Homogeneous solutions to the 3D Euler system

TL;DR

This paper investigates stationary homogeneous solutions to the 3D Euler equations, revealing new classes, rigidity properties, and implications for Onsager's conjecture and intermittency phenomena.

Contribution

It introduces new classes of solutions, characterizes irrotational and tangential flows, and establishes rigidity and exclusion results for 3D Euler solutions.

Findings

Irrotational solutions have vanishing Bernoulli function

Tangential flows are 2D axisymmetric rotations

Anomalous energy flux vanishes at Onsager-critical homogeneity

Abstract

We study stationary homogeneous solutions to the 3D Euler equation. The problem is motivated be recent exclusions of self-similar blowup for Euler and its relation to Onsager conjecture and intermittency. We reveal several new classes of solutions and prove rigidity properties in specific categories of genuinely 3D solutions. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function; and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler flow on the sphere. We further show that in the case when homogeneity corresponds to the Onsager-critical state, the anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme $0$-dimensional intermittencies in dissipative flows.