Asymptotic properties of vortex-pair solutions for incompressible Euler equations in ${\mathbb{R}}^2$

A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $${\omega }_t+{\mathrm{\nabla }}^{\perp }\mathrm{\Psi }\cdot \mathrm{\nabla }\omega =0,\mathrm{\Psi }=(-\mathrm{\Delta }{)}^{-1}\omega ,\text{in }{\mathbb{R}}^2\times (0,\mathrm{\infty })$$ is a travelling wave solution of the form $\omega (x,t)=W(x_1-ct,x_2)$ where $W(x)$ is compactly supported and odd in $x_2$. We revisit the problem of constructing solutions which are highly $\epsilon$-concentrated around points $(0,\pm q)$, more precisely with approximately radially symmetric, compactly supported bumps with radius $\epsilon$ and masses $\pm m$. Fine asymptotic expressions are obtained, and the smooth dependence on the parameters $q$ and $\epsilon$ 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.