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.

We prove finite-time existence of solutions to the 2D Euler equations where the velocity grows slower than the square root of the distance from the origin and with vorticity up to linear growth provided the initial vorticity is quasibounded, or with growth in the vorticity related to the growth in the velocity provided the initial vorticity is stable. In the quasibounded case, we provide an example with unbounded vorticity and velocity. We also prove uniqueness for solutions with stable initial vorticity and with a certain bound on the modulus of continuity of the initial velocity. This thesis expands on the recent work of Cozzi and Kelliher, which substitutes Serfati’s identity in place of the Biot-Savart law to demonstrate uniqueness and short-time existence of bounded vorticity solutions with slow growing velocity.