A new approach to compactly supported stationary Euler flows

We construct smooth, compactly supported stationary Euler flows in two and three dimensions. The three-dimensional flows are axisymmetric with swirl, and their pressure varies along some streamlines. The planar flows are not locally radial. Both constructions start from simple radial profiles, which we perturb by solving scalar elliptic equations on fixed domains. We choose these equations so that their solutions can be exactly constant on open sets and meet these constant regions smoothly. Comparison arguments produce the constant regions, while an a priori regularity theorem ensures that all derivatives vanish where a solution meets them. This gives compact support and smoothness without having to describe the boundary of the support.