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

Gui-Qiang Chen; Beixiang Fang

doi:10.3934/dcds.2009.23.85

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2009, Volume 23, Issue 1&2: 85-114. Doi: 10.3934/dcds.2009.23.85

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

Gui-Qiang Chen^1, and
Beixiang Fang^2,

1.
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730
2.
Department of Mathematics,, Shanghai Jiaotong University, Shanghai 200240

Received: October 31, 2007

Revised: June 30, 2008

Published: April 2009

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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 beformed in the flow. In this paper we analyze the stability oftransonic shock-fronts in three-dimensional steady potential flowpast a perturbed cone. We establish that the self-similar transonicshock-front solution is conditionally stable in structure withrespect to the conical perturbation of the cone boundary and theupstream flow in appropriate function spaces. In particular, it isproved that the slope of the shock-front tends asymptotically to theslope of the unperturbed self-similar shock-front downstream atinfinity.

Keywords:

Transonic shock-fronts,
stability,
steady conical flow and potential flow equations.

Mathematics Subject Classification: 35L65, 35L67, 35M10, 35B35, 76H05, 76N10.

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

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