Compactly supported solutions to the stationary 2D Euler equations with noncircular streamlines

In this talk we are interested in compactly supported solutions of the steady Euler equations. In 3D the existence of this type of solutions has been an open problem until the result of Gavrilov (2019). In 2D, instead, it is easy to construct solutions via radially symmetric stream functions. Low regularity solutions without radial symmetry have also been found in the literature, but even the $C^1$ case was left open. In this talk we construct such solutions with regularity $C^k$, for any fixed $k$ given. For the proof, we look for stream functions which are solutions to non-autonomous semilinear elliptic equations. In this framework we look for a local bifurcation around a conveniently constructed 1-parameter family of radial solutions. The linearized operator turns out to be critically singular, and is defined in anisotropic Banach spaces. This is joint work with A. Enciso (ICMAT, Madrid) and Antonio J. Fernández (UAM, Madrid).