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Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a Simplified Isentropic Relativistic Euler Equations

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Abstract

We identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions to a simplified isentropic relativistic Euler equations. Firstly, both the explicit expressions and geometric properties of the rarefaction wave curve and shock wave curve based on any left state are given with the help of Lorentz invariance, and the Riemann problem for this system is considered. Then, we rigorously prove that, as pressure vanishes, any two-shock Riemann solution tends to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between the two shocks tends to a weighted \(\delta \)-measure, which forms the delta shock wave. This describes the phenomenon of mass concentration. On the other hand, any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations and the nonvacuum intermediate state in between tends to a vacuum state, which reveals the phenomenon of cavitation. Both concentration and cavitation are fundamental and physical in fluid dynamics.

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References

  1. Anile, A.M.: Relativistic Fluids and Magneto-Fluids, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  2. Brenier, Y.: Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations. J. Math. Fluid Mech. 7, S326–S331 (2005)

    Article  MathSciNet  Google Scholar 

  3. Chen, G., Liu, H.: Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations. SIAM J. Math. Anal. 34, 925–938 (2003)

    Article  MathSciNet  Google Scholar 

  4. Chen, G., Liu, H.: Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids. Phys. D 189, 141–165 (2004)

    Article  MathSciNet  Google Scholar 

  5. Huang, F., Wang, Z.: Well-posedness for pressureless flow. Commun. Math. Phys. 222, 117–146 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  6. Keyfit, B., Kranzer, H.: Spaces of weighted measures for conservation laws with singular shock solutions. J. Differ. Equ. 118, 420–451 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  7. Li, J.: Note on the compressible Euler equations with zero temperature. Appl. Math. Lett. 14, 519–523 (2001)

    Article  MathSciNet  Google Scholar 

  8. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, San Francisco (1973)

    Google Scholar 

  9. Pant, V.: On I: Symmetry breaking under perturbations and II: Relafivistlc fluid dynamics, Ph.D. Thesis, University of Michigan (1996)

  10. Shandarin, S.F., Zeldovich, Y.B.: The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium. Rev. Mod. Phys. 61(2), 185–220 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  11. Sheng, W., Zhang, T.: The Riemann problem for transportation equation in gas dynamics. Mem. Am. Math. Soc. 137(654), 1–77 (1999)

    MathSciNet  Google Scholar 

  12. Tan, D., Zhang, T.: Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws I. Four-J cases, II. Initial data involving some rarefaction waves. J. Differ. Equ. 111, 203–282 (1994)

    Article  ADS  Google Scholar 

  13. Tan, D., Zhang, T., Zheng, Y.: Delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J. Differ. Equ. 112(1), 1–32 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  14. Taub, A.: Relativistic Rankine–Hugoniot equations. Phys. Rev. 74, 328–334 (1948)

    Article  ADS  MathSciNet  Google Scholar 

  15. Thompson, K.: The special relativistic shock tube. J. Fluid Mech. 171, 365–373 (1986)

    Article  ADS  Google Scholar 

  16. Weinan, E., Rykov, Y.G., Sinai, Y.G.: Generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Commun. Math. Phys. 177, 349–380 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  17. Yang, H.: Riemann problems for a class of coupled hyperbolic systems of conservation laws. J. Differ. Equ. 159, 447–484 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  18. Yang, H., Wang, J.: Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas. J. Math. Anal. Appl. 413, 800–820 (2014)

    Article  MathSciNet  Google Scholar 

  19. Yang, H., Wang, J.: Concentration in vanishing pressure limit of solutions to the modified Chaplygin gas equations. J. Math. Phys. 57(11), 111504 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  20. Yang, H., Zhang, Y.: New developments of delta shock waves and its applications in systems of conservation laws. J. Differ. Equ. 252, 5951–5993 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  21. Yang, H., Zhang, Y.: Delta shock waves with Dirac delta function in both components for systems of conservation laws. J. Differ. Equ. 257, 4369–4402 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  22. Yang, H., Zhang, Y.: Flux approximation to the isentropic relativistic Euler equations. Nonlinear Anal. Ser. A Theory Methods Appl. 133, 200–227 (2016)

    Article  MathSciNet  Google Scholar 

  23. Yang, H., Zhang, Y.: Pressure and flux-approximation to the isentropic relativistic Euler equations for modified Chaplygin gas. J. Math. Phys. 60(7), 071502 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  24. Yin, G., Sheng, W.: Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases. J. Math. Anal. Appl. 355, 594–605 (2009)

    Article  MathSciNet  Google Scholar 

  25. Yin, G., Sheng, W.: Delta wave formation and vacuum state in vanishing pressure limit for system of conservation laws to relativistic fluid dynamics. Z. Angew. Math. Mech. 95, 49–65 (2015)

    Article  MathSciNet  Google Scholar 

  26. Yin, G., Song, K.: Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas. J. Math. Anal. Appl. 411, 506–521 (2014)

    Article  MathSciNet  Google Scholar 

  27. Zhang, Y., Yang, H.: Flux-approximation limits of solutions to the relativistic Euler equations for polytropic gas. J. Math. Anal. Appl. 435, 1160–1182 (2016)

    Article  MathSciNet  Google Scholar 

  28. Zhang, Y., Zhang, Y.: Delta-shocks and vacuums in the relativistic Euler equations for isothermal fluids with the flux approximation. J. Math. Phys. 60(1), 011508 (2019)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by NSFC (11661015), Yunnan Applied Basic Research Projects (2018FD015), Scientific Research Foundation Project of Yunnan Education Department (2018JS150), the science and technology innovation team of variation theory and applications in Universities of Yunnan Province, the Science and Technology Foundation of Guizhou Province ([2019]1046) and the Project of High Level Creative Talents in Guizhou Province (601605005).

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Authors and Affiliations

  1. Department of Mathematics, Yunnan Normal University, Kunming, 650500, People’s Republic of China

    Yu Zhang

  2. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, People’s Republic of China

    Yicheng Pang

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  1. Yu Zhang
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  2. Yicheng Pang
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Correspondence to Yu Zhang.

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Communicated by G.-Q. G. Chen

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Zhang, Y., Pang, Y. Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a Simplified Isentropic Relativistic Euler Equations. J. Math. Fluid Mech. 23, 8 (2021). https://doi.org/10.1007/s00021-020-00526-2

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