# Global Existence for the 3D Tropical Climate Model with Small Initial Data inH˙1/2ℝ3∗

Authors: Hui Zhang; Juan Xu
Journal: Journal of Mathematics
 (2022)

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## 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 theH˙1/2ℝ3 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.


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## Body

## 1. Introduction

In this paper, we investigate the global existence of smooth solutions to the tropical climate model (TCM) inℝ3:(1)∂tu−Δu+u⋅∇u+∇p+∇⋅v⊗v=0,∂tv−Δv+v⋅∇u+u⋅∇v+∇θ=0,∂tθ−Δθ+u⋅∇θ+∇⋅v=0,∇⋅u=0,with initial data(2)ux,0=u0x,vx,0=v0x,θx,0=θ0x,where u=u1,u2,u3 and v=v1,v2,v3 denote the barotropic mode and the first baroclinic mode of the velocity, while the scalar functions θ,p 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 inℝ3. System (1) contains the 3D incompressible Navier–Stokes (NS) equations as a special case v=0,θ=0, for which the issue of global well-posedness with small initial data in the Sobolev spaces H˙1/2ℝ3 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 H˙1/2ℝ3 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 ∇⋅v=0. 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.
Lets>5/2, and u0,v0,θ0∈Hsℝ3 with ∇⋅u0=0. Then, there exists a constant c0>0, such that(3)u0H˙1/2+v0H˙1/2+ϵ+θ0H˙1/2<c0,0<ϵ<1.Then, there exists a unique global classical solutionu,v,θ∈L∞0,+∞;Hsℝ3.Remark 1.
In the case ofv=0,θ=0, 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 ϵ goes to zero. The detailed discussion of this case will be given in a separate paper.


## 2. Proof of the Theorem

The homogeneous Sobolev spaceH˙sℝ3 is defined as follows:(4)fH˙s=ΛsfL2=∫ℝ3ξ2sf^ξ2dξ1/2,where Λ=−Δ1/2.The BMO is the space of bounded mean oscillation defined by(5)f∈Lloc1ℝ3,supx,R1BR∫BRxfy−f¯BRxdy<+∞,where f¯BRx is the average of f over BRxy∈ℝ3‖x−y|<R.The following lemma is essential in the process of proof [11].Lemma 1.
Let1<r<∞, then we have the following:(6)∂αf⋅∂βgLr≤CfBMO−Δα+β/2gLr+gBMO−Δα+β/2fLr,for all f,g∈BMO∩Wα+β,r, when α=α1,α2,α3 and β=β1,β2,β3 are multi-indices with α;β≥1.We begin with the local existence and uniqueness theorem of strong solutions.Lemma 2.
Letu0,v0,θ0∈Hsℝ3; s>5/2, there exists a positive time T∗ and a unique strong solution u,v,θ on 0,T∗ to system (1) satisfying(7)u,v,θ∈L∞0,T∗;Hsℝ3∩L20,T∗;Hs+1ℝ3.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).(8)ddtΛsutL22+ΛsvtL22+ΛsθtL22+Λs+1utL22+Λs+1vtL22+Λs+1θtL22≤CΛsutL22+ΛsvL22+ΛsθL22+1ΛsutL22+ΛsvtL22+ΛsθtL22.
We set(9)Xt=ΛsutL22+ΛsvtL22+ΛsθtL22+1.
Then, from the above inequality, we have(10)ddtX≤CX2.
Then, by applying the Gronwall inequality, we have(11)Xt≤X01−C0X0t,
Now, chooseT=1/2C0X0. Then,(12)Xt≤2X0,∀t∈0,T.
By applying the same argument as [12], we can obtain the desired result.The proof of Theorem1 will be divided into two steps.Proof.
Step 1. We will derive a blow-up criterion for the strong solutions to (1).
Applying∇3 to system (1), multiplying the resultant by ∇3u,∇3v and ∇3θ, respectively, and integrating over ℝ3, we obtain the following:(13)12ddt∇3utL22+∇3vtL22+∇3θtL22+∇4utL22+∇4vtL22+∇4θtL22=−∫ℝ3∇3u⋅∇u∇3udx−∫ℝ3∇3∇⋅v⊗v∇3udx−∫ℝ3∇3v⋅∇u∇3vdx−∫ℝ3∇3u⋅∇v∇3vdx−∫ℝ3∇3∇θ∇3vdx−∫ℝ3∇3u⋅∇θ∇3θdx−∫ℝ3∇3∇⋅v∇3θdx=I1+I2+⋯+I7.Remark 2.
The operator∇3=D3/∂x1α1x2α2x3α3,α1+α2+α3=3.By using the cancellation property,(14)∫u⋅∇∇3u⋅∇3udx=0.By virtue of integrating by part and Ho¨ lder inequality and Lemma 2, we have the following:(15)I1=−∫ℝ3∇3u⋅∇u∇3u−u⋅∇∇3u⋅∇3udx,=−∑α+β=3;α≥1∫ℝ3∇αu⋅∇∇βu∇3udx≤C∑α+β=3;α≥1∇αu⋅∇∇βuL2∇3uL2≤CuBMO∇3uL2∇4uL2≤CuBMO2∇3uL22+C1∇4uL22.We rewrite it as follows:(16)∇⋅v⊗v=v⋅∇v+v∇⋅v.We chooseα=3,α=α1,α2,α3,(17)∫ℝ3v⋅∇∂αu⋅∂αvdx=∫ℝ3vi∂i∂αuj∂αvjdx=−∫ℝ3∂ivi∂αuj⋅∂αvjdx−∫ℝ3vi∂αuj⋅∂i∂αvjdx.Then, we have the following equality:(18)∫ℝ3v⋅∇∇3u⋅∇3vdx+∫ℝ3v⋅∇∇3v⋅∇3udx=−∫ℝ3∇⋅v∇3u∇3vdx.Now, we splitI2 as follows:(19)−∫ℝ3∇3∇⋅v⊗v∇3udx=−∫ℝ3∇3v⋅∇v∇3udx−∫ℝ3∇3v∇⋅v∇3udx,=I21+I22.Then we can estimateI21+I3 by using Lemma 2,(20)I21+I3≤∫ℝ3∇3v⋅∇v∇3u+∇3v⋅∇u∇3vdx−v∇3∇⋅v⋅∇3u−v⋅∇∇3v⋅∇3udx+∫ℝ3∇⋅v∇3v∇3udx,≤CuBMO∇3vL2+vBMO∇3uL2∇4uL2+∇4vL2≤CuBMO2+vBMO2∇3vL22+∇3uL22+C2∇4vL22+∇4uL22.Next, we consider the termI22(21)I22=∫ℝ3∇3v∇⋅v∇3udx≤∑α+β=3;α≥1∫ℝ3∇αu⋅∇β∇⋅v∇3udx+∫ℝ3v∇3∇⋅v∇3udx=I221+I222.We can estimateI221 by using Lemma 2(22)I221≤CuBMO∇3vL2+vBMO∇3uL2∇3uL2+∇3vL2,≤CuBMO+vBMO∇3uL22+∇3vL22.It is necessary to point out we cannot use Lemma1 to estimate I222 due to we have not ∇⋅v=0. Thus, we can estimate it by H o¨ lder inequality only,(23)I222≤CvL∞∇3uL2∇4vL2,≤CvL∞2∇3uL22+C3∇4vL22,By using the cancellation property,(24)∫u⋅∇∇3v⋅∇3vdx=0;∫u⋅∇∇3θ⋅∇3θdx=0.We can estimateI4 and I6 by using Lemma 2.(25)I4=∫ℝ3∇3u⋅∇v−u⋅∇∇3v⋅∇3v∇3vdx,≤CuBMO∇3vL2+vBMO∇3uL2∇4uL2+∇4vL2,≤CuBMO2+vBMO2∇3vL22+∇3uL22+C4∇4vL22+∇4uL22.(26)I6=∫ℝ3∇3u⋅∇θ−u⋅∇∇3θ⋅∇3θ∇3vdx,≤CuBMO∇3θL2+θBMO∇3uL2∇4uL2+∇4θL2≤CuBMO2+θBMO2∇3θL22+∇3uL22+C5∇4θL22+∇4uL22.By using the Ho¨ lder inequality, we have the following:(27)I5+I7=∫ℝ3∇3∇θ∇3vdx+∫ℝ3∇3∇⋅v∇3θdx,≤C∇3θL2+∇3vL2∇4vL2+∇4θL2≤C∇3θL22+∇3vL22+C6∇4θL22+∇4uL22.Thus, combining (15) to (27), we obtain the following:(28)ddt∇3utL22+∇3vtL22+∇3θtL22+∇4utL22+∇4vtL22+∇4θtL22,≤CuBMO2+vL∞2+θBMO2+1∇3utL22+∇3vtL22+∇3θtL22.Here, we used the inclusion relation:L∞ℝ3⊊BMOℝ3.Now, we set(29)Ht=∇3utL22+∇3vtL22+∇3θtL22.We can obtain the following inequality:(30)ddtHt≤CuBMO2+vL∞2+θBMO2+1Ht,By using Gronwall type inequality, we have the following:(31)sup0≤t≤THt≤CH0exp∫0TuBMO2+vL∞2+θBMO2dt.From the above inequalities, we obtain that if(32)∫0TuBMO2+vL∞2+θBMO2dt<+∞,then(33)sup0≤t≤T∇3utL22+∇3vtL22+∇3θtL22,+∫0T∇4uτL22+∇4vτL22+∇4θτL22dτ<+∞.Letu,v,θ be a unique local strong solutions and T∗ be the first blow-up time, then(34)T∗<+∞⇔∫0T∗uBMO2+vL∞2+θBMO2dt=+∞.Step 2.
We take the operatorΛ1/2 on both sides of u-equation and θ-equation of (1); taking the scalar product with Λ1/2u,Λ1/2θ, then(35)12ddtΛ1/2utL22+Λ1/2θtL22+Λ3/2utL22+Λ3/2θtL22=−∫ℝ3Λ1/2u⋅∇uΛ1/2udx,−∫ℝ3Λ1/2∇⋅v⊗vΛ1/2udx−∫ℝ3Λ1/2u⋅∇θΛ1/2θdx−∫ℝ3Λ1/2∇⋅vΛ1/2θdx.
We take operatorΛ1/2+ϵ on both sides of v-equation of (1); taking a scalar product with Λ1/2+ϵv, then(36)12ddtΛ1/2+ϵvtL22+Λ3/2+ϵvtL22=−∫ℝ3Λ1/2+ϵv⋅∇uΛ1/2+ϵvdx−∫ℝ3Λ1/2+ϵu⋅∇vΛ1/2+ϵvdx−∫ℝ3Λ1/2+ϵ∇θΛ1/2+ϵvdx,
By using Ho¨ lder inequality and Gagliardo–Nirenberg type interpolation inequality, we can obtain the following estimates:(37)∫ℝ3Λ1/2u⋅∇uΛ1/2udx≤Cu⋅∇uL3/2ΛuL3≤CΛ1/2uL2Λ3/2uL22.(38)∫ℝ3Λ1/2u⋅∇θΛ1/2θdx≤CΛ1/2uL2Λ3/2θL22,(39)∫ℝ3Λ1/2∇⋅v⊗vΛ1/2udx≤∇⋅v⊗vL3/2ΛuL3,≤CΛ1/2vL2Λ3/2vL2Λ3/2uL2,≤CΛ1/2+ϵvL2Λ3/2uL22+Λ3/2vL22,(40)∫ℝ3Λ1/2∇⋅vΛvθdx≤CΛ1/2vL2Λ3/2θL2,(41)∫ℝ3Λ1/2+ϵv⋅∇uΛ1/2+ϵvdx,≤CΛ1/2vL6Λ3/2uL2Λ1/2+2ϵvL3≤CΛ3/2vL2Λ3/2uL2Λ1/2+2ϵvL3≤CΛ1/2+ϵvL2Λ3/2uL2Λ3/2+ϵvL3.
Here, we used the conditionϵ<1.(42)−∫ℝ3Λ1/2+ϵu⋅∇vΛ1/2+ϵvdx,≤CΛ1/2uL6Λ3/2vL2Λ1/2+2ϵvL3,≤CΛ1/2+ϵvL2Λ3/2uL2Λ3/2+ϵvL3.
From (35) to (42), we have the following:(43)12ddtΛ1/2utL22+Λ1/2+ϵvtL22+Λ1/2θtL22+Λ3/2utL22+Λ3/2+ϵvtL22+Λ3/2θtL22,≤CΛ1/2utL2+Λ1/2+ϵvtL2+Λ1/2θtL2Λ3/2utL22+Λ3/2+ϵvtL22+Λ3/2θtL22.
Choosingc0 so small that(44)CΛ1/2u0L2+Λ1/2+ϵv0L2+Λ1/2θ0L2≤12,
Now, we have the following:(45)12ddtΛ1/2utL22+Λ1/2vtL22+Λ1/2θtL22≤0,
Integrating in time from 0 tot,(46)Λ1/2utL22+Λ1/2+ϵvtL22+Λ1/2θtL22≤Λ1/2u0L22+Λ1/2+ϵv0L22+Λ1/2θ0L22,
Applying the following inequalities(47)Λ1/2utL22+Λ1/2+ϵvtL22+Λ1/2θtL22≥13Λ1/2utL2+Λ1/2+ϵvtL2+Λ1/2θtL22,Λ1/2u0L22+Λ1/2+ϵv0L22+Λ1/2θ0L22≤Λ1/2u0L2+Λ1/2+ϵv0L2+Λ1/2θ0L22,then we yield(48)sup0≤t≤TΛ1/2utL22+Λ1/2+ϵvtL22+Λ1/2θtL22<32C.
From (48), we further know that(49)u,θ∈L∞0,T;H˙1/2ℝ3∩L20,T;H˙3/2ℝ3,v∈L∞0,T;H˙1/2+ϵℝ3∩L20,T;H˙3/2+ϵℝ3.
Applying the fact(50)H˙3/2ℝ3⟶BMOℝ3,H˙3/2+ϵℝ3⟶L∞ℝ3,then we imply that(51)u,θ∈L20,T;BMO,v∈L20,T;L∞ℝ3,for allT∈0,T∗.
Now, we supposeT<T∗ is the maximal existence time for solutions, from the blow-up criterion, the existence time can be extended after t=T which contradicts the maximality of t=T. This completes the proof of Theorem 1.


## 3. Conclusion

It is deserving to point out that our method is not adapt toϵ=0. The reason is that we have not ∇⋅v=0. We need some new ideas and methods to deal with the parameter ϵ goes to zero. The detailed discussion of this case will be given in a separate paper.



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