On attractor’s dimensions of the modified Leray-alpha equation

Abstract

The primary objective of this paper is to investigate the modified Leray-alpha equation on the two-dimensional sphere $S^{2}$ , the square torus $T^{2}$ and the three-torus $T^{3}$ . In the strategy, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor on both $S^{2}$ and $T^{2}$ . Our method is based on the estimates for the vorticity scalar equations and the stationary solutions around the invariant manifold that are constructed by using the Kolmogorov flows. Finally, we will use the results on $T^{2}$ to study the lower bound for attractor’s dimensions in the case of $T^{3}$ .

Keywords

Modified Leray-alpha equation 2-dimensional sphere square torus three-torus global attractor Hausdorff (fractal) dimension Kolmogorov flows

1. Introduction

The study about the solutions and their asymptotic behavior of the models of turbulence theory plays an important role to analyse the dynamics of the homogeneous incompressible fluid flows as well as many practical applications. In the present literature, there is a lot of interest in the three averaged turbulence equations: the Navier–Stokes-alpha, the modified Leray-alpha and the simplified Bardina equations which converge to the Navier–Stokes equation when the parameter α decreases to zero. The existence and uniqueness of weak solutions were established in [3,8,13,19,22]. The existence of the global attractors, the upper and lower bounds on their Hausdorff and fractal dimensions were studied in [6,7,13–15]. The existence of the inertial manifold for these equations was obtained recently in [9,18,20]. The algebraic decays in time were given for the Navier–Stokes-alpha equations in [2].

Besides that, there are some other works for the equations with damping coefficients such as 2-D damped-driven Navier–Stokes equations and damped 2-D, 3-D Euler-Bardina equations [14,16,17]. In these works, the authors established the global well-posedness of the weak solutions and derived the upper and lower bounds for the Hausdorff and fractal dimensions of the global attractors.

The preliminary method is used to study the upper bound on the dimension of the attractors that have aimed to combine the contemporary interest of the Lyapunov exponents and the Hausdorff (fractal) dimension of attractors (see [4,5,23]) and the Lieb–Sobolev–Thirring inequality. Initially presented for the 2-D Navier–Stokes equation in [21], the lower bound on the dimensions of the global attractors has been studied by using the Kolmogorov flow to construct the family of stationary solutions. Since then, this method has been developed for the other turbulence equations in [15,24] and the equations with damping terms in [16,17].

Concerning the study of Navier–Stokes and averaged turbulence equations on the compact manifolds, there are some works on the attractor’s dimensions of the Navier–Stokes and the turbulence equations on the 2-D closed manifolds such as the sphere $S^{2}$ and the square torus $T^{2}$ . The authors have treated the Navier–Stokes equation in [11,12], the Navier–Stokes-alpha equation in [15] and the simplified Bardina equation in [24].

In the present paper we study the modified Leray-alpha equation on the 2-D closed manifold M: $\begin{array}{l} (1.1) & \{\begin{matrix} v_{t} - ν Δ v + v \cdot \nabla u = - \nabla p + f, \\ \nabla \cdot v = 0, \\ v = u - α^{2} Δ u, \end{matrix} \end{array}$ where ν is a viscous constant, the velocity v and the filtered function u are unknown belonging to $TM$ , p is the pressure and f is an external force.

We recall that the 2-D modified Leray-alpha equation in $R^{2}$ with periodic boundary condition was studied in [9] for the global well-posedness and the existence of an inertial manifold. The 3-D modified Leray-alpha equation in $R^{3}$ with periodic boundary condition was investigated in [13]. The authors established the global well-posedness of the weak solution and derived an upper bound on the dimension of the attractor by using the Lieb–Sobolev–Thirring inequality in $R^{3}$ . Recently, the existence of the inertial manifold for the 3-D equation has been established fully in [18,20].

We will extend and apply the recent work for the simplified Bardina equation of one of the authors [24] to consider the modified Leray-alpha equations on 2-D closed manifold M with trivial harmonic forms detailed by the sphere $S^{2}$ and the square torus $T^{2}$ . We will establish the global well-posedness of the weak solutions by the Galerkin approximation scheme (see Section 3). Then, we derive an upper bound on the Hausdorff and fractal dimensions of the global attractor in both the sphere $S^{2}$ and torus $T^{2}$ . The essential method in getting upper bounds is used in views of the vorticity scalar form of the equation (1.1) and the generalized theorem of the dimension of attractors on the uniform Lyapunov exponents (see Section 4.2.1). The lower bound of the dimensions in the case of the torus $T^{2}$ is obtained by using the Kolmogorov flows to construct the stationary solutions around the invariant manifold (see also Section 4.2.1). To be precise, in this paper, we will prove that the upper and lower bounds of the attractor’s dimensions coincide with the ones of the 2-D simplified Bardina equation and they are improved in comparing with the case of the 2-D Navier–Stokes equation. Finally, we will extend the recent work of Ilyin, Zelik and Kostiano [17] to establish the lower bound of the attractor’s dimensions in the 3-D torus (see Section 5). Our approach is based on Squire’s transformation to transform the 3-D equation to the 2-D case, then we apply the results of the lower bound obtained in the 2-D case. These results, together with the results obtained in [13] complete the two-sided estimates of the global attractor’s dimensions for the modified Leray-alpha equation in the 3-D case.

This paper is organized as follows: Section 2 gives some basic formulas and the setting of the modified Leray-alpha equation, Section 3 discusses the global well-posedness of the weak solutions of the equation on the sphere and torus, Section 4 deals with the upper and lower bounds of the attractor’s dimensions on $T^{2}$ and Section 5 studies on the lower bound on $T^{3}$ .

2. Geometrical and analytical setting

2.1. Geometric formula and functional spaces

We recall some geometric formulas on the 2-dimensional closed manifold $(M, g)$ embedded in $R^{3}$ with trivial harmonic forms detailed by the two-dimensional sphere $S^{2}$ and the square torus $T^{2}$ (see for details [10,11]). We denote by $TM$ the set of tangent vector fields on M and by ${(TM)}^{⊥}$ the set of normal vector fields. We have the definitions of the following operators $\begin{matrix} {Curl}_{n} : TM \to {(TM)}^{⊥} and Curl : {(TM)}^{⊥} \to TM \end{matrix}$ in a neighbourhood of M in $R^{3}$ as follows:

Definition 2.1.
Let u be a smooth vector field on M with values in $TM$ , and let $\vec{ψ}$ be a smooth vector field on M with values in ${(TM)}^{⊥}$ , i.e. $\vec{ψ} = ψ \vec{n}$ , where $\vec{n}$ is the outward unit normal vector to M and ψ is a smooth scalar function. We then identify the vector field $\vec{ψ}$ with the scalar function ψ. Let $\hat{u}$ and $\hat{ψ}$ be smooth extensions of u and ψ into a neighbourhood of M in $R^{3}$ such that $\hat{u} |_{M} = u$ and $\hat{ψ} |_{M} = ψ$ . For $x \in M$ and $y \in R^{3}$ , we define $\begin{array}{l} {Curl}_{n} u (x) = (Curl \hat{u} (y) \cdot \vec{n} (y)) \vec{n} (y) |_{y = x}, \\ Curl \vec{ψ} (x) = Curl ψ (x) = Curl \hat{ψ} (y) |_{y = x}, \end{array}$ where the operator Curl that appears on the right-hand sides is the classical Curl operator in $R^{3}$ .

The above definitions of ${Curl}_{n} u$ and $Curl ψ$ are independent of the choice of the neighbourhood of M in $R^{3}$ . Moreover, the following formulas hold $\begin{array}{l} (2.1) & {Curl}_{n} u = - \vec{n} div (\vec{n} \times u), Curl ψ = - \vec{n} \times \nabla ψ, \\ (2.2) & Δ u = \nabla div u - Curl {Curl}_{n} u, \end{array}$ where × is the outer vector product in $R^{3}$ , $\nabla ψ$ is a gradient of the scalar function, $\nabla_{v} u$ is covariant derivative along the vector field, Δ is Laplace–de Rham operator defined on the vector fields (see the definition and formula of Δ in [10]).

Let $L^{p} (M)$ and $L^{p} (TM)$ be the $L^{p}$ -spaces of the scalar functions and of the tangent vector fields on M respectively. Let $H^{p} (M)$ and $H^{p} (TM)$ be the corresponding Sobolev spaces of scalar functions and vector fields. The inner product on $L^{2} (M)$ and $L^{2} (TM)$ are given by $\begin{array}{l} {⟨ u, v ⟩}_{L^{2} (M)} = \int_{M} u \bar{v} {dVol}_{M}, for u, v \in L^{2} (M), \\ {⟨ u, v ⟩}_{L^{2} (TM)} = \int_{M} u \cdot \bar{v} {dVol}_{M}, for u, v \in L^{2} (TM) . \end{array}$ The following integration by parts formulas will be used frequently $\begin{array}{l} {⟨ \nabla h, v ⟩}_{L^{2} (TM)} = - {⟨ h, div v ⟩}_{L^{2} (M)}, \\ {⟨ Curl \vec{ψ}, v ⟩}_{L^{2} (TM)} = {⟨ \vec{ψ}, {Curl}_{n} v ⟩}_{L^{2} (M)} . \end{array}$

By using Hodge decomposition we have $\begin{matrix} C^{\infty} (TM) = {\nabla ψ : ψ \in C^{\infty} (M)} \oplus {Curl ψ : ψ \in C^{\infty} (M)} \end{matrix}$ Putting $\begin{matrix} V = {Curl ψ : ψ \in C^{\infty} (M)}, H = {\overline{V}}^{L^{2} (TM)}, V = {\overline{V}}^{H^{1} (TM)}, \end{matrix}$ with the norms on H and V given by $\begin{matrix} ‖ u ‖_{H}^{2} = ⟨ u, u ⟩, ‖ u ‖_{V}^{2} = ⟨ A u, u ⟩ = ⟨ {Curl}_{n} u, {Curl}_{n} u ⟩ . \end{matrix}$ Since $div u = 0$ , we have the Poincaré inequality $\begin{matrix} (2.3) & ‖ u ‖_{H} ⩽ λ_{1}^{- 1 / 2} (‖ u ‖_{V} + ‖ div u ‖_{H}) = λ_{1}^{- 1 / 2} ‖ u ‖_{V} \end{matrix}$ where $λ_{1}$ is the first eigenvalue of the Stokes operator $A = Curl {Curl}_{n}$ . We know that $\begin{matrix} (2.4) & ‖ u ‖_{H^{1} (TM)} = ‖ u ‖_{L^{2} (TM)}^{2} + ‖ div u ‖_{L^{2} (M)}^{2} + ‖ {Curl}_{n} u ‖_{L^{2} (M)}^{2} . \end{matrix}$ The facts (2.3), (2.4), together with the condition $div u = 0$ on V, shows that the norms on $H^{1}$ and V are equivalent for all $u \in V$ . In the rest of this paper, we denote as usual $‖ \cdot ‖_{L^{2}} : = | \cdot |$ , $‖ \cdot ‖_{V} : = ‖ \cdot ‖$ and $‖ \cdot ‖_{H^{1}} : = ‖ \cdot ‖_{1}$ .
2.2. The modified Leray-alpha equation on 2-D closed manifolds

The modified Leray-alpha equation on M has the following form $\begin{array}{l} (2.5) & \{\begin{matrix} v_{t} - ν Δ v + v \cdot \nabla u + \nabla p = f, \\ \nabla \cdot u = \nabla \cdot v = 0, \\ v = (I - α^{2} Δ) u, \end{matrix} \end{array}$ where ν is the viscousity coefficient, p is the pressure, f is the external force and the unknown functions $u, v \in TM$ .

Using (2.2) we rewrite Equation (2.5) as $\begin{array}{l} (2.6) & \{\begin{matrix} v_{t} + ν Curl {Curl}_{n} v + v \cdot \nabla u + \nabla p = f, \\ \nabla \cdot u = \nabla \cdot v = 0, \\ v = (I - α^{2} Δ) u, \end{matrix} \end{array}$ By using Hodge projection $P$ on the space $H = {\overline{V}}^{L^{2} (TM)}$ the first equation becomes $\begin{array}{l} (2.7) & \{\begin{matrix} v_{t} + ν A v + B (v, u) = f, \\ \nabla \cdot u = \nabla \cdot v = 0, \\ v = (I + α^{2} A) u, \end{matrix} \end{array}$ where $B (v, u) = P (v \cdot \nabla u)$ .

On the other hand, if we put $u = - Curl ψ$ and take ${Curl}_{n}$ the first equation in (2.5) then we obtain the following vorticity scalar form $\begin{matrix} (2.8) & (Δ ψ_{t} - α^{2} Δ^{2} ψ_{t}) - ν Δ (Δ ψ - α^{2} Δ^{2} ψ) + J ((I - α^{2} Δ) ψ, Δ ψ) = {Curl}_{n} f . \end{matrix}$ Putting $φ = Δ ψ$ , we get $\begin{matrix} (2.9) & (φ_{t} - α^{2} Δ φ_{t}) - ν Δ (φ - α^{2} Δ φ) + J ((Δ^{- 1} (I - α^{2} Δ) φ, φ) = {Curl}_{n} f . \end{matrix}$ Consequently, $\begin{matrix} (2.10) & φ_{t} - ν Δ φ + {(I - α^{2} Δ)}^{- 1} J ((Δ^{- 1} (I - α^{2} Δ) φ, φ) = {(I - α^{2} Δ)}^{- 1} {Curl}_{n} f . \end{matrix}$

The properties of Jacobian operator $J (a, b) = n \times \nabla a \cdot \nabla b$ are given in the following proposition:

Proposition 2.2.
On the two-dimensional closed manifold M we have $\begin{matrix} J (a, b) = - J (b, a), \int_{M} J (a, b) {dVol}_{M} = \int_{M} J (a, b) b {dVol}_{M} = 0 \end{matrix}$ and $\begin{matrix} \int_{M} J (a, b) c {dVol}_{M} = \int_{M} J (b, c) a {dVol}_{M} . \end{matrix}$

3. Well-posedness and the existence of a global attractor

We consider the existence and uniqueness of the weak solution of the modified Leray-alpha equation under the vectorial form (2.7). The basic method is the Galerkin approximation scheme and the appropriate Aubin compactness theorems. It is worth noting that the well-posedness of the 2-D equation in $R^{2}$ with periodic boundary condition was established in [9] and of the 3-D equation in $R^{3}$ with periodic boundary condition was treated in [13]. Here, we can do by the same way as in [9,13] to establish the $H^{1}$ - and $H^{2}$ -estimates with the note that $\begin{matrix} ⟨ B (v, u), u ⟩ = 0 \end{matrix}$ in $H^{1}$ -estimate and the term $⟨ B (v, u), A u ⟩$ appearing in $H^{2}$ -estimate can be controlled by using Young’s inequality, namely $\begin{array}{rcl} {| ⟨ B (v, u), A u ⟩ |}_{D {(A)}^{'}} & ⩽ & c | v | ‖ v ‖^{1 / 2} | A^{3 / 2} u | ‖ u ‖ \\ ⩽ & c (λ_{1}^{- 1} + α^{2}) | A u |^{1 / 2} {| A^{3 / 2} u |}^{3 / 2} ‖ u ‖ \\ ⩽ & c {(λ_{1}^{- 1} + α^{2})}^{4} \frac{‖ u ‖^{4} | A u |^{2}}{{(ν α^{2})}^{3}} + \frac{3 ν α^{2}}{4} {| A^{3 / 2} u |}^{2} . \end{array}$ Hence, we can get the $H^{1}$ - and $H^{2}$ -estimates as follows (in details see [13]): $\begin{array}{l} (3.1) & {| u (t) |}^{2} + α^{2} {‖ u (t) ‖}^{2} ⩽ e^{- ν λ_{1} t} ({| u (0) |}^{2} + α^{2} {‖ u (0) ‖}^{2}) + \frac{K_{1}}{ν λ_{1}} (1 - e^{- ν λ_{1} t}), \\ (3.2) & t ({‖ u (t) ‖}^{2} + α^{2} {| A u (t) |}^{2}) ⩽ \frac{1}{ν} (t K_{1} + k_{1}) + t^{2} K_{2} + {(λ_{1}^{- 1} + α^{2})}^{4} \frac{2 c k_{1}^{2}}{{(ν α^{2})}^{4} α^{4}} (\frac{t^{2} K_{1}}{2} + t k_{1}) . \end{array}$ Therefore, we can derive the well-posedness of the weak solution of (2.7) as in the following theorem.

Theorem 3.1.
Let $f \in H$ . Then for any $T > 0$ and for given initial data $u_{0} = u (0) \in V$ , Equation ( 2.7 ) has a unique regular solution u in $[0, T)$ . Furthermore, this solution depends continuously on the initial values as a map from V to $C ([0, T], V)$ .
Proof.
The proof is done by taking into account $H^{1}$ -, $H^{2}$ -estimates and the Galerkin approximation scheme in the same way as in [13, Theorem 3]. □

Since the well-posedness, we get a semigroup of solution operators, denoted as ${S (t)}_{t ⩾ 0}$ which is defined by $\begin{array}{l} S (t) : V & \to V \\ u_{0} & \mapsto S (t) u_{0} : = u (t), u (t) is a unique weak solution of Equation (2.7) . \end{array}$ The $H^{1}$ -estimate (3.1) implies the existence of a bounded absorbing ball $B_{V} (0)$ in V. The compactness of the semigroup ${S (t)}_{t ⩾ 0}$ and the existence of bounded absorbing ball $B_{V}$ guarantee the existence of the nonempty compact global attractor $A$ . We now state the theorem for the existence of a global attractor generated by the semigroup of solution operators ${S (t)}_{t ⩾ 0}$ : Theorem 3.2.
There is a compact global attractor $A \subset V$ for Equation ( 2.7 ).
Proof.
Following Rellich lemma $S (t) : V ⟶ D (A) ⋐ V$ , for $t > 0$ , is a compact semigroup from V into itself. Since $S (t) B_{V} (0) \subset B_{V} (0)$ , then the set $C_{s} : = {\overline{⋃_{t ⩾ s} S (t) B_{V} (0)}}^{V}$ is nonempty and compact in V. By the monotonic property of $C_{s}$ for $s > 0$ and by the finite intersection property of compact sets, the set $\begin{matrix} A = ⋂_{s > 0} C_{s} \subset V \end{matrix}$ is a nonempty compact set, and also is the unique global attractor in V. □

4. Dimensions of global attractors on 2-D closed manifolds

4.1. Fundamental theorem on the attractor’s dimensions

Let H be a Hilbert space, X be a compact subset in H and $S_{t} : H \to H$ be the nonlinear continuous semigroup generated by the evolution equation $\begin{matrix} \partial_{t} u = F (u), u (0) = u_{0}, \end{matrix}$ and suppose that $\begin{matrix} S_{t} X = X for t ⩾ 0 . \end{matrix}$ The Hausdorff and fractal dimensions of X are estimated by using the uniform Lyapunov exponents (see [4,5]). In what follows, we recall the definition of the uniformly quasi-differentiable semigroup.

Definition 4.1.
The semigroup $S_{t}$ is uniformly quasi-differentiable on X for each t if for all $u, v \in X$ there exists a linear operator $D S_{t} (u)$ such that $\begin{matrix} ‖ S_{t} (u) - S_{t} (v) - D S_{t} (u) (u - v) ‖ ⩽ h (r) ‖ u - v ‖, \end{matrix}$ where $‖ u - v ‖ ⩽ r$ , $h (r) \to 0$ as $r \to 0$ and ${sup}_{t \in [0, 1]} {sup}_{u \in X} ‖ D S_{t} (u) ‖_{L (H, H)} < \infty$ .

The following result is established in [4, Theorem 2.1].
Theorem 4.2.
Assume that the mapping $u \mapsto S_{t} u_{0}$ is uniformly quasi-differentiable in H and its quasi-differentiation is a linear operator $L (t, u_{0}) : ζ \in H \to U (t) \in H$ , where $U (t)$ is the solution of the first variation equation $\begin{matrix} (4.1) & \partial_{t} U = L (t, u_{0}) U, U (0) = ζ . \end{matrix}$ We assume, in addition, that for a fixed t the operator $L (t, u_{0}) = D S_{t} (u)$ is compact and norm-continuous with respect to $u \in X$ .

For $N ⩾ 1, n \in N$ , we define $q_{N}$ by $\begin{matrix} (4.2) & q_{N} = \underset{t \to \infty}{lim sup} sup_{u_{0} \in X} sup_{ζ_{i} \in H, ‖ ζ_{i} ‖ ⩽ 1, i = 1, \dots, N} (\frac{1}{t} \int_{0}^{t} Tr L (τ, u_{0}) \circ Q_{N} (τ) d τ), \end{matrix}$ where $Q_{N} (τ)$ is the orthogonal projection in H into $Span {U^{1} (τ) ... U^{N} (τ)}$ , and $U^{i} (t)$ is the solution of ( 4.1 ) with $U^{i} (0) = ζ_{i}$ .

Suppose $q_{N} ⩽ f (N)$ , where f is concave. The Hausdorff and fractal dimensions of X have the same upper bound $\begin{matrix} {dim}_{H} X ⩽ {dim}_{F} X ⩽ N_{}, \end{matrix}$ where* $N_{} ⩾ 1$ is such that* $f (N_{}) = 0$ .

The concave condition of f can be replaced by the condition that the quasi-differential $D S_{t} (u)$ contracts $N_{}$ -dimensional volumes uniformly for $u \in X$ (see [5, Theorem 2.1]).
4.2. Estimate of the attractor’s dimensions

4.2.1. Upper bound

As the previous sections we denote M for both $S^{2}$ and $T^{2}$ . The upper bound on the Hausdorff and fractal dimensions of the global attractor for the 3-D modified Leray-alpha equation with periodic boundary condition was established in [13] by using the Lieb–Sobolev–Thirring inequality. However, in this section, we will derive the upper bound on dimensions of the attractors for the 2-D equation on M by another method based on the vorticity scalar equation.

Multiplying (2.9) by φ in $L^{2} (M)$ we obtain that $\begin{matrix} \frac{1}{2} \frac{d}{d t} (| φ |^{2} + | \nabla φ |^{2}) + ν (| \nabla φ |^{2} + α^{2} | Δ φ |^{2}) = ⟨ {Curl}_{n} f, φ ⟩ = ⟨ f, {Curl}_{n} φ ⟩ . \end{matrix}$ Therefore, $\begin{matrix} \frac{d}{d t} (| φ |^{2} + α^{2} | \nabla φ |^{2}) + 2 ν (| \nabla φ |^{2} + α^{2} | Δ φ |^{2}) ⩽ \frac{| f |^{2}}{ν} + ν | \nabla φ |^{2} . \end{matrix}$ Using the Poincaré and Gronwall inequalities and integrating with respect to t yields $\begin{matrix} (4.3) & \underset{t \to \infty}{lim sup} {| φ (t) |}^{2} ⩽ \frac{| f |^{2}}{λ_{1} ν^{2}} \end{matrix}$ and $\begin{matrix} (4.4) & \underset{t \to \infty}{lim sup} \frac{1}{t} \int_{0}^{t} {| \nabla φ (τ) |}^{2} d τ ⩽ \frac{| f |^{2}}{ν^{2}} . \end{matrix}$

We consider the variational equation corresponding to (2.10): $\begin{array}{l} Φ_{t} = & Δ Φ - {(I - α^{2} Δ)}^{- 1} J ((Δ^{- 1} (I - α^{2} Δ) Φ, φ) \\ (4.5) & - {(I - α^{2} Δ)}^{- 1} J ((Δ^{- 1} (I - α^{2} Δ) φ, Φ), \end{array}$ where $Φ (0) = ζ$ .

It is standard to show that this equation has a unique solution denoted by $\begin{matrix} L (t, φ (0)) ζ : = Φ (t) . \end{matrix}$ Using the general theorems in [23] we can show that the semigroup $S_{t}$ is uniformly quasi-differentiable on the attractor $A$ of the modified Leray-alpha equation.

Now we establish the Hausdorff and fractal dimensions of the attractor using (4.5) in the following theorem:

Theorem 4.3.
The Hausdorff and fractal dimensions of the attractor $A$ for the modified Leray-alpha equation on M are finite and satisfy $\begin{array}{l} (4.6) & {dim}_{H} A ⩽ {dim}_{F} A ⩽ G^{2 / 3} {(\frac{{(4 + ϵ_{G})}^{3}}{3 L (1 + α^{2} λ_{1})} (log G - \frac{1}{2} log \frac{L}{2}))}^{1 / 3}, \\ (4.7) & {dim}_{H} A ⩽ {dim}_{F} A ⩽ {(\frac{12}{\sqrt{L (1 + α^{2} λ_{1})}})}^{2 / 3} G^{2 / 3} {(log G + \frac{1}{2} + log \frac{3 \sqrt{2}}{\sqrt{L (1 + α^{2} λ_{1})}})}^{1 / 3}, \end{array}$ where L is a positive constant and $G = \frac{| f |}{ν^{2} λ_{1}}$ is the Grashof number. Here, $ϵ_{G} \to 0$ as $G \to \infty$ . In particular, the constant L equals π in the case that M is the sphere $S^{2}$ .
Proof.
Let $\begin{matrix} H = L^{2} (M) \cap {φ : \int_{M} φ {dVol}_{M} = 0} and H^{1} = H^{1} (M) \cap H . \end{matrix}$ We put $\begin{matrix} t x, y y = ⟨ x, y ⟩ - α^{2} ⟨ x, Δ y ⟩ . \end{matrix}$ In the space $Q_{N} (τ) (H)$ we take an orthonormal basis ${θ_{i}}_{i = 1}^{N} \subset H^{1}$ with norm $t \cdot, \cdot y$ . Now we have $\begin{array}{l} Tr L (τ, φ_{0}) \circ Q_{N} (τ) \\ = \sum_{i = 1}^{N} t L (τ, φ_{0}) θ_{i}, θ_{i} y \\ = - ν \sum_{i = 1}^{N} t Δ θ_{i}, θ_{i} y \\ - \sum_{i = 1}^{N} t {(I - α^{2} Δ)}^{- 1} J (Δ^{- 1} (I - α^{2} Δ) θ_{i}, φ) + {(I - α^{2} Δ)}^{- 1} J (Δ^{- 1} (I - α^{2} Δ) φ, θ_{i}), θ_{i} y \\ = - ν \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + | Δ θ_{i} |^{2}) - \sum_{i = 1}^{N} ⟨ J (Δ^{- 1} (I - α^{2} Δ) θ_{i}, φ) + J (Δ^{- 1} (I - α^{2} Δ) φ, θ_{i}), θ_{i} ⟩ \\ = - ν \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + | Δ θ_{i} |^{2}) - \sum_{i = 1}^{N} ⟨ J (Δ^{- 1} θ - α^{2} θ_{i}, φ), θ_{i} ⟩ \\ ⩽ - ν \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + | Δ θ_{i} |^{2}) - \int_{M} \sum_{i = 1}^{N} θ_{i} (n \times \nabla {(I - α^{2} Δ)}^{- 1} θ_{i}) \cdot \nabla φ d x \\ + α^{2} \sum_{i = 1}^{n} ⟨ J (θ_{i}, φ), θ_{i} ⟩ \\ ⩽ - ν \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + | Δ θ_{i} |^{2}) + \int_{M} {(\sum_{i = 1}^{N} θ_{i}^{2})}^{1 / 2} {(\sum_{i = 1}^{N} | v_{i} |^{2})}^{1 / 2} | \nabla φ | d x \\ + α^{2} \sum_{i = 1}^{n} ⟨ J (φ, θ_{i}), θ_{i} ⟩ \\ ⩽ - ν \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + | Δ θ_{i} |^{2}) + ‖ ρ ‖_{\infty}^{1 / 2} {(\sum_{i = 1}^{N} | θ_{i} |^{2})}^{1 / 2} | \nabla φ | (due to \int_{M} J (φ, θ_{i}) θ_{i} dVo l_{M} = 0) \\ (4.8) & ⩽ - ν \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + | Δ θ_{i} |^{2}) + ‖ ρ ‖_{\infty}^{1 / 2} N^{1 / 2} | \nabla φ |, \end{array}$ where $\begin{matrix} ρ (s) = \sum_{i = 1}^{N} {| v_{i} (s) |}^{2} = \sum_{i = 1}^{n} {| n \times \nabla {(Δ - α^{2} Δ^{2})}^{- 1} θ_{i} |}^{2} . \end{matrix}$ The following estimate of the function ρ on the 2-D closed manifold M is valid (for details see [24, Appendix]). $\begin{array}{l} 2 \sqrt{L (1 + α^{2} λ_{1})} ‖ ρ ‖_{\infty}^{1 / 2} \\ (4.9) & ⩽ {(2 log (k + 1) + 1)}^{1 / 2} + \sqrt{2} {(k + 1)}^{- 1} {(λ_{1}^{- 1} \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + α^{2} | Δ θ_{i} |^{2}))}^{1 / 2}, \end{array}$ where k is a positive integer and L is a positive constant ( $L = π$ in the case of $S^{2}$ ).

On $S^{2}$ the eigenvalues of Δ are $λ_{n} = n (n + 1)$ of multiplicity $2 n + 1$ for $n = 1, 2, \dots$ , we therefore have $\begin{matrix} T (t, φ_{0}) : = \sum_{i = 1}^{N} (| \nabla θ_{i} |^{2} + α^{2} | Δ θ_{i} |^{2}) ⩾ \sum_{i = 1}^{N} λ_{i} ⩾ \frac{λ_{1}}{4} N^{2} . \end{matrix}$ Hence $\begin{matrix} N ⩽ 2 {({(λ_{1})}^{- 1} T)}^{1 / 2} . \end{matrix}$ Equation (4.8) implies now, $\begin{array}{l} Tr L (τ, φ_{0}) \circ Q_{N} (τ) \\ ⩽ - ν λ_{1} (λ_{1}^{- 1} T) \\ + L^{- 1 / 2} {(1 + α^{2} λ_{1})}^{- 1 / 2} ({(2 log (k + 1) + 1)}^{1 / 2} + \sqrt{2} {(k + 1)}^{- 1} (λ_{1}^{- 1} T)) {(λ_{1}^{- 1} T)}^{1 / 4} | \nabla φ |^{2} . \end{array}$ Since we obtain the same bounded function of $Tr L (τ, φ_{0}) \circ Q_{N} (τ)$ such as the one of the simplified Bardina equation, the rest of the proof can be done by the same way in [24, Theorem 4.4] and we get the upper bounds (4.6) and (4.7) in our theorem. □
Remark 4.4.
In the above theorem, we prove that the upper bound of the Hausdorff and fractal dimensions of the global attractor coincide with the ones of the simplified Bardina equation obtained in [24]. In particular, as α tends to zero we get the same upper bound of the Hausdorff and fractal dimensions of the global attractor for the Navier–Stokes equation on $S^{2}$ (see [11,12]). Our theorem can be also extended to the two-dimensional closed manifolds which have non-trivial harmonic forms as well as [24, Theorem 4.6].

4.2.2. Lower bound

Since a global attractor is a maximal strictly invariant compact set, it follows that the attractor contains the unstable manifolds of stationary points, that is the invariant manifolds along which the solutions tend exponentially to the stationary points as t tends to negative infinity. From this point, we can establish the lower bound for the Hausdorff and fractal dimensions of the global attractor on the square torus $T^{2} = [0; 2 π] \times [0; 2 π]$ by constructing a family of stationary solutions arising from the family of Kolmogorov flows. Recall that the scalar vorticity form of the equation is $\begin{matrix} (φ_{t} - α^{2} Δ φ_{t}) - ν Δ (φ - α^{2} Δ φ) + J (Δ^{- 1} (I - α^{2} Δ) φ, φ) = {Curl}_{n} f . \end{matrix}$ Putting $ψ = φ - α^{2} Δ φ$ , then $\begin{matrix} (4.10) & ψ_{t} - ν Δ ψ + J (Δ^{- 1} ψ, {(I - α^{2} Δ)}^{- 1} ψ) = {Curl}_{n} f . \end{matrix}$

We consider the following family of forcing terms parameterized by the integer parameters s: $\begin{array}{l} f = f_{s} = \{\begin{matrix} f_{1} = \frac{1}{\sqrt{2} π} ν^{2} λ s^{2} sin s x_{2}, \\ f_{2} = 0, \end{matrix} \end{array}$ where we choose the parameter $λ : = λ (s)$ later. Then, we have $\begin{matrix} | f | = ν^{2} λ s^{2}, G = λ s^{2} \end{matrix}$ and $\begin{matrix} (4.11) & {Curl}_{n} f_{s} = F_{s} = - \frac{1}{\sqrt{2} π} ν^{2} λ s^{3} cos s x_{2}, | {Curl}_{n} f | = ν^{2} λ s^{3} . \end{matrix}$ Corresponding to the family (4.11) is the family of stationary solutions $\begin{matrix} ψ_{s} = - \frac{1}{\sqrt{2} π} ν λ s cos s x_{2} \end{matrix}$ of Equation (4.10) since $ψ_{s}$ depends only on $x_{2}$ , the nonlinear term vanishes $\begin{matrix} J (Δ^{- 1} ψ_{s}, {(I - α^{2} Δ)}^{- 1} ψ_{s}) = 0 \end{matrix}$ and the equality $- ν Δ ψ_{s} = F_{s}$ is verified directly.

We linearize (4.10) about the stationary solution (4.11) and consider the eigenvalue problem $\begin{array}{rcl} L_{s} ψ & : = & J (Δ^{- 1} ψ_{s}, {(I - α^{2} Δ)}^{- 1} ψ) \\ (4.12) & + J (Δ^{- 1} ψ, {(I - α^{2} Δ)}^{- 1} ψ_{s}) - ν Δ ψ = - σ ψ . \end{array}$ We use the orthonormal basis of trigonometric functions, which are the eigenfunctions of the Laplacian on the two-dimensional torus, $\begin{array}{l} {\frac{1}{\sqrt{2} π} sin k x, \frac{1}{\sqrt{2} π} cos k x}, k x = k_{1} x_{1} + k_{2} x_{2}, \\ k \in Z_{+}^{2} = {k \in Z_{0}^{2} | k_{1} ⩾ 0, k_{2} ⩾ 0} \cup {k \in Z_{0}^{2} | k_{1} ⩾ 1, k_{2} ⩽ 0} \end{array}$ and we rewrite ψ as a Fourier series $\begin{matrix} ψ = \frac{1}{\sqrt{2} π} \sum_{k \in Z_{+}^{2}} a_{k} cos k x + b_{k} sin k x . \end{matrix}$ Since $J (a, b) = - J (b, a)$ , we have $\begin{array}{l} J (Δ^{- 1} cos s x_{2}, {(I - α^{2} Δ)}^{- 1} cos k x) + J (Δ^{- 1} cos k x, {(I - α^{2} Δ)}^{- 1} cos s x_{2}) \\ = \frac{ν λ s}{\sqrt{2} π} (\frac{1}{s^{2}} \frac{1}{1 + α^{2} k^{2}} - \frac{1}{k^{2}} \frac{1}{1 + α^{2} s^{2}}) J (cos s x_{2}, a_{k} cos k x + b_{k} sin k x) \\ = \frac{ν λ s}{\sqrt{2} π} \frac{k^{2} - s^{2}}{(s^{2} + α^{2} s^{4}) (k^{2} + α^{2} k^{4})} J (cos s x_{2}, a_{k} cos k x + b_{k} sin k x) . \end{array}$ Plugging this into (4.12) we obtain that $\begin{array}{l} \frac{λ s}{\sqrt{2} π (s^{2} + α^{2} s^{4})} \sum_{k \in Z_{+}^{2}} (\frac{k^{2} - s^{2}}{k^{2} + α^{2} k^{4}}) J (cos s x_{2}, a_{k} cos k x + b_{k} sin k x) \\ (4.13) & + \sum_{k \in Z_{+}^{2}} (k^{2} + \hat{σ}) (a_{k} cos k x + b_{k} sin k x) = 0, \end{array}$ where $\hat{σ} = σ / ν$ .

We can calculate that $\begin{array}{rcl} J (cos s x_{2}, cos (k_{1} x_{1} + k_{2} x_{2})) & = & - k_{1} s sin s x_{2} sin (k_{1} x_{1} + k_{2} x_{2}) \\ = & \frac{k_{1} s}{2} (cos (k_{1} x_{1} + (k_{2} + s) x_{2})) - cos (k_{1} x_{1} + (k_{2} - s) x_{2}) \end{array}$ and $\begin{array}{rcl} J (cos s x_{2}, sin (k_{1} x_{1} + k_{2} x_{2})) & = & k_{1} s sin s x_{2} cos (k_{1} x_{1} + k_{2} x_{2}) \\ = & \frac{k_{1} s}{2} (sin (k_{1} x_{1} + (k_{2} + s) x_{2})) - sin (k_{1} x_{1} + (k_{2} - s) x_{2}) . \end{array}$ Substituting these equalities into (4.13) and regroup the terms with $cos (k_{1} x_{1} + k_{2} x_{2})$ , we get the following equation for the coefficients $a_{k_{1}, k_{2}}$ : $\begin{array}{l} - Λ (s) k_{1} (\frac{k_{1}^{2} + {(k_{2} + s)}^{2} - s^{2}}{k_{1}^{2} + {(k_{2} + s)}^{2} + α^{2} {(k_{1}^{2} + {(k_{2} + s)}^{2})}^{2}}) a_{k_{1} k_{2} + s} \\ + Λ (s) k_{1} (\frac{k_{1}^{2} + {(k_{2} - s)}^{2} - s^{2}}{k_{1}^{2} + {(k_{2} - s)}^{2} + α^{2} {(k_{1}^{2} + {(k_{2} - s)}^{2})}^{2}}) a_{k_{1} k_{2} - s} + (k^{2} + \hat{σ}) a_{k_{1} k_{2}} = 0, \end{array}$ where $\begin{matrix} (4.14) & Λ = Λ (s) : = \frac{s^{2} λ}{2 \sqrt{2} π (s^{2} + α^{2} s^{4})} = \frac{λ}{2 \sqrt{2} π (1 + α^{2} s^{2})} . \end{matrix}$ Similarly the equation for $b_{k_{1}, k_{2}}$ has also this form.

We put $\begin{matrix} a_{k_{1} k_{2}} (\frac{k^{2} - s^{2}}{k^{2} + α^{2} k^{4}}) = : c_{k_{1} k_{2}} . \end{matrix}$ and $\begin{array}{l} k_{1} = t, k_{2} = s n + r, and q c_{t s n + r} = e_{n}, \\ t = 1, 2, \dots, r \in Z, r_{min} < r < r_{max}, \end{array}$ where the numbers $r_{min}$ and $r_{max}$ satisfy that $r_{max} - r_{min} < s$ and will be specified below. For each t and r, we obtain the following three-term recurrence relation $\begin{matrix} (4.15) & d_{n} e_{n} + e_{n - 1} - e_{n + 1} = 0, n = 0, \pm 1, \pm 2, \dots, \end{matrix}$ where $\begin{matrix} (4.16) & d_{n} = \frac{(t^{2} + {(s n + r)}^{2} + α^{2} {(t^{2} + {(s n + r)}^{2})}^{2}) (t^{2} + {(s n + r)}^{2} + \hat{σ})}{Λ t (t^{2} + {(s n + r)}^{2} - s^{2})} . \end{matrix}$ We look for non-trivial decaying solutions ${e_{n}}$ of (4.15) and (4.16). Each nontrivial decaying solution with $Re (\hat{σ}) > 0$ produces an unstable eigenfunction ψ of the eigenvalue problem (4.12).

Theorem 4.5.
Given an integer $s > 0$ let a pair of integers t, r belong to a bounded region $A (δ)$ given by $\begin{array}{l} (4.17) & \begin{aligned} t^{2} + r^{2} ⟨ s^{2} / 3, t^{2} + {(- s + r)}^{2} ⟩ s^{2}, t^{2} + {(s + r)}^{2} > s^{2}, t ⩾ δ s, \\ r_{min} < r < r_{max}, r_{min} = - s / 6, r_{max} = s / 6, 0 < δ < 1 / \sqrt{3} . \end{aligned} \end{array}$ For any $Λ = \frac{λ}{2 \sqrt{2} π (1 + α^{2} s^{2})} > 0$ there exists a unique real eigenvalue $\hat{σ} = \hat{σ} (Λ)$ , which increases monotonically as $Λ \to \infty$ and satisfies the following inequality $\begin{matrix} (4.18) & c_{1} (α, t, r, s) Λ < \hat{σ} < c_{2} (α, t, r, s) Λ . \end{matrix}$ The unique $Λ_{0} = Λ_{0} (s)$ solving the equation $\begin{matrix} \hat{σ} (Λ_{0}) = 0 \end{matrix}$ satisfes the two-sided estimates $\begin{array}{l} (4.19) & \begin{aligned} \frac{1}{\sqrt{2}} δ^{2} s (1 + α^{2} s^{2}) < Λ_{0} < \frac{55 \sqrt{5}}{63 \sqrt{2}} \frac{s (1 + α^{2} s^{2})}{δ^{2}} for α ⩾ 0, \\ \frac{1}{\sqrt{2}} δ^{2} s < Λ_{0} < \frac{5}{3 \sqrt{3}} \frac{s}{δ^{2}} for α = 0 . \end{aligned} \end{array}$ In the term of λ, these inequalities are $\begin{array}{l} 2 π δ^{2} s {(1 + α^{2} s^{2})}^{2} < λ_{0} < \frac{110 \sqrt{5} π}{63} \frac{s {(1 + α^{2} s^{2})}^{2}}{δ^{2}} for α ⩾ 0, \\ 2 π δ^{2} s < λ_{0} < \frac{20 π}{3 \sqrt{6}} \frac{s}{δ^{2}} for α = 0 . \end{array}$
Proof.
The proof is done by a similar argument as in [24, Theorem 4.8], therefore it will be omitted. □

In the rest of this section, we give the lower bound of the attractor’s dimension by using the above theorem. Indeed, by $\begin{matrix} Λ = \frac{λ}{2 \sqrt{2} π (1 + α^{2} s^{2})}, \end{matrix}$ we rewrite (4.19) in the term of $λ (s)$ to see that for $\begin{array}{l} λ_{α ⩾ 0} = \frac{110 \sqrt{5} π}{63} s δ^{- 2} {(1 + α^{2} s^{2})}^{2}, \\ λ_{α = 0} = \frac{20 π}{3 \sqrt{6}} s δ^{- 2}, \end{array}$ each point in $(t, r)$ -plane satisfying (4.17) produces an unstable (positive) eigenvalue $\hat{σ} > 0$ of multiplicity two (the equation for the coefficients $b_{k}$ is the same). Denoting by $d (s)$ the number of points of the integer lattice inside the region $A (δ)$ we have $\begin{matrix} (4.20) & d (s) : = ♯ {(t, r) \in D (s) = Z^{2} \cap A (δ)} ≃ a (δ) s^{2} as s \to \infty, \end{matrix}$ where $a (δ) s^{2} = | A (δ) |$ is the area of the region $A (δ)$ . Therefore the dimension of the unstable manifold around the stationary solution $ψ_{s}$ is at least $2 a (δ) s^{2}$ and we obtain that $\begin{matrix} (4.21) & dim A ⩾ 2 d (s) ≃ 2 a (δ) s^{2} . \end{matrix}$

It is reasonable to consider two cases:

The case $α = 0$ .

We have $\begin{matrix} G = λ_{α = 0} s^{2} = \frac{20 π}{3 \sqrt{6}} s^{3} δ^{- 2} . \end{matrix}$ Since then, we write the estimate (4.21) in terms of the Grashof number G to obtain $\begin{array}{rcl} dim A & ⩾ & 2 a (δ) s^{2} ≃ 2 {(\frac{3 \sqrt{6}}{20 π})}^{2 / 3} a (δ) δ^{4 / 3} G^{2 / 3} \\ dim A & ⩾ & 2 {(\frac{3 \sqrt{6}}{20 π})}^{2 / 3} (max_{0 < δ < 1 / \sqrt{3}} a (δ) δ^{4 / 3}) G^{2 / 3} = 0.006 G^{2 / 3}, \end{array}$ where ${max}_{0 < δ < 1 / \sqrt{3}} a (δ) δ^{4 / 3} = 0.012$ . This is exactly the same lower bound obtained for the global attractor’s dimensions of the Navier–Stokes equation (see [15,21]).

The case $0 < α ≪ 1$ .

Here we obtain the lower bound of the attractor’s dimensions with $G ∽ {(1 / α)}^{3}$ . Let $0 < s < 1 / α$ . Then, we have $1 + α^{2} s^{2} < 2$ and $\begin{matrix} G ⩽ \frac{440 \sqrt{5} π}{63} s^{3} δ^{- 2} . \end{matrix}$ By the same way as in the case $α = 0$ we obtain that $\begin{matrix} dim A ⩾ 2 {(\frac{63}{440 \sqrt{5} π})}^{2 / 3} (max_{0 < δ < 1 / \sqrt{3}} a (δ) δ^{4 / 3}) G^{2 / 3} = 0.0018 G^{2 / 3} . \end{matrix}$ In particular, setting $s ∽ 1 / α$ we can obtain in the term of α that $\begin{matrix} C_{1} \frac{1}{α^{2}} ⩽ dim A ⩽ C_{2} \frac{1}{α^{2}} {(log \frac{1}{α})}^{1 / 3} . \end{matrix}$
5. The lower bound of global attractor on $T^{3}$

In this section, we will develop the method of Ilyin, Zelik and Kostiano in a recent work [17] to give the lower bound of the global attractor for the modified Leray-alpha equation on $T^{3} = {[0, 2 π]}^{3}$ . The method is based on Squire’s transformation which transforms the 3-D instability analysis to the instability analysis of the transformed 2-D problem and then we can use the results obtained in the previous section. To avoid confusion we denote the unknowns by $\vec{u}$ , the components by u and the covariant derivative by $\nabla_{x}$ .

5.1. The stationary solutions

Now we consider the modified Leray-alpha equation (2.5) on $T^{3}$ with the right-hand sides are given by $\begin{array}{l} (5.1) & f = f_{s} = \{\begin{matrix} f_{1} = \frac{1}{\sqrt{2} π} ν^{2} λ s^{2} sin s x_{3}, \\ f_{2} = 0, \\ f_{3} = 0, \end{matrix} \end{array}$ where $λ = λ (s)$ is chosen latter (in Theorem 5.4). Then, we have $\begin{matrix} (5.2) & | f | = ν^{2} λ s^{2}, G = λ s^{2} \end{matrix}$ and $\begin{matrix} {Curl}_{n} f_{s} = F_{s} = - \frac{1}{\sqrt{2} π} ν^{2} λ s^{3} cos s x_{3}, | {Curl}_{n} f | = ν^{2} λ s^{3} . \end{matrix}$ The family of stationary solutions of (2.5) corresponding to (5.1) are $\begin{array}{l} (5.3) & {\vec{v}}_{0} (x_{3}) = \{\begin{matrix} v_{0} (x_{3}) = \frac{1}{\sqrt{2} π} ν λ sin s x_{3}, \\ 0, \\ 0 \end{matrix} \end{array}$ Moreover, $\vec{u} = {(I - α^{2} Δ_{x})}^{- 1} {\vec{v}}_{0} = {(u_{0}, 0, 0)}^{T}$ depends only on $x_{3}$ hence ${\vec{v}}_{0} \cdot \nabla_{x} {\vec{u}}_{0} = 0$ .

We derive the linearized equation of (2.5) on the stationary solutions (5.3) as follows $\begin{array}{l} (5.4) & \{\begin{matrix} \partial_{t} ω + u_{0} \frac{\partial \bar{ω}}{\partial x_{1}} + {\bar{ω}}_{3} \frac{\partial u_{0}}{\partial x_{3}} e_{1} - Δ_{x} ω + \nabla_{x} q = 0, \\ div ω = 0, \end{matrix} \end{array}$ where $e_{1} = {(1, 0, 0)}^{T}$ and $\bar{ω} = {(I - α^{2} Δ_{x})}^{- 1} ω$ with the assumption $\begin{matrix} \int_{T^{3}} ω (x, t) d x = 0 . \end{matrix}$ We consider the solution of (5.4) in the following form $\begin{matrix} (5.5) & ω (x, t) = {(ω_{1} (x_{3}), ω_{2} (x_{3}), ω_{3} (x_{3}))}^{T} e^{i (a x_{1} + b x_{2} - a c t)} and q (t) = q (x_{3}) e^{i (a x_{1} + b x_{2} - a c t)}, \end{matrix}$ where $a, b \in Z$ satisfy that ω and q are $2 π$ -periodic in each $x_{i}$ .

If there exists a solution (5.5) of Equation (5.4), then at $t = 0$ we have that $\begin{matrix} ω (x, 0) = {(ω_{1} (x_{3}), ω_{2} (x_{3}), ω_{3} (x_{3}))}^{T} e^{i (a x_{1} + b x_{2})} \end{matrix}$ is a vector-valued eigenfunction of the stationary operator $\begin{matrix} (5.6) & L_{3} ({\vec{v}}_{0}) ω = u_{0} \frac{\partial \bar{ω}}{\partial x_{1}} + {\bar{ω}}_{3} \frac{\partial u_{0}}{\partial x_{3}} e_{1} - Δ_{x} ω + \nabla_{x} q \end{matrix}$ and $i a c$ is the corresponding eigenvalue. If $ℜ (i a c) < 0$ , then the corresponding mode is unstable.

Plugging (5.5) into (5.4) we obtaint that $\begin{array}{l} (5.7) & \{\begin{matrix} Δ_{x} ω_{1} - i a (u_{0} {\bar{ω}}_{1} - c ω_{1}) = i a q + {\bar{ω}}_{3} u_{0}^{'}, \\ Δ_{x} ω_{2} - i a (u_{0} {\bar{ω}}_{2} - c ω_{2}) = i b q, \\ Δ_{x} ω_{3} - i a (u_{0} {\bar{ω}}_{3} - c ω_{3}) = q^{'}, \\ i a ω_{1} + i b ω_{2} + ω_{3}^{'} = 0, \end{matrix} \end{array}$ where we denote $^{'} : = \partial / \partial x_{3}$ .

Lemma 5.1.
There are no unstable solutions of equation ( 5.4 ) which can be written by ( 5.5 ) at $a = 0$ .
Proof.
The proof can be seen as a slight modification of the proof of [17, Lemma 5.1] for replacing $- γ$ by $Δ_{x}$ . Let $a = 0$ we have that $\begin{matrix} ω (x, t) = {(ω_{1} (x_{3}), ω_{2} (x_{3}), ω_{3} (x_{3}))}^{T} e^{i b x_{2}} and q (t) = q (x_{3}) e^{i b x_{2}} \end{matrix}$ is solution of (5.4). Moreover, Equation (5.7) becomes $\begin{array}{l} \{\begin{matrix} Δ_{x} ω_{1} + i a c ω_{1} = i a q + {\bar{ω}}_{3} u_{0}^{'}, \\ Δ_{x} ω_{2} + i a c ω_{2} = i b q, \\ Δ_{x} ω_{3} + i a c ω_{3} = q^{'}, \\ i b ω_{2} + ω_{3}^{'} = 0, \end{matrix} \end{array}$ The final equation leads to $ω_{2} = - \frac{ω_{3}^{'}}{i b}$ . Plugging in the expression $- \frac{ω_{3}^{'}}{i b}$ for $ω_{2}$ in the second equation, we get $\begin{matrix} ω_{3}^{‴} + i a c ω_{3}^{'} = b^{2} q . \end{matrix}$ Differentiating the third with respect to $x_{3}$ we obtain $\begin{matrix} ω_{3}^{‴} + i a c ω_{3}^{'} = q^{″} . \end{matrix}$ Therefore, we have that $q^{″} = b^{2} q$ . This fact, together with that q is periodic, we get $q = 0$ .

Since we considering for unstable solutions, it follows that $ℜ (i c) < 0$ . This leads to ${Ker}_{L^{2}} (Δ_{x} + i c) = {0}$ . This gives that $ω_{2} = ω_{3} = 0$ , and, finally, $ω_{1} = 0$ .

If $a = b = 0$ , then $ω_{3}^{'} = 0$ , then $ω_{3} = 0$ by periodicity and zero mean condition. This shows that $q = 0$ and $ω_{1} = ω_{2} = 0$ . Our proof is completed. □

5.2. Transform from $T^{3}$ to $T^{2}$

Now we use Squire’s transformation to transform the eigenfunctions of $L_{3} ({\vec{v}}_{0})$ on $T^{3}$ to the ones of $L_{2} ({\vec{v}}_{0})$ on the 2-D torus. The idea and detailed techniques are given in [17].

Since Lemma (5.1), we assume that $a \neq 0$ in (5.7). Multiplying the first equation in (5.7) by a and the second one by b, then adding up the obtained results, we get $\begin{array}{l} (5.8) & \{\begin{matrix} {\hat{Δ}}_{x} ω_{1} - i \hat{a} (u_{0} {\bar{\hat{ω}}}_{1} - \hat{c} {\hat{ω}}_{1}) = i \hat{a} \hat{q} + {\bar{\hat{ω}}}_{3} u_{0}^{'}, \\ {\hat{Δ}}_{x} ω_{3} - i \hat{a} (u_{0} {\bar{\hat{ω}}}_{3} - \hat{c} {\hat{ω}}_{3}) = {\hat{q}}^{'}, \\ i \hat{a} {\hat{ω}}_{1} + {\hat{ω}}_{3}^{'} = 0, \end{matrix} \end{array}$ where $\begin{array}{l} (5.9) & \begin{aligned} {\hat{a}}^{2} = a^{2} + b^{2}, {\hat{ω}}_{1} = \frac{a ω_{1} + b ω_{2}}{\hat{a}}, {\hat{ω}}_{3} = ω_{3}, \\ \hat{Δ} = \frac{\hat{a}}{a} Δ, \hat{q} = q \frac{\hat{a}}{a}, \hat{c} = c . \end{aligned} \end{array}$ The solutions of the problem (5.8) on the 2-D torus $\begin{matrix} {\hat{T}}_{a}^{2} = {(x_{1}, x_{3}) \in [0, 2 π / | \hat{a} |] \times [0, 2 π]} \end{matrix}$ have the following form $\begin{matrix} (5.10) & \hat{ω} (x_{1}, x_{3}, t) = {({\hat{ω}}_{1} (x_{3}), {\hat{ω}}_{3} (x_{3}))}^{T} e^{i (\hat{a} x_{1} - \hat{a} \hat{c} t)}, \hat{q} (x_{1}, x_{3}, t) = q (x_{3}) e^{i (\hat{a} x_{1} - \hat{a} \hat{c} t)} . \end{matrix}$ Observe that if Equation (5.8) has the solutions (5.10), then the vector function $\begin{matrix} (5.11) & \hat{ω} (x_{1}, x_{3}, 0) = {({\hat{ω}}_{1} (x_{3}), {\hat{ω}}_{3} (x_{3}))}^{T} e^{i \hat{a} x_{1}} \end{matrix}$ is a vector-valued eigenfunction with eigenvalue $i \hat{a} \hat{c}$ of the stationary operator $\begin{matrix} (5.12) & L_{2} ({\vec{v}}_{0}) \hat{ω} = - \hat{Δ} \hat{ω} + u_{0} \frac{\partial \bar{\hat{ω}}}{\partial x_{1}} + {\bar{\hat{ω}}}_{3} \frac{\partial u_{0}}{\partial x_{3}} e_{1} + \nabla_{x} \hat{q}, div \hat{ω} = 0 \end{matrix}$ on ${\hat{T}}_{a}^{2}$ , where $u_{0} = {(I - α^{2} Δ_{x})}^{- 1} v_{0}$ . The stationary solution and the generating a family of right-hand sides are $\begin{array}{l} (5.13) & {\vec{v}}_{0} (x_{3}) = \{\begin{matrix} v_{0} (x_{3}) = \frac{1}{\sqrt{2} π} ν λ sin s x_{3} \\ 0 \end{matrix} \end{array}$ and $\begin{array}{l} (5.14) & {\tilde{f}}_{s} (x_{3}) = {\hat{Δ}}_{x} v_{0} (x_{3}) = \{\begin{matrix} f_{1} (x_{3}) = \frac{\hat{a}}{a \sqrt{2} π} ν^{2} λ s^{2} sin s x_{3} \\ 0 \end{matrix} \end{array}$ We suppose that $\hat{a} > 0$ . The result on the Squire’s reduction of the 3-D instability analysis to the 2-D case is given in the following lemma.

Lemma 5.2.
Let $\hat{ω}$ in ( 5.11 ) be an unstable eigenfunction of the operator ( 5.12 ) on the 2-D torus ${\hat{T}}_{a}^{2} = [0, 2 π / \hat{a}] \times [0, 2 π]$ . Then for any pair of integers $a, b \in Z$ with $\begin{matrix} a^{2} + b^{2} = {\hat{a}}^{2} \end{matrix}$ there exists an unstable solution of system ( 5.7 ) on three-torus $T^{3} = {[0, 2 π]}^{3}$ .
Proof.
By using the relations (5.9) we can find q, $ω_{3}$ , c. Observe that the second equation in (5.7) is $\begin{matrix} (Δ_{x} + i a c) ω_{2} - i a u_{0} {(I - α^{2} Δ)}^{- 1} ω_{2} = i b q . \end{matrix}$ This is equivalent to $\begin{matrix} (5.15) & - [Δ_{x} + i a c - i a u_{0} {(I - α^{2} Δ)}^{- 1}] ω_{2} = - i b q . \end{matrix}$ Considering the following sesquilinear form $A$ on $H_{0}^{1} ([0, 2 π], M) \times H_{0}^{1} ([0, 2 π], M)$ : $\begin{matrix} A (φ, ϕ) = - [Δ_{x} + i a c - i a u_{0} {(I - α^{2} Δ)}^{- 1} φ, ϕ] . \end{matrix}$ Clearly, $| A (φ, ϕ) |$ is bounded by $‖ φ ‖_{H_{0}^{1}} ‖ ϕ ‖_{H_{0}^{1}}$ . Moreover, we have that $\begin{matrix} | A (φ, φ) | ⩾ ‖ \nabla_{x} φ ‖_{L^{2}}^{2} - ℜ (i a c) ‖ φ ‖_{L^{2}}^{2} . \end{matrix}$ Since $\hat{ω}$ is unstable, we have $ℜ (i a c) < 0$ . This implies that the linear operator $A$ is coercive. By using Lax-Milgram theorem (in complex) (see [1, Theorem 7]), there exists a bounded and inverse operator $\tilde{A} : H_{0}^{1} ([0, 2 π], M) \to H^{- 1} ([0, 2 π], M)$ such that $\begin{matrix} A (φ, ϕ) = ⟨ \tilde{A} φ, ϕ ⟩ \end{matrix}$ for all $φ \in H_{0}^{1} ([0, 2 π], M)$ and $ϕ \in \overline{H_{0}^{1} ([0, 2 π], M)} = L^{2} ([0, 2 π], M)$ . Therefore, Equation (5.15) becomes $\tilde{A} ω_{2} = - i b q$ and it has a unique solution $ω_{2} = {\tilde{A}}^{- 1} (- i b q) \in H_{0}^{1} ([0, 2 π], M)$ . Finally, we obtain that $\begin{matrix} ω_{1} = \frac{\hat{a} {\hat{ω}}_{1} - b ω_{2}}{a} . \end{matrix}$ □

5.3. Lower bound on $T^{3}$

In this section, we apply the lower bound on the dimensions of the global attractor obtained on 2-D torus $T^{2}$ to establish the one on $T^{3}$ . We denote the second coordinate by $x_{3}$ , so that $x_{1}$ , $x_{3}$ are the coordinates on $T^{2}$ . The linearized stationary operator is (5.12) with the family of the forcing terms are (5.14), and the corresponding stationary solutions are (5.13).

Applying Curl to (5.12) we obtain the equivalent scalar operator in terms of the vorticity in the previous Section 4.2.2 on $T^{2}$ : $\begin{array}{rcl} L_{s} ω & : = & J (Δ^{- 1} ω_{s}, {(I - α^{2} Δ)}^{- 1} ω) \\ (5.16) & + J (Δ^{- 1} ω, {(I - α^{2} Δ)}^{- 1} ω_{s}) - ν Δ ω = - σ ω . \end{array}$ where $\begin{matrix} ω_{s} = {Curl}_{n} {\vec{v}}_{0} = - \frac{1}{\sqrt{2} π} ν λ s cos s x_{3} and ω = \hat{ω} . \end{matrix}$ In Section 4.2.2 we have also proved that the eigenfunctions of $L_{s}$ are $\begin{array}{l} (5.17) & \begin{aligned} ω^{1} (x_{1}, x_{3}) = \sum_{- \infty}^{\infty} a_{t, s n + r} cos (t x_{1} + (s n + r) x_{3}) \\ ω^{2} (x_{1}, x_{3}) = \sum_{- \infty}^{\infty} a_{t, s n + r} sin (t x_{1} + (s n + r) x_{3}) . \end{aligned} \end{array}$ Hence, $\begin{matrix} ω^{1} (x_{1}, x_{3}) + i ω^{2} (x_{1}, x_{3}) = e^{i t x_{1}} \sum_{n = - \infty}^{\infty} a_{t, s n + r} e^{i (s n + r) x_{3}} . \end{matrix}$ We can find an unstable vector valued eigenfunction of the operator $L_{2} ({\vec{v}}_{0})$ in the form (5.11) by applying the operator ${Curl}_{n} Δ_{x}^{- 1}$ to the above equation. Hence, we get $\begin{matrix} ω (x_{1}, x_{3}) = {(ω_{1} (x_{3}), ω_{3} (x_{3}))}^{T} e^{i t x_{1}} . \end{matrix}$

For the 3-D instability analysis we need to repeat the construction of an unstable eigenmode on the torus ${\hat{T}}_{a}^{2} = [0, 2 π / | \hat{a} |] \times [0, 2 π]$ . For this purpose we apply Theorem 4.5 on ${\hat{T}}_{a}^{2}$ to obtain the following proposition.

Proposition 5.3.

Let r and $t^{'} : = t | \hat{a} |$ belong to region $A (δ)$ : $\begin{matrix} (5.18) & t^{' 2} + r^{2} ⟨ s^{2} / 3, t^{' 2} + {(- s + r)}^{2} ⟩ s^{2}, t^{' 2} + {(s + r)}^{2} > s^{2}, t^{'} ⩾ δ s . \end{matrix}$ Taking ${\tilde{f}}_{s}$ and ${\vec{v}}_{0}$ in two dimension context as $\begin{matrix} {\tilde{f}}_{s} (x_{3}) = {(- \frac{\hat{a}}{a \sqrt{2} π} ν^{2} λ s^{2} sin s x_{3}, 0)}^{T}, \vec{v} (x_{3}) = {(\frac{1}{\sqrt{2} π} ν λ sin s x_{3}, 0)}^{T} . \end{matrix}$ Then there exists an unstable solution $\begin{matrix} (5.19) & ω (x_{1}, x_{3}) = {(ω_{1} (x_{3}), ω_{3} (x_{3}))}^{T} e^{i t ε x_{1}} where x \in {\hat{T}}_{a}^{2} \end{matrix}$ under the form ( 5.11 ) of the operator ( 5.12 ) on ${\hat{T}}_{a}^{2}$ .

Proof.

The proof is a consequence of Theorem 4.5 by substituting $t^{'} : = | \hat{a} | t$ . □

It is convenient to single out a small rectangle D in the $(t^{'}, r)$ -plane inside the region given by (5.18): $\begin{matrix} (5.20) & | r | ⩽ c_{2} s, 0 < c_{3} s ⩽ t^{'} ⩽ c_{4} s . \end{matrix}$ Here $δ = δ^{*} \in (0, 1 / \sqrt{3})$ is fixed, and all the constants $c_{i}$ are absolute constants, whose explicit values can be specified.

Theorem 5.4.

We consider the linearized equation ( 5.4 ) on the 3-torus $T^{3} = {[0, 2 π]}^{3}$ with right-hand side $f_{s}$ and stationary solution ${\vec{v}}_{0}$ given by ( 5.1 ) and ( 5.3 ), where $\begin{matrix} (5.21) & λ = λ_{3} (s) = \sqrt{2} λ_{2} (s) = \sqrt{2} c_{1} s {(1 + α^{2} s^{2})}^{2} . \end{matrix}$ (where $λ_{2} (s)$ is given in Theorem 4.5 ). Then for each triple of integers a, b, r satisfying $\begin{matrix} (5.22) & c_{3} s ⩽ \hat{a} = \sqrt{a^{2} + b^{2}} ⩽ c_{4} s, | r | ⩽ c_{2} s, | b | ⩽ a, \end{matrix}$ there exists an unstable solution of the linearized operator ( 5.6 ). The number of integers $(a, b, r)$ satisfied ( 5.22 ) is of order $c_{5} s^{3}$ , where $\begin{matrix} c_{5} = \frac{1}{4} π c_{2} (c_{4}^{2} - c_{3}^{2}) . \end{matrix}$

Proof.

The proof is a slight modification of [17, Theorem 5.5] for replacing γ by $Δ_{x}$ . We fix a, b and r satisfying (5.22). According to the first and second inequalities in (5.22), we have the pair $(t^{'}, r) \in D \subset A (δ)$ , where $t^{'} = \hat{a} .1$ (therefore, we set here $t = 1$ ). Applying Squire’s transformation we obtain a 2-D linearized problem on the torus ${\hat{T}}_{a}^{2}$ of the form (5.12) with $\hat{Δ} = \frac{\hat{a}}{a} Δ$ . Using the third inequality in (5.22), we arrive at $\begin{matrix} λ = \sqrt{2} λ_{2} (s, Δ) = \sqrt{2} \frac{a}{\hat{a}} \frac{\hat{a}}{a} λ_{2} (s) ⩾ λ_{2} (s, \hat{Δ}) . \end{matrix}$ Since Proposition 5.3, we have that the 2-D linearized problem (5.12) has an unstable eigenvalue. By using Lemma 5.2 this deduces that the 3-D linearized problem (5.6) has also unstable eigenvalue on the standard torus $T^{3} = {[0, 2 π]}^{3}$ . Our proof is completed. □

Now we give the lower bound of the attractor’s dimensions of the modified Leray-alpha equation (2.5) on the 3-D torus $T^{3} = {[0, 2 π]}^{3}$ in the following theorem.

Theorem 5.5.

Let the right-hand side in ( 2.5 ) be ( 5.1 ). The dimension of the corresponding attractor $A = A_{s}$ of ( 2.5 ) admits the lower bound $\begin{matrix} {dim}_{F} A ⩾ c_{5} \frac{1}{α^{3}}, \end{matrix}$ where $0 < α ≪ 1$ . In the term of Grashof number this is equivalent to $\begin{matrix} {dim}_{F} A ⩾ c_{6} G, \end{matrix}$ where $G = \frac{| f_{s} |}{ν^{2}}$ and $c_{6} = \frac{c_{5}}{4 \sqrt{2} c_{1}}$ .

Proof.

We consider only the case $0 < α ≪ 1$ . Since s is at our disposal, we take $s = \frac{1}{α}$ . Therefore, we obtain that $\begin{matrix} {dim}_{F} A ⩾ c_{5} s^{3} = c_{5} \frac{1}{α^{3}} . \end{matrix}$ Moreover, we obtain for $λ = λ_{3} (s)$ in (5.21) as follows $\begin{matrix} λ = 4 \sqrt{2} c_{1} \frac{1}{α} . \end{matrix}$ Plugging this value into (5.2) we get $\begin{matrix} | f_{s} | = ν^{2} λ s^{2} = 4 \sqrt{2} c_{1} \frac{ν^{2}}{α^{3}}, G = \frac{| f_{s} |}{ν^{2}} = 4 \sqrt{2} c_{1} \frac{1}{α^{3}} . \end{matrix}$ Finally, we obtain that $\begin{matrix} {dim}_{F} A ⩾ c_{6} G, \end{matrix}$ where $c_{6} = \frac{c_{5}}{4 \sqrt{2} c_{1}}$ . □

Remark 5.6.

The upper bound of the attractor’s dimension for 3-D modified Leray-alpha equation was obtained in [13, Theorem 6] as follows $\begin{matrix} {dim}_{F} A ⩽ c_{7} {\tilde{G}}^{3 / 2} {(\frac{1}{γ λ_{1} α^{2}})}^{3 / 4}, \end{matrix}$ where $\tilde{G} = \frac{| f |}{ν^{2} λ_{1}^{3 / 4}}$ and $\frac{1}{γ} = min {1, \frac{1}{λ_{1} α^{2}}}$ . Therefore, we obtain the two-sided estimate of the attractor’s dimension in the case $0 < α ≪ 1$ and $\tilde{G} ∽ \frac{1}{α^{3}}$ as follows $\begin{matrix} c_{5} \frac{1}{α^{3}} ⩽ {dim}_{F} A ⩽ c_{7} \frac{1}{α^{6}} . \end{matrix}$ Therefore, the sharp upper bound of ${dim}_{F} A$ must equivalent to $\frac{1}{α^{κ}}$ with the power $3 ⩽ κ ⩽ 6$ . The study of this sharp upper bound is an interesting question and we expect that it can be done in a forthcoming paper.

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On attractor’s dimensions of the modified Leray-alpha equation

Abstract

Keywords

1. Introduction

2. Geometrical and analytical setting

2.1. Geometric formula and functional spaces

Proposition 2.2. On the two-dimensional closed manifold M we have J ( a , b ) = − J ( b , a ) , ∫ M J ( a , b ) dVol M = ∫ M J ( a , b ) b dVol M = 0 and ∫ M J ( a , b ) c dVol M = ∫ M J ( b , c ) a dVol M . 3. Well-posedness and the existence of a global attractor

4.1. Fundamental theorem on the attractor’s dimensions

4.2.1. Upper bound

5.1. The stationary solutions

References