Existence of stationary vortex sheets for the 2D Euler equation

We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $Ω\subset \mathbb{R}^2$. Let $κ_i\not=0$, $i=1,\ldots, m$, 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 $Ω^m$ corresponding to $(κ_1,\ldots, κ_m)$, 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.

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