c → +∞.
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Anile A.M.: Relativistic fluids and Magneto-Fluids, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1989)
Bianchini S., Colombo R.M.: On the stability of the standard Riemann semigroup. Proc. Am. Math. Soc. 130, 1961–1973 (2002)
Chen G.-Q., Christoforou C., Zhang Y.: Dependence of entropy solutions in the large for the Euler equations on nonlinear flux functions. Indiana Univ. Math. J. 56, 2535–2568 (2007)
Chen G.-Q., Li Y.C.: Relativistic Euler equations for isentropic fluids: stability of Riemann solutions with large oscillation. J. Differ. Equ. 202, 332–353 (2004)
Chen G.-Q., Li Y.C.: Relativistic Euler equations for isentropic fluids: stability of Riemann solutions with large oscillation. Z. Angew. Math. Phys. 55, 903–926 (2004)
Chen J.: Conservation laws for relativistic fluid dynamics. Arch. Ration. Mech. Anal. 139, 377–398 (1997)
Chen J.: Conservation laws for relativistic p-system. Commun. PDE 20, 1605–1646 (1995)
Frid H., Perepelitsa M.: Spatially periodic solutions in relativistic isentropic gas dynamics. Commun. Math. Phys. 250, 335–370 (2004)
Glimm J.: Solutions in the large for non-linear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 95–105 (1965)
Li Y.C., Feng D., Wang Z.: Global entropy solutions to the relativistic Euler equations for a class of large initial data. Z. Angew. Math. Phys. 56, 239–253 (2005)
Li Y.C., Geng Y.C.: Non-relativistic global limits of Entropy solutions to the isentropic relativistic Euler equations. Z. Angew. Math. Phys. 57, 960–983 (2006)
Li Y.C., Ren X.: Non-relativistic global limits of entropy solutions to the relativistic euler equations with γ-law. Commun. Pure Appl. Anal. 5, 963–979 (2006)
Li Y.C., Shi Q.: Global existence of the entropy solutions to the isentropic relativistic Euler equations. Commun. Pure Appl. Anal. 4, 763–778 (2005)
Li Y.C., Wang A.: Global entropy solutions of the cauchy problem for the nonhomogeneous relativistic Euler equations. Chin. Ann. Math. 27B:5, 473–494 (2006)
Li Y.C., Wang L.: Global stability of solutions with discontinuous initial containing vaccum states for the relativistic Euler equations. Chin. Ann. Math. 26B:4, 491–510 (2005)
Liang E.P.T.: Relativistic simple waves: Shock damping and entropy production. Astrophys. J. 211, 361–376 (1977)
Li T.-T., Qin T.: Physics and partial differential equations, 2nd edn. Higher Education Press, Beijing (2005) (in Chinese)
Liu T.P.: Solutions in the large for the equations of nonisentropic gas dynamics. Indiana Univ. Math. J. 26, 147–177 (1997)
Lu M., Ukai S.: Non-relativistic global limits of weak solutions of the relativistic Euler equation. J. Math. Kyoto Univ. 38, 525–537 (1998)
Martí J.M., Müller E.: The analytical solution of the Riemann problem in relativistic hydrodynamics. J. Fluid Mech. 258, 317–333 (1994)
Makino T., Ukai S.: Local smooth solutions of the relativistic Euler equation. J. Math. Kyoto Univ. 35, 105–114 (1995)
Makino T., Ukai S.: Local smooth solutions of the relativistic Euler equation II. J. Kodai Math. 18, 365–375 (1995)
Pant V.: Global entropy solutions for isentropic relativistic fluid dynamics. Commun. PDE 21, 1609–1641 (1996)
Pant, V.: On I symmetry breaking under perturbation and II relativistic fluid dynamics. Ph.D. Thesis, University of Michigan (1996)
Ruan L., Zhu C.: Existence of global smooth solution to the relativistic Euler equations. Nonlinear Anal. 60, 993–1001 (2005)
Shi C.C.: The Relativistic Fluid Dynamics. Science Press, Beijing (1992) (in Chinese)
Smoller J., Temple B.: Global solutions of the relativistic Euler equation. Commun. Math. Phys. 156, 67–99 (1993)
Smoller J.: Shock Waves and Reaction–diffusion Equations. Springer, New York (1983)
Thorne K.S.: Relativistic shocks: the Taub adiabt. Astrophys. J. 179, 897–907 (1973)
Temple J.B.: Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics. J. Differ. Equ. 41, 96–161 (1981)
Taub A.H.: Relativistic Rankine–Hügoniot equations. Phys. Rev. 74, 328–334 (1948)
Taub A.H.: Relativistic hydrodynamics, Relativistic theory and astrophysics 1. In: Ehlers, J. (eds) Relativity and Cosmology, pp. 170–193. American Mathematical Society, Providence (1967)
Taub A.H.: Approximate solutions of the Einstein equations for isentropic motions of plane-symmetric distributions of perfect fluids. Phys. Rev. 107, 884–900 (1957)
Weinberg S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York (1972)
Xu Y., Dou Y.: Global existence of shock front solutions in 1-dimensional piston problem in the relativistic equations. Z. Angew. Math. Phys. 59, 244–263 (2008)
Yin G., Sheng W.: Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations. Chin. Ann. Math. 29:B, 611–622 (2008)
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Geng, Y., Li, Y. Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations. Z. Angew. Math. Phys. 61, 201–220 (2010). https://doi.org/10.1007/s00033-009-0031-1
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DOI: https://doi.org/10.1007/s00033-009-0031-1