A Liouville theorem for the Euler equations in a disk

Abstract

We present a symmetry result regarding stationary solutions of the 2D Euler equations in a disk. We prove that in a disk, a steady flow with only one stagnation point and tangential boundary conditions is a circular flow, which confirms a conjecture proposed by F. Hamel and N. Nadirashvili in [J. Eur. Math. Soc., 25 (2023), no. 1, 323-368]. The key ingredient of the proof is to use `local' symmetry properties for the non-negative solutions of semi-linear elliptic equations with a continuous nonlinearity in a ball, which can be established by a rearrangement technique called continuous Steiner symmetrization.