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On the Irrotational Approximation to Adiabatic Chaplygin Gas Dynamic System

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Abstract

In this paper we mainly study the difference of the weak solutions generated by a wave front tracking algorithm for the steady adiabatic Chaplygin gas dynamic system and the steady irrotational system. Under the hypothesis that the initial data are of sufficiently small total variation, we prove that the difference between the solutions to these two systems can be bounded by the cube of the total variation of the initial perturbation.

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References

  1. Bianchini, S., Colombo, R.M. On the stability of the standard Riemann Semigroup. Proc. American Math. Soc., 130: 1961–1973 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bilic, N., Tupper, G.B., Viollier, R. Dark matter, dark energy and the Chaplygin gas. arXiv: astroph/0207423

  3. Brenier, Y. Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations. J. Math. Fluid Mech., 7: 326–331 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bressan, A. The unique limit of the Glimm scheme. Arch.Rational Mech. Anal., 130: 105–230 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bressan, A. Hyperbolic systems of conservation laws. Volume 20 of Oxford Lecture Series in Mathematics and Its Applications Oxford University Press, Oxford, 2000

    MATH  Google Scholar 

  6. Bressan, A., Liu, T.P, Yang, T. L 1 stability estimates for n × n conservation laws. Arch. Rational Mech. Anal., 149: 1–22 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaplygin, S. On gas jets. Sci. Mem. Moscow Univ. Math. Phys., 21: 1–121 (1904)

    Google Scholar 

  8. Chen, G.Q., Zhang, Y.Q, Zhu, D.W. Existence and stability of supersonic Euler flows past Lipschitz Wedges. Arch. Rational Mech. Anal., 181: 261–310 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Corli, A., Tougeron, S.M. Stability of contact discontinuities under perturbations of bounded variation. Rend. Sem. Mat. Univ. Padova, 97: 35–60 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Dafermos, C.M. Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag, Berlin, 2000

    Book  MATH  Google Scholar 

  11. Glimm, J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure. Appl. Math., 18: 95–105 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gorini, V., Kamenshchik, A., Moschella, U., Pasquier, V. The Chaplygin gas as a model for dark energy. Physics Letters B, 535: 17–21 (2002)

    Article  MATH  Google Scholar 

  13. Kong, D., Yang, T. A Note on well posedness theory for hyperbolic conservation laws. Applied Mathematics Letters, 16: 143–146 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lax, P.D. Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math., 10: 537–566 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, C., Zhang, Y.Q. Isentropic approximation of the steady Euler system in two space dimensions. Commun. Pure Appl. Anal., 7(2): 277–291 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Raymond, S.L. Isentropic Approximation of the compressible Euler system in one space dimension. Arch. Rational Mech. Anal., 155: 171–199 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Serre, D. Systems of Conservation Laws. Cambridge University Press, Cambridge, 1999

    Book  MATH  Google Scholar 

  18. Setare, M.R. Holographic Chaplygin gas model. Phys. Lett. B, 648: 329–332 (2007)

    Article  MATH  Google Scholar 

  19. Smoller, J. Shock waves and reaction-diffusion equations. Springer-Verlag, New York, 1983

    Book  MATH  Google Scholar 

  20. Zhang, Y.Q. On the irrotational approximation to steady supersonic flow. Z. Angew. Math. Phys., 58(2): 209–223 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, Y.Q. Steady supersonic flow past an almost straight wedge with large vertex angle. J. Differ. Eqs., 192: 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, Y.Q. Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary. SIAM J. Math. Anal., 31: 166–183 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhu, L., Sheng, W.C. The Riemann problem of adiabatic Chaplygin Gas Dynamics system. Comm. On Appl. Math. and Comput, 24(1): 9–16 (2010) (in Chinese)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Li Wang.

Additional information

Supported by the TianYuan Special Funds of the National Natural Science Foundation of China (Grant No.11226171), Discipline construction of equipment manufacturing system optimization calculation (Grant No.13XKJC01).

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Wang, L. On the Irrotational Approximation to Adiabatic Chaplygin Gas Dynamic System. Acta Math. Appl. Sin. Engl. Ser. 34, 416–429 (2018). https://doi.org/10.1007/s10255-018-0748-8

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  • DOI: https://doi.org/10.1007/s10255-018-0748-8

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