A Two-Phase Free Boundary Problem for Axisymmetric Subsonic Euler Flows with Contact Discontinuities

Abstract

We study a free boundary problem for the three-dimensional steady compressible Euler equations in an infinitely long circular cylinder. The free boundary is a contact discontinuity separating an axisymmetric rotational subsonic flow and an axisymmetric potential subsonic flow, neither of which is prescribed a priori. The pressure continuity condition couples two unknown Euler phases through an unknown interface, leading to a genuinely two-phase free boundary problem. Using a Helmholtz decomposition, we reformulate the pressure continuity condition as nonlinear boundary conditions for the Helmholtz variables and develop a coupled iteration framework that simultaneously determines the free boundary and the two flow fields. Uniform estimates independent of the truncation length allow us to pass to the infinite-length limit and construct global solutions. We further establish the downstream asymptotic behavior of solutions, showing that the contact discontinuity becomes asymptotically cylindrical and that the radial velocity vanishes at infinity. To the best of our knowledge, this provides the first existence result and asymptotic characterization for a genuinely two-phase free boundary problem involving a contact discontinuity between an unknown rotational subsonic flow and an unknown potential subsonic flow in a three-dimensional infinitely long cylinder.