Convergence of the Euler-Voigt equations to the Euler equations in two dimensions

Abstract

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 $C([0,T];L^2(\mathbb{T}^2))$ the approximating velocity converges strongly in $C([0,T];H^1(\mathbb{T}^2))$. Moreover, for the unique Yudovich solution of the $2D$ Euler equations we provide a rate of convergence for the velocity in $C([0,T];L^2(\mathbb{T}^2))$. 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 $C([0,T];L^2(\mathbb{T}^2))$.