In this paper, we consider the two-dimensional torus and we study the convergence of solutions of the Euler-Voigt equations to solutions of the Euler equations, under several regularity settings. More precisely, we first prove that for weak solutions of the Euler equations with vorticity in C (0, T;L² (T²) ) the approximating velocity converges strongly in C (0, T;H¹ (T²) ). Moreover, for the unique Yudovich solution of the 2D Euler equations we provide a rate of convergence for the velocity in C (0, T;L² (T²) ). Finally, for classical solutions in higher-order Sobolev spaces we prove the convergence with explicit rates of both the approximating velocity and the approximating vorticity in C (0, T;L² (T²) ).
Abbate et al. (Mon,) studied this question.