Global attractor of atmospheric equations

Abstract

We consider the three-dimensional primitive equations of the atmosphere with humidity and saturation associated with suitable boundary conditions. These equations have been shown to lead to well-posed problem producing a continuous semigroup despite the discontinuities due to the change of phase between vapor water and liquid water. In this article, we aim to show the existence of a global attractor for a model problem.

Keywords

primitive equations global attractor dynamical system phase change variational inequality

1. Introduction

Geophysicists and mathematicians are interested in the understanding of global atmospheric flows and atmospheric models. It is well-known that the primitive equations are used to represent long-term weather prediction and climate changes. In [26,27], the mathematical framework of the primitive equations was formulated, and the existence of weak solutions was proved. After the pioneering research in [26,27], there was an abundant literature on the topics of the primitive equations and its applications. Concerning the issue of theoretical problems, see e.g. [5,20,22–24] for the well-posedness among many other articles. See also [4,7–9] concerning numerical analysis and computational problems.

Studies of the global attractor of a given dynamical system are a challenging and interesting topic in fluid dynamics since these can provide useful information on the future evolution of a system. For many dissipative equations including the primitive equations, the existence of a global attractor has been proved. In [6,21,28], the authors studied the primitive equations without considering the saturation of water vapor. In [14,35], the authors focus on the existence of the global attractor of some atmospheric equations such as the Boussinesq equations.

In this article, we consider the three-dimensional primitive equations of the atmosphere with humidity and saturation. The model problems were present in the physical context in e.g. [16,18,19,29,31]. Mathematical issues of existence, uniqueness and regularity of solutions are addressed in [3,11,12]. These results motivated us to conduct our present work. We consider a slightly simplified version of the problem and which keeps many of its mathematical features. In spite of the discontinuities due to the change of phase, we can construct a continuous semigroup corresponding to the initial and boundary value problem. Deriving global in time a priori estimates, we aim to prove the existence of a global attractor for these humidity equations.

This article is organized as follows. In Section 2, we introduce the 3D humidity equations and propose some physically suitable boundary conditions. Considering the variational inequality, we define the weak formulations and recall the well-posedness results essentially borrowed from [11,12]. In Section 3, we present the main theorem, the existence of the global attractor. For the proof, we look for an absorbing ball in $L^{2}$ and $H^{1}$ . We then prove the main theorem using the classical and abstract theorems in [33].

2. Mathematical setting of the primitive equations

2.1. Primitive equations

We consider the three-dimensional primitive equations of the atmosphere with humidity and saturation as in [3,4,11,12] $\begin{matrix} (2.1) & \begin{matrix} c_{p} \partial_{t} T + A_{T} T + c_{p} v \cdot \nabla T + c_{p} ω \partial_{p} T - \frac{R}{p} ω \overline{T} \\ \in \frac{L}{p} ω^{-} H (q - q_{s}) F^{+} (T) in M \times (0, \infty), \\ \partial_{t} q + A_{q} q + v \cdot \nabla q + ω \partial_{p} q \\ \in - \frac{1}{p} ω^{-} H (q - q_{s}) F^{+} (T) in M \times (0, \infty), \end{matrix} \end{matrix}$ where $M = M^{'} \times (p_{0}, p_{1})$ is a cylindrical domain and $M^{'}$ is a smooth and bounded set in $R^{2}$ . Traditionally, Eq. (2.1) are written in the $(x, y, p, t)$ coordinate system, where pressure p plays the role of the vertical coordinate. We set

$\nabla = (\partial_{x}, \partial_{y})$ is the 2D horizontal gradient.

$Δ = \partial_{x}^{2} + \partial_{y}^{2}$ is the 2D Laplacian operator.

$A_{T} = - μ_{T} Δ - ν_{T} \partial_{p} [{(\frac{g p}{R \overline{T}})}^{2} \partial_{p}]$ , diffusion operator for T.

$A_{q} = - μ_{q} Δ - ν_{q} \partial_{p} [{(\frac{g p}{R \overline{T}})}^{2} \partial_{p}]$ , diffusion operator for q.

$u = (v, ω)$ is the given velocity of the fluid and v, ω are the horizontal and vertical velocities, respectively. ω expressed in the x, y, p system.

$\overline{T} = \overline{T} (p)$ is the average temperature over the isobar with pressure p.

$F (T)$ is given by Eq. (9.13) in [18]: $\begin{matrix} (2.2) & F (T) = q_{s} T (\frac{L R - C_{p} R_{v} T}{C_{p} R_{v} T^{2} + q_{s} L^{2} (T)}), \end{matrix}$ which is a globally Lipschitz bounded function.

$q_{s}$ is the saturation specific humidity as in [18]. It is a slightly varying function of the temperature. We assume here that it is constant, which is sufficient for our purpose.

H is the Heaviside function $H (x) = \frac{1}{2} (1 + sign (x))$ with $sign (0) = [0, 1]$ .

$L (T) = 2.5008 \times 10^{6}$ is the latent heat of vaporization.

$R = 287 {J K}^{- 1} {kg}^{- 1}$ is the gas constant for dry air (p. 597 in [16]).

$R_{v} = 461.50 {J K}^{- 1} {kg}^{- 1}$ is the gas constant for water vapor (p. 597 in [16]).

$c_{p} = 1004 {J K}^{- 1} {kg}^{- 1}$ is the specific heat of dry air at constant pressure (p. 475 in [18]).

We also define the positive part and negative part of the function by setting

\begin{matrix} (2.3) & f^{+} = max {f, 0}, f^{-} = max {- f, 0} . \end{matrix}

We now define the boundary of

M

Γ_{i} \cup Γ_{u} \cup Γ_{l}

, with

\begin{matrix} (2.4) & \begin{matrix} Γ_{i} = {(x, y, p) \in \overline{M} : p = p_{1}}, \\ Γ_{u} = {(x, y, p) \in \overline{M} : p = p_{0}}, \\ Γ_{l} = {(x, y, p) \in \overline{M} : (x, y) \in \partial M, p_{0} ⩽ p ⩽ p_{1}}, \end{matrix} \end{matrix}

and impose the following physically relevant boundary conditions as in [11] and [26]: On

Γ_{i}

(on the bottom of the atmosphere), we propose the Robin boundary conditions

\begin{matrix} (2.5) & \partial_{p} T = α_{T} (T_{*} - T), \partial_{p} q = α_{q} (q_{*} - q), \end{matrix}

where

T_{*}

and

q_{*}

are typical temperature and specific humidity distributions, and

α_{T}

and

α_{q}

are positive constants. For the average temperature

\overline{T} (p)

, we assume the existence of lower and upper bounds

{\overline{T}}_{*}

and

{\overline{T}}^{*}

such that

\begin{matrix} (2.6) & 0 < {\overline{T}}_{*} ⩽ \overline{T} (p) ⩽ {\overline{T}}^{*}, \end{matrix}

and that its derivative is uniformly bounded that is

\begin{matrix} (2.7) & | \partial_{p} \overline{T} (p) | ⩽ C \end{matrix}

for some constant

C > 0

. On

Γ_{l}

(lateral boundary) and

Γ_{u}

(on the top of the atmosphere), we impose homogeneous Neumann boundary conditions

\begin{matrix} (2.8) & \begin{matrix} on Γ_{l} : \frac{\partial T}{\partial n} = 0, \frac{\partial q}{\partial n} = 0, \\ on Γ_{u} : \frac{\partial T}{\partial p} = 0, \frac{\partial q}{\partial p} = 0, \end{matrix} \end{matrix}

where n is the unit outward normal vector on

Γ_{l}

. The initial conditions for T and q are given by

\begin{matrix} (2.9) & T (x, y, p, 0) = T_{0} (x, y, p), q (x, y, p, 0) = q_{0} (x, y, p), where (x, y, p) \in M . \end{matrix}

Remark 2.1.
To guarantee the dissipation, we introduced a modified version of the humidity equations studied in [3,4,11] introducing the dissipation operators $A_{T}$ and $A_{q}$ . In (2.1)₁, we replaced (approximated) $\frac{R}{p} ω T$ by $\frac{R}{p} ω \overline{T}$ . This approximation has been introduced in [26].

2.2. Function spaces and weak formulations

We define $H = L^{2} (M)$ with scalar product $(\cdot, \cdot)$ and norm $| \cdot |$ . Regarding the proposed boundary conditions (2.5) and (2.8), we define $V = H^{1} (M)$ with the scalar product $\begin{matrix} (2.10) & ((φ, \tilde{φ})) = (\nabla φ, \nabla \tilde{φ}) + (\partial_{p} φ, \partial_{p} \tilde{φ}) + \int_{Γ_{i}} φ \tilde{φ} d Γ_{i} . \end{matrix}$ We note that the scalar product in (2.10) is equivalent to the standard $H^{1}$ -inner product thanks to the trace theorem and the Poincaré inequality. We also denote the corresponding norm by $\begin{matrix} (2.11) & ∥ φ ∥^{2} = ((φ, φ)) . \end{matrix}$ For the velocity vector field, we consider the following Hilbert spaces H and V which are usually used in the Navier–Stokes equations context; see e.g. [34] $\begin{matrix} (2.12) & \begin{matrix} H = {u \in {(L^{2} (M))}^{3}, div u = 0, u \cdot n = 0 on \partial M}, \\ V = {u \in {(H^{1} (M))}^{3}, div u = 0, u \cdot n = 0 on \partial M} . \end{matrix} \end{matrix}$ We now look for the weak formulation of (2.1) with boundary conditions (2.5) and (2.8). By integration by parts, if $T, T^{b} \in V$ are smooth, we deduce that $\begin{array}{rcl} ⟨ A_{T} T, T^{b} ⟩ & = & μ_{T} (\nabla T, \nabla T^{b}) + ν_{T} \int_{M} {(\frac{g p}{R \overline{T}})}^{2} \partial_{p} T \partial_{p} T^{b} d M \\ (2.13) & + ν_{T} \int_{Γ_{i}} {(\frac{g p_{1}}{R \overline{T}})}^{2} α_{T} (T - T_{*}) T^{b} d Γ_{i}, \end{array}$ where $⟨ \cdot, \cdot ⟩$ is the duality pairing. Similarly, if $q, q^{b} \in V$ are smooth, we derive that $\begin{array}{rcl} ⟨ A_{q} q, q^{b} ⟩ & = & μ_{q} (\nabla q, \nabla q^{b}) + ν_{q} \int_{M} {(\frac{g p}{R \overline{T}})}^{2} \partial_{p} q \partial_{p} q^{b} d M \\ (2.14) & + ν_{q} \int_{Γ_{i}} {(\frac{g p_{1}}{R \overline{T}})}^{2} α_{q} (q - q_{*}) q^{b} d Γ_{i} . \end{array}$ For $φ, \tilde{φ} \in V$ and $u \in V$ , we set the following operators: $\begin{matrix} (2.15) & \begin{matrix} b (u, φ, \tilde{φ}) = \int_{M} (v \cdot \nabla φ + ω \partial_{p} φ) \tilde{φ} d M, \\ d (ω, φ, \tilde{φ}) = \int_{M} \frac{R}{p} ω φ \tilde{φ} d M, \end{matrix} \end{matrix}$ and $\begin{array}{l} ℓ_{T} (\tilde{φ}) = ν_{T} α_{T} \int_{Γ_{i}} {(\frac{g p_{1}}{R \overline{T}})}^{2} T_{*} \tilde{φ} d Γ_{i}, \\ ℓ_{q} (\tilde{φ}) = ν_{q} α_{q} \int_{Γ_{i}} {(\frac{g p_{1}}{R \overline{T}})}^{2} q_{*} \tilde{φ} d Γ_{i}, \\ (2.16) & \begin{matrix} a_{T} (φ, \tilde{φ}) = μ_{T} (\nabla φ, \nabla \tilde{φ}) + ν_{T} \int_{M} {(\frac{g p}{R \overline{T}})}^{2} \partial_{p} φ \partial_{p} \tilde{φ} d M \\ + ν_{T} \int_{Γ_{i}} {(\frac{g p_{1}}{R \overline{T}})}^{2} α_{T} φ \tilde{φ} d Γ_{i}, \end{matrix} \\ \begin{matrix} a_{q} (φ, \tilde{φ}) = μ_{q} (\nabla φ, \nabla \tilde{φ}) + ν_{q} \int_{M} {(\frac{g p}{R \overline{T}})}^{2} \partial_{p} φ \partial_{p} \tilde{φ} d M \\ + ν_{q} \int_{Γ_{i}} {(\frac{g p_{1}}{R \overline{T}})}^{2} α_{q} φ \tilde{φ} d Γ_{i} . \end{matrix} \end{array}$ We then define a weak solution of (2.1), (2.5) and (2.8) as follows.

Definition 2.1.
Let $U_{0} = (T_{0}, q_{0}) \in H \times H$ . For almost every $t > 0$ and every $(T^{b}, q^{b}) \in V \times V$ , a vector $\begin{matrix} (2.17) & U = (T, q) \in L_{loc}^{2} (0, \infty; V \times V) \cap C ([0, \infty); H \times H), \end{matrix}$ is a solution to system (2.1) if $\partial_{t} U \in L_{loc}^{2} (0, \infty; {(V \times V)}^{})$ and we have $\begin{array}{rcl} c_{p} ⟨ \partial_{t} T, T^{b} ⟩ + a_{T} (T, T^{b}) + c_{p} b (u, T, T^{b}) \\ (2.18) & = d (ω, \overline{T}, T^{b}) + ℓ_{T} (T^{b}) + ⟨ \frac{L}{p} ω^{-} h_{T} F^{+} (T), T^{b} ⟩ \forall T^{b} \in V, \end{array}$ and $\begin{matrix} (2.19) & ⟨ \partial_{t} q, q^{b} ⟩ + a_{q} (q, q^{b}) + b (u, q, q^{b}) = ℓ_{q} (q^{b}) - ⟨ \frac{1}{p} ω^{-} h_{q} F^{+} (T), q^{b} ⟩ \forall q^{b} \in V, \end{matrix}$ for some h and $h_{q} \in L^{\infty} (M \times [0, \infty))$ which satisfy the variational inequality $\begin{matrix} (2.20) & ({(q^{b} - q_{s})}^{+}, 1) - ({(q - q_{s})}^{+}, 1) ⩾ ⟨ h, q^{b} - q ⟩ \forall q^{b} \in V, h = h_{T} or h_{q} . \end{matrix}$

We recall the definition of the subdifferential operator from e.g. [2] and [13].
Definition 2.2.
Given the lower semicontinuous, convex, proper function $φ : X \to R$ , the mapping $\partial φ : X \to X^{}$ defined by $\begin{matrix} (2.21) & \partial φ (x) = {x^{} \in X^{} : φ (x) ⩽ φ (y) + (x^{*}, x - y), \forall y \in X} \end{matrix}$ is called the subdifferential of φ at x.
Remark 2.2.
Thanks to (2.20) and (2.21), $h_{q}$ is an element of the subdifferential of $\partial ({(q - q_{s})}^{+}, 1)$ such that $\begin{matrix} (2.22) & h_{q} \in \partial ({(q - q_{s})}^{+}, 1) = H (q - q_{s}) . \end{matrix}$ Considering the physical context, we can write $\begin{matrix} (2.23) & h_{q} = \{\begin{matrix} 1 & if q > q_{s} (over saturated), \\ 0 & if q < q_{s} (under saturated), \\ \in [0, 1] & if q = q_{s} (saturation) . \end{matrix} \end{matrix}$

Existence and uniqueness results of (2.18) and (2.19) have been proved in [3,11,12], with furthermore $h = h_{T} = h_{q}$ which is essential for uniqueness. Indeed, the equations considered in [3,11] are equivalent to (2.1) except for the term $- R / p \overline{T} ω$ . However, the proof is almost exactly the same for Theorems 2.1, 2.2 and 2.3.
Theorem 2.1 (Existence and uniqueness of solutions).

We assume that $\begin{array}{rcl} (2.24) & U_{0} = (T_{0}, q_{0}) \in H \times H, T_{*}, q_{*} \in L_{loc}^{2} (0, \infty; L^{2} (Γ_{i})), \\ (2.25) & u = (v, ω) \in L_{loc}^{4} (0, \infty; V) \cap L^{\infty} (0, \infty; H), \end{array}$ and (2.6) , (2.7) hold. Then, (2.18) and (2.19) possess a unique weak solution $U = (T, q)$ such that $\begin{matrix} (2.26) & U \in L_{loc}^{2} (0, \infty; V \times V) \cap C ([0, \infty); H \times H), \end{matrix}$ and $\begin{matrix} (2.27) & \partial_{t} U \in L_{loc}^{2} (0, \infty; {(V \times V)}^{*}), \end{matrix}$ where $h = h_{q} \in L^{\infty} (M \times [0, \infty))$ is as in (2.20) .

Thanks to Theorem 2.1, we define the semigroup $S (t)$ formed by the solutions to the primitive Eqs (2.1), (2.5) and (2.8) such that $\begin{matrix} (2.28) & S (t) : (T_{0}, q_{0}) \in H \times H \to (T (t), q (t)) \in H \times H . \end{matrix}$

Theorem 2.2 (Continuity and uniqueness).

The hypotheses are those of Theorem 2.1. We assume that for $U_{1} = (T_{1}, q_{1})$ and $U_{2} = (T_{2}, q_{2})$ , $\begin{matrix} (2.29) & U_{1}, U_{2} \in L_{loc}^{2} (0, \infty; V \times V) \cap C ([0, \infty); H \times H) \end{matrix}$ are the solutions of (2.1) , (2.5) and (2.8) with the initial data $\begin{matrix} (2.30) & U_{1}^{0} = (T_{1}^{0}, q_{1}^{0}), U_{2}^{0} = (T_{2}^{0}, q_{2}^{0}) \in H \times H . \end{matrix}$ There exists a positive constant C depending on $t_{1}$ and the initial data $U_{1}^{0}$ and $U_{2}^{0}$ such that $\begin{array}{rcl} sup_{t \in [0, t_{1}]} {{| T_{1} (t) - T_{2} (t) |}^{2} + {| q_{1} (t) - q_{2} (t) |}^{2}} \\ (2.31) & ⩽ C exp (C t_{1} ∥ ω ∥_{L^{\infty} (0, t_{1}; H)}) {{| T_{1}^{0} - T_{2}^{0} |}^{2} + {| q_{1}^{0} - q_{2}^{0} |}^{2}} \end{array}$ for all $t_{1} > 0$ . Hence, the mapping $(T_{0}, q_{0}) \to (T (t), q (t))$ is continuous in $H \times H$ for all $t > 0$ .

One can obtain additional regularity results with further assumptions on the data.

Theorem 2.3 (Regularity of solutions).

Assume that $\begin{matrix} (2.32) & u = (v, ω) \in [L^{\infty} (M \times [0, \infty))]^{3}, \end{matrix}$ and the conditions (2.6) , (2.7) and (2.25) are satisfied. We also assume that $U = (T, q) \in L_{loc}^{2} (0, \infty; V \times V) \cap C ([0, \infty); H \times H)$ is the unique solution to (2.18) and (2.19) with $h_{q} \in L^{\infty} (M \times [0, \infty))$ as in (2.20). If $U_{0} = (T_{0}, q_{0}) \in V \times V$ and the boundary data $T_{*}$ , $q_{*}$ satisfy $\begin{matrix} (2.33) & T_{*}, q_{*} \in L_{loc}^{2} (0, \infty; H^{1} (Γ_{i})) and \partial_{t} T_{*}, \partial_{t} q_{*} \in L_{loc}^{2} (0, \infty; L^{2} (Γ_{i})), \end{matrix}$ then $\begin{matrix} (2.34) & T, q \in L_{loc}^{2} (0, \infty; H^{2} (M)) \cup C ([0, \infty); V) and \partial_{t} T, \partial_{t} q \in L_{loc}^{2} (0, \infty; H) . \end{matrix}$

3. The main result: Existence of a global attractor

We recall the following abstract theorem in [33] for semigroups and the existence of a global attractor to achieve our main theorem in Section 4. Concerning additional properties and developments of global attractors, see e.g. [1,10,17,25,30,32]. We borrow the following theorem from [33] (see Theorem 1.1.1).

Theorem 3.1.
We assume that X is a metric space and that the operators $S (t)$ are given and satisfy:
$S (t + s) = S (t) \cdot S (s)$ $\forall s, t ⩾ 0$ , and $S (0) = I$ .

$S (t)$ is a continuous operator from X into itself, $\forall t ⩾ 0$ .

The operators $S (t)$ are uniformly compact for t large. By this we mean that for every bounded set $B$ there exists $t_{0}$ which may depend on $B$ such that $\begin{matrix} ⋃_{t ⩾ t_{0}} S (t) B \end{matrix}$ is relatively compact in X.
We also assume that there exists an open set $U$ and a bounded set $B$ of $U$ such that $B$ is absorbing in $U$ .

Then the ω-limit set of $B$ , $A = ω (B)$ , is a compact attractor which attracts the bounded sets of $U$ . It is the maximal bounded attractor of ${S (t)}_{t ⩾ 0}$ in $U$ . Furthermore, if X is a Banach space, if $U$ is convex, and the mapping $t \mapsto S (t) u_{0}$ is continuous from $R_{+}$ into X, for every $u_{0}$ in X, then $A$ is connected, too.
Remark 3.1.
Throughout this article, we assume that $\begin{matrix} (3.1) & T_{}, q_{} \in L^{2} (Γ_{i}), u = u (x) \in V \cap {(L^{\infty} (M))}^{3} . \end{matrix}$ Thanks to the assumptions (3.1), the dynamical system that we consider is autonomous. Studies of the non-autonomous case will be performed elsewhere.

We now state and prove our main result in this article. Theorem 3.2.
Assume that (2.6) , (2.7) and (3.1) hold. Then, the semigroup ${S (t)}_{t ⩾ 0}$ defined in (2.28) is a semigroup of compact operators, that is, for any $t > 0$ , $S (t) : H \times H \to H \times H$ is compact. The semigroup ${S (t)}_{t ⩾ 0}$ possesses a global attractor $A$ . The global attractor $A$ is compact in $H \times H$ and attracts all the bounded sets in $H \times H$ in the norm of $H \times H$ . Moreover, $A$ is connected in $H \times H$ and maximal.

4. Proof of Theorem 3.2

We prove Theorem 3.2 using the abstract theorem, Theorem 3.1. Thanks to Theorems 2.1 and 2.2, the basic properties (a) and (b) in Theorem 3.1 are satisfied. It remains to show the uniform compactness property of $S (t)$ and the existence of an absorbing set. We begin with some basic estimates, which can be strictly justified via the Galerkin approximation framework.

4.1. Absorbing set in H

Choosing $T^{b} = T$ in (2.18), we find $\begin{array}{rcl} \frac{c_{p}}{2} \frac{d}{d t} | T |^{2} + a_{T} (T, T) + c_{p} b (u, T, T) \\ (4.1) & = d (ω, \overline{T}, T) + ℓ_{T} (T) + ⟨ \frac{L}{p} ω^{-} h_{q} F^{+} (T), T ⟩ . \end{array}$ Note that the bilinear form $a_{T} (\cdot, \cdot)$ is bounded and coercive on $V \times V$ and satisfies $\begin{matrix} (4.2) & \begin{matrix} | a_{T} (φ, \tilde{φ}) | ⩽ C_{1} ∥ φ ∥ ∥ \tilde{φ} ∥, \\ C_{2} ∥ φ ∥^{2} ⩽ a_{T} (φ, φ) . \end{matrix} \end{matrix}$ Here and after, the $C_{j}$ ’s are positive constants which are independent of the time t. Thanks to the property of the trilinear operator as in [34], we have $\begin{matrix} (4.3) & b (u, φ, φ) = 0 \forall φ \in V, \forall u \in V . \end{matrix}$ Since F is a globally Lipschitz bounded function and $F (0) = 0$ , we have $\begin{matrix} (4.4) & | ⟨ \frac{L}{p} ω^{-} h_{q} F^{+} (T), φ ⟩ | ⩽ C_{3} ∥ φ ∥ \forall φ \in V . \end{matrix}$ In addition, for $φ \in V$ , we have $\begin{matrix} (4.5) & | ℓ_{T} (φ) | ⩽ C_{4} ∥ φ ∥, | d (ω, \overline{T}, φ) | ⩽ C_{5} ∥ φ ∥ . \end{matrix}$ Then, (4.1) yields $\begin{matrix} (4.6) & \frac{c_{p}}{2} \frac{d}{d t} | T |^{2} + \frac{C_{2}}{2} ∥ T ∥^{2} ⩽ C_{6}, \end{matrix}$ and this implies $\begin{matrix} (4.7) & \frac{d}{d t} | T |^{2} + \frac{C_{2}}{c_{p} λ} | T |^{2} ⩽ \frac{2 C_{6}}{c_{p}}, \end{matrix}$ where λ is an usual Poincaré constant. Using the classical Gronwall lemma, we obtain $\begin{matrix} (4.8) & | T |^{2} ⩽ | T_{0} |^{2} exp (- \frac{C_{2}}{c_{p} λ} t) + \frac{2 C_{6} λ}{C_{2}} [1 - exp (- \frac{C_{2}}{c_{p} λ} t)] . \end{matrix}$ Thus, $\begin{matrix} (4.9) & \underset{t \to \infty}{lim sup} | T (t) | ⩽ ρ_{0}, ρ_{0} = \sqrt{\frac{2 C_{6} λ}{C_{2}}} . \end{matrix}$ This gives us the uniform estimate which yields an absorbing ball for T in H. From (4.6), integrating in t, we deduce that $\begin{matrix} (4.10) & \frac{C_{2}}{2} \int_{t}^{t + r} ∥ T ∥^{2} d s ⩽ r C_{6} + \frac{c_{p}}{2} {| T (t) |}^{2} \forall r > 0 . \end{matrix}$ With the use of (4.9), we obtain that $\begin{matrix} (4.11) & \underset{t \to \infty}{lim sup} \int_{t}^{t + r} ∥ T ∥^{2} d s ⩽ \frac{2 r C_{6}}{C_{2}} + \frac{c_{p} ρ_{0}^{2}}{C_{2}} . \end{matrix}$

For the $L^{2}$ -estimate for q, we utilize the same argument as in the estimate for T. We set $q^{b} = q$ in (2.19) $\begin{matrix} (4.12) & \frac{1}{2} \frac{d}{d t} | q |^{2} + a_{q} (q, q) + b (u, q, q) = ℓ_{q} (q) + ⟨ \frac{1}{q} ω^{-} h_{q} F^{+} (T), q ⟩ . \end{matrix}$ We note that $a_{q} (\cdot, \cdot)$ is bounded and coercive on $V \times V$ such that $\begin{matrix} (4.13) & \begin{matrix} | a_{q} (φ, \tilde{φ}) | ⩽ C_{7} ∥ φ ∥ ∥ \tilde{φ} ∥, \\ C_{8} ∥ φ ∥^{2} ⩽ a_{q} (φ, φ) . \end{matrix} \end{matrix}$ Bounding $\begin{matrix} (4.14) & | ℓ_{q} (q) | ⩽ C_{9} ∥ q ∥, | ⟨ \frac{1}{q} ω^{-} h_{q} F^{+} (T), q ⟩ | ⩽ C_{10} ∥ q ∥, \end{matrix}$ we derive from (4.12) that $\begin{matrix} (4.15) & \frac{1}{2} \frac{d}{d t} | q |^{2} + \frac{C_{8}}{2} ∥ q ∥^{2} ⩽ C_{11}, \end{matrix}$ and this implies $\begin{matrix} (4.16) & \frac{d}{d t} | q |^{2} + \frac{C_{8}}{λ} | q |^{2} ⩽ 2 C_{11}, \end{matrix}$ where λ is the usual Poincaré constant. Employing the classical Gronwall lemma, we deduce that $\begin{matrix} (4.17) & | q |^{2} ⩽ | q_{0} |^{2} exp (- \frac{C_{8}}{λ} t) + \frac{2 C_{11} λ}{C_{8}} (1 - exp (- \frac{C_{8}}{λ} t)) . \end{matrix}$ Thus, $\begin{matrix} (4.18) & \underset{t \to \infty}{lim sup} | q (t) | ⩽ ρ_{1}, ρ_{1} = \sqrt{\frac{2 C_{11} λ}{C_{8}}} . \end{matrix}$ This gives us the uniform estimate which yields an absorbing ball for q in H. From (4.15), integrating in t, we deduce furthermore that $\begin{matrix} (4.19) & \frac{C_{8}}{2} \int_{t}^{t + r} ∥ q ∥^{2} d s ⩽ r C_{11} + \frac{1}{2} {| q (t) |}^{2} \forall r > 0 . \end{matrix}$ Using (4.18), we obtain that $\begin{matrix} (4.20) & \underset{t \to \infty}{lim sup} \int_{t}^{t + r} ∥ q ∥^{2} d s ⩽ \frac{2 r C_{11}}{C_{8}} + \frac{ρ_{1}^{2}}{C_{8}} . \end{matrix}$

Lemma 4.1.
Assuming (3.1), there exist positive constants $ρ_{H}$ and $t_{0}$ depending only on the initial data $| T_{0} |$ and $| q_{0} |$ such that $\begin{matrix} (4.21) & | T (t) |, | q (t) | ⩽ ρ_{H} for all t > t_{0} . \end{matrix}$ Hence, there exists an absorbing set in H. Moreover, there exists a constant $I_{H} (r, ρ_{H})$ such that $\begin{matrix} (4.22) & \int_{t}^{t + r} {∥ T (s) ∥}^{2} d s, \int_{t}^{t + r} {∥ q (s) ∥}^{2} d s ⩽ I_{H} \end{matrix}$ for all $t > t_{0}$ and $r > 0$ .

4.2. Absorbing set in V

We first look for the absorbing set for T in V. Setting $T^{b} = \partial_{t} T$ in (2.18), we find that $\begin{array}{rcl} c_{p} | \partial_{t} T |^{2} + \frac{1}{2} \frac{d}{d t} a_{T} (T, T) \\ (4.23) & = - c_{p} b (u, T, \partial_{t} T) + d (ω, \overline{T}, \partial_{t} T) + ℓ_{T} (\partial_{t} T) + ⟨ \frac{L}{p} ω^{-} h_{q} F^{+} (T), \partial_{t} T ⟩ . \end{array}$ We note that $\begin{array}{rcl} (4.24) & | c_{p} b (u, T, \partial_{t} T) | ⩽ c_{p} ∥ u ∥_{L^{\infty}} ∥ T ∥ | \partial_{t} T | ⩽ C ∥ u ∥_{L^{\infty}}^{2} ∥ T ∥^{2} + \frac{c_{p}}{8} | \partial_{t} T |^{2}, \\ (4.25) & | d (ω, \overline{T}, \partial_{t} T) | ⩽ C ∥ ω ∥_{L^{\infty}}^{2} + \frac{c_{p}}{8} | \partial_{t} T |^{2}, \\ (4.26) & | ⟨ \frac{L}{p} ω^{-} h_{q} F^{+} (T), \partial_{t} T ⟩ | ⩽ C ∥ ω ∥_{L^{\infty}}^{2} + \frac{c_{p}}{8} | \partial_{t} T |^{2}, \end{array}$ and $\begin{matrix} (4.27) & ℓ_{T} (\partial_{t} T) = ν_{T} α_{T} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) T_{*} \partial_{t} T d Γ_{i}, \end{matrix}$ where $∥ \cdot ∥_{L^{\infty}}$ is the standard norm in $L^{\infty} (M)$ . Then we rewrite (4.23) as $\begin{array}{rcl} \frac{c_{p}}{2} | \partial_{t} T |^{2} + \frac{d}{d t} (\frac{1}{2} a_{T} (T, T) - ν_{T} α_{T} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) T_{*} T d Γ_{i}) \\ (4.28) & ⩽ C ∥ u ∥_{L^{\infty}}^{2} ∥ T ∥^{2} + C ∥ u ∥_{L^{\infty}}^{2} . \end{array}$ We now define $\begin{matrix} (4.29) & Θ_{T} : = \frac{1}{2} a_{T} (T, T) - ν_{T} α_{T} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) T_{*} T d Γ_{i} + M_{T} . \end{matrix}$ Here, $M_{T}$ is a time-independent constant which will be determined later. Noting that for some constant $K_{1} > 0$ , $\begin{array}{rcl} ν_{T} α_{T} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) | T_{*} | | T | d Γ_{i} \\ (4.30) & ⩽ K_{1} {(ν_{T} α_{T})}^{2} (\frac{g p_{1}}{R {\overline{T}}_{*}}) ∥ T_{*} ∥_{L^{2} (Γ_{i})} + \frac{1}{4} a_{T} (T, T), \end{array}$ we deduce that $\begin{array}{rcl} Θ_{T} & ⩾ & \frac{1}{2} a_{T} (T, T) - ν_{T} α_{T} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) | T_{*} | | T | d Γ_{i} + M_{T} \\ (4.31) & ⩾ & \frac{1}{4} a_{T} (T, T) - K_{1} {(ν_{T} α_{T})}^{2} (\frac{g p_{1}}{R {\overline{T}}_{*}}) ∥ T_{*} ∥_{L^{2} (Γ_{i})} + M_{T} . \end{array}$ Choosing $M_{T}$ such that $\begin{matrix} (4.32) & M_{T} ⩾ K_{1} {(ν_{T} α_{T})}^{2} (\frac{g p_{1}}{R {\overline{T}}_{*}}) ∥ T_{*} ∥_{L^{2} (Γ_{i})}, \end{matrix}$ we then obtain $\begin{matrix} (4.33) & \frac{1}{4} a_{T} (T, T) ⩽ Θ_{T} . \end{matrix}$ Furthermore, we find $\begin{array}{rcl} Θ_{T} & ⩽ & \frac{1}{2} a_{T} (T, T) + ν_{T} α_{T} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) | T_{*} | | T | d Γ_{i} + M_{T} \\ ⩽ & C a_{T} (T, T) + C ∥ T_{*} ∥_{L^{2} (Γ_{i})} \\ (4.34) & ⩽ & C a_{T} (T, T) + N_{1} \end{array}$ for some constant $N_{1} > 0$ which depends on $∥ T_{*} ∥_{L^{2} (Γ_{i})}$ . Collecting the results in (4.2), (4.33) and (4.34), we write $\begin{array}{rcl} \frac{C_{2}}{4} ∥ T ∥ & ⩽ & \frac{1}{4} a_{T} (T, T) ⩽ Θ_{T} \\ (4.35) & ⩽ & C a_{T} (T, T) + N_{1} ⩽ C ∥ T ∥^{2} + N_{1} . \end{array}$ From (4.28), (4.29), (4.35), we derive that $\begin{array}{rcl} \frac{d}{d t} Θ_{T} & ⩽ & C (∥ u ∥_{\infty}^{2} ∥ T ∥^{2} + ∥ u ∥_{\infty}^{2}) \\ ⩽ & (by (4.35)) \\ (4.36) & ⩽ & C (∥ u ∥_{\infty}^{2} Θ_{T} + ∥ u ∥_{\infty}^{2}) . \end{array}$ To find the absorbing ball in $H^{1}$ , we now recall the uniform Gronwall lemma as in [15] and [33]:

Lemma 4.2.

Let g, h and y be three positive locally integrable functions on $(t_{0}, \infty)$ such that $y^{'}$ is locally integrable on $(t_{0}, \infty)$ , and which satisfy $\begin{array}{l} (4.37) & \frac{d y}{d t} ⩽ g y + h for t ⩾ t_{0}, \\ (4.38) & \int_{t}^{t + r} g (s) d s ⩽ a_{1}, \int_{t}^{t + r} h (s) d s ⩽ a_{2}, \int_{t}^{t + r} y (s) d s ⩽ a_{3} \end{array}$ for $t ⩾ t_{0}$ , where r, $a_{1}$ , $a_{2}$ and $a_{3}$ are positive constants. Then $\begin{matrix} (4.39) & y (t + r) ⩽ (\frac{a_{3}}{r} + a_{2}) exp (a_{1}) \forall t ⩾ t_{0} . \end{matrix}$

We apply Lemma 4.2 to (4.36) with g, h, y replaced by $\begin{matrix} (4.40) & C ∥ u ∥_{L^{\infty}}^{2}, C ∥ u ∥_{L^{\infty}}^{2}, Θ_{T} . \end{matrix}$ Thanks to (4.34), there exist some $a_{1}$ , $a_{2}$ and $a_{3}$ such that $\begin{matrix} (4.41) & \begin{matrix} \int_{t}^{t + r} C ∥ u ∥_{L^{\infty}}^{2} d s ⩽ a_{1}, \\ \int_{t}^{t + r} C ∥ u ∥_{L^{\infty}}^{2} d s ⩽ a_{2}, \end{matrix} \end{matrix}$ and $\begin{array}{rcl} \int_{t}^{t + r} Θ_{T} (s) d s & ⩽ & \int_{t}^{t + r} C ∥ T ∥^{2} + N_{1} d s \\ ⩽ & (by Lemma 4.1) \\ (4.42) & ⩽ & C I_{H} + r N_{1} = a_{3} . \end{array}$ We then obtain $\begin{matrix} (4.43) & Θ_{T} ⩽ (\frac{a_{3}}{r} + a_{2}) exp (a_{1}) for t ⩾ t_{0} + r . \end{matrix}$ With the use of (4.35), we find that $\begin{matrix} (4.44) & ∥ T ∥^{2} ⩽ C (\frac{a_{3}}{r} + a_{2}) exp (a_{1}) for t ⩾ t_{0} + r . \end{matrix}$

We now study the $H^{1}$ -estimates for q. Choosing $q^{b} = \partial_{t} q$ in (2.19), we find that $\begin{matrix} (4.45) & \begin{matrix} | \partial_{t} q |^{2} + \frac{1}{2} \frac{d}{d t} a_{q} (q, q) \\ = - b (u, q, \partial_{t} q) + ℓ_{q} (\partial_{t} q) + ⟨ \frac{1}{p} ω^{-} h_{q} F^{+} (T), \partial_{t} q ⟩ . \end{matrix} \end{matrix}$ We note that $\begin{array}{rcl} (4.46) & | b (u, q, \partial_{t} q) | ⩽ ∥ u ∥_{L^{\infty}} ∥ q ∥ | \partial_{t} q | ⩽ C ∥ u ∥_{L^{\infty}}^{2} ∥ q ∥^{2} + \frac{1}{4} | \partial_{t} q |^{2}, \\ (4.47) & | ⟨ \frac{1}{p} ω^{-} h_{q} F^{+} (T), \partial_{t} q ⟩ | ⩽ C ∥ ω ∥_{L^{\infty}}^{2} + \frac{1}{4} | \partial_{t} q |^{2}, \end{array}$ and $\begin{matrix} (4.48) & ℓ_{q} (\partial_{t} q) = ν_{q} α_{q} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) q_{*} \partial_{t} q d Γ_{i} . \end{matrix}$ Then (4.45) becomes $\begin{matrix} (4.49) & \begin{matrix} \frac{1}{2} | \partial_{t} q |^{2} + \frac{d}{d t} (\frac{1}{2} a_{q} (q, q) - ν_{q} α_{q} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) q_{*} q d Γ_{i}) \\ ⩽ C ∥ u ∥_{L^{\infty}}^{2} ∥ q ∥^{2} + C ∥ u ∥_{L^{\infty}}^{2} . \end{matrix} \end{matrix}$ We set $\begin{matrix} (4.50) & Θ_{q} : = \frac{1}{2} a_{T} (q, q) - ν_{q} α_{q} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) q_{*} q d Γ_{i} + M_{q} . \end{matrix}$ Again, $M_{q}$ is a time-independent constant which will be determined later. Noting that for some constant $K_{2} > 0$ , $\begin{array}{rcl} ν_{q} α_{q} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) | q_{*} | | q | d Γ_{i} \\ (4.51) & ⩽ K_{2} {(ν_{q} α_{q})}^{2} (\frac{g p_{1}}{R {\overline{T}}_{*}}) ∥ q_{*} ∥_{L^{2} (Γ_{i})} + \frac{1}{4} a_{q} (q, q), \end{array}$ we deduce that $\begin{array}{rcl} Θ_{q} & ⩾ & \frac{1}{2} a_{q} (q, q) - ν_{q} α_{q} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) | q_{*} | | q | d Γ_{i} + M_{q} \\ (4.52) & ⩾ & \frac{1}{4} a_{q} (q, q) - K_{2} {(ν_{q} α_{q})}^{2} (\frac{g p_{1}}{R {\overline{T}}_{*}}) ∥ q_{*} ∥_{L^{2} (Γ_{i})} + M_{q} . \end{array}$ We now choose $M_{q}$ such that $\begin{matrix} (4.53) & M_{q} ⩾ K_{2} {(ν_{q} α_{q})}^{2} (\frac{g p_{1}}{R {\overline{T}}_{*}}) ∥ q_{*} ∥_{L^{2} (Γ_{i})}, \end{matrix}$ then we obtain $\begin{matrix} (4.54) & \frac{1}{4} a_{q} (q, q) ⩽ Θ_{q} . \end{matrix}$ Moreover, we find $\begin{array}{rcl} Θ_{q} & ⩽ & \frac{1}{2} a_{q} (q, q) + ν_{q} α_{q} \int_{Γ_{i}} (\frac{g p_{1}}{R \overline{T}}) | q_{*} | | q | d Γ_{i} + M_{q} \\ ⩽ & C a_{q} (q, q) + C ∥ q_{*} ∥_{L^{2} (Γ_{i})} \\ (4.55) & ⩽ & C a_{q} (q, q) + N_{2} \end{array}$ for some constant $N_{2} > 0$ which depends on $∥ q_{*} ∥_{L^{2} (Γ_{i})}$ . Thanks to (4.13), (4.54) and (4.55), $\begin{array}{rcl} \frac{C_{8}}{4} ∥ q ∥^{2} & ⩽ & \frac{1}{4} a_{q} (q, q) ⩽ Θ_{q} \\ (4.56) & ⩽ & C a_{q} (q, q) + N_{2} ⩽ C ∥ q ∥^{2} + N_{2} . \end{array}$ From (4.49), (4.50) and (4.56), we deduce that $\begin{array}{rcl} \frac{d}{d t} Θ_{q} & ⩽ & C ∥ u ∥_{L^{\infty}}^{2} ∥ q ∥^{2} + C ∥ u ∥_{L^{\infty}}^{2} \\ ⩽ & (by (4.56)) \\ (4.57) & ⩽ & C ∥ u ∥_{L^{\infty}}^{2} Θ_{q} + C ∥ u ∥_{L^{\infty}}^{2} . \end{array}$ We apply Lemma 4.2 to (4.57) with g, h, y replaced by $\begin{matrix} (4.58) & C ∥ u ∥_{L^{\infty}}^{2}, C ∥ u ∥_{L^{\infty}}^{2}, Θ_{q} . \end{matrix}$ Thanks to (4.55), there exist some $a_{1}$ , $a_{2}$ and $a_{3}$ such that $\begin{matrix} (4.59) & \begin{matrix} \int_{t}^{t + r} C ∥ u ∥_{L^{\infty}}^{2} d s ⩽ a_{1}, \\ \int_{t}^{t + r} C ∥ u ∥_{L^{\infty}}^{2} d s ⩽ a_{2}, \end{matrix} \end{matrix}$ and $\begin{array}{rcl} \int_{t}^{t + r} Θ_{q} (s) d s & ⩽ & \int_{t}^{t + r} C ∥ q ∥^{2} + N_{2} d s \\ ⩽ & (by Lemma 4.1) \\ (4.60) & ⩽ & C I_{H} + r N_{2} = a_{3} . \end{array}$ We obtain $\begin{matrix} (4.61) & Θ_{q} ⩽ (\frac{a_{3}}{r} + a_{2}) exp (a_{1}) for t ⩾ t_{0} + r . \end{matrix}$ From (4.56), we find that $\begin{matrix} (4.62) & ∥ q ∥^{2} ⩽ C (\frac{a_{3}}{r} + a_{2}) exp (a_{1}) for t ⩾ t_{0} + r . \end{matrix}$ We then arrive at the following lemma.

Lemma 4.3.

We assume that (3.1) is satisfied. There exist positive constants $ρ_{V}$ and $t_{0}$ depending on the initial data $| T_{0} |$ and $| q_{0} |$ such that $\begin{matrix} (4.63) & ∥ T (t) ∥, ∥ q (t) ∥ ⩽ ρ_{V} for t > t_{0} + r, \end{matrix}$ where $ρ_{V}$ is uniformly bounded in time and $r > 0$ is fixed. Hence, there exists an absorbing set in V.

From Lemmas 4.1 and 4.3, the condition (c) of Theorem 3.1 follows since the bounded sets of $H^{1}$ are compact in $L^{2}$ . Hence, the semigroup $S (t)$ associated with the system (2.1), (2.5) and (2.8) possesses an attractor $A$ which is compact, connected and maximal in $H \times H$ ; $A$ attracts the bounded sets of $H \times H$ .

Footnotes

Acknowledgements

This work was supported in part by NSF Grants DMS 1206438 and by the Research Fund of Indiana University. The author thank Professor Roger Temam for suggesting the problem and for his help in the course of this work, Dr. Michele Coti Zelati for the discussions.

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