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Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone

Published online by Cambridge University Press:  11 January 2016

Gang Xu  and
Huicheng Yin
Gang Xu
Affiliation:
Department of Mathematics and Institute of Mathematical Sciences, Nanjing University, Nanjing 210093, China, [email protected]
Huicheng Yin
Affiliation:
Department of Mathematics and Institute of Mathematical Sciences, Nanjing University, Nanjing 210093, China, [email protected]
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Abstract

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In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.

Keywords

35L7035L6535L6776N15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 199 , September 2010 , pp. 151 - 181
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[1] Asakura, F., Global solutions with a single transonic shock wave for quasilinear hyperbolic systems, Methods Appl. Anal. 4 (1997), 33–52.CrossRefGoogle Scholar
[2] Bernstein, S., Demonstration du théorème de M. Hilbert sur la nature analytique des solutions des équations du type elliptique sans l’emploi des séries normales, Math. Z. 28 (1928), 330–348.CrossRefGoogle Scholar
[3] Canic, S., Keyfitz, B. L., and Lieberman, G. M., A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), 484–511.3.0.CO;2-K>CrossRefGoogle Scholar
[4] Chen, S., A free boundary value problem of Euler system arising in supersonic flow past a curved cone, Tohoku Math. J. (2) 54 (2002), 105–120.CrossRefGoogle Scholar
[5] Chen, S., Xin, Z., and Yin, H., Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002), 47–84.CrossRefGoogle Scholar
[6] Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Interscience, New York, 1948.Google Scholar
[7] Cui, D. and Yin, H., Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2008), 101–133.Google Scholar
[8] Cui, D. and Yin, H., Global conic shock wave for the steady supersonic flow past a cone: Polytropic gas, J. Differential Equations 246 (2009), 641–669.CrossRefGoogle Scholar
[9] Friedman, A., On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech. 7 (1958), 43–59.Google Scholar
[10] Gilbarg, D. and Tudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1998.Google Scholar
[11] Kuz’min, A., Solvability of a problem for transonic flow with a local supersonic region, Nonlinear Differential Equations Appl. 8 (2001), 299–321.CrossRefGoogle Scholar
[12] Kuz’min, A., Boundary-Value Problems for Transonic Flow, Wiley, West Sussex, U.K., 2002.Google Scholar
[13] Lien, W. C. and Liu, T. P., Nonlinear stability of a self-similar 3-dimensional gas flow, Comm. Math. Phys. 204 (1999), 525–549.CrossRefGoogle Scholar
[14] Morawetz, C. S., On the nonexistence of continuous transonic flows past profiles, I, Comm. Pure Appl. Math. 9 (1956), 45–68; III, 11 (1958), 129–144; 17 (1964), 357–367; 10 (1975), 107–132.CrossRefGoogle Scholar
[15] Xin, Z. and Yin, H., Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2006), 101–132.CrossRefGoogle Scholar
[16] Xu, G. and Yin, H., Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone, J. Differential Equations 245 (2008), 3389–3432.CrossRefGoogle Scholar
[17] Xu, G. and Yin, H., Global multidimensional transonic conic shock wave for the perturbed supersonic flow past a cone, SIAM J. Math. Anal. 41 (2009), 178–218.Google Scholar
[18] Yin, H. and Zhou, C., On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges, J. Differential Equations 246 (2009), 4466–4496.CrossRefGoogle Scholar
[19] Zheng, Y., A global solution to a two-dimensional Riemann problem involving shocks as free boundaries, Acta Math. Appl. Sin. Engl. Ser. 19 (2003), 559–572.Google Scholar