A regularity criterion for a 2D tropical climate model with fractional dissipation

Tropical climate model derived by Frierson et al. (Commun Math Sci 2:591–626, 2004) and its modified versions have been investigated in a number of papers [see, e.g., Li and Titi (Discrete Contin Dyn Syst Series A 36(8):4495–4516, 2016), Wan (J Math Phys 57(2):021507, 2016), Ye (J Math Anal Appl 446:307–321, 2017) and more recently Dong et al. (Discrete Contin Dyn Syst Ser B 24(1):211–229, 2019)]. Here, we deal with the 2D tropical climate model with fractional dissipative terms in the equation of the barotropic mode u and in the equation of the first baroclinic mode v of the velocity, but without diffusion in the temperature equation, and we establish a regularity criterion for this system.

We emphasize that in this paper the equation for θ contains no dissipation, and we also assume ν = 1 and η = 1. Our goal here is mainly related to the problem of regularity in time for the solutions of system (1), in the 2D case.
Global well-posedness of solutions to a tropical climate model with dissipation in the equation of the first baroclinic mode of the velocity, under the hypotheses of small initial data, was studied by Wan [26] and Ma and Wan [17]. In fact, the issue of global regularity has been investigated in a number of articles and (partially) addressed in dependence of the values assumed by parameters α and β. In particular, in the 2D case, Ye [27] was able to prove global existence (adding 2 θ in (1) 3 ), in H s -norm, s ≥ 2, for β = 1 and α > 0. Dong et al. [4] considered the 2D model assuming α + β = 2 with 1 < β ≤ 3/2, and they proved that system (1) possesses a unique global solution when the initial data . In addition, in [4], the authors also studied the cases of α + β = 2 with 3/2 < β < 2, and α + β = 2 with α = 2 and β = 0. Let us also recall that Ma et al. [18] established the local well-posedness of strong solutions to the considered model.
In the present paper we consider problem (1) with 1/2 < α < 1, 0 < β < 1, and β + 2α = 2, which is a situation not addressed in the above mentioned works, and in any case it is new to the best of our knowledge. Taking initial data u 0 , v 0 , θ 0 ∈ H s (R 2 ), s > 2, and following an approach similar to the one given in [7] (see also [6] and [8]), in Theorem 3.1 we prove a continuation result for the solutions of (1). To be more precise, is the homogeneous Besov spaceḂ s p,q (R 2 ) with s = 0, p = ∞ and q = 2 (see below for details).

Preliminaries and basic facts
Let us recall some basic facts about the function spaces that will be used in the sequel. We also list the estimates needed to reach the claimed regularity result. Most of the controls we use in following have been established in [13]. We refer to this paper for a detailed overview (see also [14,24,25]), which is somehow aimed at our purposes, on the theory of Besov spaces B s p,q = B s p,q (R n ) (and homogeneous Besov spaceṡ B s p,q =Ḃ s p,q (R n )), with 0 < p, q ≤ ∞ and s ∈ R. Let us consider-only in this section-the case of R n as domain of the considered vector and scalar fields and recall some inequalities and embeddings in Sobolev and Besov spaces. Here,Ḃ s p,q =Ḃ s p,q (R n ), with 0 < p, q ≤ ∞ and s ∈ R indicate homogeneous Besov spaces and the non-homogeneous counterparts are B s p,q = B s p,q (R n ), 0 < p, q ≤ ∞ and s ∈ R (see, e.g., [24,25]) with norms, respectively, · Ḃs p,q and · B s p,q . In the sequel B M O = B M O(R n ) denotes the Bounded Mean oscillation space with norm · B M O (see, e.g. [12][13][14]).
For p ≥ 1, we indicate by L p = L p (R n ) the usual Lebesgue space, endowed with norm · p = · L p . Also, for L 2 , the norm is · = · 2 . We denote by W k, p = W k, p (R n ) and · k, p = · W k, p a Sobolev space and its norm, respectively (see, e.g., [1]). When p = 2 we use the notation H k = W k,2 and · H k = · W k,2 .
In the sequel we will use the symbols C (or c) to denote generic constants, which may change from line-to-line, but are not dependent on the solution. Also, we denote a generic constant by C( · ) (or by c( · )), with the meaning that the constant depends mainly on the arguments between parentheses, or alternatively by using a subscript to make explicit the quantities the constant depends on.

Further inequalities
We will also use some elementary commutator type estimates as in the following lemma concerning the operator s , s > 0 (see, e.g., [7,10,11]).
In the sequel all the function spaces are taken on R n = R 2 , and so we always have n = 2 in the used Sobolev embeddings and interpolation inequalities. In partic- As a further consequence of (3), in the case R n = R 2 , for s = 0, p = ρ = ∞, ν = 2, and also assuming f ∈ W α,2 , α > 2, we have the following logarithmic control (see [7] and [13, Theorem 2.1, p. 257]), in which we set s = α, i.e.
We also use the following interpolation inequalities (see, e.g, [3,9,15]) and (see [12,Lemma 1,p. 180]) 3 Regularity result Assume the parameters α and β in (1) are such that Let (u, v, θ) be a local solution to the system (1), defined on some time interval [0, T ), with 0 < T < ∞, and having the following regularity for any 0 <T < T . Then (u, v, θ)(t) can be extended beyond time T , with the same regularity as in (13), provided that In this paper we will not deal with the two cases (α, β) = (1/2, 1) and (α, β) = (1, 0). In fact, to try to examine them, an alternative approach to the one used here seems necessary.

Proof of Theorem 3.1
The proof consists in proving suitable a priori estimates for the considered solution (u, v, θ) showing explicitly that it can extended after time T > 0. Thus, the procedure is divided in a number of steps in which we establish the needed bounds in L 2 , H 1 , H 2 and H s , s > 2. These steps parallels the formal estimates in the global existence results given in [4] and [27], although in our case they are carried out with different techniques borrowed from [7,22] and [13, Theorem 5.1] (see also [28,29], and [2]).
Next, we consider higher order estimates.
Let us take into account K 3 . In this case, setting p 2 = q 2 = 4, we get where we used Young's inequality, Gagliardo-Nirenberg's inequality with parameter σ = (2α − 1)/2(1 − β) and control (10). Let us recall that in our case we have 1/2 < α < 1 and 0 < β < 1. Observe that and the last two conditions above are always satisfied under the assumptions in (12). Then, relation (22) reduces to Notice that which is guaranteed by the hypotheses in (12). Thus, we reach where in the last step we used (6).
For any T * ≤ t < T , T * ≥ 0, we set and applying Gronwall's inequality to (27), for any T * ≤ t < T , it follows that where > 0 is small constant, depending on T * , such that where we used (14), and the constant C * depends on (∇u, ∇v, ∇θ)(T * ) .

H 2 -estimates
Applying the operator to (1) 1 , (1) 2 and (1) 3 , and multiplying them in L 2 by u, v and θ , respectively, and adding them up, we get Let us first consider the three worst terms, i.e. I 4 + I 5 + I 6 . Thus, we have where Then, we have Consider I I 1 . By using (7) we have that with 1/ p i + 1/q i = 1/2, i = 1, 2.
Let us take into account I I 11 . Then, setting p 1 = ∞ and q 1 = 2, we have that where we used Young's inequality, and Gagliardo-Nirenberg's inequality For the term I I 12 , setting p 2 = ∞ and q 2 = 2, we reach where we applied Young and Gagliardo-Nirenberg's inequalities. In particular, parameter σ = (1 − α)/β is such that σ < 1 if and only if α + β > 1. Moreover, observe that and also this last condition is satisfied assuming (12). Hence, using Young's inequality, from (32) we get Let us take into account the integral term I 1 , to get where we used (8). We also have that Similarly, we obtain Combining (30) with (31)-to-(33), and setting where I u,v,θ (t) is defined as in (26) and, in particular, in the last inequality we used relation (6). We conclude as in (27)-to-(29) by an application of Gronwall's lemma and exploiting hypothesis (14), to get the control where y(t) is defined as in (28), > 0 is the small quantity introduced in (29), and we still use C * that, in this case, also depends on ( u, v, θ)(T * ) 2 .