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Subsonic Euler Flows with Large Vorticity Through an Infinitely Long Axisymmetric Nozzle

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Abstract

This paper is a sequel to the earlier work Du and Duan (J Diff Equ 250:813–847, 2011) on well-posedness of steady subsonic Euler flows through infinitely long three-dimensional axisymmetric nozzles. In Du and Duan (J Diff Equ 250:813–847, 2011), the authors showed the existence and uniqueness of the global subsonic Euler flows through an infinitely long axisymmetric nozzle, when the variation of Bernoulli’s function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The smallness of the variation of Bernoulli’s function in the upstream prevents the attendance of the possible singularity in the nozzles, however, at the same time it also leads that the vorticity of the ideal flow is sufficiently small in the whole nozzle and the flows are indeed adjacent to axisymmetric potential flows. The purpose of this paper is to investigate the effects of the vorticity for the smooth subsonic ideal flows in infinitely long axisymmetric nozzles. We modify the formulation of the problem in the previous work Du and Duan (J Diff Equ 250:813–847, 2011) and the existence and uniqueness results on the smooth subsonic ideal polytropic flows in infinitely long axisymmetric nozzles without the restriction on the smallness of the vorticity are shown in this paper.

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References

  1. Bae M., Feldman M.: Transonic shocks in multidimensional divergent nozzles. Arch. Ration. Mech. Anal. 201(3), 777–840 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bae M.: Stability of contact discontinuity for steady Euler System in infinite duct. Z. Angew. Math. Phys. 64, 917–936 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bae M., Duan B., Xie C.J.: Subsonic flow for multidimensional Euler-Poisson system. Arch. Ration. Mech. Anal. 220(1), 155–191 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bers L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Surveys in Applied Mathematics. Vol. 3, John Wiley and Sons, Inc., New York (1958)

    MATH  Google Scholar 

  5. Chen, C.: Subsonic non-isentropic ideal gas with large vorticity in nozzles. Math. Meth. Appl. Sci. doi:10.1002/mma.3711 (2015)

  6. Chen G.Q., Deng X., Xiang W.: Global steady subsonic flows through infinitely long nozzles for the full Euler equations. SIAM J. Math. Anal. 44, 2888–2919 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen G.Q., Feldman M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixted type. J. Am. Math. Soc. 16(3), 461–494 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen G.Q., Feldman M.: Steady transonic shocks and free boundary problems for the Euler equations in infinite cylinders. Comm. Pure Appl. Math. 57, 310–356 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen G.Q., Feldman M.: Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections. Arch. Ration. Mech. Anal. 184, 185–242 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen G.Q., Huang F.M., Wang T.Y.: Subsonic-sonic limit of approximate solutions to multidimensional steay Euler equation. Arch. Ration. Mech. Anal. 219(2), 719–740 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen C., Xie C.J.: Existence of steady subsonic Euler flows through infinitely long periodic nozzle. J. Differ. Equ. 252, 4315–4331 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Chen C., Xie C.J.: Three dimensional steady subsonic Euler flows in bounded nozzles. J. Differ. Equ. 256, 3684–3708 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Courant R., Friedrichs K.O.: Supsonic Flow and Shock Waves. Interscience Publ., New York (1948)

    MATH  Google Scholar 

  14. Dong G.C., Ou B.: Subsonic flows around a body in space. Comm. Partial Differ. Equ. 18(1-2), 355–379 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Du L.L., Duan B.: Global subsonic Euler flows in an infinitely long axisymmetric nozzle. J. Differ. Equ. 250, 813–847 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Du L.L., Duan B.: Note on the uniqueness of subsonic Euler flows in an axisymmetric nozzle. Appl. Math. Lett. 25, 153–156 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Du L.L., Xie C.J.: On subsonic Euler flows with stagnation points in two dimensional nozzles. Indiana Univ. Math. J. 63, 1499–1523 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Du L.L., Xie C.J., Xin Z.P.: Steady subsonic ideal flows through an infinitely long nozzle with large vorticity. Commun. Math. Phys. 328, 327–354 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Du L.L., Xin Z.P., Yan W.: Subsonic flows in a multi-dimensional nozzle. Arch. Ration. Mech. Anal. 201, 965–1012 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Duan B., Luo Z.: Three-dimensional full Euler flows in axisymmetric nozzles. J. Differ. Equ. 254, 2705–2731 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Finn D.G.: Asymptotic behavior and uniqueness of plane subsonic flows. Commun. Pure Appl. Math. 10, 23–63 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  22. Finn R., Gilbarg D.: Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations. Acta Math. 98, 265–296 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gilbarg D.: Comparison methods in the theory of subsonic flows. J. Ration. Mech. Anal. 2, 233–251 (1953)

    MathSciNet  MATH  Google Scholar 

  24. Gilbarg, D.: Jets and cavities. Handbuch der Physik, vol. 9, pp. 311–445. Springer-Verlag, Berlin (1960)

  25. Gilbarg D., Serrin J.: Uniqueness of axially symmetric subsonic flow past a finite body. J. Ration. Mech. Anal. 4, 169–175 (1955)

    MathSciNet  MATH  Google Scholar 

  26. Gilbarg D., Shiffman M.: On bodies achieving extreme values of the critical Mach number. I. J. Ration. Mech. Anal. 3, 209–230 (1954)

    MathSciNet  MATH  Google Scholar 

  27. Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001)

    MATH  Google Scholar 

  28. Huang F.M., Wang T.Y., Wang Y.: On multi-dimensional sonic-subsonic flow. Acta Math. Sci. 31, 2131–2140 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li J., Xin Z.P., Yin H.C.: On transonic shocks in a nozzle with variable end pressures. Commun. Math. Phys. 291, 111–150 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Li J., Xin Z.P., Yin H.C.: On transonic shocks in a conic divergent nozzle with axi-symmetric exit pressures. J. Differ. Equ. 248, 423–469 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Li J., Xin Z.P., Yin H.C.: Transonic shocks for the full compressible Euler system in a general two-dimensional De Laval nozzle. Arch. Ration. Mech. Anal. 207, 533–581 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang C.P., Xin Z.P.: Optimal Holder continuity for a class of degenarate elliptic problems with an application to subsonic-sonic flows. Commun. Partial Differ. Equ. 36(5), 873–924 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xin Z.P., Yan W., Yin H.C.: Transonic shock problem for the Euler system in a nozzle. Arch. Ration. Mech. Anal. 194, 1–47 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xin Z.P., Yin H.C.: The transonic shock in a nozzle, 2-D and 3-D complete Euler systems. J. Differ. Equ. 245, 1014–1085 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Lili Du

  2. School of Mathematical Sciences, Dalian University of Technolgoy, Dalian, 116024, People’s Republic of China

    Ben Duan

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  1. Lili Du
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  2. Ben Duan
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Correspondence to Ben Duan.

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Communicated by G.-Q. Chen

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Du, L., Duan, B. Subsonic Euler Flows with Large Vorticity Through an Infinitely Long Axisymmetric Nozzle. J. Math. Fluid Mech. 18, 511–530 (2016). https://doi.org/10.1007/s00021-016-0255-8

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  • DOI: https://doi.org/10.1007/s00021-016-0255-8

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