Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone

Pages: 85 - 114, Volume 23, Issue 1/2, January/February 2009

doi:10.3934/dcds.2009.23.85       Abstract        Full Text (422.2K)       Related Articles

Gui-Qiang Chen - Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, United States (email)
Beixiang Fang - Department of Mathematics,, Shanghai Jiaotong University, Shanghai 200240, China (email)

Abstract: For an upstream supersonic flow past a straight-sided cone in R³ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.

Keywords:  Transonic shock-fronts, stability, steady conical flow and potential flow equations
Mathematics Subject Classification:  35L65, 35L67, 35M10, 35B35, 76H05, 76N10.

Received: November 2007;      Revised: July 2008;      Published: September 2008.