Upper estimates on Hausdorff and fractal dimensions of global attractors for the 2D Navier–Stokes–Voight equations with a distributed delay

Abstract

This paper is concerned with the dynamics of 2D Navier–Stokes–Voight equations which govern an incompressible fluid flow. By virtue of energy estimates and compact embedding technique, we establish the existence of finite fractal dimensional global attractor for the 2D Navier–Stokes–Voight equations with a distributed delay.

Keywords

Navier–Stokes–Voight equations distributed delay global attractor fractal dimension

1. Introduction

From 1980s, the infinite dimensional dynamic systems for the evolution equations such as incompressible Navier–Stokes (NS) and Navier–Stokes–Voight (NSV) equations, have become a hot topic. For the autonomous case, the basic theories for global attractors were initiated in the references [1,9,11,16,24,27], which have fruitful techniques and applications for dissipative partial differential equations. While for the non-autonomous case, the related theories, such as uniform attractors, pullback and trajectory attractors, can be found in [7,8,19,22,23]. However, the dynamics for nonlinear fluid flow models has not been solved completely, such as the existence of inertial manifold.

We know that the NSV equations govern the motion of viscoelastic incompressible fluid with Kelvin-Voight damping, which were introduced by Oskolkov [21]. The infinite dimensional dynamic systems for this system become important to describe turbulence. The NSV model has a finite dimensional global attractor for generated semigroup in [12]. Under homogeneous boundary condition, Kalantarov [13] derived the existence of global attractor for the NSV equations, and further showed that the attractor lay in a bounded subset of the Sobolev space $H^{1} (Ω)$ whenever the forcing term $f \in L^{2} (Ω)$ . In 1963, Krasovskii [15] first considered the stability problem for differential equations with time delay. Barbu and Sritharan [2] solved the local solvability for the Navier–Stokes equations with a hereditary viscous term. Caraballo and Real [4–6] obtained some results relating to the existence and asymptotic behavior of solutions to the Navier–Stokes equations with variable delay and distributed delay. Taniguchi [26] established the existence of absorbing sets of the non-autonomous Navier–Stokes equations with continuous delay. Li and Qin [17] gave the existence of pullback attractors for the NSV equations with continuous delay on smooth domain $Ω \subset R^{3}$ , and the existence of pullback attractors for the NSV equations with double delay was also established on a Lipschitz domain in [25]. For theories about the estimates on dimensions of attractors, we can refer to [1,8,10,16,18,20,27]. For the applications of some fluid flow models, Brown, Perry and Shen [3] had considered the 2D Navier–Stokes equations on a Lipschitz domain and established the upper bounds of global attractors. The upper bounds for the dimension of global attractors for the 3D Navier–Stokes–Voight equations were derived by Kalantarov and Titi [14]. In [30], the explicit upper bounds for the dimension of global and exponential attractors were given. However, to our knowledge, there are few results involving the dimensional estimates on attractors for the NS or NSV equations with a delay.

The objective of this paper is to establish the upper estimates on Hausdorff and fractal dimensions of global attractors for the following 2D NSV equations with distributed delay in a bounded domain Ω with smooth boundary $\partial Ω$ : $\begin{array}{rcl} (1.1) & \{\begin{matrix} \frac{d}{d t} u - ν Δ u - α^{2} \frac{d}{d t} Δ u + (u \cdot \nabla) u + \nabla p = f (x) + g (u_{t}), & (x, t) \in Ω_{0}, \\ div u = 0, & (x, t) \in Ω_{0}, \\ u (x, t) |_{\partial Ω} = φ, φ \cdot n = 0, & (x, t) \in \partial Ω_{0}, \\ u (x, 0) = u^{0} (x), & x \in Ω, \\ u (x, t) = ϕ (t), & (x, t) \in Ω_{h}, \end{matrix} \end{array}$ where $Ω_{0} = Ω \times [0, + \infty)$ , $\partial Ω_{0} = \partial Ω \times [0, + \infty)$ , $Ω_{h} = Ω \times [- h, 0]$ , ν is the kinematic viscosity, $u = (u_{1} (x, t), u_{2} (x, t))$ is the velocity field, p is the pressure, $α > 0$ is a length scale parameter, $φ \in L^{\infty} (\partial Ω)$ , $f (x)$ is the steady external force, ϕ is the initial datum in $[- h, 0]$ where $h > 0$ is fixed. $g (u_{t}) = \int_{- h}^{0} G (s, u (t + s)) d s$ is the distributed delay where $u_{t} = u (t + s)$ , $s \in [- h, 0]$ .

Comparing with [17,25], this presented paper has the following features:

We give the existence of global attractor $A$ of 2D NSV equations with a distributed delay in the product space $X_{V}$ ;

Using the method of extension for generator appeared in [28,29], we establish the differentiability of the semigroup in $X_{V}$ , and obtain the upper boundedness of the fractal dimension of $A$ firstly.

This paper is arranged as follows. In Section 2, we shall introduce some useful notations and inequalities. In Section 3, the existence of continuous semigroup is derived in $X_{V}$ . The attracting property is obtained in Section 4, and the asymptotic compactness and existence of global attractors to (1.1) are established in Section 5. Finally, the upper estimates on Hausdorff and fractal dimensions of global attractors are also obtained.

2. Preliminaries

Assume $E : = {u | u \in {(C_{0}^{\infty} (Ω))}^{2}, div u = 0}$ . $H = {\overline{E}}^{{(L^{2} (Ω))}^{2}}$ with norm $| \cdot |$ and inner product $(\cdot, \cdot)$ respectively, where $\begin{matrix} (u, v) = \sum_{j = 1}^{2} \int_{Ω} u_{j} (x) v_{j} (x) d x, \forall u, v \in H, \end{matrix}$ and $V = {\overline{E}}^{{(H^{1} (Ω))}^{2}}$ is a Hilbert space with norm $‖ \cdot ‖$ and inner product $((\cdot, \cdot))$ respectively, where $\begin{matrix} ((u, v)) = \sum_{i, j = 1}^{2} \int_{Ω} \frac{\partial u_{j}}{\partial x_{i}} \frac{\partial v_{j}}{\partial x_{i}} d x, \forall u, v \in V . \end{matrix}$

We define the following Hilbert space $\begin{matrix} X_{K} = L^{2} ([- h, 0]; K) \times K, K = H, V \end{matrix}$ with the norm $\begin{matrix} {‖ {α, β} ‖}_{X_{K}} = {({α, β}, {α, β})}_{X_{K}}^{1 / 2}, \forall {α, β} \in X_{K} \end{matrix}$ where $\begin{matrix} {({α, β}, {ζ, η})}_{X_{K}} = \int_{- h}^{0} {(α (θ), ζ (θ))}_{K} d θ + {(β, η)}_{K}, \forall (α, β), (ζ, η) \in X_{K}, \end{matrix}$ and another space $C_{K} = C^{0} ([- h, 0]; K)$ with norm $‖ u ‖_{C_{K}} = {sup}_{θ \in [- h, 0]} ‖ u (t + θ) ‖_{K}$ .

The operator P is the Helmholz–Leray orthogonal projection of ${(L^{2} (Ω))}^{2}$ onto H, and $P_{1}, P_{2}$ denote the projections of $X_{K}$ onto $L^{2} ([- h, 0]; K)$ and K. $A : = - P Δ$ is a self-adjoint positively defined operator on H, $A^{- 1}$ is compact from H to H, and thus the orthonormal system of eigenfunctions ${ω_{j}}_{j = 1}^{\infty}$ of A exists corresponding to the eigenvalues ${λ_{j}}_{j = 1}^{\infty}$ ( $0 < λ_{1} ⩽ λ_{2} ⩽ \dots$ ).

Also we define $\begin{matrix} V_{s} = D (A^{\frac{s}{2}}), \end{matrix}$ with norm $\begin{matrix} ‖ V ‖_{s} = | A^{\frac{s}{2}} V |, s \in R . \end{matrix}$ $V^{'}$ is the dual space of V with norm $‖ \cdot ‖_{*}$ , and $⟨ \cdot ⟩$ denotes the dual product between V and $V^{'}$ .

We further define the bilinear and trilinear form operators as follows (see [27]) $\begin{array}{l} B (u, v) : = P ((u \cdot \nabla) v), \forall u, v \in V, \\ b (u, v, w) = (B (u, v), w) = \sum_{i, j = 1}^{2} \int_{Ω} u_{i} \frac{\partial v_{j}}{\partial x_{i}} w_{j} d x, \end{array}$ which satisfy $\begin{matrix} (2.1) & \{\begin{matrix} b (u, v, v) = 0, b (u, v, w) = - b (u, w, v), & \forall u, v, w \in V, \\ | b (u, v, w) | ⩽ C_{1} | u | ‖ v ‖ | w |, & \forall u, v, w \in V . \end{matrix} \end{matrix}$

Now we construct a background function ψ such that $\begin{array}{l} div ψ = 0, x \in Ω; ψ = φ, x \in \partial Ω; \\ ‖ ψ ‖ ⩽ C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)}, \end{array}$ and let $v = u - ψ$ , then (1.1) reduces to $\begin{matrix} (2.2) & \{\begin{matrix} \frac{d v}{d t} - ν Δ v - α^{2} \frac{d}{d t} Δ v_{t} + (v \cdot \nabla) v \\ + (v \cdot \nabla) ψ + (ψ \cdot \nabla) v + \nabla p = \bar{f} + g (v_{t} + ψ), & (x, t) \in Ω_{0}, \\ div v = 0, & (x, t) \in Ω_{0}, \\ v = 0, & (x, t) \in \partial Ω_{0}, \\ v (x, 0) = v^{0} (x), & x \in Ω, \\ v (x, t) = ϕ - ψ (x) = η, & (x, t) \in Ω_{h}, \end{matrix} \end{matrix}$ where $\bar{f} = f + ν Δ ψ - (ψ \cdot \nabla) ψ$ .

Let $R (u) = B (u, ψ) + B (ψ, u)$ whose characteristics can be found in [27]. Particularly, (2.2) can be written as the following abstract form $\begin{matrix} (2.3) & \{\begin{matrix} \frac{d v}{d t} + ν A v + α^{2} \frac{d}{d t} A v + B (v) + R (v) = P \bar{f} + g (v_{t} + ψ), \\ v (0) = v^{0}, x \in Ω, \\ v_{0} = η, x \in Ω . \end{matrix} \end{matrix}$ Now assume that

$g : L^{2} ([- h, 0]; H) \to H$ is twice continuously differentiable, $g (0) = 0$ , and $\begin{matrix} sup_{Φ \in L^{2} ([- h, 0]; H), i = 1, 2} | D^{i} g (Φ) | ⩽ l_{0}, 0 < l_{0} < + \infty; \end{matrix}$

∃ $L_{g} ⩾ 0$ such that for any $u_{t}, v_{t} \in C_{H}$ , $\begin{matrix} | g (u_{t}) - g (v_{t}) | ⩽ L_{g} ‖ u_{t} - v_{t} ‖_{C_{H}}, t ⩾ 0; \end{matrix}$

∃ $C_{g} > 0$ , $m_{0} ⩾ 0$ such that for any $m \in [0, m_{0}]$ and $u_{t}, v_{t} \in L^{2} ([- h, 0]; H)$ , $\begin{matrix} \int_{0}^{t} e^{m s} {| g (u_{s}) - g (v_{s}) |}^{2} d s ⩽ C_{g}^{2} \int_{- h}^{t} e^{m s} {| u (s) - v (s) |}^{2} d s, t ⩾ 0; \end{matrix}$

$ν λ_{1} > 2 C_{1} C_{2} ‖ ψ ‖_{L^{\infty} (\partial Ω)} + 3 C_{g}$ .

Remark 2.1.
The assumption (A1) is given to show the differentiability of the semigroup $S (t)$ , which is first proposed in this paper. (A2) appears in many references (see [4–6,25]), which implies for $u_{t} \in L^{2} ([- h, 0]; H)$ that $\begin{matrix} g (u_{t}) : t \in [0, T] \to {(L^{2} (Ω))}^{2} \end{matrix}$ is measurable. (A3) is to ensure the global existence of solutions.

3. Global existence of solutions

In this section, we shall prove the global existence of solutions. To achieve it, we need to introduce some lemmas below.

Lemma 3.1.
Let $\bar{f} \in H$ , conditions (A1)∼(A3) hold and ${η, v^{0}} \in X_{V}$ , then there exists a unique global solution $v (t)$ to ( 2.3 ) which satisfies $\begin{matrix} v (t) \in L^{\infty} ([0, T]; V) \cap L^{2} ([- h, T]; V) . \end{matrix}$
Proof.
We first establish the global existence of solutions to (2.3) by the standard Faedo–Galerkin method. Fix $n ⩾ 1$ , we define an approximate solution $v_{n}$ to (2.3) as $\begin{matrix} v_{n} (t) = \sum_{j = 0}^{n} a_{n j} (t) w_{j}, \end{matrix}$ which satisfies $\begin{matrix} (3.1) & \{\begin{matrix} \frac{d v_{n}}{d t} + ν A v_{n} + α^{2} A v_{n t} + B (v_{n}) + R (v_{n}) = P_{n} \bar{f} + g (v_{n t} + ψ), \\ v_{n} (0) = v_{n}^{0}, x \in Ω, \\ v_{n 0} = η_{n}, x \in Ω, \end{matrix} \end{matrix}$ and $a_{n j} (t)$ is to be determined later on.

Multiplying (3.1) by $v_{n}$ , we have $\begin{array}{l} (\frac{d v_{n}}{d t}, v_{n}) + ν (A v_{n}, v_{n}) + α^{2} (A v_{n t}, v_{n}) + (B (v_{n}), v_{n}) + (R (v_{n}), v_{n}) \\ (3.2) & = ⟨ P_{n} \bar{f}, v_{n} ⟩ + ⟨ g (v_{n t} + ψ), v_{n} ⟩, \end{array}$ and $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (| v_{n} |^{2} + α^{2} ‖ v_{n} ‖^{2}) + ν ‖ v_{n} ‖^{2} \\ ⩽ | (R (v_{n}), v_{n}) | + | ⟨ P_{n} \bar{f}, v_{n} ⟩ | + | ⟨ g (v_{n t} + ψ), v_{n} ⟩ | \\ ⩽ ‖ \bar{f} ‖ ‖ v_{n} ‖ + | g (v_{n t} + ψ) | \cdot | v_{n} | + C_{1} | v_{n} | ‖ v_{n} ‖ ‖ ψ ‖ \\ ⩽ \frac{ν ‖ v_{n} ‖^{2}}{2} + \frac{‖ \bar{f} ‖^{2}}{2 ν} + \frac{| g (v_{n t} + ψ) |^{2}}{2 C_{g}} + \frac{C_{g}}{2 λ_{1}} ‖ v_{n} ‖^{2} + C_{1} λ_{1}^{- 1} ‖ v_{n} ‖^{2} ‖ ψ ‖ \\ (3.3) & ⩽ \frac{‖ \bar{f} ‖^{2}}{2 ν} + \frac{| g (v_{n t} + ψ) |^{2}}{2 C_{g}} + (\frac{ν}{2} + C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} + \frac{C_{g}}{2 λ_{1}}) ‖ v_{n} ‖^{2}, \end{array}$ that is, $\begin{array}{l} \frac{d}{d t} (| v_{n} |^{2} + α^{2} ‖ v_{n} ‖^{2}) \\ (3.4) & ⩽ \frac{| \bar{f} ‖^{2}}{ν} + \frac{| g (v_{n t} + ψ) |^{2}}{C_{g}} - (ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - C_{g} λ_{1}^{- 1}) ‖ v_{n} ‖^{2} . \end{array}$ Choosing some constant $m > 0$ such that $ν λ_{1} > 2 C_{1} C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)} + 3 C_{g} + m + m α^{2} λ_{1}$ , we obtain $\begin{array}{l} \frac{d}{d t} [e^{m t} (| v_{n} |^{2} + α^{2} ‖ v_{n} ‖^{2})] \\ ⩽ \frac{e^{m t} ‖ \bar{f} ‖^{2}}{ν} + \frac{e^{m t} | g (v_{n t} + ψ) |^{2}}{C_{g}} \\ - e^{m t} (ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - C_{g} λ_{1}^{- 1} - m λ_{1}^{- 1} - m α^{2}) ‖ v_{n} ‖^{2} \\ ⩽ \frac{e^{m t} ‖ \bar{f} ‖^{2}}{ν} + \frac{e^{m t} | g (v_{n t} + ψ) |^{2}}{C_{g}} \\ (3.5) & - e^{m t} (ν λ_{1} - 2 C_{1} C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)} - C_{g} - m - m α^{2} λ_{1}) | v_{n} |^{2} . \end{array}$ Integrating (3.5) over $[0, t]$ , we conclude $\begin{array}{l} e^{m t} (| v_{n} |^{2} + α^{2} ‖ v_{n} ‖^{2}) \\ ⩽ {| v_{n} (0) |}^{2} + α^{2} {‖ v_{n} (0) ‖}^{2} + \frac{1}{ν} \int_{0}^{t} e^{m s} ‖ \bar{f} ‖^{2} d s + \frac{1}{C_{g}} \int_{0}^{t} e^{m s} {| g (v_{n s} + ψ) |}^{2} d s \\ (3.6) & - \int_{0}^{t} e^{m s} (ν λ_{1} - 2 C_{1} C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)} - C_{g} - m - m α^{2} λ_{1}) {| v_{n} (s) |}^{2} d s, \end{array}$ and $\begin{array}{l} e^{m t} ({| v_{n} (t) |}^{2} + α^{2} {‖ v_{n} (t) ‖}^{2}) - ({| v_{n} (0) |}^{2} + α^{2} {‖ v_{n} (0) ‖}^{2}) \\ ⩽ \frac{e^{m t}}{m ν} ‖ \bar{f} ‖^{2} + \frac{2 C_{g}}{m} (e^{m t} - 1) | ψ |^{2} + C_{g} \int_{- h}^{0} e^{m s} | ϕ_{n} |^{2} d s \\ - (ν λ_{1} - 2 C_{1} C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)} - 3 C_{g} - m - m α^{2} λ_{1}) \int_{0}^{t} e^{m s} {| v_{n} (s) |}^{2} d s \\ ⩽ \frac{e^{m t}}{m ν} ‖ \bar{f} ‖^{2} + \frac{2 C_{2}^{2} C_{g}}{m λ_{1}} e^{m t} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + 2 C_{g} ‖ η ‖_{L_{H}^{2}}^{2} \\ (3.7) & - (ν λ_{1} - 2 C_{1} C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)} - 3 C_{g} - m - m α^{2} λ_{1}) \int_{0}^{t} e^{m s} {| v_{n} (s) |}^{2} d s, \end{array}$ which implies $\begin{array}{l} {| v_{n} (t) |}^{2} + α^{2} {‖ v_{n} (t) ‖}^{2} \\ (3.8) & ⩽ \frac{‖ \bar{f} ‖^{2}}{m ν} + \frac{2 C_{2}^{2} C_{g}}{m λ_{1}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + 2 C_{g} ‖ η ‖_{L_{H}^{2}}^{2} + {| v_{n} (0) |}^{2} + α^{2} {‖ v_{n} (0) ‖}^{2} \equiv K^{2} . \end{array}$

Integrating (3.4) in t also gives $\begin{array}{l} ({| v_{n} (t) |}^{2} + α^{2} {‖ v_{n} (t) ‖}^{2}) - ({| v_{n} (s) |}^{2} + α^{2} {‖ v_{n} (s) ‖}^{2}) \\ + (ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - C_{g} λ_{1}^{- 1}) \int_{s}^{t} {‖ v_{n} (r) ‖}^{2} d r \\ ⩽ \frac{‖ \bar{f} ‖^{2}}{ν} (t - s) + \frac{1}{C_{g}} \int_{s}^{t} {| g (v_{n r} + ψ) |}^{2} d r \\ ⩽ \frac{‖ \bar{f} ‖^{2}}{ν} (t - s) + C_{g} \int_{s - h}^{t} {| v_{n} (r) + ψ |}^{2} d r \\ ⩽ \frac{‖ \bar{f} ‖^{2}}{ν} (t - s) + 2 C_{g} \int_{s - h}^{t} ({| v_{n} (r) |}^{2} + | ψ |^{2}) d r \\ ⩽ \frac{‖ \bar{f} ‖^{2}}{ν} (t - s) + \frac{2 C_{2}^{2}}{λ_{1}} (t - s + h) C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + 2 C_{g} \int_{s}^{t} {| v_{n} (r) |}^{2} d s \\ (3.9) & + 8 C_{2}^{2} h C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}, \end{array}$ and $\begin{array}{l} (ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - 3 C_{g} λ_{1}^{- 1}) \int_{s}^{t} ‖ v_{n} ‖^{2} d s \\ ⩽ (\frac{‖ \bar{f} ‖^{2}}{ν} + \frac{2 C_{2}^{2}}{λ_{1}} C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) (t - s) + 10 C_{2}^{2} h C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} \\ (3.10) & + {| v_{n} (s) |}^{2} + α^{2} {‖ v_{n} (s) ‖}^{2} . \end{array}$

Thus $v_{n} (t) \in L^{\infty} ([0, T]; V) \cap L^{2} ([- h, T]; V)$ . By the Alaoglu compactness theorem, we can find a subsequence ${v_{n}}$ such that $\begin{matrix} v_{n} \to^{*} v in L^{\infty} ([0, T]; V); v_{n} \to v in L^{2} ([0, T]; V), \end{matrix}$ i.e., $v \in L^{\infty} ([0, T]; V) \cap L^{2} ([- h, T]; V)$ .

Next, we shall prove that $\frac{d v_{n}}{d t} \in L^{2} ([0, T]; V^{'})$ . Since $\begin{matrix} \frac{d v_{n}}{d t} = - ν A v_{n} - α^{2} A v_{n t} - B (v_{n}) - R (v_{n}) + P_{n} \bar{f} + g (v_{n t} + ψ), \end{matrix}$ and $v_{n} \in L^{2} (0, T; V)$ , we have $\begin{matrix} - ν A v_{n}, α^{2} v_{n t}, g (v_{n t} + ψ) \in L^{2} (0, T; V^{'}), \end{matrix}$ and $\begin{array}{rcl} {‖ B (v_{n}) ‖}_{L^{2} (0, T; V^{'})}^{2} & = & \int_{0}^{T} {(sup_{‖ u ‖ = 1} | (v_{n} \cdot \nabla) v_{n}, u |)}^{2} d s ⩽ \int_{0}^{T} {(| (v_{n} \cdot \nabla) v_{n} | | u |)}^{2} d s \\ ⩽ & C \int_{0}^{T} {| (v_{n} \cdot \nabla) v_{n} |}^{2} ‖ u ‖^{2} d s = C \int_{0}^{T} {| (v_{n} \cdot \nabla) v_{n} |}^{2} d s \\ ⩽ & C \int_{0}^{T} | v_{n} |^{2} | \nabla v_{n} |^{2} ⩽ C ‖ v_{n} ‖_{L^{\infty} (0, T; V)}^{2} ‖ v_{n} ‖_{L^{2} (0, T; V)}^{2} . \end{array}$ Similarly, we have $\begin{matrix} {‖ R (v_{n}) ‖}_{L^{2} (0, T; V^{'})}^{2} ⩽ C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} (‖ v_{n} ‖_{L^{\infty} (0, T; V)}^{2} + ‖ v_{n} ‖_{L^{2} (0, T; V)}^{2}) . \end{matrix}$ It follows that $\frac{d v_{n}}{d t} \in L^{2} ([0, T]; V^{'})$ . By the compact embedding theorem, we conclude $\begin{array}{rcl} (3.11) & v_{n} \to v, in L^{2} ([0, T]; V); v_{n} (0) = P_{n} v_{0} \to v (0) = v_{0} . \end{array}$

In what follows, we shall prove the uniqueness and continuity of global solutions. Let $v_{1}, v_{2}$ be two solutions to (3.1) with initial data ${η_{1}, v^{1}}$ and ${η_{2}, v^{2}}$ respectively, and let $w = v_{1} - v_{2}$ , then $\begin{array}{l} \frac{d w}{d t} + ν A w + α^{2} A w_{t} + B (v_{1}, v_{1}) - B (v_{2}, v_{2}) + R (w) \\ (3.12) & = g (v_{1 t} + ψ) - g (v_{2 t} + ψ) . \end{array}$ Since $\begin{matrix} B (v_{1}, v_{1}) - B (v_{2}, v_{2}) = B (w, v_{1}) + B (v_{2}, w), \end{matrix}$ we get $\begin{array}{rcl} (3.13) & \frac{d w}{d t} + ν A w + α^{2} A w_{t} + B (w, v_{1}) + B (v_{2}, w) + R (w) = g (v_{1 t}) - g (v_{2 t}) . \end{array}$ Multiplying (3.13) by w, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (| w |^{2} + α^{2} ‖ w ‖^{2}) + ν ‖ w ‖^{2} + b (w, v_{1}, w) + b (v_{2}, w, w) \\ + (R (w), w) = ⟨ g (v_{1 t} + ψ) - g (v_{2 t} + ψ), w ⟩, \end{array}$ and $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (| w |^{2} + α^{2} ‖ w ‖^{2}) + ν ‖ w ‖^{2} \\ ⩽ | b (w, v_{1}, w) | + | (R (w), w) | + | ⟨ g (v_{1 t} + ψ) - g (v_{2 t} + ψ), w ⟩ | \\ ⩽ C | w | ‖ w ‖ ‖ v_{1} ‖ + C | w | ‖ w ‖ ‖ ψ ‖ + | g (v_{1 t} + ψ) - g (v_{2 t} + ψ) | | w | \\ ⩽ \frac{ν}{2} ‖ w ‖^{2} + \frac{C}{2 ν} | w |^{2} ‖ v_{1} ‖^{2} + \frac{ν}{2} ‖ w ‖^{2} + \frac{C}{2 ν} | w |^{2} ‖ ψ ‖^{2} \\ + \frac{| g (v_{1 t} + ψ) - g (v_{2 t} + ψ) |^{2}}{2} + \frac{| w |^{2}}{2}, \end{array}$ i.e., $\begin{array}{l} \frac{d}{d t} (| w |^{2} + α^{2} ‖ w ‖^{2}) & ⩽ (\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{C}{ν} ‖ ψ ‖^{2} + 1) | w |^{2} \\ (3.14) & + {| g (v_{1 t} + ψ) - g (v_{2 t} + ψ) |}^{2} . \end{array}$ Integrating (3.14) over $[0, t]$ , we get $\begin{array}{l} ({| w (t) |}^{2} + α^{2} {‖ w (t) ‖}^{2}) - ({| w (0) |}^{2} + α^{2} {‖ w (0) ‖}^{2}) \\ ⩽ \int_{0}^{t} (\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{C}{ν} ‖ ψ ‖^{2} + 1) | w |^{2} d s + \int_{0}^{t} {| g (v_{1 t}) - g (v_{2 t}) |}^{2} d s \\ ⩽ \int_{0}^{t} (\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{C}{ν} ‖ ψ ‖^{2} + 1) | w |^{2} d s + C_{g}^{2} \int_{- h}^{t} {| v_{1} (r) - v_{2} (r) |}^{2} d r \\ = \int_{0}^{t} (\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{C}{ν} ‖ ψ ‖^{2} + 1) | w |^{2} d s + C_{g}^{2} (\int_{- h}^{0} | η_{1} - η_{2} |^{2} d r + \int_{0}^{t} {| w (r) |}^{2} d r) \\ = \int_{0}^{t} (\frac{C}{ν} ‖ v_{1} ‖^{2} + \frac{C}{ν} ‖ ψ ‖^{2} + 1 + C_{g}^{2}) | w |^{2} d s + C_{g}^{2} ‖ η_{1} - η_{2} ‖_{L_{H}^{2}}^{2} \\ ⩽ \int_{0}^{t} C (\frac{1}{ν} ‖ v_{1} ‖^{2} + \frac{1}{ν} ‖ ψ ‖^{2} + 1 + C_{g}^{2}) (| w |^{2} + α^{2} ‖ w ‖^{2}) d s \\ (3.15) & + C_{g}^{2} ‖ η_{1} - η_{2} ‖_{L_{H}^{2}}^{2} . \end{array}$ Thus $\begin{array}{rcl} {| w (t) |}^{2} + α^{2} {‖ w (t) ‖}^{2} & ⩽ & {| w (0) |}^{2} + α^{2} {‖ w (0) ‖}^{2} + C_{g}^{2} ‖ η_{1} - η_{2} ‖_{L_{H}^{2}}^{2} \\ + \int_{0}^{t} C (\frac{1}{ν} ‖ v_{1} ‖^{2} + \frac{1}{ν} ‖ ψ ‖^{2} + 1 + C_{g}^{2}) ({| w (t) |}^{2} + α^{2} {‖ w (t) ‖}^{2}) d s, \end{array}$ which gives $\begin{array}{l} {| w (t) |}^{2} + α^{2} {‖ w (t) ‖}^{2} \\ ⩽ ({| w (0) |}^{2} + α^{2} {‖ w (0) ‖}^{2} + C_{g}^{2} ‖ η_{1} - η_{2} ‖_{L_{H}^{2}}^{2}) e^{\int_{0}^{t} C (\frac{1}{ν} ‖ v_{1} ‖^{2} + \frac{1}{ν} ‖ ψ ‖^{2} + 1 + C_{g}^{2}) d s}, \end{array}$ and hence we can obtain the global well-posedness. It follows that the semigroup in $C_{V}$ can be defined as $\begin{matrix} S (t) {η, v^{0}} = v_{t} (\cdot; {η, v^{0}}) . \end{matrix}$
Remark 3.1.
Also, we can get $\begin{array}{rcl} (3.16) & {‖ w (t) ‖}^{2} ⩽ {‖ {η_{1} - η_{2}, v^{1} - v^{2}} ‖}_{X_{V}}^{2} e^{C T}, \end{array}$ and $\begin{array}{rcl} ‖ w_{t} ‖_{L_{V}^{2}}^{2} & = & \int_{- h}^{0} {‖ w (t + s) ‖}^{2} d s ⩽ \int_{t - h}^{t} {‖ w (s) ‖}^{2} d s \\ = & \int_{[t - h, t] \cap [- h, 0]} {‖ w (s) ‖}^{2} d s + \int_{[t - h, t] ∖ [- h, 0]} {‖ w (s) ‖}^{2} d s \\ ⩽ & {‖ {η_{1} - η_{2}, v^{1} - v^{2}} ‖}_{X_{V}}^{2} e^{C T} h, \end{array}$ thus $\begin{array}{rcl} (3.17) & {‖ w (t) ‖}_{X_{V}} ⩽ C {‖ {η_{1} - η_{2}, v^{1} - v^{2}} ‖}_{X_{V}} . \end{array}$ It follows that the continuous semigroup in $X_{V}$ can also be defined as $\begin{matrix} S (t) {η, v^{0}} = {v_{t}, v (t)} . □ \end{matrix}$

4. Existence of absorbing set

To obtain the existence of global attractor, we need to derive the existence of absorbing set of $S (t)$ in $X_{V}$ , and main results are given as follows.

Lemma 4.1.
Assume $\bar{f} \in H$ , ${η, v^{0}} \in X_{V}$ , and the conditions (A1)∼(A3) hold, then the semigroup $S (t)$ to ( 2.3 ) possesses an absorbing set in $X_{V}$ .
Proof.
Assume ${η, v^{0}} \in D$ which is any bounded set of radius $d > 0$ in $X_{V}$ . Multiplying (2.3) by v, we obtain $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (| v |^{2} + α^{2} ‖ v ‖^{2}) + ν ‖ v ‖^{2} + b (v, v, v) + (R (v), v) = (P \bar{f}, v) + (g (v_{t} + ψ), v), \end{array}$ and $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (| v |^{2} + α^{2} ‖ v ‖^{2}) + ν ‖ v ‖^{2} \\ ⩽ | (R (v), v) | + | (P \bar{f}, v) | + | (g (v_{t} + ψ), v) | \\ ⩽ ‖ \bar{f} ‖ ‖ v ‖ + | g | | v | + C_{1} λ_{1}^{- 1} ‖ v ‖^{2} ‖ ψ ‖ \\ ⩽ C_{1} λ_{1}^{- 1} ‖ v ‖^{2} ‖ ψ ‖ + \frac{ν}{2} ‖ v ‖^{2} + \frac{1}{2 ν} ‖ \bar{f} ‖^{2} + \frac{1}{2 C_{g}} | g |^{2} + \frac{C_{g}}{2} | v |^{2} \\ (4.1) & ⩽ \frac{1}{2 ν} ‖ \bar{f} ‖^{2} + \frac{1}{2 C_{g}} | g |^{2} + (\frac{ν}{2} + C_{1} λ_{1}^{- 1} ‖ ψ ‖ + \frac{1}{2} C_{g} λ_{1}^{- 1}) ‖ v ‖^{2} . \end{array}$ It thus follows that $\begin{array}{l} \frac{d}{d t} (| v |^{2} + α^{2} ‖ v ‖^{2}) \\ (4.2) & ⩽ \frac{1}{ν} ‖ \bar{f} ‖^{2} + \frac{1}{C_{g}} | g |^{2} - (ν λ_{1} - 2 C_{1} ‖ ψ ‖ - C_{g}) | v |^{2}, \end{array}$ and $\begin{array}{rcl} \frac{d}{d t} [e^{m t} (| v |^{2} + α^{2} ‖ v ‖^{2})] & ⩽ & \frac{e^{m t}}{ν} ‖ \bar{f} ‖^{2} + \frac{e^{m t}}{C_{g}} | g |^{2} - e^{m t} (ν λ_{1} - 2 C_{1} C_{2} ‖ ψ ‖_{L^{\infty} (\partial Ω)} \\ (4.3) & - C_{g} - m - m α^{2} λ_{1}) | v |^{2} . \end{array}$ Integrating (4.3) over $[0, t]$ , we obtain $\begin{array}{l} {| v (t) |}^{2} + α^{2} {‖ v (t) ‖}^{2} \\ ⩽ \frac{‖ \bar{f} ‖^{2}}{m ν} + \frac{2 C_{2}^{2} C_{g}}{m λ_{1}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + e^{- m t} ({| v (0) |}^{2} + α^{2} {‖ v (0) ‖}^{2} + 2 C_{g} ‖ η ‖_{L_{H}^{2}}^{2}) \\ (4.4) & ⩽ \frac{‖ \bar{f} ‖^{2}}{m ν} + \frac{2 C_{2}^{2} C_{g}}{m λ_{1}} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} + C d^{2} e^{- m t} \end{array}$ and for any $t > 0$ , $θ \in [- h, 0]$ , $\begin{matrix} {| v (t + θ) |}^{2} + α^{2} {‖ v (t + θ) ‖}^{2} - (\frac{‖ \bar{f} ‖^{2}}{m ν} + \frac{2 C^{2} C_{g}}{m} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) ⩽ e^{- m t} e^{m h} C d^{2} . \end{matrix}$ Thus there exist constants $ρ_{V}$ and $T_{V}$ such that for any ${η, v^{0}} \in D$ , $\begin{matrix} ‖ v ‖_{C_{V}}^{2} ⩽ ρ_{V}^{2}, for all t ⩾ T_{V}, \end{matrix}$ which means $B_{V} (0, ρ_{V})$ is an absorbing set for $S (t)$ in $C_{V}$ .
Remark 4.1.
Using the technique in (3.17), we derive $\begin{matrix} ‖ v_{t} ‖_{L_{V}^{2}}^{2} ⩽ C ρ_{V}^{2}, {‖ {v_{t}, v (t)} ‖}_{X_{V}}^{2} ⩽ C ρ_{V}^{2}, \end{matrix}$ and hence the existence of absorbing set in $X_{V}$ follows. □

5. Some compactness and the existence of attractors

To prove our results, we also need some compactness of $S (t)$ in $X_{V}$ . Note that $v = S (t) {η, v^{0}}$ is the solution to (2.3) with the initial datum ${η, v^{0}} \in X_{V}$ , and $S (t)$ can be decomposed as follows $\begin{matrix} S (t) {η, v^{0}} = Y (t) {η, v^{0}} + Z (t) {η, v^{0}}, \end{matrix}$ where $Y (t)$ is the semigroup generated by the following system $\begin{matrix} (5.1) & \{\begin{matrix} \frac{d y}{d t} + ν A y + α^{2} \frac{d}{d t} A y = 0, \\ y (0) = v^{0}, x \in Ω, \\ y_{0} = η, x \in Ω, \end{matrix} \end{matrix}$ and $Z (t) {η, v^{0}}$ is the solution to the problem $\begin{matrix} (5.2) & \{\begin{matrix} \frac{d z}{d t} + ν A z + α^{2} \frac{d}{d t} A z = P \bar{f} + g (v_{t} + ψ) - B (v) - R (v), \\ z (0) = 0, x \in Ω, \\ z_{0} = 0, x \in Ω . \end{matrix} \end{matrix}$

Multiplying (5.1) by $y (t)$ , and using the Galerkin method, we can easily derive that for $t > 0$ $\begin{array}{l} {| y (t) |}^{2} + α^{2} {‖ y (t) ‖}^{2} \\ (5.3) & ⩽ e^{C t} ({| v (0) |}^{2} + α^{2} {‖ v (0) ‖}^{2}) ⩽ e^{C t} {‖ {η, v^{0}} ‖}_{X_{V}}^{2} . \end{array}$ Using Hölder’s inequality and the Gagliardo–Nirenberg inequality, we also conclude for any $u \in V$ $\begin{array}{rcl} {‖ B (v) ‖}_{- 2 / 3} & = & sup_{‖ u ‖_{2 / 3} = 1} \int_{Ω} (v \cdot \nabla) v \cdot u d x ⩽ sup_{‖ u ‖_{2 / 3} = 1} ‖ v ‖_{L^{3}} ‖ v ‖ ‖ u ‖_{L^{6}} \\ ⩽ & C sup_{‖ u ‖_{2 / 3} = 1} ‖ v ‖^{2} ‖ u ‖_{2 / 3} ⩽ C ‖ v ‖^{2}, \end{array}$ which gives $\begin{array}{rcl} (5.4) & {‖ B (v) ‖}_{- 2 / 3} \in L^{\infty} (0, T; V_{- 2 / 3}) . \end{array}$ In a similar way, we can derive $\begin{array}{rcl} (5.5) & {‖ R (v) ‖}_{- 2 / 3} \in L^{\infty} (0, T; V_{- 2 / 3}) . \end{array}$

Multiplying (5.2) by $A^{1 / 3} z$ , using (A2), (5.4), (5.5) and the Gronwall inequality, we can get $\begin{array}{rcl} {| y (t) |}_{1 / 3}^{2} + α^{2} {‖ y (t) ‖}_{4 / 3}^{2} & ⩽ & C + C \int_{0}^{T} {| g (v_{t} + ψ) |}^{2} d t \\ ⩽ & C + C \int_{- h}^{T} {| v (t) |}^{2} d t \\ (5.6) & ⩽ & C + C ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2} h + C \int_{0}^{T} {| v (t) |}^{2} d t, \end{array}$ which means the operator $Z (t)$ maps $X_{V}$ into $X_{V_{4 / 3}}$ , and from the fact that $V_{4 / 3} ↪ ↪ V$ , we derive the operator $Z (t)$ is compact for $t > 0$ . Using the decomposition method (see [16,27]), from (5.3) and the compactness of $Z (t)$ , we derive that $S (t)$ is asymptotically compact in $X_{V}$ .

In Section 3, we have established the existence of semigroup ${S (t)}$ to (2.3) in $X_{V}$ , and the existence of absorbing set $B_{V} (0, ρ_{V})$ in $X_{V}$ . Now the asymptotical compactness of ${S (t)}$ in $X_{V}$ is also established in this section, from the fundamental theory of global attractor [1,9,24,27], the existence of global attractor $A$ of ${S (t)}$ in $X_{V}$ follows naturally under the conditions in Lemma 4.1.

6. Upper estimates on dimensions of global attractors

In this section, we aim to get the upper boundedness of the fractal dimension of global attractor. To do so, we need first to show the differentiability of $S (t)$ with respect to the initial datum. Consider now the first variation equation of (2.3) $\begin{matrix} (6.1) & \{\begin{matrix} \frac{d U}{d t} + ν A U + α^{2} \frac{d}{d t} A U + B (v, U) + R (U, v) + R (U) = g (v_{t} + ψ) U_{t}, \\ {(U_{0}, U (0))} \in X_{V} . \end{matrix} \end{matrix}$ Using the usual energy estimate shown as in Lemma 3.1, we can derive the existence and uniqueness of solution to (6.1) which satisfies $\begin{matrix} U (t) \in L^{\infty} ([0, T]; V) \cap L^{2} ([- h, T]; V), U_{t} \in ([0, T]; V), \end{matrix}$ and the linear operator $Λ (t, {v_{0}, v (0)}) : X_{V} \to X_{V}$ can be defined as $\begin{matrix} Λ (t, {v_{0}, v (0)}) {U_{0}, U (0)} = {U_{t}, U (t)} . \end{matrix}$

Theorem 6.1.

Let $A$ be the global attractor of $S (t)$ to ( 2.3 ) in $X_{V}$ , then $S (t)$ is uniformly differentiable on $A$ , that is, for any ${η_{2}, v^{2}} \in A$ , there exists a linear operator $Λ (t, {η_{2}, v^{2}})$ such that for all $t ⩾ 0$ , $\begin{matrix} sup_{{η_{1}, v^{1}}, {η_{2}, v^{2}} \in A} \frac{‖ S (t) {η_{1}, v^{1}} - S (t) {η_{2}, v^{2}} - Λ (t, {η_{2}, v^{2}}) {η_{1} - η_{2}, v^{1} - v^{2}} ‖_{X_{V}}}{‖ {η_{1} - η_{2}, v^{1} - v^{2}} ‖_{X_{V}}} \to 0 \end{matrix}$ as $‖ {η_{1} - η_{2}, v^{1} - v^{2}} ‖_{X_{V}} ⩽ ε \to 0$ , and $\begin{matrix} sup_{{η_{2}, v^{2}} \in A} {‖ Λ (t, {η_{2}, v^{2}}) ‖}_{L (X_{V})} < + \infty, for all t ⩾ 0 . \end{matrix}$

Proof.

Assume that $v_{1} (t)$ and $v_{2} (t)$ are solutions to (2.3) with initial data ${η_{1}, v^{1}}$ and ${η_{2}, v^{2}}$ , $U (t) = Λ (t, {η_{2}, v^{2}}) {η_{1} - η_{2}, v^{1} - v^{2}}$ is the solution to (6.1) with initial datum ${η_{1} - η_{2}, v^{1} - v^{2}}$ . Let $w = v_{1} - v_{2}$ , then $θ = v_{1} - v_{2} - U = w - U$ satisfies $\begin{array}{l} \frac{d}{d t} (θ + α^{2} A θ) + ν A θ + B (v_{2}, θ) + B (w, w) + B (θ, v_{2}) + R (θ) \\ (6.2) & = g (v_{1 t}) - g (v_{2 t}) - g^{'} (v_{2 t}) U_{t}, \end{array}$ where $\begin{array}{l} g (v_{1 t} + ψ) - g (v_{2 t} + ψ) - g^{'} (v_{2 t} + ψ) U_{t} \\ = g (v_{1 t} + ψ) - g (v_{2 t} + ψ) - g^{'} (v_{2 t} + ψ) (v_{1 t} - v_{2 t}) + g^{'} (v_{2 t} + ψ) θ_{t} \\ = g^{″} (*) {(v_{1 t} - v_{2 t})}^{2} + g^{'} (v_{2 t} + ψ) θ_{t}, \end{array}$ with $* = p v_{1 t} + (1 - p) v_{2 t}$ , $0 ⩽ p ⩽ 1$ . Multiplying (6.2) by θ, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (| θ |^{2} + α^{2} ‖ θ ‖^{2}) + ν ‖ θ ‖^{2} + b (w, w, θ) + b (θ, v_{2}, θ) + (R (θ), θ) \\ = (g^{″} (*) {(v_{1 t} - v_{2 t})}^{2} + g^{'} (v_{2 t} + ψ) θ_{t}, θ), \end{array}$ and $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (| θ |^{2} + α^{2} ‖ θ ‖^{2}) + ν ‖ θ ‖^{2} \\ ⩽ | b (w, w, θ) | + b (θ, v_{2}, θ) + | (R (θ), θ) | + L | θ_{t} | | θ | + L | w_{t} |^{2} | θ | \\ ⩽ C (‖ w ‖^{4} + ‖ w_{t} ‖^{4}) + C (‖ θ ‖^{2} + ‖ θ_{t} ‖^{2}) + ν ‖ θ ‖^{2} . \end{array}$ Thus it follows from (3.16) that $\begin{matrix} \frac{d}{d t} ‖ θ ‖^{2} ⩽ {‖ {η_{1} - η_{2}, v^{1} - v^{2}} ‖}_{X_{V}}^{4} e^{K T} + C (‖ θ ‖^{2} + ‖ θ_{t} ‖^{2}) . \end{matrix}$ Using Gronwall’s inequality, we know $\begin{matrix} {‖ θ (t) ‖}^{2} ⩽ {‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖}_{X_{V}}^{4} T e^{K T} + C \int_{0}^{t} ({‖ θ (s) ‖}^{2} + ‖ θ_{s} ‖^{2}) d s, \end{matrix}$ and $\begin{matrix} sup_{s \in [t - h, t]} {‖ θ (s) ‖}^{2} ⩽ {‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖}_{X_{V}}^{4} T e^{K T} + C \int_{0}^{t} sup_{r \in [s - h, s]} {‖ θ (r) ‖}^{2} d s . \end{matrix}$ Thus $\begin{matrix} sup_{s \in [t - h, t]} {‖ θ (s) ‖}^{2} ⩽ {‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖}_{X_{V}}^{4} T e^{K T} (1 + e^{K T}), \end{matrix}$ which means $\begin{array}{rcl} (6.3) & {‖ θ (t) ‖}^{2} ⩽ C {‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖}_{X_{V}}^{4} . \end{array}$ Furthermore, we also get $\begin{array}{l} ‖ θ_{t} ‖_{L_{V}^{2}}^{2} & = \int_{- h}^{0} {‖ θ (t + s) ‖}^{2} d s ⩽ \int_{t - h}^{t} {‖ θ (s) ‖}^{2} d s \\ = \int_{[t - h, t] \cap [- h, 0]} {‖ θ (s) ‖}^{2} d s + \int_{[t - h, t] ∖ [- h, 0]} {‖ θ (s) ‖}^{2} d s \\ (6.4) & ⩽ 0 + \int_{0}^{t} {‖ θ (s) ‖}^{2} d s ⩽ C T {‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖}_{X_{V}}^{4}, \end{array}$ which, together with (6.3), leads to $\begin{array}{rcl} (6.5) & {‖ θ (t) ‖}_{X_{V}} ⩽ C {‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖}_{X_{V}}^{2} . \end{array}$ From the expression of θ, it follows that $\begin{array}{l} \frac{‖ S (t) {η_{1}, v^{1}} - S (t) {η_{2}, v^{2}} - Λ (t, {η_{2}, v^{2}}) {η_{1} - η_{2}, v^{1} - v^{2}} ‖_{X_{V}}}{‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖_{X_{V}}} \\ ⩽ C {‖ (η_{1}, v^{1}) - (η_{2}, v^{2}) ‖}_{X_{V}}, \end{array}$ which implies that the differentiability of $S (t)$ with respect to initial datum follows naturally and $S (t)$ is uniformly differentiable on $A$ since we have shown that the linear operator $Λ (t, {η_{2}, v^{2}})$ is bounded.

Finally, we shall establish upper boundedness of the fractal dimension of $A$ in $X_{V}$ . To do so, we rewrite (6.1) as follows $\begin{array}{rcl} \frac{d \overline{U}}{d t} & = & - \frac{ν}{α^{2}} \overline{U} + \frac{ν}{α^{2}} G^{- 2} \overline{U} - G^{- 1} B (v, G^{- 1} \overline{U}) \\ - G^{- 1} B (G^{- 1} \overline{U}, v) - G^{- 1} R (G^{- 1} \overline{U}) + G^{- 2} g^{'} (v_{t}) {\overline{U}}_{t}, \end{array}$ where $G = {(I + α^{2} A)}^{1 / 2}$ , and $\overline{U} = G U$ . The corresponding form is written as $\begin{array}{rcl} (6.6) & \frac{d}{d t} \overline{U} = L (t) \overline{U} + G^{- 2} g^{'} (v_{t} + ψ) {\overline{U}}_{t}, \end{array}$ where $\begin{matrix} L (t) * = - \frac{ν}{α^{2}} * + \frac{ν}{α^{2}} G^{- 2} * - G^{- 1} B (v, G^{- 1} *) - G^{- 1} B (G^{- 1} *, v) - G^{- 1} R (G^{- 1} *) . \end{matrix}$

Define $\overline{L} : D (\overline{L}) \subset X_{V} \to X_{V}$ by $\begin{array}{l} \begin{matrix} D (\overline{L}) & = {{α, β} \in X_{V} : α is absolutely continuous on [- h, 0], \\ \dot{α} \in L^{2} ([- h, 0]; V) and β = α (0) \in D (L)}, \end{matrix} \\ \overline{L} {α, β} = {\dot{α}, L β}, {α, β} \in D (\overline{L}), \end{array}$ and $D (\overline{L})$ is dense in $X_{V}$ . Let $W = {{\overline{U}}_{t}, \overline{U} (t)}$ , (6.6) is reformulated as an evolution system on $X_{V}$ $\begin{matrix} (6.7) & \{\begin{matrix} \frac{d}{d t} W = R (t) W, \\ W (0) \in X_{V}, \end{matrix} \end{matrix}$ where $R (t) : X_{V} \to X_{V}$ is defined as $\begin{matrix} R (t) W = \overline{L} (t) W + {0, G^{- 2} g^{'} (v_{t} + ψ) P_{1} W} . \end{matrix}$

From (6.6)–(6.7), we can extend the generator L (in V) to $\overline{L}$ (in $X_{V}$ ) naturally so that the variational equation can be reformulated as an evolution system on $X_{V}$ . Only in this way, can we apply directly the method in [27] to establish the upper boundedness. This idea comes from papers of Webb [28,29] in which they constructed a generator for a nonlinear semigroup and we also find that $\begin{matrix} u_{t} = P_{1} W (t) if u (t) = P_{2} W (t) . \end{matrix}$ □

Theorem 6.2.

Under the assumptions in Lemma 3.1 , the Hausdorff and fractal dimensions of $A$ to ( 2.3 ) are less than or equal to $\begin{matrix} \frac{C_{3}^{1 / 2} α λ_{1}^{1 / 2} | Ω |^{1 / 2}}{{(2 π ν)}^{1 / 2} {(\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{2 l_{0}}{λ_{1}})}^{1 / 2}} {(\frac{\frac{‖ \bar{f} ‖^{2}}{ν} + \frac{2 C_{2}^{2}}{λ_{1}} C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - 3 C_{g} λ_{1}^{- 1}} + C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2})}^{1 / 2} + 1 . \end{matrix}$

Proof.

For any ${U_{j 0}, U_{j} (0)}_{j = 1, 2, \dots, m} \in X_{V}$ , let ${U_{j t}, U_{j} (t)} = Λ (t, {v_{0}, v (0)}) {U_{j 0}, U_{j} (0)}$ , and $Q_{m} (τ)$ and $Q_{m}^{'} (τ)$ denote the projectors of $X_{H}$ to the spans ${{U_{j τ}, U_{j} (τ)}_{j = 1, 2, \dots, m}}$ and ${{G U_{j τ}, G U_{j} (τ)}_{j = 1, 2, \dots, m}}$ respectively. Let ${φ_{j τ}, φ_{j} (τ)}_{j = 1, 2, \dots, m}$ be an orthonormal basis for span ${U_{j τ}, U_{j} (τ)}_{j = 1, 2, \dots, m}$ . Since ${U_{j τ}, U_{j} (τ)} \in X_{V}$ , ${φ_{j τ}, φ_{j} (τ)} \in X_{V}$ , we conclude $\begin{array}{l} Tr (R (τ) \cdot Q_{m}^{'} (τ)) \\ = \sum_{j = 1}^{m} ⟨ G (τ) {R φ_{j τ}, G φ_{j} (τ)}, {G φ_{j τ}, G φ_{j} (τ)} ⟩ \\ = \sum_{j = 1}^{m} ⟨ \overline{L} {G φ_{j τ}, G φ_{j} (τ)} + {0, G^{- 2} g^{'} (v_{t} + ψ) G φ_{j τ}}, {G φ_{j τ}, G φ_{j} (τ)} ⟩ \\ = \sum_{j = 1}^{m} ⟨ {\dot{G φ_{j τ}}, L G φ_{j} (τ)} + {0, G^{- 2} g^{'} (v_{t} + ψ) G φ_{j τ}}, {G φ_{j τ}, G φ_{j} (τ)} ⟩, \end{array}$ by (6.6) and (A1) we obtain $\begin{array}{l} ⟨ {\dot{G φ_{j τ}}, L G φ_{j} (τ)} + {0, G^{- 2} g^{'} (v_{t} + ψ) G φ_{j τ}}, {G φ_{j τ}, G φ_{j} (τ)} ⟩ \\ = \int_{- h}^{0} (\frac{d}{d s} G φ_{j} (τ + s), G φ_{j} (τ + s)) d s + (L G φ_{j} (τ), G φ_{j} (τ)) \\ + (G^{- 2} g^{'} (v_{t} + ψ) G φ_{j τ}, G φ_{j} (τ)) \\ = {| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2} - \frac{ν}{α^{2}} {| G φ_{j} (τ) |}^{2} + \frac{ν}{α^{2}} {| φ_{j} (τ) |}^{2} \\ - b (φ_{j} (τ), v, φ_{j} (τ)) - (R (φ_{j} (τ)), φ_{j} (τ)) + (g^{'} (v_{t} + ψ) φ_{j τ}, φ_{j} (τ)) \\ ⩽ {| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2} - \frac{ν}{α} {‖ φ_{j} (τ) ‖}^{2} + \frac{ν}{α^{2}} {| φ_{j} (τ) |}^{2} \\ + | b (φ_{j} (τ), v, φ_{j} (τ)) | + | b (φ_{j} (τ), ψ, φ_{j} (τ)) | + l_{0} | φ_{j τ} | | φ_{j} (τ) | \\ ⩽ {| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2} - \frac{ν}{α} {‖ φ_{j} (τ) ‖}^{2} + \frac{ν}{α^{2} λ_{1}} {‖ φ_{j} (τ) ‖}^{2} \\ + | b (φ_{j} (τ), v, φ_{j} (τ)) | + | b (φ_{j} (τ), ψ, φ_{j} (τ)) | + \frac{l_{0}}{λ_{1}} ‖ φ_{j τ} ‖^{2} + \frac{l_{0}}{λ_{1}} {‖ φ_{j} (τ) ‖}^{2} \\ ⩽ {| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2} - (\frac{ν}{α} - \frac{ν}{α^{2} λ_{1}} - \frac{l_{0}}{λ_{1}}) {‖ φ_{j} (τ) ‖}^{2} \\ + | b (φ_{j} (τ), v, φ_{j} (τ)) | + | b (φ_{j} (τ), ψ, φ_{j} (τ)) | + \frac{l_{0}}{λ_{1}} ‖ φ_{j τ} ‖^{2}, \end{array}$ and $\begin{array}{l} Tr (R (τ) \cdot Q_{m}^{'} (τ)) \\ ⩽ - (\frac{ν}{α} - \frac{ν}{α^{2} λ_{1}} - \frac{l_{0}}{λ_{1}}) \sum_{j = 1}^{m} {‖ φ_{j} (τ) ‖}^{2} + \frac{l_{0}}{λ_{1}} \sum_{j = 1}^{m} ‖ φ_{j τ} ‖^{2} \\ + \sum_{j = 1}^{m} ({| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2}) + \sum_{j = 1}^{m} | b (φ_{j} (τ), v, φ_{j} (τ)) | \\ (6.8) & + \sum_{j = 1}^{m} | b (φ_{j} (τ), ψ, φ_{j} (τ)) | . \end{array}$ From the Lieb–Thirring inequality $\begin{matrix} {| \sum_{j = 1}^{m} {| φ_{j} (τ) |}^{2} |}^{2} ⩽ C_{3} \sum_{j = 1}^{m} {‖ φ_{j} (τ) ‖}^{2}, \end{matrix}$ we know the last two terms in (6.8) are both bounded and $\begin{array}{l} \sum_{j = 1}^{m} | b (φ_{j} (τ), v, φ_{j} (τ)) | \\ ⩽ | \sum_{j = 1}^{m} {| φ_{j} (τ) |}^{2} | \cdot ‖ v ‖ ⩽ \sqrt{C_{3}} {(\sum_{j = 1}^{m} {‖ φ_{j} (τ) ‖}^{2})}^{1 / 2} \cdot ‖ v ‖ \\ (6.9) & ⩽ \frac{ν}{2 α^{2} λ_{1}} \sum_{j = 1}^{m} {‖ φ_{j} (τ) ‖}^{2} + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ v ‖^{2}, \\ (6.10) & \sum_{j = 1}^{m} | b (φ_{j} (τ), ψ, φ_{j} (τ)) | ⩽ \frac{ν}{2 α^{2} λ_{1}} \sum_{j = 1}^{m} {‖ φ_{j} (τ) ‖}^{2} + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ ψ ‖^{2} . \end{array}$ Using (6.9), (6.10) and the variational principle in (6.8), we get $\begin{array}{l} Tr (R (τ) \cdot Q_{m}^{'} (τ)) \\ ⩽ - (\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{l_{0}}{λ_{1}}) \sum_{j = 1}^{m} {‖ φ_{j} (τ) ‖}^{2} + \frac{l_{0}}{λ_{1}} \sum_{j = 1}^{m} ‖ φ_{j τ} ‖^{2} \\ + \sum_{j = 1}^{m} ({| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2}) + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ ψ ‖^{2} + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ v ‖^{2} \\ ⩽ - (\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{l_{0}}{λ_{1}}) \sum_{j = 1}^{m} λ_{j} + \frac{l_{0}}{λ_{1}} \sum_{j = 1}^{m} ‖ φ_{j τ} ‖^{2} \\ + \sum_{j = 1}^{m} ({| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2}) + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ ψ ‖^{2} + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ v ‖^{2} \\ ⩽ - (\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{l_{0}}{λ_{1}}) \frac{π m^{2}}{| Ω |} + \frac{l_{0}}{λ_{1}} \sum_{j = 1}^{m} ‖ φ_{j τ} ‖^{2} \\ (6.11) & + \sum_{j = 1}^{m} ({| G φ_{j} (τ) |}^{2} - {| G φ_{j} (τ - h) |}^{2}) + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ ψ ‖^{2} + \frac{C_{3} α^{2} λ_{1}}{2 ν} ‖ v ‖^{2} . \end{array}$ Noting that $\begin{array}{rcl} q_{m} (t) & = & sup_{v \in A} sup_{{U_{0}, U (0)} \in X_{H}, j = 1 \dots m} {\frac{1}{t} \int_{0}^{t} Tr (R (τ) \cdot Q_{m}^{'} (τ)) d τ} \\ ⩽ & - (\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{2 l_{0}}{λ_{1}}) \frac{π m^{2}}{| Ω |} + \frac{C_{3} α^{2} λ_{1}}{2 ν} sup_{v \in A} \frac{1}{t} \int_{0}^{t} (‖ ψ ‖^{2} + ‖ v ‖^{2}) d τ, \end{array}$ then it follows from (3.10) that $\begin{array}{rcl} q_{m} & \equiv & Lim sup_{t \to \infty} q_{m} (t) \\ ⩽ & - (\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{2 l_{0}}{λ_{1}}) \frac{π m^{2}}{| Ω |} + \frac{C_{3} α^{2} λ_{1}}{2 ν} sup_{v \in A} \frac{1}{t} \int_{0}^{t} (‖ ψ ‖^{2} + ‖ v ‖^{2}) d τ \\ ⩽ & - (\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{2 l_{0}}{λ_{1}}) \frac{π m^{2}}{| Ω |} \\ + \frac{C_{3} α^{2} λ_{1}}{2 ν} (\frac{\frac{‖ \bar{f} ‖^{2}}{ν} + \frac{2 C_{2}^{2}}{λ_{1}} C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - 3 C_{g} λ_{1}^{- 1}} + ‖ ψ ‖^{2}) \\ ⩽ & - (\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{2 l_{0}}{λ_{1}}) \frac{π m^{2}}{| Ω |} \\ (6.12) & + \frac{C_{3} α^{2} λ_{1}}{2 ν} (\frac{\frac{‖ \bar{f} ‖^{2}}{ν} + \frac{2 C_{2}^{2}}{λ_{1}} C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - 3 C_{g} λ_{1}^{- 1}} + C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}) . \end{array}$ Using the assumption

$\begin{matrix} \frac{ν}{α} > \frac{2 ν}{α^{2} λ_{1}} + \frac{2 l_{0}}{λ_{1}}, \end{matrix}$

and Theorem V 3.3 in [27], from (6.12) we derive the upper boundedness of the fractal dimension of

A

as follows

\begin{matrix} \frac{C_{3}^{1 / 2} α λ_{1}^{1 / 2} | Ω |^{1 / 2}}{{(2 π ν)}^{1 / 2} {(\frac{ν}{α} - \frac{2 ν}{α^{2} λ_{1}} - \frac{2 l_{0}}{λ_{1}})}^{1 / 2}} {(\frac{\frac{‖ \bar{f} ‖^{2}}{ν} + \frac{2 C_{2}^{2}}{λ_{1}} C_{g} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2}}{ν - 2 C_{1} C_{2} λ_{1}^{- 1} ‖ φ ‖_{L^{\infty} (\partial Ω)} - 3 C_{g} λ_{1}^{- 1}} + C_{2} ‖ φ ‖_{L^{\infty} (\partial Ω)}^{2})}^{1 / 2} + 1 . \end{matrix}

□

Footnotes

Acknowledgements

This paper is partly supported by NSFC of China (Grant number 11671075).

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