In this paper, we investigate the well-posedness theory of compressible jet flows for two dimensional steady Euler system with non-zero vorticity. One of the key observations is that the stream function formulation for two dimensional compressible steady Euler system with non-zero vorticity enjoys a variational structure, so that the jet problem can be reformulated as a domain variation problem. This allows us to adapt the framework developed by Alt, Caffarelli and Friedman for the one-phase free boundary problems to obtain the existence and uniqueness of smooth solutions to the subsonic jet problem with non-zero vorticity. We also show that there is a critical mass flux, such that as long as the incoming mass flux does not exceed the critical value, the well-posedness theory holds true.
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