Strong convergence of the vorticity and conservation of the energy for the α-Euler equations

In this paper, we study the convergence of solutions of the alpha-Euler equations to solutions of the Euler equations on the two-dimensional torus. In particular, given an initial vorticity omega(0 )in L-x(p) for p is an element of (1,infinity), we prove strong convergence in L-t infinity L-x(p) of the vorticities q alpha , solutions of the alpha-Euler equations, towards a Lagrangian and energy-conserving solution of the Euler equations. Furthermore, if we consider solutions with bounded initial vorticity, we prove a quantitative rate of convergence of q(alpha) to omega in L-p , for p is an element of (1,infinity).