Transonic Shocks in 3-d Compressible Flow Passing a Duct with a General Section for Euler Systems

This paper is devoted to the study of a transonic shock in threedimensional 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 elliptichyperbolic 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.