Modeling error of α-models of turbulence on a two-dimensional torus

This paper is devoted to study the rate of convergence of the weak solutions uα of α-regularization models to the weak solution u of the Navier- Stokes equations in the two-dimensional periodic case, as the regularization parameter α goes to zero. More specifically, we will consider the Leray-α, the simplified Bardina, and the modified Leray-α models. Our aim is to improve known results in terms of convergence rates and also to show estimates valid over long-time intervals. The results also hold in the case of bounded domain with homogeneous Dirichlet boundary conditions.