Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1239–1279 | Cite as

Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows

  • Gui-Qiang G. Chen
  • Matthew R. I. Schrecker


We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted L p norms for the whole range of physical adiabatic exponents \({\gamma\in (1, \infty)}\), so that the viscosity approximate solutions satisfy the general L p compensated compactness framework. The viscosity method is designed to incorporate artificial viscosity terms with the natural Dirichlet boundary conditions to ensure the uniform estimates. Then such estimates lead to both the convergence of the approximate solutions and the existence theory of globally defined finite-energy entropy solutions to the Euler equations for transonic flows that may have different end-states in the class of nozzles with general cross-sectional areas for all \({\gamma\in (1, \infty)}\). The approach and techniques developed here apply to other problems with similar difficulties. In particular, we successfully apply them to construct globally defined spherically symmetric entropy solutions to the Euler equations for all \({\gamma\in (1, \infty)}\).


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  1. 1.
    Chen, G.-Q.: Remarks on spherically symmetric solutions of the compressible Euler equations. Proc. R. Soc. Edinb. 127A, 243–259 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, G.-Q.: Weak continuity and compactness for nonlinear partial differential equations. Chin. Ann. Math. 36B, 715–736 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, G.-Q., Perepelitsa, M.: Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow. Commun. Pure. Appl. Math. 63, 1469–1504 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, G.-Q., Perepelitsa, M.: Vanishing viscosity solutions of the compressible Euler equations with spherical symmetry and large initial data. Commun. Math. Phys. 338, 771–800 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Springer, New York (1962)zbMATHGoogle Scholar
  6. 6.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 4th edn. Springer, Berlin (2016)zbMATHGoogle Scholar
  7. 7.
    DiPerna, R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Embid, P., Goodman, J., Majda, A.: Multiple steady states for 1-D transonic flow. SIAM J. Sci. Stat. Comput. 5, 21–41 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Germain, P., LeFloch, P.G.: Finite energy method for compressible fluids: the Navier-Stokes-Korteweg model. Commun. Pure Appl. Math. 69, 3–61 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Glaz, H., Liu, T.-P.: The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow. Adv. Appl. Math. 5, 111–146 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Glimm, J., Marshall, G., Plohr, B.: A generalized Riemann problem for quasi-one-dimensional gas flow. Adv. Appl. Math. 5, 1–30 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guderley, G.: Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung 19(9), 302–311 (1942)Google Scholar
  14. 14.
    Huang, F., Li, T., Yuan, D.: Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry. Preprint arXiv:1711.04430 (2017)
  15. 15.
    LeFloch, P.G., Westdickenberg, M.: Finite energy solutions to the isentropic Euler equations with geometric effects. J. Math. Pures Appl. 88, 389–429 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, T., Wang, D.: Blowup phenomena of solutions to the Euler equations for compressible fluid flow. J. Differ. Equ. 221, 91–101 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lions, P.-L., Perthame, B., Tadmor, E.: Kinetic formulation for the isentropic gas dynamics and p-system. Commun. Math. Phys. 163, 415–431 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ladyzhenskaja, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. LOMI-AMS, Providence (1968)CrossRefGoogle Scholar
  19. 19.
    Liu, T.-P.: Quasilinear hyperbolic systems. Commun. Math. Phys. 68, 141–172 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu, T.-P.: Nonlinear stability and instability of transonic flows through a nozzle. Commun. Math. Phys. 83, 243–260 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, T.-P.: Nonlinear resonance for quasilinear hyperbolic equation. J. Math. Phys. 28, 2593–2602 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Makino, T., Mizohata, K., Ukai, S.: The global weak solutions of compressible Euler equation with spherical symmetry. Jpn. J. Ind. Appl. Math. 9, 431–449 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Makino, T., Takeno, S.: Initial-boundary value problem for the spherically symmetric motion of isentropic gas. Jpn. J. Ind. Appl. Math. 11, 171–183 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Murat, F.: Compacité par compensation. Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat. 5, 489–507 (1978)Google Scholar
  25. 25.
    Rosseland, S.: The Pulsation Theory of Variable Stars. Dover Publications Inc., New York (1964)zbMATHGoogle Scholar
  26. 26.
    Slemrod, M.: Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit. Proc. R. Soc. Edinb. 126A, 1309–1340 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tartar, L.: Compensated compactness and applications to partial differential equations, In: Knops R.J. (ed.) Nonlinear Analysis and Mechanics, Herriot-Watt Symposium, Research Notes in Mathematics, vol. 4. Pitman Press (1979)Google Scholar
  28. 28.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)zbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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