Two-dimensional Euler flows with concentrated vorticities

Abstract

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 + ε 2 εψε =4π αδ pa ε in Ω,ψ ε = 0 on ω, for a suitable point p = p a , ε ε 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. © 2010 American Mathematical Society.