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

A vortex pair solution of the incompressible 2d Euler equation in vorticity form ω _t+∇ ^⊥Ψ⋅∇ω=0,Ψ=(−Δ) ⁻¹ω,in R ²×(0,∞) is a travelling wave solution of the form ω(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 ε and masses ±m. Fine asymptotic expressions are obtained, and the smooth dependence on the parameters q and ε 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.