In 1963 V. I. Yudovich proved the existence and uniqueness of weak solutions of the incompressible 2D Euler equations assuming that the vorticity, which is the curl of velocity, is bounded and integrable in the full plane. A few extensions of this result have been established, most notably by Yudovich himself and, also, by M. Vishik, always assuming some decay of vorticity at infinity. Paradoxically, if the vorticity is doubly-periodic then there is no difficulty in establishing well-posedness of weak solutions, as long as the vorticity is also bounded. In this talk I will report on work in progress aimed at extending, for 2D flows in a domain exterior to an island, well-posedness of weak solutions to include all vorticities which are bounded and are the curl of a bounded velocity field. This work is related to recent results by Taniuchi, Tashiro and Yoneda and it builds on previous, albeit incomplete, work due to Ph. Serfati, where flow in the full plane was considered.

This is joint work with J. P. Kelliher (UCR) and M. C. Lopes Filho (UNICAMP).