Archive for Rational Mechanics and Analysis

, Volume 219, Issue 2, pp 701–718 | Cite as

The Compressible to Incompressible Limit of One Dimensional Euler Equations: The Non Smooth Case

  • Rinaldo M. ColomboEmail author
  • Graziano Guerra
  • Veronika Schleper


We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal one dimensional Euler equations.


Euler Equation Riemann Problem Interaction Point Entropy Solution Compressible Euler Equation 
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  1. 1.
    Borsche R., Colombo R.M., Garavello M.: Mixed systems: ODEs—balance laws. J. Differ. Equ. 252(3), 2311–2338 (2012)zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Borsche, R., Colombo, R.M., Garavello, M.: On the interactions between a solid body and a compressible inviscid fluid. Interfaces Free Bound. 15(3), 381–403 (2013)Google Scholar
  3. 3.
    Bressan, A.: Hyperbolic systems of conservation laws. Oxford Lecture Series in Mathematics and its Applications, Vol. 20. Oxford University Press, Oxford, 2000. (The one-dimensional Cauchy problem)Google Scholar
  4. 4.
    Colombo, R.M., Schleper, V.: Two-phase flows: non-smooth well posedness and the compressible to incompressible limit. Nonlinear Anal. Real World Appl. 13(5), 2195–2213 (2012)Google Scholar
  5. 5.
    Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 325, 3rd edn. Springer, Berlin, 2010Google Scholar
  6. 6.
    Ebin, D.G.: The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. Math. (2), 105(1), 141–200 (1977)Google Scholar
  7. 7.
    Feireisl, E., Novotný, A.: Singular limits in thermodynamics of viscous fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel, 2009Google Scholar
  8. 8.
    Hoff D.: The zero-Mach limit of compressible flows. Commun. Math. Phys. 192(3), 543–554 (1998)zbMATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    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)zbMATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Klainerman, S., Majda, A.: Compressible and incompressible fluids. Commun. Pure Appl. Math. 35(5), 629–651 (1982)Google Scholar
  11. 11.
    Métivier, G., Schochet, S.: The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158(1), 61–90 (2001)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Schochet, S.: The mathematical theory of low Mach number flows. M2AN Math. Model. Numer. Anal. 39(3), 441–458 (2005)Google Scholar
  14. 14.
    Secchi, P.: On the singular incompressible limit of inviscid compressible fluids. J. Math. Fluid Mech. 2(2), 107–125 (2000)Google Scholar
  15. 15.
    Xu, J., Yong, W.-A.: A note on incompressible limit for compressible Euler equations. Math. Methods Appl. Sci. 34(7), 831–838 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Rinaldo M. Colombo
    • 1
    Email author
  • Graziano Guerra
    • 2
  • Veronika Schleper
    • 3
  1. 1.INDAM UnitUniversity of BresciaBresciaItaly
  2. 2.Department of Mathematics and ApplicationsUniversity of Milano-BicoccaMilanItaly
  3. 3.Institute for Applied Analysis and Numerical SimulationsUniversity of StuttgartStuttgartGermany

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