We study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations. From the numerical experiments implemented in [7, 16, 17] it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle [14, 18], however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments. The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.
Keywords: Euler equations, steady vortex pairs, desingularization.
@article{JEDP_2016____A6_0, author = {Hmidi, Taoufik and Mateu, Joan}, title = {Corotating and counter-rotating vortex pairs for {Euler} equations}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, note = {talk:6}, pages = {1--16}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2016}, doi = {10.5802/jedp.647}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jedp.647/} }
TY - JOUR AU - Hmidi, Taoufik AU - Mateu, Joan TI - Corotating and counter-rotating vortex pairs for Euler equations JO - Journées équations aux dérivées partielles N1 - talk:6 PY - 2016 SP - 1 EP - 16 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.647/ DO - 10.5802/jedp.647 LA - en ID - JEDP_2016____A6_0 ER -
%0 Journal Article %A Hmidi, Taoufik %A Mateu, Joan %T Corotating and counter-rotating vortex pairs for Euler equations %J Journées équations aux dérivées partielles %Z talk:6 %D 2016 %P 1-16 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.647/ %R 10.5802/jedp.647 %G en %F JEDP_2016____A6_0
Hmidi, Taoufik; Mateu, Joan. Corotating and counter-rotating vortex pairs for Euler equations. Journées équations aux dérivées partielles (2016), Talk no. 6, 16 p. doi : 10.5802/jedp.647. http://www.numdam.org/articles/10.5802/jedp.647/
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