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Stabilization Effect of Frictions for Transonic Shocks in Steady Compressible Euler Flows Passing Three-Dimensional Ducts

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Abstract

Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces. This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.

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References

  1. Yuan H R. On transonic shocks in two-dimensional variable-area ducts for steady Euler system. SIAM J Math. Anal, 2006, 38: 1343–1370

    Article  MathSciNet  Google Scholar 

  2. Yuan H R. Transonic shocks for steady Euler flows with cylindrical symmetry. Nonlinear Anal, 2007, 66: 1853–1878

    Article  MathSciNet  Google Scholar 

  3. Chen S X, Yuan H R. Transonic shocks in compressible flow passing a duct for three-dimensional Euler systems. Arch Ration Mech Anal, 2008, 187: 523–556

    Article  MathSciNet  Google Scholar 

  4. Yuan H R. A remark on determination of transonic shocks in divergent nozzles for steady compressible Euler flows. Nonlinear Anal, Real World Appl, 2008, 9: 316–325

    Article  MathSciNet  Google Scholar 

  5. Liu L, Yuan H R. Stability of cylindrical transonic shocks for the two-dimensional steady compressible Euler system. J Hyperbolic Differ Equ, 2008, 5: 347–379

    Article  MathSciNet  Google Scholar 

  6. Chen G Q, Yuan H R. Local uniqueness of steady spherical transonic shock-fronts for the three-dimensional full Euler equations. Commun Pure Appl Anal, 2013, 12: 2515–2542

    Article  MathSciNet  Google Scholar 

  7. Fang B X, Liu L, Yuan H R. Global uniqueness of transonic shocks in two-dimensional steady compressible Euler flows. Arch Ration Mech Anal, 2013, 207: 317–345

    Article  MathSciNet  Google Scholar 

  8. Liu L, Xu G, Yuan H R. Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations. Adv Math, 2016, 291: 696–757

    Article  MathSciNet  Google Scholar 

  9. Chen G Q, Feldman M. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J Amer Math Soc, 2003, 16: 461–494

    Article  MathSciNet  Google Scholar 

  10. Xin Z P, Yin H C. Transonic shock in a nozzle I: Two-dimensional case. Comm Pure Appl Math, 2005, 58: 999–1050

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. 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, 2013, 207: 533–581 Springer

    Article  MathSciNet  Google Scholar 

  13. Chen G Q, Huang F M, Wang T Y, Xiang W. Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles. Adv Math, 2019, 346: 946–1008

    Article  MathSciNet  Google Scholar 

  14. Huang F M, Kuang J, Wang D H, Xiang W. Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle. J Differential Equations, 2019, 266: 4337–4376

    Article  MathSciNet  Google Scholar 

  15. Rathakrishnan E. Applied gas dynamics. John Wiley & Sons (Asia) Pte Ltd, 2010

    Google Scholar 

  16. Van Dyke M. An album of fluid motion. California: The Parabolic Press, 1982

    Google Scholar 

  17. Yuan H R, Zhao Q. Subsonic flow passing a duct for three-dimensional steady compressible Euler system with friction (in Chinese). To appear in Sci Sin Math, 2021, 51: 1–23. doi: 10.1360/N012019-00103

    Google Scholar 

  18. Liu T P. Transonic gas flow in a duct of varying area. Arch Rational Mech Anal, 1982, 80: 1–18

    Article  MathSciNet  Google Scholar 

  19. Liu T P. Nonlinear stability and instability of transonic flows through a nozzle. Comm Math Phys, 1982, 83: 243–260

    Article  MathSciNet  Google Scholar 

  20. Rauch J, Xie C J, Xin Z P. Global stability of steady transonic Euler shocks in quasi-one-dimensional nozzles. J Math Pures Appl, 2013, 99: 395–408

    Article  MathSciNet  Google Scholar 

  21. Tsuge N. Existence of global solutions for isentropic gas flow in a divergent nozzle with friction. J Math Anal Appl, 2015, 426: 971–977

    Article  MathSciNet  Google Scholar 

  22. Chou S W, Hong J M, Huang B C, Quita R. Global bounded variation solutions describing Fanno-Rayleigh fluid flows in nozzles. Math Models Methods Appl Sci, 2018, 28: 1135–1169

    Article  MathSciNet  Google Scholar 

  23. Sun Q Y, Lu Y G, Klingenberg C. Global L8 Solutions to System of Isentropic Gas Dynamics in a Divergent Nozzle with Friction. Acta Mathematica Scientia, 2019, 39(2): 1213–1218

    Article  MathSciNet  Google Scholar 

  24. Huang F M, Marcati P, Pan R H. Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch Ration Mech Anal, 2005, 176: 1–24

    Article  MathSciNet  Google Scholar 

  25. Huang F M, Pan R H, Wang Z. L1 convergence to the Barenblatt solution for compressible Euler equations with damping. Arch Ration Mech Anal, 2011, 200: 665–689

    Article  MathSciNet  Google Scholar 

  26. Chen C, Xie C J. Three dimensional steady subsonic Euler flows in bounded nozzles. J Differential Equations, 2014, d256: 684–3708

    MathSciNet  MATH  Google Scholar 

  27. Weng S. A new formulation for the 3-D Euler equations with an application to subsonic flows in a cylinder. Indiana Univ Math J, 2015, 64: 1609–1642

    Article  MathSciNet  Google Scholar 

  28. Shapiro A H. The dynamics and thermodynamics of compressible fluid flow. Vol 1. New York: Ronald Press Co, 1953

  29. Courant R, Friedrichs K O. Supersonic flow and shock waves. New York-Heidelberg: Springer-Verlag, 1976

    Book  Google Scholar 

  30. Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Berlin Heidelberg: Springer-Verlag, 2010

    Book  Google Scholar 

  31. Benzoni-Gavage S, Serre D. Multidimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications. Oxford: Clarendon Press, 2007

    MATH  Google Scholar 

  32. Venttsel A D. On boundary conditions for multi-dimensional diffusion processes. Theor Probability Appl, 1959, 4: 164–177

    Article  MathSciNet  Google Scholar 

  33. Apushkinskaya D E, Nazarov A I. A survey of results on nonlinear Venttsel problems. Appl Math, 2000, 45: 69–80

    Article  MathSciNet  Google Scholar 

  34. Luo Y, Trudinger N S. Linear second order elliptic equations with Venttsel boundary conditions. Proc Roy Soc Edinburgh Sect A, 1991, 118: 193–207

    Article  MathSciNet  Google Scholar 

  35. Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order. Berlin: Springer-Verlag, 2001

    MATH  Google Scholar 

  36. Walter W. Ordinary differential equations. New York: Springer-Verlag, 1998

    Book  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, China

    Hairong Yuan

  2. Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, 430070, China

    Qin Zhao

Authors
  1. Hairong Yuan
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  2. Qin Zhao
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Corresponding author

Correspondence to Qin Zhao.

Additional information

This work was supported in part by National Nature Science Foundation of China (11371141 and 11871218), and by Science and Technology Commission of Shanghai Municipality (18dz2271000).

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Yuan, H., Zhao, Q. Stabilization Effect of Frictions for Transonic Shocks in Steady Compressible Euler Flows Passing Three-Dimensional Ducts. Acta Math Sci 40, 470–502 (2020). https://doi.org/10.1007/s10473-020-0212-8

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  • DOI: https://doi.org/10.1007/s10473-020-0212-8

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