On a critical Leray- model of turbulence

Approximate deconvolution model in a bounded domain with vertical regularization

“…For this reason one can replace the classical Laplace operator by the Laplace operator with fractional regularization θ and seek for the limiting case where we can prove global existence and uniqueness of regular solutions. In a series of papers [10,4,1,2], it was shown that the LES models which are derived by using instead of the operator (1.1), the operator A θ = I + α 2θ (−∆) θ for suitable (large enough) θ < 1, are still well-posed. In this paper, we consider LES models with fractional filter acting only in one variable…”

Mathematical Results for Some α Models of Turbulence with Critical and Subcritical Regularizations

“…Fractionnal order Laplace operator has been used in another α models of turbulence in [18,3,10]. Existence and uniqueness of solutions of other modifications of the Navier-Stokes equations have been studied by Ladyzhenskaya [12] Lions [16], Málek et al [17].…”

“…Our task is to find the critical relation between the regularizations θ 1 and θ 2 (see We note that the α family considered here is a particular case of the general study in [10] where the results do not recover the critical case 2θ 1 + θ 2 = 1 2 . The Leray-α model with critical regularization is studied in [3]. We know, thanks to the works [11,7] that for θ 1 = 0, θ 2 = 1 or θ 1 = 1, θ 2 = 1, that their exist a unique weak solution to the model (1.1)-(1.4).…”

Navier-Stokes equations in the whole space with an eddy viscosity

“…Surprisingly, Leray has planted a seed that germinates these last two decades in the field of modern LES. See for instance in Ali [1,2], Berselli-Iliescu-Layton [6], Foias-Holm-Titi [16], Gibbon-Holm [20,21], Geurt-Holm [19], llyin-Lunasin-Titi [22], Layton-Rebholz [26], Rebholz [36], this list being non exhaustive. These models are based on a regularization calculated by the Helmlhoz filter determined by:…”