Skip to main content
SpringerLink
Log in

Global existence and stability of a stationary Mach reflection

  • Articles
  • Progress of Projects Supported by NSFC
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We prove the global existence and stability of a wave structure containing a stationary Mach configuration, which occurs when an incident shock front hits a wall with a large incident angle. Our result shows that the data of the upstream flow and the pressure at downstream part jointly determine the whole flow, as well as the wave structure. Particularly, we show that the height of the Mach stem depends not only on the data of upstream flow, but also on the pressure at downstream flow. The flow with the assigned wave structure is governed by a free boundary value problem for the Euler system. In the problem the location of the triple point, the shock fronts and the contact discontinuity are all unknown, they are finally determined together with the solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agmon S, Douglis S, Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Comm Pure Appl Math, 1959, 12: 623–727

    Article  MATH  MathSciNet  Google Scholar 

  2. Agmon S, Douglis S, Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II. Comm Pure Appl Math, 1964, 17: 35–92

    Article  MATH  MathSciNet  Google Scholar 

  3. Azevedo D J, Liu C. Engineering approach to the prediction of shock patterns in bounded high-speed flows. AIAA J, 1993, 31: 83–90

    Article  MATH  Google Scholar 

  4. Ben-Dor G. Shock Waves Reflection Phenomena. New York: Springer-Verlag, 1992

    Book  Google Scholar 

  5. Canic S, Kerfitz B, Kim E H. A free boundary problems for unsteady transonic small disturbance equation: Transonic regular reflection. Methods Appl Anal, 2000, 7: 313–336

    MATH  MathSciNet  Google Scholar 

  6. Canic S, Kerfitz B, Lieberman G. A proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm Pure Appl Math, 2000, 53: 484–511

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen G Q, Feldman M. Global solution to shock reflection by large-angle wedges for potential flow. Ann Math, 2010, 171: 1067–1182

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen S X. Linear approximation of shock reflection at a wedge with large angle. Comm Partial Diff Eqs, 1996, 21: 1103–1118

    Article  MATH  Google Scholar 

  9. Chen S X. Stability of a Mach Configuration. Comm Pure Appl Math, 2006, 59: 1–35

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen S X. Mach configuration in pseudo-stationary compressible flow. J Amer Math Soc, 2008, 21: 63–100

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen S X. E-H type Mach configuration and its stability. Comm Math Phys, 2012, 315: 563–602

    Article  MATH  MathSciNet  Google Scholar 

  12. Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Interscience Publishers Inc., 1948

    MATH  Google Scholar 

  13. Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order, 2nd ed. Berlin-New York: Springer, 1983

    Book  MATH  Google Scholar 

  14. Grisvard P. Elliptic Problems in Nonsmooth Domains. London: Pitman, 1985

    MATH  Google Scholar 

  15. Li H, Ben-Dor G. A parametric study of Mach reflection in steady flows. J Fluid Mech, 1997, 341: 101–125

    Article  MATH  MathSciNet  Google Scholar 

  16. Li Y, Vogelius M. Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch Rat Mech Anal, 2000, 153: 91–151

    Article  MATH  MathSciNet  Google Scholar 

  17. Lieberman G M. Mixed boundary value problems for elliptic and parabolic differential equation of second order. J Math Anal Appl, 1985, 113: 329–352

    Google Scholar 

  18. Majda A. The Stability of Multi-dimensional Shock Fronts. Providence, RI: Amer Math Soc, 1983

    Google Scholar 

  19. Majda A. One perspective on open problems in multi-dimensional conservation laws. IMA Math Appl, 1991, 29: 217–237

    MathSciNet  Google Scholar 

  20. Morawetz C S. Potential theory for regular and Mach reflection of a shock at a wedge. Comm Pure Appl Math, 1994, 47: 593–624

    Article  MATH  MathSciNet  Google Scholar 

  21. Smoller J. Shock Waves and Reaction-diffusion Equations. New York: Springer-Verlag, 1983

    Book  MATH  Google Scholar 

  22. Von Neumann J. Oblique Reflection of Shocks. Washigton, DC: US Dept Comm Off of Tech Serv, PB–37079, 1943

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    ShuXing Chen

Authors
  1. ShuXing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ShuXing Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, S. Global existence and stability of a stationary Mach reflection. Sci. China Math. 58, 11–34 (2015). https://doi.org/10.1007/s11425-014-4909-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4909-9

Keywords

MSC(2010)

Search

Navigation