LOCAL WELL-POSEDNESS FOR THE TROPICAL CLIMATE MODEL WITH FRACTIONAL VELOCITY DIFFUSION

. This paper deals with the Cauchy problem for tropical climate model with the fractional velocity diﬀusion which was derived by Frierson- Majda-Pauluis in [16]. We establish the local well-posedness of strong solutions to this generalized model.


1.
Introduction. This paper considers the following tropical climate model with fractional velocity diffusion: ∂ t θ + (u · ∇)θ + divv = 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), θ(x, 0) = θ 0 (x), where x ∈ R d with d ≥ 2, u = (u 1 (x, t), u 2 (x, t), · · · , u d (x, t)) and v = (v 1 (x, t), v 2 (x, t), · · · , v d (x, t)) are vector fields representing the barotropic mode and the first baroclinic mode of the velocity, respectively, while p = p(x, t) and θ = θ(x, t) denote the scalar pressure and temperature, respectively. α ≥ 0, β ≥ 0 are real parameters. We identify the ideal case α = β = 0 as the original system derived in [16] stands for none of the diffusion terms of the barotropic mode and the first baroclinic mode of the velocity. Fractional Laplacian operator Λ = (−∆) 1/2 is defined in terms of the Fourier transform, The tropical climate model is derived from the primitive equations by performing a Galerkin truncation to the first baroclinic mode. In 1990s, Lions, Temam and Wang originally in [24]- [26] established the global existence of weak solutions for the viscous primitive equations (but the uniqueness is still unknow for 3D case). Meanwhile, the global existence of strong solutions for the viscous primitive equations is given in [9] and [20]. For the case of the inviscid primitive equations, in [21] and [22] the authors established the local existence of solutions on a bounded domain. [10] and [36] showed that the corresponding smooth solutions of the inviscid primitive equations blow up in finite time for certain class of initial data. Very recently, global well-posedness of strong solutions for the 2D equations (1)-(5) with α = β = 1 were obtained in [23]. The problem of global well-posedness of the n-dimensional (n ≥ 3) equations (1)- (5) is an outstanding challenge problem for obvious reason that it is the model includes the Navier-Stokes (or Euler when α = β = 0) equations as a special case. Thus, it is a natural question to consider the local well-posedness for this model.
We survey the local existence and uniqueness of strong solutions to (1)-(5) with any α ≥ 1, β ≥ 0 or 1 > α ≥ 0, β ≥ 1. More precisely, our main result is the following theorem. Remark 1. In recent years there has been a surge of activity focused on the nonlocal (especially fractional diffusion) operators to replace the standard Laplace operator because of their connection with many real-world phenomena. The new operators do not act by point wise differentiation but by a global integration with respect to a singular kernel. If 0 < α < 2, we can also use the integral representation where P.V. denotes principal value and C d, is a normalization constant. In the limits C d,α ≈ α as α → 0 and C d,α ≈ 2 − α as α → 2 it is possible to recover respectively the identity or the standard Laplacian, cf. [5].
Remark 2. The motivation of this paper comes from [15], [17] for MHD, because of divv = 0, we have to split the nonlinear term div(v ⊗ v) in (1) into two parts (v · ∇)v and vdivv. Then the difficulty occurs when we estimate the high order norm of vdivv.
The rest of this paper is divided into two sections followed by an appendix. In Section 2, we review some elementary results and prove the result for the local a priori estimates. In Section 3, we complete the proof of Theorem 1.1. The definition and related proposition of Littlewood-Paley decomposition Besov spaces used in this paper are provided in the appendix.
2. Preliminaries and local a priori estimates. In this section we recall some elementary results which will be used in this paper. Then we establish a local a priori estimates for smooth solutions of (1)-(5), which play a key role in the proof of Theorem 1.1.
The following lemma is about Fourier truncation estimates, where the inequalities are proved in [15].
Lemma 2.1. Let S R be the Fourier truncation operator and it is defined as follows: where B R denotes the closed ball of radius R centered at 0 and 1 B R denotes the characteristic functions on B R . Then the following estimations are satisfied For self-consistence of our paper, we will give another proof of the Lemma 2.2 in [17] by the Littlewood-Palay decomposition and Besov space techniques in the appendix.
Lemma 2.2. (Commutator Estimates) Let α ≥ 0, for any s > max d 2 here ·, · denotes the L 2 (R d ) inner product and C is a constant depending on d, s only.
The result for the local a priori bound can be stated as follows the corresponding solution of equations (1)- (5). Then, there exists a time Proof of Proposition 1. We first prove the basic energy estimate. Taking the L 2 (R d ) inner product to equations (1), (2), (3) with (u, v, θ) , after integrating by parts and taking the divergence free property into account, we have 1 2 Next, we prove the H s estimate. Applying the operator Λ s to (1), (2), (3) and taking the L 2 (R d ) inner product to the resultants with (Λ s u, Λ s v, Λ s θ) , after integrating by parts and using the cancellation property where Using the fact that u is divergence free, we have combining with commutator estimate (8) and (9) yields where The estimate of I 41 is given by (9).
We recall the Kato-Ponce type commutator estimate (refer to [18] for details) Using the Kato-Ponce type commutator estimate together with the Sobolev inequality, the I 42 can be estimated as follows Here we used the following Sobolev embedding and the following Gagliardo-Nirenberg interpolation inequalities The most difficult term I 43 can be split into three parts Similar to (9), I 431 can be estimated as follows Using the Sobolev inequality (12), Gagliardo-Nirenberg interpolation inequalities (13), we have and the second term can be estimated as follows Inserting all the estimates above in (11) and combining this with basic energy estimate (10), we obtain Using Young's inequality and Gronwall's inequality, we deduce that . This completes the proof of Proposition 1.
From (15) we can immediately obtain the following results. 3. Local existence and uniqueness. In this section, we prove Theorem 1.1 through an approximation procedure that is given by Friedrichs. Now, we consider the following truncated tropical climate model We introduce the function space Thus we can reduce the truncated tropical climate model (16)- (20) to the following where For each fixed R, Using (6), (7) and the fact of S R f H s ≤ C f L 2 , we can verify We denote the solution of (16)- (20) by (u R , v R , θ R ). Similarly as the proof of Proposition 1 and due to (u R , v R , θ R ) H s ≤ (u 0 , v 0 , θ 0 ) H s , we can obtain a uniform local bounds of the solution as follows The above inequalities are uniform in R.
Furthermore, these uniform bounds allow us to show that (u R , v R , θ R ) converge strongly in L ∞ ([0, T 0 ]; L 2 (R d )), and one has the following proposition.
Proof of Proposition 2. Taking the difference between the equations (16)-(18) for R and R , using we have Taking the L 2 (R d ) inner product to equations (24)-(26) with u R − u R , v R − v R and θ R − θ R respectively, summing the resultants up and using the cancellation property We can split each term into three parts and without loss of generality, we assume R > R ≥ 1. For I 4 , we have For the first term I 41 here we have used the following inequality (refer to [6] for details) . Then use the Gagliardo-Nirenberg inequalities s , here we need s > 1 and suitably small. Using the Sobolev inequality (12), Gagliardo-Nirenberg interpolation inequalities (13) and Young's inequality, we have After integrating by parts and taking the divergence free property into account, we have I 43 = 0.

CAOCHUAN MA, ZAIHONG JIANG AND RENHUI WAN
Hence we obtain that Similarly, the J 1 and J 6 can be estimated as follows

Now, we estimate other terms
Here choosing > 0 so small such that 0 < 1 + < s, and we have used Gagliardo-Nirenberg interpolation inequalities (13) and the following estimates where we have used the Hausdorff-Young inequality, and choosing > 0 so small such that Similarly, the I 31 and I 51 can be estimated as follows Using the Sobolev inequality (12), Gagliardo-Nirenberg interpolation inequalities (13) and Young's inequality, we have Similarly, the I 32 can be estimated as follows

Integration by parts, we estimate other terms
When α ≥ 1, integration by part, we have :=I 331 + I 332 .
Similar to I 22 , the I 331 can be estimated as follows Next, we estimate the last term I 332 where we used the Sobolev inequality (12) and Gagliardo-Nirenberg interpolation inequalities When 1 > α ≥ 0, β ≥ 1, we have Inserting all the estimates above in (27) and using the bound
Proof of Theorem 1.1. It is straightforward to use the estimate in the proof above to . Combining Propositions 1 and 2 and using Sobolev interpolation inequality, for any 0 < s < s, , respectively. Nonlinear terms are also strongly convergent in a suitable space. Thus (u, v, θ) is indeed a strong solution of (1).
Using the uniform bounds proposition, one may use the Banach-Alaoglu theorem to extract a weakly- * convergent subsequence such that hence the limit satisfies The proof of uniqueness for (1) is similar to the proof of Proposition 2 and is omitted.
In addition, the uniform bound of (u, v, θ) allows us to show the weak time continuity. We have u, v, θ ∈ C W ([0, T * ]; H s (R n )).
Appendix. In this appendix, we will prove Lemma 2.2. Firstly, we recall the definitions and some properties of the Besov spaces. This kind of space plays an important role in studying nonlinear partial differential equations (see [2,3,29,31,35] and references therein). Let Choose two nonnegative smooth radial functions χ, ϕ supported, respectively, in B and C such that We denote ϕ j = ϕ(2 −j ξ), h = F −1 ϕ andh = F −1 χ, where F −1 stands for the inverse Fourier transform. Then the dyadic blocks ∆ j and S j can be defined as follows Formally, ∆ j = S j − S j−1 is a frequency projection to annulus {ξ : C 1 2 j ≤ |ξ| ≤ C 2 2 j }, and S j is a frequency projection to the ball {ξ : |ξ| ≤ C2 j }. One can easily verifies that with our choice of ϕ With the introduction of ∆ j and S j , let us recall the definition of the Besov space. Let s ∈ R, (p, q) ∈ [1, ∞] 2 , the homogeneous spaceḂ s p,q is defined bẏ and S denotes the dual space of S = {f ∈ S(R d ); ∂ αf (0) = 0; ∀ α ∈ N d multi-index} and can be identified by the quotient space of S /P with the polynomials space P. Let s > 0, and (p, q) ∈ [1, ∞] 2 , the inhomogeneous Besov space B s p,q is defined by Additionally, we have the following equivalence relations.
f Ḃs Bernstein's inequalities are useful tools when dealing with Fourier localized functions and these inequalities trade integrability for derivatives. The following proposition provides Bernstein type inequalities for fractional derivatives.
for some integer j and a constant K > 0, then .
2) If f satisfies for some integer j and constants 0 < K 1 ≤ K 2 , then where C 1 and C 2 are constants depending on α, p and q only.
By Hölder inequality and Bernstein's inequality, we estimate I 1 , I 2 , I 3 , I 4 as follows ≤C u H s+α v 2 H s . Finally, we prove the commutator estimate (9). Using inhomogeneous Bony's decomposition, we have By Hölder inequality and Bernstein's inequality, we estimate J 1 , J 2 , J 3 , J 4 as follows ≤C u H s+α v 2 H s , and H s . This completes the proof of Lemma 2.2.