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Uniqueness of the 1D compressible to incompressible limit

  • Rinaldo M. Colombo
  • Graziano Guerra
Article
Part of the following topical collections:
  1. Hyperbolic PDEs, Fluids, Transport and Applications: Dedicated to Alberto Bressan for his 60th birthday

Abstract

Consider two compressible immiscible fluids in 1D in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, the coupled dynamics of the two fluids results as the compressible to incompressible limit and is known to satisfy an ODE–PDE system. Below, a characterization of this limit is provided, ensuring its uniqueness.

Keywords

Compressible to incompressible limit Hyperbolic conservation laws Uniqueness of the zero Mach number limit 

Mathematics Subject Classification

35L65 35Q35 76N99 

References

  1. 1.
    Borsche, R., Colombo, R.M., Garavello, M.: Mixed systems: ODEs—balance laws. J. Differ. Equ. 252(3), 2311–2338 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bressan, A.: On the Cauchy problem for systems of conservation laws. In: Actes du 29ème Congrès d’Analyse Numérique: CANum’97 (Larnas, 1997), pp. 23–36 (electronic). Soc. Math. Appl. Ind., Paris (1998)Google Scholar
  3. 3.
    Bressan, A.: Hyperbolic Systems of Conservation Laws, Volume 20 of Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2000). The one-dimensional Cauchy problemGoogle Scholar
  4. 4.
    Chen, G.-Q., Christoforou, C., Zhang, Y.: Continuous dependence of entropy solutions to the Euler equations on the adiabatic exponent and Mach number. Arch. Ration. Mech. Anal. 189(1), 97–130 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Colombo, R.M., Guerra, G.: On the stability functional for conservation laws. Nonlinear Anal. 69(5–6), 1581–1598 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Colombo, R.M., Guerra, G.: Differential equations in metric spaces with applications. Discrete Contin. Dyn. Syst. 23(3), 733–753 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colombo, R.M., Guerra, G.: On general balance laws with boundary. J. Differ. Equ. 248(5), 1017–1043 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colombo, R.M., Guerra, G.: BV solutions to 1D isentropic Euler equations in the zero Mach number limit. J. Hyperbolic Differ. Equ. 13(4), 685–718 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Colombo, R.M., Guerra, G.: Characterization of the solutions to ODE–PDE systems. Appl. Math. Lett. 62, 69–75 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Colombo, R.M., Guerra, G.: A coupling between a non-linear 1D compressible–incompressible limit and the 1D \(p\)-system in the non smooth case. Netw. Heterog. Media 11(2), 313–330 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Colombo, R.M., Guerra, G., Herty, M., Schleper, V.: Optimal control in networks of pipes and canals. SIAM J. Control Optim. 48(3), 2032–2050 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Colombo, R.M., Guerra, G., Schleper, V.: The compressible to incompressible limit of one dimensional Euler equations: the non smooth case. Arch. Ration. Mech. Anal. 219(2), 701–718 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (2010)Google Scholar
  14. 14.
    Guerra, G., Schleper, V.: A coupling between a 1D compressible–incompressible limit and the 1D \(p\)-system in the non smooth case. Bull. Braz. Math. Soc. N. Ser. 47(1), 381–396 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34(4), 481–524 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Klainerman, S., Majda, A.: Compressible and incompressible fluids. Commun. Pure Appl. Math. 35(5), 629–651 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Macdonald, J.R.: Some simple isothermal equations of state. Rev. Mod. Phys. 38, 669–679 (1966)CrossRefGoogle Scholar
  18. 18.
    Métivier, G., Schochet, S.: The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158(1), 61–90 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schochet, S.: The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit. Commun. Math. Phys. 104(1), 49–75 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schochet, S.: The mathematical theory of low Mach number flows. M2AN Math. Model. Numer. Anal. 39(3), 441–458 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Xu, J., Yong, W.-A.: A note on incompressible limit for compressible Euler equations. Math. Methods Appl. Sci. 34(7), 831–838 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.INDAM UnitUniversity of BresciaBresciaItaly
  2. 2.Department of Mathematics and Its ApplicationsUniversity of Milano-BicoccaMilanItaly

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