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

Abstract

We show the existence of an inertial manifold (ie, a globally invariant, exponentially attracting, finite‐dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.

Remark on a Regularity Criterion in Terms of Pressure for the 3D Inviscid Boussinesq–Voigt Equations

“…Moreover, by Gronwall's lemma, we obtain where C =C(‖u 0 ‖ 1,2 , ‖ 0 ‖ p , T) . This relation, together with (6), (5) and (19), implies which gives (7). Now, set…”

“…From (1), when = 0 , we formally recover the inviscid Boussinesq equations, which are often used for the determination of the coupled flow and temperature field in natural convection (see, e.g., [22]). These equations are also used, for instance, as a mathematical scheme to describe Newtonian fluids whenever salinity concentration or density stratification -according to the meaning of -play a significant role (see, e.g., [3][4][5]7] for some recent papers on this subject). Moreover, they are extensively employed in studying oceanographic and atmospheric phenomena (see [26,27,30]).…”

On the exact controllability of a Galerkin scheme for 3D viscoelastic fluids with fractional Laplacian viscosity and anisotropic filtering

“…Among the others, the large eddy simulation (LES) community has manifested interest in models involving such a kind of filtering procedure (see, e.g., Refs. [2,7,12,[14][15][16][17][18]55]) and, in particular, the connection between anisotropic 𝛼-models and turbulence has been investigated by Berselli in Ref. [9] for the 3D Navier-Stokes equations, even if, in fact, the first use of anisotropic filters in turbulence dates back to the approach of Germano [29] (see also Refs.…”

On the convergence rates for the three‐dimensional filtered Boussinesq equations

“…System (1.4)–(1.6) appears in the description of various geophysical and atmospheric phenomena (see, e.g., [35, 36, 38]). Also, due to the apparent connection with the Navier–Stokes equations, it has received wide attention in the field of mathematical fluid dynamics for incompressible flows (see, e.g., [9, 13, 30, 34, 39] for some recent studies on the topic).…”