Courant-Friedrichs' Hypothesis and Stability of the Weak Shock Wave Satisfying the Lopatinski Condition

Dmitry Tkachev, Alexander Blokhin

Mechanics and Mathematics Department, Novosibirsk State University, Sobolev Institute of Mathematics, Novosibirsk, Russia.

DOI: 10.4236/ojapps.2013.31B1016                

Abstract

We are studying the problem of a stationary supersonic flow of an inviscid non-heat-conducting gas in thermodynamical equilibrium onto a planar infinite wedge. It is known that theoretically this problem has two solutions: the solution with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and the solution with a weak shock wave (when, generally speaking, the velocity behind the front of the shock wave is supersonic). In the present paper, the case of a weak shock wave is studied. It is proved that if the Lopatinski condition for the shock wave is satisfied (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed, and its classical solution is found. In this case, unlike the case when the uniform Lopatinski condition holds, additional plane waves appear. It is shown that for compactly supported initial data the solution of the linearized problem converges in finite time to the zero solution. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds in a weak sense these results complete the verification of the well-known Courant-Friedrichs' conjecture that the strong shock wave solution is unstable whereas the weak shock wave solution is stable.

Keywords

Weak Shock Wave; Asymptotic Stability (in the Sense of Lyapunov)

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. Courant and K. O. Friedrichs, “Supersonic Flow and Shock Waves,” Intersc. Publ., New York, 1948.
[2]	L. V. Ovsyannikov, “Lectures on Basic Gas Dynamics,” Institute of Computer Studies, Moscow, Izhevsk, 2003.
[3]	G. G. Chernyi, “Gas Dynamics,” Nauka, Moscow, 1988.
[4]	A. I. Rylov, “On the Possible Modes of Flow Round Tapered Bodies of Finite Thickness at Arbitrary Supersonic Velocities of the Approach Stream,” Journal of Applied Mathematic and Mechanics, Vol. 55, No. 1,1991, pp. 78-85.doi:10.1016/0021-8928(91)90065-3
[5]	A. A. Nikolski, “Plane Vorticity Gas Flows,” Teor. Issled. Mekhk. Zhidkosti i Gaza. Tr. Tsentral. Aero- i Gidrodinamich. Inst., Vol. 2122, 1984, pp. 74-85.
[6]	B. L. Rozhdestvenskii, “Corrected Theory of Supersonic Flow of a Nonviscous Gas about a Wedge,” Math Modeling Comput. Experiment 1, No. 1, 1993, pp. 15-18.
[7]	B. M. Bulakh, “Nonlinear Conical Flow,” Delft Univ. Press, Delft, 1985.
[8]	A. M. Blokhin, D. L. Tkachev and L. O. Baldan, “Study of the Stability in the Problem on Flowing Around a Wedge. The Case of Strong Wave,” Journal of Mathematical Analysis and Applications, Vol. 319, No. 1, 2006, pp. 248-277. doi:10.1016/j.jmaa.2005.10.023
[9]	A. M. Blokhin, D. L. Tkachev and Y. Y. Pashinin, “Stability Condition for Strong Shock Waves in the Problem of Flow around an Infinite Plane Wedge,” Nonlinear Analysis: Hybrid Systems, Vol. 2, 2008, pp. 1-17. doi:10.1016/j.nahs.2006.10.012
[10]	A. M. Blokhin and D. L. Tkachev, “Stability of a Supersonic Flow about a Wedge with Weak Shock Wave,” Sbornik: Mathematics, Vol. 200, No. 2, 2009, pp. 157-184.
[11]	A. M. Blokhin, D. L. Tkachev and Y. Y. Pashinin, “The Strong Shock Wave in the Problem on Flow around Infinite Plane Wedge,” Proceedings of the 11th interna-tional conference on hyperbolic problems, Springer-Verlag, Berlin, 2008, pp. 1037-1044.
[12]	D. L. Tkachev and A. M. Blokhin, “Courant - Friedrich's Hypothesis and Stability of the Weak Shock,” Proceedings of the 12th international conference on hyperbolic problems, Vol. 67, No. 2, 2009, pp. 958-966.
[13]	R. Sakamoto, “Hyperbolic Boundary Value Problems,” Iwanami Shoten, Tokyo, 1978. PMC-id:1537516
[14]	A. M. Blokhin, “Energy Integrals and their Applications to Problems of Gas Dynamics,” Nauka, Novosi-birsk, 1986.
[15]	A. M. Blokhin, R. S. Bushmanov and D. L. Tkachev, “The Lopatinski Condition in the Problem of Normal Gas Flow around the Wedge,” Preprint 271, Sobolev Institute of Mathematics, Novosibirsk, 2011. PMCid:3308588