Global Attractor of Atmospheric Circulation Equations with Humidity Effect

and Applied Analysis 3 The paper is organized as follows. In Section 2, we recall preliminary results. In Section 3, we present uniqueness of the solution to the atmospheric circulation equations. In Section 4, we obtain global attractor of the equations. ‖ · ‖X denote norm of the space X; C and Ci are variable constants. Let H {φ u, T, q ∈ L2 Ω, R4 | φ satisfy 1.4 , 1.6 }, and H1 {φ u, T, q ∈ H1 Ω, R4 | φ satisfy 1.4 , 1.6 }. 2. Preliminaries Let X and X1 be two Banach spaces, X1 ⊂ X a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on X, given by du dt Lu G u , u x, 0 u0, 2.1 where u t is an unknown function, L : X1 → X a linear operator, and G : X1 → X a nonlinear operator. A family of operators S t : X → X t ≥ 0 is called a semigroup generated by 2.1 if it satisfies the following properties: 1 S t : X → X is a continuous map for any t ≥ 0; 2 S 0 id : X → X is the identity; 3 S t s S t · S s , for all t, s ≥ 0. Then, the solution of 2.1 can be expressed as u t, u0 S t u0. 2.2 Next, we introduce the concepts and definitions of invariant sets, global attractors, and ωlimit sets for the semigroup S t . Definition 2.1. Let S t be a semigroup defined on X. A set Σ ⊂ X is called an invariant set of S t if S t Σ Σ, for all t ≥ 0. An invariant set Σ is an attractor of S t if Σ is compact, and there exists a neighborhood U ⊂ X of Σ such that for any u0 ∈ U, inf v∈Σ ‖S t u0 − v‖X −→ 0, as t −→ ∞. 2.3 In this case, we say that Σ attractsU. Particularly, if Σ attracts any bounded set of X, Σ is called a global attractor of S t in X. For a set D ⊂ X, we define the ω-limit set of D as follows: ω D ⋂ s≥0 ⋃ t≥s S t D, 2.4 where the closure is taken in the X-norm. Lemma 2.2 is the classical existence theorem of global attractor by Temam 13 . 4 Abstract and Applied Analysis Lemma 2.2. Let S t : X → X be the semigroup generated by 2.1 . Assume that the following conditions hold: 1 S t has a bounded absorbing set B ⊂ X, that is, for any bounded set A ⊂ X there exists a time tA ≥ 0 such that S t u0 ∈ B, for all u0 ∈ A and t > tA; 2 S t is uniformly compact, that is, for any bounded set U ⊂ X and some T > 0 sufficiently large, the set ⋃ t≥T S t U is compact in X. Then theω-limit setA ω B of B is a global attractor of 2.1 , andA is connected providing B is connected. Definition 2.3 see 19 . We say that S t : X → X satisfies C-condition, if for any bounded set B ⊂ X and ε > 0, there exist tB > 0 and a finite dimensional subspace X1 ⊂ X such that {PS t B} is bounded, and ‖ I − P S t u‖X < ε, ∀t ≥ tB, u ∈ B, 2.5 where P : X → X1 is a projection. Lemma 2.4 see 19 . Let S t : X → X (t ≥ 0) be a dynamical systems. If the following conditions are satisfied: 1 there exists a bounded absorbing set B ⊂ X; 2 S t satisfies C-condition, then S t has a global attractor in X. From Linear elliptic equation theory, one has the following. Lemma 2.5. The eigenvalue equation: −ΔT x1, x2 βT x1, x2 , x1, x2 ∈ 0, 2π × 0, 1 , T 0, x2 0, 1, T 0, x2 T 2π, x2 2.6 has eigenvalue {βk}k 1, and 0 < β1 ≤ β2 ≤ · · · , βk −→ ∞, as k −→ ∞. 2.7 3. Uniqueness of Global Solution Theorem 3.1. If σ̃β1 ≥ max{ R 1 , R̃ − 1 2 /Le }, and β1 is the first eigenvalue of elliptic equation 2.6 , then the weak solution to 1.1 – 1.7 is unique. Abstract and Applied Analysis 5 Proof. From 1 , u, T, q ∈ L∞ 0, T ,H ∩ L2 0, T ,H1 , 0 < T < ∞ is the weak solution to 1.1 – 1.7 . Then for all v, S, z ∈ H1, 0 ≤ t ≤ T , we have 1 Pr ∫and Applied Analysis 5 Proof. From 1 , u, T, q ∈ L∞ 0, T ,H ∩ L2 0, T ,H1 , 0 < T < ∞ is the weak solution to 1.1 – 1.7 . Then for all v, S, z ∈ H1, 0 ≤ t ≤ T , we have 1 Pr ∫