The two dimensional Euler equations on singular exterior domains (Q887923)

The existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets is established. These open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with \(L^p\) vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called \(\gamma\)-convergence of open sets. The authors conclude with a discussion of Sobolev capacity and Hausdorff convergence.