Self-similar solutions of the relativistic Euler system with spherical symmetry
Authors:
Bing-Ze Lu, Chou Kao and Wen-Ching Lien
Journal:
Quart. Appl. Math. 82 (2024), 705-734
MSC (2020):
Primary 35L65, 35L67, 76N10, 83A05
DOI:
https://doi.org/10.1090/qam/1680
Published electronically:
October 24, 2023
Full-text PDF
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Additional Information
Abstract: We consider the spherical piston problem in relativistic fluid dynamics. When the spherical piston expands at a constant speed, we show that the self-similar solution with a shock front exists under the relativistic principle that all velocities are bounded by the light speed. The equation of state is given by $P= \sigma ^2 \rho$, where $\sigma$, the sound speed, is a constant. It is an important model describing the evolution of stars. Also, we present the global existence of BV solutions for the relativistic Euler system given that the piston speed is perturbed around a constant in a finite time interval. The analysis is based on the modified Glimm scheme and the smallness assumption is required on the initial data.
References
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- Tai Ping Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979), no. 2, 141–172. MR 543196, DOI 10.1007/BF01418125
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References
- A. M. Anile, Relativistic fluids and magneto-fluids, with applications in astrophysics and plasma physics, Cambridge University Press, New York, 1989.
- Grigory Isaakovich Barenblatt, Scaling, self-similarity, and intermediate asymptotics, Cambridge Texts in Applied Mathematics, vol. 14, Cambridge University Press, Cambridge, 1996. With a foreword by Ya. B. Zeldovich. MR 1426127, DOI 10.1017/CBO9781107050242
- R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience Publishers, Inc., New York, 1948. MR 29615
- James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
- James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, RI, 1970. MR 265767
- Seung-Yeal Ha, Hsiu-Chuan Huang, and Wen-Ching Lien, Nonlinear stability of spherical self-similar flows to the compressible Euler equations, Quart. Appl. Math. 72 (2014), no. 1, 109–136. MR 3185135, DOI 10.1090/S0033-569X-2013-01329-9
- Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 419901
- Cheng-Hsiung Hsu, Song-Sun Lin, and Tetu Makino, On spherically symmetric solutions of the relativistic Euler equation, J. Differential Equations 201 (2004), no. 1, 1–24. MR 2057536, DOI 10.1016/j.jde.2004.03.003
- 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
- Chen-Chang Peng and Wen-Ching Lien, Self-similar solutions of the Euler equations with spherical symmetry, Nonlinear Anal. 75 (2012), no. 17, 6370–6378. MR 2959812, DOI 10.1016/j.na.2012.07.019
- Tai Ping Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), no. 2, 135–148. MR 470508
- Tai Ping Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979), no. 2, 141–172. MR 543196
- P. L. Sachdev, K. T. Joseph, and M. Ejanul Haque, Exact solutions of compressible flow equations with spherical symmetry, Stud. Appl. Math. 114 (2005), no. 4, 325–342. MR 2131550, DOI 10.1111/j.0022-2526.2005.01552.x
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- Joel Smoller and Blake Temple, Global solutions of the relativistic Euler equations, Comm. Math. Phys. 156 (1993), no. 1, 67–99. MR 1234105
- G. I. Taylor, The air wave surrounding an expanding sphere, Proc. Roy. Soc. London Ser. A 186 (1946), 273–292. MR 18519, DOI 10.1098/rspa.1946.0044
- Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2nd ed., Springer-Verlag, Berlin, 1999. A practical introduction. MR 1717819, DOI 10.1007/978-3-662-03915-1
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 483954
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Additional Information
Bing-Ze Lu
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan City 70101, Taiwan
Email:
l18081028@gs.ncku.edu.tw
Chou Kao
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan City 70101, Taiwan
MR Author ID:
1485635
Email:
l18101030@gs.ncku.edu.tw
Wen-Ching Lien
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan City 70101, Taiwan
MR Author ID:
340649
Email:
wlien@mail.ncku.edu.tw
Keywords:
Special relativity,
relativistic Euler equations,
shock waves,
supersonic flows,
self-similarity,
the spherical piston problem
Received by editor(s):
April 16, 2023
Received by editor(s) in revised form:
September 17, 2023
Published electronically:
October 24, 2023
Additional Notes:
The first author would like to acknowledge Science College of National Cheng Kung University (NCKU Science) and Ministry of Science and Technology (MOST), Taiwan, Republic of China, for the fellowship to support his PhD study in mathematics.
Article copyright:
© Copyright 2023
Brown University
