Rigidity results for finite energy solutions to the stationary 2D Euler equations

TL;DR

The paper studies finite-energy stationary solutions to the 2D Euler equations in $\\mathbb{R}^2$ with connected stagnation sets and shows that the associated stream function $u$ satisfies an autonomous semilinear elliptic equation $-\\Delta u=f(u)$ on the plane. Using energy estimates and a carefully adapted continuous Steiner symmetrization, it proves that $u$ is locally symmetric in every direction and, under a monotonicity condition at infinity, that the flow is circular (concentric streamlines) with decay of $|v|$ and integrability of $\\omega$ and $\\!B$. The results provide rigidity for finite-energy steady Euler flows, clarifying when such flows must be circular or radially symmetric. The combination of PDE techniques for semilinear equations and rearrangement methods yields new structural insight into the Euler dynamics under natural energy and stagnation-set assumptions.

Abstract

In this paper we prove rigidity results for classical solutions to the stationary 2D Euler equations in $\mathbb{R}^2$. Assuming that the velocity field has finite energy and that the stagnation set is connected, we prove that the corresponding stream function solves an autonomous semilinear elliptic equation. Under some extra conditions on the vorticity near infinity we can also prove that the streamlines are concentric circles. The proofs include several energy estimates on the behavior of the stream function at infinity, as well as an adaptation of the continuous Steiner symmetrization to our setting.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['prop:L2grad']}
• Corollary 2.4
• proof
• proof : Proof of Theorem \ref{['main:thm0']}