Rigidity of stationary and uniformly rotating planar Euler flows in the $C^1$ Yudovich class

Abstract

We prove a rigidity theorem for stationary and uniformly rotating solutions of the two-dimensional incompressible Euler equation with vorticity $ ω_0\in C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2). $ If the angular velocity $Ω$ satisfies \[ Ω\leq\frac12\inf_{\mathbb{R}^2}ω_0 \qquad\text{or}\qquad Ω\geq\frac12\sup_{\mathbb{R}^2}ω_0, \] then $ω_0$ is radially symmetric, in particular, every stationary vorticity of one sign in the class $C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$ must be radially symmetric about some point. Furthermore, for $Ω\neq0$, the center is necessarily the origin; The proof is based on the analysis for level-set geometry of a normalized stream function. A componentwise Bernoulli formula, together with the isoperimetric inequality and Pohozaev identities, yields a nonnegative defect measure encoding both geometric defects and topological branching. An analysis at infinity forces this measure to vanish, reducing the problem to a semilinear elliptic equation with a bounded nonnegative Borel nonlinearity. Radial symmetry then follows from a theorem of P.-L.\ Lions.