Inertial Manifolds for Certain Subgrid-Scale $\alpha$-Models of Turbulence

Abstract

Abstract. In this note we prove the existence of an inertial manifold, i.e., a global invariant, exponentially attracting, finite-dimensional smooth manifold, for two different sub-grid scale α-models of turbulence: the simplified Bardina model and the modified Leray-α model, in two-dimensional space. That is, we show the existence of an exact rule that parameterizes the dynamics of small spatial scales in terms of the dynamics of the large ones. In particular, this implies that the long-time dynamics of these turbulence models is equivalent to that of a finite-dimensional system of ordinary differential equations.MSC Classification: 35Q30, 37L30, 76BO3, 76D03, 76F20, 76F55, 76F65

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

“…In reproducing the argument used in Hamed to prove point (i) in Hamed,, proposition 3 (see also Hamed, proposition 9), ie, the “cone invariance property,” which is stated in Proposition (i) below, we introduce the following truncated nonlinearities: Here, χ is a smooth cutoff function outside the ball of radius ρ =2 in (L 2 (Ω)) 3 . Indeed, let with χ ( r )=1 for 0≤ r ≤1, χ ( r )=0 for r ≥2, and | χ ′ ( r )|≤2 for r ≥0 (which is null outside the ball of radius ).…”

“…Proof of Proposition Let V 1 =( V 1 , ϑ 1 ) and V 2 =( v 2 , ϑ 2 ) be 2 solutions of to . To show the cone invariance, ie, the condition given in Proposition (i) (see also Hamed, proposition 3‐(i)), it is enough to show that cannot pass through the boundary of the cone if the dynamics starts inside the cone. More precisely, we will prove that whenever , where stands for the boundary of the cone .…”

“…As remarked in Hamed, the original motivations for the development of the theory of inertial manifolds were mainly related to the analysis of the NSEs. Nevertheless, as far as we know, the problem about the existence of inertial manifolds for the 2D NSEs is still open.…”

“…In performing our analysis, we follow closely what has been done in Hamed 4 where the existence of inertial manifolds has been proved in the case of 2 regularizing models for the 2D Navier-Stokes equations (NSEs). Different from Hamed, 4 here, we consider a turbulence model that converges to 2D mean Navier-Stokes-like equations, namely, the 2D mean Boussinesq system, when the deconvolution parameter N goes to infinity (see (5) below). Further, we take into account a double filtered system, that is, a model in which both equations for u and are somehow regularized by an application of the inverse of the Helmholtz operator (see (6)-(9) below).…”

“…where u ⊗ u ∶= (u 1 u, u 2 u), and we supply this problem, which is equivalent to (1) to (4), with periodic boundary conditions (ie, the torus Ω = T 2 is the considered domain).…”