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Mach configuration in pseudo-stationary compressible flow


Author: Shuxing Chen
Journal: J. Amer. Math. Soc. 21 (2008), 63-100
MSC (2000): Primary 35L65, 35L67, 76N10
DOI: https://doi.org/10.1090/S0894-0347-07-00559-0
Published electronically: March 5, 2007
MathSciNet review: 2350051
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Abstract: This paper is devoted to studying the local structure of Mach reflection, which occurs in the problem of the shock front hitting a ramp. The compressible flow is described by the full unsteady Euler system of gas dynamics. Because of the special geometry, the motion of the fluid can be described by self-similar coordinates, so that the unsteady flow becomes a pseudo-stationary flow in this coordinate system. When the slope of the ramp is less than a critical value, the Mach reflection occurs. The wave configuration in Mach reflection is composed of three shock fronts and a slip line bearing contact discontinuity. The local existence of a flow field with such a configuration under some assumptions is proved in this paper. Our result confirms the reasonableness of the corresponding physical observations and numerical computations in Mach reflection.

In order to prove the result, we formulate the problem to a free boundary value problem of a pseudo-stationary Euler system. In this problem two unknown shock fronts are the free boundary, and the slip line is also an unknown curve inside the flow field. The proof contains some crucial ingredients. The slip line will be transformed to a fixed straight line by a generalized Lagrange transformation. The whole free boundary value problem will be decomposed to a fixed boundary value problem of the Euler system and a problem to updating the location of the shock front. The Euler system in the subsonic region is an elliptic-hyperbolic composite system, which will be decoupled to the elliptic part and the hyperbolic part at the level of principal parts. Then some sophisticated estimates and a suitable iterative scheme are established. The proof leads to the existence and stability of the local structure of Mach reflection.


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  • 1. S. Agmom, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623-777. MR 0125307 (23:A2610)
  • 2. S. Agmom, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964), 35-92. MR 0162050 (28:5252)
  • 3. G. Ben-Dor, Shock Waves Reflection Phenomena, Springer-Verlag, New York, 1992.
  • 4. M. Brio and J. K. Hunter, Mach reflection for the two-dimensional Burgers equation, Phys. D 60 (1992), 148-207. MR 1195600 (93i:35119)
  • 5. M. Costabel and M. Dauge, Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems, Math. Nachr. 162 (1993), 209-237. MR 1239587 (94k:35090)
  • 6. M. Costabel and M. Dauge, Stable asymptotics for elliptic systems on plane domains with corners, Comm. P.D.E. 19 (1991), 1677-1766. MR 1294475 (95g:35051)
  • 7. S. Canic and B. Kerfitz, Riemann problems for the two-dimensional unsteady transonic small disturbance equation, SIAM J. Appl. Math. 58 (1998), 636-665. MR 1617618 (99f:35127)
  • 8. S. Canic, B. Kerfitz and G. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), 484-511. MR 1733695 (2001m:76056)
  • 9. S. Canic, B. Kerfitz and E. H. Kim, A free boundary problems for unsteady transonic small disturbance equation: transonic regular reflection, Methods and Appl. Anal. 7(2000), 313-336. MR 1869288 (2002h:76077)
  • 10. S. Canic, B. Kerfitz and E. H. Kim, A free boundary problems for a quasilinear degenerate elliptic equation: transonic regular reflection, Comm. Pure Appl. Math. 55 (2002), 71-92. MR 1857880 (2003a:35206)
  • 11. Shuxing Chen, Linear approximation of shock reffection at a wedge with large angle, Comm. P.D.E. 21 (1996), 1103-1114. MR 1399192 (97f:35136)
  • 12. Shuxing Chen, Stability of Transonic Shock Front in M-D Euler System, Trans. Amer. Math. Soc. 307 (2005), 287-308. MR 2098096 (2005h:35278)
  • 13. Shuxing Chen, Stability of Mach Configuration, Comm. Pure Appl. Math. 59 (2006), 1-35. MR 2180082 (2006i:35294)
  • 14. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, 1948. MR 0029615 (10:637c)
  • 15. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, Heiderberg, New York, 2000. MR 1763936 (2001m:35212)
  • 16. M. Dauge, Elliptic Boundary Value Problems in Corner Domains - Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341, Springer-Verlag, Berlin, 1988. MR 961439 (91a:35078)
  • 17. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24, Pitman, London 1987. MR 775683 (86m:35044)
  • 18. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin, New York, 1983. MR 737190 (86c:35035)
  • 19. J. K. Hunter and M. Brio, Weak shock reflection, J. Fluid Mech. 410 (2000), 235-261. MR 1761754 (2001a:76094)
  • 20. L. F. Henderson, Region and Boundaries for Diffracting Shock Wave Systems, Zeitshrift für Angewandte Mathematik und Mechanik 67 (1987), 73-88.
  • 21. Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rat. Mech. Anal. 153 (2000), 91-151. MR 1770682 (2001m:35083)
  • 22. T. T. Li and W. C. Yu, Boundary value problem for quasi-linear hyperbolic systems, Duke Univ. Math. Ser. 5 (1985). MR 0823237 (88g:35115)
  • 23. A. Majda, One perspective on open problems in multi-dimensional conservation laws, IMA Math. Appl. 29(1991), 217-237. MR 1087085 (91m:35142)
  • 24. C. S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, Comm. Pure Appl. Math. 47(1994), 593-624. MR 1278346 (95g:76030)
  • 25. John von Neumann, Oblique Reflection of Shocks, U.S. Dept. Comm. Off. of Tech. Serv., Washigton, D.C., PB-37079 (1943).
  • 26. D. Serre, Ecoulements de fluides parfaits en deux variables indépendantes de type espace. Réflexion d'un choc plan par un dièdre compressif, Arch. Rational Mech. Anal. 132 (1995), 15-36. MR 1360078 (96i:76106)
  • 27. D. Serre, Shock reflection in gas dynamics, Handbook of Fluid Dynamics, 4, Elsevier, North-Holland, 2005.
  • 28. J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1983. MR 688146 (84d:35002)

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Additional Information

Shuxing Chen
Affiliation: School of Mathematical Sciences, Fudan University, and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University, Ministry of Education), Shanghai 200433, People’s Republic of China
Email: sxchen@public8.sta.net.cn

DOI: https://doi.org/10.1090/S0894-0347-07-00559-0
Keywords: Euler system, Mach configuration, pseudo-stationary flow, free boundary problem, elliptic-hyperbolic composed system
Received by editor(s): November 14, 2005
Published electronically: March 5, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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