Singularities in Steady Axisymmetric Euler Flows With Swirl

ABSTRACT This paper investigates the formation of singularities in Bernoulli‐type free boundary problems for steady axially symmetric inviscid incompressible flows with general vorticity and swirl. We focus on the asymptotic behavior near degenerate points and provide a complete characterization of singular wave profiles on the free boundary. Degenerate points are classified into two types: Type 1, which occur away from the axis of symmetry, and Type 2, which occur on the axis. By developing a unified framework that combines variational methods for semilinear elliptic equations with novel analytical techniques for axially symmetric Euler systems, we show that the singular profiles at Type 1 degenerate points are fundamentally limited to three canonical forms: Stokes corners, cusps, or horizontal flat profiles. Our analysis employs monotonicity formulas to construct blow‐up limits at degenerate points and utilizes concentration‐compactness arguments. A key contribution is the derivation of a frequency formula that rigorously excludes the possibility of horizontal flat singularities. For Type 2 degenerate points, we prove that the singular profiles must be cusps.