Justification of the Courant–Friedrich's hypothesis in the case of a weak shock. Part I. Presentation of solution to linear problem

• Published in: Journal of mathematical analysis and applications Vol. 355; no. 1; pp. 41 - 52
• Main Authors: Blokhin, A.M., Tkachev, D.L.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.07.2009; Elsevier
• Subjects: Gas dynamics; Lyapunov's stability; Steady-state flow onto planar wedge; Weak shock; Weak shock; Gas dynamics; Lyapunov's stability; Steady-state flow onto planar wedge
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2009.01.040

As is well known, two solutions of the problem of a supersonic stationary inviscid nonheatconducting gas flow onto a planar infinite wedge are theoretically possible: the solution with a strong shock (the flow speed behind the shock is subsonic) and the solution with a weak shock (the flow speed behind the shock is supersonic). Unlike the well-studied case of a strong shock that is generically unstable [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Study of the stability in the problem on flowing around a wedge. The case of strong wave, J. Math. Anal. Appl. 319 (2006) 248–277; A.M. Blokhin, D.L. Tkachev, Yu.Yu. Pashinin, Stability condition for strong shock waves in the problem of flow around an infinite plane wedge, Nonlinear Anal. Hybrid Syst. 2 (2008) 1–17], R. Courant and K.O. Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948] assumed that the solution with a weak shock is asymptotically stable by Lyapunov. Presentation of classical solution to the corresponding problem which is found in the present paper is the first step on the way to justification of Courant–Friedrichs hypothesis on linear level.