Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems

Chen, Shuxing

doi:10.1090/S0002-9947-08-04493-0

Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems
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Trans. Amer. Math. Soc. 360 (2008), 5265-5289 Request permission

Abstract:

This paper is devoted to the study of a transonic shock in three-dimensional steady compressible flow passing a duct with a general section. The flow is described by the steady full Euler system, which is purely hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region. The upstream flow at the entrance of the duct is a uniform supersonic one adding a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic-hyperbolic system. The new ingredients of our paper contain the decomposition of the elliptic-hyperbolic system, the determination of the shock front by a pair of partial differential equations coupled with the three-dimensional Euler system, and the regularity analysis of solutions to the boundary value problems introduced in our discussion.

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