Uniformly rotating Euler flows with compactly supported velocity

TL;DR

The paper proves the existence of nontrivial $C^k$-smooth uniformly rotating solutions to the 2D Euler equations with compact spatial support, for any $k$ and any angular velocity $\Omega$, by gluing compactly supported stationary flows to the exterior radial profile $-\frac{\Omega}{2}|x|^2$, yielding finite-energy, non-locally radial flows. The authors recast the problem in terms of a stream function $\phi$ satisfying $\nabla^{\perp}\phi \cdot \nabla \Delta\phi = 0$ inside the support and $\phi(x) = -\frac{\Omega}{2}|x|^2$ outside a ball, leveraging stationary flows from prior work (EFR) to build the interior, then attaching the exterior. The work also establishes rigidity constraints: if a boundary circle is non-isolated in the radial-dependence set, then $\nabla\phi$ must vanish on that circle, and in a semilinear setting $\Delta\phi+f(\phi)=0$ with exterior data, $\phi$ is locally radial (radial symmetry when $f\in C^1$). These results reveal that, beyond radial or patch configurations, the geometry of such rotating flows is highly constrained, yet the paper provides the first smooth, finite-energy examples with compact support that are not locally radial.

Abstract

For any positive integer $k$, we prove the existence of nontrivial $C^k$-smooth uniformly rotating solutions to the 2D incompressible Euler equations with compact spatial support. These solutions, which can be chosen to be small perturbations of radial flows, are the first example of smooth rotating flows with finite energy which are not locally radial. We also prove new rigidity results for rotating solutions which show that the geometric structure of these flows is severely constrained.

Figures (1)

• Figure 1: An illustrative diagram of the graph of a locally radial, but nonradial, function $\phi$ satisfying \ref{['E.Euler2']}. Observe that $\phi(x)$ coincides with $- \frac{\Omega}{2}|x|^2$ (dashed) outside a ball, so the velocity has compact support.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Theorem 5
• Remark 6
• proof : Proof of Theorem \ref{['T.main1']}
• Remark 7
• Lemma 8
• proof