Abstract
The existence of an inertial manifold for the modified Leray-alpha model with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averaging principle. Moreover, an adaptation of the Perron method for constructing inertial manifolds in the particular case of zero spatial averaging is suggested.
Highlights
This system was introduced in [12] and it was shown there that consideration of the ML-α model as a closure model to Reynolds averaged equations in turbulent channels and pipes leads to the similar reduced system as the Leray-α and the Navier-Stokes-α models
More precisely we are going to show the existence of an inertial manifold (IM) for the ML-α system subject to periodic boundary conditions
The notion of an inertial manifold was introduced exactly to formulate in a rigorous mathematical way what it means a finite-dimensionality for the infinite-dimensional system
Summary
We recall some basic definitions and standard results which will be used thought out the paper. Due to the Plancherel theorem, the norm in the space (Hs(T3)) is defined by (2.2). The Leray-Helmholtz orthoprojector P : (L2(T3))3 → H := P (L2(T3)) to divergent free vector fields with zero mean can be defined as follows (2.3). We may define the Stokes operator A := −P ∆ as the restriction of the Laplacian to the divergence free vector fields. Hs := D(As/2) = (Hs(T3))3 ∩ {∇ · u = 0} ∩ { u = 0} (see, e.g., [19]) and, due to the Parseval equality, the norm in this space can be defined by (2.6). For u1, u2 ∈ H1 we define the standard bilinear form associated with the Navier-Stokes equation:. Let us define the main object of this paper, namely, the inertial manifold (IM) associated with the modified-Leray-α model. S(t)u0 − S(t)v0 H ≤ Ce−γt u0 − v0 H , t ≥ 0
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