This paper is devoted to study the rate of convergence of the weak solutions $ {\bf u}_\alpha $ of $ \alpha $-regularization models to the weak solution $ {\bf u} $ of the Navier-Stokes equations in the two-dimensional periodic case, as the regularization parameter $ \alpha $ goes to zero. More specifically, we will consider the Leray-$ \alpha $, the simplified Bardina, and the modified Leray-$ \alpha $ 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.
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