Skip to main content
SpringerLink
Log in

Two-Dimensional Regular Shock Reflection for the Pressure Gradient System of Conservation Laws

  • Original Papers
  • Published:
Acta Mathematicae Applicatae Sinica Aims and scope Submit manuscript

Abstract

We establish the existence of a global solution to a regular reflection of a shock hitting a ramp for the pressure gradient system of equations. The set-up of the reflection is the same as that of Mach’s experiment for the compressible Euler system, i. e., a straight shock hitting a ramp. We assume that the angle of the ramp is close to 90 degrees. The solution has a reflected bow shock wave, called the diffraction of the planar shock at the compressive corner, which is mathematically regarded as a free boundary in the self-similar variable plane. The pressure gradient system of three equations is a subsystem, and an approximation, of the full Euler system, and we offer a couple of derivations.

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. Agarwal, R.K., Halt, D.W. A modified CUSP scheme in wave/particle split form for unstructured grid Euler flows. In Frontiers of Computational Fluid Dynamics, Caughey, D.A. and Hafez, M.M. eds., Chichester, Wiley, New York, 1994

  2. Azzam, A. Smoothness properties of mixed boundary value problems for elliptic equations in sectionally smooth n-dimensional domains. Ann. Polon. Math., 40: 81–93 (1981)

    MATH  MathSciNet  Google Scholar 

  3. Bazhenova, T.V., Gvozdeva, L.G., Nettleton, M.A. Unsteady interactions of shock waves. Prog. Aero. Sci., 21: 249–331 (1984)

    Article  Google Scholar 

  4. Ben-Dor, G. Steady, pseudo-steady and unsteady shock wave reflections. Prog. Aero. Sci., 25: 329–412 (1988)

    Article  Google Scholar 

  5. Bitsadze, A.V. Boundary value problems for second order elliptic equations. Translated from the Russian by Scripta Technica, ltd., Translation edited by Laird, M.J. North-Holland Publishing Company, Amsterdam, John Wiley & Sons Inc., New York (Intersciences Publishers Division), 1968

  6. Borrelli, R. L. The singular, second order oblique derivative problem. J. Math. Mech., 16: 51–81 (1966)

    MATH  MathSciNet  Google Scholar 

  7. Borrelli, R.L. The I-Fredholm property for oblique derivative problems. Math. Z., 101: 103–122 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brio, M., Hunter, J.K. Irregular reflection of weak shocks, Proceedings of the 5th Inter. Conference on Hyperbolic Problems: Theory, Numerics, Applications. Editors Glimm, J., Graham, M.J., Grove, J.W., Plohr, B.J. World Scientific, 1996, 347–353

  9. Brio, M., Hunter, J.K. Mach reflection for the two-dimensional Burgers equation. Physica D, 60: 194–207 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Čanić, S., Keyfitz, B., Kim, E. Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection. Methods and Applications of Analysis, 7: 313–336 (2000)

    MathSciNet  Google Scholar 

  11. Čanić, S., Keyfitz, B., Kim, E. A free boundary problem for quasilinear degenerate elliptic equation: Regular reflection of weak shocks. Commun. Pure Appl. Math., LV: 71–92 (2002)

    Google Scholar 

  12. Čanić, S., Keyfitz, B., Kim, E. Free boundary problems for nonlinear wave systems: Interacting shocks, preprint 2003

  13. Čanić, S., Keyfitz, B., Lieberman, G. A proof of existence of perturbed steady transonic shocks via a free boundary problem. Commun. Pure Appl. Math., 53: 484–511 (2000)

    Google Scholar 

  14. Chen, Gui-Qiang, Feldman, M. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Amer. Math. Soc., 16: 461–494 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chen, Gui-Qiang, Feldman, M. Steady transonic shock and free boundary problems in infinite cylinders for the Euler equations. Comm. Pure Appl. Math., 57: 310–356 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chen, Shu-xing. Existence of stationary supersonic flow past a pointed body. Arch Rat. Mech. Anal., 156: 141–181 (2001)

    Article  MATH  Google Scholar 

  17. Chen, Shuxing: A free boundary problem of elliptic equation arising in supersonic flow past a conical body, Zeitschrift fur Angewandt Mathematik und Physik, 54: 1–23 (2003)

  18. Chen, Shu-xing. Stability of transonic shock front in multidimensional Euler system. preprint October 24, 2003

  19. Chen, Shu-xing, Xin, Zhou-ping, Yin, Hui-cheng. Global shock waves for the supersonic flow past a perturbed cone. preprint 2004

  20. Chen, Shu-xing, Xin, Zhou-ping, Yin, Hui-cheng. Unsteady supersonic flow past a wedge. preprint 2004

  21. Cole, J.D., Cook, L.P. Transonic Aerodynamics. North-Holland, Amsterdam, 1986

  22. Courant, R., Friedrichs, K.O. Supersonic Flow and Shock Waves. Applied Mathematical Sciences, Volume 21. Springer-Verlag, New York, Heidelberg, Berlin, 1948, 1976

  23. Dai, Zi-huan, Zhang, Tong. Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics. Arch. Ration. Mech. Anal., 155: 277–298 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Egorov, Ju.V., Kondrat’ev, V.A. The oblique derivative problem, Mathematics of the ussr-sbornik, 7: 139–169 (1969)

    Google Scholar 

  25. Elling, V., Liu, T.P. The ellipticity principle for steady and selfsimilar polytropic potential flow, preprint, Stanford, 2004

  26. Gamba, I.M. Rosales, R.R., Tabak, E.G. Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations. Comm. Pure Appl. Math., 52: 763–779 (1999)

    Article  MathSciNet  Google Scholar 

  27. Gilbarg, D., L. Hömander. Intermediate Schauder estimates. Arch. Rat. Mech. Anal., 74: 297–318 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  28. Gilbarg, D., Trudinger, N. Elliptic partial differential equations of second order. Springer, New York, 2nd ed, 1983

  29. Glass, I.I., Sislian, J.P. Nonstationary flows and shock waves. Oxford Engineering Science series, 39, 1994

  30. Glimm, J., Majda, A. Multidimensional Hyperbolic Problems and Computations. Springer-Verlag, Berlin, 1991

  31. Grisvard, P. Elliptic problems in nonsmooth domains. Pitman Advanced Publishing Program, Boston, 1985

  32. Guan, Peng-fei, Sawyer, E. Regularity estimates for the oblique derivative problem on non-smooth domains (I). Chin. Ann. of Math., 16B: 3: 299–324 (1995)

    Google Scholar 

  33. Hadamard, J. Leçons sur la propagation des ondes et les équations de l’hydrodynamique. Hermann, Paris, 1903

  34. Henderson, L.F. Regions and boundaries for diffracting shock wave systems. Z. Angew. Math. Mech., 67: 73–86 (1987)

    Google Scholar 

  35. Hörmander, L. Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math., 83: 129–209 (1966)

    MATH  MathSciNet  Google Scholar 

  36. Hornung, H. Regular and Mach reflection of shock waves. Ann. Rev. Fluid Mech., 18: 33–58 (1985)

    Google Scholar 

  37. Hunter, J. K. Private communication, November, 2003

  38. Hunter, J.K. Irregular reflection of weak shocks II. Derivation of the asymptotic equations, unpublished manuscript (1995)

  39. Hunter, J.K., Keller, J.B. Weak shock diffraction Wave Motion, 6: 79–89 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  40. Hunter, J.K., Tesdall, A.M. Self-similar solutions for weak shock reflection. SIAM Journal on Applied Mathematics, 63: 42–61 (2002)

    MATH  MathSciNet  Google Scholar 

  41. Keller, J.B., Blank, A. Diffraction and reflection of pulses by wedges and corners. Comm. Pure Appl. Math., 4: 75–94 (1951)

    MATH  MathSciNet  Google Scholar 

  42. Kerimov, T.M. The classical solution of the mixed problem for second order elliptic equations (Russian). Uspehi Mat. Nauk, 32(6) (198): 255–256 (1977)

    MATH  MathSciNet  Google Scholar 

  43. Kim, E.H., Song, K. Classical solutions for the pressure-gradient equation in the non-smooth and nonconvex domain. J. Math. Anal. Appl., 293: 541–550 (2004)

    MATH  MathSciNet  Google Scholar 

  44. Li, Jie-quan, Zhang, T., Yang, Shu-li. The two-dimensional Riemann problem in gas dynamics. Pitman monographs and surveys in pure and applied mathematics 98. Addison Wesley Longman Limited, 1998

  45. Li, Yin-fan, Cao, Yiming. Second order “large particle” difference method. Sciences in China (series A), 8: 1024–1035 (1985) (in Chinese)

    Google Scholar 

  46. Lieberman, G.M. The Perron process applied to oblique derivative problems. Advances in Mathematics, 55: 161–172 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  47. Lieberman, G.M. Mixed boundary value problems for elliptic and parabolic differential equations of second order. Journal of Mathematical Analysis and Applications, 113: 422–440 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  48. Lieberman, G.M. Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data. Journal für die Reine und Angewandte Mathematik, 389: 1–21 (1988)

    MATH  MathSciNet  Google Scholar 

  49. Lieberman, G.M. Optimal Hölder regularity for mixed boundary value problems. Journal of Mathematical Analysis and Applications, 143: 572–586 (1989)

    Article  MathSciNet  Google Scholar 

  50. Maugeri, A., Palagachev, D.K., Vitanza, C. A singular boundary value problem for uniformly elliptic operators. Journal of Mathematical Analysis and Applications, 263: 33–48 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  51. Maz’ya, V., and Shaposhnikova, T. Jacques Hadamard, A Universal Mathematician, American Mathematical Society, London Mathematical Society, History of Mathematics, Vol. 14, 1998

  52. Miranda, C. Partial differential equations of elliptic type, 2nd revised edition. Springer-Verlag, Berlin, 1970

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

    MATH  MathSciNet  Google Scholar 

  54. Pack, D.C. The reflexion and diffraction of shock waves. J. Fluid Mech., 18: 549–576 (1964)

    MATH  Google Scholar 

  55. Paneah, B.P. The Oblique Derivative Problem The Poincaré Problem, Wiley-VCH, 2000

  56. Popivanov, P., Kutev, N. The tangential oblique derivative problem for nonlinear elliptic equations. Comm. in Partial Differential Equations, 14: 413–428 (1989)

    MATH  MathSciNet  Google Scholar 

  57. Popivanov, P., Kutev, N. Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations. C. R. Acad. Sci. Paris (Series I), 334: 661–666 (2002)

    MATH  MathSciNet  Google Scholar 

  58. Popivanov, P.R., Palagachev, D.K. The degenerate oblique derivative problem for elliptic and parabolic equations. Academic Verlag, VCH Publishers, Inc., Berlin, New York, 1997

  59. Rosales, R.R., Tabak, E.G. Caustics of weak shock waves. Phys. Fluids, 10: 206–222 (1998)

    Article  MathSciNet  Google Scholar 

  60. Serre, D. Écoulements de fluides parfaits en deux variables indépendantes de type espace. Réflexion d’un choc plan par un dièdre compressif. Arch. Rat. Mech. Anal., 132: 15–36 (1995) (English rewriting, see “Handbook of Mathematical Fluid Dynamics” eds Serre and Friedlander, Elsevier, to appear, 2006)

    Article  MATH  MathSciNet  Google Scholar 

  61. Song, Kyungwoo. The pressure-gradient system on non-smooth domains. Comm. in Partial Differential Equations, 28: 199–221 (2003)

    MATH  Google Scholar 

  62. Tabak, E.G., Rosales, R.R. Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection. Phys. Fluids A, 6: 1874–1892 (1994)

    MATH  MathSciNet  Google Scholar 

  63. Winzell, B. A boundary value problem with an oblique derivative. Comm. in Partial Differential Equations, 6: 305–328 (1981)

    MATH  MathSciNet  Google Scholar 

  64. Xin, Zhou-ping, Yin, Huicheng. Transonic shock in a nozzle I, 2-D case. Preprint, 2003

  65. Xin, Zhou-ping, Yin, Huicheng. Transonic shock in a nozzle II, 3-D case. IMS of CUHK, Preprint, 2003

  66. Xin, Zhou-ping, Yin, Huicheng. Transonic flow for the supersonic gas past a wedge. Preprint, 2003

  67. Zaremba, S. Sur un problème mixte relatif a l’équation de Laplace. Bull. Int. Acad. Cracovie, Classe Math. Nat. (series A): 313–344 (1910)

  68. Zhang, P., Li, Jiequan; Zhang, Tong: On two-dimensional Riemann problem for pressure-gradient equations of Euler system. Discrete and Continuous Dynamical Systems, 4: 609–634 (1998)

    MATH  MathSciNet  Google Scholar 

  69. Zhang, Yongqian. Steady supersonic flow past an almost straight wedge with large vertex angle. Journal of Differential Equations, 192: 1–46 (2003)

    Article  MathSciNet  Google Scholar 

  70. Zheng, Yuxi. Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions. Communications in Partial Differential Equations, 22: 1849–1868 (1997)

    MATH  MathSciNet  Google Scholar 

  71. Zheng, Yuxi. Systems of Conservation Laws: Two-Dimensional Riemann Problems. 38 PNLDE, Birkhäuser, Boston, 2001

  72. Zheng, Yuxi. A global solution to a two-dimensional Riemann problem involving shocks as free boundaries. Acta Mathematicae Applicatae Sinica, English Series, 19: 559–572 (2003)

    MATH  Google Scholar 

  73. Zheng, Yuxi. Shock reflection for the Euler system, Proceedings for the Hyperbolic Problem, Osaka, Japan, September 13–17, 2004

  74. Zheng, Yuxi. Regular shock reflection for the Euler system, work in progress.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

    Yuxi Zheng

Authors
  1. Yuxi Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuxi Zheng.

Additional information

Partially supported by NSF-DMS-0305497 and 0305114.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zheng, Y. Two-Dimensional Regular Shock Reflection for the Pressure Gradient System of Conservation Laws. Acta Mathematicae Applicatae Sinica, English Series 22, 177–210 (2006). https://doi.org/10.1007/s10255-006-0296-5

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-006-0296-5

Keywords

2000 MR Subject Classification

Search

Navigation