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(e) for it fluid in a bounded region Omega, with concentrated vorticities w(e) for epsilon > 0 small. More precisely, given a positive integer a and a sufficiently small complex number a, we find a family of stream functions psi(epsilon) which solve the Liouville equation with Dirac mass source, Delta psi(epsilon) + epsilon(2)e(psi epsilon) = 4 pi alpha delta(pn,epsilon) in Omega, psi(epsilon) = 0 on partial derivative Omega, for a suitable point p = p(a,epsilon) is an element of Omega, The vorticities w(epsilon) := -Delta(psi epsilon), concentrate in the sense that w(epsilon) + 4 pi alpha delta(pa,epsilon) - 8 pi Sigma(alpha+1)(j=1) delta(pa,epsilon) + aj -> 0 as epsilon -> 0, where the satellites a1,...,a(a+1) denote the complex (alpha + 1)-roots of a. The point p(a,epsilon) lies close to a zero point of a vector field explicitly built upon derivatives of order <= a + 1 of the regular part of Green's function of the domain.