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STABILITY AND EXPONENTIAL DECAY FOR THE 2D

ANISOTROPIC BOUSSINESQ EQUATIONS WITH HORIZONTAL

DISSIPATION

BOQING DONG1, JIAHONG WU2, XIAOJING XU3 AND NING ZHU4

Abstract. The hydrostatic equilibrium is a prominent topic in fluid dynamicsand astrophysics. Understanding the stability of perturbations near the hydro-static equilibrium of the Boussinesq system helps gain insight into certain weatherphenomena. The 2D Boussinesq system focused here is anisotropic and involvesonly horizontal dissipation and horizontal thermal diffusion. Due to the lack ofthe vertical dissipation, the stability and precise large-time behavior problem isdifficult. When the spatial domain is R2, the stability problem in a Sobolev settingremains open. When the spatial domain is T × R, this paper solves the stabil-ity problem and specifies the precise large-time behavior of the perturbation. Bydecomposing the velocity u and temperature θ into the horizontal average (u, θ)

and the corresponding oscillation (u, θ), and deriving various anisotropic inequal-ities, we are able to establish the global stability in the Sobolev space H2. In

addition, we prove that the oscillation (u, θ) decays exponentially to zero in H1

and (u, θ) converges to (u, θ). This result reflects the stratification phenomenonof buoyancy-driven fluids.

1. Introduction

The goal of this paper is to understand the stability and large-time behaviorproblem on perturbations near the hydrostatic equilibrium of buoyancy-driven flu-ids. Being capable of capturing the key features of buoyancy-driven fluids such asstratification, the Boussinesq equations have become the most frequently used mod-els for these circumstances (see, e.s., [36, 40]). The Boussinesq system concernedhere assumes the form

∂tu+ u · ∇u = −∇P + ν∂11u+Θe2,

∂tΘ+ u · ∇Θ = κ∂11Θ,

∇ · u = 0,

(1.1)

where u = (u1, u2) denotes the velocity field, P the pressure, Θ the temperature, e2 =(0, 1) (the unit vector in the vertical direction), and ν > 0 and κ > 0 are the viscosityand the thermal diffusivity, respectively. (1.1) involves only horizontal dissipationand horizontal thermal diffusion, and governs the motion of anisotropic fluids whenthe corresponding vertical dissipation and thermal diffusion are negligible (see, e.g.,[40]).

2020 Mathematics Subject Classification. 35B35, 35B40, 35Q35, 76D03, 76D50.Key words and phrases. Boussinesq equations; Partial dissipation; Stability; Decay.

1

http://arxiv.org/abs/2009.13445v2

2 DONG, WU, XU AND ZHU

The Boussinesq systems have attracted considerable interests recently due to theirbroad physical applications and mathematical significance. The Boussinesq systemsare the most frequently used models for buoyancy-driven fluids such as many large-scale geophysical flows and the Rayleigh-Benard convection (see, e.g., [14,18,36,40]).The Boussinesq equations are also mathematically important. They share manysimilarities with the 3D Navier-Stokes and the Euler equations. In fact, the 2DBoussinesq equations retain some key features of the 3D Euler and Navier-Stokesequations such as the vortex stretching mechanism. The inviscid 2D Boussinesqequations can be identified as the Euler equations for the 3D axisymmetric swirlingflows [37].

Many efforts have been devoted to understanding two fundamental problemsconcerning the Boussinesq systems. The first is the global existence and regularityproblem. Substantial progress has been made on various Boussinesq systems, espe-cially those with only partial or fractional dissipation (see, e.g., [1–4,7,9–13,15–17,21–35, 38, 39, 41, 42, 47–57]). The second is the stability problem on perturbationsnear several physically relevant steady states. The investigations on the stabilityproblem are relatively more recent. One particular important steady state is thehydrostatic equilibrium. Mathematically the hydrostatic equilibrium refers to thestationary solution (uhe,Θhe, Phe) with

uhe = 0, Θhe = x2, Phe =1

2x22.

The hydrostatic equilibrium is one of the most prominent topics in fluid dynamics,atmospherics and astrophysics. In fact, our atmosphere is mostly in the hydrostaticequilibrium with the upward pressure-gradient force balanced out by the downwardgravity. The work of Doering, Wu, Zhao and Zheng [20] initiated the rigorous studyon the stability problem near the hydrostatic equilibrium of the 2D Boussinesqequations with only velocity dissipation. A followup work of Tao, Wu, Zhao andZheng establishes the large-time behavior and the eventual temperature profile [44].The paper of Castro, Cordoba and Lear successfully established the stability andlarge time behavior on the 2D Boussinesq equations with velocity damping insteadof dissipation [8]. There are other more recent work ( [5, 46, 52, 58]). Anotherimportant steady state is the shear flow. Linear stability results on the shear flowfor several partially dissipated Boussinesq systems are obtained in [43] and [58] whilethe nonlinear stability problem on the shear flow of the 2D Boussinesq equationswith only vertical dissipation was solved by [19].

The goal of this paper is to assess the stability and the precise large-time behav-ior of perturbations near the hydrostatic equilibrium. To understand the stabilityproblem, we write the equations for the perturbation (u, p, θ) where

p = P − 1

2x22 and θ = Θ− x2.

STABILITY AND EXPONENTIAL DECAY FOR BOUSSINESQ EQUATIONS 3

It is easy to check that (u, p, θ) satisfies

∂tu+ u · ∇u = −∇p+ ν ∂11u+ θe2,

∂tθ + u · ∇θ + u2 = κ ∂11θ,

∇ · u = 0,

u(x, 0) = u0(x), θ(x, 0) = θ0(x).

(1.2)

We have observed a new phenomenon on (1.2). It appears that the type of the spatialdomain plays a crucial role in the resolution of the stability problem concerned here.

When the spatial domain is the whole plane R2, the stability problem remainsan open problem. Given any sufficiently smooth initial data (u0, θ0) ∈ H2(R2),(1.2) does admit a unique global solution. But the solution could potentially growrather rapidly in time. In fact, the best upper bounds one could obtain on ‖u(t)‖H1

and ‖θ(t)‖H1 grow algebraically in time. The only way to possibly establish upperbounds that are uniform in time is to combine the equation of ∇u and that of ∇θ,or equivalently the equation of ω with ∇θ, where ω = ∇× u is the vorticity. Whenwe combine the equations of ω and ∇θ,

∂tω + u · ∇ω = ν ∂11ω + ∂1θ,

∂t∇θ + u · ∇(∇θ) +∇u2 = κ∇∂11θ −∇u · ∇θ (1.3)

and estimate ‖ω‖L2 and ‖∇θ‖L2 simultaneously, we can eliminate the term fromthe buoyancy, namely ∂1θ. In fact, a simple energy estimate on (1.3) yields, aftersuitable integration by parts,

d

dt

(‖ω(t)‖2L2 + ‖∇θ(t)‖2L2

)+ 2ν ‖∂1ω(t)‖2L2 + 2κ ‖∂1∇θ(t)‖2L2

= −∫

R2

∇θ · ∇u · ∇θ dx. (1.4)

The difficulty is how to obtain a suitable upper bound on the term on the right-handside of (1.4). To make full use of the anisotropic dissipation, we naturally dividethis term further into four component terms

−∫

R2

∇θ · ∇u · ∇θ dx = −∫

R2

∂1u1(∂1θ)2 dx−

∫

R2

∂1u2 ∂1θ ∂2θ dx

−∫

R2

∂2u1 ∂1θ ∂2θ dx−∫

R2

∂2u2(∂2θ)2 dx. (1.5)

Due to the lack of dissipation or thermal diffusion in the vertical direction, the lasttwo terms on (1.5) prevents us from bounding them suitably. This is one of thedifficulties that keep the stability problem on (1.2) open when the spatial domainis the whole plane R2.

When the spatial domain isΩ = T× R

with T = [0, 1] being a 1D periodic box and R being the whole line, this paper isable to solve the desired stability problem on (1.2). In fact, we are able to prove thefollowing result.


Theorem 1.1. Let T = [0, 1] be a 1D periodic box and let Ω = T × R. Assumeu0, θ0 ∈ H2(Ω) and ∇ · u0 = 0. Then there exists ε > 0 such that, if

‖u0‖H2 + ‖θ0‖H2 ≤ ε, (1.6)

then (1.2) has a unique global solution that remains uniformly bounded for all time,

‖u(t)‖2H2 + ‖θ(t)‖2H2 + ν

∫ t

0

‖∂1u(τ)‖2H2 dτ + κ

∫ t

0

‖∂1θ(τ)‖2H2 dτ ≤ C20ε

2

for some pure constant C0 > 0 and for all t > 0.

How does this domain makes a difference? The key point is that Ω allows usto separate the horizontal average (or the zeroth horizontal Fourier mode) fromthe corresponding oscillation part. These two different parts have different physicalbehavior. In fact, this decomposition is partially motivated by the stratification phe-nomenon observed in numerical results [20]. The numerical simulations performedin [20] show that the temperature becomes horizontally homogeneous and stratify inthe vertical direction as time evolves. Mathematically we do not expect the horizon-tal average to decay in time since it is associated with the zeroth horizontal Fouriermode and the dissipative effect at this mode vanishes. The oscillation part coulddecay exponentially. In addition, this decomposition and the oscillation part possessseveral desirable mathematical properties such as a strong Poincare type inequality.The two difficult terms in (1.5) are now handled by decomposing both u and θ intothe aforementioned two parts, and different terms induced by the decomposition areestimated differently. This is the main reason why the impossible stability problemin the R2 case becomes solvable when the domain is Ω = T× R.

To make the idea described above more precise, we introduce a few notations.Since the functional setting for our solution (u, θ) is H2(Ω), it is meaningful to definethe horizontal average,

u(x2, t) =

∫

T

u(x1, x2, t) dx1.

We set u to be the corresponding oscillation part

u = u− u or u = u+ u.

θ and θ are similarly defined. This decomposition is orthogonal,

(u, u) :=

∫

Ω

u u dx = 0, ‖u‖2L2(Ω) = ‖u‖2L2(Ω) + ‖u‖2L2(Ω).

A crucial property of u is that it obeys a strong version of the Poincare type in-equality,

‖u‖L2(Ω) ≤ C ‖∂1u‖L2(Ω), (1.7)

where the full gradient in the standard Poincare type inequality is replaced by ∂1.With this decomposition at our disposal, we are ready to handle the two difficultterms in (1.5). For the sake of conciseness, we focus on the first term. By invokingthe decomposition, we can further split it into four terms,∫

Ω

∂2u1 ∂1ω ∂2ω dx =

∫

Ω

∂2(u1 + u1) ∂1(ω + ω) ∂2(ω + ω) dx


=

∫

Ω

∂2u1 ∂1ω ∂2ω dx+

∫

Ω

∂2u1 ∂1ω ∂2ω dx

+

∫

Ω

∂2u1 ∂1ω ∂2ω dx+

∫

Ω

∂2u1 ∂1ω ∂2ω dx, (1.8)

where we have used ∂1ω = 0. The first term in (1.8) is clearly zero,∫

Ω

∂2u1 ∂1ω ∂2ω dx = 0.

The other three terms in (1.8) can all be bounded suitably by applying (1.7) andseveral other anisotropic inequalities, as stated in the lemmas in Section 2. We leavemore technical details to the proof of Theorem 1.1 in Section 3.

Our second main result states that the oscillation part (u, θ) decays to zero expo-nentially in time in the H1-norm. This result reflects the stratification phenomenonof buoyancy driven fluids. It also rigorously confirms the observation of the numer-ical simulations in [20], the temperature eventually stratifies and converges to thehorizontal average. As we have explained before, the horizontal average (u, θ) is notexpected to decay in time.

Theorem 1.2. Let u0, θ0 ∈ H2(Ω) with ∇ · u0 = 0. Assume that (u0, θ0) satisfies(1.6) for sufficiently small ε > 0. Let (u, θ) be the corresponding solution of (1.2).

Then the H1 norm of the oscillation part (u, θ) decays exponentially in time,

‖u(t)‖H1 + ‖θ(t)‖H1 ≤ (‖u0‖H1 + ‖θ0‖H1)e−C1t,

for some pure constant C1 > 0 and for all t > 0.

As a special consequence of this decay result, the solution (u, θ) approaches thehorizontal average (u, θ) asymptotically, and the Boussinesq system (1.2) evolves tothe following 1D system

∂tu+ u · ∇u+

(0

∂2p

)=

(0

θ

),

∂tθ + u · ∇θ = 0,

as given in (4.2). The proof of Theorem 1.2 starts with the system governing the

oscillation (u, θ),∂tu+ u · ∇u+ u2∂2u− ν∂21 u+∇p = θe2,

∂tθ +˜u · ∇θ + u2∂2θ − κ∂21 θ + u2 = 0.

By performing separate energy estimates for ‖(u, θ)‖2L2 and ‖(∇u,∇θ)‖2

L2 and care-fully evaluating the nonlinear terms with the strong Poincare type inequality andother anisotropic tools, we are able to establish the inequality

d

dt‖(u, θ)‖2H1 + (2ν − C1 ‖(u, θ)‖H2) ‖∂1u‖2H1

+(2κ− C1 ‖(u, θ)‖H2) ‖∂1θ‖2H1 ≤ 0. (1.9)


When the initial data (u0, θ0) is taken to be sufficiently small in H2, say

‖(u0, θ0)‖H2 ≤ ε

for sufficiently small ε > 0, then ‖(u, θ)‖H2 ≤ C0 ε and

2ν − C1 ‖(u, θ)‖H2 ≥ ν, 2η − C1 ‖(u, θ)‖H2 ≥ η.

Applying the strong Poincare inequality to (1.9) yields the desired exponential decay.

The rest of this paper is divided into three sections. Section 2 serves as a prepa-ration. It presents several anisotropic inequalities and some fine properties relatedto the orthogonal decomposition. Section 3 proves Theorem 1.1 while Section 4 isdevoted to verifying Theorem 1.2.

2. Anisotropic inequalities

This section presents several anisotropic inequalities to be used extensively inthe proofs of Theorem 1.1 and Theorem 1.2. In addition, several key propertiesof the horizontal average and the corresponding oscillation are also listed for theconvenience of later applications.

We first recall the horizontal average and the corresponding orthogonal decom-position. For any function f = f(x1, x2) that is integrable in x1 over the 1D periodicbox T = [0, 1], its horizontal average f is given by

f(x2) =

∫

T

f(x1, x2) dx1. (2.1)

We decompose f into f and the corresponding oscillation portion f ,

f = f + f . (2.2)

The following lemma collects a few properties of f and f to be used in the subsequentsections. These properties can be easily verified via (2.1) and (2.2).

Lemma 2.1. Assume that the 2D function f defined on Ω = T × R is sufficiently

regular, say f ∈ H2(Ω). Let f and f be defined as in (2.1) and (2.2).

(a) f and f obey the following basic properties,

∂1f = ∂1f = 0, ∂2f = ∂2f , f = 0, ∂2f = ∂2f .

(b) If f is a divergence-free vector field, namely ∇· f = 0, then f and f are alsodivergence-free,

∇ · f = 0 and ∇ · f = 0.

(c) f and f are orthogonal in L2, namely

(f , f) :=

∫

Ω

f f dx = 0, ‖f‖2L2(Ω) = ‖f‖2L2(Ω) + ‖f‖2L2(Ω).

In particular, ‖f‖L2 ≤ ‖f‖L2 and ‖f‖L2 ≤ ‖f‖L2.


Proof. (a) follows from the definitions of f and f directly. If ∇ · f = 0, then

0 = ∂1f + ∂2f = ∂1f + ∂2f = ∇ · f = ∇ · f −∇ · f = −∇ · f ,which gives (b). For (c), according to the definitions of f and f ,

(f , f) =

∫

Ω

f f dx =

∫

R

f

(∫

T

f(x1, x2) dx1

)dx2 −

∫

R

|f |2 dx2 = 0.

This completes the proof of Lemma 2.1.

We now present several anisotropic inequalities. Basic 1D inequalities play a rolein these anisotropic inequalities. We emphasize that 1D inequalities on the wholeline R are not always the same as the corresponding ones on bounded domainsincluding periodic domains. For any 1D function f ∈ H1(R),

‖f‖L∞(R) ≤√2‖f‖

1

2

L2(R) ‖f ′‖1

2

L2(R). (2.3)

For a bounded domain such as T and f ∈ H1(T),

‖f‖L∞(T) ≤√2 ‖f‖

1

2

L2(T) ‖f ′‖1

2

L2(T) + ‖f‖L2(T). (2.4)

However, if a function has mean zero such as the oscillation part f , the 1D inequality

for f is the same as the whole line case, that is, for f ∈ H1(T),

‖f‖L∞(T) ≤ C ‖f‖1

2

L2(T) ‖(f)′‖1

2

L2(T). (2.5)

These basic inequalities are incorporated into the anisotropic inequalities stated inthe following lemmas.

Lemma 2.2. Let Ω = T × R. For any f, g, h ∈ L2(Ω) with ∂1f ∈ L2(Ω) and∂2g ∈ L2(Ω), then

∣∣∣∣∫

Ω

f g h dx

∣∣∣∣ ≤ C ‖f‖1

2

L2 (‖f‖L2 + ‖∂1f‖L2)1

2 ‖g‖1

2

L2 ‖∂2g‖1

2

L2 ‖h‖L2 . (2.6)

For any f ∈ H2(Ω), we have

‖f‖L∞(Ω) ≤ C ‖f‖1

4

L2(Ω)

(‖f‖L2(Ω) + ‖∂1f‖L2(Ω)

) 1

4 ‖∂2f‖1

4

L2(Ω)

×(‖∂2f‖L2(Ω) + ‖∂1∂2f‖L2(Ω)

) 1

4 . (2.7)

Proof. The upper bound for the triple product in (2.6) on R2 was stated and provenin [6], but (2.6) for the domain Ω includes an extra lower-order term. For theconvenience of the readers, we provide the proofs of (2.6) and (2.7). ApplyingHolder’s inequality in each direction, Minkowski’s inequality, and (2.3) and (2.4),we have ∣∣∣∣

∫

Ω

f g h dx

∣∣∣∣ ≤ ‖f‖L2x2

L∞

x1

‖g‖L∞

x2L2x1

‖h‖L2

≤ ‖f‖L2x2

L∞

x1

‖g‖L2x1

L∞

x2

‖h‖L2

≤ C∥∥∥‖f‖

1

2

L2x1

‖∂1f‖1

2

L2x1

+ ‖f‖L2x1

∥∥∥L2x2


×∥∥∥‖g‖

1

2

L2x2

‖∂2g‖1

2

L2x2

∥∥∥L2x1

‖h‖L2

≤ C ‖f‖1

2

L2 (‖f‖L2 + ‖∂1f‖L2)1

2 ‖g‖1

2

L2 ‖∂2g‖1

2

L2 ‖h‖L2 .

Here ‖f‖L2x2

L∞

x1

represents the L∞-norm in the x1-variable, followed by the L2-norm

in the x2-variable. To prove (2.7), we again use Holder’s inequality, Minkowski’sinequality, and (2.3) and (2.4),

‖f‖L∞

x1L∞

x2

≤ C∥∥∥‖f‖

1

2

L2x2

‖∂2f‖1

2

L2x2

∥∥∥L∞

x1

≤ C∥∥∥‖f‖L∞

x1

∥∥∥1

2

L2x2

∥∥∥‖∂2f‖L∞

x1

∥∥∥1

2

L2x2

≤ C∥∥∥‖f‖

1

2

L2x1

‖∂1f‖1

2

L2x1

+ ‖f‖L2x1

∥∥∥1

2

L2x2

×∥∥∥‖∂2f‖

1

2

L2x1

‖∂1∂2f‖1

2

L2x1

+ ‖∂2f‖L2x1

∥∥∥1

2

L2x2

≤ C ‖f‖1

4

L2 (‖f‖L2 + ‖∂1f‖L2)1

4 ‖∂2f‖1

4

L2

× (‖∂2f‖L2 + ‖∂1∂2f‖L2)1

4 .


If we replace f by the oscillation part f , some of the lower-order parts in (2.6)and (2.7) can be dropped, as the following lemma states.

Lemma 2.3. Let Ω = T × R. For any f, g, h ∈ L2(Ω) with ∂1f ∈ L2(Ω) and∂2g ∈ L2(Ω), then

∣∣∣∣∫

Ω

f g h dx

∣∣∣∣ ≤ C ‖f‖1

2

L2 ‖∂1f‖1

2

L2 ‖g‖1

2

L2 ‖∂2g‖1

2

L2 ‖h‖L2. (2.8)

For any f ∈ H2(Ω), we have

‖f‖L∞(Ω) ≤ C ‖f‖1

4

L2(Ω)‖∂1f‖1

4

L2(Ω) ‖∂2f‖1

4

L2(Ω) ‖∂1∂2f‖1

4

L2(Ω).

Proof. The two inequalities in this lemma can be shown similarly as those in Lemma2.2. The only modification here is to use (2.5) instead of (2.4). Since (2.5) doesnot contain the lower-order part, the inequalities in this lemma do not have thelower-order terms.

The next lemma assesses that the oscillation part f obeys a strong Poincare type

inequality with the upper bound in terms of ∂1f instead of ∇f .

Lemma 2.4. Let f be a smooth function, f and f be defined as in (2.1) and (2.2).

If ‖∂1f‖L2(Ω) <∞, then

‖f‖L2(Ω) ≤ C‖∂1f‖L2(Ω), (2.9)


where C is a pure constant. In addition, if ‖∂1f‖H1(Ω) <∞, then

‖f‖L∞(Ω) ≤ C‖∂1f‖H1(Ω). (2.10)

Proof. For fixed x2 ∈ R,∫ 1

0

f(x1, x2) dx1 = 0.

According to the mean-value theorem, there exists η ∈ [0, 1] such that f(η, x2) = 0.Therefore, by Holder inequality,

|f(x1, x2)| =∣∣∣∣∫ x1

η

∂y1 f(y1, x2) dy1

∣∣∣∣ ≤(∫ 1

0

(∂y1 f(y1, x2))2 dy1

) 1

2

. (2.11)

Taking the L2-norm of (2.11) over Ω yields

‖f‖L2(Ω) ≤ C‖∂1f‖L2(Ω).

To prove (2.10), we use (2.4), namely, for any fixed x2 ∈ R,

‖f(x1, x2)‖L∞

x1(T) ≤ C‖f‖

1

2

L2x1

(T)‖∂1f‖1

2

L2x1

(T).

Taking the L∞-norm in x2, and using (2.3) and (2.9), we find

‖f‖L∞(Ω) ≤ C‖∂1f‖H1(Ω).


As an application of Lemma 2.4, the inequality in (2.8) can be converted to∣∣∣∣∫

Ω

f g h dx

∣∣∣∣ ≤ C ‖∂1f‖L2 ‖g‖1

2

L2 ‖∂2g‖1

2

L2 ‖h‖L2.

3. Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. Since the local well-posednesscan be shown via standard methods such as Friedrichs’ Fourier cutoff, our focus ison the global a priori bound on the solution in H2(Ω).

The framework for proving the global H2-bound is the bootstrapping argument(see, e.g., [45, p.21]). By selecting a suitable energy functional at the H2-level, wedevote our main efforts to showing that this energy functional obeys a desirableenergy inequality. This process is lengthy and involves establishing suitable upperbounds for several nonlinear terms such as the one in (1.8). As described in the

introduction, we invoke the orthogonal decomposition u = u + u and θ = θ + θ,apply various anisotropic inequalities in the previous section and make use of the

fine properties of u and θ. More details are given in the proof of Theorem 1.1.

Proof of Theorem 1.1. We define the natural energy functional,

E(t) := sup0≤τ≤t

(‖u(τ)‖2H2 + ‖θ(τ)‖2H2

)+ ν

∫ t

0

‖∂1u(τ)‖2H2 dτ + κ

∫ t

0

‖∂1θ(τ)‖2H2 dτ.


Our main efforts are devoted to proving that, for a constant C uniform for all t > 0,

E(t) ≤ E(0) + C E(t)2 + C E(t)3. (3.1)

Once (3.1) is established, the bootstrapping argument implies that, if

E(0) = ‖u0‖2H2 + ‖θ0‖2H2 ≤ ε2, for ε2 < min 1

16C,

1

8√C,

then E(t) admits the desired uniform global bound E(t) ≤ C ε2. To initiate thebootstrapping argument, we make the ansatz

E(t) ≤ min 1

4C,

1

2√C. (3.2)

(3.1) will allow us to conclude that E(t) actually admits an even smaller bound. Infact, if (3.2) holds, then (3.1) implies

E(t) ≤ E(0) +1

4E(t) +

1

4E(t),

or

E(t) ≤ 2E(0) ≤ 2ε2 ≤ 1

2min 1

4C,

1

2√C

with the bound being half of the one in (3.2). The bootstrapping argument thenasserts that E(t) is bounded uniformly for all time,

E(t) ≤ C ε2. (3.3)

Then we can deduce the global existence as well as the stability result from theglobal bound in (3.3). Now we show (3.1). A L2-estimate yields

‖u(t)‖2L2 + ‖θ(t)‖2L2 + 2ν

∫ t

0

‖∂1u(τ)‖2L2 dτ + 2κ

∫ t

0

‖∂1θ(τ)‖2L2 dτ

= ‖u0‖2L2 + ‖θ0‖2L2 . (3.4)

To estimate the H1-norm, we make use of the vorticity equation associated with thevelocity equation in (1.2),

∂tω + u · ∇ω = ∂1θ,

where ω = ∇×u. Taking the inner product of (ω,∆θ) with the equations of vorticityand temperature, we have, due to ∇ · u = 0,

1

2

d

dt(‖ω(t)‖2L2 + ‖∇θ(t)‖2L2) + ν‖∂1ω‖2L2 + κ‖∂1∇θ‖2L2

= −∫

∇(u · ∇θ) · ∇θ dx :=M, (3.5)

where we have used∫∂1θ ω dx = −

∫θ ∂1ω dx = −

∫θ∆u2 dx = −

∫∆θ u2 dx.

We further write M in (3.5) as

M = −∫

∇(u · ∇θ) · ∇θ dx


= −2∑

i,j=1

∫∂j(ui∂iθ) ∂jθ dx

= −2∑

i,j=1

∫∂jui∂iθ ∂jθ dx−

2∑

i,j=1

∫ui∂i∂jθ ∂jθ dx

= −∫∂1u1 ∂1θ ∂1θ dx−

∫∂1u2 ∂2θ ∂1θ dx

−∫∂2u1 ∂1θ ∂2θ dx−

∫∂2u2 ∂2θ ∂2θ dx

:= M1 +M2 +M3 +M4.

Here, due to ∇ · u = 0, we have used

2∑

i,j=1

∫ui∂i∂jθ ∂jθ dx = 0.

By Lemma 2.1, Lemma 2.3, Lemma 2.4 and Young’s inequality,

M1 = −∫∂1u1 ∂1θ ∂1θ dx

≤ C‖∂1θ‖L2‖∂1θ‖1

2

L2‖∂1∂1θ‖1

2

L2‖∂1u1‖1

2

L2‖∂2∂1u1‖1

2

L2

≤ C‖∂1θ‖2L2(‖∂1u‖2L2 + ‖∂1θ‖2L2) + δ(‖∂1∇u‖2L2 + ‖∂1∇θ‖2L2),

where δ > 0 is a small fixed constant to be specified later. Similarly,

M2 = −∫∂1u2 ∂2θ ∂1θ dx

≤ C‖∂2θ‖L2‖∂1θ‖1

2

L2‖∂1∂1θ‖1

2

L2‖∂1u2‖1

2

L2‖∂2∂1u2‖1

2

L2

≤ C‖∂2θ‖2L2(‖∂1u‖2L2 + ‖∂1θ‖2L2) + δ(‖∂1∇u‖2L2 + ‖∂1∇θ‖2L2).

To deal with M3, we invoke the decompositions u = u+ u and θ = θ + θ to write itinto four terms,

M3 = −∫∂2u1 ∂1θ ∂2θ dx

= −∫∂2u1 ∂1θ ∂2θ dx−

∫∂2u1 ∂1θ ∂2θ dx

−∫∂2u1 ∂1θ ∂2θ dx−

∫∂2u1 ∂1θ ∂2θ dx

:= M31 +M32 +M33 +M34.

According to Lemma 2.1, it is easy to see M31 = 0. To bound M32 and M33, we useLemma 2.1, Lemma 2.3, Lemma 2.4 and Young’s inequality to obtain

M32 = −∫∂2u1 ∂1θ ∂2θ dx

≤ C‖∂2u1‖L2‖∂1θ‖1

2

L2‖∂2∂1θ‖1

2

L2‖∂2θ‖1

2

L2‖∂1∂2θ‖1

2

L2


≤ C‖∂2u‖L2‖∂1θ‖1

2

L2‖∂1∂2θ‖3

2

L2

≤ C‖∂2u‖4L2‖∂1θ‖2L2 + δ‖∂1∇θ‖2L2

and

M33 = −∫∂2u1 ∂1θ ∂2θ dx

≤ C‖∂2θ‖L2‖∂2u1‖1

2

L2‖∂1∂2u1‖1

2

L2‖∂1θ‖1

2

L2‖∂2∂1θ‖1

2

L2

≤ C‖∂2θ‖L2‖∂1θ‖1

2

L2‖∂1∂2u1‖L2‖∂1∂2θ‖1

2

L2

≤ C‖∂2θ‖4L2‖∂1θ‖2L2 + δ(‖∂1∇u‖2L2 + ‖∂1∇θ‖2L2).

M34 can be similarly bounded as M32. For M4, we also write it as

M4 =

∫∂1u1 ∂2θ ∂2θ dx

= 2

∫∂1u1 ∂2θ ∂2θ dx+

∫∂1u1 ∂2θ ∂2θ dx

:= M41 +M42.

Similar as M33, we can bound M41 and M42 by

M41, M42 ≤ C‖∂2θ‖4L2‖∂1u‖2L2 + δ(‖∂1∇u‖2L2 + ‖∂1∇θ‖2L2).

Collecting the estimates for M and taking δ > 0 to be small, say

δ ≤ 1

16minν, η,

we obtain

‖ω(t)‖2L2 + ‖∇θ(t)‖2L2 + ν

∫ t

0

‖∂1ω(τ)‖2L2τ + κ

∫ t

0

‖∂1∇θ(τ)‖2L2 dτ

≤ C

∫ t

0

(‖∇θ‖2L2 + ‖∇θ‖4L2 + ‖∇u‖4L2)× (‖∂1u‖2L2 + ‖∂1θ‖2L2) dτ. (3.6)

Next we estimate the H2-norm of (u, θ). We take the inner product of ∆ω and ∆2θ

with the equations of vorticity and temperature, respectively. Due to ∇ · u = 0 andafter integrating by parts, we have

1

2

d

dt(‖∇ω(t)‖2L2 + ‖∆θ(t)‖2L2) + ν‖∂1∇ω‖2L2 + κ‖∂1∆θ‖2L2

= −∫

∇ω · ∇u · ∇ω dx−∫

∆(u · ∇θ)∆θ dx. (3.7)

For the first term on the right hand side, we can decompose it as

N := −∫

∇ω · ∇u · ∇ω dx

= −∫∂1u1 (∂1ω)

2 dx−∫∂1u2 ∂1ω ∂2ω dx

−∫∂2u1 ∂1ω ∂2ω dx−

∫∂2u2 (∂2ω)

2 dx


:= N1 +N2 +N3 +N4.

N1 and N2 can be bounded directly. According to Lemma 2.1, ∂1u = 0 and ∂1u =∂1u. By Lemma 2.3,

N1 = −∫∂1u1 ∂1ω ∂1ω dx

≤ C ‖∂1u1‖1

2

L2 ‖∂1∂1u1‖1

2

L2 ‖∂1ω‖1

2

L2 ‖∂2∂1ω‖1

2

L2 ‖∂1ω‖L2

≤ C ‖∂1u‖H1 ‖∂1u‖H2‖∂1∇ω‖L2

≤ C ‖∂1u‖2H1 ‖∂1u‖2H2 + δ′‖∂1∇ω‖2L2

and

N2 ≤ C ‖∂1u2‖1

2

L2 ‖∂1∂1u2‖1

2

L2 ‖∂1ω‖1

2

L2 ‖∂2∂1ω‖1

2

L2 ‖∂2ω‖L2

≤ C ‖u‖2H2 ‖∂1u‖2H2 + δ′‖∂1∇ω‖2L2,

where δ′ > 0 is a small but fixed parameter. The estimate of N3 is slightly moredelicate.

N3 = −∫∂2u1 ∂1ω ∂2ω dx

= −∫∂2(u1 + u1) ∂1ω ∂2(ω + ω) dx

= −∫∂2u1 ∂1ω ∂2ω dx−

∫∂2u1 ∂1ω ∂2ω dx

−∫∂2u1 ∂1ω ∂2ω dx−

∫∂2u1 ∂1ω ∂2ω dx

:= N31 +N32 +N33 +N34.

The first term N31 is clearly zero,

N31 = −∫

R

∂2u1 ∂2ω

∫

T

∂1ω1 dx1 dx2 = 0.

To bound N32 and N33, we first use (2.8) of Lemma 2.3 and then Lemma 2.4 toobtain

N32 ≤ C ‖∂2u1‖L2 ‖∂1ω‖1

2

L2 ‖∂2∂1ω‖1

2

L2 ‖∂2ω‖1

2

L2‖∂1∂2ω‖1

2

L2

≤ C ‖∂2u1‖L2 ‖∂1ω‖1

2

L2 ‖∂1∂2ω‖3

2

L2

≤ C ‖u‖4H1 ‖∂1u‖2H2 + δ′‖∂1∇ω‖2L2

and

N33 ≤ C ‖∂2ω‖L2 ‖∂2u1‖1

2

L2 ‖∂1∂2u1‖1

2

L2 ‖∂1ω‖1

2

L2 ‖∂2∂1ω‖1

2

L2

≤ C ‖u‖2H2 ‖∂1u‖2H2 + δ′‖∂1∇ω‖2L2,

N34 can be similarly bounded as N32. N4 can also be bounded similarly.

N4 = −∫∂1u1 (∂2ω + ∂2ω)

2 dx


= −2

∫∂1u1 ∂2ω ∂2ω dx−

∫∂1u1 (∂2ω)

2 dx

≤ C (‖∂2ω‖L2 + ‖∂2ω‖L2)‖∂1u1‖1

2

L2 ‖∂2∂1u1‖1

2

L2 ‖∂2ω‖1

2

L2 ‖∂1∂2ω‖1

2

L2

≤ C ‖u‖2H2 ‖∂1u‖2H2 + δ′‖∂1∇ω‖2L2.

Thus we have shown that

|N | ≤ C (‖u‖2H2 + ‖u‖4H2) ‖∂1u‖2H2 + 6δ′‖∂1∇ω‖2L2. (3.8)

For the last term of (3.7), we can write it as

−∫

∆(u · ∇θ)∆θ dx = −∫

∆u · ∇θ∆θ dx− 2

∫∇u · ∇2θ∆θ dx

= −∫

∆u1 ∂1θ∆θ dx−∫

∆u2 ∂2θ∆θ dx

−2

∫∂1u1 ∂1∂1θ∆θ dx− 2

∫∂1u2 ∂2∂1θ∆θ dx

−2

∫∂2u1 ∂1∂2θ∆θ dx− 2

∫∂2u2 ∂2∂2θ∆θ dx

:= P1 + P2 + P3 + P4 + P5 + P6.

According to Lemma 2.1, we can divide P1 into four terms,

P1 = −∫

∆u1 ∂1θ∆θ dx−∫


−∫

∆u1 ∂1θ∆θ dx−∫


:= P11 + P12 + P13 + P14.

It is clear that P11 = 0. For P12, we can bound it using Lemma 2.1, Lemma 2.3 andLemma 2.4,

P12 ≤ C‖∆u1‖L2‖∂1θ‖1

2

L2‖∂2∂1θ‖1

2

L2‖∆θ‖1

2

L2‖∂1∆θ‖1

2

L2

≤ C‖∆u‖L2‖∂1θ‖1

2

L2‖∂1∇θ‖1

2

L2‖∂1∆θ‖L2

≤ C‖∆u‖2L2(‖∂1θ‖2L2 + ‖∂1∇θ‖2L2) + δ′‖∂1∆θ‖2L2 .

Similarly, P14 shares the same bounded with P12. For P13, we can bound it by

P13 ≤ C‖∆θ‖L2‖∆u1‖1

2

L2‖∂1∆u1‖1

2

L2‖∂1θ‖1

2

L2‖∂2∂1θ‖1

2

L2

≤ C‖∆θ‖L2‖∂1θ‖1

2

L2‖∂1∆u‖L2‖∂1∆θ‖1

2

L2

≤ C‖∆θ‖4L2‖∂1θ‖2L2 + δ′(‖∂1∆u‖2L2 + ‖∂1∆θ‖2L2).

Therefore,

P1 ≤ C(‖∆u‖2L2 + ‖∆θ‖4L2)× ‖∂1θ‖2H1 + δ′(‖∂1∆u‖2L2 + ‖∂1∆θ‖2L2).

According to the relation ∆u2 = ∂1ω, we can decompose P2 as follows,

P2 = −∫∂1ω ∂2θ∆θ dx


= −∫∂1ω ∂2θ∆θ dx−

∫∂1ω ∂2θ∆θ dx

−∫∂1ω ∂2θ∆θ dx−

∫∂1ω ∂2θ∆θ dx

:= P21 + P22 + P23 + P24.

Clearly, P21 = 0. Making use of Lemma 2.1, Lemma 2.3 and Lemma 2.4, we obtain

P22, P24 ≤ C‖∂2θ‖L2‖∂1ω‖1

2

L2‖∂2∂1ω‖1

2

L2‖∆θ‖1

2

L2‖∂1∆θ‖1

2

L2

≤ C‖∇θ‖L2‖∂1ω‖1

2

L2‖∂1∇ω‖1

2

L2‖∂1∆θ‖L2

≤ C‖∇θ‖4L2‖∂1ω‖2L2 + δ′(‖∂1∇ω‖2L2 + ‖∂1∆θ‖2L2)

and

P23 ≤ C‖∆θ‖L2‖∂1ω‖1

2

L2‖∂2∂1ω‖1

2

L2‖∂2θ‖1

2

L2‖∂1∂2θ‖1

2

L2

≤ C‖∆θ‖L2‖∂1ω‖1

2

L2‖∂1∇ω‖1

2

L2‖∂1∆θ‖L2

≤ C‖∆θ‖4L2‖∂1ω‖2L2 + δ′(‖∂1∇ω‖2L2 + ‖∂1∆θ‖2L2).

Thus,

P2 ≤ C(‖∇u‖4L2 + ‖∆θ‖4L2)× ‖∂1ω‖2L2 + δ′(‖∂1∆u‖2L2 + ‖∂1∆θ‖2L2).

For P3, we can bound it by

P3 = −2

∫∂1u1 ∂1∂1θ∆θ dx

≤ C‖∆θ‖L2‖∂1u1‖1

2

L2‖∂2∂1u1‖1

2

L2‖∂1∂1θ‖1

2

L2‖∂1∂1∂1θ‖1

2

L2

≤ C‖∆θ‖L2‖∂1u‖1

2

L2‖∂1∆u‖1

2

L2‖∂1∆θ‖L2

≤ C‖∆θ‖4L2‖∂1u‖2L2 + δ′(‖∂1∆u‖2L2 + ‖∂1∆θ‖2L2).

P4 can be bounded in the same way. Using Lemma 2.1, we can write P5 as

P5 = −2

∫∂2u1 ∂1∂2θ∆θ dx− 2

∫∂2u1 ∂1∂2θ∆θ dx

−2

∫∂2u1 ∂1∂2θ∆θ dx− 2

∫∂2u1 ∂1∂2θ∆θ dx

:= P51 + P52 + P53 + P54.

It is easy to check that P51 = 0. P52 and P54 can be bounded by

P52, P54 ≤ C‖∇u‖4L2‖∂1∇θ‖2L2 + δ′‖∂1∆θ‖2L2 ,

and P53 have the bound

P53 ≤ C‖∆θ‖4L2‖∂1∇u‖2L2 + δ′(‖∂1∆u‖2L2 + ‖∂1∆θ‖2L2).

Finally we estimate P6, which can be written as

P6 = 2

∫∂1u1 ∂2∂2θ∆θ dx


= 2

∫∂1u1 ∂2∂2θ∆θ dx+ 2

∫∂1u1 ∂2∂2θ∆θ dx

+2

∫∂1u1 ∂2∂2θ∆θ dx+ 2

∫∂1u1 ∂2∂2θ∆θ dx

:= P61 + P62 + P63 + P64.

As in the estimate of P1, we have P61 = 0 and

P62, P63, P64 ≤ C‖∆θ‖4L2‖∂1u‖2L2 + δ′(‖∂1∆u‖2L2 + ‖∂1∆θ‖2L2).

Inserting (3.8) and the estimates for P1 through P6 in (3.7), and choosing δ′ > 0sufficiently small, we obtain

‖∇ω(t)‖2L2 + ‖∆θ(t)‖2L2 + ν

∫ t

0

‖∂1∇ω(τ)‖2L2 dτ + κ

∫ t

0

‖∂1∆θ(τ)‖2L2 dτ

≤ C

∫ t

0

(‖∂1u‖2H2 + ‖∂1θ‖2H2)× (‖u‖2H2 + ‖u‖4H2 + ‖θ‖2H2 + ‖θ‖4H2) dτ. (3.9)

Combining (3.9), (3.4) and (3.6) leads to the desired inequality in (3.1). This com-pletes the proof of Theorem 1.1.

4. Proof of Theorem 1.2

This section proves Theorem 1.2. We work with the equations of (u, θ) andmake use of the properties of the orthogonal decomposition and various anisotropicinequalities.

Proof of Theorem 1.2. . We first write the equation of (u, θ). Making use of Lemma2.1, we have ∂1u = 0 and

u · ∇u = u1∂1u+ u2∂2u = u2∂2u.

Since ∇ · u = 0 in Ω, there exists a stream function ψ such that

u = ∇⊥ψ := (−∂2ψ, ∂1ψ).

Then

u2 = ∂1ψ = 0

and

u · ∇u = 0. (4.1)

Taking the average of (1.2) and making use of (4.1) yield

∂tu+ u · ∇u+

(0

∂2p

)=

(0

θ

),

∂tθ + u · ∇θ = 0.

(4.2)


Taking the difference of (1.2) and (4.2), we find∂tu+ u · ∇u+ u2∂2u− ν∂21 u+∇p = θe2,

∂tθ +˜u · ∇θ + u2∂2θ − κ∂21 θ + u2 = 0.

(4.3)

The L2-estimate gives

1

2

d

dt(‖u(t)‖2L2 + ‖θ(t)‖2L2) + ν‖∂1u‖2L2 + κ‖∂1θ‖2L2

= −∫u · ∇u · u dx−

∫u2 ∂2u · u dx−

∫˜u · ∇θ θ dx−

∫u2 ∂2θ θ dx

:= A1 + A2 + A3 + A4.

For A1, according to the divergence-free condition of u and Lemma 2.1, we have

A1 = −∫u · ∇u · u dx = −

∫u · ∇u · u dx+

∫u · ∇u · u dx = 0.

Similarly, A3 = 0. Then we estimate A2. By Lemma 2.1, Lemma 2.3 and Lemma2.4,

A2 = −∫u2 ∂2u · u dx

≤ C‖∂2u‖L2‖u2‖1

2

L2‖∂2u2‖1

2

L2‖u‖1

2

L2‖∂1u‖1

2

L2

≤ C‖∂2u‖L2‖∂1u2‖1

2

L2‖∂1u1‖1

2

L2‖∂1u‖1

2

L2‖∂1u‖1

2

L2

≤ C‖u‖H1‖∂1u‖2L2 .

Similarly,

A4 = −∫u2 ∂2θ θ dx ≤ C‖θ‖H1(‖∂1u‖2L2 + ‖∂1θ‖2L2).

Collecting the estimates for A1 through A4, we obtain

d

dt(‖u(t)‖2L2 + ‖θ(t)‖2L2) + (2ν − C‖(u, θ)‖H1)‖∂1u‖2L2

+(2κ− C‖(u, θ)‖H1)‖θ‖2L2 ≤ 0.

By Theorem 1.1, if ε > 0 is sufficiently small and ‖u0‖H2+ ‖θ0‖H2

≤ ε, then‖(u(t), θ(t))‖H2 ≤ Cε and

2ν − C‖(u(t), θ(t))‖H2 ≥ ν, 2κ− C‖(u(t), θ(t))‖H2 ≥ κ.

Invoking the Poincare type inequality in Lemma 2.4 leads to the desired exponential

decay for ‖(u, θ)‖L2 ,

‖u(t)‖L2 + ‖θ(t)‖L2 ≤ (‖u0‖L2 + ‖θ0‖L2)e−C1t, (4.4)


where C1 = C1(ν, η) > 0. We now turn to the exponential decay of ‖(∇u(t),∇θ(t))‖L2.Applying ∇ to (4.3) yields

∂t∇u+∇(u · ∇u) +∇(u2∂2u)− ν∂21∇u+∇∇p = ∇(θe2),

∂t∇θ +∇(˜u · ∇θ) +∇(u2∂2θ)− κ∂21∇θ +∇u2 = 0.

(4.5)

Taking the L2 inner product of system (4.5) with (∇u, ∇θ), we have

1

2

d

dt(‖∇u(t)‖2L2 + ‖∇θ(t)‖2L2) + ν‖∂1∇u‖2L2 + κ‖∂1∇θ‖2L2

= −∫

∇(u · ∇u) · ∇u dx−∫

∇(u2∂2u) · ∇u dx

−∫

∇(˜u · ∇θ) · ∇θ dx−

∫∇(u2∂2θ) · ∇θ dx

:= B1 +B2 +B3 +B4. (4.6)

By Lemma 2.1, B1 can be further written as

B1 = −∫

∇(u · ∇u) · ∇u dx+∫

∇(u · ∇u) · ∇u dx

= −∫∂1u1 ∂1u · ∂1u dx+

∫∂1u2 ∂2u · ∂1u dx

−∫∂2u1 ∂1u · ∂2u dx+

∫∂2u2 ∂2u · ∂2u dx

:= B11 +B12 +B13 +B14.

Using Lemma 2.3 and Lemma 2.4, B11 can be bounded by

B11 ≤ C‖∂1u1‖L2‖∂1u‖1

2

L2‖∂2∂1u‖1

2

L2‖∂1u‖1

2

L2‖∂1∂1u‖1

2

L2

≤ C‖∂1u1‖L2‖∂1u‖1

2

L2‖∂2∂1u‖1

2

L2‖∂1u‖1

2

L2‖∂1∂1u‖1

2

L2

≤ C‖u‖H1‖∂1∇u‖2L2 .

B12 and B13 can be bounded similarly and

B12, B13 ≤ C‖u‖H1‖∂1∇u‖2L2 .

For B14, according to the divergence-free condition of u and similar as B11, we have

B14 = −∫∂1u1 ∂2u · ∂2u dx =

∫∂1u1 ∂2u · ∂2u dx

≤ C‖∂2u‖L2‖∂1u1‖1

2

L2‖∂2∂1u1‖1

2

L2‖∂2u‖1

2

L2‖∂1∂2u‖1

2

L2

≤ C‖u‖H1‖∂1∇u‖2L2.

Therefore, B1 is bounded by

|B1| ≤ C‖u‖H1‖∂1∇u‖2L2.

Similarly, we can bound B3 by

|B3| ≤ C(‖u‖H1 + ‖θ‖H1)× (‖∂1∇u‖2L2 + ‖∂1∇θ‖2L2).


To bound B2, we first write it explicitly as

B2 = −∫

∇(u2∂2u) · ∇u dx

= −∫∂1u2 ∂2u · ∂1u dx−

∫u2 ∂2∂1u · ∂1u dx

−∫∂2u2 ∂2u · ∂2u dx−

∫u2 ∂2∂2u · ∂2u dx

:= B21 +B22 +B23 +B24.

By Lemma 2.1, Lemma 2.3 and Lemma 2.4,

B21 = −∫∂1u2 ∂2u · ∂1u dx

≤ C‖∂2u‖L2‖∂1u2‖1

2

L2‖∂2∂1u2‖1

2

L2‖∂1u‖1

2

L2‖∂1∂1u‖1

2

L2

≤ C‖u‖H1‖∂1∇u‖2L2.

According to the definition of u,

B22 = −∫u2 ∂2∂1u · ∂1u dx = 0.

Similarly, B23 and B24 can be bounded by

B23 =

∫∂1u1 ∂2u · ∂2u dx

≤ C‖∂2u‖L2‖∂1u1‖1

2

L2‖∂2∂1u1‖1

2

L2‖∂2u‖1

2

L2‖∂1∂2u‖1

2

L2

≤ C‖u‖H1‖∂1∇u‖2L2

and

B24 = −∫u2 ∂2∂2u · ∂2u dx

≤ C‖∂2∂2u‖L2‖u2‖1

2

L2‖∂2u2‖1

2

L2‖∂2u‖1

2

L2‖∂1∂2u‖1

2

L2

≤ C‖u‖H2‖∂1∇u‖2L2 .

Thus we obtain the bound for B2,

|B2| ≤ C‖u‖H2‖∂1∇u‖2L2.

Similarly, B4 is bounded by

|B4| ≤ C‖θ‖H2 × (‖∂1∇u‖2L2 + ‖∂1∇θ‖2L2).

Inserting the estimates for B1 through B4 in (4.6), we obtain

d

dt(‖∇u(t)‖2L2 + ‖∇θ(t)‖2L2) + (2ν − C‖(u, θ)‖H2)‖∂1∇u‖2L2

+(2κ− C‖(u, θ)‖H2)‖∂1∇θ‖2L2 ≤ 0.


Choosing ε sufficiently small and invoking the Poincare type inequality in Lemma

2.4, we obtain the exponential decay result for ‖(∇u,∇θ)‖L2 ,

‖∇u(t)‖L2 + ‖∇θ(t)‖L2 ≤ (‖∇u0‖L2 + ‖∇θ0‖L2)e−C1t. (4.7)

Combining the estimates (4.4) and (4.7), we obtain the desired decay result. Thiscompletes the proof of Theorem 1.2.

Acknowledgments. Dong is partially supported by the National Natural ScienceFoundation of China (No.11871346), the Natural Science Foundation of GuangdongProvince (No. 2018A030313024), the Natural Science Foundation of Shenzhen City(No. JCYJ 20180305125554234) and Research Fund of Shenzhen University (No.2017056). Wu was partially supported by NSF under grant DMS 1624146 and theAT&T Foundation at Oklahoma State University. X. Xu was partially supported bythe National Natural Science Foundation of China (No. 11771045 and No.11871087).N. Zhu was partially supported by the National Natural Science Foundation of China(No. 11771043 and No. 11771045).

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1 College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060,

P.R. China

Email address : [email protected]

2 Department of Mathematics, Oklahoma State University, Stillwater, OK 74078,

United States


3 Laboratory of Mathematics and Complex Systems, Ministry of Education,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P.R.

China




4 School of Mathematical Sciences, Peking University, Beijing 100871, P.R.

China


stability and exponential decay for the 2d …

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