A steady Euler flow with compact support

A steady smooth Euler flow with support in the vicinity of a helix

In this article we construct a smooth Euler flow supported in a neighborhood of a helix. It may be considered a generalization of a similar solution found by the author for a circle.

Steady Euler flows and the Faddeev-Skyrme model with mass term

We point out a duality between steady incompressible Euler flows and solutions of the strongly coupled Faddeev-Skyrme sigma model with potential (mass) term. We supplement this result with various applications and several explicit examples.

A characterization of 3D steady Euler flows using commuting zero-flux homologies

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.

Steady Euler flows on the 3-sphere and other Sasakian 3-manifolds

We present new steady Euler solutions on the (round) 3-sphere, that bifurcate from an ansatz proposed by Khesin, Kuksin and Peralta-Salas, showing that these previously known solutions are not isolated. We also extend this ansatz to any Sasakian 3-manifold, such as the Heisenberg group and $SL(2, \mathbb{R})$.

The periodic orbit conjecture for steady Euler flows

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a counterexample by Sullivan. However, it is satisfied under the geometric condition of being geodesible. In this work, we use the recent characterization of Eulerisable flows (or more generally flows admitting a strongly adapted one-form) to prove that the conjecture remains true for this larger class of vector fields.

三维球面上的稳态欧拉流及其关联的Faddeev-Skyrme解

A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution

本文构造了三维球面上具有两个独立首次积分且与到二维球面的近浸没纤维相切的稳态欧拉流，并证明该浸没是带标准势的四次Faddeev-Skyrme模型的临界点。