Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere

Under general boundary conditions we consider the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solution of the 3D moist primitive equations with viscosity. Firstly, we obtain time-uniform estimates of the first-order time derivative of the strong solutions in $L^2(\mho)$. Then, to prove the finiteness of the Hausdorff and fractal dimensions of the global attractor, the common method is to obtain the uniform boundedness of the strong solution in $H^2(\mho)$ to establish the squeezing property of the solution operator. But it is difficult to achieve due to the boundary conditions and complicated structure of the 3D moist primitive equations. To overcome the difficulties, we try to use the uniform boundedness of the derivative of the strong solutions with respect to time $t$ in $L^2(\mho)$ to prove the uniform continuity of the global attractor. Finally, using the uniform continuity of the global attractor we establish the squeezing property of the solution operator which implies the finiteness of the Hausdorff and fractal dimensions of the global attractor.


Introduction
The paper is concerned with the 3-dimensional viscous primitive equations in the pressure coordinate system (see e.g. [19,36,38,39] and the references therein).

4)
∂ t q + ∇ v q + w∂ ξ q + L 3 q = Q 2 . (1.5) The unknowns for the primitive equations are the fluid velocity field (v, w) = (v θ , v ϕ , w) ∈ R 3 with v = (v θ , v ϕ ) and v ⊥ = (−v ϕ , v θ ) being horizontal, the temperature T , q the mixing ratio of water vapor in the air and the geopotential Φ. f = 2cosθ is the given Coriolis parameter, Q 1 corresponds to the sum of the heating of the sun and the heat added or removed by condensation or evaporation, Q 2 represents the amount of water added or removed by condensation or evaporation, a and b are positive constants with a ≈ 0.618, R 0 is the Rossby number, P stands for an approximate value of pressure at the surface of the earth, p 0 is the pressure of the upper atmosphere with p 0 > 0 and the variable ξ satisfies p = (P − p 0 )ξ + p 0 where 0 < p 0 ≤ p ≤ P. The viscosity, the heat and the water vapor diffusion operators L 1 , L 2 and L 3 are given respectively as the following: Here the positive constants ν 1 , µ 1 are the horizontal and vertical viscosity coefficients; the positive constant ν 2 , µ 2 are the horizontal and vertical heat diffusivity coefficients; while the positive constant ν 3 , µ 3 are the horizontal and vertical water vapor diffusivity coefficients. The definitions of ∇ v v, ∆v, ∆T, ∆q, ∇ v q, ∇ v T, divv, gradΦ will be given in section 2.
The space domain of equations: (1.1) − (1.5) is where S 2 is two-dimensional unit sphere. The boundary value conditions are given by where α s , β s are positive constants, T s is the given temperature on the surface of the earth, q s is the given mixing ratio of water vapor on the surface of the earth. To simplify the notations, we set T s = 0 and q s = 0 without losing any generality. For the case T s = 0 and q s = 0, we can homogenize the boundary value conditions for T, q; see [19] for detailed discussion on this issue. Moreover, using (1.  where Φ s (t; θ, ϕ) is a certain unknown function at the isobaric surface ξ = 1. In this article, we assume that the constants v i = µ i = 1, i = 1, 2, 3. For the general case, the results will still be valid. Then using (1.8) − (1.10), we obtain the following equivalent formulation for system (1.1) − (1.7) with initial condition In order to understand the mechanism of long-term weather prediction, one can take advantage of the historical records and numerical computations to detect the future weather. Alternatively, one should also study the long time behavior mathematically for the equations and models governing the motion. The primitive equations represent the classic model for the study of climate and weather prediction, describing the motion of the atmosphere when the hydrostatic assumption is enforced [18,23,24,42,45]. But the resulting flow or the atmosphere is rich in its organization and complexity (see [18,23,24]), the full governing equations are too complicated to be treatable both from the theoretical and the computational side. To overcome this difficulty, some simple numerical models were introduced. The 2-D and 3-D quasi-geostrophic models have been the subject of analytical mathematical study (see e.g., [3,5,10,11,15,16,41,50,51,52] and references therein). To the best of our knowledge, the mathematical framework of primitive equations was formulated in [38,39,40], where the definitions of weak and strong solutions were given and the existence of weak solution was proven, leaving the uniqueness of weak solution as an open problem for now. Local well-posedness of strong solutions was obtained in [21,49]. If the domains was thin, the global well-posedness of 3D primitive equations was shown in [26]. Taking advantage of the fact that the pressure is essentially two-dimensional in the primitive equations, global wellposedness of the full three-dimensional case was established in [12] and independently in [30,31]. In the subsequent work [32] a different proof was developed which allows one to treat non-rectangular domains. Recently, the results were improved in [7,8,9,13] by considering the system with partial dissipation, i.e. , with only partial viscosities or only partial diffusion. For the inviscid primitive equations, finite-time blowup was established in [6]. To study the long term behavior of primitive equations, the existence of global attractor was established in [27] and dimensions were proven to be finite in [29]. When moisture is included, an equation for the conservation of water must be added, which is the case in e.g. [19,20,38,44]. In [53], global well-posedness of quasi-strong and strong solutions was obtained for the primitive equations of atmosphere in presence of vapour saturation. The understanding of asymptotic behavior of dynamical system is one of the most important topics of modern mathematical physics. One way to solve the problem for dissipative deterministic dynamical system is to consider its global attractor (see its definition in section 2). Thus, in order to capture the dynamical features of moist primitive equations, Guo and Huang in [20] proved the existence of universal attractor. Recently, the authors in [54] proved the existence of the global attractor of the strong solutions to the 3D moist primitive equations.
In this paper, we investigate the finiteness of the the Hausdorff and fractal dimensions of the global attractor for the strong solutions to the 3D moist primitive equations with viscosity. The common method is to obtain uniformly estimates for the strong solution in H 2 (℧) space and then use the squeezing property of the solution operator to show our main result. But it is difficult to get the time-uniform estimates for the strong solution in the space H 2 (℧). Because, firstly, the boundary conditions in this work is general and consists with the boundary conditions assuring the global well-posedness of the moist primitive equations without any extra conditions which help us to use integration by parts formula to obtain a priori estimates in space with higher regularity. Secondly, the structure of the moist primitive equations is even more complicated than the oceanic primitive equations studied in [12]. For example, in the horizontal momentum equation, there is a challenging term of the gradient of temperature times the mixing ratio of water vapor. The temperature equations and the mixing ratio of water vapor equations also have the similar challenging terms which present essential difficulties for a priori estimates in the H 2 (℧) norm. To overcome the difficulties, inspired by [28] we try to use the uniform continuity of the global attractor to prove the squeezing property of the solution operator which implies the finiteness of the the Hausdorff and fractal dimensions of the global attractor for the strong solutions to the 3D moist primitive equations. This method can be applied to other dissipative equations with physical boundary conditions. The remaining of the paper is organized as follows. In section 2, we present the notations and recall some important facts which are crucial to later analysis. Absorbing ball of the first-order time derivatives of the solution in L 2 (℧) is obtained in section 3. Section 4 is for the uniform continuity and the dimensions of the global attractor. As usual, the positive constants c may change from one line to the next, unless, we give a special declaration.

Preliminaries
In this section we collect some preliminary results that will be used in the rest of this paper, and we start with the following notations which will be used throughout this work. Denotē Now we give the definitions of some differential operators. Firstly, the natural generalization of the directional derivative on the Euclidean space to the covariant derivative on S 2 is given as where C ∞ (T ℧|T S 2 ) is the first two components of smooth vector fields on ℧. We define the covariant derivative of u, T and q with respect to v as follows We give the definition of the horizontal gradient ∇ = grad for T and Φ s on S 2 by We define the divergence of v by The horizontal Laplace-Beltrami operator of scalar functions T and q are We define the horizontal Laplace-Beltrami operator ∆ for vector functions on S 2 as Consequently, by integration by parts, we have Taking the average of equations (1.11) in the z direction, over the interval (0, 1) and using (2.18) − (2.20) and the boundary conditions (1.15) − (1.16), we arrive at By subtracting (2.21) from (1.11), we obtain the following equation Let e θ , e ϕ , e ξ be the unite vectors in θ, ϕ and ξ directions of the space domain ℧ respectively, The inner product and norm on T (θ,ϕ,ξ) ℧ (the tangent space of ℧ at the point (θ, ϕ, ξ) ) are defined by be the usual Lebesgue spaces with the norm | · | p and | · | L p (S 2 ) respectively. If there is no confusion, we will write | · | p instead of | · | L p (S 2 ) . L 2 (T Ω|T S 2 ) is the first two components of is the Sobolev space of functions which are in L 2 , together with all their covariant derivatives with respect to e θ , e ϕ , e ξ of order ≤ m, with the norm We will conduct our work in the following functional spaces. Let We denote by V 1 , V 2 and V 3 the closure spaces of V 1 , V 2 and V 3 in H 1 (℧) under H 1 − topology, respectively. In addition, we denote by H 1 , H 2 and H 3 the closure of By definition, the inner products and norms on V 1 , V 2 and V 3 are given by Let V ′ i (i = 1, 2, 3) be the dual space of V i with , being the inner products between V ′ i and V i . Without confusion, we also denote by , the inner product in L 2 (℧) and L 2 (S 2 ). Define the linear operator according to the classic spectral theory we can define the power A s i for any s ∈ R. Then we have where D(A i ) ′ is the dual space of D(A i ) and the embeddings above are all compact. In the following, we state some lemmas including integrations by parts and uniform Gronwall lemma, which are frequently used in our paper. For the proof of Lemma 2.1-Lemma 2.3, we can see [19]. The proof of uniform Gronwall lemma was given in [17,48].
In our article, we will frequently use the following inequalities. So we state them as the lemmas below. For their proof, one can refer to [20] and [54].
Then, there exists a positive constant c independent of v, µ and ν such that Then, there exists a positive constant c independent of v, µ and ν such that In the process of obtaining absorbing ball for the strong solution U in H 2 (T ℧|T S 2 )×H 2 (℧)×H 2 (℧), the uniform Gronwall lemma is used extensively. Therefore, for sake of convenience, we cite it here.
Lemma 2.6 Let f, g and h be three non-negative locally integrable functions on (t 0 , ∞) such that where r, a 1 , a 2 , a 3 are positive constants. Then Before considering the dimensions of the global attractor for the strong solutions to the moist primitive equations, we recall the definitions of strong solution to (1.11) − (1.17).
Now we state the global well-posedness theorem for the strong solution as follows. For the proof of the therem, one can refer to [20].
Then for any τ > 0 given, the global strong solution U of the system (1.11) − (1.17) is unique on the interval [0, τ ]. Moreover, the strong solution U is continuous with respect to initial data in H.
For the reader's convenience, we introduce the definition of global attractor in the following. For more details, we refer to [22,48] and other references. Let (X, d) be a separable metric space and S(t) : X → X, 0 ≤ t < ∞, be a semigroup satisfying: Typically, S(t) is associated with a autonomous differential equation; S(t)x is the state at time t of the solution whose initial data is x.

Definition 2.2 A subset
A in X is said to be a global attractor if it satisfies the following properties: In [54], we prove the result below about the existence of global attractor for the solutions to 3D moist primitive equations.
Theorem 2.1 Assume Q 1 , Q 2 ∈ L 2 (℧) and Q 1 | ξ=1 , Q 2 | ξ=1 ∈ L 2 (S 2 ). Then, for t ≥ 0, the solution operator {S(t)} t≥0 of the 3D viscous PEs of large-scale moist atmosphere (1.11) − (1.17) : ) defines a semigroup in the space V. Furthermore, the results below hold: (1) For any (2) For any t > 0, S(t) is a continuous map in V.  Therefore, the main result of this work is the finite Hausdorff and fractal dimensions of the global attractor in space V. To prove the properties of global attractor, we use a theorem in [33]. For reader's convenience, we cite it below.
Theorem 2.2 Let X be a Hilbert space with norm · X . S : X → X be a map and A ⊂ X be a compact set such that S(A) = A. Suppose that there exist a positive constant c and δ ∈ (0, 1), such that ∀a 1 , a 2 ∈ A.
where Q N is the projection in X onto some subspace (X N ) ⊥ of co-dimension N ∈ N. Then where d H (A) and d F (A) are the Hausdorff and fractal dimensions of A respectively and G a is the Gauss constant:

Uniform estimates and absorbing balls for ∂ t U in the space H
Then, there exists a unique strong solution U := (v, T, q) of (1.11) − (1.17) such that Furthermore, there exists a bounded absorbing ball for U t := ∂ t U in space H.
Proof. The uniqueness of such solutions follows by Proposition 2.1. To obtain uniform estimates of ∂ t U in H, we introduce some notations denoted by To prove ∂ t U is uniformly bounded in H with respect to t, we first prove that there exists a positive constant c independent of t such that t+1 t |∂ s U (s)| 2 2 ds < c (3.24) for arbitrary t ≥ 0. By Lemma 2.1, we have By the boundary conditions (1.15) − (1.16), we arrive at

27)
where the first inequality follows by Lemma 2.4. Then (3.27) implies that for t > 0 It is proved in [54] that there exists a positive constant c independent of t such that for arbitrary t ≥ 0. Therefore, by (3.28) and (3.29), we obtain the uniform boundedness for t+1 t |η(s)| 2 2 ds with respect t. Taking an similar argument of (3.27), we have 1 2 where the first inequality follows by Lemma 2.4. For t > 0, integrating (3.30) with respect to t yields Then taking inner product of (1.11) with v t and using equalities (3.32) − (3.34) yields Using uniform estimates (3.29) and integrating (3.25) with respect to t, we prove that t+1 t |u(s)| 2 2 ds is uniformly bounded with respect to t(t ≥ 0). With these time-uniform a priori estimates of we prove (3.24). In the following, we will show the existence of the absorbing ball of U t in H. Taking derivative with respect to t in (1.13), we have Taking inner product of (3.36) with η and using boundary conditions (1.15) − (1.16), we have where the second equality follows by Lemma 2.3. Therefore, by Hölder inequality, Lemma 2.4, interpolation inequality and Young's inequality we obtain ≤ ε η 2 1 + ε u 2 1 + c|η| 2 2 + c q 8 1 |u| 2 2 + c q 2 1 q 2 2 |η| 2 2 . (3.37) Taking derivative with respect to t in (1.12), we have Since by Lemma 2.3, Consequently, multiplying (3.38) by θ and integrating over ℧ yields By Hölder inequality, Lemma 2.4, interpolation inequality and Young's inequality we have To estimate I 2 , using Hölder inequality, interpolation inequality and Young's inequality we reach Taking an analogous argument as I 2 , we deduce Substituting the estimates of I 1 − I 3 into (3.39), we reach Taking derivative with respect to t in (1.11), we have By the aid of Lemma 2.3 and Lemma 2.1, it follows that Therefore, multiplying (3.41) by u and integrating over ℧ yields
Substituting these estimates of J 1 − J 4 into (3.42), we have For t ≥ 0, denote by Then, it follows from the time-uniform estimates (3.29) that t+1 t f (s)ds is uniformly bounded with respect to t. Summing (3.37), (3.40) and (3.43), we obtain

Dimensions of the global attractor
Theorem 4.1 Assume Q 1 , Q 2 ∈ L 2 (℧) and Q 1 | ξ=1 , Q 2 | ξ=1 ∈ L 2 (S 2 ). Let U 0 ∈ A. Then there is a positive constant c independent of initial condition U 0 and time τ (τ > 0) such that Proof. By direct calculation, we reach Similarly, Taking inner product of (1.13) with A 3 q and integrating over [0, τ ] with respect to t yields Consequently, we have Similarly, we deduce that From [20], we have which combined with the uniform-time estimates proved in [54] implies In view of the energy estimates proved in [54], it follows that According to [54], we know the following estimates Since it is proved in [54] that |T | ξ=1 | 4 and |q| ξ=1 | 4 are uniformly bounded with respect to t, summing the above estimates about T and q and integrating with respect to t over [0, τ ] yields (4.47) Combining (4.45) − (4.47), we prove the last inequality of the theorem. Next, we state our main result of this paper.
Proof. Recalling that in [54] we prove that the strong solution U to (1.11) − (1.17) is Lipschitz continuous with respect to initial data in space V. Therefore, in order to prove our theorem, according to Theorem 2.2, we only need to show the condition (ii) of Theorem 2.2.
Let initial data U i = (v i , T i , q i ), i = 1, 2 be two strong solutions to (1.11) − (1.17) with initial data U i (0) = (v i,0 , T i,0 , q i,0 ) ∈ A, where A is the global attractor given by Theorem 2.1. In view of the invariance property of global attractor and Proposition 2.1, we can assume U i (0) ∈ D(A 1 ) × D(A 2 ) × D(A 3 ) with i = 1, 2. By virtue of energy estimates we can easily show that |∂ t U i (0)| 2 are finite for i = 1, 2. Consequently, according to Theorem 3.1, we know that |∂ t U i (t)| are uniformly bounded with respect to t(t ≥ 0). Meanwhile, we also have Then we derive from (1.11) − (1.17) that µ, ϑ and ρ satisfy Since operators A i , i = 1, 2, 3 are positive selfadjoint with compact resolvent, we denote by {λ i,k } k≥1 the corresponding eigenvalues of A i . Obviously 0 ≤ λ i,1 < λ i,2 < λ i,3 · · · and λ i,k → ∞ as k → ∞ for arbitrary i ∈ {1, 2, 3}. Let P i,n , i = 1, 2, 3 be the orthogonal projector in H i onto the subspace spanned by first n eigenvectors of A i . Denote Q i,n = I − P i,n , i = 1, 2, 3 and n ∈ N.
Consequently, for small T and large λ n , we have ϕ(T ) ≤ δψ(0), which combined with Theorem 2.2 proves this theorem.