Geodesic
Existence of stationary vortex sheets for the 2D incompressible Euler equation
Canadian journal of mathematics, Tome 75 (2023) no. 3, pp. 828-853

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DOI

We construct a new type of planar Euler flows with localized vorticity. Let $\kappa _i\not =0$, $i=1,\ldots , m$, be m arbitrarily fixed constants. For any given nondegenerate 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 concentrate on curves perturbed from small circles centered near $x_{0,i}$, $i=1,\ldots ,m$. The proof is accomplished via the implicit function theorem with suitable choice of function spaces.
DOI : 10.4153/S0008414X22000190
Mots-clés : Euler equation, vortex sheets, non-degenerate, the Birkhoff–Rott operator, implicit function theorem 
Cao, Daomin; Qin, Guolin; Zou, Changjun. Existence of stationary vortex sheets for the 2D incompressible Euler equation. Canadian journal of mathematics, Tome 75 (2023) no. 3, pp. 828-853. doi: 10.4153/S0008414X22000190
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     title = {Existence of stationary vortex sheets for the {2D} incompressible {Euler} equation},
     journal = {Canadian journal of mathematics},
     pages = {828--853},
     year = {2023},
     volume = {75},
     number = {3},
     doi = {10.4153/S0008414X22000190},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X22000190/}
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