Ladder theorem and length-scale estimates for a Leray alpha model of turbulence

Abstract

In this paper, we study the Modified Leray alpha model with periodic boundary conditions. We show that the regular solution verifies a sequence of energy inequalities that is called "ladder inequalities". Furthermore, we estimate some quantities of physical relevance in terms of the Reynolds number.MSC:76B03; 76F05; 76D05; 35Q30.

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

“…. 1 Leray states his Lemma 8 as a consequence of a uniqueness result. If the uniqueness result is entirely proved, there is in his paper [27] no proof of thislemma 8, although it is quite reasonnable.…”

“…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:…”

“…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],…”