Yanqiu Guo. INERTIAL MANIFOLDS FOR THE HYPERVISCOUS Ciprian G Gal
email: cgal@fiu.edu

Yanqiu Guo
email: yanguo@fiu.edu

Navier- Stokes Equations

INERTIAL MANIFOLDS FOR THE HYPERVISCOUS NAVIER-STOKES EQUATIONS

Keywords: inertial manifold, Navier-Stokes equations, global attractor, hyperviscosity

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Introduction

Let us consider the following hyperviscous version of the Navier-Stokes equations of incompressible fluid flow on a torus,

u t + ν(-∆) β u + (u • ∇)u + ∇p = f, (t, x) ∈ R + × T d , div u = 0, u| t=0 = u 0 , (1.1)
where β ≥ d/4 + 1/2 for the spatial dimension d ≥ 3, and β ≥ 1 for d = 2. Here,

T d = [-π
, π] d is endowed with the periodic boundary condition. It is well-known that the regularized system (1.1) is globally well-posed in the L 2 space (see, e.g., [START_REF] Cholewa | Fractional Navier-Stokes equations[END_REF]). It has been used, among others, by numerical analysts as a substitute model for the standard case β = 1 and plays a key role in understanding turbulent phenomena in science (cf. Avrin [START_REF] Avrin | Singular initial data and uniform global bounds for the hyper-viscous Navier-Stokes equations with periodic boundary conditions[END_REF], Borue and Orszag [START_REF] Borue | Local energy flux and subgrid-scale statistics in three-dimensional turbulence[END_REF], Basdevant et. al. [START_REF] Basdevant | A study of barotropic model flows: intermittency, waves, and predictability[END_REF], Browning and Kreiss [START_REF] Browning | Comparison of numerical methods for the calculation of twodimensional turbulence[END_REF], Cerruto et. al. [START_REF] Cerruto | Spectral and hyper-eddy viscosity in high-Reynolds-number turbulence[END_REF], Fornberg [START_REF] Fornberg | A numerical study of two-dimensional turbulence[END_REF] and McWilliams [START_REF] Mcwilliams | The emergence of isolated coherent vortices in turbulent flows[END_REF]). Besides, the system (1.1) has some physical meaning (see again [START_REF] Cerruto | Spectral and hyper-eddy viscosity in high-Reynolds-number turbulence[END_REF]). Given the nonlinear nature of turbulent incompressible viscous flows and the ensuing multiscale interactions, the direct numerical simulation of the Navier-Stokes equations is still presently lacking apart from some investigations performed on regularized systems which still retain the basic nonlinear structure and the essential features of the full hydrodynamic Navier-Stokes equations (cf. Holst et. al. [START_REF] Holst | Analysis of a general family of regularized Navier-Stokes and MHD models[END_REF], Gal and Medjo [START_REF] Gal | On a regularized family of models for homogeneous incompressible two-phase flows[END_REF]). Among such regularized models, it is worth noting the following globally well-posed systems in three dimensions: the Leray-α model (cf. Cheskidov et. al. [START_REF] Cheskidov | On a Leray-α model of turbulence[END_REF]), the simplified Bardina model (cf. Cao et. al. [START_REF] Cao | Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models[END_REF]), the Navier-Stokes-α equations (cf. Foias et. al. [START_REF] Foias | The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory[END_REF]) and the Navier-Stokes-Voight equations (cf. Kalantarov and Titi [START_REF] Kalantarov | Global attractors and determining modes for the 3D Navier-Stokes-Voight equations[END_REF], Coti-Zelati and Gal [START_REF] Zelati | Singular limits of Voigt models in fluid dynamics[END_REF]). One advantage of using the hyperviscous Navier-Stokes equations (1.1) with β ≥ 5/4 in the three-dimensional domain is that the system only modifies the spectral distribution of energy, and the global well-posedness (i.e., existence, uniqueness and stability with respect to the initial data) of the solutions can be rigorously proven unlike for the three-dimensional Navier-Stokes equations (see [START_REF] Avrin | Singular initial data and uniform global bounds for the hyper-viscous Navier-Stokes equations with periodic boundary conditions[END_REF][START_REF] Cholewa | Fractional Navier-Stokes equations[END_REF]). On the other hand, (1.1) has also been proposed and investigated for the purpose of direct numerical simulations of turbulent incompressible viscous flows, although orders of dissipation β ≥ 2 have typically been used (see again [START_REF] Browning | Comparison of numerical methods for the calculation of twodimensional turbulence[END_REF][START_REF] Cerruto | Spectral and hyper-eddy viscosity in high-Reynolds-number turbulence[END_REF][START_REF] Fornberg | A numerical study of two-dimensional turbulence[END_REF][START_REF] Mcwilliams | The emergence of isolated coherent vortices in turbulent flows[END_REF]). The existence of finite-dimensional global attractors for the regularized family (1.1) can been proven by employing standard theory of attractors. For the sake of completeness, we provide a brief proof of the global well-posedness and existence of global attractors of (1.1) in the appendix. Furthermore, by directly verifying a spectral gap condition it was shown in [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF] that the hyperviscous Navier-Stokes equations (1.1) possess an inertial manifold in L 2 provided β > d for any spatial dimension d ≥ 2. Exploiting the same strategy and a more refined analysis, the existence of inertial manifolds for (1.1) with β > 5/2 in three-dimensional domains was established in [START_REF] Avrin | The asymptotic finite-dimensional character of a spectrally-hyperviscous model of 3D turbulent flow[END_REF] (the results in [START_REF] Avrin | The asymptotic finite-dimensional character of a spectrally-hyperviscous model of 3D turbulent flow[END_REF] actually hold for a general family of hyperviscous operators). The purpose of this manuscript is to prove the existence of an inertial manifold for (1.1) with β ≥ 3/2 in two or three dimensions when the spectral gap condition is not necessarily fulfilled. The motivation for this line of research is the longstanding open problem concerning the existence of inertial manifolds for the Navier-Stokes equation (β = 1).

We recall that the existence of an inertial manifold guarantees that, in the long-term, the system resembles a finite-dimensional system of ordinary differential equations, which describes the limit dynamics of the original system as time goes to infinity (see, e.g. [START_REF] Foias | Inertial manifolds for nonlinear evolutionary equations[END_REF][START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF]). Indeed, spectral gap conditions have been widely used in the literature to establish the existence of inertial manifolds for many dissipative evolution equations (cf. [START_REF] Hamed | Inertial manifolds for certain subgrid-scale α-models of turbulence[END_REF][START_REF] Constantin | Spectral barriers and inertial manifolds for dissipative partial differential equations[END_REF][START_REF] Foias | Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension[END_REF][START_REF] Foias | Inertial manifolds for nonlinear evolutionary equations[END_REF][START_REF] Foias | Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations[END_REF][START_REF] Sell | Inertial manifolds: the non-selfadjoint case[END_REF][START_REF] Vukadinovic | Inertial manifolds for a Smoluchowski equation on the unit sphere[END_REF]). However, for a system that lacks the spectral gap condition, in their pioneering work [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF], Mallet-Paret and Sell have introduced the so-called spatial averaging method to prove the existence of inertial manifolds for a three-dimensional reaction-diffusion equation. This technique was further simplified by Zelik [START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF], and extended by Kostianko and Zelik to the three-dimensional Cahn-Hilliard equation [START_REF] Kostianko | Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions[END_REF], and by Kostianko to the three-dimensional modified Leray-α model [START_REF] Kostianko | Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions[END_REF].

In the present work, we employ the spatial averaging method to establish the existence of inertial manifolds for the system (1.1) in the hyperdissipation range β ≥ 3/2 in three dimensions when the spectral gap condition fails, whereas in two dimensions an arbitrarily large spectral gap is available. We draw techniques from the contributions [START_REF] Foias | Inertial manifolds for nonlinear evolutionary equations[END_REF][START_REF] Kostianko | Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions[END_REF][START_REF] Kostianko | Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions[END_REF][START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF][START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF]. In particular, we would like to mention the article [START_REF] Kostianko | Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions[END_REF] by Kostianko and Zelik in which they studied the existence of inertial manifolds for a model problem that can be associated with the Cahn-Hilliard equation subject to periodic boundary conditions on T 3 . The model problem in [START_REF] Kostianko | Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions[END_REF] reads

u t + A 2 u + AF (u) = 0, u| t=0 = u 0 , (1.2)
where A is a positive self-adjoint operator in some Hilbert space H, with compact inverse.

The Cahn-Hilliard equation on T 3 is an example of (1.2) when A = -∆ with periodic boundary conditions on T 3 and F (u) = u 3 -u. It is well known that the Cahn-Hilliard equation can be also obtained as the conserved gradient flow in H -1 (i.e., the dual of a proper subspace of H 1 ) associated with the Fréchet derivative of some free energy functional.

Indeed, the Cahn-Hilliard equation is naturally dissipative in H -1 in the sense that the solution semigroup is well-defined in H -1 , possessing an absorbing ball in H -1 . Based on this interpretation, one may then employ analytical techniques that are similar to ones for the reaction-diffusion equation u t + Au + F (u) = 0. Roughly speaking, this is also the approach used in [START_REF] Kostianko | Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions[END_REF] where in order to verify the existence of an inertial manifold for the Cahn-Hilliard equation, one constructs invariant cones in H -1 and adopts the spatial averaging method that was initially developed for the 3D reaction-diffusion equation in [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF].

Inspired by [START_REF] Kostianko | Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions[END_REF][START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF][START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF], we study an abstract model in some Hilbert space H,

u t + A β u + A 1/2 F (u) = f, u| t=0 = u 0 , (1.3)
for β ≥ 3/2. Notice that the hyperviscous Navier-Stokes equations (1.1) with ν = 1 coincides with the model problem (1.3) when A = -∆ and

F (u) = A -1/2 B(u, u), where B (u, v) = P σ ((u • ∇)v)
is the bilinear form associated with the advection term in (1.1).

Here P σ is the Helmholtz-Leray orthogonal projection operator. Then one needs to extend the aforementioned invariant cone and spatial averaging method to our model (1.3) for establishing the existence of a Lipschitz inertial manifold. Our argument also relies on number theoretic results from [START_REF] Richards | On the gaps between numbers which are sums of two squares[END_REF] by Richards, and from [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF] by Mallet-Paret and Sell. Specifically, we explicitly use the fact that the size of the gap among certain quadratic forms of integers is logarithmic growing (See Theorem 4.3). The most technical and essential content of this paper is contained in the proof of Theorem 2.4. This manuscript is divided into sections as follows. Section 2 is devoted to a treatment of the model problem (1.3) by finding sufficient conditions that guarantee the existence of an inertial manifold. In particular, in Subsection 2.1, we state a well-known optimal spectral gap condition which immediately implies the existence of inertial manifolds for (1.3). Next, in Subsection 2.2 we adapt the method of spatial averaging developed in [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF] and further exploited by [START_REF] Kostianko | Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions[END_REF], to treat the critical case β = 3/2 of (1.3), when the spectral gap condition is violated. Section 3 is devoted to an application of the theory established on the abstract model (1.3) to the hyperviscous Navier-Stokes equations (1.1). Specifically, in Subsection 3.1 we briefly verify the existence of a compact global attractor for (1.1) with β ≥ 3/2 by showing the existence of an absorbing ball in H 4 . Then, in Subsection 3.2, by adopting the technique in [START_REF] Kostianko | Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions[END_REF], we modify (1.1) outside of this absorbing ball in such a way that the new equation possesses an inertial manifold. As usual, this inertial manifold will still be invariant with respect to the solution semigroup associated with (1.1) at least in the neighborhood of the global attractor and therefore will contain all of its non-trivial dynamics. All the corresponding conditions of our abstract theorem are verified directly in Subsection 3.3. The appendix contains some supporting material from number theory (see Appendix 4.1), and the proof of a standard result which says the cone invariance property and the decay property together imply the existence of inertial manifolds for the model problem (1.3) (see Appendix 4.2). For the sake of completeness, in the appendix we also provide the verification of the strong cone property for (1.3) when the optimal spectral gap condition is fulfilled (see Appendix 4.3), as well as a brief proof of the global wellposedness of weak solutions and the existence of the global attractor for the hyperviscous Navier-Stokes equations (1.1) for β ≥ 5/4 in three dimensions (see Appendix 4.4).

The abstract model

First, let H be a Hilbert space. We study the following abstract model in H,

u t + A β u + A 1/2 F (u) = f, u| t=0 = u 0 , (2.1)
for β ≥ 3/2. Here F is an operator mapping from H to H, and A : D(A) → H is a linear self-adjoint positive operator with compact inverse. The operator A possesses the complete orthonormal system of eigenvectors {e j } ∞ j=1 in H which corresponds to eigenvalues λ j such that λ j → ∞ as j → ∞ and

Ae j = λ j e j , 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ • • • (2.2)
Throughout, we denote by | • | the norm of H and (•, •) denotes the inner product in H. For N ∈ N, let us now define the following projection operators

P N v = N j=1 v j e j , Q N v = ∞ j=N +1 v j e j , (2.3)
where v j = (v, e j ). Our goal in this section is to find sufficient conditions on the operator A and F such that the model problem (2.1) possesses an inertial manifold in H. An application to the hyperviscous Navier-Stokes equations (1.1) shall be discussed in the subsequent section. Let us first recall the notion of inertial manifold for a dynamical system (see, e.g., [START_REF] Foias | Inertial manifolds for nonlinear evolutionary equations[END_REF][START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF]). Definition 2.1. Let H be a Hilbert space. A subset M ⊂ H is called an inertial manifold for a dynamical system in H associated with the semigroup S(t) if the following conditions are satisfied:

(1) M is invariant with respect to the semigroup S(t), i.e. S(t)M = M, for all t ≥ 0;

(2) M is a finite-dimensional Lipschitz manifold, i.e. there exists a Lipschitz continuous function Φ :

P N H → Q N H such that M := {u + + Φ(u + ), u + ∈ P N H}.
(3) The exponential tracking property holds, namely, there exist C, α > 0 such that for every u 0 ∈ H, there exists v 0 ∈ M such that

|S(t)u 0 -S(t)v 0 | ≤ Ce -αt |u 0 -v 0 |, for all t ≥ 0.

2.1.

A spectral gap condition. We state the following well-known optimal spectral gap condition which guarantees the existence of inertial manifolds for (2.1) (see [START_REF] Miklavcic | A sharp condition for existence of an inertial manifold[END_REF][START_REF] Romanov | Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations[END_REF][START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF]).

Theorem 2.2. Let the above assumptions on the operator A hold, and assume F : H → H is globally Lipschitz continuous with Lipschitz constant L. Suppose there exists N ∈ N such that the following spectral gap condition is satisfied:

λ β N +1 -λ β N λ 1/2 N +1 + λ 1/2 N > L. (2.4)
Then (2.1) possesses an N -dimensional inertial manifold in the sense of Definition 2.1.

Remark 2.3. If β > 3/2, then the spectral gap condition (2.4) is satisfied if we assume that λ j → ∞ as j → ∞, and λ N +1 -λ N ≥ 1 for some large N . However, in the critical case β = 3/2, the condition (2.4) demands a large spectral gap. Indeed, if λ N +1 -λ N ≥ 2L, then (2.4) is valid for β = 3/2. This result with sharp spectral gap condition (2.4) was derived by Miklavcic [START_REF] Miklavcic | A sharp condition for existence of an inertial manifold[END_REF] and Romanov [START_REF] Romanov | Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations[END_REF] (see also Zelik [35]). The condition (2.4) is optimal in the sense that one can find globally Lipschitz nonlinearity F such that (2.1) does not possess an inertial manifold if (2.4) is violated (cf. [START_REF] Eden | Counterexamples to the regularity of Mane projections in the attractors theory[END_REF][START_REF] Romanov | Three counterexamples in the theory of inertial manifolds[END_REF][START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF]).

For the reader's convenience, in Appendix 4.3 we prove a strong cone property provided the optimal spectral gap (2.4) is satisfied, which eventually leads to the existence of inertial manifolds for the model (2.1).

2.2. The spatial averaging scheme. In the above subsection, we have pointed out that, regarding the critical case β = 3/2 for the model (2.1), the spectral gap condition (2.4) demands a sufficiently large spectral gap for the operator A, which may not be available. This motivates us to consider the model (2.1) in the critical case

u t + A 3/2 u + A 1/2 F (u) = f, u| t=0 = u 0 , (2.5)
under the scenario when the eigenvalues of A do not have a sufficiently large gap.

We define the projection operators on the low, high and intermediate Fourier modes, respectively, as follows:

P k,N u := λ j <λ N -k u j e j , Q k,N u := λ j >λ N +k u j e j , R k,N u := λ N -k≤λ j ≤λ N +k u j e j ,
where u j = (u, e j ), for some k < λ N .

The following theorem states that if the spectral gap condition for (2.5) (i.e. λ 3/2

N +1 -λ 3/2 N > L(λ 1/2 N +1 + λ 1/2 N )
) is not fulfilled, but the spatial averaging condition (2.6) is valid, then a strong cone property can be obtained, which implies the existence of an inertial manifold for (2.5).

Theorem 2.4. Let u 1 and u 2 be two solutions of (2.5) H,H) ≤ L for all u ∈ H, and for some L ≥ 1. Suppose there exist N ∈ N and k ∈ [γ log λ N , λ N ) for some γ ∈ (0, 1] such that λ N ≥ e 40L 2 /γ with 1 ≤ λ N +1 -λ N ≤ 2L, and the spatial averaging condition holds:

in H. Set v = u 1 -u 2 . Define V (t) = |q| 2 -|p| 2 where p = P N v and q = Q N v. Assume F : H → H is Gateaux differentiable with |F (u)| L(
|R k,N F (u)R k,N v| ≤ δ|v|, for all u ∈ H, (2.6)
for some δ ≤ 1 50 . Then the following strong cone property is valid,

d dt V (t) + λ 3/2 N +1 + λ 3/2 N V (t) ≤ - λ 1/2 N 4 |v(t)| 2
, for all t ≥ 0.

(2.7)

Proof. Let u 1 and u 2 be two solutions of (2.5) and set v = u 1 -u 2 , then one has

v t + A 3/2 v + A 1/2 [F (u 1 ) -F (u 2 )] = 0. ( 2.8)
Set p = P N v and q = Q N v. Take the scalar product of (2.8) with p and q respectively:

1 2 d dt |p| 2 + |A 3/4 p| 2 + (F (u 1 ) -F (u 2 ), A 1/2 p) = 0, 1 2
d dt |q| 2 + |A 3/4 q| 2 + (F (u 1 ) -F (u 2 ), A 1/2 q) = 0.
(2.9)

By subtracting the two equations in (2.9), we obtain

d dt V (t) = -2(|A 3/4 q| 2 -|A 3/4 p| 2 ) + 2(F (u 1 ) -F (u 2 ), A 1/2 p -A 1/2 q).
(2.10)

Due to the fundamental theorem of calculus for the Gateaux derivative, we have

F (u 1 ) -F (u 2 ) = 1 0 F (su 1 + (1 -s)u 2 )vds. (2.11)
Therefore, by setting α :=

λ 3/2 N +1 +λ 3/2 N

2

, and due to (2.10) and (2.11), we have

d dt V (t) + 2αV (t) = αV (t) -(|A 3/4 q| 2 -|A 3/4 p| 2 ) -(|A 3/4 q| 2 -α|q| 2 ) -(α|p| 2 -|A 3/4 p| 2 ) + 2 1 0 F (su 1 + (1 -s)u 2 )v, A 1/2 p -A 1/2 q ds. (2.
12)

The idea is to properly bound each of the summands on the right-hand side of (2.12). Since

|v| 2 = |p| 2 + |q| 2 and A 3/4 q 2 ≥ λ 3/2 N +1 |q| 2 , A 3/4 p 2 ≤ λ 3/2
N |p| 2 , we have

αV (t) -(|A 3/4 q| 2 -|A 3/4 p| 2 ) ≤ λ 3/2 N +1 + λ 3/2 N 2 (|q| 2 -|p| 2 ) -λ 3/2 N +1 |q| 2 + λ 3/2 N |p| 2 = - λ 3/2 N +1 -λ 3/2 N 2 |v| 2 . (2.13)
For any real numbers a ≥ b ≥ 0, the following elementary inequality holds:

a 3 -b 3 = (a -b)(a 2 + ab + b 2 ) ≥ 1 2 (a -b)(a 2 + 2ab + b 2 ) = 1 2 (a 2 -b 2 )(a + b). (2.14)
Hence,

λ 3/2 N +1 -λ 3/2 N 2 ≥ 1 4 (λ N +1 -λ N )(λ 1/2 N +1 + λ 1/2 N ) ≥ 1 2 λ 1/2 N , (2.15)
owing to the assumption λ N +1 -λ N ≥ 1. Applying (2.15) to (2.13) yields

αV (t) -(|A 3/4 q| 2 -|A 3/4 p| 2 ) ≤ - 1 2 λ 1/2 N |v| 2 . (2.16) Next notice that p = P k,N v + R k,N p. Thus, since P k,N A 3/4 v 2 ≤ (λ N -k) 3/2 |P k,N v| 2 and R k,N A 3/4 p 2 ≤ λ 3/2 N |R k,N p| 2 , we have α|p| 2 -|A 3/4 p| 2 = λ 3/2 N +1 + λ 3/2 N 2 |P k,N v| 2 + |R k,N p| 2 -|P k,N A 3/4 v| 2 + |R k,N A 3/4 p| 2 ≥ λ 3/2 N +1 + λ 3/2 N 2 -(λ N -k) 3/2 |P k,N v| 2 + λ 3/2 N +1 -λ 3/2 N 2 |R k,N p| 2 .
(2.17)

Since λ N +1 ≥ λ N and due the elementary inequality (2.14), one has

λ 3/2 N +1 + λ 3/2 N 2 -(λ N -k) 3/2 ≥ λ 3/2 N -(λ N -k) 3/2 ≥ 1 2 [λ N -(λ N -k)][λ 1/2 N + (λ N -k) 1/2 ] ≥ k 2 λ 1/2 N . (2.18)
Thus, applying (2.18) and (2.15) to (2.17) shows that

α|p| 2 -|A 3/4 p| 2 ≥ k 2 λ 1/2 N |P k,N v| 2 + 1 2 λ 1/2 N |R k,N p| 2 . (2.19)
Since q = Q k,N v + R k,N q, we write |A 3/4 q| 2 -α|q| 2 as follows:

|A 3/4 q| 2 -α|q| 2 = (|Q k,N A 3/4 v| 2 -α|Q k,N v| 2 ) + (|R k,N A 3/4 q| 2 -α|R k,N q| 2 ). (2.20) Clearly, |R k,N A 3/4 q| 2 ≥ λ 3/2
N +1 |R k,N q| 2 , then by using (2.15), we have

|R k,N A 3/4 q| 2 -α|R k,N q| 2 ≥ λ 3/2 N +1 -λ 3/2 N 2 |R k,N q| 2 ≥ 1 2 λ 1/2 N |R k,N q| 2 . (2.21)
By the assumption k ≥ γ log λ N and λ N ≥ e 40L 2 /γ , one has

k ≥ 40L 2 . (2.22) Moreover, note that |Q k,N A 3/4 v| 2 ≥ (λ N + k) 3/2 |Q k,N v| 2 . Then using α = λ 3/2 N +1 +λ 3/2 N 2 ≤ λ 3/2
N +1 , as well as the elementary inequality (2.14), we obtain

|Q k,N A 3/4 v| 2 -α|Q k,N v| 2 ≥ (λ N + k) 3/2 -λ 3/2 N +1 |Q k,N v| 2 ≥ 1 2 (λ N + k -λ N +1 ) (λ N + k) 1/2 + λ 1/2 N +1 |Q k,N v| 2 ≥ (k -2L)λ 1/2 N +1 |Q k,N v| 2 (2.23)
by the assumption λ N +1 -λ N ≤ 2L. Also we have used

λ N + k ≥ λ N + 2L ≥ λ N +1
, where the first inequality is due to (2.22) as well as L ≥ 1.

However, we may find another lower bound for |Q k,N A 3/4 v| 2 -α|Q k,N v| 2 different from (2.23). Indeed,

|Q k,N A 3/4 v| 2 -α|Q k,N v| 2 ≥ γ log λ N λ N |Q k,N A 3/4 v| 2 + 1 - γ log λ N λ N |Q k,N A 3/4 v| 2 -α|Q k,N v| 2 ≥ γ log λ N λ 1/2 N |Q k,N A 1/2 v| 2 + 1 - γ log λ N λ N |Q k,N A 3/4 v| 2 -α|Q k,N v| 2 . (2.24)
We shall show that the second term on the right-hand side of (2.24) is nonnegative. By the assumptions λ N > k ≥ γ log λ N for some γ ∈ (0, 1] and λ N +1 ≤ λ N + 2L, we evaluate

1 - γ log λ N λ N |Q k,N A 3/4 v| 2 -α|Q k,N v| 2 ≥ 1 - γ log λ N λ N (λ N + k) 3/2 |Q k,N v| 2 -λ 3/2 N +1 |Q k,N v| 2 ≥ 1 - γ log λ N λ N (λ N + γ log λ N ) 3/2 -(λ N + 2L) 3/2 |Q k,N v| 2 . (2.25)
Now, we calculate

1 - γ log λ N λ N 2 (λ N + γ log λ N ) 3 = λ 3 N + λ 2 N γ log λ N -2λ N (γ log λ N ) 2 -2(γ log λ N ) 3 + (γ log λ N ) 4 λ N + (γ log λ N ) 5 λ 2 N ≥ λ 3 N + 1 2 λ 2 N γ log λ N ≥ λ 3 N + 20L 2 λ 2 N ≥ (λ N + 2L) 3 , (2.26)
since we assume λ N ≥ e 40L 2 /γ , where L ≥ 1. The inequality (2.26) implies that the righthand side of (2.25) is non-negative, and then due to (2.24), we obtain

|Q k,N A 3/4 v| 2 -α|Q k,N v| 2 ≥ γ log λ N λ 1/2 N |Q k,N A 1/2 v| 2 . (2.27)
We apply the estimates (2.21), (2.23) and (2.27) to the equality (2.20). Then

|A 3/4 q| 2 -α|q| 2 ≥ γ log λ N 2λ 1/2 N |Q k,N A 1/2 v| 2 + k -2L 2 λ 1/2 N +1 |Q k,N v| 2 + 1 2 λ 1/2 N |R k,N q| 2 . (2.28)
In order to deal with the nonlinear term in (2.12), we evaluate (F (u)v, A 1/2 p -A 1/2 q) for any u ∈ H as follows.

(F (u)v, A 1/2 p -A 1/2 q) = (R k,N F (u)v, A 1/2 p -A 1/2 q) + (P k,N F (u)v, A 1/2 p -A 1/2 q) + (Q k,N F (u)v, A 1/2 p -A 1/2 q) = (R k,N F (u)R k,N v, A 1/2 p -A 1/2 q) + (R k,N F (u)P k,N v, A 1/2 p -A 1/2 q) + (R k,N F (u)Q k,N v, A 1/2 p -A 1/2 q) + (P k,N F (u)v, A 1/2 p -A 1/2 q) + (Q k,N F (u)v, A 1/2 p -A 1/2 q) = (R k,N F (u)R k,N v, A 1/2 p -A 1/2 q) + (F (u)P k,N v, R k,N A 1/2 p) -(F (u)P k,N v, R k,N A 1/2 q) + (F (u)Q k,N v, R k,N A 1/2 p) -(F (u)Q k,N v, R k,N A 1/2 q) + (F (u)v, P k,N A 1/2 v) -(F (u)v, Q k,N A 1/2 v).
(2.29)

We shall estimate every term on the right-hand side of (2.29). Since

|F (u)| L(H,H) ≤ L, we notice that |(F (u)P k,N v, R k,N A 1/2 p)| + |(F (u)Q k,N v, R k,N A 1/2 p)| ≤ L(|P k,N v| + |Q k,N v|)|R k,N A 1/2 p| ≤ Lλ 1/2 N (|P k,N v| + |Q k,N v|)|R k,N p| ≤ L 2 λ 1/2 N |P k,N v| 2 + |Q k,N v| 2 + λ 1/2 N 4 |R k,N p| 2 .
(2.30)

Similarly,

|(F (u)P k,N v, R k,N A 1/2 q)| + |(F (u)Q k,N v, R k,N A 1/2 q)| ≤ L(|P k,N v| + |Q k,N v|)|R k,N A 1/2 q| ≤ L(|P k,N v| + |Q k,N v|)(λ N + k) 1/2 |R k,N q| ≤ L(|P k,N v| + |Q k,N v|)(2λ N ) 1/2 |R k,N q| ≤ 4L 2 λ 1/2 N |P k,N v| 2 + |Q k,N v| 2 + λ 1/2 N 8 |R k,N q| 2 , (2.31)
where we have used the assumption k < λ N . Next,

|(F (u)v, P k,N A 1/2 v)| ≤ L|v||P k,N A 1/2 v| ≤ L|v| (λ N -k) 1/2 |P k,N v| ≤ L |v| λ 1/2 N |P k,N v| ≤ 5L 2 λ 1/2 N |P k,N v| 2 + λ 1/2 N 20 |v| 2 . (2.32) Also, |(F (u)v, Q k,N A 1/2 v)| ≤ L|v||Q k,N A 1/2 v| ≤ 2L 2 λ 1/2 N γ log λ N |v| 2 + γ log λ N 8λ 1/2 N |Q k,N A 1/2 v| 2 . (2.33)
Moreover, by the spatial averaging condition

|R k,N F (u)R k,N v| ≤ δ|v| (see (2.6
)), we obtain

|(R k,N F (u)R k,N v, A 1/2 p -A 1/2 q)| ≤ δ|v||A 1/2 p -A 1/2 q| ≤ δ|v||A 1/2 p| + δ|v||A 1/2 q| ≤ δλ 1/2 N |v||p| + δ|v||Q k,N A 1/2 v| + δ|v||R k,N A 1/2 q| ≤ δλ 1/2 N |v| 2 + δ|v||Q k,N A 1/2 v| + δ(2λ N ) 1/2 |v||R k,N q| ≤ δλ 1/2 N |v| 2 + 2δ 2 λ 1/2 N γ log λ N |v| 2 + γ log λ N 8λ 1/2 N |Q k,N A 1/2 v| 2 + 4δ 2 λ 1/2 N |v| 2 + λ 1/2 N 8 |R k,N q| 2 = δ + 2δ 2 γ log λ N + 4δ 2 λ 1/2 N |v| 2 + γ log λ N 8λ 1/2 N |Q k,N A 1/2 v| 2 + λ 1/2 N 8 |R k,N q| 2 . (2.34) Substituting (2.30)-(2.34) into (2.29) yields |(F (u)v,A 1/2 p -A 1/2 q)| ≤ δ + 2(δ 2 + L 2 ) γ log λ N + 4δ 2 + 1 20 λ 1/2 N |v| 2 + γ log λ N 4λ 1/2 N |Q k,N A 1/2 v| 2 + 5L 2 λ 1/2 N |Q k,N v| 2 + 10L 2 λ 1/2 N |P k,N v| 2 + λ 1/2 N 4 |R k,N q| 2 + λ 1/2 N 4 |R k,N p| 2 , ( 2
d dt V (t) + 2αV (t) ≤ - 1 2 -2δ - 4(δ 2 + L 2 ) γ log λ N -8δ 2 - 1 10 λ 1/2 N |v| 2 - k -2L 2 -10L 2 λ 1/2 N |Q k,N v| 2 - k 2 -20L 2 λ 1/2 N |P k,N v| 2 . (2.36) By (2.22) we know that k ≥ 40L 2 , where L ≥ 1, then clearly k -2L 2 -10L 2 > 0 and k 2 -20L 2 ≥ 0. Since δ ≤ 1/50 and γ log λ N ≥ 40L 2 with L ≥ 1, then 1 2 -2δ - 4(δ 2 + L 2 ) γ log λ N -8δ 2 - 1 10 ≥ 1 4 .
(2.37) By (2.36)-(2.37), we conclude that

d dt V (t) + 2αV (t) ≤ - λ 1/2 N 4 |v| 2 , for all t ≥ 0, where α = λ 3/2 N +1 +λ 3/2 N 2
. This completes the proof.

The following result shows that the strong cone property (2.7) implies the cone invariance property and the decay property.

Proposition 2.5. Let u 1 and u 2 be two solutions of (2.5)

in H. Set v = u 1 -u 2 . Define V (t) = |q| 2 -|p| 2 where p = P N v and q = Q N v. Assume F : H → H is globally Lipschitz continuous with Lipschitz constant L. Suppose the strong cone property d dt V (t) + αV (t) ≤ -µ|v(t)| 2 , for all t ≥ 0, (2.38)
holds for some constants α, µ > 0. Then, the following properties hold.

• Cone invariance property:

If V (0) ≤ 0, then V (t) ≤ 0 for all t ≥ 0. • Decay property: If V (T ) > 0 for some T > 0, then |v(t)| 2 ≤ Ce -αt |v(0)| 2 , for all t ∈ [0, T ], (2.39)
where C > 0. (This is also called the squeezing property.)

Remark 2.6. Notice that, the cone invariance property implies that, if V (T ) > 0 for some

T > 0, then V (t) > 0 for all t ∈ [0, T ].
Proof. The argument adopts the proof of [START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF]Corollary 2.22]. The cone invariance property follows directly from (2.38). It remains to show the decay property. Indeed, taking the scalar product of (2.8) with v yields 1 2

d dt |v| 2 + |A 3/4 v| 2 ≤ |(F (u 1 ) -F (u 2 ), A 1/2 v)| ≤ L 2 2 |v| 2 + |A 1/2 v| 2 2 ,
by using the Lipschitz continuity |F (u 1 ) -F (u 2 )| ≤ L|v|. Without loss of generality, we assume the eigenvalues λ j of A satisfy λ j ≥ 1 for all j ∈ N, so that |A 3/4 v| ≥ |A

d dt V (t) + αV (t) ≤ [ (α + L 2 ) -µ]|v| 2 = 0, by choosing = µ α+L 2 . It follows that V (t) ≤ e -αt V (0), for t ≥ 0. (2.41)
If V (T ) > 0 for some T > 0, then by the cone invariance property, we have

V (t) > 0 for all t ∈ [0, T ]. Also, notice that V (t) = |q| 2 -|p| 2 ≤ |q| 2 + |p| 2 = |v| 2 . Therefore, due to (2.41), |v(t)| 2 ≤ |v(t)| 2 + V (t) ≤ e -αt |v(0)| 2 + V (0) ≤ ( + 1)e -αt |v(0)| 2 , (2.42) for all t ∈ [0, T ]. Then, |v(t)| 2 ≤ (1 + 1 )e -αt |v(0)| 2 , for all t ∈ [0, T ].
It is expected that the cone invariance property along with the decay property should imply the existence of inertial manifolds. This result is stated in Theorem 2.7 below. For the sake of completeness, we refer the reader for a proof of Theorem 2.7 in Appendix 4.2.

Theorem 2.7. Assume F : H → H is globally Lipschitz and |A -1/4 F (u)| ≤ C for all u ∈ H. Assume that the solutions of (2.5) satisfy the cone invariance property and the decay property stated in Proposition 2.5. Then problem (2.5) possesses an N -dimensional inertial manifold M in the sense of Definition 2.1.

Finally, owing to Proposition 2.5 and Theorem 2.7, we conclude that the spatial averaging condition of Theorem 2.2 guarantees the existence of an inertial manifold for the model problem (2.5). In particular, we have the following corollary.

Application to the hyperviscous Navier-Stokes equations

We aim to employ the theory that has been established on the abstract model (2.1) in Section 2 to prove the existence of an inertial manifold for the hyperviscous Navierstokes equation (1.1) in two or three dimensions. We focus on the scenario when the hyperviscosity ν(-∆) β u has the critical power β = 3/2, because for the case of β > 3/2 the spectral gap condition (2.4) is satisfied. Moreover, we shall concentrate on the case of the three-dimensional spatial domain. Indeed, the two-dimensional setting is much easier to study since the required spectral gap condition (2.4) in Theorem 2.2 is fulfilled by a number theoretical result (Theorem 4.2) due to Richards [START_REF] Richards | On the gaps between numbers which are sums of two squares[END_REF].

Specifically, we consider the incompressible hyperviscous Navier-Stokes equation on the three-dimensional torus:

u t + ν(-∆) 3/2 u + (u • ∇)u + ∇p = f, (t, x) ∈ R + × T 3 ; div u = 0, u| t=0 = u 0 . (3.1)
Set the phase space H as

H := {u ∈ (L 2 (T 3 )) 3 : T 3 udx = 0, div u = 0}. We denote by | • | the norm of H. Notice that, if u ∈ (L 2 (T 3 )) 3 , then u = j∈Z 3 ûj e ij•x
where ûj are Fourier coefficients. Thus,

T 3 udx = 0 is equivalent to û0 = 0. Therefore, if u ∈ H, then u = j∈Z 3 \{0} ûj e ij•x .
We denote by P σ : (L 2 (T 3 )) 3 → H the Helmholtz-Leray orthogonal projection operator, and by A = -P σ ∆ the Stokes operator. Since we work with the periodic space, it is known that Au = -P σ ∆u = -∆u, for all u ∈ D(A).

(3.

2) The operator A -1 is a self-adjoint positive definite compact operator from H to H.

As usual, the Sobolev spaces H

s := D(A s/2 ), s > 0, with the norm | • | H s = |A s 2 • |, are defined as H s =    u ∈ H : |u| 2 H s = j∈Z 3 \{0} |j| 2s |û j | 2 < ∞    .
For s < 0, one defines H s as the completion of H with respect to the corresponding norm

|•| H s .
For w 1 , w 2 ∈ H 1 , we define the bilinear form

B(w 1 , w 2 ) = P σ ((w 1 • ∇)w 2 ).
Notice that (3.1) can be written in the equivalent form in H:

u t + νA 3/2 u + B(u, u) = f, u| t=0 = u 0 . (3.3)
Since the exact value of ν is not essential to our mathematical analysis, we assume without loss of generality that ν = 1. Thus in the following work we consider the equation

u t + A 3/2 u + B(u, u) = f, u| t=0 = u 0 . (3.4) Given the vectors u = (u 1 , u 2 , u 3 ), v = (v 1 , v 2 , v 3 ), w = (w 1 , w 2 , w 3 ), we denote b(u, v, w) := (B(u, v), w) = 3 m,n=1 T 3 u m ∂ m v n w n dx (3.5)
whenever the integrals in (3.5) make sense. It is known that b(u, v, w) has the property

b(u, v, w) = -b(u, w, v), (3.6)
which implies that b(u, v, v) = 0. We state the following global well-posedness result for problem (3.4). The solvability of (3.4) will become clear from the dissipative estimates that we obtain in the subsequent subsection. We choose to omit the proof of Theorem 3.1 since it follows from classical arguments employed for the Navier-Stokes equations (see, e.g., [START_REF] Constantin | Navier-Stokes equations[END_REF][START_REF] Cholewa | Fractional Navier-Stokes equations[END_REF]).

Theorem 3.1. Let f ∈ H -3/2 . Then for every u 0 ∈ H, problem (3.4) possesses a unique weak solution such that u ∈ C ([0, T ] ; H) ∩ L 2 (0, T ; H 3/2 ) for an arbitrary time T > 0. The weak solution depends continuously on the initial datum in a Lipschitz way. Proof. We recall that the existence of a finite dimensional global attractor in H for a slightly more general problem than (3.4) is also proven in [START_REF] Avrin | The asymptotic finite-dimensional character of a spectrally-hyperviscous model of 3D turbulent flow[END_REF]. Here, for the sake of simplicity we provide the a priori estimates for the existence of a regular absorbing set for the semigroup. The conclusion of the theorem then follows in a standard way (see, e.g. [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF]).

In the following estimates, we repeatedly use Hölder's inequality, Young's inequality and the Sobolev imbedding H s → L

Step 1 (L 2 -estimate). Take scalar product of (3.4) with u. We get 1 2

d dt |u| 2 + |A 3/4 u| 2 = (f, u) = (A -3/4 f, A 3/4 u) ≤ 1 2 |f | 2 H -3/2 + 1 2 |A 3/4 u| 2 .
It follows that Due to (3.9) there exists t 0 > 0 such that

d dt |u| 2 + |A 3/4 u| 2 ≤ |f | 2 H -3/2 . (3.7) Therefore, d dt |u| 2 + |u| 2 ≤ |f | 2 H -3/2 . This implies |u| 2 ≤ |f | 2 H -3/2 (1 -e -t ) + e -t |u 0 | 2 . ( 3
|u(t)| ≤ 2|f | H -3/2 , for t ≥ t 0 . (3.10)
This shows that all trajectories of (3.4) enter an absorbing ball of radius 2|f | H -3/2 in H for t ≥ t 0 .

Step 2 (H 1 -estimate). Take scalar product of (3.4) with Au. We get 1 2

d dt |A 1/2 u| 2 + |A 5/4 u| 2 + b(u, u, Au) = (f, Au) ≤ |f | 2 H -1/2 + 1 4 |A 5/4 u| 2 . (3.11)
We estimate

|b(u, u, Au)| ≤ C|u| L 6 |u| H 1 |Au| L 3 ≤ C|u| 2 H 1 |A 5/4 u| ≤ 1 4 |A 5/4 u| 2 + C|u| 4 H 1 .
(3.12) By (3.11) and (3.12), we see that

d dt |u| 2 H 1 + |A 5/4 u| 2 ≤ C|u| 4 H 1 + 2|f | 2 H -1/2 . (3.13)
Notice that, by integrating (3.7) from t to t + 1, we obtain

t+1 t |u(τ )| 2 H 3/2 dτ ≤ |u(t)| 2 + |f | 2 H -3/2 ≤ 5|f | 2 H -3/2 , (3.14)
for t ≥ t 0 by using (3.10). By (3.13) and (3.14) along with the uniform Gronwall inequality (see Appendix 4.1, Lemma 4.1), we have

|u(t)| 2 H 1 ≤ ρ 2 1 := exp C|f | 2 H -3/2 5|f | 2 H -3/2 + 2|f | 2 H -1/2 , for t ≥ t 0 + 1. (3.15)
Therefore, all trajectories of the dynamical system enter an absorbing ball of radius ρ 1 in H 1 whenever t ≥ t 0 + 1.

Step 3 (H 2 -estimate). Take the scalar product of (3.4) with A 2 u. We deduce 1 2

d dt |Au| 2 + |A 7/4 u| 2 ≤ |b(u, u, A 2 u)| + |(f, A 2 u)| ≤ |b(u, u, A 2 u)| + |f | 2 H 1/2 + 1 4 |A 7/4 u| 2 . (3.16)
We estimate

|b(u, u, A 2 u)| ≤ C |u| H 1 |A 1/2 u| L 6 |A 3/2 u| L 3 + |u| L 6 |Au||A 3/2 u| L 3 ≤ C|u| H 1 |Au||A 7/4 u| ≤ Cρ 2 1 |Au| 2 + 1 4 |A 7/4 u| 2 , (3.17)
for t ≥ t 0 + 1, on account of (3.15). We obtain from (3.16) and (3.17) that, for t ≥ t 0 + 1,

d dt |u| 2 H 2 + |u| 2 H 7/2 ≤ Cρ 2 1 |u| 2 H 2 + 2|f | 2 H 1/2 . (3.18)
In order to find an absorbing ball for u in H 2 , we drop the term |u| 2 H 7/2 in (3.18) and then integrate the inequality from s to t + 1 for t 0 + 1 ≤ t ≤ s ≤ t + 1. One has

|u(t + 1)| 2 H 2 ≤ |u(s)| 2 H 2 + Cρ 2 1 t+1 t |u(τ )| 2 H 2 dτ + 2|f | 2 H 1/2 . ( 3
|u(t + 1)| 2 H 2 ≤ Cρ 2 1 + 1 t+1 t |u(τ )| 2 H 2 dτ + 2|f | 2 H 1/2 . (3.20)
By integrating (3.13) from t to t + 1 and using (3.15), we obtain that

t+1 t |u(τ )| 2 H 5/2 dτ ≤ ρ 2 1 + Cρ 4 1 + 2|f | 2 H -1/2 , (3.21)
for t ≥ t 0 + 1. By virtue of (3.20) and (3.21), we see that

|u(t)| 2 H 2 ≤ ρ 2 2 := C ρ 6 1 + ρ 2 1 |f | 2 H -1/2 + 2|f | 2 H 1/2 , for t ≥ t 0 + 2. (3.22)
This implies that any solution of (3.4) enters an absorbing ball of radius ρ 2 in H 2 after a sufficiently large time t ≥ t 0 + 2.

Step 4 (H 4 -estimate). Recall our assumption f ∈ H 1 . Then, from equation (3.4), we have

A 1/2 u t + A 2 u + A 1/2 B(u, u) = A 1/2 f, in H. (3.23) Since |A 1/2 B(u, u)| ≤ C|u| 2 H 2 , we obtain from (3.23) and (3.22) that |u| H 4 = |A 2 u| ≤ |u t | H 1 + C|u| 2 H 2 + |f | H 1 ≤ |u t | H 1 + Cρ 2 2 + |f | H 1 , for t ≥ t 0 + 2. (3.
24) It remains to show that |u t | H 1 has a uniform bound for large time. To this end, we take the scalar product of (3.4) with Au t :

|u t | 2 H 1 + 1 2 d dt |u| 2 H 5/2 = -b(u, u, Au t ) + (f, Au t ) ≤ C|u| 2 H 2 |u t | H 1 + |f | H 1 |u t | H 1 ≤ 1 2 |u t | 2 + Cρ 4 2 + |f | 2 H 1 , for t ≥ t 0 + 2. Hence, |u t | 2 H 1 + d dt |u| 2 H 5/2 ≤ Cρ 4 2 + 2|f | 2 H 1 , for t ≥ t 0 + 2. ( 3
u tt + A 3/2 u t + B(u t , u) + B(u, u t ) = 0. (3.29)
By taking the scalar product of (3.29) with Au t , we have 1 2

d dt |u t | 2 H 1 + |u t | 2 H 5/2 = -b(u t , u, Au t ) -b(u, u t , Au t ) ≤ C|u t | H 5/2 |u t | H 1 |u| H 1 .
It follows that

d dt |u t | 2 H 1 + |u t | 2 H 5/2 ≤ C|u t | 2 H 1 |u| 2 H 1 ≤ Cρ 2 1 |u t | 2 H 1 , for t ≥ t 0 + 1. (3.30)
Owing to (3.30) and (3.28), we deduce

|u t | 2 H 1 ≤ (Cρ 2 1 + 1)(ρ 2 3 + Cρ 4 2 + 2|f | 2 H 1 ), for t ≥ t 0 + 4. (3.31)
Finally, by virtue of (3.24) and (3.31), we conclude

|u| H 4 ≤ ρ := C(ρ 1 + 1)(ρ 3 + ρ 2 2 + |f | H 1 ), for t ≥ t 0 + 4. (3.32)
That is, all trajectories of (3.4) enter an absorbing ball in H 4 with the radius ρ when

t ≥ t 0 + 4 if f ∈ H 1 .
The proof is complete.

Remark 3.3. The existence of an absorbing ball in H 4 is crucial for the construction of an inertial manifold for (3.4) as we modify the nonlinearity outside the absorbing ball and show that the modified nonlinearity satisfies the spatial averaging condition in the subsequent sections. On the other hand, the global well-posedness of weak solutions and the existence of a global attractor for equation (1.1) for any β ≥ 5/4 in three dimensions can be shown by using the standard energy method. For the sake of completion, we briefly discuss in Appendix 4.4 the case β ≥ 5/4 for (1.1) in 3D.

3.2.

Modification of the nonlinearity outside the absorbing ball. Since we are concerned with the large-time behavior of the hyperviscous Navier-Stokes equations, we may freely modify the nonlinearity outside the absorbing ball in H 4 . In this section we adopt the method by Kostianko [START_REF] Kostianko | Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions[END_REF] to truncate the nonlinearity of our system (3.4). To this end, let us introduce a smooth cut-off function θ ∈ C ∞ 0 (C) satisfying θ(ξ) = ξ for |ξ| ≤ 1 and |θ(ξ)| ≤ 2 for all ξ ∈ C. Also, we define the corresponding vector-valued cut-off function as

θ := (θ(ξ 1 ), θ(ξ 2 ), θ(ξ 3 )) ∈ C 3 , for ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ C 3 .
Recall that H denotes the L2 divergence-free vector field with zero mean value. The Helmholtz-Leray orthogonal projector P σ : (L 2 (T 3 )) 3 → H is defined as

P σ u = j∈Z 3 \{0} P j ûj e ij•x
where the 3 × 3 matrices P j are given by

P j := 1 |j| 2   j 2 2 + j 2 3 -j 1 j 2 -j 1 j 3 -j 1 j 2 j 2 1 + j 2 3 -j 2 j 3 -j 1 j 3 -j 2 j 3 j 2 1 + j 2 2   .
P j are orthonormal projectors in C 3 . Following the idea in [START_REF] Kostianko | Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions[END_REF], we introduce the operator U : H → H defined as

U (u) := j∈Z 3 \{0} ρ |j| 4 P j θ |j| 4 ûj ρ e ij•x , (3.33)
where ρ is the radius of the absorbing ball in H 4 established in (3.32) for the system (3.4). We remark that, in two dimensions, the operator U can be defined in a similar way. In order to modify the nonlinearity B(u, u) in (3.4) outside the absorbing ball in H 4 , we shall replace B(u, u) by B(U (u), U (u)), and then prove the existence of an inertial manifold in H for the so-called "prepared" equation for (3.4), namely

u t + A 3/2 u + B(U (u), U (u)) = f, (3.34)
for f ∈ H 1 . Indeed we shall see that the original equation (3.4) and the "prepared" equation (3.34) have the same large time behavior in the absorbing ball in H 4 .

For the sake of preparation, we prove a few useful properties of the operator U .

Lemma 3.4. The function U : H → H has the following properties.

(1) U (u) = u, for all u ∈ D(A 2 ) with |u| H 4 ≤ ρ.

(2) U is a regularization operator: for any δ > 0, U : H → H Moreover, the map U : H → H

(3) U is Gateaux differentiable from H to H, and its derivative U has the expression

U (u)v = j∈Z 3 \{0} P j θ |j| 4 ûj ρ vj e ij•x , for u, v ∈ H. (3.36)
Moreover, there exists L 1 > 0 such that

|U (u)| L(H,H) ≤ L 1 , for all u ∈ H. (3.37) Furthermore, for each v ∈ H, the map u → U (u)v is continuous from H to H.
Proof. The proof is similar to the arguments in [START_REF] Kostianko | Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions[END_REF].

(1) Let u ∈ D(A 2 ) such that

|u| 2 H 4 = j∈Z 3 \{0} |j| 8 |û j | 2 ≤ ρ 2 ,
then obviously,

|j| 4 |û j | ρ ≤ 1, for all j ∈ Z 3 \{0}.
Since θ(ξ) = ξ for |ξ| ≤ 1, one has θ

|U (u)| 2 H 5 2 -δ ≤ C j∈Z 3 \{0} |j| 5-2δ ρ 2 |j| 8 ≤ Cρ 2 j∈Z 3 \{0} 1 |j| 3+2δ ≤ C δ . Next we show that U : H → H 5 2 -δ is continuous. Indeed, given > 0, let u, v ∈ H such that |u -v| ≤ σ. We estimate |U (u) -U (v)| 2 H 5 2 -δ = j∈Z 3 \{0} ρ 2 |j| 3+2δ θ |j| 4 ûj ρ -θ |j| 4 vj ρ 2 = 1≤|j|≤N ρ 2 |j| 3+2δ θ |j| 4 ûj ρ -θ |j| 4 vj ρ 2 + C |j|>N ρ 2 |j| 3+2δ , (3.38)
where we have used the fact that θ is uniformly bounded in C 3 . Now we choose N sufficiently large such that C |j|>N ρ 2 |j| 3+2δ ≤ 2 . Next, since |u -v| ≤ σ, one has |û j -vj | ≤ σ for every j ∈ Z 3 \{0}, then by the uniform continuity of θ, we may select σ > 0 small enough such that the 1st summation on the right-hand side of (3.38) is bounded by 2 .

(3) Let u, v ∈ H. We calculate

U (u + εv) -U (u) ε = 1 ε j∈Z 3 \{0} ρ |j| 4 P j θ |j| 4 (û j + εv j ) ρ -θ |j| 4 ûj ρ e ij•x = j∈Z 3 \{0} P j vj 1 0 θ |j| 4 (û j + τ εv j ) ρ dτ e ij•x , (3.39)
where we have used the fundamental theorem of calculus.

If U (u)v is given by (3.36), we aim to show that the L 2 -norm of the difference of (3.36) and (3.39) converges to zero if ε → 0. Indeed,

U (u + εv) -U (u) ε -U (u)v 2 H = j∈Z 3 \{0} P j vj 1 0 θ |j| 4 (û j + τ εv j ) ρ -θ |j| 4 ûj ρ dτ e ij•x 2 H ≤ j∈Z 3 \{0} |v j | 2 1 0 θ |j| 4 (û j + τ εv j ) ρ -θ |j| 4 ûj ρ 2 dτ ≤ 1≤|j|≤N |v j | 2 1 0 θ |j| 4 (û j + τ εv j ) ρ -θ |j| 4 ûj ρ 2 dτ + C |j|>N |v j | 2 , (3.40)
where we have used θ is uniformly bounded in C 3 . Let δ > 0. Since v ∈ H, we choose N sufficiently large such that the C |j|>N |v j | 2 ≤ δ 2 . Also, since θ is uniformly continuous, we can select small enough such that the 1st summation is less than δ 2 . This implies

U (u + εv) -U (u) ε -U (u)v 2 H ≤ δ,
for sufficiently small , provided U (u)v is given by (3.36). Therefore, we conclude that (3.36) is the correct expression for U (u)v. Next, by (3.36) and the uniform boundedness of θ , there exists

L 1 > 0 such that |U (u)v| ≤ L 1 |v|, for all u, v ∈ H, i.e., |U (u)| L(H,H) ≤ L 1 for all u ∈ H.
Finally, let v ∈ H, we show that the map u → U (u)v is continuous from H to H. Indeed, given > 0, we let u, w ∈ H such that |u -w| ≤ σ. Due to (3.36), we have

|(U (u) -U (w))v| 2 = j∈Z 3 \{0} θ |j| 4 ûj ρ -θ |j| 4 ŵj ρ 2 |v j | 2 ≤ 1≤|j|≤N θ |j| 4 ûj ρ -θ |j| 4 ŵj ρ 2 |v j | 2 + C |j|>N |v j | 2 , ( 3

3.3.

Verification of the spatial averaging condition. This subsection is devoted to verifying the spatial averaging condition for the "prepared" equation (3.34). In fact, in order to employ our theory established on the abstract model (2.5), we write the hyperviscous Navier-Stokes equations (3.4) in the form of (2.5):

u t + A 3/2 u + A 1/2 F (u) = f, where F (u) := A -1/2 B(u, u). (3.42)
Notice that, for the operator A = -∆, the eigenvector e ij•x corresponds to the eigenvalue |j| 2 . We modify F (u) outside the absorbing ball in H 4 by replacing u by U (u). Namely, we define F : H → H by

F (u) := F (U (u)) = A -1/2 B(U (u), U (u)). ( 3
F (u) = F (U (u)) = F (u) if |u| H 4 ≤ ρ.
Hence, after the trajectory u(t) enters the absorbing ball in H 4 with radius ρ, the equation (3.42) is equivalent to

u t + A 3/2 u + A 1/2 F (u) = f, where F (u) = A -1/2 B(U (u), U (u)), ( 3
(F (u)v, w) = lim ε→0 F (u + εv) -F (u) ε , w = lim ε→0 B(U (u + εv), U (u + εv)) -B(U (u), U (u)) ε , A -1/2 w = (B(U (u), U (u)v), A -1/2 w) + (B(U (u)v, U (u)), A -1/2 w) = -b(U (u), A -1/2 w, U (u)v) -b(U (u)v, A -1/2 w, U (u)), ( 3
(F (u)v, w) = -b(U (u), A -1/2 w, U (u)v) -b(U (u)v, A -1/2 w, U (u)), (3.46)
for u, v, w ∈ H. In addition, there exists L > 0 such that

|F (u)| L(H,H) ≤ L, for all u ∈ H.
Proof. To show that F is uniformly bounded from H to H 2 , we take an arbitrary w ∈ H and estimate

(AF (u), w) = (A 1/2 B(U (u), U (u)), w) ≤ C|w| |A 1/2 U (u)| 2 L 4 + |U (u)| L ∞ |AU (u)| ≤ C|w||U (u)| 2 H 2 . Thus |AF (u)| ≤ |U (u)| 2 H 2 ≤ C
, for all u ∈ H, by virtue of (3.35). Next, we show that F is Gateaux differentiable from H to H. Indeed, for any u, v, w ∈ H, we calculate

F (u + εv) -F (u) ε , w = B(U (u + εv), U (u + εv)) -B(U (u), U (u)) ε , A -1/2 w = -b U (u + εv), A -1/2 w, U (u + εv) -U (u) ε -b U (u + εv) -U (u) ε , A -1/2 w, U (u) , (3.47)
where we have used (3.5) and (3.6).

In order to show that b U (u + εv), A -1/2 w, U (u+εv)-U (u)

ε → b(U (u), A -1/2 w, U (u)v) as ε → 0, we evaluate the difference: b U (u + εv), A -1/2 w, U (u + εv) -U (u) ε -b(U (u), A -1/2 w, U (u)v) ≤ b U (u + εv) -U (u), A -1/2 w, U (u + εv) -U (u) ε + b U (u), A -1/2 w, U (u + εv) -U (u) ε -U (u)v . (3.48)
We shall show that the right-hand side of (3.48) converges to zero. Since U is Gateaux differentiable and the map u → U (u)v is continuous from H to H due to Lemma 3.4, by the fundamental theorem of calculus for the Gateaux derivative, we have

U (u + εv) -U (u) = 1 0 U (u + τ εv)εvdτ. (3.49) As a result, b U (u + εv) -U (u), A -1/2 w, U (u + εv) -U (u) ε ≤ b U (u + εv) -U (u), A -1/2 w, 1 0 U (u + τ εv)vdτ ≤ C|U (u + εv) -U (u)| L ∞ |w| 1 0 U (u + τ εv)vdτ H ≤ C|U (u + εv) -U (u)| L ∞ |w| 1 0 U (u + τ εv)v H dτ ≤ CL 1 |U (u + εv) -U (u)| H 2 |w||v| (3.50)
where we have applied (3.37). Also, by Lemma 3.

U (u), A -1/2 w, U (u + εv) -U (u) ε -U (u)v ≤ b U (u), A -1/2 w, 1 0 (U (u + τ εv) -U (u))vdτ ≤ C|U (u)| L ∞ |w| 1 0 (U (u + τ εv) -U (u))vdτ H ≤ C|U (u)| H 2 |w| 1 0 |(U (u + τ εv) -U (u))v| H dτ. (3.51) According to Lemma 3.4, the map u → U (u)v is continuous from H to H. Therefore, sup τ ∈[0,1] |(U (u + τ εv) -U (u)
)v| H → 0 as ε → 0. It follows that the right-hand side of (3.51) converges to zero as ε → 0. We have shown that both terms on the right-hand side of (3.48) converge to zero as

ε → 0. Hence lim ε→0 b U (u + εv), A -1/2 w, U (u + εv) -U (u) ε = b(U (u), A -1/2 w, U (u)v). (3.52)
Analogously, we may show that lim

ε→0 b U (u + εv) -U (u) ε , A -1/2 w, U (u) = b(U (u)v, A -1/2 w, U (u)). ( 3
F (u + εv) -F (u) ε , w = -b(U (u), A -1/2 w, U (u)v) -b(U (u)v, A -1/2 w, U (u)),
for all u, v, w ∈ H. This implies (3.46).

Finally, we show that |F (u)| L(H,H) is uniformly bounded. By (3.46) we estimate

(F (u)v, w) ≤ |b(U (u), A -1/2 w, U (u)v)| + |b(U (u)v, A -1/2 w, U (u))| ≤ C|U (u)| L ∞ |w||U (u)v| ≤ C|U (u)| H 2 |w||U (u)v| ≤ CL 1 |w||v| for all w ∈ H,
where we have used (3.35) and (3.37) to obtain the last inequality. It follows that

|F (u)| L(H,H) ≤ CL 1 , for all u ∈ H.
This completes the proof.

Next, we verify the spatial averaging condition (see Theorem 2.4) for the "prepared" equation (3.44) in T 3 . Proposition 3.6. Let F be the operator defined by (3.43). Given δ > 0. There exist arbitrarily large

N ∈ N and k ∈ [γ log λ N , λ N ) for some constant γ ∈ (0, 1] independent of N , such that |R k,N F (u)R k,N v| ≤ δ|v|, for any u, v ∈ H. Proof. Let u = (u 1 , u 2 , u 3 ), v = (v 1 , v 2 , v 3
), w = (w 1 , w 2 , w 3 ) be vectors in H. Then, due to (3.46), we have

(R k,N F (u)R k,N v, w) = (F (u)R k,N v, R k,N w) = -b(U (u), A -1/2 R k,N w, U (u)R k,N v) -b(U (u)R k,N v, A -1/2 R k,N w, U (u)). ( 3
(U (u), A -1/2 R k,N w, U (u)R k,N v) = 3 m,n=1 T 3 U (u) m R k,N [U (u)v] n (R k,N ∂ m A -1/2 w n )dx = 3 m,n=1 T 3 R k,N U (u) m R k,N [U (u)v] n ∂ m A -1/2 w n dx. It follows that |b(U (u), A -1/2 R k,N w, U (u)R k,N v)| ≤ |w| 3 m,n=1 R k,N U (u) m R k,N [U (u)v] n .
The same estimate holds for the second term on the right-hand side of (3.54). Therefore,

(R k,N F (u)R k,N v, w) ≤ 2|w| 3 m,n=1 R k,N U (u) m R k,N [U (u)v] n . (3.55)
In order to estimate the right-hand side of (3.55), we let ϕ, ψ ∈ L 2 (T 3 ) such that ϕ and ψ have mean value zero. Let r > 0 and then denote Thus, ϕ = ϕ >r + ϕ <r . Then

R k,N (ϕR k,N ψ) = R k,N (ϕ >r R k,N ψ) + R k,N (ϕ <r R k,N ψ).
(3.56) By Proposition 4.5 (see Appendix 4.1), there exist arbitrarily large λ > 0 and κ ∈ [γ log λ, λ) for some γ > 0 independent of the choice of λ, satisfying: whenever

|n| 2 , |l| 2 ∈ [λ -κ, λ + κ]
with distinct n and l ∈ Z 3 one has |n -l| > r. Without loss of generality, we assume γ ∈ (0, 1]. Therefore, due to Weyl's law λ N ∼ CN 2 3 , there exist arbitrarily large

λ N > 0 and k ∈ [γ log λ N , λ N ) satisfying: whenever |n| 2 , |l| 2 ∈ [λ N -k, λ N + k] with distinct n and l ∈ Z 3 one has |n -l| > r. It follows that R k,N (ϕ <r R k,N ψ) = λ N -k≤|n| 2 ≤λ N +k      λ N -k≤|l| 2 ≤λ N +k 1≤|n-l|≤r φn-l ψl      e in•x = 0. (3.57)
Due to (3.56) and (3.57), for the chosen N and k, we have

R k,N (ϕR k,N ψ) = R k,N (ϕ >r R k,N ψ).
Consequently, by assuming ϕ ∈ H 2 with zero mean and using a basic interpolation inequality, we obtain

|R k,N (ϕR k,N ψ)| ≤ |ϕ >r R k,N ψ| ≤ |ϕ >r | L ∞ |ψ| ≤ |ϕ >r | 1 4 |ϕ| 3 4 H 2 |ψ|. (3.58) Notice that |ϕ >r | 2 = |j|>r | φj | 2 = |j|>r 1 |j| 4 |j| 4 | φj | 2 ≤ 1 r 4 |ϕ| 2 H 2 . (3.59)
By (3.58) and (3.59), it follows that

|R k,N (ϕR k,N ψ)| ≤ 1 √ r |ϕ| H 2 |ψ|. (3.60)
By (3.55) and (3.60), one has

(R k,N F (u)R k,N v, w) ≤ 2|w| √ r 3 m,n=1 |U (u) m | H 2 |[U (u)v] n |. ( 3
(R k,N F (u)R k,N v, w) ≤ CL 1 √ r |v||w|, for all u, v, w ∈ H.
This implies that

|R k,N F (u)R k,N v| ≤ CL 1 √ r |v| ≤ δ|v|, for all u, v, w ∈ H,
by choosing r sufficiently large. Notice that the value of r does not depend on N and k. This completes the proof.

3.4.

Existence of an inertial manifold. Finally, we state and prove the main result of the manuscript: the existence of inertial manifolds for the hyperviscous Navier-Stokes equations (1.1) for the critical case β = 3/2. Recall that, if f ∈ H 1 , all solutions of u t + A 3/2 u + B(u, u) = f enter an absorbing ball B in H 4 for large time, which coincide with solutions of the "prepared" equation (3.62) in B.

Theorem 3.7. Consider the "prepared" equation for the incompressible hyperviscous Navier-Stokes equations on T d , d = 2 or 3, namely,

u t + A 3/2 u + B(U (u), U (u)) = f, (3.62)
for f ∈ H 1 . The equation (3.62) possesses an inertial manifold in H in the sense of Definition 2.1.

Proof. First, we write (3.62) in the form

u t + A 3/2 u + A 1/2 F (u) = f,
where

F (u) = A -1/2 B(U (u), U (u)
). By Proposition 3.5 (whose statement is valid in both two and three dimensions),

F is Gateaux differentiable from H to H such that |F (u)| L(H,H) ≤ L for all u ∈ H. Also |F (u)| H 2 ≤ C for all u ∈ H, for some constant C > 0. Let {λ n = |j| 2 : j ∈ Z d \{0}}, d = 2 or 3, such that 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ • • • , correspond to the eigenvectors {e ij•x : j ∈ Z d \{0}} of A in H.
Case 1 : If there exists a sufficiently large spectral gap, namely, there exists N ∈ N such that λ N +1 -λ N ≥ 2L, then by Theorem 2.2, there exists an N -dimensional inertial manifold for (3.62). In particular, on the two-dimensional periodic domain T 2 , an arbitrary large gap for eigenvalues of A = -∆ is guaranteed by Theorem 4.2 (see the Appendix), due to Richards [START_REF] Richards | On the gaps between numbers which are sums of two squares[END_REF].

Case 2 : In three dimensions, there does not necessarily exist a sufficiently large spectral gap. But, by Proposition 3.6, there exist N ∈ N and k ∈ [γ log λ N , λ N ) for some γ ∈ (0, 1] such that λ N ≥ e 40L 2 /γ and 1 ≤ λ N +1 -λ N ≤ 2L, and

|R k,N F (u)R k,N v| ≤ δ|v|, for all u, v ∈ H,
for some δ ≤ 1 50 . Consequently, the spatial averaging condition (2.6) is fulfilled, so we conclude from Corollary 2.8 that equation (3.62) possesses an N -dimensional inertial manifold in H.

Appendix

In this appendix, we include a series of well-known theoretical results (see Appendix 4.1) and then provide the complete proof of Theorem 2.7 (see Appendix 4.2). We also provide a proof of the strong cone property when the optimal spectral gap condition is satisfied (see Appendix 4.3), and a proof of the global well-posedness of weak solutions and the existence of global attractors for the hyperviscous Navier-Stokes equations (1.1) when β ≥ 5/4 in three-dimensional domains (see Appendix 4.4). 4.1. Supporting results. First let us recall the uniform Gronwall inequality. Lemma 4.1 (Temam [33]). Let g, h and y be nonnegative locally integrable functions defined for t ≥ t 0 . Suppose dy dt ≤ gy + h, for t ≥ t 0 .

Then, it holds

y(t + 1) ≤ e t+1 t g(τ )dτ t+1 t y(τ )dτ + t+1 t h(τ )dτ , for t ≥ t 0 .
Next we state a few number theoretic results which are related to the construction of an inertial manifold. Theorem 4.2 (Richards [START_REF] Richards | On the gaps between numbers which are sums of two squares[END_REF]). The sequence

{s n = k 2 1 + k 2 2 : k 1 , k 2 ∈ Z and s n+1 ≥ s n } satisfies lim sup n→∞ s n+1 -s n log s n ≥ δ,
for some δ > 0.

Theorem 4.3 (Mallet-Paret and Sell [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF]). Let D ⊂ Z be a finite nonempty set of integers with the property that d∈A d is not a perfect square whenever A ⊂ D has odd cardinality. There exist arbitrarily large m ∈ N and h ≥ C log m for some constant C > 0 independent of m, satisfying: if T is any quadratic form

T (k 1 , k 2 ) = ak 2 1 + bk 1 k 2 + ck 2 2 , a, b, c ∈ Z, with discriminant d = b 2 -4ac ∈ D, then T (k 1 , k 2 ) ∈ [m, m + h] for each k 1 , k 2 ∈ Z.
Proof. Theorem 4.3 has been proved by Mallet-Paret and Sell in [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF]. The estimate of the size of the gap h, i.e., h ≥ C log m has been claimed in [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF] without a detailed argument. For the sake of completion, we present the proof for the estimate h ≥ C log m as follows. The idea is from the proof of Theorem 4.2 by Richards in [START_REF] Richards | On the gaps between numbers which are sums of two squares[END_REF]. Indeed, since we consider the discriminant d = b 2 -4ac ∈ D, without loss of generality, one may assume that every element d in D satisfying d ≡ 0 or 1 (mod 4). It has been proved in [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF] Define P be the product

P := p 1+α (4.3)
where the product is taken over all primes p with p δ and p α ≤ A < p 1+α for some integer α > 0. It has been shown in [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF] that

T (k 1 , k 2 ) ∈ [m, m + h] for any k 1 , k 2 ∈ Z.
We argue that h ≥ C log m. Indeed, the number of primes p ≤ A is asymptotic to A log A . By (4.3), we obtain that

P ≤ A 2(1+ε)A log A
= e 2(1+ε)A , for some ε > 0.

This implies that

A ≥ log P 2(1 + ε) ≥ log m 2(1 + ε) (4.6)
since P ≥ m. Moreover, since r and δ are fixed integers, then h and A can be chosen arbitrarily large such that (4.2) is valid and

h ≥ A 2δ . (4.7)
In view of (4.6) and (4.7), we conclude that h ≥ log m 4δ (1 + ε) .

This demonstrates the size of the gap stated in Theorem 4.3.

The following result is a corollary of Theorem 4.3.

Corollary 4.4 (Mallet-Paret and Sell [START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF]). Let T be a finite collection of functions T of the form

T (k 1 , k 2 ) = ak 2 1 + bk 1 k 2 + ck 2 2 + sk 1 + tk 2 + u, with rational coefficients and negative discriminant, i.e., a, b, c, s, t, u ∈ Q, b 2 -4ac < 0.
Then, there exist arbitrarily large m ∈ N and h ≥ C log m for some constant C > 0 independent of m, such that 4.2. Proof of Theorem 2.7. It is standard that the cone invariance property and the decay property together imply the existence of an inertial manifold in H for the model problem (2.5). We present the proof of Theorem 2.7 for the sake of completion.

T (k 1 , k 2 ) ∈ [m, m + h] for each T ∈ T and k 1 , k 2 ∈ Z.
Proof of Theorem 2.7. We adopt the proof by Zelik [START_REF] Zelik | Inertial manifolds and finite-dimensional reduction for dissipative PDEs[END_REF] (see also Kostianko and Zelik [25]). The argument can be split into four steps.

Step 1 : We aim to prove that the boundary-value problem

u t + A 3/2 u + A 1/2 F (u) = f P N u(0) = u 0 , Q N u(-T ) = 0 (4.9)
has a unique solution for any T > 0 and u 0 ∈ P N H.

In fact, let us denote by S(t) the semigroup induced by (2.5) and consider the map G T : P N H → P N H given by G T (ϕ) = P N S(T )ϕ, for ϕ ∈ P N H.

First notice that G T is Lipschitz continuous. To see it, we let ϕ 1 and ϕ 2 ∈ P N H. Set u 1 (t) = S(t)ϕ 1 and u 2 (t) = S(t)ϕ 2 . Due to (2.40), one has

d dt |u 1 (t) -u 2 (t)| 2 ≤ L 2 |u 1 (t) -u 2 (t)| 2 , for all t ≥ 0. Therefore, |u 1 (t) -u 2 (t)| 2 ≤ e tL 2 |ϕ 1 -ϕ 2 | 2 , for all t ≥ 0. It follows that |G T (ϕ 1 ) -G T (ϕ 2 )| 2 ≤ e T L 2 |ϕ 1 -ϕ 2 | 2 , for any ϕ 1 , ϕ 2 ∈ P N H.
This shows that G T : P N H → P N H is Lipschitz continuous.

Next we shall justify that the map G T is injective. Put v(t) = u 1 (t)-u 2 (t). Also set p(t) = P N v(t) and q(t) = Q N v(t). Taking the inner product of v t +A 3/2 v+A 1/2 (F (u 1 )-F (u 2 )) = 0 with p yields 1 2 d dt |p| 2 + |A 3/4 p| 2 + (F (u 1 ) -F (u 2 ), A 1/2 p) = 0, which implies 1 2

d dt |p| 2 + λ 3/2 N |p| 2 + Lλ 1/2 N |p||v| ≥ 0, (4.10)
where we have used the Lipschitz continuity .11) This implies that G T : P N H → P N H is injective, and G -1 T is Lipschitz continuous on its domain.

|F (u 1 ) -F (u 2 )| ≤ L|u 1 -u 2 | = L|v|. Since v(0) = ϕ 1 -ϕ 2 ∈ P N H,
|ϕ 1 -ϕ 2 | ≤ e λ 3/2 N +2Lλ 1/2 N T |G T (ϕ 1 ) -G T (ϕ 2 )|. ( 4
Since P N H is finite dimensional and G T is continuous and injective, by using the theorem of invariance of domain [START_REF] Brouwer | Zur Invarianz des n-dimensionalen Gebiets[END_REF], we conclude that G T is a homeomorphism from P N H to an open set G T (P N H). Due to (4.11), we see that the open set G T (P N H) has no boundary. For, if ψ 0 is a point on the boundary of the open set G T (P N H), then there exists a sequence {ψ n } ⊂ G T (P N H) such that ψ n → ψ 0 , and by (4.11), G -1 T ψ n is a Cauchy sequence and has a limit point φ 0 in P N H, then G T (φ 0 ) = ψ 0 by the continuity of G T , which contracts the assumption that ψ 0 ∈ G T (P N H). As a result, G T (P N H) = P N H, i.e., G T is a homeomorphism from P N H to P N H. Therefore, G T (ϕ) = u 0 is uniquely solvable for all u 0 ∈ P N H. It follows that u(t) = S(t + T )G -1 T (u 0 ) is the unique solution of the boundary-value problem (4.9).

Step 2 : Given u 0 ∈ P N H. Let u T (t) be the solution of the boundary-value problem (4.9). We aim to prove that the limit ũ(t) = lim

T →∞ u T (t) (4.12)
exists for all t ≤ 0 and is a backward solution of the problem

u t + A 3/2 u + A 1/2 F (u) = f with P N u(0) = u 0 . Indeed, let T 2 > T 1 > 0, and set v(t) = u T 1 (t)-u T 2 (t), for t ∈ [-T 1 , 0]. Put p(t) = P N v(t) and q(t) = Q N v(t). Notice that p(0) = P N u T 1 (0) -P N u T 2 (0) = u 0 -u 0 = 0.
|v(t)| ≤ Ce -α(t+T 1 ) |v(-T 1 )| ≤ 2Ce -α(t+T 1 ) |q(-T 1 )| = 2Ce -α(t+T 1 ) |Q N u T 2 (-T 1 )|, (4.13)
for all t ∈ [-T 1 , 0], where the last equality is due to the fact that Q N u T 1 (-T 1 ) = 0. In order to find a uniform bound for |Q N u T 2 (-T 1 )| for any T 2 > T 1 > 0, we consider an arbitrary solution u T of problem (4.9). One has 1 2

d dt |Q N u T | 2 + |A 3/4 Q N u T | 2 ≤ |(F (u T ), A 1/2 Q N u T )| + |(f, Q N u T )| ≤ |A -1/4 F (u T )| 2 + 1 4 |A 3/4 Q N u T | 2 + |A -3/4 f | 2 + 1 4 |A 3/4 Q N u T | 2 , for t ∈ [-T, 0]. It follows that d dt |Q N u T | 2 + |A 3/4 Q N u T | 2 ≤ 2|A -1/4 F (u T )| 2 + 2|A -3/4 f | 2 ≤ C,
for t ∈ [-T, 0], due to the assumption that |A -1/4 F (u)| is uniformly bounded for all u ∈ H. As a result, we have

d dt |Q N u T | 2 + λ 3/2 N +1 |Q N u T | 2 ≤ C, for t ∈ [-T, 0]. (4.14)
Since u T is a solution of (4.9), we know that Q N u T (-T ) = 0, then by (4.14), one has

|Q N u T (t)| 2 ≤ C, (4.15)
for some constant C > 0 and all t ∈ [-T, 0] and for any T > 0. It follows that

|Q N u T 2 (T 1 )| 2 ≤ C.
Therefore, by (4.13), we obtain that

|u T 1 (t) -u T 2 (t)| = |v(t)| ≤ Ce -α(t+T 1 ) , for all t ∈ [-T 1 , 0],
for any T 2 > T 1 > 0. Consequently, for any t * > 0, as T 1 , T 2 → ∞, one has

|u T 1 (t) -u T 2 (t)| → 0, uniformly on [-t * , 0].
This implies that u T is a Cauchy sequence in C([-t * , 0]; H) for any t * > 0. Hence, ũ = lim T →∞ u T exists in C([-t * , 0]; H) for any t * > 0, where u T is the solution of (4.9). This shows that ũ(t) is a backward solution of u t + A 3/2 u + A 1/2 F (u) = f with P N u(0) = u 0 and it has a unique extension ũ which solves u t + A 3/2 u + A 1/2 F (u) = f for all t ∈ R. Also, we remark that, due to (4.15), |Q N ũ(t)| is uniformly bounded for all t ∈ R.

Step 3 : Define the set A as the set of all solutions ũ(t), t ∈ R, of (2.5) constructed in Step 2 using the limiting process (4.12). By the construction, the set A is invariant in time, i.e, T (τ )A = A, where (T (τ )ũ)(t) = ũ(t + τ ), τ ∈ R. Also, for two trajectories ũ1 and ũ2 ∈ A, we claim that

|Q N (ũ 1 (t) -ũ2 (t))| ≤ |P N (ũ 1 (t) -ũ2 (t))|, for all t ∈ R. (4.16)
Indeed, by the limiting process (4.12), there exist u 1 0 and u 2 0 ∈ P N H such that lim T →∞ u T,u 1 0 = ũ1 and lim T →∞ u T,u 2 0 = ũ2 , where u T,u j 0 , j = 1, 2, are solutions of (2.5) with the boundary

condition P N u(0) = u j 0 , Q N u(-T ) = 0. Since Q N u T,u 1 0 (-T ) = Q N u T,u 2
0 (-T ) = 0, then by the cone invariance property, we obtain that

|Q N (u T,u 1 0 (t) -u T,u 2 0 (t))| ≤ |P N (u T,u 1 0 (t) -u T,u 2 0 (t))|, for all t ≥ -T. (4.17)
By passing to the limit on (4.17) as T → ∞, we obtain (4.16). We define the map Φ :

P N H → Q N H by Φ(u 0 ) = Q N ũ(0) where ũ ∈ A with P N ũ(0) = u 0 .
Due to (4.16), we have

|Q N (ũ 1 (0) -ũ2 (0))| ≤ |P N (ũ 1 (0) -ũ2 (0))|. That is, |Φ(u 1 0 ) -Φ(u 2 0 )| ≤ |u 1 0 -u 2 0 |, for any u 1 0 , u 2 0 ∈ H. This shows that Φ : P N H → Q N H is Lipschitz continuous with Lipschitz constant 1.
Now we construct the Lipschitz manifold of dimension N as

M := {u 0 + Φ(u 0 ) : u 0 ∈ P N H}.
The invariance of A implies the invariance of M with respect to the semigroup S(t).

Step 4 : We aim to show that any trajectory of the solution of (2.5) approaches the finitedimensional Lipschitz manifold M exponentially fast. Indeed, let u(t), t ≥ 0, be a solution of (2.5) such that its trajectory does not belong to M. For each T > 0, there exists ũT ∈ A such that P N u(T ) = P N ũT (T ). Then by the cone invariance property, we have

|P N (u(t) -ũT (t))| < |Q N (u(t) -ũT (t))|, for t ∈ [0, T ], (4.18)
and by the decay property, we have

|u(t) -ũT (t)| ≤ Ce -αt |u(0) -ũT (0)|, for t ∈ [0, T ]. ( 4
(u(0) -ũT (0))| < |Q N (u(0) -ũT (0))|.
As a result, |ũ T (0)| is uniformly bounded for all T > 0. Since the Lipschitz manifold M is finite dimensional and ũT (0) ∈ M, there exists a sequence ũT j (0) → ũ(0) ∈ M as T j → ∞ and the corresponding trajectory ũ(t) ∈ A. Since ũT j and ũ are both solutions of (2.5), due to (2.40), we obtain that

|ũ T j (t) -ũ(t)| 2 ≤ e L 2 t |ũ T j (0) -ũ(0)| 2 , for all t ≥ 0.
This implies that |ũ T j (t) -ũ(t)| → 0 as T j → ∞ for each t ≥ 0. Therefore, by virtue of (4.19), we conclude that

|u(t) -ũ(t)| ≤ Ce -αt |u(0) -ũ(0)|, for t ≥ 0.
This shows that any trajectory u(t) outside the inertial manifold M exponentially approaches a trajectory ũ(t) in M.

4.3.

Derivation of the cone property under the optimal spectral gap condition.

For the self-containedness of the paper, we derive a strong cone property for the model (2.1) with the assumption of the sharp spectral gap condition (2.4).

Proposition 4.6. Let u 1 and u 2 be two solutions of (2.1).

Set v = u 1 -u 2 . Denote p = P N v and q = Q N v. Define V(t) = |q| 2 H -1/2 -|p| 2 H -1/2 .
Assume F : H → H is globally Lipschitz continuous with Lipschitz constant L. Suppose the spectral gap condition (2.4) is satisfied. Then, the following strong cone property holds:

1 2 d dt V(t) + αV(t) ≤ -µ|v(t)| 2 , for all t ≥ 0, (4.20)
where α =

λ β N λ 1/2 N +1 +λ β N +1 λ 1/2 N λ 1/2 N +1 +λ 1/2 N and µ = λ β N +1 -λ β N λ 1/2 N +1 +λ 1/2 N -L > 0.
Proof. Let u 1 and u 2 be two solutions of (2.1) and set v = u 1 -u 2 , then one has

v t + A β v + A 1/2 [F (u 1 ) -F (u 2 )] = 0, (4.21)
where β ≥ 3/2. Set p = P N v and q = Q N v. Take the scalar product of (4.21) with A -1/2 p and A -1/2 q respectively: It is standard that the above cone invariance and the decay properties lead to the existence of an inertial manifold in H -1/2 , and thanks to the parabolic smoothing property, it immediately becomes an inertial manifold in the phase space H.

1 2 d dt |p| 2 H -1/2 + |A β/2 p| 2 H -1/2 + (F (u 1 ) -F (u 2 ), p) = 0, 1 2 d dt |q| 2 H -1/2 + |A β/2 q| 2 H -1/2 + (F (u 1 ) -F (u 2 ), q) = 0. (4.22) Define V(t) = |q| 2 H -1/2 -|p| 2 H -1/2

4.4.

Global well-posedness and global attractors of equation (1.1). For the reader's convenience, we provide a brief analysis of the hyperviscous Navier-Stokes equations (1.1) in three dimensions when β ≥ 5/4. For the sake of clarity, we consider the borderline case that β = 5/4. Our analysis can readily be adapted to treat the general case β ≥ 5/4. (The reader may also refer to [START_REF] Cholewa | Fractional Navier-Stokes equations[END_REF][START_REF] Holst | Analysis of a general family of regularized Navier-Stokes and MHD models[END_REF] for well-posedness and the existence of global attractors for some general families of problems including (1.1)). Without loss of generality, we assume ν = 1. So in the following analysis, we consider u t + A 5/4 u + B(u, u) = f in T 3 , u| t=0 = u 0 .

(4.27)

Theorem 4.7. Let f ∈ H -5/4 . Then for every u 0 ∈ H, problem (4.27) possesses a unique weak solution such that u ∈ C ([0, T ] ; H) ∩ L 2 (0, T ; H 5/4 ) with u t ∈ L 2 (0, T ; H -5/4 ) for an arbitrary time T > 0. The weak solution depends continuously on the initial data.

Proof. We provide the following a priori estimates. By taking the inner product of ( Estimate (4.37) shows that all trajectories of the dynamics enter an absorbing ball of the radius ρ in H 1 for large time. Therefore, due to the standard theory of attractors (see [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF]), the dynamics possesses a compact global attractor in the phase space H.

Corollary 2 . 8 .

 28 Suppose the assumptions of Theorem 2.4 are satisfied and |A -1/4 F (u)| ≤ C for all u ∈ H. Then problem (2.5) possesses an N -dimensional inertial manifold M in the sense of Definition 2.1.

3. 1 .

 1 Dissipative estimates. This section is devoted to showing that the dynamics of (3.4) has an absorbing ball in H 4 , such that (3.4) possesses a global attractor. Let S (t) : H → H be the solution semigroup generated by the system (3.4). Theorem 3.2. Let f ∈ H 1 . Then (S, H) possesses a compact global attractor G in the phase space H, which is a bounded subset of H 4 .

5 2 - 5 2

 25 δ and there exists a constant C δ such that|U (u)| H -δ ≤ C δ , for all u ∈ H. (3.35)

|j| 4 ûj ρ = |j| 4 ( 2 )

 42 ûj ρ for all j ∈ Z 3 \{0}. Hence by the definition (3.33), we see that U (u) = u provided |u| H 4 ≤ ρ. Let u ∈ H. Since |θ(ξ)| ≤ 2 for all ξ ∈ C, then by (3.33), we calculate

  .43) By Lemma 3.4, for any u ∈ D(A 2 ) such that |u| H 4 ≤ ρ, one has U (u) = u, and thus

  .54) We denote by U (u) n the n-th component of the vector U (u), and denote by [U (u)v] n the n-th component of the vector U (u)v, where n = 1, 2, 3. By applying (3.5) we write b

ϕ

  >r := |j|>r φj e ij•x , and ϕ <r := 1≤|j|≤r φj e ij•x .

  that there exists r = 0 such that gcd (d, r) = 1 and d r = -1, for each d ∈ D, (4.1) where d r is the Jacobi symbol. Define δ := lcm {|d| : d ∈ D}. By (4.1), one has gcd (δ, r) = 1. Let h > 0 and set A := sup 0≤j≤h |r + δj|.(4.2)

(4. 4 )

 4 Recall gcd(δ, r) = 1. Set the integer m ∈ [1, P ] satisfying δm ≡ r (mod P ).(4.5)

(4. 8 )

 8 Corollary 4.4 implies the following proposition.

Proposition 4 . 5 (

 45 Mallet-Paret and Sell[START_REF] Mallet-Paret | Inertial manifolds for reaction diffusion equations in higher space dimensions[END_REF]). Let r > 0. There exist arbitrarily large λ > 0 and κ > C log λ for some constant C > 0 independent of λ, satisfying: whenever |n| 2 , |l| 2 ∈ [λ -κ, λ + κ] with distinct n and l ∈ Z 3 one has |n -l| > r.

  .25) Integrate (3.25) over [s, t + 1] for t 0 + 2 ≤ t ≤ s ≤ t + 1:

	|u(t + 1)| 2 H 5/2 ≤ |u(s)| 2 H 5/2 + Cρ 4 2 + 2|f | 2 H 1 .	(3.26)
	Integrating (3.26) over [t, t + 1] with respect to s yields
			t+1
	|u(t + 1)| 2 H 5/2 ≤	t	|u(s)| 2 H 5/2 ds + Cρ 4 2 + 2|f | 2 H 1
	≤ ρ 2 3 := ρ 2 1 + Cρ 4 1 + 2|f | 2 H -1/2 + Cρ 4 2 + 2|f | 2 H 1 , for t ≥ t 0 + 2,	(3.27)
	due to estimate (3.21). Now integrating (3.25) over [t, t + 1] for t ≥ t 0 + 3 and using (3.27),
	we obtain
	t+1
	t	|u t | 2 H 1 dτ ≤ ρ 2 3 + Cρ 4 2 + 2|f | 2 H 1 , for t ≥ t 0 + 3.	(3.28)
	Next we differentiate (3.4) in time,

  .41) where we have used θ is uniformly bounded in C 3 . Since v ∈ H, we may choose N sufficiently large such that C |j|>N |v j | 2 ≤ 2 . Also, since |u -w| ≤ σ, one has |û j -ŵj | ≤ σ for any j ∈ Z 3 \{0}, and then by the uniform continuity of θ , we may select σ > 0 small enough such

	that 1≤|j|≤N θ	|j| 4 ûj ρ	-θ	|j| 4 ŵj ρ	2	|v

j | 2 ≤ 2 . Therefore, we have |(U (u)-U (w))v| 2 ≤ if |u -w| ≤ σ.

  .44) for f ∈ H 1 . Recall that (3.44) is named the "prepared" equation for (3.42), since equations (3.42) and (3.44) have the same large time behavior in the absorbing ball in H 4 .Before showing that F is Gateaux differentiable from H to H (see Proposition 3.5 below), we formally compute the Gateaux derivative of F : H → H at the point u ∈ H in the direction of v ∈ H. In fact, for any w ∈ H,

  Let F be the operator defined by(3.43). Then F is uniformly bounded from H to H 2 , i.e, there exists C > 0 with |F (u)| H 2 ≤ C for all u ∈ H. Moreover, F is Gateaux differentiable from H to H, and its derivative F has the expression:

	where we have used (3.5) and (3.6).
	Proposition 3.5.
	.45)

  .61) By (3.35) and (3.37) in Lemma 3.4, we see that |U (u)| H 2 ≤ C for all u ∈ H and |U (u)v| ≤ L 1 |v|. Therefore, we derive from (3.61) that

  by the cone invariance property, |q(t)| ≤ |p(t)|, for all t ≥ 0. Thus |v(t)| ≤ 2|p(t)|, for all t ≥ 0. Then, due to (4.10), one has

	d dt	|p| 2 + 2 λ 3/2 N + 2Lλ	1/2 N	|p| 2 ≥ 0, for all t ≥ 0.
	As a result, |p(0)| ≤ e	λ 3/2 N +2Lλ 1/2 N	t |p(t)| for all t ≥ 0. Consequently, we have

  Then by the cone invariance property, we have |q(t)| ≥ |p(t)| for all t ∈ [-T 1 , 0], and thus 2|q(t)| ≥ |v(t)| for all t ∈ [-T 1 , 0]. Then by the decay property, we obtain that

  |A β/2 q| 2 H -1/2 -|A β/2 p| 2 H -1/2 = (F (u 1 ) -F (u 2 ), p -q) ≤ L|v| 2 ,due to the assumption that F :H → H is globally Lipschitz with Lipschitz constant L. + αV(t) ≤ -|A β/2 q| 2 H -1/2 -α|q| 2 H -1/2 -α|p| 2 H -1/2 -|A β/2 p| 2 H -1/2 + L|v| 2 .-L > 0 owing to the spectral gap condition(2.4).By adopting the proof of Proposition 2.5 and using the estimates d dt |v(t)| 2 ≤ L 2 |v(t)| 2 as well as d dt |v(t)| 2 H -1/2 ≤ L|v(t)| 2 , it can be readily shown that the strong cone property (4.20) implies • Cone invariance property: If V(0) ≤ 0, then V(t) ≤ 0 for all t ≥ 0; • Decay property: If V(T ) > 0 for some T > 0, then |v(t)| 2 ≤ Ce -αt |v(0)| 2 and |v(t)| 2 H -1/2 ≤ Ce -αt |v(0)| 2 H -1/2 for all t ∈ [0, T ]. (4.26)

	Since α =	λ β N λ λ 1/2 N +1 +λ β N +1 λ 1/2 N N +1 +λ 1/2	1/2 N	, we calculate
		α|p| 2 H -1/2 -|A β/2 p| 2 H -1/2 =	N j=1	α -λ β j λ 1/2 j	|v j | 2 ≥	α -λ β N λ 1/2 N	|p| 2 =	λ β N +1 -λ β N λ 1/2 N N +1 + λ 1/2	|p| 2 .	(4.24)
	Also,
	|A β/2 q| 2 H -1/2 -α|q| 2 H -1/2 =	∞ j=N +1	λ β j -α λ 1/2 j	|v j | 2 ≥	λ β N +1 -α λ 1/2 N +1	|q| 2 =	λ β N +1 -λ β N λ 1/2 N N +1 + λ 1/2	|q| 2 . (4.25)
	Substituting (4.24) and (4.25) into (4.23), we obtain
							1 2	d dt	V(t) + αV(t) ≥ -	λ β N +1 -λ β N λ 1/2 N N +1 + λ 1/2	-L |v| 2 ,
	where µ =	λ β N +1 -λ β N λ 1/2 1/2 N +1 +λ N
										. Subtracting the two equations in (4.22) yields
			1 2	d dt	V(t) + It
	follows that
	1 2	d dt	V(t) (4.23)

  4.27) with u, it is easy to deduce that d dt |u| 2 + |u| 2 H 5/4 ≤ |f | 2 H -5/4 . (4.28) From (4.28), we see that |u(t)| 2 ≤ e -t |u 0 | 2 + |f | 2 H -5/4 (1 -e -t ), for t ≥ 0, (4.29) dτ ≤ |u 0 | 2 + t|f | 2 H -5/4 , for t ≥ 0. (4.30) Using the Sobolev imbedding H s → L 6 3-2s in three dimensions for s ∈ (0, 3/2), we deduce (B(u, u), φ) = b(u, u, φ) ≤ C|φ| L 12 |∇u| L 12/5 |u| ≤ C|φ| H 5/4 |u| H 5/4 |u|. It follows that |B(u, u)| H -5/4 ≤ C|u| H 5/4 |u|. Then due to (4.29) and (4.30), we have B(u, u) is bounded in L 2 (0, T ; H -5/4 ), and thus u t is bounded in L 2 (0, T ; H -5/4 ). (4.31)On one hand, the existence of global weak solutions can be easily obtained by standard Galerkin method, using the a priori estimates (4.29)-(4.31) and a compactness argument.On the other hand, it is crucial to verify the uniqueness of weak solutions, which explains why β = 5/4 is the borderline case for the well-posedness of 3D hyperviscous Navier-Stokes equations. To show the uniqueness, we take two solutions u 1 and u 2 of (4.27) and set v = u 1 -u 2 . Then, we havev t + A 5/4 v + B(u 1 , v) + B(v, u 2 ) = 0. (4.32) Take the L 2 inner product of (4.32) with v, then ≤ |b(v, u 2 , v)| = |b(v, v, u 2 )| ≤ C|u 2 | L 12 |∇v| L 12/5 |v| ≤ C|u 2 | H 5/4 |v| H 5/4 |v| ≤ 1 2 |v| 2 H 5/4 + C|u 2 | 2 H 5/4 |v| 2 . C|u 2 | 2 H 5/4 |v| 2 . (4.33) Thanks to the Gronwall inequality, we obtain from (4.33) and (4.30) that|v(t)| 2 ≤ exp C t 0 |u 2 | 2 H 5/4 dτ |v(0)| 2 ≤ C exp |u 2 (0)| 2 + t|f | 2 H -5/4 |v(0)| 2 , (4.34) for t ≥ 0. If v(0) = u 1 (0) -u 2 (0) = 0, then (4.34) shows that v(t) = u 1 (t) -u 2 (t) = 0 for all t ≥ 0,so weak solutions are unique. The inequality (4.34) also implies the continuous dependence on initial data. Theorem 4.8. Let f ∈ H -1/4 . Then the dynamics induced by (4.27) possesses a compact global attractor in the phase space H. Proof. Estimate (4.29) implies that all trajectories of the dynamics induced by (4.27) enter an absorbing ball of radius 2|f | H -5/4 in H. Namely, |u(t)| ≤ 2|f | H -5/4 , for t ≥ t 0 . (4.35) To construct an absorbing ball in H 1 , we take the inner product of (4.27) with Au, + |u| 2 H 9/4 ≤ |b(u, u, Au)| + |(f, Au)| ≤ C|u| L 12 |Au| L 12/5 |u| H 1 + |u| H 9/4 |f | H -1/4 ≤ C|u| H 5/4 |u| H 9/4 |u| H 1 + H 5/4 dτ ≤ |u(t)| 2 + |f | 2 H -5/4 ≤ 5|f | 2 H -5/4 for t ≥ t 0 owing to (4.28) and (4.35). Therefore, by using the uniform Gronwall inequality on (4.36), we obtain |u(t)| 2 H 1 ≤ ρ 2 := exp C|f | 2 H -5/4 (5|f | 2 H -5/4 + 2|f | 2 H -1/4 ), for t ≥ t 0 + 1. (4.37)

	and H 5/4 It follows that t 0 H 5/4 1 |u| 2 2 d dt |v| 2 + |v| 2 d dt |v| 2 + |v| 2 2 d dt |u| 2 H 1 1 4 |u| 2 H 9/4 + |f | 2 H -1/4 ≤ C|u| 2 H 5/4 |u| 2 H 1 + Hence, d dt |u| 2 H 1 + |u| 2 H 9/4 ≤ C|u| 2 H 5/4 |u| 2 H 1 + 2|f | 2 H -1/4 . 1 2 H 5/4 ≤ 1 Notice that t+1 t |u| 2	|u| 2 H 9/4 + |f | 2 H -1/4 . (4.36)

 3-2s in three dimensions for s ∈ (0, 3/2).

-δ is continuous.

Acknowledgment. The authors are grateful to the reviewer for providing valuable comments and suggestions which further improved the quality of the paper.


Patterns of Knowledge Outflow from Industrial Marketing Management to

Introduction

Journals often mark special, commemorative occasions and major milestones by engaging in healthy introspective evaluations of their history, strategy, evolution, and impact (e.g., [START_REF] Schrock | JPSSM since the beginning: intellectual cornerstones, knowledge structure, and thematic developments[END_REF][START_REF] Huber | A topical history of JMR[END_REF][START_REF] Hustad | PDMA History, Publications, and Developing a Future Research Agenda: Appreciating Our PDMA Accomplishments -Celebrating People, Lasting Friendships and Our Collective Accomplishments[END_REF][START_REF] Chintagunta | Marketing Science: A strategic review[END_REF]Calabretta, Durisin and Ogiliengo 2011). In this rich tradition, we mark the changing of the guard at Industrial Marketing Management, and Peter Laplaca's long tenure as Editor of IMM by tracing the impact Industrial Marketing Management has had on major marketing and specialized B2B marketing journals from 1999-2013. This period of interest overlaps significantly with Peter's stewardship of the journal.

Recent studies have examined levels of citation flows to assess import and export of knowledge between disciplines and domains (e.g. [START_REF] Clark | The intellectual ecology of mainstream marketing research: An inquiry into the place of marketing in the family of business disciplines[END_REF][START_REF] Shafiq | Thinking inside the box? Intellectual structure of the knowledge base of innovation research (1988-2008)[END_REF]. We use a similar approach, but at the level of a particular journal. Specifically, we take a dynamic, yearby-year look at the impact Industrial Marketing Management (IMM) has had in terms of its Impact Factor, and citations in relevant major marketing and journals focusing on industrial/business-to-business marketing. In particular, from 1999-2013 we take a longitudinal look at:

1) the level of self-citation by IMM, and the growth in its impact factor; 2) the level of citation of IMM by top tier marketing journals; 3) the level of citation of IMM by second tier marketing journals, and specialized B2B/industrial marketing journals; and 4) the effect of self-and cross-citations on the 2-year Impact Factor of IMM from 1999-2013.

IMM's Impact Factor, Self-and Cross-Citation Rates in Top Tier, Second Tier and Specialized Marketing Journals (1999 -2013)

The journals under consideration differ with respect to the number of issues published per year, the number of articles published per issue, and the number of references cited per article.

In order to account for these differences, all citation data were normalized, and are presented here in terms of the percentage of all references cited in that journal in that year.

Figure 1 presents the change in the 2-year impact factor of IMM along with its selfcitation rates from 1999-2013. After decreasing from 1999 to 2006, the self-citation rates grew consistently, reaching nearly 11% in 2013. All this while, the 2-year impact factor of the journal increased steadily from 1999-2013. Figure 2 shows the growth in IMM's 2-year impact factor relative to its cross-citations in the top-tier, and second-tier and specialized B2B marketing journals. The figure shows that IMM's citations in top-tier marketing journals remains consistently low over the 15-year period under examination. However, the impact factor of the journals tracks the citations in second-tier and specialized B2B marketing journals quite well over this period.

Next, we examined the effect of self-and cross-citation (top tier, second tier and specialized marketing journals) rates on the 2-year impact factor of IMM. The correlation matrix shows only a weak positive correlation between IMM self-citation and it 2-year impact factor (r = .49; p <.10). This is could be because the relationship between the two may not be strictly linear (see Figure 1). IMM self-citations, on the other hand, were strongly correlated with citations in second-tier and specialized B2B marketing journals (r = .88; p <.01). Top-tier marketing journals have a significant and positive correlation with both IMM self-citation (r = .59; p <.05) and citations in second tier and specialized marketing journals (r = .54; p <.05).IMM self-citation and second tier and specialized marketing journals' citations were not significantly correlated.

In order to further ascertain the effect of citation rates on impact factors, we regressed IMM's 2-year impact factor on self-and cross-citation rates (top-tier, second-tier and specialized B2B marketing journals) over the 1999-2013 period. Table 2 reports the results of this multiple regression analysis. Multiple regression show that only second-tier and specialized B2B marketing journals citations have a significant positive effect on the 2-year impact factor (β = .79; p < .00). Neither self-citation rates, nor citations in top-tier marketing journals, have any significant effect on the 2-year impact factor of IMM from 1999-2013. Overall, the model explained 75% of the variance in IMM's impact factor. All the variance inflation factors (VIFs) were less than 2, suggesting that multicollinearity was not an issue in the analysis. Given the self-citation patterns displayed in Figure 1, we also ran a regression analysis to test for a potential non-linear effect of selfcitation on impact factor. The results of the non-linear analysis were consistent with those seen in Table 2, suggesting that the effects obtained were stable and robust.

Discussion

The overall pattern of our results indicates that IMM self-citations followed a very shallow u-shaped pattern from 1999-2013 (See Figure 1). While the self-citation rates initially decrease, before starting to climb again, the IMM impact factor grew steadily over this period. The citations patterns show very low citation rates for IMM in the top-tier marketing journals (see Figure 2), indicating that IMM's share of knowledge imported by top marketing journals remains relatively small. The IMM impact factor seems to track citation rates in second-tier and specialized B2B marketing journals more closely.

Regressing the IMM impact factor over the years on the rates of self-citation, citation in toptier marketing journals, and citation rates in second-tier and specialized journals, indicates that self-citation and citation in top marketing journals had no significant effect on the impact factor of IMM (Table 2). For the period under examination, by far the biggest driver of IMM's 2-year impact factor were citations in second-tier and specialized journals. The proposed model explained 75% of the variance in IMM impact factor of IMM from 1999-2013.

However, in order to tease out this effect even further, we conducted post-hoc analyses where we separated the citation rates in the second-tier journals from those in specialized B2B marketing journals. This post-hoc analysis presented in Figure 3 shows that when separated, the IMM citation pattern in second-tier marketing journals resemble those in top-tier marketing journals, with both tiers citing IMM at a fairly low rate relative to their overall levels of citations (i.e., knowledge imports). In fact, Figure 3 shows that the pattern of growth of IMM impact factor closely follows its citation rates in specialized B2B marketing journals (e.g., JBIM, JBBM) 5 .

5 Our database contained citation data for JBBM and JBIM for only 9 years, which leaves us with insufficient data points to estimate a new regression model with specialized B2B journals included as a separate predictor. While at first the results related to second-tier marketing journals may seem to contradict the findings presented in Figure 3, it is important to note the distinction between the two analyses. The findings reported in Figure 3 pertain to the rate of citations in the top-tier and second-tier marketing journals. That is the fraction of the total citations or knowledge imported in these journals. The analysis presented in Table 3, by contrast, reports the raw numbers of actual IMM citations in these journals. These results together indicate that while IMM's share of knowledge cited in the top-tier (Rank 1-5) and second tier (Rank 5-10) marketing journals remains low, second tier marketing journals (outside the top 8 rankings) constitute some of the most frequent citers of IMM.

Thus while IMM's impact on the top 7-8 marketing journals remains light, it is a reflection of the decreasing emphasis being placed on marketing strategy research in general (and consequently B2B/industrial marketing research) in these journals during 1999-2013.

However, IMM's impact factor has continued to grow at an impressive rate over this period.

Much of this growth has come from sources other than self-citation. Our post-hoc analysis

shows that IMM's appeal is growing, and it is having an ever increasing influence on the related domains of supply chain management (including operations management and logistics), and technology and innovation management.

These results attest to IMM's growing influence on fields and domains beyond marketing. Thus the partial vacuum created by a reduced emphasis on B2B/industrial marketing topics in top marketing journals is being filled by the second tier marketing journals (ranked 8 and beyond), and specialized journals in related fields of supply chain management, and technology and innovation management, which show some of the highest citations of research published in IMM from 1999-2013. The results concerning IMM's influence on fields outside the marketing discipline, in fact, corroborate the conclusions of [START_REF] Touzani | Ranking marketing journals using the search engine Google Scholar[END_REF], who ranked marketing journals using Google Scholar data. They noted that IMM was one of the journals that ranked very high using their methodology, and speculated that one reason for this finding was the popularity of IMM among academics from other disciplines such as strategy, innovation, management, and product development. Our results show that much of the increase in IMM cites is in journals that correspond precisely to these research disciplines (e.

Conclusions

This study presents a citation analysis of the impact of Industrial Marketing Management on the marketing literature. We assessed the impact of IMM on top-tier, second-tier, and specialized B2B marketing journals, and in post hoc analysis, increased the scope of analysis to include journals in related disciplines as well. The period of data availability corresponded closely to the years during which Peter LaPlaca guided the journal, so the results present a form of external validation of the impact of IMM on research in B2B marketing and elsewhere.

The results showed relatively low citation patterns for IMM in some of the top-tier marketing journals -not an unexpected finding, given the relatively lower emphasis on marketing strategy research in recent years among the top marketing journals [START_REF] Houston | Is strategy a dirty word?[END_REF]).

Figure 3 -

 3 Figure 3 -IMM Impact Factor, Self-Citation Rates, and Cross-Citation Rates in Top-Tier,Second-Tier, and Specialized B2B Marketing Journals (1999-2013)

  g., Journal of Product Innovation Management, International Journal of Technology Management, IEEE Transactions on Engineering Management) or to related disciplines such as supply chain management and operations (e.g., Journal of Supply Chain Management, Journal of Purchasing and Supply Management, Journal of Operations Management).

  Nevertheless, one first-tier journal, Journal of the Academy of Marketing Science, shows strong citation patterns, indicating that it may be the top-tier journal most frequently (or most successfully) targeted by B2B marketing researchers. IMM has retained a strong and growing presence in the second-tier and specialized B2B marketing journals, suggesting that articles published in IMM are viewed as highly influential by authors working in B2B marketing research and related streams. The post hoc analysis showed IMM's influence increasing in other fields as well, as judging by high citation numbers in innovation and product development journals such as Journal of Product Innovation Management, and operations and supply chain management journals such as Journal of Operations Management. The impressive citation counts of IMM over the years in B2B marketing and general marketing publications, as well as in related disciplines such as innovation, indicate that, under the thoughtful stewardship of Peter LaPlaca, Industrial Marketing Management has consistently grown in quality, relevance, and influence on ever larger numbers of business scholars. IMM has been the leading journal for published research in B2B and industrial marketing management and strategy for many years, and thanks to the sure-handed and insightful leadership and guidance of Editor-in-Chief Peter LaPlaca, is poised to maintain this position for years to come.

Table 1

 1

	presents the

Table 2 -

 2 Regression Results of the Effect of Self-and Cross-Citation (Top-Tier, Second-Tier and Specialized B2B Marketing Journals) on 2-year Impact Factor of IMM

	Unstandardized	Standardized	VIF
	Coefficients	Coefficients
	(p-values)

* p < .10, ** p < .05, *** p < .01

Table 3

 3

	presents a list of these journals with year-by-

 Journals in Technology-Innovation Management (TIM) domain (i.e., Journal of Product Innovation Management, International Journal of Technology Management, IEEE Transactions on Engineering Management, etc.); and  Journals in operations management/logistics/supply chain management (OM/SCM) domains (i.e., Journal of Supply Chain Management, International Journal of Operations and Production Management, Journal of Purchasing and Supply Management, Journal of Operations Management, Journal of Physical Distribution and Logistics Management, International Journal of Logistics Management, etc.).

Table 3 -

 3 Journals Citing IMM Highest Number of Times from 1999-2013

	Year

Not all the journal records included in this database started in 1999. Some of the journals were incorporated in the database at later dates.

Journal of Advertising Research was tied with JPIM in[START_REF] Guidry | Surveying the cites: A ranking of marketing journals using citation analysis[END_REF] ranking, but it was excluded from this analysis because of its lack of relevance to industrial/business marketing.

Data for these two journals were only available from 2005-2013 in the database.

While the Journal of Personal Selling and Sales Management (JPSSM) would also have been of interest in the specialized journals, we unfortunately did not have data on JPSSM in the database.

Data