  This paper aims to study a family of Leray-$\alpha$ models with periodic
bounbary conditions. These models are good approximations for the Navier-Stokes
equations. We focus our attention on the critical value of regularization
"$\theta$" that garantees the global well-posedness for these models. We
conjecture that $\theta= 1/4$ is the critical value to obtain such results.
  When alpha goes to zero, we prove that the Leray-$\alpha$ solution, with
critical regularization, gives rise to a suitable solution to the Navier-Stokes
equations. We also introduce an interpolating deconvolution operator that
depends on "$\theta$". Then we extend our results of existence, uniqueness and
convergence to a family of regularized magnetohydrodynamics equations.

Analysis of PDEs