Communications in Mathematical Physics

, Volume 328, Issue 1, pp 327–354 | Cite as

Steady Subsonic Ideal Flows Through an Infinitely Long Nozzle with Large Vorticity

Article

Abstract

In this paper, the existence, uniqueness, and far field behavior of a class of subsonic flows with large vorticity for the steady Euler equations in infinitely long nozzles are established. More precisely, for any given convex horizontal velocity of incoming flow in the upstream, there exists a critical value mcr, if the mass flux is larger than mcr, then there exists a unique smooth subsonic Euler flow through the infinitely long nozzle. This well-posedness result is proved by a new observation for the method developed in Xie and Xin (SIAM J Math Anal 42:751–784, 2010) to deal with the Euler system. Furthermore, the optimal convergence rates of the subsonic flows at far fields are obtained via the maximum principle and an elaborate choice of the comparison functions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bers, L.: Mathematical aspects of subsonic and transonic gas dynamics. In: Surveys in Applied Mathematics, Vol. 3, New York: Wiley and Sons, Inc., 1958Google Scholar
  2. 2.
    Bers L.: Existence and uniqueness of a subsonic flow past a given profile. Comm. Pure Appl. Math. 7, 441–504 (1954)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Chen C., Xie C.J.: Existence of steady subsonic Euler flows through infinitely long periodic nozzles. J. Differ. Equ. 252, 4315–4331 (2012)ADSCrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chen G.Q., Dafermos C., Slemrod M., Wang D.H.: On two-dimensional sonic-subsonic flow. Comm. Math. Phys. 271, 635–647 (2007)ADSCrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Chen G.Q., Deng X.M., Xiang W.: Global steady subsonic flows through infinitely long nozzles for the full Euler equations. SIAM J. Math. Anal. 44, 2888–2919 (2012)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chen J.: Subsonic flows for the full Euler equations in half plane. J. Hyperbolic Differ. Equ. 6, 207–228 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dong G.C., Ou B.: Subsonic flows around a body in space. Comm. Partial Differ. Equ. 18(1-2), 355–379 (1993)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Du L.L., Duan B.: Global subsonic Euler flows in an infinitely long axisymmetric nozzle. J. Differ. Equ. 250, 813–847 (2011)ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Du, L.L., Xie, C.J.: On subsonic Euler flows with stagnation points in two dimensional nozzles, preprintGoogle Scholar
  10. 10.
    Du L.L., Xin Z.P., Yan W.: Subsonic flows in a multi-dimensional nozzle. Arch. Ration. Mech. Anal. 201, 965–1012 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Finn R., Gilbarg D.: Asymptotic behavior and uniqueness of plane subsonic flows. Comm. Pure Appl. Math. 10, 23–63 (1957)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Finn R., Gilbarg D.: Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations. Acta Math. 98, 265–296 (1957)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Gilbarg D.: Comparison methods in the theory of subsonic flows. J. Ration. Mech. Anal. 2, 233–251 (1953)MATHMathSciNetGoogle Scholar
  14. 14.
    Gilbarg D., Serrin J.: Uniqueness of axially symmetric subsonic flow past a finite body. J. Ration. Mech. Anal. 4, 169–175 (1955)MATHMathSciNetGoogle Scholar
  15. 15.
    Gilbarg D., Shiffman M.: On bodies achieving extreme values of the critical Mach number. I. J. Ratio. Mech. Anal. 3, 209–230 (1954)MATHMathSciNetGoogle Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Berlin, Springer, 2001Google Scholar
  17. 17.
    Huang F.M., Wang T.Y., Wang Y.: On multi-dimensional sonic-subsonic flow. Acta. Math. Sci. 31, 2131–2140 (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Wang C.P., Xin Z.P.: Optimal Holder continuity for a class of degenarate elliptic problems with an application to subsonic-sonic flows. Comm. Partial Differ. Eq. 36(5), 873–924 (2011)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Wang C.P., Xin Z.P.: On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle. Arch. Ration. Mech. Anal. 208, 911–975 (2012)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Xie C.J., Xin Z.P.: Global subsonic and subsonic-sonic flows through infinitely long nozzles. Indiana Univ. Math. J. 56(6), 2991–3023 (2007)MATHMathSciNetGoogle Scholar
  21. 21.
    Xie C.J., Xin Z.P.: Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles. J. Differ. Equ. 248, 2657–2683 (2010)ADSMATHMathSciNetGoogle Scholar
  22. 22.
    Xie C.J., Xin Z.P.: Existence of global steady subsonic Euler flows through infinitely long nozzle. SIAM J. Math. Anal. 42(2), 751–784 (2010)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduPeople’s Republic of China.
  2. 2.Department of Mathematics, Institute of Natural Sciences, Ministry of Education Key Laboratory of Scientific and Engineering ComputingShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  3. 3.Institute of Mathematical Science and Department of MathematicsThe Chinese University of Hong KongShatinHong Kong

Personalised recommendations