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 $L_x^p$ for $p\in (1,\mathrm{\infty })$, we prove strong convergence in $L_t^{\mathrm{\infty }}{L}_x^p$ 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 $p\in (1,\mathrm{\infty })$.