No abstract

In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simplyconnected bounded domain. The domain and the stationary point vortex must satisfy a condition expressed in terms of the conformal mapping from the domain to the unit disk. Explicit examples are discussed at the end.

In this paper, we prove that in bounded planar domains with C 2,α boundary, for almost every initial condition in the sense of the Lebesgue measure, the point vortex system has a global solution, meaning that there is no collision between two pointvortices or with the boundary. This extends the work previously done in [13] for the unit disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. In this paper, we make extensive use of the estimates given in [7]. We establish our relevant inequalities first in simply connected domains using conformal maps, then in multiply connected domains.