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 ws for a fluid in a bounded region Q, with concentrated vorticities w_ε for _ε ≥ 0 small. More precisely, given α positive integer a 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, Δ ψ_ε + ε²_εψε =4π αδ_paε in Ω,ψ_ε = 0 on ω, for a suitable point p = p_a,_εε ω. The vorticities W_ε = - Δφ_ε concentrate in the sense that [Eqation Present] where the satellites ai,⋯,a_α+1 denote the complex (a + 1)-roots of a.The point pa<s 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.