2023/11/17 by Dávila, Juan, del Pino, Manuel, Musso, Monica +1 · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.12039
A \em vortex pair solution of the incompressible 2d Euler equation in vorticity form ωt + ∇^⊥ Ψ⋅ ∇ ω= 0 , Ψ= (-Δ)-1 ω, \hboxin 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.