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

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