Two-dimensional Euler flows with concentrated vorticities

For a planar model of Euler flows proposed by Tur and Yanovsky (2004), we construct a family of velocity fields w_ε for a fluid in a bounded region Ω, with concentrated vorticities ω_ε for ε>0 small. More precisely, given a positive integer α and a sufficiently small complex number a, we find a family of stream functions ψ_ε which solve the Liouville equation with Dirac mass source, Δψ_ε +ε^2 e^{ψ_ε} =4πα δ_{p_{a,ε}} in Ω, ψ_ε =0 on ∂Ω, for a suitable point p=p_{a,ε}∈Ω. The vorticities ω_ε :=−Δψ_ε concentrate in the sense that ω_ε+4πα δ_{p_{a,ε}} −8π \sum_{j=1}^{α+1} δ_{p_{a,ε}+a_j} ⇀0 as ε→0, where the satellites a_1, . . . , a_{α+1} denote the complex (α + 1)-roots of a. The point p_{a,ε} lies close to a zero point of a vector field explicitly built upon derivatives of order ≤ α + 1 of the regular part of Green’s function of the domain.