Abstract
The well-posedness problem is an important but challenging research topic in nonlinear partial differential equations. In this paper, we establish a global-in-time existence result of strong solutions for small initial data in terms of the norm on three-dimensional tropical climate model with viscosities by derive a blow-up criterion combine with energy estimates. This result can be regard as a generalization of the famous Fujita–Kato result to 3D Navier–Stokes equations.
1. Introduction
In this paper, we investigate the global existence of smooth solutions to the tropical climate model (TCM) in :with initial datawhere and denote the barotropic mode and the first baroclinic mode of the velocity, while the scalar functions denote the temperature and pressure, respectively.
It should be pointed out that the original system derived in [1] has no viscous terms because it is derived from the inviscid primitive equations. In many research studies of tropical atmospheric dynamics, Gill and Matsuno [2, 3] first used the baroclinic mode models. Majda-Biello [4] pointed out that the transport of momentum between the barotropic and baroclinic model is an important effect, it is necessary to retain both the barotropic and baroclinic modes of the velocity. There are geophysical circumstances in which the Laplacian may arise. Fundamental issues such as the global existence and regularity of solutions have attracted considerable attention. Some important results in terms of the global existence and uniqueness of classical solutions in 2D have been obtained, see [5–7] etc. In addition, some results on the global well-posedness issue for the 3D tropical climate model with fractional dissipation and damping terms can be referred to [8, 9].
This paper focuses on the global existence of the classical solutions for TCM in . System (1) contains the 3D incompressible Navier–Stokes (NS) equations as a special case , for which the issue of global well-posedness with small initial data in the Sobolev spaces has been solved by Fujita and Kato [10]. Our goal of this paper is to extend similar results to TCM. We should point out the space is a critical space with respect to the scaling invariance in terms of Navier–Stokes equations, but to system (1) which is not scaling-invariant. System (1) contains the Navier–Stokes equations as a subsystem, so in general, we cannot expect any better results than those for the Navier-Stokes equations. Therefore, a natural problem is whether we can establish the global existence and regularity of (1) under the same assumption. It should be noted that the result for (1) is not completely parallel to that for the Navier–Stokes equations. The difficulty is coming from without . Thus, it seems difficult to obtain the desired results by using the same process. Here, we use the trick from a series of works on the Hall-magnetohydrodynamcis equations by Dongho Chae and his collaborators [12, 13] etc. and by utilizing the structure of the system to overcome this difficulty. The main results of this paper are the following:
Theorem 1. Let , and with . Then, there exists a constant , such that
Then, there exists a unique global classical solution .
Remark 1. In the case of , system (1) can be read as the incompressible Navier–Stokes equations, and what proved in [10] is a straightforward consequence of Theorem 1. An interesting point is that the parameter goes to zero. The detailed discussion of this case will be given in a separate paper.
2. Proof of the Theorem
The homogeneous Sobolev space is defined as follows:where .
The BMO is the space of bounded mean oscillation defined bywhere is the average of over .
The following lemma is essential in the process of proof [11].
Lemma 1. Let , then we have the following:for all , when and are multi-indices with .
We begin with the local existence and uniqueness theorem of strong solutions.
Lemma 2. Let ; , there exists a positive time and a unique strong solution on to system (1) satisfying
Proof. The main part of the proof consists of a priori estimates. To make the proof rigorous requires some approximation procedure such as that employed in [12], to which we refer for details. Nevertheless, we would like to point out two key observations in the proof for the convenience of the reader. We can establish priori estimates (the process is similar as a part of the proof of Theorem 1).We setThen, from the above inequality, we haveThen, by applying the Gronwall inequality, we haveNow, choose . Then,By applying the same argument as [12], we can obtain the desired result.
The proof of Theorem 1 will be divided into two steps.
Proof. Step 1. We will derive a blow-up criterion for the strong solutions to (1).
Applying to system (1), multiplying the resultant by and , respectively, and integrating over , we obtain the following:
Remark 2. The operator .
By using the cancellation property,
By virtue of integrating by part and H lder inequality and Lemma 2, we have the following:
We rewrite it as follows:
We choose ,
Then, we have the following equality:
Now, we split as follows:
Then we can estimate by using Lemma 2,
Next, we consider the term
We can estimate by using Lemma 2
It is necessary to point out we cannot use Lemma 1 to estimate due to we have not . Thus, we can estimate it by H lder inequality only,
By using the cancellation property,
We can estimate and by using Lemma 2.
By using the H lder inequality, we have the following:
Thus, combining (15) to (27), we obtain the following:
Here, we used the inclusion relation: .
Now, we set
We can obtain the following inequality:
By using Gronwall type inequality, we have the following:
From the above inequalities, we obtain that ifthen
Let be a unique local strong solutions and be the first blow-up time, then
Step 2. We take the operator on both sides of u-equation and -equation of (1); taking the scalar product with , thenWe take operator on both sides of v-equation of (1); taking a scalar product with , thenBy using H lder inequality and Gagliardo–Nirenberg type interpolation inequality, we can obtain the following estimates:Here, we used the condition .From (35) to (42), we have the following:Choosing so small thatNow, we have the following:Integrating in time from 0 to ,Applying the following inequalitiesthen we yieldFrom (48), we further know thatApplying the factthen we imply thatNow, we suppose is the maximal existence time for solutions, from the blow-up criterion, the existence time can be extended after which contradicts the maximality of . This completes the proof of Theorem 1.
3. Conclusion
It is deserving to point out that our method is not adapt to . The reason is that we have not . We need some new ideas and methods to deal with the parameter goes to zero. The detailed discussion of this case will be given in a separate paper.
Data Availability
The data used to support the working are cited within the article as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors contributed equally and significantly in writing this article. All the authors have read and approved the final manuscript.
Acknowledgments
Xu was supported by the Anhui Education Bureau (No. KJ2020A0505). Zhang was partially supported by the NSFC (No. 31971185).