Inertial manifolds for the incompressible Navier-Stokes equations

Abstract

In this article, we devote to the existence of an N -dimensional inertial manifold for the incompressible Navier-Stokes equations in T d (d = 2, 3).Our results can be summarized as two aspects: Firstly, we construct an N -dimensional inertial manifold for the Navier-Stokes equations in T 2 ; Secondly, we extend slightly the spatial averaging method to the abstract case: ∂tuoperator with compact inverse and F is Lipschitz from a Hilbert space H to H), and then verify the existence of an N -dimensional inertial manifold for the hyperviscous Navier-Stokes equation with the hyperviscous index 5/4 in T 3 .

Asymptotic regularity for the generalized MHD‐Boussinesq equations

“…The existence of inertial manifolds for the hyperviscous Navier-Stokes equations was obtained for 𝛼 ≥ 3 2 by using the spatial averaging method in Gal and Guo [16]. Meanwhile, Li and Sun [17] proved the existence of inertial manifolds for the hyperviscous Navier-Stokes equations for 𝛼 ≥ 5 4 by using the extend slightly the spatial averaging method. The well-posedness of strong solutions for the hyperviscous magneto-micropolar equation was proved in Liu et al [18].…”

“…In order to get the well-posedness and regularity for the 3D generalized MHD-Boussinesq equations, we should overcome the main difficulties for the estimations of the nonlinear terms (u • ∇)u, (u • ∇)b, (b • ∇)u, (b • ∇)b, and (u • ∇)𝜃. Based on Gal and Guo [16] and Li and Sun [17], by using the delicate estimates, we improve their results and get the better estimations. Moreover, we will prove the regularity of system (1.1) in H 9 2 × H 9 2 × H 9 2 and H a × H a × H a (a ≥ 9 2 ).…”