On a critical Leray$-\alpha$ model of turbulence

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.