  We prove the existence of nonradial classical solutions to the 2D
incompressible Euler equations with compact support. More precisely, for any
positive integer $k$, we construct compactly supported stationary Euler flows
of class $C^k(\mathbb{R}^2)$ which are not locally radial. The proof uses a
degree-theory-based bifurcation argument which hinges on three key ingredients:
a novel approach to stationary Euler flows through elliptic equations with
non-autonomous nonlinearities; a set of sharp regularity estimates for the
linearized operator, which involves a potential that blows up as the inverse
square of the distance to the boundary of the support; and overcoming a serious
problem of loss of derivatives by the introduction of anisotropic weighted
functional spaces between which the linearized operator is Fredholm.
