Existence of a strong solution to moist atmospheric equations with the effects of topography

In this paper, we consider the primitive three-dimensional viscous equations for large-scale atmosphere dynamics with topography effects and water vapor phase transition process. This modified climate model is commonly used in weather and climate predictions, and few theoretical analyses have been performed on them. The existence and uniqueness of a global strong solution to this climate model is established based on the initial data assumptions.


Introduction
Applying the Boussinesq and hydrostatic approximation, the so-called primitive atmospheric equations can be derived, and this system is formulated in terms of the Navier-Stokes equations with the Coriolis force, the thermodynamic equations and the diffusion equation for vapor [19,22,28]. However, few studies have considered the effects of topography, changes of the external forcing with time and water vapor phase transitions, which have remarkable influences on climate dynamics. Based on realistic conditions, Zeng [29] showed the modified climate dynamics model in the following ways: (1) the effects of topography on the climate dynamics are considered; (2) the phase change of water vapor is studied; (3) the upper atmospheric pressure is set to zero; and (4) the anelastic approximation is not required in the system.
Then we show a moving frame (θ , λ, ζ , t), where θ ∈ [0, π] is the colatitude, λ ∈ [0, 2π] is the longitude, ζ = p/p s ∈ [0, 1], p ∈ [0, p s ] is the atmospheric pressure, where p s (θ , λ, t) is the atmospheric pressure on the surface of the Earth, and t is the time. The atmospheric state functions can be defined by the atmospheric horizontal velocity V = (v θ , v λ ), vertical velocityζ , temperature deviation T , geopotential deviation Φ and Earth surface pressure deviation p s , the specific humidity q, and the liquid water content m w . As the reference standard temperature T(ζ ), the reference standard geopotential Φ(ζ ) and the reference standard Earth surface pressure p s (θ , λ, t) are determined, we can find that T(ζ ) + T (θ , λ, ζ , t) is the atmospheric temperature T(θ , λ, ζ , t), Φ(ζ ) + Φ (θ , λ, ζ , t) is the geopotential Φ(θ , λ, ζ , t) and p s (θ , λ, t)+p s (θ , λ, t) is the surface pressure of Earth p s (θ , λ, t). All of these state functions satisfy the following system: where ω is the Earth's rotation angular velocity; g is the gravitational acceleration; c 0 , c p and R are thermodynamic parameters; μ i and ν i (i = 1, 2, 3, 4) are the diffusion coefficients; 2ω cos θ 0 -1 1 0 V stands for the Coriolis force on the atmosphere. Moreover, we introduce dQ/dt, which is composed of the radiant heating H 1 and the latent heating H 2 caused by the water vapor phase transition, and they give the forms Here, we simply assume that Newtonian cooling holds, where κ a is a positive constant. H 2 should be closely related to the microphysics processes of condensation and evaporation, and we have (1.4) where L is the latent heat constant, and which represents the mass of water that is added by condensation or removed by evaporation. δ 21  where q m is the saturation special humidity. We assume that W (T) is a globally Lipschitz bounded function, namely where R v is the gas constant for water vapor. The term P r is the precipitation rate, which takes the following form: where (1.9) and 0 < α < 1, β > 0.
In this work, we show that the differential operators grad := ∇, div := ∇· and on the spherical surface are Moveover, we choose the modified smooth velocity field (V * ,ζ * ) [18]. Here we set V := 1 0 V (θ , λ, ζ , t) dζ , and decompose p s V into the three parts via where the function Ψ satisfies and V * can be obtained as follows: Meanwhile, we getζ * as the solution of with the boundary conditioṅ ζ * = 0, as ζ = 0, (1.15) which implies thaṫ Then from the definition of V , we know and we havė Then we can give the boundary conditions without relief are as follows: All functions are π periodical with respect to θ , 2π periodical with respect to λ, and where the given function T s (θ , λ) is the reference standard surface temperature of Earth, q * m is the reference standard surface saturation special humidity of Earth, the given function V 10 (θ , λ) is the 10-m wind speed, k s1 , k s2 and k s3 are positive constants and the chilling coefficient f (|V |) is a positive function of |V |.
There is a huge literature on the study of various atmospheric problems; e.g., in the 1990s, Lions et al. [19][20][21] gave the new formulation for the primitive equations of largescale atmosphere and ocean, and proved the existence of global weak solutions to the initial boundary value problem. The existence of global attractors to the primitive equations associated with the atmospheric evolution process have been studied by Chepzhov and Vishik [9]. Wu et al. [24] obtained the existence of global weak solutions to the climate model with the effects of topography. Furthermore, Huang and Guo [14,15] proved the existence of the atmospheric global attractors of the atmospheric motion model without or with the effects of topography, respectively, and they also obtained the existence and the asymptotic behaviors of a weak solution. Recently, Lian et al. [16,18] addressed the L 1 -stability of weak solutions to the atmospheric equations with or without the effects of topography.
Meanwhile, there are many studies of strong solution for viscous primitive large-scale ocean and atmosphere equations. Taking the values of the initial data to be sufficiently small, the global existence of a strong solution to primitive equations was investigated by Guillén-González et al. [10], and the local existence of strong solution to the system for all initial data was also proved. Furthermore, Temam and Ziane [23] considered the coupled atmosphere-ocean equations and showed the local existence of strong solution. Cao and Titi [5] proved the global well-posedness and finite-dimensional global attractors of the 3D planetary geostrophic model. The well-posedness and long-time behavior of the strong solution to the horizontal hyper-diffusion 3D thermocline planetary geostrophic model were also obtained by Cao et al. [8]. In particular, for all initial data, Cao and Titi [6] proved the global existence of strong solution to three-dimensional viscous primitive equations. Furthermore, Cao et al. [1,7] investigated the global well-posedness of threedimensional viscous primitive equations with only vertical diffusion, and proved the global existence of strong solution using H 2 initial data. Cao et al. [2] also considered the initial boundary value problem of the primitive equations with only horizontal diffusion in the temperature equation, and they obtained the global existence of strong solutions using H 2 initial data. Recently, the initial boundary value problem of the primitive equations with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation were addressed by Cao et al. [3]. Cao et al. [4] also studied the 3D primitive equations with only horizontal eddy viscosity in the horizontal momentum equations and only vertical diffusivity in the temperature equation. In these studies, Cao et al. gave the H 2 regularity estimates of the strong solution. However, the upper atmospheric pressure in the references above is treated as a positive constant, in particular, Lian and Zeng [17] proved the existence of strong solution and trajectory attractor to a modified climate dynamic model, while the upper atmospheric pressure is treated as zero.
Up to now, there are some important studies about the well-posedness of the moist atmosphere. For example, Guo and Huang [11] proved the existence of global weak solutions and attractors to the primitive equations of a moist atmosphere. In addition, Guo and Huang [12] investigated the existence and uniqueness of global strong solution, and they addressed the existence of the universal attractor for the large-scale moist atmosphere. Under a physically reasonable assumption, Zelati and Temam [27] derived the existence and uniqueness of solution for the specific humidity equation. They also studied the coupling of the specific humidity equation with the temperature equation. Based on [27], Zelati et al. [25] addressed the uniqueness of solution for the system of moist atmosphere with saturation. Additional properties and regularity results of the solutions were also proven. Zelati et al. [26] proved the global existence of quasi-strong and strong solutions of primitive equations in the interest of thinking over vital phase transition phenomena due to air saturation and condensation. Using the ideas of Cao and Titi [6], Hittmeir et al. [13] showed the well-posedness for the full moist atmospheric flow model, where the moisture model is coupled to the large-scale atmosphere equations.
In this paper, we investigate the initial boundary value problem for the climate dynamics model with effects of topography and water vapor phase transition. Moreover, the heating is caused by internal sources depending on the atmospheric motion rather than by an external one, and the internal heating function due to the phase change of water vapor is approximated by some properly analytical functions suitable for mathematical analyses. Meanwhile, the upper atmospheric pressure is treated as zero. Based on the initial data assumptions, the existence and uniqueness of a global strong solution to this modified climate model is established.
We organize this paper as follows. In Sect. 2, we present the major results associated with the existence of a unique global strong solution. Section 3 gives some useful lem-mas. In Sect. 4, we will provide some useful proofs, followed by several important a priori estimates. Finally, the conclusions are drawn in Sect. 5.

Main results
We will show the global well-posedness of strong solution to (1.1) and give a simple version of system (1.1) firstly. From the boundary conditions (1.19), we have and 3) into (1.1) and define the unknown function U := (V , T , q, m w ) T , then we give the simplification of the system (1.1) Then we can state the main results of the present paper as follows.

Theorem 2.1
For any M > 0, we assume that where W (s) is a globally Lipschitz bounded function, T 0 , C 1 , C 2 , C 3 and C 4 are positive constants. (2.12)

Some lemmas
and and Furthermore, we recall some useful interpolation inequalities as follows: (1) For f ∈ H 1 (S 2 ),

The a priori estimates
Next, we will consider the a priori estimates for the strong solution U to the system (2.4).
Using definition of the fluctuation V in [17], we can give the equations of the fluctuation V as follows: We let which also implies Note that and we have where From (4.1), (4.6) and (4.7), we get Subtracting (4.11) from (2.4) 1 , we also find that the fluctuation V satisfies the following equation: with the boundary conditions 14) Then we have the usual energy inequality as follows.
Proof Multiplying (2.4) by p s U and using the boundary conditions, we obtain Thanks to the Young inequality, the Hardy inequality and the Hölder inequality, we get where C > 0 denotes a constant independent of time M. Using (4.18)-(4.22) and the Young inequality, we deduce by applying Gronwall inequality, we can prove (4.16).
Remark 4.2 Note that, from [17], for the strong solution V , T to the system (2.4) 1,2 , and we omit the details of proof here.

Lemma 4.3
Under the assumptions of Theorem 2.1, for any M > 0 given, the specific humidity q to the system (2.4) 3 satisfies By virtue of (3.5) and (4.16), the Cauchy-Schwarz inequality, the Hardy inequality, the Gagliardo-Nirenberg-Sobolev inequality and the Young inequality, we find that Ω V * · ∇ q +ζ * ∂q ∂ζ p s |q|q dσ dζ = 0, (4.32) where C(M) > 0 denotes a constant dependent of time M and ε > 0 is a small constant such Thanks to (3.6), the Cauchy-Schwarz inequality, the Hardy inequality, the Gagliardo-Nirenberg-Sobolev inequality and the Young inequality, we know that Ω V * · ∇ m w +ζ * ∂m w ∂ζ p s |m w |m w dσ dζ = 0, (4.40) where C(M) > 0 denotes a constant dependent of time M and ε > 0 is a small constant such that Using (3.5), the Cauchy-Schwarz inequality, the Hardy inequality, the Gagliardo-Nirenberg-Sobolev inequality and the Young inequality, we know that Ω V * · ∇ q +ζ * ∂q ∂ζ p s q 3 dσ dζ = 0, (4.47) Thanks to (3.6), the Cauchy-Schwarz inequality, the Hardy inequality, the Gagliardo-Nirenberg-Sobolev inequality and the Young inequality, we get Ω V * · ∇ m w +ζ * ∂m w ∂ζ p s m 3 w dσ dζ = 0, (4.55) where C(M) > 0 denotes a constant dependent of time M and ε > 0 is a small constant such that by applying the Gronwall inequality, we infer (4.53).

Lemma 4.7
Under the assumptions of Theorem 2.1, for any M > 0 given, the specific humidity q to the system (2.4) 3 satisfies Proof Taking the derivative with respect to ζ of (2.4) 3 , we find that Multiplying (4.61) by p s q ζ , we have Similarly to (4.32), we know -Ω V * · ∇ q ζ +ζ * q ζ ζ p s q ζ dσ dζ = 0.
where ε > 0 is a small constant.
By applying the Hardy inequality, we find that where ε > 0 is a small constant, by (4.65), Thanks to (2.4) 3 and by the boundary conditions, we know that By virtue of (4.16) and (4.26), we obtain Using (4.63)-(4.71), we have which combining with (4.16), (4.26), (4.45) and the Gronwall inequality shows (4.60), where we use the fact that Proof Taking the derivative with respect to ζ of (2.4) 4 , we find that Multiplying (4.75) by p s m wζ , we find that By virtue of (4.16), (4.24)-(4.26), (4.53), the Gagliardo-Nirenberg-Sobolev inequality, the Young inequality and the fact that V * ζ = V ζ , we deduce that where ε > 0 is a small constant.
Using (4.16), the Young inequality and the Hardy inequality, we find that

Lemma 4.9
Under the assumptions of Theorem 2.1, for any M > 0 given, the specific humidity q to the system (2.4) 3 satisfies Proof Taking the inner product of (2.4) 3 with q, we find that Thanks to (4.16) and (4.25), we get where ε > 0 is a small constant. By (4.16), (4.28) and (4.60), we infer where ε > 0 is a small constant. Applying (4.16), (4.60), the Hardy inequality and the Young inequality, we also obtain where ε > 0 is a small constant. By (4.85)-(4.90), we obtain  Proof Taking the inner product of (2.4) 4 with m w , we know that By (4.16) and (4.25), we obtain where ε > 0 is a small constant. Thanks to (4.16), (4.28) and (4.74), we have Ωζ * m wζ m w dσ dζ where ε > 0 is a small constant. By (4.28), the Cauchy-Schwarz inequality, the Hardy inequality and the Young inequality, we get where C(M) > 0 denotes a constant dependent of time M and ε > 0 is a small constant such that (4.97) thanks to the Gronwall inequality, we deduce (4.92).

Proof of Theorem 2.1
Proof By (4.16), (4.26)-(4.29), (4.60), (4.74), (4.83), (4.92) and the proof of the short time existence in Refs. [10,23], we can extend the strong solution U to the system (2.4) beyond M * , contradicting the fact that M * is a finite maximal time of existence. This contradiction means that M * = +∞; then we get the global existence of strong solution.
Next, we will show the uniqueness of global strong solution as follows: let (V 1 , T 1 , q 1 , m w1 ) and (V 2 , T 2 , q 2 , m w2 ) be two strong solutions of system (2.4) on the time interval [0, M] with the initial data (V 01 , T 01 , q 01 , m w01 ) and (V 02 , T 02 , q 02 , m w02 ), respectively. De- Then V , T , q, m w satisfy the following system: with the initial data and boundary conditions as follows: Note that, from [17], we can find that Next, multiplying (5.1) 3 by p s q, we know  3 4 × q 1 4 L 2 (Ω) q L 2 (Ω) + ∇q L 2 (Ω) + q ζ L 2 (Ω) Similarly, taking the inner product of (5.1) 4 with p s m w , we get   applying the Gronwall inequality, we can complete the proof.