Energy Conservation for 3D Tropical Climate Model in Periodic Domains

In this paper, we prove the energy conservation for the weak solutions of the 3D tropical climate model under some sufficient conditions. Our results are similar to Onsager’s conjecture which is on energy conservation for weak solutions of Euler equations.


Introduction
e incompressible Euler equations in R 3 are read as follows: where u is the velocity and p is the pressure. Given that the solution to system (1) is sufficiently smooth, it is easy to see that the total kinetic energy of the flow is conserved. However, turbulent flows are highly irregular; one major character of turbulent fluids is the energy dissipation. In the context of hydrodynamics, this leads to the famous Onsager's conjecture [1]. Specifically, Onsager's conjecture that every weak solution to the Euler equations with Hölder continuity exponent α > (1/3) conserves energy; and anomalous dissipation of energy occurs when α < (1/3). e first part of the conjecture was proved by Eyink [2] and extended to Besov space in subsequent work of Constantin et al. [3], while significant progress has recently been made on the second part, for example, see [4,5], and some prior work on the compressible Onsager's conjecture can be found in the literature, for example, see [6,7]. e object of this study is to address the relation between the energy conservation and the degree of regularity of the solution for the tropical climate model in Ω � [0, 1] 3 with periodic boundary conditions with initial data where u and v denote the vector fields and θ and p denote the temperature and the pressure, respectively. e system (2) is derived from the inviscid primitive equations by Frierson et al. [8]. For more details of the related background about the tropical climate model, for instance, see [9,10] and the references therein. Here, it is worth particularly mentioning the solution of (2) with enough regularity to justify integration by parts, and it is clear that the following total energy is conserved: So, it seems very reasonable to investigate the analogue of Onsager's conjecture about system (2). To the best of our knowledge, there is no available result for the weak solutions of the tropical climate model. Our study is motivated by [3,11]. However, we are unable to obtain the desired result using the same processes as [3,12]. e obstacle is due to ∇ · v ≠ 0 and the nonlinear term u · ∇θ. In fact, we require a stronger regularity about v due to the presence of the term u∇ · v. Now, we are ready to state the main results.
) be a weak solution of systems (2) and (3), i.e., then we have

Preliminary Remarks
In this section, we briefly discuss some properties of the Besov space B α,∞ 3 (Ω).
From the above definition, we have the following inequalities [3]: Let η ∈ C ∞ c (R 3 ) be a standard mollifying kernel and set With the notation we have the following inequalities [3]: Now, we define en, it is easy to check the following equalities, it is pointed out that the first equality was discovered in [3]:

Proof of the Main Theorem
For the sake of simplicity, we will proceed as if the solution is differentiable in time. e extra arguments need to be modifed in time are straightforward.
Proof. First, we apply the test function φ � (u ϵ ) ϵ on the first equation and then we have (17) We point out that the I 1 term can be estimated as same as that in [3], and we omit the details: Now, we estimate I 2 : 2 Journal of Mathematics Using the following inequality and (12) and (13), we obtain We can also obtain the following estimate through the above inequalities: Because ∇ · v ≠ 0, I 21 cannot be zero, and we will estimate this term later.

(39)
We can also see that Finally, we can establish the following inequality through the above estimates and condition (6): as ϵ ⟶ 0, which completes the proof.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest.