## Abstract

A vortex pair solution of the incompressible 2d Euler equation in vorticity form

ωt+∇⊥Ψ⋅∇ω=0,Ψ=(−Δ)−1ω,in ℝ2×(0,∞)

is a travelling wave solution of the form ω(x,t)=W(x1−ct,x2) where W(x) is compactly supported and odd in x2. We revisit the problem of constructing solutions which are highly ε-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.

ωt+∇⊥Ψ⋅∇ω=0,Ψ=(−Δ)−1ω,in ℝ2×(0,∞)

is a travelling wave solution of the form ω(x,t)=W(x1−ct,x2) where W(x) is compactly supported and odd in x2. We revisit the problem of constructing solutions which are highly ε-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.

Original language | English |
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Publisher | arXiv |

Number of pages | 26 |

Publication status | Published - 17 Nov 2023 |