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

Abstract

Abstract In this paper, we study the convergence of solutions of the α-Euler equations to solutions of the Euler equations on the two-dimensional torus. In particular, given an initial vorticity ω 0 in L x p for p ∈ ( 1 , ∞ ) , we prove strong convergence in L t ∞ L x p of the vorticities q α , solutions of the α-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 α to ω in Lp , for p ∈ ( 1 , ∞ ) .