Strong convergence of the vorticity and conservation of the energy for the $\alpha$-Euler equations

In this paper, we study the convergence of solutions of the $\alpha$-Euler equations to solutions of the Euler equations on the $2$-dimensional torus. In particular, given an initial vorticity $\omega_0$ in $L^p_x$ for $p \in (1,\infty)$, we prove strong convergence in $L^\infty_tL^p_x$ 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 $p \in (1, \infty)$.