UC Riverside

The two-dimensional Euler equations describe the velocity of an inviscid incompressible fluid. A classical vortex patch is a solution to the two-dimensional Euler equations whose initial vorticity is the indicator function of a bounded simply connected open region in the plane. Properties of the flow maps and vorticity transport in two dimensions ensure that the vorticity at any time will be the indicator function of the the image of the region, which remains simply connected and bounded. In 1991, Chemin proved in [Che91] that, in the whole plane, a vortex patch with an initially Holder continuous boundary maintains that boundary regularity for all time. A few years later, Serfati published an alternate strategy in [Ser94b] that simplifies certain aspects of Chemin's argument. Here, we prove that, for 0 < α < 1, C^{1,α} regularity of a vortex patch boundary persists for all time for fluids in a simply connected bounded domain that itself has a smooth boundary, as long as the patch is initially not touching the boundary. The proof reproduces a 1998 result of Depauw ([Dep98]) using simpler methods inspired by Serfati's approach, which is more easily adaptable to a bounded domain than the methods of Chemin and Depauw.