Rigidity of stationary and uniformly rotating planar Euler flows in the C1 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≤1/2infℝ2ω0 or Ω≥1/2supℝ2ω0, then ω0 is radially symmetric, in particular, every stationary vorticity of one sign in the class C1(ℝ2)∩ L1(ℝ2) ∩ L∞(ℝ2) must be radially symmetric about some point. Furthermore, for Ω≠0, 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.
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BibTeX
@misc{fan2026rigidity,
title = {{Rigidity of stationary and uniformly rotating planar Euler flows in the $C^1$ Yudovich class}},
author = {Boquan Fan and Yuchen Wang and Chunjing Xie and Weicheng Zhan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30664},
eprint = {2609.30664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30664},
note = {CC0},
}
RIS
TY - UNPB
TI - Rigidity of stationary and uniformly rotating planar Euler flows in the C^1 Yudovich class
AU - Boquan Fan
AU - Yuchen Wang
AU - Chunjing Xie
AU - Weicheng Zhan
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30664
UR - https://arxiv.org/abs/2609.30664
AB - 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≤1/2inf_ℝ^2ω_0 or Ω≥1/2sup_ℝ^2ω_0, then ω_0 is radially symmetric, in particular, every stationary vorticity of one sign in the class C^1(ℝ^2)∩ L^1(ℝ^2) ∩ L^∞(ℝ^2) must be radially symmetric about some point. Furthermore, for Ω≠0, 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.
ER -