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

Abstract

Abstract. In this paper, we establish the existence of a unique "regular" weak solution to turbulent flows governed by a general family of α models with critical regularizations. In particular this family contains the simplified Bardina model and the modified Leray-α model. When the regularizations are subcritical, we prove the existence of weak solutions and we establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result in the critical case.

Large Eddy Simulation for turbulent flows with critical regularization

“…The LES for NSE with θ = 1 can be also addressed as the zeroth order Approximate Deconvolution Model referring to the family of models in [1] . The value θ = 1 6 is consistent with the critical regularization value needed to get existence and uniqueness to the simplified Bardina model studied in [3]. We note also that fractionnal order Laplace operator has been used in another α models of turbulence in [21,2,13].…”

“…Let us mention that the idea to consider the LES for MHD with critical regularization is a new feature for the present work. The Approximate Deconvolution Model for Navier-Stokes equations with θ > 3 4 is studied in [6] and the Approximate Deconvolution Model for Magnetohydrodynamics equations with θ = 1 is studied in [14]. As mentioned in [6] the value θ > 3 4 is not optimal in order to prove the existence and the uniqueness of the solutions in the deconvolution case.…”

Partial and full hyper-viscosity for Navier-Stokes and primitive equations

“…Another class of modifications of the Navier-Stokes equations are the so-called α-models, see [1] and the references therein. An example is the simplified Bardina turbulence model, which involves some additional smoothing, and global existence and uniqueness of weak solutions and its global attractor are studied in [39].…”