Asymptotic properties of vortex-pair solutions for incompressible Euler equations in R2

A vortex pair solution of the incompressible 2 d Euler equation in vorticity form co t + del perpendicular to II/ <middle dot> del co = 0 , = ( - 4 ) - 1 co , in R 2 x ( 0 , infinity ) is a travelling wave solution of the form co(x, t) = W (x 1 - ct, x 2 ) where W(x) is compactly supported and odd in x 2 . We revisit the problem of constructing solutions which are highly concentrated around points ( 0 , +/- q) , more precisely with approximately radially symmetric, compactly supported bumps with radius e and masses +/- m . Fine asymptotic expressions are obtained, and the smooth dependence on the parameters q and e for the solution and its propagation speed c are established. These results improve constructions through variational methods in [14] and in [5] for the case of a bounded domain. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).