On the flexibility of 2D Euler steady states

TL;DR

This paper demonstrates that smooth steady states of the 2D Euler equations can exhibit greater flexibility than previously thought, allowing for vorticity functions that are not globally single-valued, even near stable configurations.

Contribution

It shows that the previously assumed functional relationship between stream function and vorticity does not hold for smooth steady states, revealing new structural possibilities.

Findings

Smooth steady states can have multiple critical points with non-single-valued vorticity functions.

There exist isolated branches of stable steady states not connected to analytic solutions.

Flexibility results are also established near the degenerate cellular flow on the flat torus.

Abstract

We consider steady states of the incompressible Euler equation on two-dimensional domains. For non-radial analytic steady states on bounded simply connected domains, it was shown previously that there must be a global functional relationship between the stream function and the vorticity. We show that this does not extend to smooth functions, even under further structural assumptions such as the Morse condition or Arnold's stability criterion. More precisely, we show that a broad class of steady states with multiple critical points can be perturbed to smooth steady states for which the vorticity is not a single-valued function of the stream function. We also establish an analogous flexibility result near the cellular flow on the flat torus, which is a degenerate case. As a consequence of our constructions, there are "branches" of smooth steady states that are isolated from analytic ones. In some cases, the resulting isolated branches can even consist entirely of linearly stable steady states.