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

Abstract

A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$\ddot{I}{‰}_t+{\nabla }^{\perp }\hat{I}¨\cdot \nabla \ddot{I}‰=0,\hat{I}¨=(-\hat{I}”{)}^{-1}\ddot{I}‰,in{\mathbb{R}}^2\times (0,\infty )$$ is a travelling wave solution of the form $\ddot{I}‰(x,t)=W(x_1-ct,x_2)$ 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.

To appear in Journal of Differential Equations. arXiv admin note: substantial text overlap with arXiv:2310.07238