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Existence of stationary vortex sheets for the 2D incompressible Euler equation

Part of: Incompressible inviscid fluids Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  05 May 2022

Daomin Cao ,
Guolin Qin Open the ORCID record for Guolin Qin [Opens in a new window]  and
Changjun Zou
Daomin Cao
Affiliation:
Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, P.R. China, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, P.R. China e-mail: dmcao@amt.ac.cn
Guolin Qin*
Affiliation:
Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, P.R. China, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, P.R. China e-mail: dmcao@amt.ac.cn
Changjun Zou
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, P.R. China e-mail: zouchangjun@amss.ac.cn
*
e-mail: qinguolin18@mails.ucas.ac.cn
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Abstract

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.

Keywords

Euler equationvortex sheetsnon-degeneratethe Birkhoff–Rott operatorimplicit function theorem

MSC classification

Primary: 76B47: Vortex flows
Secondary: 35Q31: Euler equations 76B03: Existence, uniqueness, and regularity theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 3 , June 2023 , pp. 828 - 853
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work was supported by the NNSF of China Grant (No. 11831009).

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