Global Well-Posedness of the Fractional Tropical Climate Model

In this paper, we consider the Cauchy problem of 3-dimensional tropical climate model. This model reflects the interaction and coupling among the ba-rotropic mode u, the first baroclinic mode v of the velocity and the temperature θ . The systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances. Mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization. The aim here is the global strong solution with the least dissipation. By energy estimate and delicate analysis, we prove the existence of global solution under three different cases: first, with the help of damping terms, the global strong solution of the system with 2 u


Introduction
In this paper, we consider the following tropical climate model with fractional diffusion and nonlinear damping terms ( ) ( )  x ∈  , the vector fields ( ) ( ) ( ) ( ) ( ) , , , , , , u x t u x t u x t u x t = and ( ) ( ) ( ) ( ) ( ) , , , , , , v x t v x t v x t v x t = denote the barotropic mode and the first baroclinic mode of the velocity, respectively.The scalar functions ( ) , p x t and ( ) represent the pressure and the temperature, respectively.0  ( ) ( ) The tropical climate model (1.1) -(1.5) was originally introduced by Frierson-Majda-Pauluis in [1] without any dissipation terms ( 0 µ ν η = = = ) in order to perform a Galerkin truncation to the hydrostatic Boussinesq equations.For more background about the tropical climate model, we refer to [2].If the effect of temperature is ignored, the system is similar in form to the generalized MHD equation with divergence free condition both on u and v.
Firstly, we recall some global existence results for the tropical climate model without any damping terms.Ye [3] obtained global regularity for a class of 2-dimensional tropical climate model with 0 α > , 1 β γ = = .Li and Titi [4]   established the 2-dimensional global well-posedness of strong solution for the system with 1 α β = = , 0 η = by introducing a combined quantity called pseudo baroclinic velocity.Wan [5] proved the global well-posedness of the classical solutions to the climate model with the dissipation of the first baroclinic model of the velocity and some damping terms ( Dong et al. [6] investigated the case when Next, we will give some global existence results for the system.The global existence and uniqueness of a strong solution is established provided 1 4 by Yuan and Chen [11].Yuan and Zhang [12] proved the global regularity assuming that one of the following three condition holds true: 1) 1 2 , 4 ρ ρ α ≥ − with 1 α β γ = = = .Berti, Bisconti and Catania in [13] provided a regularity criterion to obtain the smoothness of the solutions < , and ( ) Since the specific values of , , µ ν η do not play a special role in our discussions, for the sake of simplicity, we set It should be noted that all the above mentioned works for the system require the restriction that at least one of the , , α β γ must greater than or equal to 1.A natural question is that what would happen if they were all less than 1.In this paper, the focus of our work is to discuss the global exitence when all , , α β γ are less than 5 4 . Also, we improved the previous global solution when , , α β γ , which is meaningful.
Our main results are stated as follows: Theorem 1.1.Assume ( ) ( )  ) ( ) and ( ) ( ) , , Throughout the whole paper, we use ( ) H  denote the homogeneous Sobolev space with the norm and nonhomogeneous Sobolev space with the norm ( ) , respectively.C denotes a generic positive constant, and it may be different from line to line.
We find that when , , α β γ are relatively large, with the help of dissipative terms, the global existence is relatively easy to obtain.But when all , , α β γ are relatively small, the global existence is not easy to obtain, and it needs to be controlled by damping terms.

Preliminaries
We state the Gagliardo-Nirenberg inequality in Lemma 2.1 and the Kato-Ponce type inequality in Lemma 2.2.
Lemma 2.1.( [14]) Let u belongs to q L in n  and its derivatives of order m, where ( ) for all α in the interval

Proof of the Theorem 1.1
Proof.Multiplying (1.1), (1.2), (1.3) respectively by , , u v θ , after integrating by parts and taking the divergence free property into account, we have ( ) Next, applying the operator ∇ to (1.1), (1.2), (1.3) and taking the ( ) L  inner product to the resultants with ( ) Integration by parts implies Because of the divergence-free condition of u, the estimates for 1 I , 2 I , 3 I and 6 I are similar, and we take the detailed estimate for 3 I as an example.
For 3 I , when 3 1 4 α < ≤ , using Kato-Ponce type inequality and Young's inequality, we can get here we have used the following Gagliardo-Nirenberg inequality Therefore, we need 3 4 . Then, we get Similarly, for the terms 1 I , 2 I and 6 I , we can obtain the following estimates ( ) ( ) ( ) It remains to estimate the term 5 I .However, this term can not be treated as above due to the non-divergence free property of v, so we estimate 5 I as follows ( ) , div , , .
For 51 I , similarly to 3.9, we obtain ( ) and for 52 I , we have Therefore, we have ( ) Taking the above estimations (3.9) -(3.13) into (3.6),we obtain that This completes the proof of the Theorem 1.1 by Gronwall's inequality and energy estimate (3.5).

I I + = (4.17)
We take the detailed estimate for 3 I as an example.
For 3 I , when 5 1 4 α ≤ ≤ , using Kato-Ponce type inequality, we can get ( ) ( ) Here, we can yeild Then, we have Due to the non-divergence free condition we have used, so we can obtain other terms using the same way.Then we have ( ) ( ) Taking the above estimations (4.17) -(4.22) into (4.16),we obtain that ( ) This completes the proof of the Theorem 1.2 by Gronwall's inequality and energy estimate (4.15).
Integration by parts implies ( ) ( ) ( ) here, we need 5 4 α ≥ and have used the following inequalities Similarly, we can obtain 5 I as follows ( ) ( ) Though above estimate for 3 I and 5 I , we know that we don't need to use the divergence free condition.Therefore, we can obtain ) , 16 16 Taking the above estimations (5.25) -(5.38) into (5.24),we obtain that For the .
For 2 E , we can estimate it as follows ( ) 1 .
For 1 E , we have with damping terms; finally, the global strong solution of the system for 5 4 real parameters.For s ∈  , the fractional Laplacian operator arbitrarily large and obtained the global smooth solution of d-dimensional tropical climate model.Yu, Li and Yin establish the global regularity for the system with 0 and Wang [10] dealt with the global well-posedness and large-time behavior of the 2D tropical climate model with small initial data for 1 α γ = = , 0 β = and 0 µ > , 0 ν = , 0 η = .Journal of Applied Mathematics and Physics the system (1.1) -(1.5) has a global strong solution

(
= − Journal of Applied Mathematics and Physics then, for any 0 T > , the system (1.1) -(1.5) has a global strong solution any 0 T > , the system (1.1) -(1.5) has a global strong solution ( )
5.32) Journal of Applied Mathematics and Physics