Desingularization of multiscale solutions to planar incompressible Euler equations

In this paper, we consider the desingularization of multiscale solutions to 2D steady incompressible Euler equations. When the background flow ψ₀ is nontrivial, we construct a family of solutions which has nonzero vorticity in small neighborhoods of a given collection of points. One prescribed set of points comprises minimizers of the Kirchhoff-Routh function, while another part of points is on the boundary determined by both ψ₀ and Green's function. Moreover, heights and circulation of solutions have two kinds of scale. We prove the results by considering maximization problem for the vorticity and analyzing the asymptotic behavior of the maximizers.