Global regularity and convergence of a Birkhoff-Rott-α 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 the convergence of the solutions of Euler-α equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-α equation, remains Lipschitz for all times, (ii) an initially Hölder C , 0 ≤ β < 1, chord arc curve remains in C for all times, and finally, (iii) an initially Hölder C , n ≥ 1, 0 < β < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-α and the BR-α descriptions of the vortex sheet motion are equivalent.