GLOBAL REGULARITY RESULTS FOR THE CLIMATE MODEL WITH FRACTIONAL DISSIPATION

. This paper studies the global well-posedness problem on a tropical climate model with fractional dissipation. This system allows us to simultane- ously examine a family of equations characterized by the fractional dissipative terms ( − ∆) α u in the equation of the barotropic mode u and ( − ∆) β v in the equation of the ﬁrst baroclinic mode v . We establish the global existence and regularity of the solutions when the total fractional power is 2, namely α + β = 2.


1.
Introduction. This paper focuses on the global existence and regularity of solutions to the initial-value problem on a tropical climate model with fractional dissipation where the vector fields u = (u 1 , u 2 ) and v = (v 1 , v 2 ) denote the barotropic mode and the first baroclinic mode of the velocity, respectively, and the scalar p denotes Theorem 1.1. Let s > 2. Assume the initial data (u 0 , v 0 , θ 0 ) satisfies Consider (1) with α and β satisfying α + β = 2, 1 < β ≤ 3 2 .
We remark that the case α + β = 2 and 3 2 < β < 2 is no more difficult than the case presented here and will be worked out later. In addition, we also examine a special case of α + β = 2, namely α = 2 and β = 0. We establish the global existence and uniqueness of the solutions when the initial data is in H s with s > 2.
Our approach for proving Theorem 1.1 is new and different from those in [10] and [14]. The proof of Theorem 1.1 boils down to proving global a priori bounds. The global bound for the L 2 -norm of (u, v, θ) follows directly from (1). The proof of the global H 1 -bound is more difficult and is at the core of the proof of Theorem 1.1. For notational convenience, we set ν = η = 1. Our approach for proving the global H 1 -bound is new. We take the structure of (1) into full account and reformulate (1) in terms of the variables where Λ = (−∆) 1 2 and the notation A : B for two matrices A = (a ij ) and B = (b ij ) is defined as The advantage of (2) is that the equations of ω and j do not involve θ. Due to the lack of dissipation in the equation of θ, it is difficult to deal with the term ∆θ in the equation for h. This motivates us to consider the following combined quantity which satisfies (by combining the equations of h and θ) We work with (2) and (3) to prove the global bound for (ω, j, H), which reads This global bound does not immediately translate into a global bound for t 0 Λ β ∇v 2 L 2 dτ < ∞.
As in Lemma 1.3 stated below, the L 2 -norm of the gradient can be represented in terms of the L 2 -norms of the curl and the divergence, namely , but the trouble is that we do not have control of Λ 2−β θ L 2 . Since there is no dissipation in the equation of θ, we need to control ∇u L ∞ or its equivalent such as ω H 1 in order to control any derivative of θ.
This prompts us to prove global bounds for ω in more regular settings. We first show a global bound for ω L 2 α , which allows us to further prove the global bound for t 0 ∇ω 2 L 2 dτ < ∞. This global bound is enough for us to control the nonlinear terms in the estimate of the H s via the logarithmic Sobolev inequality, for s > 2, The proof of Theorem 1.2 is achieved through a two-stage process. The first stage proves a global H 1 -bound via energy estimates and an improved version of Gronwall's inequality while the second stage establishes the global H s bound by making use of the global H 1 bound.
Finally we supply a basic fact that relates ∇F to ∇ × F and ∇ · F for a vector field F . As we know, for a divergence-free vector ∇ · F = 0, If F is not divergence-free, ∇ · F = 0, in general ∇F L 2 = ∇ × F L 2 . The following lemma relates the L 2 -norms of ∇F , ∇ × F and ∇ · F and provides a bound for the L q -norm of ∇F . This lemma will be used repeatedly throughout the rest of this paper. Lemma 1.3. For any vector field F , and, for 2 < q < ∞, (4) follows from the identity Plancherel's theorem and a direct calculation. (5) follows from a variant of (6), the Calderon-Zygmund inequality on singular integral operators. The rest of this paper is divided into two sections followed by an appendix. Section 2 provides the proof of Theorem 1.1 while Section 3 proves Theorem 1.2. The appendix supplies some of the inequalities as well as the definition of Besov spaces.
2. Proof of Theorem 1.1. This section proves Theorem 1.1. As we know, the proof of Theorem 1.1 boils down to the global a priori bounds in H s . The proof is achieved in several steps, which successively establish the global bounds in more and more regular functional settings.

2.1.
Global H 1 -bound. The subsection proves the following global H 1 bound. Proposition 1. Assume (u 0 , v 0 , θ 0 ) obeys the assumptions stated in Theorem 1.1. Assume α + β = 2 and 1 < β ≤ 3 2 . Let (u, v, θ) be the corresponding solution. Then (u, v, θ) obeys the following global H 1 -bound, for any t > 0, In order to prove Proposition 1, we first state the following global L 2 bound for (u 0 , v 0 , θ 0 ) and the global L q -bound for θ. Lemma 2.1. Assume (u 0 , v 0 , θ 0 ) obeys the assumptions stated in Theorem 1.1. Let (u, v, θ) be the corresponding solution. Then (u, v, θ) obeys the following global L 2 -bound, for any t > 0, In addition, for any q satisfying 2 ≤ q ≤ 2 2−β and for any t > 0, Proof of Lemma 2.1. The global L 2 -bound in (7) follows from a standard energy estimate involving integration by parts and the application of ∇ · u = 0. The global bound in (8) follows from the fact that, for q = 2 2−β > 2, by taking the scalar product of θ|θ| q−1 with the equation for θ in (1.2), The global bound for θ L q with 2 ≤ q ≤ 2 2−β follows from a simple interpolation inequality.
We now turn to the proof of Proposition 1.
Proof of Proposition 1. Dotting the equation of ω with ω, the equation of j with j and the equation of H with H, we obtain, after integration by parts, where We now estimate the terms on the right-hand side. By Hölder's inequality and Sobolev embedding inequality To estimate I 2 , we recall that h = H + Λ 2−2β θ and write I 2 as By Hölder's inequality and Sobolev's embedding inequality, To bound I 22 , we shift the derivatives away from θ to obtain

BOQING DONG, WENJUAN WANG, JIAHONG WU AND HUI ZHANG
By Hölder's inequality and Sobolev's embedding inequality, Again, by Hölder's inequality and Sobolev embedding inequality, where p and q are defined in (9). Thanks to 1 < β ≤ 3 2 , by an interpolation inequality, By Lemma A.1, where, due to β > 1, we have 5 − 3β ≤ 1 + α and 4 − 2β ≤ 1 + α, and Further, by Young's inequality and Lemma 2.1, Combining all the estimates above, we obtain Gronwall's inequality then implies the global H 1 -bound in Proposition 1 2.2. Global W 1,p bound for v. This subsection proves a global bound for θ L q with any q ∈ [2, ∞] and for ∇v L q for any 2 ≤ q < ∞. (1). Then, for any t > 0, To prove Proposition 2, we need to consider θ Ḣ−1 . It appears reasonable to consider theḢ −1 -norm of θ. The equation of θ The following asserts that θ Ḣ−1 remains bounded for all time.
Proof of Lemma 2.2. We write θ = ∆ θ. Inserting this in the equation for θ and dotting the equation by θ, we obtain 1 2 The embedding inequality and the global bound in Proposition 1 then implies that This completes the proof of Lemma 2.2.
We now prove Proposition 2.
Proof of Proposition 2. First we show that, for any 2 ≤ q ≤ ∞, In fact, for any 2 ≤ q < q ≤ ∞ satisfying we have, from the equation of θ and the Hardy-Littlewood-Sobolev inequality,

BOQING DONG, WENJUAN WANG, JIAHONG WU AND HUI ZHANG
Because of the embedding inequality L 2 , the global bounds in Proposition 1 and Lemma 2.1, we use (11) to control θ(t) L q by θ L q and an iterative process leads to a global bound on θ(t) L q for all q.
We now establish a global bound on ∇v L q . We rewrite the equation for v in the integral form where g is the kernel function associated with the operator e −t (−∆) β , or By Young's inequality, for q 1 , q 2 ∈ [1, ∞] and 1 + 1 Noticing that In addition, and, due to β ≤ 3 2 , by an interpolation inequality and Lemma 2.
By Proposition 1, for any t > 0, and thus This completes the proof of Proposition 2. Then ω = ∇ × u satisfies, for any t > 0, Proof of Proposition 3. For notational convenience, we write q = 2 α . Recalling the equation of ω, we obtain 1 q The dissipative part admits the lower bound, We write v in terms of j and h. According to (6), Inserting this representation in (14) and applying Hölder's inequality yield By Lemma A.5,

BOQING DONG, WENJUAN WANG, JIAHONG WU AND HUI ZHANG
where we have used a Besov embedding inequality and a simple identity Inserting (15) and (17) in (14), we obtain, due to α ≥ 1 2 because α + β = 2 and Applying Λ γ to (13) and then dotting with Λ γ ω, we find where To bound J 1 , we write it as, due to ∇ · u = 0, By Lemma A.1, To bound J 2 , we invoke (16) and apply Hölder's inequality where we have written 3 − 2β + γ = α. Inserting the bounds for J 1 and J 2 in (18) and invoking the global bound for ω L 2 α , we obtain Noticing α + γ = 1 finishes the proof of (12) and the proof of Proposition 3.

2.4.
Global H s bound. This subsection establishes the global H s bound for the solution. More precisely, we prove the following proposition.  Let (u, v, θ) be the corresponding solution. Then (u, v, θ) obeys, for any t > 0, where C(u 0 , v 0 , θ 0 , t) depends on t and the initial data.
Proof of Proposition 4. Let J = (I − ∆) 1 2 denote the inhomogeneous differentiation operator. Taking the inner product of (1) with (J 2s u, J 2s v, J 2s+2−2β θ), we have To bound J 1 , we apply ∇ · u = 0 to write it as By Lemma A.1 and Sobolev's inequality, We estimate J 2 and J 3 together.

BOQING DONG, WENJUAN WANG, JIAHONG WU AND HUI ZHANG
By Lemma A.1, Sobolev's inequality and α + β = 2, By Hölder's inequality, To estimate J 6 , we write, due to ∇ · u = 0, By Lemma A.1, We invoke the logarithmic Sobolev inequality, Due to α + β = 2, Then, for any t > 0, (∇u, ∇v, ∇θ) 2 where C depends on t and (u 0 , v 0 , θ 0 ) 2 H 1 . Proof. Taking the L 2 inner product of equations (1) with (u, v, θ) and integrating by parts, we have 1 2 Integrating in time yields the assertion (20). To prove (21), we take the inner product of (1) with (∆u, ∆v, ∆θ) and integrate by parts to obtain Invoking the logarithmic Sobolev inequality, can be rewritten as This completes the proof of Proposition 5. Then, for any t > 0, Proof. Taking the inner product of (1) with (J 2s u, J 2s v, J 2s θ) and integrating by parts, we have 1 2 As in the proof of Proposition 4, by Lemma A.1, we have L 4 and L 5 can be bounded by Inserting the estimates above in the right hand side of (24) and applying Gronwall's inequality, we have This completes the proof of 6.
Appendix A. Inequalities and Besov spaces. This appendix supplies several inequalities and some facts on the Besov spaces used in the previous sections. First we recall two calculus inequalities involving fractional derivatives. Second we provide an improved Gronwall type inequality. Third we describe the definition of the Littlewood-Paley decomposition and the definition of Besov spaces. Some related facts used in the previous sections are also included. The material presented in this appendix can be found in several books and many papers (see, e.g., [2,3,12,13]). Let J = (I − ∆) 1 2 denote the inhomogeneous differentiation operator. We recall following calculus inequalities (see, e.g., [7, p.334]).
Then, for two constants C 1 and C 2 , These estimates still hold if we replace J s by the homogeneous operator Λ s .
Lemma A.2. Assume that Y, Z, A and B are non-negative functions satisfying Let T > 0. Assume A ∈ L 1 (0, T ) and B ∈ L 2 (0, T ). Then, for any t ∈ [0, T ], Proof. Setting Using the simple fact that, for f ≥ 0, we obtain Gronwall's inequality then implies We now describe the Littlewood-Paley decomposition and the Besov spaces. We start with several notations. S denotes the usual Schwarz class and S its dual, the space of tempered distributions. S 0 denotes a subspace of S defined by and S 0 denotes its dual. S 0 can be identified as S 0 = S /S ⊥ 0 = S /P, where P denotes the space of multinomials. We also recall the standard Fourier transform and the inverse Fourier transform, To introduce the Littlewood-Paley decomposition, we write for each j ∈ Z The Littlewood-Paley decomposition asserts the existence of a sequence of functions Therefore, for a general function ψ ∈ S, we have In addition, if ψ ∈ S 0 , then in the sense of weak- * topology of S 0 . For notational convenience, we definė We now choose Ψ ∈ S such that Then, for any ψ ∈ S, in S for any f ∈ S . We set For notational convenience, we write ∆ j for∆ j when there is no confusion. They are different for j ≤ −1. As provided below, the homogeneous Besov spaces are defined in terms of∆ j while the inhomogeneous Besov spaces are defined in ∆ j . Besides the Fourier localization operators ∆ j , the partial sum S j is also a useful notation. For an integer j, where ∆ k is given by (30). For any f ∈ S , the Fourier transform of S j f is supported on the ball of radius 2 j and In addition, for two tempered distributions u and v, we also recall the notion of paraproducts and Bony's decomposition, see e.g. [2], In addition, the notation ∆ k , defined by ∆ k = ∆ k−1 + ∆ k + ∆ k+1 , is also useful.
Definition A.3. For s ∈ R and 1 ≤ p, q ≤ ∞, the homogeneous Besov spaceḂ s p,q consists of f ∈ S 0 satisfying f Ḃs p,q ≡ 2 js ∆ j f L p l q < ∞.
An equivalent norm of the the homogeneous Besov spaceḂ s p,q with s ∈ (0, 1) is given by Definition A.4. The inhomogeneous Besov space B s p,q with 1 ≤ p, q ≤ ∞ and s ∈ R consists of functions f ∈ S satisfying f B s p,q ≡ 2 js ∆ j f L p l q < ∞. Many frequently used function spaces are special cases of Besov spaces. The following proposition lists some useful equivalence and embedding relations.
Proposition 7. For any s ∈ R, H s ∼ B s 2,2 . For any s ∈ R and 1 < q < ∞, B s q,min{q,2} → W s q → B s q,max{q,2} . For any non-integer s > 0, the Hölder space C s is equivalent to B s ∞,∞ . Bernstein's inequalities are useful tools in dealing with Fourier localized functions. These inequalities trade integrability for derivatives. The following proposition provides Bernstein type inequalities for fractional derivatives. The upper bounds also hold when the fractional operators are replaced by partial derivatives.
We have also used the following inequality. It is a generalization of the Kato-Ponce inequality, which requires m to be an integer (see, e.g., [8]). This lemma extends it to any real number m ≥ 2. A proof for this lemma can be found in [4].