Global regularity for a Birkhoff–Rott- approximation of the dynamics of vortex sheets of the 2D Euler equations

We present an α -regularization of the Birkhoff–Rott equation, induced by the two-dimensional Euler- α equations, for the vortex sheet dynamics. We show that an initially smooth self-avoiding vortex sheet remains smooth for all times under the α -regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure.