Open Access Published by De Gruyter November 28, 2017

Finite-Dimensionality and Determining Modes of the Global Attractor for 2D Boussinesq Equations with Fractional Laplacian

Aimin Huang , Wenru Huo and Michael Jolly

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2017-6036

Abstract

We prove the finite dimensionality of the global attractor and estimate the numbers of the determining modes for the 2D Boussinesq system in a periodic domain with fractional Laplacian in the subcritical case.

Keywords: Boussinesq System; Fractional Laplacian; Hausdorff and Fractal Dimensions, Determining Modes

MSC 2010: 35Q86; 35R11; 34D45

1 Introduction

Recently, the 2D Boussinesq equations and their fractional generalizations have attracted considerable attention due to their physical applications and mathematical challenges. When α=β=1, system (2.1) is called the standard 2D Boussinesq system, which is widely used to model the geophysical flows such as atmospheric fronts and oceanic circulation and which also plays an important role in the study of Rayleigh–Bénard convection (cf. [12]). Flows which travel upward in the middle atmosphere encounter varied conditions in different strata. This phenomenon can be modeled by using a fractional Laplacian. Moreover, some models with a fractional Laplacian such as the surface quasi-geostrophic equations and the Boussinesq equations have very significant applications. In the mathematical respect, the global well-posedness, global regularity of the standard 2D Boussinesq system as well as the existence of the global attractor have been widely studied, see, for example, [4, 14, 15, 17].

This paper estimates the number of determining modes and the dimension of the global attractor for the two-dimensional (2D) incompressible Boussinesq equations with subcritical dissipation. This work is motivated by [8], where the finite dimensionality of the global attractor for 3D primitive equations has been proved, and it is a natural continuation of [6], where we proved the existence of a global attractor for the 2D Boussinesq equations. The aim of this article is twofold. We first prove the finite dimensionality of the global attractor of system (2.1) by showing that the strong solutions of (2.1) on the global attractor satisfy the Ladyzhenskaya squeezing property. The second goal is to prove an estimate for the number of determining modes for system (2.1). In particular, we prove that there exists a finite number m>0 such that each trajectory (θ⁢(t),𝐮⁢(t)) of strong solutions on the global attractor is uniquely determined by its projection Pm⁢(θ⁢(t),𝐮⁢(t)) onto the space spanned by the first m Fourier modes.

The fractal (Hausdorff) dimension of the global attractor and the determining Fourier modes are both measures of the number of degrees of freedom of a turbulence. As α,β decrease, the dissipation in the 2D Boussinesq system becomes weaker, and the non-linear terms are harder to control. We derive a sharp upper bound for the fractal dimension of the global attractor and the number of determining Fourier modes, which are both decreasing functions of the fractional dissipation powers α, β, the viscosity ν, and the diffusivity κ.

The structure of this article is as follows. In Section 2, we introduce the notation, some preliminary results, and state our main results, as well as the results from [6] about the existence of the global attractor in a certain Sobolev space. Section 3 is devoted to proving that the global attractor 𝒜 has finite Hausdorff and fractal dimensions. In Section 4, we prove the existence of the absorbing ball in H2⁢β×H2⁢α, and estimate the number of determining modes on the global attractor.

2 Notations and Preliminaries

2.1 The System

The 2D Boussinesq equations read

(2.1) { ∂ t ⁡ 𝐮 + 𝐮 ⋅ ∇ ⁡ 𝐮 + ν ⁢ ( - Δ ) α ⁢ 𝐮 = - ∇ ⁡ π + θ ⁢ 𝐞 2 , x ∈ Ω , t > 0 , ∇ ⋅ 𝐮 = 0 , x ∈ Ω , t > 0 , ∂ t ⁡ θ + 𝐮 ⋅ ∇ ⁡ θ + κ ⁢ ( - Δ ) β ⁢ θ = f , x ∈ Ω , t > 0 ,

where Ω=[0,2⁢π]2 is the periodic domain, ν>0 the fluid viscosity, and κ>0 the diffusivity. Moreover, 𝐮=𝐮⁢(x,t)=(u1⁢(x,t),u2⁢(x,t)) denotes the velocity, π=π⁢(x,t) the pressure, θ=θ⁢(x,t) a scalar function which may, for instance, represent the temperature variation in the context of thermal convection, 𝐞2=(0,1) the unit vector in the vertical direction, and f=f⁢(x) a time-independent forcing term. We associate to (2.1) the following initial data:

(2.2) 𝐮 ⁢ ( x , 0 ) = 𝐮 0 ⁢ ( x ) , θ ⁢ ( x , 0 ) = θ 0 ⁢ ( x ) , x ∈ Ω .

Since in this article we consider the 2D Boussinesq equations with subcritical dissipation, we assume that the exponents α and β satisfy

α , β ∈ ( 1 2 , 1 ) .

Additionally, as in [6], we also assume that s1 and s2 are numbers such that

(2.3) s 1 > 2 ⁢ max ⁡ { 1 - α , 1 - β } , s 2 ≥ 1 ,

and

(2.4) 0 ≤ s 2 - s 1 < α + β .

Moreover, integrating (2.1) over Ω and applying integration by parts yields

d d ⁢ t ⁢ 𝐮 ¯ = 1 | Ω | ⁢ d d ⁢ t ⁢ ∫ Ω 𝐮 ⁢ d x = θ ¯ ⁢ 𝐞 2 , d d ⁢ t ⁢ θ ¯ = 1 | Ω | ⁢ d d ⁢ t ⁢ ∫ Ω θ ⁢ d x = f ¯ ,

where 𝐮¯,θ¯,f¯ are the mean of 𝐮,θ,f over Ω, respectively, that is,

𝐮 ¯ ≡ 1 | Ω | ⁢ ∫ Ω 𝐮 ⁢ d x , θ ¯ ≡ 1 | Ω | ⁢ ∫ Ω θ ⁢ d x , f ¯ ≡ 1 | Ω | ⁢ ∫ Ω f ⁢ d x .

Therefore, without loss of generality, we assume that 𝐮,θ,f are all of mean zero. Otherwise, we can replace 𝐮-𝐮¯,θ-θ¯,f-f¯ by 𝐮,θ,f, respectively.

2.2 Notation and Function Spaces

Here and throughout this article, we will not distinguish the notations for vector and scalar function spaces whenever they are self-evident from the context. Let Lp⁢(Ω) (1≤p≤∞) be the classical Lebesgue space with norm ∥⋅∥Lp, and let 𝒞⁢([0,T];X) be the space of all continuous functions from the interval [0,T] to some normed space X. We denote by Lp⁢(0,T;X) (1≤p≤∞) the space of all measurable functions u:[0,T]→X with the norm

∥ u ∥ L p ⁢ ( 0 , T ; X ) p = ∫ 0 T ∥ u ∥ X p d t , ∥ u ∥ L ∞ ⁢ ( 0 , T ; X ) = ess ⁢ sup t ∈ [ 0 , T ] ∥ u ∥ X .

For f∈L1⁢(Ω) and k=(k1,k2)∈ℤ2, the Fourier coefficient f^⁢(k) of f is defined as

f ^ ⁢ ( k ) = 1 ( 2 ⁢ π ) 2 ⁢ ∫ Ω f ⁢ ( x ) ⁢ e - i ⁢ k ⋅ x ⁢ d x .

We denote the square root of the Laplacian (-Δ)12 by Λ and have

Λ ⁢ f ^ ⁢ ( k ) = | k | ⁢ f ^ ⁢ ( k ) ,

where |k|=k12+k22. More generally, for s∈ℝ, the fractional Laplacian Λs⁢f can be defined by the Fourier series

Λ s ⁢ f := ∑ k ∈ ℤ 2 | k | s ⁢ f ^ ⁢ ( k ) ⁢ e i ⁢ k ⋅ x .

We denote by Hs⁢(Ω) the space of all the functions f of mean zero with ∥f∥Hs<∞, where the norm ∥⋅∥Hs is defined as

∥ f ∥ H s 2 = ∥ Λ s f ∥ L 2 2 = ∑ k ∈ ℤ 2 | k | 2 ⁢ s | f ^ ( k ) | 2 .

For 1≤p≤∞ and s∈ℝ, the space Hs,p⁢(Ω) consists of the functions f such that f=Λ-s⁢g for some g∈Lp⁢(Ω). The Hs,p-norm of f is defined by

∥ f ∥ H s , p = ∥ Λ s ⁢ f ∥ L p .

By the classic spectral theory of compact operators, we denote by {λj}j=1∞ (0<λ1=1≤λ2≤λ3≤⋯) the eigenvalues of the operator Λ, which are repeated according to their multiplicities and arranged in non-decreasing order corresponding to the eigenfunctions {ωj}j=1∞. For the sake of simplicity, we use ∥⋅∥ to stand for the L2-norm and write Lp, Hs and Hs,p to denotes the spaces Lp⁢(Ω), Hs⁢(Ω) and Hs,p⁢(Ω), respectively, for 1≤p≤∞ and s∈ℝ.

Remark 2.1.

Since the first eigenvalue λ1 of the operator Λ is 1, the constant in the Poincaré inequality is also 1. If s1≤s2, then

∥ Λ s 1 ⁢ g ∥ ≤ ∥ Λ s 2 ⁢ g ∥ for all ⁢ g ∈ H s 2 .

2.3 Some Preliminary Results

We first recall the sharp fractional Sobolev inequality. See [1].

Lemma 2.2 (The Sobolev Inequality).

For 0<s<1 and p=21-s, we have

( ∫ Ω | u ( x ) | p d x ) 2 p ≤ C s ∥ u ( x ) ∥ H s 2 for all u ∈ H s ⁢ ( Ω ) ,

where the best constant Cs is given by Cs=Γ⁢(1-s)(4⁢π)s⁢(π)s/2⁢Γ⁢(1+s).

Next, we recall the interpolation inequality and the uniform Gronwall lemma, which are used frequently in this article; for their proofs, see [13].

Lemma 2.3 (The Interpolation Inequality).

For any s1≤s≤s2 and g∈Hs2, we have

∥ Λ s ⁢ g ∥ ≤ ∥ Λ s 1 ⁢ g ∥ δ ⁢ ∥ Λ s 2 ⁢ g ∥ 1 - δ ,

where s=δ⁢s1+(1-δ)⁢s2 for some 0≤δ≤1.

Lemma 2.4 (Uniform Gronwall Lemma).

Let g, h and y be non-negative locally integrable functions on (t0,+∞) such that

d ⁢ y ⁢ ( t ) d ⁢ t ≤ g ⁢ ( t ) ⁢ y ⁢ ( t ) + h ⁢ ( t ) for all ⁢ t ≥ t 0 ,

and

∫ t t + r g ⁢ ( s ) ⁢ d s ≤ a 1 , ∫ t t + r h ⁢ ( s ) ⁢ d s ≤ a 2 , ∫ t t + r y ⁢ ( s ) ⁢ d s ≤ a 3 for all ⁢ t ≥ t 0 ,

where r,a1,a2 and a3 are positive constants. Then

y ⁢ ( t + r ) ≤ ( a 3 r + a 2 ) ⁢ e a 1 for all ⁢ t ≥ t 0 .

We will use the following Kato–Ponce and commutator inequalities from [9], see also [16, 7].

Lemma 2.5.

Suppose that g,h∈Cc∞⁢(Ω). Then

(2.5) ∥ Λ s ⁢ ( g ⁢ h ) ∥ ≤ C ⁢ ( ∥ Λ s ⁢ g ∥ L p 1 ⁢ ∥ h ∥ L p 2 + ∥ Λ s ⁢ h ∥ L q 1 ⁢ ∥ g ∥ L q 2 ) ,

where s>0, 2≤p1,p2,q1,q2≤∞ and 12=1p1+1p2=1q1+1q2.

Lemma 2.6.

Suppose that g∈(Cc∞⁢(Ω))2 and h∈Cc∞⁢(Ω). Then

(2.6) ∥ Λ s ⁢ ( 𝐠 ⋅ ∇ ⁡ h ) - 𝐠 ⋅ ( Λ s ⁢ ∇ ⁡ h ) ∥ ≤ C ⁢ ( ∥ ∇ ⁡ 𝐠 ∥ L p 1 ⁢ ∥ Λ s ⁢ h ∥ L p 2 + ∥ Λ s ⁢ 𝐠 ∥ L q 1 ⁢ ∥ ∇ ⁡ h ∥ L q 2 ) ,

where s>0, 2<p1,p2,q1,q2≤∞ and 12=1p1+1p2=1q1+1q2.

Remark 2.7.

We remark that inequalities (2.5) and (2.6) are also valid for those g (or 𝐠) and h belonging to certain Sobolev spaces which make the right-hand sides of (2.5) and (2.6) finite.

We now recall the following existence and uniqueness results from [6] for the 2D Boussinesq system (2.1).

Theorem 2.8.

Let

H ˙ 0 = { θ ∈ L 2 : ∫ Ω θ ⁢ d x = 0 }

and

H ˙ 1 = { 𝐮 ∈ L 2 : ∇ ⋅ 𝐮 = 0 , ∫ Ω u 1 ⁢ d x = ∫ Ω u 2 ⁢ d x = 0 } .

Suppose that f∈H-β and (θ0,u0)∈H˙0×H˙1. Then there exists at least one weak solution (θ⁢(t),u⁢(t)), defined on [0,T], which satisfies the 2D Boussinesq equations (2.1) in the sense of distributions. Moreover, θ∈L∞⁢(0,T;H˙0)∩L2⁢(0,T;Hβ) and u∈L∞⁢(0,T;H˙1)∩L2⁢(0,T;Hα). Furthermore, if we assume that s1, s2 satisfy (2.3) and (2.4), (θ0,u0)∈Hs1×Hs2 and f∈Hs1-β∩Lp0, where

(2.7) r 0 = { s 1 , 2 ⁢ max ⁡ { 1 - α , 1 - β } < s 1 < 1 , any number in ⁢ ( 2 ⁢ max ⁡ { 1 - α , 1 - β } , 1 ) , s 1 ≥ 1 , p 0 = 2 1 - r 0 ,

then the Boussinesq system (2.1)–(2.2) has a unique strong solution (θ,u), defined on [0,T], satisfying

( θ , 𝐮 ) ∈ 𝒞 ⁢ ( [ 0 , T ] , H s 1 ) × 𝒞 ⁢ ( [ 0 , T ] , H s 2 ) , ( θ t , 𝐮 t ) ∈ L 2 ⁢ ( 0 , T ; H s 1 - β ) × L 2 ⁢ ( 0 , T ; H s 2 - α ) .

The existence of the global attractor for the 2D Boussinesq system (2.1) was also proved in [6].

Theorem 2.9 (Existence of a Global Attractor).

Assume that ν>0, κ>0, s1, s2 satisfy (2.3) and (2.4), and f∈Hs1-β∩Lp0, where p0 is defined in (2.7). Then the solution operator {S⁢(t)}t≥0 of the 2D Boussinesq system, S⁢(t)⁢(θ0,u0)=(θ⁢(t),u⁢(t)), defines a semigroup in the space Hs1×Hs2 for all t∈R+. Moreover, the following statements are valid:

For any ( θ 0 , 𝐮 0 ) ∈ H s 1 × H s 2 , t↦S⁢(t)⁢(θ0,𝐮0) is a continuous function from ℝ+ into Hs1×Hs2.
For any fixed t > 0 , S⁢(t) is a continuous and compact map in Hs1×Hs2.
{ S ⁢ ( t ) } t ≥ 0 possesses a global attractor 𝒜 in the space H s 1 × H s 2 . The global attractor 𝒜 is compact and connected in H s 1 × H s 2 and it is the maximal bounded attractor and the minimal invariant set in H s 1 × H s 2 in the sense of the set inclusion relation.

3 Dimensions of the Global Attractor

The aim of this section is to prove that the global attractor 𝒜 in Theorem 2.9 has finite Hausdorff and fractal dimensions, and to apply the Ladyzhenskaya squeezing property to estimate the dimension of the global attractor 𝒜. We mention that an alternative approach would be to use Lyapunov exponents to estimate the Hausdorff and fractal dimensions of the global attractor 𝒜, see [13, 10] for details. The main result in this section is the following.

Theorem 3.1.

Under the assumptions of Theorem 2.9, the global attractor A has finite Hausdorff and fractal dimensions measured in the Hs1×Hs2 space.

In order to prove Theorem 3.1, we first recall the following result from Ladyzhenskaya, see [11, 8].

Theorem 3.2.

Let X be a Hilbert space, S:X→X a map and A⊂X a compact set such that S⁢(A)=A. Suppose that there exist l∈[1,+∞) and δ∈(0,1) such that, for all a1,a2∈A,

∥ S ⁢ ( a 1 ) - S ⁢ ( a 2 ) ∥ X ≤ l ⁢ ∥ a 1 - a 2 ∥ X ,

∥ Q m ⁢ [ S ⁢ ( a 1 ) - S ⁢ ( a 2 ) ] ∥ X ≤ δ ⁢ ∥ a 1 - a 2 ∥ X ,

where Qm is the projection in X onto some subspace (Xm)⊥ of co-dimension m∈N. Then

d H ⁢ ( 𝒜 ) ≤ d F ⁢ ( 𝒜 ) ≤ m ⁢ ln ⁡ ( 8 ⁢ G a 2 ⁢ l 2 1 - δ 2 ) ln ⁡ ( 2 1 + δ 2 ) ,

where dH⁢(A) and dF⁢(A) are the Hausdorff and fractal dimensions of A, respectively, and Ga is the Gauss constant: Ga=2π⁢∫01d⁢x1-x4=0.8346268⁢….

Proof of Theorem 3.1.

Suppose (θ1,𝐮1,π1), (θ2,𝐮2,π2) are two strong solutions of the 2D Boussinesq system (2.1) with two initial data (θ10,𝐮10),(θ20,𝐮20)∈𝒜, respectively. Let η=θ1-θ2 and 𝐰=𝐮1-𝐮2. Then (η,𝐰) satisfies the following equations:

(3.1)

(3.1a) ∂ t ⁡ 𝐰 + 𝐮 1 ⋅ ∇ ⁡ 𝐰 + 𝐰 ⋅ ∇ ⁡ 𝐮 2 + ν ⁢ ( - Δ ) α ⁢ 𝐰 = - ∇ ⁡ ( π 1 - π 2 ) + η ⁢ 𝐞 2 ,

(3.1b) ∂ t ⁡ η + 𝐰 ⋅ ∇ ⁡ θ 1 + 𝐮 2 ⋅ ∇ ⁡ η + κ ⁢ ( - Δ ) β ⁢ η = 0 .

Let Pm be the projection onto the subspace spanned by the first m Fourier modes such that

P m ⁢ 𝐮 = ∑ | k | ≤ m 𝐮 ^ k ⁢ e i ⁢ k ⋅ x , P m ⁢ θ = ∑ | k | ≤ m θ ^ k ⁢ e i ⁢ k ⋅ x ,

and set Qm=I-Pm. Taking the inner product with Λ2⁢s2⁢Qm⁢𝐰 and Λ2⁢s1⁢Qm⁢η on the equations (3.1a) and (3.1b) in L2, respectively, we obtain

(3.2) { 1 2 ⁢ d d ⁢ t ⁢ ∥ Λ s 2 ⁢ Q m ⁢ 𝐰 ∥ 2 + ν ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ 2 = - 〈 𝐮 1 ⋅ ∇ ⁡ 𝐰 , Λ 2 ⁢ s 2 ⁢ Q m ⁢ 𝐰 〉 - 〈 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Λ 2 ⁢ s 2 ⁢ Q m ⁢ 𝐰 〉 + 〈 η ⁢ 𝐞 2 , Λ 2 ⁢ s 2 ⁢ Q m ⁢ 𝐰 〉 , 1 2 ⁢ d d ⁢ t ⁢ ∥ Λ s 1 ⁢ Q m ⁢ η ∥ 2 + κ ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ 2 = - 〈 𝐰 ⋅ ∇ ⁡ θ 1 , Λ 2 ⁢ s 1 ⁢ Q m ⁢ η 〉 - 〈 𝐮 2 ⋅ ∇ ⁡ η , Λ 2 ⁢ s 1 ⁢ Q m ⁢ η 〉 .

We see from condition (2.3) that

s 1 > 2 ⁢ max ⁡ { 1 - α , 1 - β } ≥ 1 - α + 1 - β = 2 - α - β .

Hence, we could fix an α1∈(12,α) such that s1≥2-α1-β, and we also fix a β1∈(12,β). Since s2≥1, we have

s 2 ≥ 1 > 2 - α 1 - α , s 2 ≥ 1 > 2 - β 1 - β .

Therefore, by the Sobolev embedding theorem, we have

(3.3) H s 1 ⊂ ⊂ H 2 - α 1 - β ⊂ ⊂ L 2 α 1 + β - 1

and

(3.4) H s 2 ⊂ ⊂ H 2 - α 1 - α ⊂ ⊂ L 2 α 1 + α - 1 , H s 2 ⊂ ⊂ H 2 - β 1 - β ⊂ ⊂ L 2 β + β 1 - 1 .

We now estimate the term 〈𝐮1⋅∇⁡𝐰,Λ2⁢s2⁢Qm⁢𝐰〉. Since 𝐰 is divergence free, we have

| 〈 𝐮 1 ⋅ ∇ ⁡ 𝐰 , Λ 2 ⁢ s 2 ⁢ Q m ⁢ 𝐰 〉 | = 〈 Λ s 2 - α ⁢ ( 𝐮 1 ⋅ ∇ ⁡ 𝐰 ) , Λ s 2 + α ⁢ Q m ⁢ 𝐰 〉 ≤ ∥ Λ s 2 + 1 - α ⁢ ( 𝐮 1 ⊗ 𝐰 ) ∥ ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ .

Let p,q>2 satisfy 1p+1q=12 and choose

r = 2 - α - α 1 , p = 2 r = 2 2 - α - α 1 , q = 2 1 - r = 2 α + α 1 - 1 .

Applying Lemmas 2.2 and 2.5, we have

∥ Λ s 2 + 1 - α ⁢ ( 𝐮 1 ⊗ 𝐰 ) ∥ ≤ C ⁢ ( ∥ Λ s 2 + 1 - α ⁢ 𝐮 1 ∥ L p ⁢ ∥ 𝐰 ∥ L q + ∥ Λ s 2 + 1 - α ⁢ 𝐰 ∥ L p ⁢ ∥ 𝐮 1 ∥ L q )

≤ C ⁢ ( ∥ Λ s 2 + 2 - α - r ⁢ 𝐮 1 ∥ ⁢ ∥ 𝐰 ∥ L 2 1 - r + ∥ Λ s 2 + 2 - α - r ⁢ 𝐰 ∥ ⁢ ∥ 𝐮 1 ∥ L 2 1 - r )

≤ C ⁢ ( ∥ Λ s 2 + α 1 ⁢ 𝐮 1 ∥ ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ + ∥ Λ s 2 + α 1 ⁢ 𝐰 ∥ ⁢ ∥ Λ s 2 ⁢ 𝐮 1 ∥ ) ,

where we used (3.4) for the last inequality. Therefore, by the interpolation inequality in Lemma 2.3 and the Cauchy–Schwarz inequality, we have

| 〈 𝐮 1 ⋅ ∇ ⁡ 𝐰 , Λ 2 ⁢ s 2 ⁢ Q m ⁢ 𝐰 〉 | ≤ C ⁢ ∥ Λ s 2 + α ⁢ 𝐮 1 ∥ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐮 1 ∥ 1 - α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥

+ C ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 1 - α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐮 1 ∥ ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥

≤ C ⁢ ∥ Λ s 2 + α ⁢ 𝐮 1 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐮 1 ∥ 2 ⁢ ( α - α 1 ) α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2

(3.5) + C ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 ⁢ ( α - α 1 ) α ⁢ ∥ Λ s 2 ⁢ 𝐮 1 ∥ 2 + ν 6 ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ 2 .

Similar to (3.5), we have the estimate

| 〈 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Λ 2 ⁢ s 2 ⁢ Q m ⁢ 𝐰 〉 | ≤ C ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 ⁢ ( α - α 1 ) α ⁢ ∥ Λ s 2 ⁢ 𝐮 2 ∥ 2

(3.6) + C ⁢ ∥ Λ s 2 + α ⁢ 𝐮 2 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐮 2 ∥ 2 ⁢ ( α - α 1 ) α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 + ν 6 ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ 2 .

Next, we estimate the term 〈𝐰⋅∇⁡θ1,Λ2⁢s1⁢Qm⁢η〉. Since 𝐰 is divergence free,

| 〈 𝐰 ⋅ ∇ ⁡ θ 1 , Λ 2 ⁢ s 1 ⁢ Q m ⁢ η 〉 | = 〈 Λ s 1 - β ⁢ ( 𝐰 ⋅ ∇ ⁡ θ 1 ) , Λ s 1 + β ⁢ Q m ⁢ η 〉 ≤ ∥ Λ s 1 + 1 - β ⁢ ( 𝐰 ⋅ θ 1 ) ∥ ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ .

Let p1,q1,p2,q2>2 satisfy 1p1+1q1=1p2+1q2=12 and choose

r 1 = 2 - α 1 - β , p 1 = 2 r 1 = 2 2 - α 1 - β , q 1 = 2 1 - r 1 = 2 α 1 + β - 1 ,

and

r 2 = 2 - β - β 1 , p 2 = 2 r 2 = 2 2 - β - β 1 , q 2 = 2 1 - r 2 = 2 β + β 1 - 1 .

Applying Lemmas 2.2 and 2.5, since s1≤s2, we have

∥ Λ s 1 + 1 - β ⁢ ( 𝐰 ⋅ θ 1 ) ∥ ≤ C ⁢ ( ∥ Λ s 1 + 1 - β ⁢ 𝐰 ∥ L p 1 ⁢ ∥ θ 1 ∥ L q 1 + ∥ Λ s 1 + 1 - β ⁢ θ 1 ∥ L p 2 ⁢ ∥ 𝐰 ∥ L q 2 )

≤ C ⁢ ( ∥ Λ s 1 + 2 - β - r 1 ⁢ 𝐰 ∥ ⁢ ∥ θ 1 ∥ L 2 1 - r 1 + ∥ Λ s 1 + 2 - β - r 2 ⁢ θ 1 ∥ ⁢ ∥ 𝐰 ∥ L 2 1 - r 2 )

≤ C ⁢ ( ∥ Λ s 2 + α 1 ⁢ 𝐰 ∥ ⁢ ∥ Λ s 1 ⁢ θ 1 ∥ + ∥ Λ s 1 + β 1 ⁢ θ 1 ∥ ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ ) ,

where we used (3.3)–(3.4) for the last inequality. Thus, applying the interpolation inequality in Lemma 2.3 and the Cauchy–Schwarz inequality, we obtain

| 〈 𝐰 ⋅ ∇ ⁡ θ 1 , Λ 2 ⁢ s 1 ⁢ Q m ⁢ η 〉 | ≤ C ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 1 - α 1 α ⁢ ∥ Λ s 1 ⁢ θ 1 ∥ ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥

+ C ⁢ ∥ Λ s 1 + β ⁢ θ 1 ∥ β 1 β ⁢ ∥ Λ s 1 ⁢ θ 1 ∥ 1 - β 1 β ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥

≤ C ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 ⁢ ( α - α 1 ) α ⁢ ∥ Λ s 1 ⁢ θ 1 ∥ 2

(3.7) + C ⁢ ∥ Λ s 1 + β ⁢ θ 1 ∥ 2 ⁢ β 1 β ⁢ ∥ Λ s 1 ⁢ θ 1 ∥ 2 ⁢ ( β - β 1 ) β ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 + κ 6 ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ 2 .

Similar to (3.7), we have the estimate

| 〈 𝐮 2 ⋅ ∇ ⁡ η , Λ 2 ⁢ s 1 ⁢ Q m ⁢ η 〉 | ≤ C ⁢ ∥ Λ s 2 + α ⁢ 𝐮 2 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐮 2 ∥ 2 ⁢ ( α - α 1 ) α ⁢ ∥ Λ s 1 ⁢ η ∥ 2

(3.8) + C ⁢ ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ β 1 β ⁢ ∥ Λ s 1 ⁢ η ∥ 2 ⁢ ( β - β 1 ) β ⁢ ∥ Λ s 2 ⁢ 𝐮 2 ∥ 2 + κ 6 ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ 2 .

Finally, applying the interpolation inequality in Lemma 2.3 and the Cauchy–Schwarz inequality, we have

| 〈 η ⁢ 𝐞 2 , Λ 2 ⁢ s 2 ⁢ Q m ⁢ 𝐰 〉 | = | 〈 Λ s 1 + β ⁢ Q m ⁢ η ⁢ 𝐞 2 , Λ 2 ⁢ s 2 - s 1 - β ⁢ Q m ⁢ 𝐰 〉 |

≤ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ ⁢ ∥ Λ 2 ⁢ s 2 - s 1 - β ⁢ Q m ⁢ 𝐰 ∥

≤ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ 1 - r * ⁢ ∥ Λ s 2 ⁢ Q m ⁢ 𝐰 ∥ r *

(3.9) ≤ κ 6 ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ 2 + ν 6 ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ 2 + C 1 ⁢ ∥ Λ s 2 ⁢ Q m ⁢ 𝐰 ∥ 2 ,

where r~=s1+α+β-s2α>0 and C1=1κr~⁢ν1/r~-1.

It was shown in [6] that when the solutions (θi,𝐮i) belong to 𝒜⊂Hs1×Hs2 for i=1,2, it follows that ∥Λs1+β⁢θi∥ and ∥Λs2+α⁢𝐮i∥ are uniformly bounded and independent of t for i=1,2. Therefore, summing (3.2) and (3.5)–(3.9) together, we obtain

d d ⁢ t ⁢ ( ∥ Λ s 2 ⁢ Q m ⁢ 𝐰 ∥ 2 + ∥ Λ s 1 ⁢ Q m ⁢ η ∥ 2 ) + ν ⁢ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ 2 + κ ⁢ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ 2

≤ C ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 ⁢ ( α - α 1 ) α ⁢ C ⁢ ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ β 1 β ⁢ ∥ Λ s 1 ⁢ η ∥ 2 ⁢ ( β - β 1 ) β

(3.10) + C ⁢ ( ∥ Λ s 2 ⁢ 𝐰 ∥ 2 + ∥ Λ s 1 ⁢ η ∥ 2 ) + C 1 ⁢ ∥ Λ s 2 ⁢ Q m ⁢ 𝐰 ∥ 2 .

By the Poincaré inequality, we have

λ m 2 ⁢ α ⁢ ∥ Λ s 2 ⁢ Q m ⁢ 𝐰 ∥ 2 ≤ ∥ Λ s 2 + α ⁢ Q m ⁢ 𝐰 ∥ 2 and λ m 2 ⁢ β ⁢ ∥ Λ s 1 ⁢ Q m ⁢ η ∥ 2 ≤ ∥ Λ s 1 + β ⁢ Q m ⁢ η ∥ 2 .

We set

y ⁢ ( t ) := ∥ Λ s 2 ⁢ 𝐰 ⁢ ( t ) ∥ 2 + ∥ Λ s 1 ⁢ η ⁢ ( t ) ∥ 2 , z ⁢ ( t ) := ∥ Λ s 2 ⁢ Q m ⁢ 𝐰 ⁢ ( t ) ∥ 2 + ∥ Λ s 1 ⁢ Q m ⁢ η ⁢ ( t ) ∥ 2 ,

and let ρm=12⁢min⁡{ν⁢λmα,κ⁢λmβ}. Then we can choose m large enough, so that C1≤ρm. Hence, from (3.10), it follows that

(3.11) z ′ ⁢ ( t ) + ρ m ⁢ z ⁢ ( t ) ≤ C ⁢ y ⁢ ( t ) + C ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 ⁢ ( α - α 1 ) α + C ⁢ ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ β 1 β ⁢ ∥ Λ s 1 ⁢ η ∥ 2 ⁢ ( β - β 1 ) β .

Now, integrating (3.11) with respect to t∈[0,T], we have

z ⁢ ( T ) ≤ e - ρ m ⁢ T ⁢ z ⁢ ( 0 ) + C ⁢ e - ρ m ⁢ T ⁢ ∫ 0 T e ρ m ⁢ t ⁢ y ⁢ ( t ) ⁢ d t + C ⁢ e - ρ m ⁢ T ⁢ ∫ 0 T e ρ m ⁢ t ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ ∥ Λ s 2 ⁢ 𝐰 ∥ 2 ⁢ ( α - α 1 ) α ⁢ d t

+ C ⁢ e - ρ m ⁢ T ⁢ ∫ 0 T e ρ m ⁢ t ⁢ ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ β 1 β ⁢ ∥ Λ s 1 ⁢ η ∥ 2 ⁢ ( β - β 1 ) β ⁢ d t

= : I 1 + I 2 + I 3 + I 4 .

First, we notice that

(3.12) I 1 ≤ e - ρ m ⁢ T ⁢ y ⁢ ( 0 ) .

Next, we recall the results from [6, Section 4.2] that if (θi,𝐮i) are two strong solutions in 𝒜⊂Hs1×Hs2 for i=1,2, then, for all t≥0,

∥ Λ s 2 ⁢ 𝐰 ⁢ ( t ) ∥ 2 + ∥ Λ s 1 ⁢ η ⁢ ( t ) ∥ 2 + ν ⁢ ∫ 0 t ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ d s + κ ⁢ ∫ 0 t ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ d s

≤ C ( ∥ Λ s 2 𝐰 ( 0 ) ∥ 2 + ∥ Λ s 1 η ( 0 ) ∥ 2 ) exp { ∫ 0 t ∥ Λ s 2 + α 𝐮 1 ( s ) ∥ 2 + ∥ Λ s 2 + α 𝐮 2 ( s ) ∥ 2 + ∥ Λ s 1 + β θ 2 ( s ) ∥ 2 d s } .

Thus,

(3.13) y ⁢ ( t ) + σ ⁢ ∫ 0 t ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 + ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ d ⁢ s ≤ y ⁢ ( 0 ) ⁢ K ⁢ ( t ) for all ⁢ t ≥ 0 ,

where σ=min⁢{ν,κ} and

K ( t ) = C exp { ∫ 0 t ∥ Λ s 2 + α 𝐮 1 ( s ) ∥ 2 + ∥ Λ s 2 + α 𝐮 2 ( s ) ∥ 2 + ∥ Λ s 1 + β θ 2 ( s ) ∥ 2 d s } ,

which is a positive continuous non-decreasing function on [0,∞), independent of the initial data. Therefore,

(3.14) I 2 ≤ C ⁢ e - ρ m ⁢ T ⁢ y ⁢ ( 0 ) ⁢ K ⁢ ( T ) ⁢ ∫ 0 T e ρ m ⁢ t ⁢ d t ≤ C ⁢ ρ m - 1 ⁢ K ⁢ ( T ) ⁢ y ⁢ ( 0 ) .

Finally, we have

I 3 ≤ C ⁢ e - ρ m ⁢ T ⁢ ∫ 0 T e ρ m ⁢ t ⁢ y ⁢ ( t ) α - α 1 α ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ d t

≤ C ⁢ e - ρ m ⁢ T ⁢ y ⁢ ( 0 ) α - α 1 α ⁢ K ⁢ ( T ) α - α 1 α ⁢ ∫ 0 T e ρ m ⁢ t ⁢ ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ α 1 α ⁢ d t

≤ C ⁢ e - ρ m ⁢ T ⁢ y ⁢ ( 0 ) α - α 1 α ⁢ K ⁢ ( T ) α - α 1 α ⁢ ( ∫ 0 T e ρ m ⁢ t ⋅ α α - α 1 ⁢ d t ) α - α 1 α ⁢ ( ∫ 0 T ∥ Λ s 2 + α ⁢ 𝐰 ∥ 2 ⁢ d t ) α 1 α

(3.15) ≤ C ⁢ ρ m - α - α 1 α ⁢ K ⁢ ( T ) ⁢ y ⁢ ( 0 )

and, similarly,

I 4 ≤ C ⁢ e - ρ m ⁢ T ⁢ ∫ 0 T e ρ m ⁢ t ⁢ y ⁢ ( t ) β - β 1 β ⁢ ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ β 1 β ⁢ d t

≤ C ⁢ e - ρ m ⁢ T ⁢ y ⁢ ( 0 ) β - β 1 β ⁢ K ⁢ ( T ) β - β 1 β ⁢ ∫ 0 T e ρ m ⁢ t ⁢ ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ β 1 β ⁢ d t

≤ C ⁢ e - ρ m ⁢ T ⁢ y ⁢ ( 0 ) β - β 1 β ⁢ K ⁢ ( T ) β - β 1 β ⁢ ( ∫ 0 T e ρ m ⁢ t ⋅ β β - β 1 ⁢ d t ) β - β 1 β ⁢ ( ∫ 0 T ∥ Λ s 1 + β ⁢ η ∥ 2 ⁢ d t ) β 1 β

(3.16) ≤ C ⁢ ρ m - β - β 1 β ⁢ K ⁢ ( T ) ⁢ y ⁢ ( 0 ) .

Summing the estimates in (3.12) and (3.14)–(3.16), we have

(3.17) z ⁢ ( T ) ≤ ( e - ρ m ⁢ T + ρ m - 1 ⁢ K ⁢ ( T ) + ρ m - α - α 1 α ⁢ K ⁢ ( T ) + ρ m - β - β 1 β ⁢ K ⁢ ( T ) ) ⁢ y ⁢ ( 0 ) .

Therefore, for any fixed T>0, l=K⁢(T)∈[1,∞), some δ∈(0,1), and two given strong solutions (θi,𝐮i)∈𝒜, combining the results from (3.13) and (3.17), we can choose m large enough so that

y ⁢ ( T ) ≤ l ⁢ y ⁢ ( 0 ) and z ⁢ ( T ) ≤ δ ⁢ y ⁢ ( 0 ) .

Hence, Theorem 3.1 immediately follows from Theorem 3.2. ∎

4 Determining Modes on the Attractor

We next consider the concept of determining modes (the number of the first Fourier modes), introduced in [5]. This notion offers another measure of the finite number of degrees of freedom.

In this section, we are going to prove that there exists a positive large enough number m such that if the projections on the space spanned by the first m eigenvectors of the operator Λ of two different trajectories on the attractor 𝒜 coincide for all t∈ℝ, then these two trajectories actually coincide for all t∈ℝ.

4.1 The Definition of Determining Modes

Let us consider two vectors (θ1,𝐮1)=(θ1⁢(x,t),𝐮1⁢(x,t)) and (θ2,𝐮2)=(θ2⁢(x,t),𝐮2⁢(x,t)) satisfying 2D Boussinesq systems corresponding to the same force f=f⁢(x). Now we give the definition of determining modes for trajectories on the global attractor. See [3, 2].

Definition 4.1.

The first m modes associated with Pm are said to be determining if for two trajectories (θ1⁢(x,t),𝐮1⁢(x,t)) and (θ2⁢(x,t),𝐮2⁢(x,t)) on the global attractor 𝒜, the condition

P m ⁢ ( θ 1 ⁢ ( x , t ) , 𝐮 1 ⁢ ( x , t ) ) = P m ⁢ ( θ 2 ⁢ ( x , t ) , 𝐮 2 ⁢ ( x , t ) ) for all ⁢ t ∈ ℝ

implies

( θ 1 ⁢ ( t ) , 𝐮 1 ⁢ ( t ) ) = ( θ 2 ⁢ ( t ) , 𝐮 2 ⁢ ( t ) ) for all ⁢ t ∈ ℝ .

4.2 H 2 ⁢ β × H 2 ⁢ α -Estimates for (θ,𝐮)

From [6, Proposition 3.4], we could readily see that ∥Λ2⁢β⁢θ∥ and ∥Λ2⁢α⁢θ∥ are uniformly bounded. We would like to find the explicit dependence on the viscosity ν, the diffusivity κ, and the fractional dissipation powers α and β for the bounds of ∥Λ2⁢β⁢θ∥ and ∥Λ2⁢α⁢𝐮∥. Hence, throughout this section, we emphasize that the constant C below is independent of the viscosity ν, the diffusivity κ, and the fractional dissipation powers α and β.

Proposition 4.2.

Under the assumption of Theorem 2.9, suppose that (θ,u) is on the global attractor A. If f∈L22⁢β1-1∩Hβ, where β1∈(12,β), then

(4.1) ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M ,

where M is defined below in (4.7) and the constant C is independent of ν, κ, α and β.

Proof.

Let us first consider the case β≤α. Taking the inner product of the third equation in (2.1) with Λ4⁢β⁢θ in L2, we have

1 2 ⁢ d d ⁢ t ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + 〈 𝐮 ⋅ ∇ ⁡ θ , Λ 4 ⁢ β ⁢ θ 〉 + κ ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 = 〈 Λ β ⁢ f , Λ 3 ⁢ β ⁢ θ 〉 ≤ C κ ⁢ ∥ Λ β ⁢ f ∥ 2 + κ 4 ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 .

Let β1 be a fixed number such that 12<β1<β. Since 𝐮 is divergence free, applying Lemma 2.5 and choosing

p 1 = q 1 = 1 1 - β 1 , p 2 = q 2 = 2 2 ⁢ β 1 - 1 ,

we have

| 〈 𝐮 ⋅ ∇ ⁡ θ , Λ 4 ⁢ β ⁢ θ 〉 | ≤ ∥ Λ 2 ⁢ β - β 1 ⁢ ( 𝐮 ⋅ ∇ ⁡ θ ) ∥ ⁢ ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥

≤ C ⁢ ∥ Λ 1 + 2 ⁢ β - β 1 ⁢ ( 𝐮 ⋅ θ ) ∥ ⁢ ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥

≤ C ⁢ ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥ ⁢ ∥ Λ 2 ⁢ β + β 1 ⁢ 𝐮 ∥ ⁢ ∥ θ ∥ L 2 2 ⁢ β 1 - 1 + C ⁢ ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥ 2 ⁢ ∥ 𝐮 ∥ L 2 2 ⁢ β 1 - 1

(4.2) ≤ C ⁢ ( ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ β + β 1 ⁢ 𝐮 ∥ 2 ) ⁢ ∥ θ ∥ L 2 2 ⁢ β 1 - 1 + C ⁢ ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥ 2 ⁢ ∥ 𝐮 ∥ L 2 2 ⁢ β 1 - 1 .

Since (θ⁢(t),𝐮⁢(t)) is on the attractor 𝒜 for all t∈ℝ, we have the uniform Lp-estimates for (θ,𝐮), in [6, Sections 3.1 and 3.3], that for all p≥2, θ,𝐮∈L∞⁢(0,∞;Lp), with

(4.3) ∥ θ ⁢ ( t ) ∥ L p ≤ C ⁢ p ⁢ ∥ f ∥ L p κ and ∥ 𝐮 ⁢ ( t ) ∥ L p ≤ C ⁢ ∥ 𝐮 ⁢ ( t ) ∥ H 1 ≤ C ⁢ e ν ⁢ ( 1 + κ ) ν 2 ⁢ κ 3 ⁢ ∥ f ∥ 2 .

Hence, applying (4.3), the interpolation inequality, Young’s inequality, and the assumption β≤α, we find from (4.2) that

| 〈 𝐮 ⋅ ∇ ⁡ θ , Λ 4 ⁢ β ⁢ θ 〉 | ≤ C ⁢ κ - 1 ⁢ ( 2 ⁢ β 1 - 1 ) - 1 ⁢ ∥ f ∥ L 2 2 ⁢ β 1 - 1 ⁢ ( ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ β + β 1 ⁢ 𝐮 ∥ 2 ) + C ⁢ κ - 3 ⁢ ν - 2 ⁢ ∥ f ∥ 2 ⁢ ∥ Λ 2 ⁢ β + β 1 ⁢ θ ∥ 2

≤ C ⁢ κ - 3 ⁢ ν - 2 ⁢ ( 2 ⁢ β 1 - 1 ) - 1 ⁢ ∥ f ∥ L 2 2 ⁢ β 1 - 1 ⁢ ∥ f ∥ 2 ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 - 2 ⁢ β 1 β ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 ⁢ β 1 β

+ C ⁢ κ - 1 ⁢ ( 2 ⁢ β 1 - 1 ) - 1 ⁢ ∥ Λ 2 ⁢ β ⁢ 𝐮 ∥ 2 - 2 ⁢ β 1 β ⁢ ∥ Λ 3 ⁢ β ⁢ 𝐮 ∥ 2 ⁢ β 1 β

≤ C 1 ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + κ 2 ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 + C 2 ⁢ ∥ Λ 2 ⁢ β ⁢ 𝐮 ∥ 2 + ν 6 ⁢ ∥ Λ 3 ⁢ β ⁢ 𝐮 ∥ 2

≤ C 1 ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + κ 2 ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 + C 2 ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 + ν 6 ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 ,

where

C 1 = O ⁢ ( κ - 3 ⁢ β + β 1 β - β 1 ⁢ ν - 2 ⁢ β β - β 1 ⁢ ( 2 ⁢ β 1 - 1 ) - β β - β 1 ⁢ ∥ f ∥ L 2 2 ⁢ β 1 - 1 β β - β 1 ⁢ ∥ f ∥ 2 ⁢ β β - β 1 )

and

C 2 = O ⁢ ( κ - β + β 1 β - β 1 ⁢ ( 2 ⁢ β 1 - 1 ) - β β - β 1 ) .

Therefore, we obtain

(4.4) 1 2 ⁢ d d ⁢ t ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + κ 2 ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 ≤ C 1 ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + C 2 ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 + ν 4 ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 + C κ ⁢ ∥ Λ β ⁢ f ∥ 2 .

Next, we take the inner product of the first equation in (2.1) with Λ4⁢α⁢𝐮 in L2. Since 𝐮 is divergence free, we have

1 2 ⁢ d d ⁢ t ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 + ν ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 = 〈 θ ⁢ 𝐞 2 , Λ 4 ⁢ α ⁢ 𝐮 〉 - 〈 𝐮 ⋅ ∇ ⁡ 𝐮 , Λ 4 ⁢ α ⁢ 𝐮 〉 .

Similar to estimate (4.2), let α1 be a fixed number such that 12<α1<α. Then applying the interpolation inequality, we find

| 〈 𝐮 ⋅ ∇ ⁡ 𝐮 , Λ 4 ⁢ α ⁢ 𝐮 〉 | ≤ C ⁢ ∥ Λ 2 ⁢ α + α 1 ⁢ 𝐮 ∥ 2 ⁢ ∥ 𝐮 ∥ L 2 2 ⁢ α 1 - 1 ≤ C ⁢ κ - 3 ⁢ ν - 2 ⁢ ∥ Λ 2 ⁢ α + α 1 ⁢ 𝐮 ∥ 2 ≤ C 3 ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 + ν 6 ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 ,

where C3=O⁢(κ-3⁢αα-α1⁢ν-2⁢α+α1α-α1). In addition, applying the Poincaré and the Cauchy–Schwarz inequalities, we have

| 〈 θ ⁢ 𝐞 2 , Λ 4 ⁢ α ⁢ 𝐮 〉 | = | 〈 Λ α ⁢ θ ⁢ 𝐞 2 , Λ 3 ⁢ α ⁢ 𝐮 〉 | ≤ ∥ Λ α ⁢ θ ∥ ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ ≤ ∥ Λ 2 ⁢ β ⁢ θ ∥ ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ ≤ C ν ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ν 6 ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 .

Thus, we obtain

(4.5) 1 2 ⁢ d d ⁢ t ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 + 2 ⁢ ν 3 ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C 3 ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 + C ν ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 .

Summing equations (4.4) and (4.5), we have

(4.6) d d ⁢ t ⁢ ( ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ) + κ ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 + ν ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C 1 ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ( C 2 + C 3 ) ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 + C κ ⁢ ∥ Λ β ⁢ f ∥ 2 .

We set

(4.7) M := κ - max ⁡ { 3 ⁢ β + β 1 β - β 1 , 3 ⁢ α α - α 1 } ⁢ ν - max ⁡ { 2 ⁢ α + α 1 α - α 1 , 2 ⁢ β β - β 1 } ⁢ ( 2 ⁢ β 1 - 1 ) - β β - β 1 ⁢ ∥ f ∥ L 2 2 ⁢ β 1 - 1 β β - β 1 ⁢ ∥ f ∥ 2 ⁢ β β - β 1 .

It follows from (4.6) that

(4.8) d d ⁢ t ⁢ ( ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ) + κ ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 + ν ⁢ ∥ Λ 3 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C ⁢ M ⁢ ( ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ) + C κ ⁢ ∥ Λ β ⁢ f ∥ 2 .

In order to apply the uniform Gronwall inequality and obtain the uniform bounds for Λ2⁢β⁢θ and Λ2⁢α⁢𝐮, we have to find the uniform time average bounds for Λ2⁢β⁢θ and Λ2⁢α⁢𝐮. Taking the inner product of (2.1) with (Λβ⁢θ,Λα⁢𝐮) in L2 and using analogous arguments as for (4.8), we have

(4.9) d d ⁢ t ⁢ ( ∥ Λ β ⁢ θ ∥ 2 + ∥ Λ α ⁢ 𝐮 ∥ 2 ) + κ ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ν ⁢ ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C ⁢ M ⁢ ( ∥ Λ β ⁢ θ ∥ 2 + ∥ Λ α ⁢ 𝐮 ∥ 2 ) + C κ ⁢ ∥ f ∥ 2 .

It has been shown in [6, Sections 3.1 and 3.3] that the time averages of ∥Λβ⁢θ∥2 and ∥Λα⁢𝐮∥2 are uniformly bounded. That is, for t≥t1⁢(θ0,𝐮0) large enough,

∫ t t + 1 ∥ Λ β ⁢ θ ∥ 2 ⁢ d s ≤ C ⁢ 1 + κ κ 3 ⁢ ∥ f ∥ 2

and

∫ t t + 1 ∥ Λ α ⁢ 𝐮 ∥ 2 ⁢ d s ≤ 1 ν ⁢ ∥ 𝐮 ∥ 2 + 1 ν 2 ⁢ ∫ t t + 1 ∥ θ ∥ 2 ⁢ d s ≤ C ⁢ 1 + ν κ 3 ⁢ ν 2 ⁢ ∥ f ∥ 2 .

Applying the uniform Gronwall inequality on the differential inequality (4.9), with a1=C⁢M, a2=Cκ⁢∥f∥2 and a3=C⁢(1+κ)⁢(1+ν)2κ3⁢ν2⁢∥f∥2, for t≥t2⁢(θ0,𝐮0), we have

∥ Λ β ⁢ θ ∥ 2 + ∥ Λ α ⁢ 𝐮 ∥ 2 ≤ C ⁢ ( ( 1 + κ ) ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 + 1 κ ) ⁢ ∥ f ∥ 2 ⁢ e M ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 ) ⁢ ∥ f ∥ 2 ⁢ e M .

In addition, for t≥t2⁢(θ0,𝐮0),

∫ t t + 1 ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ⁢ d ⁢ s ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 ) ⁢ ∥ f ∥ 2 ⁢ e M + C κ ⁢ ∥ f ∥ 2 ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 ) ⁢ ∥ f ∥ 2 ⁢ e M .

Applying the uniform Gronwall inequality on the differential inequality (4.8), with a1=C⁢M, a2=Cκ⁢∥Λβ⁢f∥2 and a3=C⁢(1+κ)2⁢(1+ν)2κ3⁢ν2⁢∥f∥2⁢eM, for t≥t3⁢(θ0,𝐮0), we have

∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C ⁢ ( 1 κ ⁢ ∥ Λ β ⁢ f ∥ 2 + ( 1 + κ ) 2 ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 ⁢ ∥ f ∥ 2 ⁢ e M ) ⁢ e M

≤ C ⁢ ( ( 1 + κ ) 2 ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M .

Since we assume that the solution (θ,𝐮) is on the global attractor, we can shift the initial time so that

(4.10) ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M for all ⁢ t ∈ ℝ .

For the case β>α, similar to equation (4.8), we find

(4.11) d d ⁢ t ⁢ ( ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ β + α ⁢ 𝐮 ∥ 2 ) + κ ⁢ ∥ Λ 3 ⁢ β ⁢ θ ∥ 2 + ν ⁢ ∥ Λ 2 ⁢ β + 2 ⁢ α ⁢ 𝐮 ∥ 2 ≤ C ⁢ M ⁢ ( ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ β + α ⁢ 𝐮 ∥ 2 ) + C κ ⁢ ∥ Λ β ⁢ f ∥ 2 .

In addition, we have

(4.12) d d ⁢ t ⁢ ( ∥ Λ β ⁢ θ ∥ 2 + ∥ Λ β + α 2 ⁢ 𝐮 ∥ 2 ) + κ ⁢ ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ν ⁢ ∥ Λ 2 ⁢ β + α ⁢ 𝐮 ∥ 2 ≤ C ⁢ M ⁢ ( ∥ Λ β ⁢ θ ∥ 2 + ∥ Λ β + α 2 ⁢ 𝐮 ∥ 2 ) + C κ ⁢ ∥ f ∥ 2 .

We recall the results from [6, Sections 3.1 and 3.3] that the time averages of ∥Λβ⁢θ∥2 and ∥Λβ+α2⁢𝐮∥2 are uniformly bounded. That is, for t≥t4⁢(θ0,𝐮0) large enough,

∫ t t + 1 ∥ Λ β ⁢ θ ∥ 2 ⁢ d s ≤ C ⁢ 1 + κ κ 3 ⁢ ∥ f ∥ 2

and

∫ t t + 1 ∥ Λ β + α 2 ⁢ 𝐮 ∥ 2 ⁢ d s ≤ ∫ t t + 1 ∥ Λ 1 + α ⁢ 𝐮 ∥ 2 ⁢ d s ≤ C ⁢ e 2 ⁢ ν ⁢ ( 1 + κ ) κ 3 ⁢ ν 2 ⁢ ∥ f ∥ 2 .

Applying the uniform Gronwall inequality to the differential inequality (4.12), with a1=C⁢M, a2=Cκ⁢∥f∥2 and a3=C⁢(1+κ)⁢e2⁢νκ3⁢ν2⁢∥f∥2, for t≥t5⁢(θ0,𝐮0), we have

∥ Λ β ⁢ θ ∥ 2 + ∥ Λ β + α 2 ⁢ 𝐮 ∥ 2 ≤ C ⁢ ( ( 1 + κ ) ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 + 1 κ ) ⁢ ∥ f ∥ 2 ⁢ e M ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ) ⁢ ∥ f ∥ 2 ⁢ e M .

Then applying the uniform Gronwall inequality again to the differential inequality (4.11), with a1=C⁢M, a2=Cκ⁢∥Λβ⁢f∥2 and a3=C⁢(1+κ)2⁢e2⁢νκ3⁢ν2⁢∥f∥2⁢eM, for t≥t6⁢(θ0,𝐮0), we have

∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ β + α ⁢ 𝐮 ∥ 2 ≤ C ⁢ ( 1 κ ⁢ ∥ Λ β ⁢ f ∥ 2 + ( 1 + κ ) 2 ⁢ ( 1 + ν ) 2 κ 3 ⁢ ν 2 ⁢ ∥ f ∥ 2 ⁢ e M 2 ) ⁢ e M ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M .

Since we assume that the solution (θ,𝐮) is on the global attractor, we can shift the initial time so that

(4.13) ∥ Λ 2 ⁢ β ⁢ θ ∥ 2 + ∥ Λ 2 ⁢ β + α ⁢ 𝐮 ∥ 2 ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M for all ⁢ t ∈ ℝ .

Since α,β∈(12,1), we have that H2⁢α⊂⊂H2⁢β+α. Hence, we can conclude (4.1), based on equations (4.10) and (4.13). ∎

4.3 Main Results

Theorem 4.3.

Under the assumptions of Theorem 2.9, let (θ1,u1) and (θ2,u2) be two trajectories of system (2.1) on the attractor A. If Pm⁢(θ1⁢(t),u1⁢(t))=Pm⁢(θ2⁢(t),u2⁢(t)) for all t∈R, and for some large enough integer m>0,

(4.14) λ m + 1 α - 1 2 ≥ 2 ⁢ C ⁢ ( κ ⁢ N 1 2 + N + 1 ) κ ⁢ ν ,

where the number N is defined in (4.17) below, then we have (θ1⁢(t),u1⁢(t))=(θ2⁢(t),u2⁢(t)) for all t∈R.

Proof.

Let (θ1,𝐮1), (θ2,𝐮2) be the solutions on the attractor 𝒜 and (η,𝐰)=(θ1-θ2,𝐮1-𝐮2). Then (η,𝐰) satisfies:

(4.15)

(4.15a) ∂ t ⁡ 𝐰 + 𝐮 1 ⋅ ∇ ⁡ 𝐰 + 𝐰 ⋅ ∇ ⁡ 𝐮 2 + ν ⁢ ( - Δ ) α ⁢ 𝐰 = - ∇ ⁡ ( π 1 - π 2 ) + η ⁢ 𝐞 2 ,

(4.15b) ∂ t ⁡ η + 𝐰 ⋅ ∇ ⁡ θ 1 + 𝐮 2 ⋅ ∇ ⁡ η + κ ⁢ ( - Δ ) β ⁢ η = 0 .

We now take the inner product of the equation (4.15a) and (4.15b) with Qm⁢𝐰 and Qm⁢η in L2, respectively, where Qm=I-Pm. Since Pm⁢(η,𝐰)=0, applying integration by parts, we obtain

〈 𝐮 1 ⋅ ∇ ⁡ 𝐰 , Q m ⁢ 𝐰 〉 = 〈 𝐮 1 ⋅ ∇ ⁡ P m ⁢ 𝐰 , Q m ⁢ 𝐰 〉 + 〈 𝐮 1 ⋅ ∇ ⁡ Q m ⁢ 𝐰 , Q m ⁢ 𝐰 〉 = 0

and

〈 𝐮 2 ⋅ ∇ ⁡ η , Q m ⁢ η 〉 = 〈 𝐮 2 ⋅ ∇ ⁡ P m ⁢ η , Q m ⁢ η 〉 + 〈 𝐮 2 ⋅ ∇ ⁡ Q m ⁢ η , Q m ⁢ η 〉 = 0 .

Then we find

1 2 ⁢ d d ⁢ t ⁢ ∥ Q m ⁢ 𝐰 ∥ 2 + ν ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 = - 〈 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Q m ⁢ 𝐰 〉 + 〈 η ⁢ 𝐞 2 , Q m ⁢ 𝐰 〉 ,

1 2 ⁢ d d ⁢ t ⁢ ∥ Q m ⁢ η ∥ 2 + κ ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 = - 〈 𝐰 ⋅ ∇ ⁡ θ 1 , Q m ⁢ η 〉 .

We now estimate the term 〈𝐰⋅∇⁡𝐮2,Qm⁢𝐰〉. Since Pm⁢𝐰=0,

〈 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Q m ⁢ 𝐰 〉 = 〈 P m ⁢ 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Q m ⁢ 𝐰 〉 + 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Q m ⁢ 𝐰 〉 = 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Q m ⁢ 𝐰 〉 ,

and since 𝐮2 is divergence free, we have

| 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Q m ⁢ 𝐰 〉 | = | 〈 Λ - α ⁢ ( Q m ⁢ 𝐰 ⋅ ∇ ⁡ 𝐮 2 ) , Λ α ⁢ Q m ⁢ 𝐰 〉 |

≤ ∥ Λ - α ⁢ ( Q m ⁢ 𝐰 ⋅ ∇ ⁡ 𝐮 2 ) ∥ ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥

(4.16) ≤ C ⁢ ∥ Λ 1 - α ⁢ ( Q m ⁢ 𝐰 ⊗ 𝐮 2 ) ∥ ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ .

Let p1=43-2⁢α, p2=42⁢α-1, q1=4 and q2=4 be such that

1 p 1 + 1 p 2 = 1 q 1 + 1 q 2 = 1 2 .

Applying Lemma 2.5 and the Sobolev inequality in Lemma 2.2, we get

∥ Λ 1 - α ⁢ ( Q m ⁢ 𝐰 ⊗ 𝐮 2 ) ∥ ≤ C ⁢ ( ∥ Λ 1 - α ⁢ Q m ⁢ 𝐰 ∥ L p 1 ⁢ ∥ 𝐮 2 ∥ L p 2 + ∥ Λ 1 - α ⁢ 𝐮 2 ∥ L q 1 ⁢ ∥ Q m ⁢ 𝐰 ∥ L q 2 )

≤ C ⁢ ∥ Λ 1 2 ⁢ Q m ⁢ 𝐰 ∥ ⁢ ∥ Λ 3 2 - α ⁢ 𝐮 2 ∥ .

Applying the Poincaré inequality and using (4.1), we find

∥ Λ 3 2 - α ⁢ 𝐮 2 ∥ 2 ≤ ∥ Λ 2 ⁢ α ⁢ 𝐮 2 ∥ 2 ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M

and

∥ Λ 1 2 ⁢ Q m ⁢ 𝐰 ∥ 2 ≤ λ m + 1 1 - 2 ⁢ α ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 .

Hence, we deduce that

| 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ 𝐮 2 , Q m ⁢ 𝐰 〉 | ≤ C ⁢ ∥ Λ 1 2 ⁢ Q m ⁢ 𝐰 ∥ ⁢ ∥ Λ 3 2 - α ⁢ 𝐮 2 ∥ ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥

≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M ) 1 2 ⁢ λ m + 1 1 2 - α ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 .

Next, we estimate 〈𝐰⋅∇⁡θ1,Qm⁢η〉. Since Pm⁢𝐰=0, we have

〈 𝐰 ⋅ ∇ ⁡ θ 1 , Q m ⁢ η 〉 = 〈 P m ⁢ 𝐰 ⋅ ∇ ⁡ θ 1 , Q m ⁢ η 〉 + 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ θ 1 , Q m ⁢ η 〉 = 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ θ 1 , Q m ⁢ η 〉 .

Similarly to (4.16), since 𝐰 is divergence free, we obtain

| 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ θ 1 , Q m ⁢ η 〉 | ≤ ∥ Λ 1 - β ⁢ ( Q m ⁢ 𝐰 ⋅ θ 1 ) ∥ ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ .

Let p1=43-2⁢β, p2=42⁢β-1, q1=4 and q2=4 be such that

1 p 1 + 1 p 2 = 1 q 1 + 1 q 2 = 1 2 .

Then we have

∥ Λ 1 - β ⁢ ( Q m ⁢ 𝐰 ⋅ θ 1 ) ∥ ≤ C ⁢ ( ∥ Λ 1 - β ⁢ Q m ⁢ 𝐰 ∥ L p 1 ⁢ ∥ θ 1 ∥ L p 2 + ∥ Λ 1 - β ⁢ θ 1 ∥ L q 1 ⁢ ∥ Q m ⁢ 𝐰 ∥ L q 2 )

≤ C ⁢ ∥ Λ 1 2 ⁢ Q m ⁢ 𝐰 ∥ ⁢ ∥ Λ 3 2 - β ⁢ θ 1 ∥ .

Applying the Poincaré inequality and (4.1), we find

∥ Λ 3 2 - β ⁢ θ 1 ∥ 2 ≤ ∥ Λ 2 ⁢ β ⁢ θ 1 ∥ 2 ≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M .

Hence, by the Cauchy–Schwarz inequality, we deduce

| 〈 Q m ⁢ 𝐰 ⋅ ∇ ⁡ θ 1 , Q m ⁢ η 〉 | ≤ C ⁢ ∥ Λ 1 2 ⁢ Q m ⁢ 𝐰 ∥ ⁢ ∥ Λ 3 2 - β ⁢ θ 1 ∥ ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥

≤ C κ ⁢ ∥ Λ 1 2 ⁢ Q m ⁢ 𝐰 ∥ 2 ⁢ ∥ Λ 3 2 - β ⁢ θ 1 ∥ 2 + κ 4 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2

≤ C ⁢ ( ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 4 ⁢ ν 2 ) ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M ⁢ λ m + 1 1 - 2 ⁢ α ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 + κ 4 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 .

Finally, we estimate the term 〈η⁢𝐞2,Qm⁢𝐰〉. Since Pm⁢η=0, we have

〈 η ⁢ 𝐞 2 , Q m ⁢ 𝐰 〉 = 〈 P m ⁢ η ⁢ 𝐞 2 , Q m ⁢ 𝐰 〉 + 〈 Q m ⁢ η ⁢ 𝐞 2 , Q m ⁢ 𝐰 〉 = 〈 Q m ⁢ η ⁢ 𝐞 2 , Q m ⁢ 𝐰 〉

and

| 〈 Q m ⁢ η ⁢ 𝐞 2 , Q m ⁢ 𝐰 〉 | = | 〈 Λ β ⁢ Q m ⁢ η ⁢ 𝐞 2 , Λ - β ⁢ Q m ⁢ 𝐰 〉 |

≤ ∥ Λ β ⁢ Q m ⁢ η ∥ ⁢ ∥ Λ - β ⁢ Q m ⁢ 𝐰 ∥

≤ κ 4 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 + 1 κ ⁢ ∥ Λ - β ⁢ Q m ⁢ 𝐰 ∥ 2

≤ κ 4 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 + 1 κ ⁢ λ m + 1 - 2 ⁢ ( α + β ) ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 .

We set

(4.17) N := ( 1 + κ ) 2 ⁢ e 2 ⁢ ν κ 3 ⁢ ν 2 ⁢ ∥ Λ β ⁢ f ∥ 2 ⁢ e 2 ⁢ M .

Therefore, we arrive at the differential inequalities

1 2 ⁢ d d ⁢ t ⁢ ∥ Q m ⁢ 𝐰 ∥ 2 + ν ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 ≤ ( C ⁢ N 1 2 ⁢ λ m + 1 1 2 - α + 1 κ ⁢ λ m + 1 - 2 ⁢ ( α + β ) ) ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 + κ 4 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 ,

1 2 ⁢ d d ⁢ t ⁢ ∥ Q m ⁢ η ∥ 2 + κ ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 ≤ C ⁢ N ⁢ 1 κ ⁢ λ m + 1 1 - 2 ⁢ α ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 + κ 4 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 .

Summing the above two differential inequalities, we obtain

1 2 ⁢ d d ⁢ t ⁢ ( ∥ Q m ⁢ 𝐰 ∥ 2 + ∥ Q m ⁢ η ∥ 2 ) + κ 2 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 + ( ν - C ⁢ N 1 2 ⁢ λ m + 1 1 2 - α - 1 κ ⁢ λ m + 1 - 2 ⁢ ( α + β ) - C ⁢ N ⁢ 1 κ ⁢ λ m + 1 1 - 2 ⁢ α ) ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 ≤ 0 .

Since λm+1>1, we have λm+1-2⁢(α+β)<λm+11-2⁢α. Hence,

(4.18) 1 2 ⁢ d d ⁢ t ⁢ ( ∥ Q m ⁢ 𝐰 ∥ 2 + ∥ Q m ⁢ η ∥ 2 ) + κ 2 ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 + ( ν - C ⁢ λ m + 1 1 2 - α ⁢ ( N 1 2 + 1 κ + N κ ) ) ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 ≤ 0 .

Under condition (4.14), (4.18) implies

d d ⁢ t ⁢ ( ∥ Q m ⁢ 𝐰 ∥ 2 + ∥ Q m ⁢ η ∥ 2 ) + ν ⁢ ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 + κ ⁢ ∥ Λ β ⁢ Q m ⁢ η ∥ 2 ≤ 0 .

Hence,

(4.19) d d ⁢ t ⁢ ( ∥ Q m ⁢ 𝐰 ∥ 2 + ∥ Q m ⁢ η ∥ 2 ) + σ ⁢ ( ∥ Λ α ⁢ Q m ⁢ 𝐰 ∥ 2 + ∥ Λ β ⁢ Q m ⁢ η ∥ 2 ) ≤ 0 ,

where σ=min⁢{κ,ν}. Now, integrating (4.19) from t0 to t, we have

∥ Q m ⁢ 𝐰 ⁢ ( t ) ∥ 2 + ∥ Q m ⁢ η ⁢ ( t ) ∥ 2 ≤ ( ∥ Q m ⁢ 𝐰 ⁢ ( t 0 ) ∥ 2 + ∥ Q m ⁢ η ⁢ ( t 0 ) ∥ 2 ) ⁢ e r ⁢ ( t 0 - t ) ,

which provides Qm⁢𝐰⁢(t)=Qm⁢η⁢(t)=0 for all t∈ℝ, by taking t0→-∞. ∎

Remark 4.4.

It is well known that for m→∞, we have λm∼c⁢λ112⁢m12, where c is a nondimensional constant, see [3]. We can conclude that (4.14) provides the number of determining modes m such that

m 2 ⁢ α - 1 ≥ C ⁢ N ⁢ κ - 1 ⁢ ν - 1 ,

where N is defined in (4.17). Moreover, m tends to infinity as κ,ν→0 and α,β→12.

Communicated by Kenneth Palmer

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1418911

Funding source: Leverhulme Trust

Award Identifier / Grant number: VP1-2015-036

Funding statement: Michael Jolly was supported by the NSF grant DMS-1418911 and the Leverhulme Trust grant VP1-2015-036.

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Received: 2017-03-23

Revised: 2017-10-10

Accepted: 2017-10-10

Published Online: 2017-11-28

Published in Print: 2018-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Finite-Dimensionality and Determining Modes of the Global Attractor for 2D Boussinesq Equations with Fractional Laplacian

Abstract

1 Introduction

2 Notations and Preliminaries

2.1 The System

2.2 Notation and Function Spaces

Remark 2.1.

2.3 Some Preliminary Results

Lemma 2.2 (The Sobolev Inequality).

Lemma 2.3 (The Interpolation Inequality).

Lemma 2.4 (Uniform Gronwall Lemma).

Lemma 2.5.

Lemma 2.6.

Remark 2.7.

Theorem 2.8.

Theorem 2.9 (Existence of a Global Attractor).

3 Dimensions of the Global Attractor

Theorem 3.1.

Theorem 3.2.

Proof of Theorem 3.1.

4 Determining Modes on the Attractor

4.1 The Definition of Determining Modes

Definition 4.1.

4.2 H 2 ⁢ β × H 2 ⁢ α -Estimates for (θ,𝐮)

Proposition 4.2.

Proof.

4.3 Main Results

Theorem 4.3.

Proof.

Remark 4.4.

References

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