Local structure of the set of steady-state solutions to the 2D incompressible Euler equations

TL;DR

This paper investigates the geometric structure of steady-state solutions to the 2D incompressible Euler equations, establishing a local one-to-one correspondence with co-adjoint orbits under certain conditions, enhancing understanding of their solution space.

Contribution

It proves a local one-to-one correspondence between steady-states and co-adjoint orbits for 2D Euler equations under non-degeneracy assumptions, linking geometric structures to steady solutions.

Findings

Established a local bijection between steady-states and co-adjoint orbits.

Provided geometric insights into the structure of steady solutions.

Extended finite-dimensional geometric concepts to the infinite-dimensional Euler setting.

Abstract

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler's equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.