Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation

2025/06/05 by Boquan Fan, Yuchen Wang, Fan, Boquan +3 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.05034

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that any uniformly rotating solution of the 2D incompressible Euler equation with compactly supported vorticity ω must be radially symmetric whenever its angular velocity satisfies Ω∈ (-∞,inf ω/ 2] ∪ [ sup ω/ 2, +∞ ), in both the patch and smooth settings. This result extends the rigidity theorems established in \citeGom2021MR4312192 (Duke Math. J.,170(13):2957-3038, 2021), which were confined to the case of non-positive angular velocities and non-negative vorticity. Moreover, our results do not impose any regularity conditions on the patch beyond requiring that its boundary consists of Jordan curves, thereby refining the previous result to encompass irregular vortex patches.

Citations

• Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc
• On the rigidity of the 2D incompressible Euler equations
• Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches
• Existence of non-trivial non-concentrated compactly supported stationary solutions of the 2D Euler equation with finite energy
• Symmetry and stability of non-negative solutions to degenerate elliptic equations in a ball
• Symmetry in stationary and uniformly-rotating solutions of active scalar equations
• A Liouville theorem for the Euler equations in the plane
• Symmetry for a general class of overdetermined elliptic problems
• Degenerate bifurcation of the rotating patches
• On the trivial solutions for the rotating patch model
• Doubly connected V-states for the planar Euler equations
• Weak solutions to the stationary incompressible Euler equations
• The h-principle and the equations of fluid dynamics
• Radial symmetry of solutions to diffusion equations with discontinuous\n nonlinearities

Cited by

• Uniformly rotating Euler flows with compactly supported velocity

Related

• Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc
• Symmetry in stationary and uniformly-rotating solutions of active scalar equations
• Uniformly rotating Euler flows with compactly supported velocity
• On radial symmetry of rotating vortex patches in the disc
• Symmetry of uniformly rotating solutions for the vortex-wave system