Theory for the rotational deconvolution model of turbulence with fractional regularization

Abstract

We study a regularization of the rotational Navier-Stokes equations that we call the Rotational Approximate Deconvolution Model (RADM). We generalize the deconvolution model, studied by Berselli and Lewandowski in (Convergence of Approximate Deconvolution Models to the mean Navier-Stokes equations. Annales de l'Institut Henri Poincare (C), NonLinear Analysis. 2012;29:171-198), to the RADM with fractional regularization, where the convergence of the solution is studied with weaker conditions on the parameter regularization.

Global well-posedness for deconvolution magnetohydrodynamics models with fractional regularization

“…The question of the limit behavior of the ADM solutions when N tends to infinity is studied also in [3] when θ > 3 4 . The above uniqueness and convergence results are improved in [2] to cover the range when θ ≥ 1 6 . The deconvolution operator, for different values of θ, was used in [7] to study the rate of convergence of the ADM model to the mean Navier-Stokes Equations.…”

“…Applying A 1 2 θ to the first equation of (4.73) we obtain Since ∂ t δB ∈ L 2 (0, T ; H −1 ), by using Lions-Magenes Lemma [11] we may justifiably write Proof. The first part of this lemma is given in [4], see also in [2] for an alternative proof. The second part is a direct consequence from the property (2.9) of the operator D N,θ , the relation (2.4) and Poincaré inequality.…”

“…Proof. The first part of this lemma is given in [4], see also in [2] for an alternative proof. The second part is a direct consequence from the property (2.9) of the operator D N,θ , the relation (2.4) and Poincaré inequality.…”

“…Proof of Theorem 5.1. The proof of Theorem 5.1 follows the lines of the proof of Theorem 4.1 in [2]. First, we need to reconstruct a uniform estimates for (w N , B N , q N ) with respect to N .…”