Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering

Abstract

We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator.It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.

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

“…where A 2 ∶= − 2 Δ + . Further, taking ∈ H s , the expression of in terms of Fourier series can be retrieved, formally, by (13) substituting in place of w.…”

“…In what follows, in order to keep the notation compact, we use the same symbol A for the operators A 1 and A 2 , distinguishing the 2 situations only when it is required by the context. According to the above facts, the deconvolution operator D N is actually given by (11), when it is acting on the elements of H s and, by relation (13), in the case of the scalar functions in H s .…”

“…Here, we consider the approximate deconvolution model, introduced by Adams and Stolz (see also previous studies); by following this scheme, we approximate the filtered bilinear terms as follows: where v and φ play the role of the variables u and θ , respectively, and the filtering operator G α is defined by the Helmholtz filter (see, eg, other studies; see also Bisconti and Catania for an analogous case involving an anisotropic horizontal filter), with and G α :=( I − α 2 Δ) −1 . Here, D N is the deconvolution operator, which is constructed using the Van Cittert algorithm (see, eg, Lewandowski) and is formally defined by …”

Exponential Attractors for the 3D Fractional-order Bardina Turbulence Model With Memory and Horizontal Filtering

“…In fact, despite the smoothing created by the horizontal filter, the regularity gained by the system, and therefore by the weak solutions, does not guarantee neither a direct use of the standard Sobolev inequalities nor the validity of compact embeddings, which are standard tools to prove dissipation and the existence of an absorbing set for the dynamical system associated with the considered system of PDEs. In order to recover, at least partially, these properties and to be able to exploit their consequences, our estimates make explicit use of a splitting of both vector fields and differential operators into horizontal and vertical components (see, e.g., [5,13]).…”

Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region

“…For a function w, we introduce the horizontal filter (related to the horizontal Helmholtz operator), given by [3,16,17], from the point of view of the numerical simulations, this filter is less memory consuming with respect to the standard one. Further, another interesting feature of this filter is that, even in the case of domains which are not periodic in the vertical direction, there is no need to introduce artificial boundary conditions for the Helmholtz operator (see, e.g., [3,8,9,10,11]).…”