Existence of stationary vortex sheets for the 2D Euler equation

Abstract

We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $\Omega \subset {\mathbb{R}}^2$. Let ${\kappa }_i\ne 0$, $i=1,\dots ,m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point ${\mathbf{x}}_0=(x_{0,1},\dots ,x_{0,m})$ of the Kirchhoff-Routh function defined on ${\Omega }^m$ corresponding to $({\kappa }_1,\dots ,{\kappa }_m)$, we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and are perturbations of small circles centered near $x_i$, $i=1,\dots ,m$. 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.

References in corpus (6)

• Existence of corotating and counter-rotating vortex pairs for active scalar equations
• Gluing methods for vortex dynamics in Euler flows
• Uniformly rotating smooth solutions for the incompressible 2D Euler equations
• The Morse property for functions of Kirchhoff-Routh path type
• Remarks on stationary and uniformly-rotating vortex sheets: Rigidity results
• Rotating Equilibria of Vortex Sheets