Existence of stationary vortex sheets for the 2D Euler equation

TL;DR

This paper proves the existence of stationary vortex sheet solutions for the 2D Euler equations in bounded or unbounded domains, using the implicit function theorem to construct flows near critical points of the Kirchhoff-Routh function.

Contribution

It is the first nontrivial proof of stationary vortex sheets in domains, constructing solutions near critical points of the Kirchhoff-Routh function.

Findings

Existence of stationary vortex sheet solutions in 2D Euler flows.

Construction of solutions with large vorticity amplitude.

Perturbation of small circular vortex sheets near critical points.

Abstract

We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $\Omega\subset \mathbb{R}^2$. Let $\kappa_i\not=0$, $i=1,\ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point $\mathbf{x}_0=(x_{0,1},\ldots,x_{0,m})$ of the Kirchhoff-Routh function defined on $\Omega^m$ corresponding to $(\kappa_1,\ldots, \kappa_m)$, we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and are perturbations of small circles centered near $x_i$, $i=1,\ldots,m$. The proof is accomplished via the implicit function theorem with suitable choice of function spaces. This seems to be the first nontrivial result on the existence of stationary vortex sheets in domains.