We discuss several issues on the large time behavior of solutions of the incompressible Euler equations in dimension two. The point-vortex system, a discrete version of the Euler equations, gives a good indication on what this large time behavior should be. Of particular interest are the so-called self-similar configurations of point vortices which either collapse to a point or, when reversing time, grow to infinity like the square root of the time. We consider such a self-similar configuration of point vortices and we find a condition on the point vortices such that a vorticity initially confined around one point vortex will remain "confined" around the point vortex. We will also discuss its relevance to the large time behavior of the Euler equations. This is joint work with Carlo Marchioro.