Inertial Manifolds for the 3D Modified-Leray- $$\alpha $$ α Model with Periodic Boundary Conditions

Inertial manifolds for the hyperviscous Navier–Stokes equations

“…Let ρ represent the radius of the absorbing ball in H 3+ǫ for the HNSE (3.42), as indicated in Proposition 3.6. In line with Kostianko's approach [15], we define the operator W : H → H as…”

“…Mallet-Paret and Sell pioneered the application of this method to show the existence of IMs for 3D reactiondiffusion equations [20]. Subsequently, it has been adapted to acquire IMs for other models, including the 3D Cahn-Hilliard equation [16] and 3D regularized NSEs [15,12,19]. Using a combination of spatial and time averaging mechanisms, an IM has been obtained for the 3D complex Ginzburg-Landau equations [18].…”

Global finite‐dimensional invariant manifolds of the Boissonade system

“…The proof was based on an abstract invariant manifold theorem for dynamical systems on a Banach space. Note that in [21][22][23], the spectral gap condition may fail and the so-called spatial averaging principle (PSA) is used to construct the manifolds. Now, let us return to (1.1).…”

“…By directly verifying a spectral gap condition, Abu‐Hamed et al [20] obtained an inertial manifold for two different subgrid scale $$$ \alpha $$$‐models of turbulence: the simplified Bardina model and the modified Leray‐ $$$ \alpha $$$ model, in two‐dimensional torus. Kostianko [21] proved the existence of an inertial manifold for the modified Leray‐ $$$ \alpha $$$ model in three‐dimensional torus. Lu et al [22] proved the existence of a finite‐dimensional manifold of global type for a generalized phase‐field system on the rectangular or cubic spacial domains.…”

Attractors. Then and now

“…The first result in this direction was obtained by Mallet-Paret and Sell in [166], where inertial manifolds were constructed in the case of scalar reaction-diffusion equations in 3D with periodic boundary conditions, by means of the method of spatial averaging; see also [167] where it was shown that this method does not work in spaces of dimension higher than three. Taking into account the recent progress in this area (for instance, the extension of methods of spatial averaging to the 3D Cahn-Hilliard equation: see [131], various truncated or regularized versions of the 3D Navier-Stokes equations: see [82], [126], [128], the development of the method of spatio-temporal averaging and its applications to the 3D complex Ginzburg-Landau equation: see [127], [130]), we give a brief exposition of this method in § 7.…”