Singularity \& Regularity Issues for Simplified Models of Turbulence

We consider a family of Leray-$\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\theta$, of the Navier-Stokes equations. In particular, they share with the original equation (NS) the property of existence of global weak solutions. We establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions when $\theta$ is subcritical. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result proved in \cite{A01}.