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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
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