# zbMATH — the first resource for mathematics

On a critical Leray-$$\alpha$$ model of turbulence. (English) Zbl 1261.35098
Summary: This 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 $$\theta=\frac{1}{4}$$ 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.

##### MSC:
 35Q30 Navier-Stokes equations 76Fxx Turbulence 76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: