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03 Nov 2007
Transonic Shocks in Compressible Flow Passing a Duct for ThreeDimensional Euler Systems
 Shuxing Chen,
 Hairong Yuan
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In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the threedimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a threedimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a freeboundary value problem. In order to solve the freeboundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the threedimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the threedimensional Euler system.
Communicated by T.P. Liu
The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902, and the Doctorial Foundation of National Educational Ministry 20050246001.
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 Title
 Transonic Shocks in Compressible Flow Passing a Duct for ThreeDimensional Euler Systems
 Journal

Archive for Rational Mechanics and Analysis
Volume 187, Issue 3 , pp 523556
 Cover Date
 20080301
 DOI
 10.1007/s002050070079z
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 Shuxing Chen ^{(1)}
 Hairong Yuan ^{(1)}
 Author Affiliations

 1. School of Mathematical Sciences and Key Laboratory for Nonlinear Sciences (Fudan University, Ministry of Education), Fudan University, Shanghai, 200433, China