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

International audienceThis paper aims to study a family of Leray-$\alpha$ models with periodic boundary conditions. These models are good approximations for the Navier-Stokes equations. We focus our attention on the critical value of regularization "\$theta$" that guarantees the global well-posedness for these models. We conjecture that View the MathML source $\theta = 1/4$ 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