COMMUN. MATH. SCI. c 2019 International Press

COMMUN. MATH. SCI. c© 2019 International Press

Vol. 17, No. 8, pp. 2325–2352

STABILITY AND DECAY RATES FORA VARIANT OF THE 2D BOUSSINESQ-BENARD SYSTEM∗

JIAHONG WU† , XIAOJING XU‡ , AND NING ZHU§

Abstract. This paper investigates the stability and large-time behavior of perturbations near thetrivial solution to a variant of the 2D Boussinesq-Benard system. This system does not involve thermaldiffusion. Our research is partially motivated by a recent work [C.R. Doering, J. Wu, K. Zhao, and X.Zheng, Phys. D, 376/377:144–159, 2018] on the stability and large-time behavior of solutions near thehydrostatic balance concerning the 2D Boussinesq system. Due to the lack of thermal diffusion, thesestability problems are difficult. The energy method and classical approaches are no longer effective indealing with these partially dissipated systems. This paper presents a new approach that takes intoaccount the special structure of the linearized system. The linearized parts of the vorticity equationand the temperature equation both obey a degenerate damped wave-type equation. By representingthe nonlinear system in an integral form and carefully crafting the functional setting for the initial dataand solution spaces, we are able to establish the long-term stability and global (in time) existence anduniqueness of smooth solutions. Simultaneously, we also obtain exact decay rates for various derivativesof the perturbations.

Keywords. Boussinesq-Benard equations; global solution; large-time behavior; stability; velocitydamping.

AMS subject classifications. 35Q35; 76D03; 76D05.

1. IntroductionThis paper focuses on the following 2D Boussinesq-Benard system

∂tu+u ·∇u+u+∇P =θe2,

∂tθ+u ·∇θ= ∆u · e2,

∇·u= 0,

(1.1)

where u(t,x,y), (u1(t,x,y),u2(t,x,y)) denotes the 2D velocity field, P =P (t,x,y) thepressure and θ=θ(t,x,y) the temperature. Here e2 = (0,1) and the term θe2 representsthe buoyancy forcing due to the gravity in the vertical direction. The first equation isthe 2D damped Euler equation with a buoyancy forcing and the second equation simplystates that the temperature is transported by the velocity field with a diffusion termof the second component of the velocity. Equation (1.1) is a variant of the standardBoussinesq-Benard system. The standard Boussinesq-Benard system contains ∆u andthe term u ·e2 instead of the damping u and ∆u · e2 in (1.1). Equation (1.1) may bephysically relevant when the diffusive effect of u2 plays a more dominant role in theequation of temperature.

The aim here is to understand the large-time stability property of perturbationsnear the trivial steady state (u,θ) = (0,0). Equivalently, we explore the global existence

∗Received: October 29, 2018; Accepted (in revised form): September 3, 2019. Communicated byYaguang Wang.†Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA (jiahong.wu@

okstate.edu).‡School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of

Education, Beijing Normal University, Beijing 100875, China ([email protected]).§Corresponding Author. School of Mathematical Sciences, Laboratory of Mathematics and Com-

plex Systems, Ministry of Education, Beijing Normal University, Beijing 100875, China ([email protected]).

2325

mailto:[email protected]





2326 STABILITY AND DECAY RATES FOR BOUSSINESQ-BENARD SYSTEM

and uniqueness of small solutions and the precise large-time behavior of these solutions.This study is partially motivated by a recent work of Doering, Wu, Zhao and Zheng [18]and a followup paper [50] on the global stability and large-time behavior of perturbationsnear the hydrostatic balance concerning the Boussinesq equations. The 2D Boussinesqsystem

∂tu+u ·∇u+∇P = ∆u+θe2,

∂tθ+u ·∇θ= 0,

∇·u= 0

(1.2)

admits a special class of solutions

uhe= 0, θhe=βy, Phe=1

2βy2, (1.3)

where β>0 is a parameter. The special solution in (1.3) is the so-called hydrostaticbalance, which refers to the geophysical circumstance when the fluid is at rest and whenthe pressure gradient is balanced out by the buoyancy force in the direction of gravity.The perturbations

u=u−uhe, θ=θ−θhe, P =P −Phe

satisfy ∂tu+ u ·∇u+∇P = ∆u+ θe2,

∂tθ+ u ·∇θ+βu2 = 0,

∇· u= 0.

(1.4)

The difference between (1.1) and (1.4) is that the velocity equation in (1.4) involvesdissipation instead of damping, and the temperature equation contains u2 instead of∆u2 as in (1.1). The recent papers [18] and [50] have significantly advanced our un-derstanding of the stability problem involving (1.4). [18] obtains the global stability

and large-time behavior of the perturbation (u,P , θ). It is proven, for β>0, that theL2 norms of the velocity perturbation (not necessarily small) and its first-order spatialand temporal derivatives converge to zero as t→∞. Consequently it is found that thepressure and temperature functions stratify in the vertical direction in a weak topology.Remarkably, the second-order spatial derivatives of the velocity perturbation (not nec-essarily small) are shown to be bounded uniformly in time for all time. In addition, [18]contains extensive numerical simulations illustrating the analytic results and investigat-ing unsolved problems. [50] furthers the studies of [18]. [50] obtains precise large-timedecay rates for the velocity field of the linearized equations and explicit eventual profilefor the temperature. Conditional decay rates are also obtained for the nonlinear system.There are still many interesting open issues concerning (1.4). One issue is whether or

not ‖∇θ‖L2 is bounded uniformly in time. The lack of thermal diffusion in (1.4) makesit extremely difficult to answer this question.

In contrast, (1.1) has a very special structure and solutions of (1.1) appear to behavedifferently from those of (1.4). We are able to prove the global stability of the solutionsof (1.1) in a highly regular setting and extract precise decay rates for various derivativesof the solutions. We explain in some detail our main idea. We take advantage of thevorticity formulation. The vorticity ω=∇×u,∂xu2−∂yu1 measures how fast the fluid

JIAHONG WU, XIAOJING XU AND NING ZHU 2327

rotates and its control is very important in the study of fluid stability problem. Takingthe curl of the velocity equation in (1.1) yields

∂tω+u ·∇ω+ω=∂xθ.

Due to the divergence-free condition, we have

∆u ·e2 = (∂xx+∂yy)u2 =∂xxu2−∂xyu1 =∂xω.

Equation (1.1) is then reduced to the following system∂tω+u ·∇ω+ω=∂xθ,

∂tθ+u ·∇θ=∂xω,

u=∇⊥∆−1ω.

(1.5)

Since (1.5) is translation invariant, we assume, without loss of generality, that the initialdata start with t= 1, namely

u(1,x,y) =u0(x,y), ω(1,x,y) =ω0(x,y), θ(1,x,y) =θ0(x,y). (1.6)

Differentiating (1.5) in t yields{∂ttω+∂tω−∂xxω=F1,

∂ttθ+∂tθ−∂xxθ=F2,(1.7)

and {F1 =−∂x(u ·∇θ)−∂t(u ·∇ω),

F2 =−∂x(u ·∇ω)−∂t(u ·∇θ)−u ·∇θ.(1.8)

We observe that the linear parts ω and θ obey exactly the same degenerate dampedwave equation. It follows from (1.5) and (1.6) that

ω1(1,x,y), (∂tω)(1,x,y) =−u0 ·∇ω0−ω0−∂xθ0,θ1(1,x,y), (∂tθ)(1,x,y) =−u0 ·∇θ0 +∂xω0.

(1.9)

We have derived an equivalent system of (1.1) and our attention will be focused on thisequivalent initial-value problem

∂ttω+∂tω−∂xxω=F1,

∂ttθ+∂tθ−∂xxθ=F2,

ω(1,x,y) =ω0(x,y), (∂tω)(1,x,y) =ω1(x,y),

θ(1,x,y) =θ0(x,y), (∂tθ)(1,x,y) =θ1(x,y).

(1.10)

Before we analyze the nonlinear system, we first understand the linearized system of(1.5), which is given by {

∂tω+ω=∂xθ,

∂tθ=∂xω.


Clearly, for any integer k∈N,

‖Λkω(t)‖2L2 +‖Λkθ(t)‖2L2 +

∫ t

0

‖Λkω(τ)‖2L2 dτ =‖Λkω0‖2L2 +‖Λkθ0‖2L2 , (1.11)

which implies the linear stability in any Sobolev space Hk(R2). Here Λ =√−∆ denotes

the Zygmund operator. Λ and more general fractional Laplacian operators Λα withα∈R are defined via the Fourier transform

Λαf(ξ,η) = (ξ2 +η2)α2 f(ξ,η),

where the Fourier transform is defined by

f(ξ,η) =

∫R2

e−ixξ−iyηf(x,y)dxdy.

However, the energy estimate (1.11) does not give us any information on the decayrates of ω or θ, due to the lack of thermal diffusion. The Fourier-splitting method ofSchonbek has been very effective on large-time decay problems (see, e.g., [9,47,48]), butit does not apply here when there is no thermal diffusion or damping in θ. Therefore,new ideas and different approaches appear to be necessary in order to handle the large-time behavior problem here.

To establish the long-term stability and large-time behavior of (ω,θ) of (1.10), wetake advantage of the special structure of the linear portion in (1.10). Our first step isto represent the solution of the degenerate damped wave equation{

∂ttf+∂tf−∂xxf =F,

f(1,x,y) =f0(x,y), ∂tf(1,x,y) =f1(x,y)(1.12)

to be the integral form

f(t,x,y) =K0(t,∂x)f0 +K1(t,∂x)

(1

2f0 +f1

)+

∫ t

1

K1(t−s,∂x)F (s,x,y)ds, (1.13)

where K0 and K1 are Fourier multiplier operations,

K0(t,∂x) =1

2

(e

(− 1

2+√

14+∂

2x

)(t−1) +e

(− 1

2−√

14+∂

2x

)(t−1)

),

K1(t,∂x) =1

2√

14 +∂2x

(e

(− 1

2+√

14+∂

2x

)(t−1)−e

(− 1

2−√

14+∂

2x

)(t−1)

).

A natural second step is to obtain the precise large-time behavior of K0 and K1 andsharp bounds for their action on L1 and L2 functions. Once these preparations aremade, we then write (1.10) into an integral form and apply the continuity argument.We need a suitable functional setting for the initial data. Let a∈N be a sufficientlylarge positive integer, say a>8 and define X0 to be the Sobolev space, equipped withthe norm

‖(ω0,θ0)‖X0,‖〈∇〉a(ω0,θ0)‖L2 +‖〈∇〉4(ω0,θ0)‖L1 +‖〈∇〉4(ω1,θ1)‖L1

+‖Λ−1(ω0,θ0)‖L1 +‖Λ−1(ω1,θ1)‖L1 +‖Λ−2θ0‖L1 +‖Λ−2θ1‖L1 ,


where (ω1,θ1) is given by (1.9) and 〈∇〉 denotes the inhomogeneous derivative,

〈∇〉, (I−∆)12 .

We set the Banach space X as the working space for the solution (ω,θ) to system (1.5)equipped with the following norm

‖(ω,θ)‖X = supt≥1

{t−ε‖〈∇〉aω,〈∇〉aθ‖L2 + t

14 ‖〈∇〉2θ‖L2 + t

14 ‖Λ−1ω‖L2

+ t14 ‖〈∇〉2Λ−1θ‖L2 + t

34 ‖∂x〈∇〉ω‖L2 + t

34 ‖∂x〈∇〉2θ‖L2

+ t54 ‖∂xx〈∇〉θ‖L2 + t

34 ‖∂xΛ−1θ‖L2 + t

78 ‖∂xxΛ−2θ‖L2

+ t54 ‖∂t〈∇〉2ω‖L2 + t

54 ‖∂tθ‖L2 + t

54 ‖∂tΛ−1ω‖L2

}. (1.14)

The time weights in (1.14) except the first term are based on the decay properties ofthe kernels K0 and K1. With these preparations at our disposal, we can state our mainresult as follows. For notational convenience, we write A.B for the statement thatA≤CB for some absolute constant C>0.

Theorem 1.1. There exists a small number ε0>0 such that, if the initial data (ω0,θ0)satisfies

‖(ω0,θ0)‖X0≤ ε0,

then there exists a unique global solution (ω,θ) to system (1.5) or (1.10) with

(ω,θ)∈X, P ∈C([1,∞);Ha(R2)). (1.15)

Moreover, the following decay estimates hold:

‖ω(t)‖L∞xy . ε0t−1, ‖θ(t)‖L∞xy . ε0t

− 12 , ‖P (t)‖L∞xy . ε0t

− 14 . (1.16)

To prove Theorem 1.1, we make use of the following continuity argument.

Lemma 1.1. Assume the initial data (ω0,θ0)∈X0 and the solution (ω,θ) given by(1.10) fulfills the following condition

‖(ω,θ)‖X .‖(ω0,θ0)‖X0+Q(‖(ω,θ)‖X), (1.17)

where Q(z)≥Czβ for z.1 and β>1. Then there exists ε0>0 such that, if

‖(ω0,θ0)‖X0 . ε0,

then (1.10) has a unique global solution (ω,θ)∈X and satisfies

‖(ω,θ)‖X . ε0.

To facilitate the proof of Lemma 1.1, we introduce a working space Y equippedwith the norm

‖(u,ω,θ)‖Y =‖(ω,θ)‖X +supt≥1

{t34 ‖〈∇〉ω‖L2 + t

34 ‖u1‖L2 + t

78 ‖u2‖L2

+t‖ω‖L∞+ t54 ‖∂x〈∇〉ω‖L2

}


and verify that the norm ‖(u,ω,θ)‖Y is bounded by ‖(ω,θ)‖X and Q(‖(ω,θ)‖X) withQ(z) being of the form Czβ with β>1, as stated in the following lemma.

Lemma 1.2. Let X and Y be the Banach spaces with their norms being defined as theabove. Then

‖(u,ω,θ)‖Y .‖(ω,θ)‖X +Q(‖(ω,θ)‖X). (1.18)

As a consequence of Lemma 1.2, to prove (1.17), it suffices to verify

‖(ω,θ)‖X .‖(ω0,θ0)‖X0+Q(‖(u,ω,θ)‖Y ). (1.19)

So the proof of Theorem 1.1 is reduced to establishing (1.18) in Lemma 1.2 and (1.19).

The Boussinesq equations have attracted considerable interest recently and there aresubstantial developments. Physically the Boussinesq equations are important modelsin the study of geophysical fluids and the Rayleigh-Benard convection (see, e.g., [14,17,22, 23, 27, 42, 43, 45, 51]). Mathematically they share many similar properties with the3D Navier-Stokes and the 3D Euler equations such as the vortex stretching mechanism,which is believed to be the primary reason for any potential finite-time blowup [41].The significance of the Boussinesq equations has made them the subject of numerousinvestigations. The global regularity problem and the issue of stability near physicallyrelevant equilibria are among the most prominent topics on the Boussinesq equations,especially when there is only partial or fractional dissipation. Important progress hasbeen made on both topics (see, e.g., [1–5, 8–21, 24–26, 28–40, 44, 46, 49, 50, 52–54, 56–63]). It is hoped that the results presented in this paper will help lead to a betterunderstanding of the hydrostatic equilibrium.

We remark that Wu, Wu and Xu [55] studied the global solution of a magnetohy-drodynamic (MHD) system near a background magnetic field and skillfully solved thesmall data global well-posedness problem there. We take advantage of some of the toolsdeveloped in [55]. Although there are resemblances between the linear part of (1.10)and that of the MHD system, there are a few differences between the Boussinesq-Benardsystem and the MHD system in [55]. First, the equation of θ in (1.1) involves ∆u2 whilethe magnetic stream function in [55] depends on u2. As a consequence, the functionalsetting for θ involves Sobolev spaces of negative indices, as in the definitions of X0 andX. Second, ω and θ naturally form a self-contained system, as in (1.10) and it sufficesto understand this system of two scalar functions. There are some other differences aswell such as the nonlinear terms.

The rest of this paper is divided into five sections. The second section provides thederivation of the integral representation (1.13) for the convenience of the readers. Inaddition, decay rates of K0 and K1 and some estimates of K0 and K1 acting on L1 andL2 functions are also listed here. The third section verifies (1.18). The fourth sectionshows that

supt≥1

t−ε‖〈∇〉a(ω,θ)‖L2 .‖(ω0,θ0)‖X0 +Q(‖(u,ω,θ)‖Y ),

which fulfills part of the proof for (1.19). Section 5 serves as a preparation for Section6. It estimates the nonlinear terms and provides bounds for various Sobolev norms ofthese terms. Section 6 completes the proof of (1.19) via the integral representation in(1.13) and thus finishes the proof of Theorem 1.1.


2. Integral representation and properties of kernel functionsThis section serves as a preparation. We derive the integral representation in (1.13)

and list several estimates on the kernel functions K0 and K1.

Lemma 2.1. The solution of the degenerate damped wave equation{∂ttf+∂tf−∂xxf =F,

f(1,x,y) =f0(x,y), ∂tf(1,x,y) =f1(x,y)

can be written in the integral form

f(t,x,y) =K0(t,∂x)f0 +K1(t,∂x)

(1

2f0 +f1

)+

∫ t

1

K1(t−s,∂x)F (s,x,y)ds,

where K0 and K1 are Fourier multiplier operators given by

K0(t,ξ) =1

2

(e

(− 1

2+√

14−ξ2

)(t−1) +e

(− 1

2−√

14−ξ2

)(t−1)

), (2.1)

K1(t,ξ) =1

2√

14−ξ2

(e

(− 1

2+√

14−ξ2

)(t−1)−e

(− 1

2−√

14−ξ2

)(t−1)

). (2.2)

Proof. We decompose the second-order operator into first-order time operators,

∂ttf+∂tf−∂xxf =

(∂t+

1

2− 1

2

√1+4∂2x

)(∂t+

1

2+

1

2

√1+4∂2x

)f = 0.

Therefore ∂ttf+∂tf−∂xxf = 0 can be written as(∂t+

12−

12

√1+4∂2x

)g= 0,(

∂t+12 + 1

2

√1+4∂2x

)f =g

or (∂t+

12 + 1

2

√1+4∂2x

)h= 0,(

∂t+12−

12

√1+4∂2x

)f =h.

Clearly,

g(t,x,y) =e

(− 1

2+12

√1+4∂2

x

)(t−1)g(1,x,y),

h(t,x,y) =e

(− 1

2−12

√1+4∂2

x

)(t−1)h(1,x,y).

Therefore,

f(t,x,y) =1√

1+4∂2x(g(t,x,y)−h(t,x,y))

=1√

1+4∂2x

(e

(− 1

2+12

√1+4∂2

x

)(t−1)g(1,x,y)−e

(− 1

2−12

√1+4∂2

x

)(t−1)h(1,x,y)

)


with

g(1,x,y) =f1 +

(1

2+

1

2

√1+4∂2x

)f0, h(1,x,y) =f1 +

(1

2− 1

2

√1+4∂2x

)f0.

Regrouping the terms in the representation of f above yields the desired formula for f .This completes the proof of Lemma 2.1.

The integral representation stated in Lemma 2.1 will be applied to (1.10) and alsoto the system for (Λ−1ω,Λ−1θ), which satisfies{

∂ttΛ−1ω+∂tΛ

−1ω−∂xxΛ−1ω=F3,

∂ttΛ−1θ+∂tΛ

−1θ−∂xxΛ−1θ=F4,(2.3)

where {F3 =−∂xΛ−1∇·(uθ)−∂tΛ−1∇·(uω),

F4 =−∂xΛ−1∇·(uω)−∂tΛ−1∇·(uθ)−Λ−1∇·(uθ).(2.4)

As a preparation for the estimates in the subsequent sections, the following lemmaprovides some decay estimates on K0 and K1. These estimates are derived in [55].

Lemma 2.2. Let K0,K1 be defined as above. Then for any α>0, 1≤ q≤∞, i= 0,1,

(1) ‖|ξ|αKi(t,·)‖Lqξ(|ξ|≤ 12 ). 〈t〉−

12 (

1q+α),

(2) ‖∂tKi(t,·)‖Lqξ(|ξ|≤ 12 ). 〈t〉−1−

12q ,

(3) |Ki(t,ξ)|.e−12 t, for any |ξ|≥ 1

2,

(4) |〈ξ〉−1∂tK0(t,ξ)|, |∂tK1(ξ)|.e− 12 t, for any |ξ|≥ 1

2.

(2.5)

We list several bounds on K0 and K1 when acting on L1 and L2 functions. Thesebounds can be found in [55].

Lemma 2.3. Assume that ‖K(t,·)‖L∞ is bounded. Then, for any Schwartz functionf , we have

‖K(t,∂x)f‖L2xy

.‖K(t,ξ)‖L∞ξ ‖f‖L2xy.

Lemma 2.4. Assume that ‖K(t,·)‖L2 is bounded. Then, for any Schwartz function f ,we have


.‖K(t,ξ)‖L2ξ‖Λ 1

2−ε〈∇〉2εf‖L1xy.

Lemma 2.5. Assume that ‖K(t,·)‖L∞ is bounded. Then, for any Schwartz functionf , we have


.‖K(t,ξ)‖L∞ξ ‖Λ12−ε〈∇〉 12+2εf‖L1

xy.

As a consequence of Lemma 2.4 and Lemma 2.5, we have the following corollary.


Corollary 2.1. Assume the Fourier multiplier operator K(t,∂x) satisfies, for someα≥0,

‖∂αxK(t,ξ)‖L2(|ξ|≤ 12 )<∞, ‖K(t,ξ)‖L∞(|ξ|≥ 1

2 )<∞.

Then, for any Schwartz function f and any ε>0,

‖∂αxK(t,∂x)f‖L2xy

. (‖∂αxK(t,ξ)‖L2ξ(|ξ|≤

12 )

+‖K(t,ξ)‖L∞ξ (|ξ|≥ 12 )

)

×‖〈∇〉α+ 12+2εΛ

12−εf‖L1

xy.

3. Verifying Lemma 1.2This section and the subsequent three sections are devoted to the proof of Theorem

1.1. As aforementioned, the proof of Theorem 1.1 is reduced to the proof of (1.18) inLemma 1.2 and (1.19). This section verifies (1.18), namely

‖(u,ω,θ)‖Y .‖(ω,θ)‖X +Q(‖(ω,θ)‖X).

Due to the definitions of X and Y , it suffices to check that

supt≥1

{t34 ‖〈∇〉ω‖L2 + t

34 ‖u1‖L2 + t

78 ‖u2‖L2 + t‖ω‖L∞+ t

54 ‖∂x〈∇〉ω‖L2

}.‖(ω,θ)‖X +Q(‖(ω,θ)‖X). (3.1)

We start with the first term in (3.1). Noticing that u=∇⊥∆−1ω, we have

‖u‖L2 =‖∇⊥∆−1ω‖L2 =‖Λ−1ω‖L2 .

It follows from the basic inequality

‖f‖L∞(R)≤√

2‖f‖12

L2(R)‖f′‖

12

L2(R)

that

‖u‖L∞(R2).‖u‖14

L2‖∂xu‖14

L2‖∂yu‖14

L2‖∂xyu‖14

L2

.‖Λ−1ω‖14

L2‖ω‖12

L2‖∂xω‖14

L2 .

where L∞ and L2 are abbreviated for L∞(R2) and L2(R2), respectively. Unless oth-erwise stated, this convention is assumed throughout the rest of the paper. By aninterpolation inequality and Young’s inequality,

‖ω‖L2 ≤‖∂tω‖L2 +‖u ·∇ω‖L2 +‖∂xθ‖L2

≤‖∂tω‖L2 +‖u‖L∞‖∇ω‖L2 +‖∂xθ‖L2

≤‖∂tω‖L2 +‖Λ−1ω‖14

L2‖ω‖12

L2‖∂xω‖14

L2‖ω‖1− 1

a

L2 ‖∇aω‖1a

L2 +‖∂xθ‖L2

≤‖∂tω‖L2 +‖Λ−1ω‖14

L2‖ω‖78

L2‖∂xω‖14

L2‖ω‖58−

1a

L2 ‖∇aω‖1a

L2 +‖∂xθ‖L2

≤‖∂tω‖L2 +1

2‖ω‖L2 +‖Λ−1ω‖2L2‖∂xω‖2L2‖〈∇〉aω‖5L2 +‖∂xθ‖L2 .

By the definition of X,

‖ω‖L2 .‖∂tω‖L2 +‖Λ−1ω‖2L2‖∂xω‖2L2‖〈∇〉aω‖5L2 +‖∂xθ‖L2

.s−54 ‖(ω,θ)‖X +s(−

14×2−

34×2+5ε)‖(ω,θ)‖7X +s−

34 ‖(ω,θ)‖X

.s−34 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)),


where − 12−

32 +5ε≤− 3

4 and Q(b) = b7. Therefore,

‖ω‖L∞ ≤‖ω‖14

L2‖∂xω‖14

L2‖∂yω‖14

L2‖∂xyω‖14

L2

≤‖ω‖14

L2‖∇ω‖12

L2‖∇2ω‖14

L2

≤‖ω‖14

L2‖ω‖12 (1−

1a )

L2 ‖∇aω‖12a

L2‖ω‖14 (1−

2a )

L2 ‖∇aω‖12a

L2

≤‖ω‖1−1a

L2 ‖∇aω‖1a

L2

.s−34 (1−

1a )sε

1a (‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−58 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)),

where a and ε satisfy − 18 + 1

a ( 34 +ε)≤0. This is a preliminary estimate for ‖ω‖L∞ and a

better decay rate will be provided later. By an interpolation inequality and the Sobolevembedding,

‖∇ω‖L∞ .‖ω‖1−1a−2

L∞ ‖∇a−2ω‖1a−2

L∞

.‖ω‖1−1a−2

L∞ ‖〈∇〉aω‖1a−2

L∞

.s−58 (1−

1a−2 )+ε

1a−2 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−12 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)), (3.2)

when − 58 (1− 1

a−2 )+ε 1a−2 ≤−

12 . Similarly,

‖∇2ω‖L∞ .s−12 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)).

To bound ‖∂xω‖L2 , we apply ∂x to the vorticity equation in (1.5) to get

∂t∂xω+∂xu ·∇ω+u ·∇∂xω+∂xω=∂xxθ.

Therefore,

‖∂xω‖L2 ≤‖∂t∂xω‖L2 +‖∂xu ·∇ω‖L2 +‖u ·∇∂xω‖L2 +‖∂xxθ‖L2

≤‖∂t∂xω‖L2 +‖∂xu‖L2‖∇ω‖L∞+‖u‖L∞‖∂x∇ω‖L2 +‖∂xxθ‖L2

≤‖∂t∂xω‖L2 +‖ω‖L2‖∇ω‖L∞+‖Λ−1ω‖14

L2‖ω‖12

L2‖∂xω‖14

L2‖ω‖1− 2

a

L2 ‖∇aω‖2a

L2

+‖∂xxθ‖L2

≤‖∂t∂xω‖L2 +‖ω‖L2‖∇ω‖L∞+‖Λ−1ω‖13

L2‖ω‖23

L2‖ω‖43−

83a

L2 ‖〈∇〉aω‖83a

L2

+1

2‖∂xω‖L2 +‖∂xxθ‖L2 .

Then, by the definition of ‖(ω,θ)‖X , we get

‖∂xω‖L2

.‖∂t∇ω‖L2 +‖ω‖L2‖∇ω‖L∞+‖Λ−1ω‖13

L2‖ω‖2− 8

3a

L2 ‖〈∇〉aω‖83a

L2 +‖∂xxθ‖L2

. (s−54 +s−

34−

12 +s−

112−

54 +s−

54 )(‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−54 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)).


Applying ∇ to the vorticity equation in (1.5), we obtain

∂t∇ω+∇u ·∇ω+u ·∇2ω+∇ω=∂x∇θ.

By Young’s inequality,

‖∇ω‖L2 ≤‖∂t∇ω‖L2 +‖∇u ·∇ω‖L2 +‖u ·∇2ω‖L2 +‖∂x∇θ‖L2

≤‖∂t∇ω‖L2 +‖∇u‖L2‖∇ω‖L∞+‖u‖L∞‖∇2ω‖L2 +‖∂x∇θ‖L2

≤‖∂t∇ω‖L2 +‖ω‖L2‖∇ω‖L∞+‖Λ−1ω‖14

L2‖ω‖12

L2‖∂xω‖14

L2‖ω‖1− 2

a

L2 ‖∇aω‖2a

L2

+‖∂x∇θ‖L2

≤‖∂t∇ω‖L2 +‖ω‖L2‖∇ω‖L∞+‖Λ−1ω‖13

L2‖ω‖23

L2‖ω‖43−

83a

L2 ‖〈∇〉aω‖83a

L2

+1

2‖∇ω‖L2 +‖∂x∇θ‖L2 .

Then by the definition of ‖(ω,θ)‖X , we obtain

‖∇ω‖L2

.‖∂t∇ω‖L2 +‖ω‖L2‖∇ω‖L∞+‖Λ−1ω‖13

L2‖ω‖2− 8

3a

L2 ‖〈∇〉aω‖83a

L2 +‖∂x∇θ‖L2

. (s−54 +s−

34−

12 +s−

112−

54 +s−

34 )(‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−34 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)). (3.3)

Next we deal with the term ‖∂x∇ω‖L2 . Applying ∂x∇ to the vorticity equation in (1.5),we have

∂t∂x∇ω+∂x∇u ·∇ω+∇u ·∇∂xω+∂xu ·∇2ω+u ·∇2∂xω+∂x∇ω=∂xx∇θ.

Taking the L2-norm yields

‖∂x∇ω‖L2 ≤‖∂t∂x∇ω‖L2 +‖∂x∇u ·∇ω‖L2 +‖∇u ·∇∂xω‖L2 +‖∂xu ·∇2ω‖L2

+‖u ·∇2∂xω‖L2 +‖∂xx∇θ‖L2 . (3.4)

By the definition of ‖(ω,θ)‖X , we have

‖∂t∂x∇ω‖L2 .‖∂t∇2ω‖L2 .s−54 ‖(ω,θ)‖X , ‖∂xx∇θ‖L2 .s−

54 ‖(ω,θ)‖X ,

‖∂x∇u ·∇ω‖L2 .‖∂xω‖L2‖∇ω‖L∞

.s−54−

12 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−74 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)),

‖∇u ·∇∂xω‖L2 .‖ω‖L2‖∇2ω‖L∞

.s−34−

12 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−54 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)),

‖∂xu ·∇2ω‖L2 .‖ω‖L2‖∇2ω‖L∞

.s−34−

12 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−54 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X))


and

‖u ·∇2∂xω‖L2 .‖u‖L∞‖∂x∇2ω‖L2

.‖Λ−1ω‖14

L2‖ω‖12

L2‖∂xω‖14

L2‖∂xω‖a−3a−1

L2 ‖∂x〈∇〉a−1ω‖a−3a−1

L2

.s−54 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)).

Inserting these estimates in (3.4) leads to

‖∂x∇ω‖L2 .s−54 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)).

Using the estimate obtained above, we can calculate the decay estimate of ‖ω‖L∞ againwith a better decay rate,

‖ω‖L∞ ≤‖ω‖14

L2‖∂xω‖14

L2‖∂yω‖14

L2‖∂xyω‖14

L2

.s14×(−

34−

54−

34−

54 )(‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−1(‖(ω,θ)‖X +Q(‖(ω,θ)‖X)). (3.5)

In order to obtain the decay estimate of ‖u‖L2 , we make use of the structure of system(1.5) and the boundedness of Riesz transforms. Applying Λ−1 to the first equation of(1.5), we have

∂tΛ−1ω+Λ−1∇·(uω)+Λ−1ω=∂xΛ−1θ. (3.6)

By the boundedness of the Riesz transform in L2, we have

‖Λ−1ω‖L2 ≤‖∂tΛ−1ω‖L2 +‖Λ−1∇·(uω)‖L2 +‖∂xΛ−1θ‖L2

≤‖∂tΛ−1ω‖L2 +C‖uω‖L2 +‖∂xΛ−1θ‖L2

≤‖∂tΛ−1ω‖L2 +C‖Λ−1ω‖L2‖ω‖L∞+‖∂xΛ−1θ‖L2

≤‖∂tΛ−1ω‖L2 +C‖Λ−1ω‖L2‖ω‖L∞+‖∂xΛ−1θ‖L2

. (s−54 +s−

14−1 +s−

34 )(‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

.s−34 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)).

Therefore,

‖u‖L2 =‖∇⊥∆−1ω‖L2 =‖Λ−1ω‖L2 .s−34 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)).

Noticing that u2 =∂x∆−1ω and using the first equation of (1.5), we have

u2 =−∂t∂xΛ−2ω+∂xΛ−2(u ·∇ω)+∂xxΛ−2θ.

According to the boundedness of Riesz transform, divergence-free condition of u andHolder inequality, we obtain

‖u2‖L2 .‖∂tΛ−1ω‖L2 +‖uω‖L2 +‖∂xxΛ−2θ‖L2

.‖∂tΛ−1ω‖L2 +‖u‖L∞‖ω‖L2 +‖∂xxΛ−2θ‖L2

.s−78 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)).


A special consequence of the estimates above is the following bound for ‖u‖L∞ , whichwill be used in a subsequent section.

‖u‖L∞ .‖u‖14

L2‖∂xu‖14

L2‖∂yu‖14

L2‖∂xyu‖14

L2

.s−14 (

34+

34+

34+

54 ) (‖(ω,θ)‖X +Q(‖(ω,θ)‖X))

=s−78 (‖(ω,θ)‖X +Q(‖(ω,θ)‖X)). (3.7)

Thus we have verified (3.1).

4. Estimate for ‖〈∇〉a(ω,θ)‖L2

This section and the rest two sections continue the proof of Theorem 1.1. The goalis to prove

‖(ω,θ)‖X .‖(ω0,θ0)‖X0 +Q(‖(u,ω,θ)‖Y ).

This section verifies

supt≥1

t−ε‖〈∇〉a(ω,θ)‖L2 .‖(ω0,θ0)‖X0+Q(‖(u,ω,θ)‖Y ).

To do so, we first derive the L2 bound of (ω,θ). Taking the L2-inner product of the firsttwo equations of (1.5) with (ω, θ) yields

1

2

d

dt(‖ω(t)‖2L2 +‖θ(t)‖2L2)+‖ω‖2L2 =

∫R2

∂xθω+∂xωθ dxdy= 0.

Integrating from 1 to t with respect to the time variable leads to

supt≥1

t−ε‖(ω,θ)‖2L2 .‖(ω0,θ0)‖2L2 .‖(ω0,θ0)‖X0 . (4.1)

Next we establish the high-order energy estimate. Applying ∇a to (1.5), we have{∂t∇aω+[∇a,u ·∇]ω+u ·∇∇aω+∇aω=∂x∇aθ,∂t∇aθ+[∇a,u ·∇]θ+u ·∇∇aθ=∂x∇aω,

(4.2)

where we have used the commutator notation

[f,g] =fg−gf.

Taking the L2-inner product of (4.2) with (∇aω,∇aθ) yields

1

2

d

dt(‖∇aω‖2L2 +‖∇aθ‖2L2)+‖∇aω‖2L2

=−∫R2

[∇a,u ·∇]ω ·∇aω dxdy− [∇a,u ·∇]θ ·∇aθ dxdy

.‖[∇a,u ·∇]ω‖L2‖∇aω‖L2 +‖[∇a,u ·∇]θ‖L2‖∇aθ‖L2

. (‖∇au‖L2‖∇ω‖L∞+‖∇aω‖L2‖∇u‖L∞)‖∇aω‖L2

+(‖∇au‖L2‖∇θ‖L∞+‖∇aθ‖L2‖∇u‖L∞)‖∇aθ‖L2 .

.‖∇aω‖a−1a

L2 ‖ω‖1a

L2(‖∇ω‖L∞‖∇aω‖L2 +‖∇θ‖L∞‖∇aθ‖L2)

+1

4‖∇aω‖2L2 +(‖∇u‖2L∞+‖∇u‖L∞)(‖∇aω‖2L2 +‖∇aθ‖2L2)

.‖ω‖2a+1

L2 (‖∇ω‖L∞‖∇aω‖L2 +‖∇θ‖L∞‖∇aθ‖L2)2aa+1

+1

2‖∇aω‖2L2 +(‖∇u‖2L∞+‖∇u‖L∞)(‖∇aω‖2L2 +‖∇aθ‖2L2).


Here we have used the standard commutator estimate

‖[∇a,f ·∇]g‖L2 .‖∇f‖L∞‖∇ag‖L2 +‖∇g‖L∞‖∇af‖L2

and the interpolation formula

‖∇au‖L2 ≈‖∇a−1ω‖L2 .‖∇aω‖1−1a

L2 ‖ω‖1a

L2 .

Thus

1

2

d

dt(‖∇aω‖2L2 +‖∇aθ‖2L2).‖ω‖

2a+1

L2 (‖∇ω‖L∞‖∇aω‖L2 +‖∇θ‖L∞‖∇aθ‖L2)2aa+1

+(‖∇u‖2L∞+‖∇u‖L∞)(‖∇aω‖2L2 +‖∇aθ‖2L2).

Integrating from 1 to t with respect to the time variable, we have

‖∇aω‖2L2 +‖∇aθ‖2L2

.‖∇aω0‖2L2 +‖∇aθ0‖2L2 +

∫ t

1

(‖∇u‖2L∞+‖∇u‖L∞)(‖∇aω‖2L2 +‖∇aθ‖2L2) ds

+

∫ t

1

‖ω‖2a+1

L2 (‖∇ω‖L∞‖∇aω‖L2 +‖∇θ‖L∞‖∇aθ‖L2)2aa+1 ds.

By the definition of ‖(u,ω,θ)‖Y ,

‖∇θ‖L∞ .‖∇θ‖14

L2‖∂x∇θ‖14

L2‖∂y∇θ‖14

L2‖∂xy∇θ‖14

L2

.‖〈∇〉2θ‖12

L2‖∂x〈∇〉2θ‖12

L2

.s−14×

12−

34×

12 ‖(u,ω,θ)‖Y

.s−12 ‖(u,ω,θ)‖Y . (4.3)

Invoking the estimates in (3.2), (3.3) and (4.3), we find∫ t

1

‖ω‖2a+1

L2 (‖∇ω‖L∞‖∇aω‖L2 +‖∇θ‖L∞‖∇aθ‖L2)2aa+1 ds

.∫ t

1

s−3

2(a+1) (s−1+ε+s−12+ε)

2aa+1 ds(‖(u,ω,θ)‖4−

2a+1

Y +Q(‖(u,ω,θ)‖Y ))

.∫ t

1

s−1+2εds(‖(u,ω,θ)‖4−2a+1

Y +Q(‖(u,ω,θ)‖Y ))

. t2ε(‖(u,ω,θ)‖4−2a+1

Y +Q(‖(u,ω,θ)‖Y )). (4.4)

As for the term ‖∇u‖L∞ , the definition of ‖(u,ω,θ)‖Y yields that

‖∇u‖L∞ .‖∇u‖14

L2‖∂x∇u‖14

L2‖∂y∇u‖14

L2‖∂xy∇u‖14

L2

.‖ω‖14

L2‖∂xω‖14

L2‖∇ω‖14

L2‖∂x∇ω‖14

L2

.s14×(−

34−

54−

34−

54 )‖(u,ω,θ)‖Y

.s−1‖(u,ω,θ)‖Y .


Thus ∫ t

1

(‖∇u‖2L∞+‖∇u‖L∞)(‖∇aω‖2L2 +‖∇aθ‖2L2) ds

. t2ε(‖(u,ω,θ)‖3Y +‖(u,ω,θ)‖4Y ). (4.5)

Therefore

‖∇aω‖2L2+‖∇aθ‖2L2 .‖∇aω0‖2L2 +‖∇aθ0‖2L2

+ t2ε(‖(u,ω,θ)‖3Y +‖(u,ω,θ)‖4−2a+1

Y +‖(u,ω,θ)‖4Y +Q(‖(u,ω,θ)‖Y )). (4.6)

Then we deduce that

supt≥1

t−ε‖∇a(ω,θ)‖L2 .‖(ω0,θ0)‖X0 +Q(‖(u,ω,θ)‖Y ). (4.7)

Combining (4.1) and (4.7), we obtain

supt≥1

t−ε‖〈∇〉a(ω,θ)‖L2 .‖(ω0,θ0)‖X0+Q(‖(u,ω,θ)‖Y ). (4.8)

5. Estimates for the nonlinear termsThis section serves as a preparation for the estimates to be presented in the subse-

quent section. We bound various Sobolev norms of the nonlinear terms Fi for i= 1,2,3,4defined in (1.8) and (2.4). The Littlewood-Paley techniques will be employed. We givea brief introduction to Littlewood-Paley theory. Let φ(ξ) be a smooth bump functionsupported in the ball |ξ|≤2 and equal to 1 on the ball |ξ|≤1. For any real numberN >0 and f ∈S ′ (tempered distributions), projection operators are defined as follows:

P≤Nf(ξ), φ

(ξ

N

)f(ξ),

P>Nf(ξ),

(1−φ

(ξ

N

))f(ξ),

PNf(ξ),

(φ

(ξ

N

)−φ(

2ξ

N

))f(ξ).

P<N and P≥N are similarly defined. The following Bernstein inequalities play an im-portant role in the process of estimating the nonlinear terms (see, e.g., [6, 7]).

Lemma 5.1 (Bernstein Inequalities). Let α≥0 and N >0 be real numbers and let1≤p≤ q≤∞. Then there exist three constants C1,C2,C3 such that

‖ΛαP≤Nf‖Lq(Rd)≤C1Nα+d( 1

p−1q )‖P≤Nf‖Lp(Rd),

C2Nα‖PNf‖Lq(Rd)≤‖ΛαPNf‖Lq(Rd)≤C3N

α‖PNf‖Lq(Rd).

Our first lemma bounds F1.

Lemma 5.2. For any s≥1,

‖〈∇〉5Λ12−εF1(s,·)‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ), (5.1)

where ε is same as in Corollary 2.1 and ε is same as in the definition of X-norm definedin (1.14).


Proof. We recall that F1 is given by

F1 =−∂x(u ·∇θ)−∂t(u ·∇ω)

=−∂xu ·∇θ−u ·∇∂xθ−∂tu ·∇ω−u ·∇∂tω,F11 +F12 +F13 +F14.

For F11(u,θ), we divide it into three parts via the frequency decomposition definedabove

F11 =−∂x(1−P≤sδ)u ·∇θ−∂xP≤sδu ·∇(1−P≤sδ)θ−∂xP≤sδu ·∇P≤sδθ. (5.2)

For the first high-frequency part, we have

‖〈∇〉5Λ12−ε(∂x(1−P≤sδ)u ·∇θ)‖L1

xy

.‖〈∇〉6(∂xP&sδu ·∇θ)‖L1xy

.‖〈∇〉6∂xP&sδu‖L2xy‖〈∇〉6∇θ‖L2

xy

.‖〈∇〉7P&sδu‖L2xy‖〈∇〉7θ‖L2

xy

.s−(a−7)δ‖〈∇〉aP&sδu‖L2xy‖〈∇〉aθ‖L2

xy

.s−32−εQ(‖(u,ω,θ)‖Y )

for large a and small δ. For the second high-frequency part in (5.2), we can bound it inthe same way. For sufficiently large a,

‖〈∇〉5Λ12−ε(∂xP≤sδu ·∇(1−P≤sδ)θ)‖L1

xy

.‖〈∇〉6(∂xP.sδu ·∇P&sδθ)‖L1xy

.‖〈∇〉6∂xP.sδu‖L2xy‖〈∇〉6∇P&sδθ‖L2

xy

.‖〈∇〉7P.sδu‖L2xy‖〈∇〉7P&sδθ‖L2

xy

.s−(a−7)δ‖〈∇〉aP.sδu‖L2xy‖〈∇〉aP&sδθ‖L2

xy

.s−32−εQ(‖(u,ω,θ)‖Y ).

For the low-frequency part in (5.2), we can bound it by

‖〈∇〉5Λ12−ε(∂xP≤sδu ·∇P≤sδθ)‖L1

xy

.‖〈∇〉6P≤4sδ∂xP≤sδu ·∇P≤sδθ‖L1xy

.s6δ(‖(P≤sδ∂xu1∂xP≤sδθ)‖L1xy

+‖(P≤sδ∂xu2∂yP≤sδθ)‖L1xy

)

.s6δ‖ω‖L2xy‖∂xθ‖L2

xy+s7δ‖u2‖L2

xy‖∂yθ‖L2

xy

. (s6δs−34−

34 +s7δs−

78−

14 )Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y )

for δ small. Thus we proved

‖〈∇〉5Λ12−εF11‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ). (5.3)

Along the same lines, we can also prove that

‖〈∇〉5Λ12−εF1i‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ), i= 2,3,4. (5.4)


For F12(u,θ), we again divide it into three parts

F12 =−(1−P≤sδ)u ·∇∂xθ−P≤sδu ·∇(1−P≤sδ)∂xθ−P≤sδu ·∇P≤sδ∂xθ.

Then, for the first high-frequency part, we have

‖〈∇〉5Λ12−ε((1−P≤sδ)u ·∇∂xθ)‖L1

xy

.‖〈∇〉6(P&sδu ·∇∂xθ)‖L1xy

.‖〈∇〉6P&sδu‖L2xy‖〈∇〉6∇∂xθ‖L2

xy

.‖〈∇〉6P&sδu‖L2xy‖〈∇〉8θ‖L2

xy


xy

.s−32−εQ(‖(u,ω,θ)‖Y ),

for large enough a and small δ. For the second high-frequency part, we can bound it by

‖〈∇〉5Λ12−ε(P≤sδu ·∇(1−P≤sδ)∂xθ)‖L1

xy

.‖〈∇〉6(P.sδu ·∇P&sδ∂xθ)‖L1xy

.‖〈∇〉6P.sδu‖L2xy‖〈∇〉6∇P&sδ∂xθ‖L2

xy

.‖〈∇〉6P.sδu‖L2xy‖〈∇〉6P&sδθ‖L2

xy

.s−(a−8)δ‖〈∇〉aP.sδu‖L2xy‖〈∇〉aP&sδθ‖L2

xy

.s−32−εQ(‖(u,ω,θ)‖Y ).

For the low-frequency part, we can bound it by

‖〈∇〉5Λ12−ε(P≤sδu ·∇P≤sδ∂xθ)‖L1

xy

.‖〈∇〉6P≤4sδ(P≤sδu ·∇P≤sδ∂xθ)‖L1xy

.s6δ‖(P≤sδu ·∇P≤sδ∂xθ)‖L1xy

.s6δ‖P≤sδu‖L2xy‖∇P≤sδ∂xθ‖L2

xy

.s7δ‖Λ−1ω‖L2xy‖∂xθ‖L2

xy

.s7δs−34−

34Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

Thus we have shown

‖〈∇〉5Λ12−εF12‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ). (5.5)

For F13(u,ω), we also divide it into three parts

F13 =−(1−P≤sδ)∂tu ·∇ω−P≤sδ∂tu ·∇(1−P≤sδ)ω−P≤sδ∂tu ·∇P≤sδω.

For the first high-frequency part, we have

‖〈∇〉5Λ12−ε((1−P≤sδ)∂tu ·∇ω)‖L1

xy

.‖〈∇〉6(P&sδ∂tu ·∇ω)‖L1xy

.‖〈∇〉6P&sδ∂tu‖L2xy‖〈∇〉6∇ω‖L2

xy.


By Lemma 5.1 and (1.5), we obtain

‖〈∇〉6P&sδ∂tu‖L2xy

.s−(a−3)δ‖〈∇〉a−2∂tω‖L2xy

.s−(a−3)δ‖〈∇〉a−2(∂xθ−ω−u ·∇ω)‖L2xy

.s−(a−3)δsCε(‖(u,ω,θ)‖Y +Q(‖(u,ω,θ)‖Y )).

Therefore, combining two estimates above, we obtain

‖〈∇〉5Λ12−ε((1−P≤sδ)∂tu ·∇ω)‖L1

xy.s

32−εQ(‖(u,ω,θ)‖Y ).

for large enough a and small δ. Similarly, for the second high-frequency part,

‖〈∇〉5Λ12−ε(P≤sδ∂tu ·∇(1−P≤sδ)ω)‖L1

xy.s−

32−εQ(‖(u,ω,θ)‖Y ).

The low-frequency part is bounded by

‖〈∇〉5Λ12−ε(P≤sδ∂tu ·∇P≤sδω)‖L1

xy

.‖〈∇〉6P≤4sδ(P≤sδ∂tu ·∇P≤sδω)‖L1xy

.s6δ‖(P≤sδ∂tu ·∇P≤sδω)‖L1xy

.s6δ‖P≤sδ∂tu‖L2xy‖∇P≤sδω‖L2

xy

.s7δ‖∂tΛ−1ω‖L2xy‖ω‖L2

xy

.s7δs−54−

34Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

Thus we have proven

‖〈∇〉5Λ12−εF13‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ). (5.6)

For F14(u,θ), we also divide it into three parts,

F14 =−(1−P≤sδ)u ·∇∂tω−P≤sδu ·∇(1−P≤sδ)∂tω−P≤sδu ·∇P≤sδ∂tω.

Then, for the first high-frequency part, we have

‖〈∇〉5Λ12−ε((1−P≤sδ)u ·∇∂tω)‖L1

xy

.‖〈∇〉6(P&sδu ·∇∂tω)‖L1xy

.‖〈∇〉6P&sδu‖L2xy‖〈∇〉6∇∂tω‖L2

xy

.‖〈∇〉6P&sδu‖L2xy‖〈∇〉7(∂xθ−ω−u ·∇ω)‖L2

xy

.s−(a−6)δ‖〈∇〉aP&sδu‖L2xysCε(‖(u,ω,θ)‖Y +Q(‖(u,ω,θ)‖Y ))

.s−32−εQ(‖(u,ω,θ)‖Y ),

for large enough a and small δ. Similarly, for the second high-frequency part,

‖〈∇〉5Λ12−ε(P≤sδu ·∇(1−P≤sδ)∂tω)‖L1

xy.s−

32−εQ(‖(u,ω,θ)‖Y ).


For the low-frequency part, we can bound it by

‖〈∇〉5Λ12−ε(P≤sδu ·∇P≤sδ∂tω)‖L1

xy

.‖〈∇〉6P≤4sδ(P≤sδu ·∇P≤sδ∂tω)‖L1xy

.s6δ‖(P≤sδu ·∇P≤sδ∂tω)‖L1xy

.s6δ‖P≤sδu‖L2xy‖∇P≤sδ∂tω‖L2

xy

.s7δ‖Λ−1ω‖L2xy‖∂tω‖L2

xy

.s7δs−34−

54Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

Thus we have proven

‖〈∇〉5Λ12−εF14‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ). (5.7)

This completes the proof of Lemma 5.2.

F2 admits a similar bound.


‖〈∇〉5Λ12−εF2(s,·)‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ), (5.8)

where ε is same as in Corollary 2.1 and ε is same as in the definition of X-norm in(1.14).

Proof. We can rewrite F2 explicitly as

F2 =−∂x(u ·∇ω)−∂t(u ·∇θ)−u ·∇θ=−∂xu ·∇ω−u ·∇∂xω−∂tu ·∇θ−u ·∇∂tθ−u ·∇θ,F21 +F22 +F23 +F24 +F25.

It suffices to analyze u ·∇θ, which is the worst term in F2. Other terms can be treated inthe same way as in the proof of Lemma 5.2. We again use the frequency-decompositiontechnique to handle this term. Firstly we can divide it into three parts

u ·∇θ=−(1−P≤sδ)u ·∇θ−P≤sδu ·∇(1−P≤sδ)θ−P≤sδu ·∇P≤sδθ.

So for the first high-frequency part, we have

‖〈∇〉5Λ12−ε((1−P≤sδ)u ·∇θ)‖L1

xy

.‖〈∇〉6(P&sδu ·∇θ)‖L1xy

.‖〈∇〉6P&sδu‖L2xy‖〈∇〉6∇θ‖L2

xy

.‖〈∇〉6P&sδu‖L2xy‖〈∇〉7θ‖L2

xy


xy

.s−32−εQ(‖(u,ω,θ)‖Y )

for large enough a and small δ, say (a−6)δ≥ 32 +ε. Using the same method, one can

obtain the estimate for the second high-frequency part,

‖〈∇〉5Λ12−ε(P≤sδu ·∇(1−P≤sδ)θ)‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ).


For the low-frequency part,

‖〈∇〉5Λ12−ε(P≤sδu ·∇P≤sδθ)‖L1

xy

.‖〈∇〉6P≤4sδ(P≤sδu ·∇P≤sδθ)‖L1xy

.s6δ(‖P≤sδu1∂xP≤sδθ‖L1xy

+‖P≤sδu2∂yP≤sδθ‖L1xy

)

.s6δ(‖u1‖L2xy‖∂xθ‖L2

xy+‖u2‖L2

xy‖∂yθ‖L2

xy)

.s6δ(s−34−

34 +s−

78−

14 )Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

This finishes the proof of Lemma 5.3.

According to the boundness of Riesz transforms, we can also bound F3 and F4 inthe same way. The main result can be stated as follows.


‖〈∇〉5Λ12−εF3‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ), (5.9)

‖〈∇〉5Λ12−εF4‖L1

xy.s−1−εQ(‖(u,ω,θ)‖Y ), (5.10)

where ε is same as in Corollary 2.1 and ε is same as in the definition of X-norm in(1.14).

In order to bound Λ−2θ, we need a decay estimate for the nonlinear term Λ−1F4,which can be stated as follows.


‖|∂x|14 Λ−1F4(s,·)‖L2

xy.s−1−εQ(‖(u,ω,θ)‖Y ), (5.11)

where |∂x|γ with a fractional power γ≥0 is defined via the Fourier transform

|∂x|γf(ξ,η) = |ξ|γ f(ξ,η)

and ε is same as in Corollary 2.1 and ε is same as in the definition of X-norm in(1.14).

Proof. F4 is given by

F4 =−∂xΛ−1∇·(uω)−∂tΛ−1∇·(uθ)−Λ−1∇·(uθ)=−Λ−1∇·(∂xuω)−Λ−1∇·(u∂xω)−Λ−1∇·(∂tuθ)−Λ−1∇·(u∂tθ)−Λ−1∇·(uθ)

,F41 +F42 +F43 +F44 +F45. (5.12)

By the standard bounds for the Riesz transforms, Hardy-Littlewood-Sobolev inequality,Holder inequality and interpolation inequality,

‖|∂x|14 Λ−1F41‖L2

xy.‖Λ− 3

4 (∂xuω)‖L2xy

.‖∂xuω‖L

87xy

.‖∂xu‖L2xy‖ω‖

L83xy

.‖∂xu‖L2xy‖ω‖

34

L2xy‖ω‖

14

L∞xy.s−

34−

34×

34−

14Q(‖(u,ω,θ)‖Y )


.s−1−εQ(‖(u,ω,θ)‖Y ).

Invoking the bound for ‖u‖L∞ in (3.7), we have

‖|∂x|14 Λ−1F42‖L2

xy.‖Λ− 3

4 (u∂xω)‖L2xy

.‖u∂xω‖L

87xy

.‖∂xω‖L2xy‖u‖

L83xy

.‖∂xω‖L2xy‖u‖

34

L2xy‖u‖

14

L∞xy.s−

34−

34×

34−

14Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

To bound F43, we invoke the bound

‖θ‖L∞ .‖θ‖14

L2 ‖∂xθ‖14

L2 ‖∂yθ‖14

L2 ‖∂xyθ‖14

L2

.s−14 (

14+

34+

14+

34 )Q(‖(u,ω,θ)‖Y ) =s−

12 Q(‖(u,ω,θ)‖Y )

to obtain

‖|∂x|14 Λ−1F43‖L2

xy.‖Λ− 3

4 (∂tuθ)‖L2xy

.‖∂tuθ‖L

87xy

.‖∂tu‖L2xy‖θ‖

L83xy

.‖∂tu‖L2xy‖θ‖

34

L2xy‖θ‖

14

L∞xy.s−

54−

14×

34−

12×

14Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

‖|∂x|14 Λ−1F44‖L2

xy.‖Λ− 3

4 (u∂tθ)‖L2xy

.‖u∂tθ‖L

87xy

.‖∂tθ‖L2xy‖u‖

L83xy

.‖∂tθ‖L2xy‖u‖

34

L2xy‖u‖

14

L∞xy.s−

54−

34×

34−

14Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

‖|∂x|14 Λ−1F45‖L2

xy.‖Λ− 3

4 (uθ)‖L2xy

.‖uθ‖L

87xy

.‖u‖L2xy‖θ‖

L83xy

.‖u‖L2xy‖u‖

34

L2xy‖θ‖

14

L∞xy.s−

34−

34×

34−

12×

14Q(‖(u,ω,θ)‖Y )

.s−1−εQ(‖(u,ω,θ)‖Y ).

Collecting these estimates yield the desirable bound (5.11), which finishes the proof ofLemma 5.5.

6. Proof of Equation (1.19)This section completes the proof of Theorem 1.1. We continue to verify

‖(ω,θ)‖X .‖(ω0,θ0)‖X0+Q(‖(u,ω,θ)‖Y ).

For the sake of clarity, we divide this section into several subsections, each dealing withsome of the terms in ‖(ω,θ)‖X .

6.1. Estimates of ‖〈∇〉2θ‖L2 , ‖Λ−1ω‖L2 and ‖〈∇〉2Λ−1θ‖L2 . Using Duhamel’sformula,

θ(t,x,y) =K0(t,∂x)θ0 +K1(t,∂x)

(1

2θ0 +θ1

)+

∫ t

1

K1(t−s,∂x)F2(s)ds. (6.1)


For notational convenience, we may sometimes write K0(t) for K0(t,∂x) and K1(t) forK1(t,∂x). Then by Lemma 2.2, Corollary 2.1 and Lemma 5.3, we have

‖〈∇〉2θ‖L2 .‖K0(t)〈∇〉2θ0‖L2 +‖K1(t)〈∇〉2(

1

2θ0 +θ1

)‖L2

+‖∫ t

1

K1(t−s)〈∇〉2F2(s)ds‖L2

. t−14 (‖〈∇〉3+εθ0‖L1

xy+‖〈∇〉3+εθ1‖L1

xy)+

∫ t

1

(‖K1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖K1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖Λ 12−ε〈∇〉 52+2εF2(s)‖L1

xyds

. t−14 ‖(ω0,θ0)‖X0

+

∫ t

1

〈t−s〉− 14 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−14 (‖(ω0,θ0)‖X0

+Q(‖(u,ω,θ)‖Y )).

Again by Duhamel’s formula,

Λ−1ω(t,x,y) =K0(t,∂x)Λ−1ω0 +K1(t,∂x)

(1

2Λ−1ω0 +Λ−1ω1

)+

∫ t

1

K1(t−s,∂x)F3(s)ds,

Λ−1θ(t,x,y) =K0(t,∂x)Λ−1θ0 +K1(t,∂x)

(1

2Λ−1θ0 +Λ−1θ1

)+

∫ t

1

K1(t−s,∂x)F4(s)ds.

Thus, by Lemma 2.2, Corollary 2.1 and Lemma 5.4,

‖Λ−1ω‖L2 .‖K0(t)Λ−1ω0‖L2 +‖K1(t)

(1

2Λ−1ω0 +Λ−1ω1

)‖L2

+‖∫ t

1

K1(t−s)F3(s)ds‖L2

. t−14 (‖〈∇〉1+εΛ−1ω0‖L1

xy+‖〈∇〉1+εΛ−1ω1‖L1

xy)

+

∫ t

1

(‖K1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖K1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖〈∇〉 12+2εΛ12−εF3(s)‖L1

xyds

. t−14 ‖(ω0,θ0)‖X0

+

∫ t

1

〈t−s〉− 14 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−14 (‖(ω0,θ0)‖X0

+Q(‖(u,ω,θ)‖Y )).

Similarly, by Lemma 2.2, Corollary 2.1 and Lemma 5.4,

‖〈∇〉2Λ−1θ‖L2 . t−14 (‖(ω0,θ0)‖X0

+Q(‖(u,ω,θ)‖Y )).


6.2. Estimates of ‖∂x〈∇〉ω‖L2 , ‖∂x〈∇〉2θ‖L2 , ‖∂xx〈∇〉2θ‖L2 and ‖∂xΛ−1θ‖L2 .To estimate ‖∂x〈∇〉ω‖L2 , we start with Duhamel’s formula

ω(t,x,y) =K0(t,∂x)ω0 +K1(t,∂x)

(1

2ω0 +ω1

)+

∫ t

1

K1(t−s,∂x)F1(s)ds. (6.2)

By Lemma 2.2, Corollary 2.1 and Lemma 5.3,

‖∂x〈∇〉ω‖L2 .‖∂xK0(t)〈∇〉ω0‖L2 +‖∂xK1(t)〈∇〉(

1

2ω0 +ω1

)‖L2

+‖∫ t

1

∂xK1(t−s)〈∇〉F1(s)ds‖L2

. t−34 (‖〈∇〉2+εω0‖L1

xy+‖〈∇〉2+εω1‖L1

xy)

+

∫ t

1

(‖∂xK1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖∂xK1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖Λ 12−ε〈∇〉 32+2εF1(s)‖L1

xyds

. t−34 ‖(ω0,θ0)‖X0

+

∫ t

1

〈t−s〉− 34 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−34 (‖(ω0,θ0)‖X0 +Q(‖(u,ω,θ)‖Y )).

Now we consider ‖∂x〈∇〉2θ‖L2 . As in the estimates above,

‖∂x〈∇〉2θ‖L2 .‖∂xK0(t)〈∇〉2θ0‖L2 +‖∂xK1(t)〈∇〉2(

1

2θ0 +θ1

)‖L2

+‖∫ t

1

∂xK1(t−s)〈∇〉2F2(s)ds‖L2

. t−34 (‖〈∇〉3+εθ0‖L1

xy+‖〈∇〉3+εθ1‖L1

xy)+

∫ t

1

(‖∂xK1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖∂xK1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖Λ 12−ε〈∇〉 52+2εF2(s)‖L1

xyds

. t−34 ‖(ω0,θ0)‖X0 +

∫ t

1

〈t−s〉− 34 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−34 (‖(ω0,θ0)‖X0 +Q(‖(u,ω,θ)‖Y )).

As for ‖∂xx〈∇〉2θ‖L2 , using Lemma 2.2, Corollary 2.1 and Lemma 5.3 again, we obtain

‖∂xx〈∇〉2θ‖L2 .‖∂xxK0(t)〈∇〉2θ0‖L2 +‖∂xxK1(t)〈∇〉2(

1

2θ0 +θ1

)‖L2

+‖∫ t

1

∂xxK1(t−s)〈∇〉2F2(s)ds‖L2

. t−54 (‖〈∇〉3+εθ0‖L1

xy+‖〈∇〉3+εθ1‖L1

xy)+

∫ t

1

(‖∂xxK1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖∂xxK1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖Λ 12−ε〈∇〉 52+2εF2(s)‖L1

xyds

. t−54 ‖(ω0,θ0)‖X0

+

∫ t

1

〈t−s〉− 54 s−1−εds ·Q(‖(u,ω,θ)‖Y )


. t−54 (‖(ω0,θ0)‖X0

+Q(‖(u,ω,θ)‖Y )).

Then we deal with the term ‖∂xΛ−1θ‖L2 . We proceed in the same way as in the previousestimate,

‖∂xΛ−1θ‖L2 .‖∂xK0(t)Λ−1θ0‖L2 +‖∂xK1(t)

(1

2Λ−1θ0 +Λ−1θ1

)‖L2

+‖∫ t

1

∂xK1(t−s)F4(s)ds‖L2

. t−34 (‖〈∇〉1+εΛ−1θ0‖L1

xy+‖〈∇〉1+εΛ−1θ1‖L1

xy)

+

∫ t

1

(‖∂xK1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖∂xK1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖〈∇〉 12+2εΛ12−εF4(s)‖L1

xyds

. t−34 ‖(ω0,θ0)‖X0 +

∫ t

1

〈t−s〉− 34 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−34 (‖(ω0,θ0)‖X0

+Q(‖(u,ω,θ)‖Y )).

6.3. Estimates of ‖∂t〈∇〉2ω‖L2 , ‖∂tΛ−1ω‖L2 , ‖∂tθ‖L2 and ‖∂xxΛ−2θ‖L2 . ByLemma 2.2, Corollary 2.1 and Lemma 5.2, we have

‖∂t〈∇〉2ω‖L2 .‖∂tK0(t)〈∇〉2ω0‖L2 +‖∂tK1(t)〈∇〉2(

1

2ω0 +ω1

)‖L2

+‖∫ t

1

∂tK1(t−s)〈∇〉2F1(s)ds‖L2

. t−54 (‖〈∇〉3+εω0‖L1

xy+‖〈∇〉3+εω1‖L1

xy)+

∫ t

1

(‖∂tK1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖∂tK1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖Λ 12−ε〈∇〉 52+2εF1(s)‖L1

xyds

. t−54 ‖(ω0,θ0)‖X0 +

∫ t

1

〈t−s〉− 54 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−54 (‖(ω0,θ0)‖X0 +Q(‖(u,ω,θ)‖Y )).

Next we bound ‖∂tΛ−1ω‖L2 . By Lemma 2.2, Corollary 2.1 and Lemma 5.4, we have

‖∂tΛ−1ω‖L2 .‖∂tK0(t)Λ−1ω0‖L2 +‖∂tK1(t)Λ−1(

1

2ω0 +ω1

)‖L2

+‖∫ t

1

∂tK1(t−s)Λ−1F3(s)ds‖L2

. t−54 (‖〈∇〉1+εΛ−1ω0‖L1

xy+‖〈∇〉1+εΛ−1ω1‖L1

xy)

+

∫ t

1

(‖∂tK1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖∂tK1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖Λ 12−ε〈∇〉 12+2εF3(s)‖L1

xyds

. t−54 ‖(ω0,θ0)‖X0 +

∫ t

1

〈t−s〉− 54 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−54 (‖(ω0,θ0)‖X0

+Q(‖(u,ω,θ)‖Y )).


For ‖∂tθ‖L2 , by (6.1), Lemma 2.2, Corollary 2.1 and Lemma 5.3, we have

‖∂tθ‖L2 .‖∂tK0(t)θ0‖L2 +‖∂tK1(t)

(1

2θ0 +θ1

)‖L2

+‖∫ t

1

∂tK1(t−s)F2(s)ds‖L2

. t−54 (‖〈∇〉1+εθ0‖L1

xy+‖〈∇〉1+εθ1‖L1

xy)+

∫ t

1

(‖∂tK1(t−s,ξ)‖L2ξ(|ξ|≤

12 )

+‖∂tK1(t−s,ξ)‖L∞ξ (|ξ|≥ 12 )

)‖Λ 12−ε〈∇〉 12+2εF2(s)‖L1

xyds

. t−54 ‖(ω0,θ0)‖X0

+

∫ t

1

〈t−s〉− 54 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−54 (‖(ω0,θ0)‖X0 +Q(‖(u,ω,θ)‖Y )).

Now we consider the last term ‖∂xxΛ−2θ‖L2 . By Duhamel’s formula,

Λ−2θ(t,x,y) =K0(t,∂x)Λ−2θ0 +K1(t,∂x)

(1

2Λ−2θ0 +Λ−2θ1

)+

∫ t

1

K1(t−s,∂x)Λ−1F4(s)ds.

Using Lemma 2.2, Lemma 2.3 and Lemma 5.5, we deduce that

‖∂xxΛ−2θ‖L2 .‖∂xxK0(t)Λ−2θ0‖L2 +‖∂xxK1(t)Λ−2(

1

2θ0 +θ1

)‖L2

+‖∫ t

1

|∂x|74K1(t−s)|∂x|

14 Λ−1F4(s)ds‖L2

.‖∂xxK0(t)‖L2‖Λ−2θ0‖L1xy

+‖∂xxK1(t)‖L2‖Λ−2(

1

2θ0 +θ1

)‖L1

xy

+

∫ t

1

‖ |∂x|74K1(t−s)‖L∞‖|∂x|

14 Λ−1F4(s)‖L2

xyds

. t−54 (‖Λ−2θ0‖L1

xy+‖Λ−2θ1‖L1

xy)+

∫ t

1

〈t−s〉− 78 s−1−εds ·Q(‖(u,ω,θ)‖Y )

. t−78 (‖(ω0,θ0)‖X0

+Q(‖(u,ω,θ)‖Y )).

Finally we estimate the pressure P . According to the divergence-free condition, we have

P = ∆−1∂yθ−∆−1∇·(∇·(u⊗u)).

Therefore, for any t≥1, by the boundedness of Riesz transforms and the definition ofY -norm,

‖P (t)‖Ha .‖∆−1∂yθ‖Ha +‖∆−1∇·(∇·(u⊗u))‖Ha.‖∆−1∂yθ‖L2 +‖Λa∆−1∂yθ‖L2 +‖∆−1∇·(∇·(u⊗u))‖Ha.‖Λ−1θ‖L2 +‖Λa−1‖L2 +‖u‖Ha‖u‖L∞. tε(‖(u,ω,θ)‖Y +Q(‖(u,ω,θ)‖Y ))

. tεε0.


This implies that P ∈C([1,+∞);Ha(R2)). Moreover,

‖P (t)‖L∞ .‖∆−1∂yθ‖L∞+‖∆−1∇·(∇·(u⊗u))‖L∞.‖〈∇〉2∆−1∂yθ‖L2 +‖〈∇〉2∆−1∇·(∇·(u⊗u))‖L2

.‖〈∇〉2Λ−1θ‖L2 +‖〈∇〉2(u⊗u)‖L2

.‖〈∇〉2Λ−1θ‖L2 +‖〈∇〉2u‖L2‖u‖L∞

. t−14 ε0.

This finishes the proof of Theorem 1.1.

Acknowledgements. J. Wu was partially supported by NSF grant DMS 1624146,by the AT&T Foundation at Oklahoma State University (OSU) and by NSFC (No.11471103, a grant awarded to Professor Baoquan Yuan). X. Xu was partially supportedby NSFC (No. 11771045, No. 11871087), BNSF (No. 2112023) and by the FundamentalResearch Funds for the Central Universities of China. N. Zhu was partially supportedby NSFC (No. 11771045, No. 11771043). The authors would like to thank ProfessorYifei Wu for helpful discussions.

REFERENCES

[1] H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Diff. Eqs.,233:199–220, 2007. 1

[2] D. Adhikari, C. Cao, H. Shang, J. Wu, X. Xu, and Z. Ye, Global regularity results for the 2DBoussinesq equations with partial dissipation, J. Diff. Eqs., 260:1893–1917, 2016. 1

[3] D. Adhikari, C. Cao, and J. Wu, The 2D Boussinesq equations with vertical viscosity and verticaldiffusivity, J. Diff. Eqs., 249:1078–1088, 2010. 1

[4] D. Adhikari, C. Cao, and J. Wu, Global regularity results for the 2D Boussinesq equations withvertical dissipation, J. Diff. Eqs., 251:1637–1655, 2011. 1

[5] D. Adhikari, C. Cao, J. Wu, and X. Xu, Small global solutions to the damped two-dimensionalBoussinesq equations, J. Diff. Eqs., 256:3594–3613, 2014. 1

[6] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial DifferentialEquations, Springer, 2011. 5

[7] J. Bergh and J. Lofstrom, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976. 5

[8] A. Biswas, C. Foias, and A. Larios, On the attractor for the semi-dissipative Boussinesq equa-tions, Ann. Inst. H. Poincare Anal. Non Lineaire, 34:381–405, 2017. 1

[9] L. Brandolese and M. Schonbek, Large time decay and growth for solutions of a viscous Boussi-nesq system, Trans. Amer. Math. Soc., 364:5057–5090, 2012. 1, 1

[10] J.R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations withdata in Lp, in R. Rautmann (ed.), Approximation Methods for Navier-Stokes Problems(Proc. Sympos., Univ. Paderborn, 1979), Lecture Notes in Math., Springer, Berlin, 771:129–144, 1980. 1

[11] C. Cao and J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with verticaldissipation, Arch. Ration. Mech. Anal., 208:985–1004, 2013. 1

[12] A. Castro, D. Cordoba, and D. Lear, On the asymptotic stability of stratified solutions for the2D Boussinesq equations with a velocity damping term, Math. Models Methods Appl. Sci.,29(7):1227–1277, 2019. 1

[13] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv.Math., 203:497–513, 2006. 1

[14] P. Constantin and C.R. Doering, Heat transfer in convective turbulence, Nonlinearity, 9:1049–1060, 1996. 1

[15] D. Cordoba, C. Fefferman, and R. de la Llave, On squirt singularities in hydrodynamics, SIAMJ. Math. Anal., 36:204–213, 2004. 1

[16] R. Danchin and M. Paicu, Global existence results for the anisotropic Boussinesq system indimension two, Math. Models Meth. Appl. Sci., 21:421–457, 2011. 1

[17] C.R. Doering and P. Constantin, Variational bounds on energy dissipation in incompressibleflows. III. Convection, Phys. Rev. E, 53:5957–5981, 1996. 1


[18] C.R. Doering, J. Wu, K. Zhao, and X. Zheng, Long time behavior of the two-dimensionalBoussinesq equations without buoyancy diffusion, Phys. D, 376/377:144–159, 2018. 1, 1, 1

[19] W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6:49–58,1994. 1

[20] T.M. Elgindi and I.-J. Jeong, Finite-time singularity formation for strong solutions to theBoussinesq system, arXiv:1708.0272v5. 1

[21] F. Hadadifard and A. Stefanov, On the global regularity of the 2D critical Boussinesq systemwith α>2/3, Commun. Math. Sci., 15:1325–1351, 2017. 1

[22] A.V. Getling, Rayleigh-Benard Convection: Structures and Dynamics, Advanced Series Nonlin-ear Dynamics: Volume 11, World Scientific, 1998. 1

[23] A.E. Gill, Atmosphere-Ocean Dynamics, International Geophysics Series, Academic Press, 30,1982. 1

[24] L. He, Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domainswith applications, J. Funct. Anal., 262:3430-3464, 2012. 1

[25] T. Hmidi, S. Keraani, and F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokessystem with critical dissipation, J. Diff. Eqs., 249:2147–2174, 2010. 1

[26] T. Hmidi, S. Keraani, and F. Rousset, Global well-posedness for Euler-Boussinesq system withcritical dissipation, Comm. Part. Diff. Eqs., 36:420–445, 2011. 1

[27] J. Holton, An Introduction to Dynamic Meteorology, International Geophysics Series, FourthEdition, Elsevier Academic Press, 2004. 1

[28] T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin.Dyn. Syst., 12:1–12, 2005. 1

[29] L. Hu and H. Jian, Blow-up criterion for 2-D Boussinesq equations in bounded domain, Front.Math. China, 2:559–581, 2007. 1

[30] W. Hu, I. Kukavica, and M. Ziane, Persistence of regularity for a viscous Boussinesq equationswith zero diffusivity, Asymptot. Anal., 91(2):111–124, 2015. 1

[31] W. Hu, I. Kukavica, and M. Ziane, On the regularity for the Boussinesq equations in a boundeddomain, J. Math. Phys., 54:1637–1655, 2013 . 1

[32] W. Hu, Y. Wang, J. Wu, B. Xiao, and J. Yuan, Partially dissipated 2D Boussinesq equationswith Navier type boundary conditions, Phys. D, 376/377:39–48, 2018. 1

[33] Q. Jiu, C. Miao, J. Wu, and Z. Zhang, The 2D incompressible Boussinesq equations with generalcritical dissipation, SIAM J. Math. Anal., 46:3426–3454, 2014. 1

[34] Q. Jiu, J. Wu, and W. Yang, Eventual regularity of the two-dimensional Boussinesq equationswith supercritical dissipation, J. Nonlinear Sci., 25:37–58, 2015. 1

[35] M. Lai, R. Pan, and K. Zhao, Initial boundary value problem for 2D viscous Boussinesq equa-tions, Arch. Ration. Mech. Anal., 199:739–760, 2011. 1

[36] A. Larios, E. Lunasin, and E.S. Titi, Global well-posedness for the 2D Boussinesq system withanisotropic viscosity and without heat diffusion, J. Diff. Eqs., 255:2636–2654, 2013. 1

[37] D. Li and X. Xu, Global wellposedness of an inviscid 2D Boussinesq system with nonlinearthermal diffusivity, Dyn. Par. Diff. Eqs., 10:255–265, 2013. 1

[38] J. Li, H. Shang, J. Wu, X. Xu, and Z. Ye, Regularity criteria for the 2D Boussinesq equationswith supercritical dissipation, Commun. Math. Sci., 14:1999–2022, 2016. 1

[39] J. Li and E.S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipa-tion, Arch. Ration. Mech. Anal., 220:983–1001, 2016. 1

[40] S.A. Lorca and J.L. Boldrini, The initial value problem for a generalized Boussinesq model,Nonlinear Anal., 36:457–480, 1999. 1

[41] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant LectureNotes in Mathematics, AMS/CIMS, 9, 2003. 1

[42] A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press,Cambridge, UK, 2002. 1

[43] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. 1[44] P. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Benard problem,

Arch. Ration. Mech. Anal., 29:32–57, 1968. 1[45] R. Salmon, Lectures on Geophysical Fluid Dynamics, Oxford Univ. Press, 1998. 1[46] A. Sarria and J. Wu, Blow-up in stagnation-point form solutions of the inviscid 2D Boussinesq

equations, J. Diff. Eqs., 259:3559–3576, 2015. 1[47] M.E. Schonbek, L2 decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech.

Anal., 88:209–222, 1985. 1[48] M.E. Schonbek and T. Schonbek, Moments and lower bounds in the far-field of solutions to

quasi-geostrophic flows, Discrete Contin. Dyn. Syst., 13:1277–1304, 2005. 1[49] A. Stefanov and J. Wu, A global regularity result for the 2D Boussinesq equations with critical

dissipation, J. Anal. Math., 137:269–290, 2019. 1


[50] L. Tao, J. Wu, K. Zhao, and X. Zheng, Stability near hydrostatic equilibrium to the 2D Boussi-nesq equations without thermal diffusion, submitted for publication. 1, 1, 1

[51] G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, Cambridge,U.K., 2006. 1

[52] R. Wan, Global well-posedness for the 2D Boussinesq equations with a velocity damping term,Discrete Contin. Dyn. Syst., 39(5):2709–2730, 2019. 1

[53] X. Wang and J.P. Whitehead, A bound on the vertical transport of heat in the ‘ultimate’ stateof slippery convection at large Prandtl numbers, J. Fluid Mech., 729:103–122, 2013. 1

[54] J.P. Whitehead and C.R. Doering, Ultimate state of two-dimensional Rayleigh-Benard convec-tion between free-slip fixed-temperature boundaries, Phys. Rev. Lett., 106:293–301, 2011.1

[55] J. Wu, Y. Wu, and X. Xu, Global small solution to the 2D MHD system with a velocity dampingterm, SIAM J. Math. Anal., 47:2630–2656, 2015. 1, 2, 2

[56] J. Wu and X. Xu, Well-posedness and inviscid limits of the Boussinesq equations with fractionalLaplacian dissipation, Nonlinearity, 27:2215–2232, 2014. 1

[57] J. Wu, X. Xu, L. Xue, and Z. Ye, Regularity results for the 2D Boussinesq equations with criticaland supercritical dissipation, Commun. Math. Sci., 14:1963–1997, 2016. 1

[58] K. Yamazaki, On the global regularity of N-dimensional generalized Boussinesq system, Appl.Math., 60:109–133, 2015. 1

[59] W. Yang, Q. Jiu, and J. Wu, Global well-posedness for a class of 2D Boussinesq systems withfractional dissipation, J. Diff. Eqs., 257:4188–4213, 2014. 1

[60] Z. Ye, Global smooth solution to the 2D Boussinesq equations with fractional dissipation, Math.Meth. Appl. Sci., 40:4595–4612, 2017. 1

[61] Z. Ye and X. Xu, Global well-posedness of the 2D Boussinesq equations with fractional Laplaciandissipation, J. Diff. Eqs., 260:6716–6744, 2016. 1

[62] Z. Ye, X. Xu, and L. Xue, On the global regularity of the 2D Boussinesq equations with fractionaldissipation, Math. Nachr., 290:1420–1439, 2017. 1

[63] K. Zhao, 2D inviscid heat conductive Boussinesq system in a bounded domain, Michigan Math.J., 59:329–352, 2010. 1

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