Oussama Ben

Mona Ben Said

STABILITY AND LARGE-TIME BEHAVIOR FOR THE 2D BOUSSINESQ SYSTEM WITH VERTICAL DISSIPATION AND HORIZONTAL THERMAL DIFFUSION

This paper addresses the stability and large-time behavior problem on the perturbations near the hydrostatic balance of the two dimensional Boussinesq system, taking into account vertical dissipation and horizontal thermal diffusion. The spatial framework Ω is defined as T × R, where T spans [0, 1], representing the 1D periodic box, while R denotes the whole line. The results outlined in this article confirm the fact that the temperature can actually have a stabilizing effect on the buoyancy-driven fluids. The stability and long-time behavior issues discussed here are difficult due to the lack of the horizontal dissipation and vertical thermal diffusion. By formulating in the appropriate energy functional and implementing the orthogonal decomposition of the velocity and the temperature into their horizontal averages and oscillation parts, we are able to make up for the missing regularization and establish the nonlinear stability in the Sobolev space H 2 (Ω) and acheive the algebraic decay rates for the oscillation parts in the H 1 -norm.

Introduction

This paper focuses on the following 2D anisotropic Boussinesq system     

∂ t U + U • ∇U = -∇P + ν ∂ 22 U + g 0 Θe 2 , x ∈ Ω, t > 0, ∂ t Θ + U • ∇Θ = η ∂ 11 Θ, ∇ • U = 0, (1.1)
where U denotes the fluid velocity, P the pressure, Θ the temperature, ν > 0 and η > 0 are parameters representing the kinematic viscosity and the thermal diffusivity, respectively. Here e 2 = (0, 1) is the unit vector in the vertical direction, g 0 is a non zero constant and the spatial domain Ω is taken to be

Ω = T × R,
with T = [0, 1] being a 1D periodic box and R being the whole line. This partially dissipated system models anisotropic buoyancy-driven fluids in the circumstance when the horizontal dissipation and the vertical thermal diffusion are negligible [START_REF]Pedlosky Geophysical Fluid Dynamics[END_REF].

The Boussinesq systems stand out as the most commonly employed models for studying atmospheric and oceanographic flows (see, e.g., [START_REF] Bluestein | Severe Convective Storms and Tornadoes: Observations and Dynamics[END_REF], [START_REF] Holton | An Introduction to Dynamic Meteorology[END_REF], [START_REF] Majda | Introduction to PDEs and Waves for the Atmosphere and Ocean[END_REF]). Recent research has been focused on addressing two fundamental challenges related to these equations, namely, global existence and regularity problem and the stability problem on perturbations near various physically relevant equilibrium states (see, e.g., [START_REF] Ben Said | The stabilizing effect of the temperature on buoyancydriven fluids[END_REF], [START_REF] Adhikari | Stability and large-time behavior for the 2D Boussinesq system with horizontal dissipation and vertical thermal diffusion[END_REF], [START_REF] Castro | On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term[END_REF], [START_REF] Doering | Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion[END_REF], [START_REF] Dong | Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation[END_REF], [START_REF] Kiselev | Small scale creation for solutions of the incompressible twodimensional Euler equation[END_REF], [START_REF] Lukaszewicz | On nonstationary flows of asymmetric fluids[END_REF], [START_REF] Lukaszewicz | On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids[END_REF], [START_REF] Lukaszewicz | Micropolar Fluids Theory and Applications[END_REF], [START_REF] Ladyzhenskaya | Solution "in the large" of the nonstationary boundary value problem for the Navier-Stokes system with two space variables[END_REF], [START_REF] Li | Global regularity for the viscous Boussinesq equations[END_REF], [START_REF] Larios | Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion[END_REF], [START_REF] Lai | Optimal decay estimates for the 2D Boussinesq equations with partial dissipation[END_REF], [START_REF] Lai | Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation[END_REF], [START_REF] Tao | Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion[END_REF]). This work intends to show the H 2 (Ω)-stability and examine the the large-time behavior of perturbations near the hydrostatic equilibrium (U he , Θ he ) with U he = 0, Θ he = g 0 x 2 .

For the velocity U he , the momentum equation is fulfilled when the pressure gradient is balanced by the buoyancy force, namely -∇P he + g 0 Θ he e 2 = 0 or P he = 1 2 g 2 0 x 2 2 . To examine the stability problem, we need first to write down the equations for the perturbation (u, p, θ), where u = U -U he , p = P -P he and θ = Θ -Θ he .

It is evident from equations (1.1) that (u, p, θ) satisfies the following anisotropic Boussinesq equations with vertical dissipation and horizontal thermal diffusion

         ∂ t u + u • ∇u = -∇p + ν ∂ 22 u + g 0 θe 2 , x ∈ Ω, t > 0, ∂ t θ + u • ∇θ + g 0 u 2 = η ∂ 11 θ, ∇ • u = 0, u (x
, 0) = u 0 (x), θ(x, 0) = θ 0 (x).

(1.

2)

The difference between the original system (1.1) and the system governing the perturbations (1.2) is that the temperature equation in (1.2) contains g 0 u 2 . Without this extra term, the L 2 -norm of the velocity u to (1.1) can grow in time due to the buoyancy forcing term g 0 θe 2 . With even full dissipation and thermal diffusion, as taken in [START_REF] Brandolese | Large time decay and growth for solutions of a viscous Boussinesq system[END_REF], solutions of the 3D Boussinesq equations can actually grow in time. This term in (1.2) contributes to balancing g 0 θe 2 in the energy estimates. Consequently, the buoyancy forcing ceases to have a destabilizing impact in (1.2). In cases where dissipation is degenerate and is only one-directional as in (1.1), it is not clear how the solution would behave.

When the spacial domain is the whole space R 2 , the lack of the horizontal dissipation complicates the control of the growth of the vorticity ω = ∇ × u, satisfying

∂ t ω + u • ∇ω = ν ∂ 22 ω + g 0 ∂ 1 θ, x ∈ R 2 , t > 0. (1.3)
More precisely, it is feasible to derive a uniform bound on the L 2 -norm of the vorticity ω itself. Nonetheless, controlling the L 2 -norm of the gradient of the vorticity, ∇ω, does not seem achievable. In particular, if the temperature θ is zero, (1.3) reduces to the 2D Navier-Stokes equation with degenerate dissipation,

∂ t ω + u • ∇ω = ν ∂ 22 ω, x ∈ R 2 , t > 0.
(1.4)

While (1.4) always has a unique global solution ω for any initial data ω 0 ∈ H 1 (R 2 ), the question of whether ∇ω(t) L 2 for the solution ω of (1.4) grows with respect to t remains an open problem.

Furthermore, when there is no dissipation at all, namely when ν = 0, (1.4) takes the form of the 2D Euler vorticity equation

∂ t ω + u • ∇ω = 0, x ∈ R 2 , t > 0.
(1.5)

As pointed out in many works (see, e.g., [START_REF] Denisov | Double-exponential growth of the vorticity gradient for the two-dimensional Euler equation[END_REF], [START_REF] Kiselev | Small scale creation for solutions of the incompressible twodimensional Euler equation[END_REF], [START_REF] Zlatos | Exponential growth of the vorticity gradient for the Euler equation on the torus[END_REF]), ∇ω(t) of (1.5) can grow even double exponentially in time. Particularly, the velocity of the 2D Euler equations in the Sobolev space H 2 is not stable. Conversely, solutions to the 2D Navier-Stokes equations with full dissipation

∂ t ω + u • ∇ω = ν ∆ω, x ∈ R 2 , t > 0
decays algebraically in time, as shown by Schonbek (see, e.g., [START_REF] Schonbek | L 2 decay for weak solutions of the Navier-Stokes equations[END_REF], [START_REF] Schonbek | On the decay of higher-order norms of the solutions of Navier-Stokes equations[END_REF]). The abscence of the horizontal dissipation in (1.4) hinders our ability to follow the approach used for the fully dissipative Navier-Stokes equations. Specifically, when applying the energy method to bound ∇ω(t) L 2 , namely 1 2

d dt ∇ω(t) 2 L 2 + ν ∂ 2 ∇ω(t) 2 L 2 = -∇ω • ∇u • ∇ω dx, (1.6)
the one-directional dissipation is not enough to control the nonlinearity. The challenge lies in acquiring a suitable upper bound for the term on the right-hand side of (1.6). To effectively leverage the anisotropic dissipation, we naturally decompose this term further into four component terms.

∇ω • ∇u • ∇ω dx = ∂ 1 u 1 (∂ 1 ω) 2 dx + ∂ 1 u 2 ∂ 1 ω ∂ 2 ω dx (1.7)
+ ∂ 2 u 1 ∂ 1 ω ∂ 2 ω dx + ∂ 2 u 2 (∂ 2 ω) 2 dx.
Without horizontal dissipation, establishing a time-integrable upper bound for the first two terms in (1.7) is not possible.

When dealing with the stability problem on (1.2), we come across the same nonlinear term presented in (1.7). Fortunately, the smoothing and stabilization effect of the temperature through the coupling and interaction allows us to solve the stability problem in (1.2). To reveal these effects, we start by eliminating the pressure term in (1.2). Applying the Helmholtz-Leray projection P = I -∇∆ -1 ∇• to the velocity equation in (1.2), we get ∂ t u = ν∂ 22 u + P(g 0 θe 2 ) -P(u • ∇u).

(1.8)

Using the definition of the Leray projection P,

P(g 0 θe 2 ) = g 0 θe 2 -∇∆ -1 ∇ • (g 0 θe 2 ) = g 0 -∂ 1 ∂ 2 ∆ -1 θ θ -∂ 2 2 ∆ -1 θ
.

(1.9)

Substituting (1.9) into (1.8) and expressing (1.8) in terms of its component equations, yields

∂ t u 1 = ν ∂ 22 u 1 -g 0 ∂ 1 ∂ 2 ∆ -1 θ + N 1 , ∂ t u 2 = ν ∂ 22 u 2 + g 0 ∂ 1 ∂ 1 ∆ -1 θ + N 2 , (1.10)
where N 1 and N 2 represent the nonlinear terms,

N 1 = -(u • ∇u 1 -∂ 1 ∆ -1 ∇ • (u • ∇u)), N 2 = -(u • ∇u 2 -∂ 2 ∆ -1 ∇ • (u • ∇u)).
Differentiating the first equation of (1.10) with respect to t, we get

∂ tt u 1 = ν∂ 22 ∂ t u 1 -g 0 ∂ 1 ∂ 2 ∆ -1 ∂ t θ + ∂ t N 1 .
Using the equation of θ in (1.2), we substitute ∂ t θ in the above equation with η ∂ 11 θg 0 u 2u • ∇θ to write

∂ tt u 1 = ν∂ 22 ∂ t u 1 + g 2 0 ∂ 1 ∂ 2 ∆ -1 u 2 -g 0 η ∂ 11 ∂ 1 ∂ 2 ∆ -1 θ + g 0 ∂ 1 ∂ 2 ∆ -1 (u • ∇θ) + ∂ t N 1 . Additionally, replacing g 0 ∂ 1 ∂ 2 ∆ -1 θ by the first component equation of (1.10), namely -g 0 ∂ 1 ∂ 2 ∆ -1 θ = ∂ t u 1 -ν ∂ 22 u 1 -N 1 ,
we find

∂ tt u 1 = ν∂ 22 ∂ t u 1 + g 2 0 ∂ 1 ∂ 2 ∆ -1 u 2 + η ∂ 11 (∂ t u 1 -ν ∂ 22 u 1 -N 1 ) +g 0 ∂ 1 ∂ 2 ∆ -1 (u • ∇θ) + ∂ t N 1 ,
which in turn gives, due to the divergence-free condition

∂ 2 u 2 = -∂ 1 u 1 , ∂ tt u 1 -(η∂ 11 + ν∂ 22 )∂ t u 1 + νη∂ 11 ∂ 22 u 1 + g 2 0 ∂ 11 ∆ -1 u 1 = N 3 , (1.11)
where N 3 is the nonlinear term,

N 3 = (∂ t -η∂ 11 )N 1 + g 0 ∂ 1 ∂ 2 ∆ -1 (u • ∇θ).
Following the same procedure, we can easily show that u 2 and θ satisfy

∂ tt u 2 -(η∂ 11 + ν∂ 22 )∂ t u 2 + νη∂ 11 ∂ 22 u 2 + g 2 0 ∂ 11 ∆ -1 u 2 = N 4 , (1.12)
∂ tt θ -(η∂ 11 + ν∂ 22 )∂ t θ + νη∂ 11 ∂ 22 θ + g 2 0 ∂ 11 ∆ -1 θ = N 5 with N 4 = (∂ t -η∂ 11 )N 2 -g 0 ∂ 1 ∂ 1 ∆ -1 (u • ∇θ), N 5 = -(∂ t -ν∂ 22 )(u • ∇θ) -g 0 N 2 .
Then, merging (1.11) and (1.12) and expressing them into the velocity vector form, we have reformulated (1.2) into the following new system

∂ tt u -(η∂ 11 + ν∂ 22 )∂ t u + νη∂ 11 ∂ 22 u + g 2 0 ∂ 11 ∆ -1 u = N 6 , ∂ tt θ -(η∂ 11 + ν∂ 22 )∂ t θ + νη∂ 11 ∂ 22 θ + g 2 0 ∂ 11 ∆ -1 θ = N 5 , (1.13)
where

N 6 = (N 3 , N 4 ) = -(∂ t -η∂ 11 )P(u • ∇u) + g 0 ∇ ⊥ ∂ 1 ∆ -1 (u • ∇θ) with ∇ ⊥ = (∂ 2 , -∂ 1 )
. By applying the curl of the velocity equation, we can likewise transform (3.44) into a system of ω and θ,

∂ tt ω -(η∂ 11 + ν∂ 22 )∂ t ω + νη∂ 11 ∂ 22 ω + g 2 0 ∂ 11 ∆ -1 ω = N 7 , ∂ tt θ -(η∂ 11 + ν∂ 22 )∂ t θ + νη∂ 11 ∂ 22 θ + g 2 0 ∂ 11 ∆ -1 θ = N 5 , with N 7 = -(∂ t -η∂ 11 )(u • ∇ω) -g 0 ∂ 1 (u • ∇θ).
Remarkably, we observe that all physical quantities u, θ and ω obey the same damped degenerate wave equation, differing only in their respective nonlinear terms. Compared to the original system of (u, θ) in (1.2), the wave equations (1.13) reveal the underlying smoothing and stabilization hidden in (1.2). In (1.2), where horizontal dissipation is absent in the velocity field, the wave structure implies that the temperature can stabilize the fluids by creating the horizontal regularization through the coupling and interaction. By taking advantage of these effects, the stability problem on (1.2) was recently established by Ben Said and al in [START_REF] Ben Said | The stabilizing effect of the temperature on buoyancydriven fluids[END_REF] when the spacial domain is the whole plane R 2 . However, the large time behaviour of the solution in R 2 remains a mystery. When the spatial domain is Ω = T × R, this paper also proves the stability of (1.2). Additionally, we analyze the large-time behavior of the solutions. The core idea involves breaking down a physical quantity into its horizontal average and the associated oscillation. Specifically, for a function f = f (x 1 , x 2 ) defined on T × R and integrable in x 1 over the 1D periodic box T = [0, 1], we define its horizontal average f by

f (x 2 ) = T f (x 1 , x 2 )dx 1 , (1.14)
and we write,

f = f + f . (1.15)
Note here that, the horizontal average f corresponds to the zeroth Fourier mode of f while f contains all non-zero Fourier modes. The decomposition (1.15) possesses distinct properties. To begin with, this decomposition is orthogonal in the Sobolev space H k (Ω) for any non-negative integer. This implies that the H k -norms of f and f are bounded by the H k -norm of f . Furthermore, this decomposition commutes with derivatives, and f and f of a divergence-free vector field f are also divergence-free. An essential property to be frequently used in our estimates is that f admits strong versions of the Poincaré type inequality

f L 2 (Ω) ≤ C ∂ 1 f L 2 (Ω) , f L ∞ (Ω) ≤ C ∂ 1 f H 1 (Ω) .
Applying this decomposition to the velocity field u and the temperature θ, namely writing

u = u + u, θ = θ + θ
and exploiting the aforementioned properties we can effectively handle the nonlinear terms in (1.7) appropriately, even when there is only vertical dissipation. More precisely, the following theorems hold. Theorem 1.1 establishes the H 2 -stability while Theorem 1.2 specifies the decay rates of the oscillation part ( u, θ).

Theorem 1.1. Let T = [0, 1] be a 1D periodic box and let Ω = T × R. Assume u 0 , θ 0 ∈ H 2 (Ω) and ∇ • u 0 = 0. Then there exists ε = ε(ν, η) > 0 such that, if u 0 H 2 + θ 0 H 2 ≤ ε, then (1.
2) has a unique global solution (u, θ) that remains uniformly bounded for all time, for any t ≥ 0,

u(t) 2 H 2 + θ(t) 2 H 2 + 2ν t 0 ∂ 2 u(τ ) 2 H 2 dτ + 2η t 0 ∂ 1 θ(τ ) 2 H 2 dτ + C(ν, η) t 0 g 0 ∂ 1 u 2 2 L 2 dτ ≤ Cε 2
where C(ν, η) and C > 0 are constants.

Theorem 1.1 states that any small initial perturbation, in the H 2 -sense, leads to a unique global, in time, solution of (1.2) that remains small in H 2 for all time t. Furthermore, it implies that the time-integral of ∂ 1 u 2 (τ ) 2 L 2 is finite. The following Theorem asserts that the oscillation portion ( u, θ) decays to zero algebraically in time in the H 1 -norm. This result aligns with the stratification phenomenon of buoyancy driven fluids. Additionally, it affirms the observation derived from the numerical simulations presented in [START_REF] Doering | Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion[END_REF], indicating that the temperature becomes horizontally homogeneous and stratify in the vertical direction over time.

Theorem 1.2. Let u 0 , θ 0 ∈ H 2 (Ω) with ∇ • u 0 = 0 . Assume that (u 0 , θ 0 ) satisfies u 0 H 2 + θ 0 H 2 ≤ ε,
for sufficiently small ε > 0. Let (u, θ) be the corresponding solution of (1.2) with g 0 negative constant. Then the oscillation part ( u, θ) satisfies the following algebraic decay in time,

u H 1 + θ H 1 ≤ c(1 + t) -1 2 ,
for some constant c > 0 and for all t ≥ 0. In addition, ( u, θ) has the asymptotic behavior, as t → ∞, t ( u(t) 2

H 1 + θ(t) 2 H 1 ) → 0.
According to Theorem 1.2, the solution (u, θ) of (1.2) approaches its horizontal average (u, θ) asymptotically, and eventually, the Boussinesq equations (1.2) evolves to the following 1D system

     ∂ t u + u • ∇ u + 0 ∂ 2 p = g 0 0 θ + ν∂ 2 2 u, ∂ t θ + u • ∇ θ = 0.
We briefly outline the proofs for Theorem 1.1 and Theorem 1.2. As the local, in time, well-posedness on (1.2) in the Sobolev setting H 2 (Ω) can be established using standard approaches such as Friedrichs' Fourier cutoff (see, e.g., [START_REF] Majda | Vorticity and Incompressible Flow[END_REF]), the proof of Theorem 1.1 is essentially reduced to demonstrating the global, in time, a priori bound on the solution in H 2 (Ω). To do so, we make use of the bootstrapping argument (see [START_REF] Tao | Nonlinear Dispersive Equations: Local and Global Analysis[END_REF], p 20). To set it up, we introduce the following energy functional for the H 2 -solution,

E(t) = max 0≤τ ≤t ( u(τ ) 2 H 2 + θ(τ ) 2 H 2 ) + 2ν t 0 ∂ 2 u 2 H 2 dτ +2η t 0 ∂ 1 θ 2 H 2 dτ + δ t 0 g 0 ∂ 1 u 2 2 L 2 dτ, (1.16)
where δ > 0 is a suitably selected small parameter. Our central objective, is to show that, for a constant C uniform and for all t > 0,

E(t) ≤ C E(0) + C E(t) 3 2 .
(1.17)

To prove (1.17), we should make full use of the extra regularization resulting from the wave structure in (1.13). Furthermore, the control on the time integral of the horizontal derivative of the velocity field, namely

t 0 g 0 ∂ 1 u 2 (τ ) 2 L 2 dτ (1.18)
plays an improtant role our proof. Note here, that the uniform boundedness of (1.18) is not a consequence of the vertical dissipation in the velocity equation but due to the interaction with the temperature equation. In fact, using the equation of θ in (3.61), we represent g 0 ∂ 1 u 2 as,

g 0 ∂ 1 u 2 = -∂ t ∂ 1 θ -∂ 1 (u • ∇θ) + η∂ 111 θ, then g 0 ∂ 1 u 2 2 L 2 = -g 0 ∂ t ∂ 1 θ ∂ 1 u 2 dx -g 0 ∂ 1 u 2 ∂ 1 (u • ∇θ) dx + ηg 0 ∂ 1 u 2 ∂ 111 θ dx.
Hence, the time integrability of g 0 ∂ 1 u 2 2 L 2 is converted to the time integrability of other terms. This phenomenon of extra regularization and time integrability, resulting from the coupling, is also observed in other models of partial differential equations, such as the Oldroyd-B system (see [START_REF] Constantin | High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation[END_REF], [START_REF] Elgindi | Global regularity for some Oldroyd-B type models[END_REF]).

Once (1.17) is proven, the bootstrapping argument then gives that, if

E(0) = (u 0 , θ 0 ) 2 H 2 ≤ ε 2
for some sufficiently small ε > 0, then E(t) remains uniformly small for all time, namely

E(t) ≤ C ε 2 (1.19)
for a constant C > 0 and for all t ≥ 0. In particular, (1.19) yields the desired global H 2 -bound on the solution (u, θ). We leave details on the application of the bootstrapping argument in the proof of Theorem 1.1 in Section 3.

To demonstrate the algebraic decay rates on the H 1 -norm of the oscillation component, as stated in Theorem 1.2, we initially take the difference of (1.2) and its horizontal average, to write down the system governing the oscillation part ( u, θ)

   ∂ t u + u • ∇ u + u 2 ∂ 2 u -ν∂ 2 2 u + ∇ p = g 0 θe 2 , ∂ t θ + u • ∇ θ + u 2 ∂ 2 θ -η∂ 2 1 θ + g 0 u 2 = 0.
(1.20)

Controling the H 1 -norm of ( u, θ) naturally involves estimating the L 2 -norms ( u, θ) L 2 and (∇ u, ∇ θ) L 2 . Here, one major difficulty is that the equation of u has only vertical dissipation, however, the aforementioned Poincaré inequality can only bound a function in terms of its horizontal derivatives. Consequently, some of the nonlinear parts associated with u can not be bounded suitably and require the upper bounds involving u 2 L 2 . To handle these terms, we seek extra smoothing and stabilizing effect on u 2 by exploiting the coupling in (1.20). Specifically, we introduce the following extra term along with the H 1 -norm to form a Lyapunov functional,

-δ( u 2 , θ),
where δ > 0 is a small constant and ( u 2 , θ) denotes the L 2 -inner product. The time derivative of this terme produces δ u 2 2 L 2 , which help balance u 2 2 L 2 from the nonlinearity. Then, applying anisotropic inequalities presented in section 2, we demonstrate the following energy inequality.

d dt u 2 H 1 + θ 2 H 1 -δ( u 2 , θ) + ν ∂ 2 u 2 H 1 + η ∂ 1 θ 2 H 1 + δ 4 u 2 2
L 2 ≤ 0, resulting the desired algebraic decay stated in Theorem 1.2. More details are given in Section 4.

The subsequent sections are organized as follows: Section 2 presents various anisotropic inequalities and some crucial properties related to the orthogonal decomposition, including the Poincaré type inequality for the oscillation portion f . Section 3 is dedicated to the proof of Theorem 1.1 and Section 4 proves Theorem 1.2.

Preliminaries

This Section serves as preparation for the proof of Theorems 1.1 and 1.2. Lemma 2.1 through Lemma 2.5 provide several frequently used facts on the orthogonal decomposition. While Lemma 2.6 presents a precise decay rate for a nonnegative integrable function, which is also monotonic in a generalized sense.

We start first, by presenting some basic properties of f and f . Lemma 2.1. Let Ω = T × R. Assume that f defined on Ω is sufficiently regular, say f ∈ H 2 (Ω). Let f and f be defined as in (1.14) and (1.15). Then (a) The average operator f and the oscillation operator f commute with partial derivatives,

∂ 1 f = ∂ 1 f = 0, ∂ 2 f = ∂ 2 f , ∂ 1 f = ∂ 1 f , ∂ 2 f = ∂ 2 f , f = 0. (b) If f is a divergence-free vector field, namely ∇ • f = 0, then f and f are also divergence-free, ∇ • f = 0 and ∇ • f = 0.
(c) f and f are orthogonal in Ḣk for any integer k ≥ 0, namely

(f , f ) Ḣk (Ω) := Ω D k f • D k f dx = 0, f 2 Ḣk (Ω) = f 2 Ḣk (Ω) + f 2 Ḣk (Ω) .
In particular,

f Ḣk (Ω) ≤ f Ḣk (Ω) and f Ḣk (Ω) ≤ f Ḣk (Ω) .
The orthogonality is actually more general and holds for any integrable functions,

Ω f • g dx = 0.
Lemma 2.1 can be proven easily using the definition of f and f .

The next Lemma compares the 1D Sobolev inequalities on the whole line R and on bounded domains.

Lemma 2.2. For any 1D function f ∈ H 1 (R), f L ∞ (R) ≤ √ 2 f 1 2 L 2 (R) f ′ 1 2 L 2 (R) . For any bounded domain such as T = [0, 1] and f ∈ H 1 (T), f L ∞ (T) ≤ √ 2 f 1 2 L 2 (T) f ′ 1 2 L 2 (T) + f L 2 (T)
, in particular, if the function f has mean zero such as the oscillation part f ,

f L ∞ (T) ≤ C f 1 2 L 2 (T) f ′ 1 2 L 2 (T) .
The following lemma presents anisotropic upper bounds for triple products as well as for the L ∞ -norm on the domain Ω. Anisotropic Sobolev inequalities are powerful tools for dealing with anisotropic models. The whole space version of these type of inequalities has previously been used in [START_REF] Cao | Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion[END_REF] in the 2D cases and in [START_REF] Wu | Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium[END_REF] in the 3D case.

Lemma 2.3. Let Ω = T × R. For any f, g, h ∈ L 2 (Ω) with ∂ 1 f ∈ L 2 (Ω) and ∂ 2 g ∈ L 2 (Ω), then Ω f gh dx ≤ C f 1 2 L 2 ( f L 2 + ∂ 1 f L 2 ) 1 2 g 1 2 L 2 ∂ 2 g 1 2 L 2 h L 2 .
(2.1)

For any f ∈ H 2 (Ω), we have f L ∞ (Ω) ≤C f 1 4 L 2 ( f L 2 + ∂ 1 f L 2 ) 1 4 ∂ 2 f 1 4 L 2 × ( ∂ 2 f L 2 + ∂ 1 ∂ 2 f L 2 ) 1 4 .
Replacing f in Lemma 2.3 by its oscillation portion f , the lower-order part in (2.1) can be dropped, as presented in the next Lemma.

Lemma 2.4. Let Ω = T × R. For any f, g, h ∈ L 2 (Ω) with ∂ 1 f ∈ L 2 (Ω) and ∂ 2 g ∈ L 2 (Ω), then Ω f gh dx ≤ C f 1 2 L 2 ∂ 1 f 1 2 L 2 g 1 2 L 2 ∂ 2 g 1 2 L 2 h L 2 .
(2.2)

For any f ∈ H 2 (Ω), we have f L ∞ (Ω) ≤ C f 1 4 L 2 ∂ 1 f 1 4 L 2 ∂ 2 f 1 4 L 2 ∂ 1 ∂ 2 f 1 4 L 2 .
The subsequent Lemma states that the oscillation component f verifies a strong Poincaré type inequality with the upper bound expressed in terms of ∂ 1 f rather than ∇ f . Lemma 2.5. Let f and f be defined as in (1.14) and (1.15)

. If ∂ 1 f L 2 (Ω) < ∞, then f L 2 (Ω) ≤ C ∂ 1 f L 2 (Ω) ,
where C is a pure constant. In addition, if

∂ 1 f H 1 (Ω) < ∞, then f L ∞ (Ω) ≤ C ∂ 1 f H 1 (Ω) .
As a direct consequence of Lemma 2.5 and the inequality (2.2), one has

Ω f gh dx ≤ C ∂ 1 f L 2 g 1 2 L 2 ∂ 2 g 1 2 L 2 h L 2 .
(2.3)

We refer the readers to [START_REF] Dong | Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation[END_REF] for detailed proofs of Lemmas 2.1, 2.3, 2.4 and 2.5.

The last lemma precises an explicit decay rate in (2.5) for functions that are integrable and are decreasing in a general sense, namely (2.4).

Lemma 2.6. Let f = f (t) be a nonnegative function satisfying , for two constants

C 0 > 0 and C 1 > 0, ∞ 0 f (τ )dτ < C 0 and f (t) ≤ C 1 f (s) for any 0 ≤ s < t.
(2.4)

Then, for C 2 = max{2C 1 f (0), 4C 0 C 1 } and for any t > 0,

f (t) ≤ C 2 (1 + t) -1 . (2.5)
Furthermore, f (t) has the following large-time asymptotic behavior,

lim t→+∞ t f (t) = 0.
A detailed proof of Lemma 2.6 can be found in [START_REF] Lai | Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation[END_REF].

The H 2 Nonlinear Stability

This section proves Theorem. 1.1.

Proof. The proof is naturally divided into two major parts. The first part is for the existence, while the second part is for the uniqueness of solutions to (1.2).

To prove the global existence of solutions, it suffices to establish the energy inequality in (1.17) with E(t) being defined in (1.16). This process consists of two main parts. The first is to estimate the H 2 -norm of (u, θ) while the second is to estimate ∂ 1 u 2 2 L 2 and its time integral. Note that, for a divergence-free vector field u, namely ∇ • u = 0, we have

∇u L 2 = ω L 2 , ∆u L 2 = ∇ω L 2 ,
where ω = ∇ × u is the vorticity. Then, the H 2 -norm of u is equivalent to the sum of the L 2 -norms of u, ω and ∇ω.

Taking the L 2 -inner product of (u, θ) with the first two equations in (1.2), we find that the L 2 -norm of (u, θ) obeys

u(t) 2 L 2 + θ(t) 2 L 2 + 2ν t 0 ∂ 2 u(τ ) 2 L 2 dτ + 2η t 0 ∂ 1 θ(τ ) 2 L 2 dτ = u 0 2 L 2 + θ 0 2 L 2 . (3.1)
Next, we estimate the L 2 -norm of (ω, ∇θ). We make use of the vorticity equation and the temperature equation,

∂ t ω + u • ∇ω = ν∂ 22 ω + g 0 ∂ 1 θ, ∂ t θ + u • ∇θ + g 0 u 2 = η∂ 11 θ. (3.2)
Dotting the equations of ω and ∇θ by (ω, ∇θ), yields 1 2

d dt ( ω 2 L 2 + ∇θ 2 L 2 ) + ν ∂ 2 ω 2 L 2 + η ∂ 1 ∇θ 2 L 2 = I 1 + I 2 , (3.3)
where

I 1 = g 0 (∂ 1 θ ω -∇u 2 • ∇θ) dx, I 2 = -∇θ • ∇u • ∇θ dx.
Then, expressing ω and u in terms of the stream function ψ, namely ω = ∆ψ and u = ∇ ⊥ ψ := (-∂ 2 ψ, ∂ 1 ψ), we get

I 1 = g 0 (∂ 1 θ ω -∇u 2 • ∇θ) dx = g 0 (∂ 1 θ∆ψ -∇∂ 1 ψ • ∇θ) dx = g 0 (-θ ∆∂ 1 ψ + ∆∂ 1 ψ θ) dx = 0.
We further write I 2 into four terms,

I 2 = -(∂ 1 u 1 (∂ 1 θ) 2 + ∂ 1 u 2 ∂ 1 θ∂ 2 θ + ∂ 2 u 1 ∂ 1 θ∂ 2 θ + ∂ 2 u 2 (∂ 2 θ) 2 ) dx := I 21 + I 22 + I 23 + I 24 . (3.4)
The key point here is to obtain upper bounds for the terms on the right-hand side of (3.4) that are time integrable. By Lemmas 2.1, 2.4 and Young's inequality, I 21 , I 22 and I 23 can be bounded as follows

I 21 : = -∂ 1 u 1 (∂ 1 θ) 2 dx = -∂ 1 u 1 (∂ 1 θ) 2 dx ≤ c ∂ 1 θ 1 2 L 2 ∂ 2 ∂ 1 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ∂ 1 u 1 L 2 ≤ c u H 2 ∂ 1 θ 2 H 2 , (3.5)
I 22 : = -∂ 1 u 2 ∂ 1 θ∂ 2 θdx = -∂ 1 u 2 ∂ 1 θ∂ 2 θdx ≤ c ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ∂ 2 θ 1 2 L 2 ∂ 2 ∂ 2 θ 1 2 L 2 ∂ 1 u 2 L 2 ≤ c ∂ 1 θ H 2 θ H 2 ∂ 1 u 2 L 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 , (3.6)
I 23 : = -∂ 2 u 1 ∂ 1 θ∂ 2 θdx = -∂ 2 u 1 ∂ 1 θ∂ 2 θdx ≤ c ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ∂ 2 u 1 1 2 L 2 ∂ 2 ∂ 2 u 1 1 2 L 2 ∂ 2 θ L 2 ≤ c ∂ 1 θ H 2 ∂ 2 u H 2 θ H 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u H 2 . (3.7)
Using the divergence-free condition ∇ • u = 0, integration by parts and Lemmas 2.1 and 2.5, we obtain

I 24 : = -∂ 2 u 2 (∂ 2 θ) 2 dx = ∂ 1 u 1 (∂ 2 θ) 2 dx = -2 u 1 ∂ 2 θ ∂ 1 ∂ 2 θ dx ≤ c ∂ 2 θ L 2 u 1 1 2 L 2 ∂ 1 u 1 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 2 ∂ 1 ∂ 2 θ 1 2 L 2 ≤ c θ H 2 ∂ 1 u 1 L 2 = ∂ 2 u 2 L 2 ∂ 1 θ H 2 ≤ c θ H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . (3.8)
Hence, collecting the upper bounds on I 2 and inserting them in (3.3), we find

d dt ( ∇u 2 L 2 + ∇θ 2 L 2 ) + 2ν ∂ 2 ∇u 2 L 2 + 2η ∂ 1 ∇θ 2 L 2 ≤ c (u, θ) H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 .
(3.9) Thus, integrating (3.9) over [0, t] and combining with (3.1), we get

(u, θ) 2 H 1 + 2ν t 0 ∂ 2 u(s) 2 H 1 ds + 2η t 0 ∂ 1 θ(s) 2 H 1 ds ≤ (u 0 , θ 0 ) 2 H 1 + c t 0 (u, θ) H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 dτ ≤ E(0) + c E(t) 3 2 . (3.10)
To bound the H 2 -norm of (u, θ), it then remains to control the L 2 -norm of (∇ω, ∆θ). Applying ∇ to the first equation of (3.2) then dotting with ∇ω, and applying ∆ to the second equation of (3.2) then dotting with ∆θ, we find

1 2 d dt ( ∇ω 2 L 2 + ∆θ(t) 2 L 2 ) + ν ∂ 2 ∇ω 2 L 2 + η ∂ 1 ∆θ 2 L 2 = J 1 + J 2 + J 3 , (3.11)
with

J 1 = g 0 (∇∂ 1 θ • ∇ω -∆u 2 ∆θ) dx, J 2 = -∇ω • ∇u • ∇ω dx, J 3 = -∆θ • ∆(u • ∇θ)dx.
Similarly, we need to obtaining an upper bound for that is time integrable for each term in (3.11). Writing ω and u in terms of the stream function ψ, namely ω = ∆ψ and u = ∇ ⊥ ψ := (-∂ 2 ψ, ∂ 1 ψ), we have

J 1 = g 0 (∇∂ 1 θ • ∇ω -∆u 2 ∆θ) dx = g 0 (∇∂ 1 θ • ∇ω -∆∂ 1 ψ∆θ) dx = g 0 (∇∂ 1 θ • ∇ω -∂ 1 ω ∆θ) dx = g 0 (∇∂ 1 θ • ∇ω + ∂ 1 ∇ω • ∇θ) dx = g 0 ∂ 1 (∇θ • ∇ω) dx = 0.
After integration by parts, we decompose J 3 it into four pieces,

J 3 = -∆θ ∆u 1 ∂ 1 θ dx -∆θ ∆u 2 ∂ 2 θ dx -2 ∆θ ∇u 1 • ∂ 1 ∇θ dx -2 ∆θ ∇u 2 • ∂ 2 ∇θ dx := J 31 + J 32 + J 33 + J 34 . (3.12)
To deal with J 31 , we make use of the orthogonal decompositions u = u + u and θ = θ + θ to write

J 31 : = -∆θ∆u 1 ∂ 1 θdx = -∆θ∆u 1 ∂ 1 θdx = -∆θ∂ 11 u 1 ∂ 1 θdx -∆θ∂ 22 u 1 ∂ 1 θdx = ∆θ∂ 12 u 2 ∂ 1 θdx -∆θ∂ 22 u 1 ∂ 1 θdx := J 311 + J 312 . (3.13)
Applying Lemma 2.4 we obtain,

J 311 := ∆θ∂ 12 u 2 ∂ 1 θdx ≤ c ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ∂ 12 u 2 1 2 L 2 ∂ 2 ∂ 12 u 2 1 2 L 2 ∆θ L 2 ≤ c θ H 2 ∂ 1 θ H 2 ∂ 2 u H 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 , (3.14)
J 312 := -∆θ∂ 22 u 1 ∂ 1 θdx ≤ c ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ∂ 22 u 1 1 2 L 2 ∂ 2 ∂ 22 u 1 1 2 L 2 ∆θ L 2 ≤ c θ H 2 ∂ 1 θ H 2 ∂ 2 u H 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . (3.15)
Inserting the upper bounds for J 311 and J 312 in (3.13) yields

J 31 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . (3.16)
To deal with J 32 , we divide it first into two terms,

J 32 = -∆θ∆u 2 ∂ 2 θdx = -∂ 1 ∂ 1 θ ∆u 2 ∂ 2 θdx -∂ 2 ∂ 2 θ ∆u 2 ∂ 2 θ dx = -∂ 1 ∂ 1 θ ∆u 2 ∂ 2 θ dx + 1 2 ∆∂ 2 u 2 (∂ 2 θ) 2 dx = -∂ 1 ∂ 1 θ ∆u 2 ∂ 2 θ dx - 1 2 ∆∂ 1 u 1 (∂ 2 θ) 2 dx = -∂ 1 ∂ 1 θ ∆u 2 ∂ 2 θ dx + ∆u 1 ∂ 2 θ ∂ 1 ∂ 2 θdx = J 321 + J 322 . (3.17)
Invoking the decompositions of u and θ, we can rewrite J 321 as,

J 321 := -∂ 1 ∂ 1 θ ∆u 2 ∂ 2 θ dx = -∂ 1 ∂ 1 θ ∂ 11 u 2 ∂ 2 θ dx -∂ 1 ∂ 1 θ ∂ 11 u 2 ∂ 2 θ dx -∂ 1 ∂ 1 θ ∂ 22 u 2 ∂ 2 θ dx := J 3211 + J 3212 + J 3213 . (3.18)
The three terms in J 321 can be bounded as follows. By interation by parts, Lemma 2.1, Hölder's inequality, Lemma 2.2 and Young's inequality,

J 3211 := -∂ 1 ∂ 1 θ ∂ 11 u 2 ∂ 2 θ dx = ∂ 1 ∂ 11 θ ∂ 1 u 2 ∂ 2 θ dx = R ∂ 2 θ T ∂ 1 ∂ 11 θ ∂ 1 u 2 dx 1 dx 2 ≤ R |∂ 2 θ| ∂ 1 u 2 L 2 x 1 ∂ 1 ∂ 11 θ L 2 x 1 dx 2 ≤ ∂ 2 θ L ∞ x 2 ∂ 1 u 2 L 2 x 2 L 2 x 1 ∂ 1 ∂ 11 θ L 2 x 2 L 2 x 1 ≤ c ∂ 2 θ H 1 ∂ 1 u 2 L 2 ∂ 1 ∂ 11 θ L 2 ≤ c θ H 2 ∂ 1 u 2 2 L 2 + ∂ 1 θ 2 H 2 . (3.19)
By lemma 2.4 and then lemma 2.5

J 3212 := -∂ 1 ∂ 1 θ ∂ 11 u 2 ∂ 2 θ dx ≤ c ∂ 2 θ 1 2 L 2 ≤ ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 11 θ 1 2 L 2 ∂ 2 ∂ 11 θ 1 2 L 2 ∂ 11 u 2 L ≤ c u H 2 ∂ 1 θ 2 H 2 . (3.20)
Making use of the divergence-free condition of u, Lemmas 2.1 and 2.4, we have

J 3213 := -∂ 1 ∂ 1 θ ∂ 22 u 2 ∂ 2 θ dx = -∂ 1 ∂ 1 θ ∂ 21 u 1 ∂ 2 θ dx ≤ c ∂ 21 u 1 1 2 L 2 ∂ 1 ∂ 21 u 1 1 2 L 2 ∂ 11 θ 1 2 L 2 ∂ 2 ∂ 11 θ 1 2 L 2 ∂ 2 θ L ≤ c θ H 2 ∂ 1 θ H 2 ∂ 2 u H 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . ( 3
J 321 ≤ c (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 . (3.22)
We now turn to J 322 . We further decompose it into two terms,

J 322 := ∆u 1 ∂ 2 θ ∂ 1 ∂ 2 θdx = ∂ 11 u 1 ∂ 2 θ ∂ 1 ∂ 2 θdx + ∂ 22 u 1 ∂ 2 θ ∂ 1 ∂ 2 θdx = J 3221 + J 3222 . (3.23)
Due to the divergence-free condition of u and Lemma 2.4,

J 3221 := ∂ 11 u 1 ∂ 2 θ ∂ 12 θdx = -∂ 12 u 2 ∂ 2 θ ∂ 12 θdx ≤ c ∂ 12 u 2 1 2 L 2 ∂ 1 ∂ 12 u 2 1 2 L 2 ∂ 12 θ 1 2 L 2 ∂ 2 ∂ 12 θ 1 2 L 2 ∂ 2 θ L ≤ c θ H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c θ H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . (3.24)
By Lemma 2.4,

J 3222 := ∂ 22 u 1 ∂ 2 θ ∂ 12 θdx ≤ c ∂ 12 θ 1 2 L 2 ∂ 1 ∂ 12 θ 1 2 L 2 ∂ 22 u 2 1 2 L 2 ∂ 2 ∂ 22 u 2 1 2 L 2 ∂ 2 θ L ≤ c θ H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c θ H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . (3.25)
Combining the estimates (3.24) and (3.25) and inserting them in (3.23) we find

J 322 ≤ c θ H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . ( 3
J 32 ≤ c (u, θ) H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 . (3.27)
The next term J 33 is naturally split into two parts,

J 33 := -2 ∆θ∇u 1 • ∂ 1 ∇θdx = -2 ∆θ∂ 1 u 1 ∂ 1 ∂ 1 θdx -2 ∆θ∂ 2 u 1 ∂ 1 ∂ 2 θdx := J 331 + J 332 . (3.28)
All terms can be bounded suitably. In fact, due to the divergence-free condition of u and Lemma 2.4,

J 331 := -2 ∆θ∂ 1 u 1 ∂ 11 θdx = 2 ∆θ∂ 2 u 2 ∂ 1 ∂ 1 θdx ≤ c ∂ 11 θ 1 2 L 2 ∂ 1 ∂ 11 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ∆θ L 2 ≤ c θ H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c θ H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . (3.29)
J 332 can be bounded similarly, by ∂ 1 θ = ∂ 1 θ and Lemma 2.4,

J 332 := -2 ∆θ∂ 2 u 1 ∂ 12 θdx = -2 ∆θ∂ 2 u 1 ∂ 12 θdx ≤ c ∂ 12 θ 1 2 L 2 ∂ 1 ∂ 12 θ 1 2 L 2 ∂ 2 u 1 1 2 L 2 ∂ 2 ∂ 2 u 1 1 2 L 2 ∆θ L 2 ≤ c θ H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c θ H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . (3.30)
Inserting these upper bounds in (3.28) we get

J 33 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . (3.31)
To estimate J 34 , we first invoke the decompositions u = u + u, θ = θ + θ and Lemma 2.1, to write J 34 as

J 34 := -2 ∆θ∇u 2 • ∂ 2 ∇θdx = -2 (∂ 1 u 2 ∂ 1 ∂ 2 θ∆θ + ∂ 2 u 2 ∂ 2 ∂ 2 θ∆θ)dx = -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∆θ -2 ∂ 2 u 2 ∂ 2 ∂ 2 θ∆θdx = -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 11 θdx -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 22 θdx -2 ∂ 2 u 2 ∂ 2 ∂ 2 θ∆θdx := J 341 + J 342 + J 343 . (3.32)
We start with J 341 . By integration by parts, Lemmas 2.1 and 2.4 we have

J 341 := -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 11 θdx = 2 u 2 ∂ 1 ∂ 1 ∂ 2 θ∂ 11 θdx ≤ c ∂ 11 θ 1 2 L 2 ∂ 1 ∂ 11 θ 1 2 L 2 ∂ 12 θ 1 2 L 2 ∂ 2 ∂ 11 θ 1 2 L 2 ∂ 1 u 2 L 2 ≤ c u H 2 ∂ 1 θ 2 H 2 . (3.33)
Using the decomposition θ = θ + θ we write J 342 as,

J 342 := -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 22 θdx = -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 22 θdx -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 22 θdx := J 3421 + J 3422 . (3.34)
We start with J 3421 . Due to integration by parts, Lemma 2.1, Hölder's inequality, Lemma 2.2 and Young's inequality,

J 3421 := -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 22 θdx = 2 (∂ 2 ∂ 1 u 2 ∂ 1 ∂ 2 θ + ∂ 1 u 2 ∂ 2 ∂ 1 ∂ 2 θ)∂ 2 θ dx = 2 R ∂ 2 θ T (∂ 2 ∂ 1 u 2 ∂ 1 ∂ 2 θ + ∂ 1 u 2 ∂ 2 ∂ 1 ∂ 2 θ)dx 1 dx 2 ≤ c R |∂ 2 θ| ∂ 2 ∂ 1 u 2 L 2 x 1 ∂ 1 ∂ 2 θ L 2 x 1 + ∂ 1 u 2 L 2 x 1 ∂ 2 ∂ 1 ∂ 2 θ L 2 x 1 dx 2 ≤ c ∂ 2 θ L ∞ x 2 ∂ 2 ∂ 1 u 2 L 2 x 2 L 2 x 1 ∂ 1 ∂ 2 θ L 2 x 2 L 2 x 1 + ∂ 1 u 2 L 2 x 2 L 2 x 1 ∂ 2 ∂ 1 ∂ 2 θ L 2 x 2 L 2 x 1 ≤ c ∂ 2 θ H 1 ∂ 2 ∂ 1 u 2 L 2 ∂ 1 ∂ 2 θ L 2 + ∂ 1 u 2 L 2 ∂ 2 ∂ 1 ∂ 2 θ L 2 ≤ c θ H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 . (3.35)
For J 3422 , we apply Lemma 2.4 then Young's inequality,

J 3422 := -2 ∂ 1 u 2 ∂ 1 ∂ 2 θ∂ 22 θdx ≤ c ∂ 22 θ 1 2 L 2 ∂ 1 ∂ 22 θ 1 2 L 2 ∂ 12 θ 1 2 L 2 ∂ 2 ∂ 12 θ 1 2 L 2 ∂ 1 u L 2 ≤ c θ 1 2 H 2 u 1 2 H 2 ∂ 1 θ 3 2 H 2 ∂ 1 u 2 1 2 L 2 ≤ c θ 1 2 H 2 u 1 2 H 2 ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 . (3.36)
In view of (3.34), (3.35) and (3.36) we have

J 342 ≤ c (u, θ) H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L .
(3.37)

Writing J 343 more explicitly and using ∂ 1 θ = ∂ 1 θ, we have

J 343 := -2 ∂ 2 u 2 ∂ 2 ∂ 2 θ∆θdx = -2 ∂ 2 u 2 ∂ 22 θ∂ 11 θdx -2 ∂ 2 u 2 ∂ 22 θ∂ 22 θdx := J 3431 + J 3432 . (3.38)
From Lemma 2.4, J 3431 can be bounded as,

J 3431 := -2 ∂ 2 u 2 ∂ 22 θ∂ 11 θdx ≤ c ∂ 11 θ 1 2 L 2 ∂ 1 ∂ 11 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ∂ θ L 2 ≤ c θ H 2 ∂ 1 θ H 2 ∂ 2 u H 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . (3.39)
The estimate of J 3432 is slightly more delicate. Due to the decomposition θ = θ + θ, we write J 3432 as,

J 3432 := -2 ∂ 2 u 2 ∂ 22 θ∂ 22 θdx = -2 ∂ 2 u 2 ∂ 22 θ∂ 22 θdx -4 ∂ 2 u 2 ∂ 22 θ∂ 22 θdx -2 ∂ u 2 ∂ 22 θ∂ 22 θdx := J 34321 + J 34322 + J 34323 . (3.40)
By ∇ • u = 0 and Lemma 2.1, the first term J 34321 is clearly zero,

J 34321 = -2 ∂ 2 u 2 ∂ 22 θ∂ 22 θdx = 2 ∂ 1 u 1 ∂ 22 θ∂ 22 θdx = 0. (3.41)
Applying Lemmas 2.4 and 2.5 and Young's inequality,

J 34322 := -4 ∂ 2 u 2 ∂ 22 θ∂ 22 θdx ≤ c ∂ 22 θ L 2 ∂ 22 θ 1 2 L 2 ∂ 1 ∂ 22 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ≤ c ∂ 22 θ L 2 ∂ 1 ∂ 22 θ 1 2 L 2 ∂ 1 ∂ 22 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ u 2 1 2 L 2 ≤ c θ H 2 ∂ 1 θ H 2 ∂ 2 u H 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 , (3.42)
J 34323 := -4 ∂ 2 u 2 ∂ 22 θ∂ 22 θdx ≤ c ∂ 22 θ L 2 ∂ 22 θ 1 2 L 2 ∂ 1 ∂ 22 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ≤ c ∂ 22 θ L 2 ∂ 1 ∂ 22 θ 1 2 L 2 ∂ 1 ∂ 22 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ≤ c θ H 2 ∂ 1 θ H 2 ∂ 2 u H 2 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . (3.43)
The bounds for J 3432 in (3.41), (3.42) and (3.43) lead to,

J 3432 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . ( 3
J 343 ≤ c θ H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 . ( 3
J 34 ≤ c (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 . ( 3
J 3 ≤ c (u, θ) H 2 ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 + ∂ 1 θ 2 H 2 . (3.47)
As outlined in the introduction, we need the help of an extra regularization term to bound J 2 , namely,

t 0 ∂ 1 u 2 2 L 2 dτ. (3.48)
In order to make efficient use of the anisotropic dissipation, we express J 2 as follows

J 2 = -∂ 1 u 1 (∂ 1 ω) 2 dx -∂ 1 u 2 ∂ 1 ω ∂ 2 ω dx -∂ 2 u 1 ∂ 1 ω ∂ 2 ωdx -∂ 2 u 2 (∂ 2 ω) 2 dx = ∂ 2 u 2 (∂ 1 ω) 2 dx -∂ 1 u 2 ∂ 1 ω ∂ 2 ω dx -∂ 2 u 1 ∂ 1 ω ∂ 2 ω dx -∂ 2 u 2 (∂ 2 ω) 2 dx :=J 21 + J 22 + J 23 + J 24 .
(3.49)

The terms J 21 through J 24 can be bounded in the following manner. Due to ∇•u = 0, integration by parts and Lemmas 2.1 and 2.4,

J 21 := -∂ 1 u 1 (∂ 1 ω) 2 dx = ∂ 2 u 2 (∂ 1 ω) 2 dx = -2 u 2 ∂ 1 ω∂ 2 ∂ 1 ω dx ≤ c u 2 1 2 L 2 ∂ 1 u 2 1 2 L 2 ∂ 2 ∂ 1 ω 1 2 L 2 ∂ 2 ∂ 1 ω 1 2 L 2 ∂ 2 ∂ 1 ω L 2 ≤ c u H 2 ∂ 1 u 2 1 2 L 2 ∂ 2 u 3 2 H 2 ≤ c u H 2 ∂ 1 u 2 2 L 2 + ∂ 2 u 2 H 2 . (3.50)
According to Lemmas 2.1 and 2.4,

J 22 := -∂ 1 u 2 ∂ 1 ω∂ 2 ω dx = -∂ 1 u 2 ∂ 1 ω∂ 2 ω dx ≤ c ∂ 1 u 2 1 2 L 2 ∂ 1 ∂ 1 u 2 1 2 L 2 ∂ 1 ω 1 2 L 2 ∂ 2 ∂ 1 ω 1 2 L 2 ∂ 2 ω L 2 ≤ c u H 2 ∂ 1 u 2 1 2 L 2 ∂ 2 u 3 2 H 2 ≤ c u H 2 ∂ 1 u 2 2 L 2 + ∂ 2 u 2 H 2 . (3.51)
To bound J 23 , we first use the orthogonal decomposition of u 1 and ω and Lemma 2.1, to write J 23 as

J 23 := -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx = -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx = -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx = J 231 + J 232 + J 233 . (3.52)
According to Lemma 2.1, the first term J 231 is clearly zero,

J 231 := -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx = - R ∂ 2 u 1 ∂ 2 ω T ∂ 1 ω dx 1 dx 2 = 0. (3.53)
The terms J 232 and J 233 can be bounded directly. By Lemma 2.4,

J 232 := -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx ≤ c ∂ 2 ω 1 2 L 2 ∂ 1 ∂ 2 ω 1 2 L 2 ∂ 1 ω 1 2 L 2 ∂ 2 ∂ 1 ω 1 2 L 2 ∂ 2 u 1 L 2 ≤ c u H 2 ∂ 2 u 2 H 2 , (3.54)
J 233 := -∂ 2 u 1 ∂ 1 ω∂ 2 ω dx ≤ c ∂ 2 u 1 1 2 L 2 ∂ 1 ∂ 2 u 1 1 2 L 2 ∂ 1 ω 1 2 L 2 ∂ 2 ∂ 1 ω 1 2 L 2 ∂ 2 ω L 2 ≤ c u H 2 ∂ 2 u 2 H 2 . (3.55)
Inserting these upper bounds in (3.52) yields

J 23 ≤ c u H 2 ∂ 2 u 2 H 2 . (3.56)
To deal with J 24 we use the divergence-free condition of u, Lemma 2.1, and the inequality (2.3) in Lemma 2.5

J 24 := -∂ 2 u 2 (∂ 2 ω) 2 dx = -∂ 1 u 1 (∂ 2 ω + ∂ 2 ω) 2 dx = -2 ∂ 1 u 1 ∂ 2 ω∂ 2 ω dx -2 ∂ 1 u 1 (∂ 2 ω) 2 dx ≤ c ∂ 2 ω L 2 + ∂ 2 ω L 2 ∂ 1 u 1 1 2 L 2 ∂ 2 ∂ 1 u 1 1 2 L 2 ∂ 1 ∂ 2 ω L 2 ≤ c u H 2 ∂ 2 u 2 H 2 . ( 3
d dt ( ∆u 2 L 2 + ∆θ 2 L 2 ) + 2ν ∂ 2 ∆u 2 L 2 + 2η ∂ 1 ∆θ 2 L 2 ≤ c (u, θ) H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 . (3.59)
Integrating (3.59) over [0, t], we get

∆u(t) 2 L 2 + ∆θ(t) 2 L 2 + 2ν t 0 ∂ 2 ∆u 2 L 2 dτ + 2η t 0 ∆∂ 1 θ 2 L 2 dτ ≤ ∆u 0 2 L 2 + ∆θ 0 2 L 2 + c t 0 (u, θ) H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 ≤ E(0) + c E(t) 3 2 .
(3.60)

The subsequent step is to control the last piece in E(t) defined by (1.16), namely

t 0 g 0 ∂ 1 u 2 2 L 2 dτ. (3.61)
Our strategy is to make use of the special structure of the equation for θ in (1.2) and replace g 0 ∂ 1 u 2 in (3.61) via the equation of θ,

g 0 ∂ 1 u 2 = -∂ t ∂ 1 θ -∂ 1 (u • ∇θ) + η∂ 111 θ. (3.62)
Multiplying (3.62) by g 0 ∂ 1 u 2 and then integrating over Ω, we obtain

g 0 ∂ 1 u 2 2 L 2 = -g 0 ∂ t ∂ 1 θ ∂ 1 u 2 dx -g 0 ∂ 1 u 2 ∂ 1 (u • ∇θ) dx + g 0 η ∂ 1 u 2 ∂ 111 θ dx := K 1 + K 2 + K 3 . (3.63)
We bound K 3 as follows,

|K 3 | ≤ η g 0 ∂ 1 u 2 L 2 ∂ 111 θ L 2 ≤ 1 2 g 0 ∂ 1 u 2 2 L 2 + c ∂ 1 θ 2 H 2 , (3.64)
the term with unfavorable derivative ∂ 1 u 2 will be then absorbed by the left-hand side of (3.64).

For K 1 , we first shift the time derivative

K 1 = -g 0 d dt ∂ 1 θ ∂ 1 u 2 dx + g 0 ∂ 1 θ ∂ 1 ∂ t u 2 dx := K 11 + K 12 . (3.65)
Using the equation for the second component of the velocity, we write

K 12 = -g 0 ∂ 1 ∂ 1 θ ∂ t u 2 dx (3.66) = -g 0 ∂ 11 θ(-(u • ∇u 2 ) -∂ 2 p + ν∂ 22 u 2 + g 0 θ) dx = g 0 ∂ 11 θ (u • ∇u 2 ) dx + g 0 ∂ 11 θ ∂ 2 p dx -g 0 ν ∂ 11 θ ∂ 22 u 2 dx -g 2 0 ∂ 11 θ θ dx. (3.67)
Then, we apply the divergence operator to the velocity equation to express the pressure term as

p = -∆ -1 ∇ • (u • ∇u) + g 0 ∆ -1 ∂ 2 θ.
(3.68) Inserting (3.68) in (3.67), we obtain

K 12 = g 0 ∂ 11 θ (u • ∇u 2 ) dx + g 0 ∂ 11 θ (-∂ 2 ∆ -1 ∇ • (u • ∇u)) dx -g 0 ν ∂ 11 θ ∂ 22 u 2 dx -g 2 0 ∂ 11 θ ∂ 11 ∆ -1 θ dx := K 121 + K 122 + K 123 + K 124 .
(3.69)

Due to Hölder's inequality and the fact that the double Riesz transform ∂ 11 ∆ -1 is bounded on L q for any 1 < q < ∞ (see, e.g., [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF]), we have

K 124 := -g 2 0 ∂ 1 θ ∂ 11 ∆ -1 ∂ 1 θ dx ≤ c ∂ 1 θ L 2 ∂ 11 ∆ -1 ∂ 1 θ L 2 ≤ c ∂ 1 θ 2 L 2 . (3.70)
Thanks to Hölder's inequality,

K 123 := -g 0 ν ∂ 11 θ ∂ 22 u 2 dx ≤ c ∂ 11 θ L 2 ∂ 22 u 2 L 2 . (3.71)
By integration by parts, Hölder's inequality and the boundedness of the double Riesz transform,

K 122 := g 0 ∂ 11 θ (-∂ 2 ∆ -1 ∇ • (u • ∇u)) dx = g 0 ∂ 1 θ ∂ 12 ∆ -1 ∇ • (u • ∇u) dx ≤ c ∂ 1 θ L 2 ∆ -1 ∂ 12 ∇ • (u • ∇u) L 2 ≤ c ∂ 1 θ L 2 ∂ 2 (u • ∇u) L 2 ≤ c ∂ 1 θ L 2 ∂ 2 u • ∇u + u • ∇∂ 2 u L 2 ≤ c ∂ 1 θ L 2 ( ∂ 2 u L 4 ∇u L 4 + u ∞ ∇∂ 2 u L 2 ) ≤ c ∂ 1 θ L 2 ∂ 2 u H 1 ∇u H 1 + c ∂ 1 θ L 2 u H 2 ∇∂ 2 u L 2 . (3.72)
To deal with K 121 , we rewrite it as

K 121 = g 0 ∂ 11 θ(u 1 ∂ 1 u 2 + u 2 ∂ 2 u 2 )dx = g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx + g 0 ∂ 11 θ u 2 ∂ 2 u 2 dx = g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx + g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx + g 0 ∂ 11 θ u 2 ∂ 2 u 2 dx := K 1211 + K 1212 + K 1213 . (3.73)
By Lemma 2.4, the divergence-free condition of u and Lemma 2.5,

K 1211 := g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx ≤ c ∂ 11 θ 1 2 L 2 ∂ 2 ∂ 11 θ L 2 u 1 1 2 L 2 ≤ ∂ 1 u 1 1 2 L 2 ∂ 1 u 1 1 2 L 2 = ∂ 2 u 2 1 2 L 2 ∂ 1 u 2 L 2 ≤ c u H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c u H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . (3.74)
Due to Lemma 2.1, Hölder's inequality, Lemma 2.2 and then Young's inequality,

K 1212 := g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx = g 0 R u 1 T ∂ 11 θ∂ 1 u 2 dx 1 dx 2 ≤ c R |u 1 | ∂ 11 θ L 2 x 1 ∂ 1 u 2 L 2 x 1 dx 2 ≤ c u 1 L ∞ x 2 ∂ 11 θ L 2 x 2 L 2 x 1 ∂ 1 u 2 L 2 x 2 L 2 x 1 ≤ c u 1 H 1 ∂ 11 θ L 2 ∂ 1 u 2 L 2 ≤ c u H 2 ∂ 1 u 2 2 L 2 + ∂ 1 θ 2 H 2 . (3.75)
According to Lemma 2.4,

K 1213 := g 0 ∂ 11 θ u 2 ∂ 2 u 2 dx ≤ c ∂ 11 θ 1 2 L 2 ∂ 1 ∂ 11 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 u 2 L 2 ≤ c u H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c u H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . ( 3
K 121 ≤ c u H 2 ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 + ∂ 1 θ 2 H 2 . ( 3
|K 12 | ≤ c u H 2 ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 + ∂ 1 θ 2 H 2 .
(3.78)

We now need to bound K 2 . We first split it into four terms,

K 2 : = -g 0 ∂ 1 u 2 ∂ 1 (u • ∇θ) dx = -g 0 ∂ 1 u 2 ∂ 1 u 1 ∂ 1 θ dx -g 0 ∂ 1 u 2 u 1 ∂ 1 ∂ 1 θ dx -g 0 ∂ 1 u 2 ∂ 1 u 2 ∂ 2 θ dx -g 0 ∂ 1 u 2 u 2 ∂ 1 ∂ 2 θ dx : = K 21 + K 22 + K 23 + K 24 . (3.79) Due to ∂ 1 θ = ∂ 1 θ, Lemma 2.

and Young's inequality

K 21 := -g 0 ∂ 1 u 2 ∂ 1 u 1 ∂ 1 θ dx = -g 0 ∂ 1 u 2 ∂ 1 u 1 ∂ 1 θ dx ≤ c ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ∂ 1 u 2 L 2 ≤ c u H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c u H 2 ∂ 2 u 2 H 2 + ∂ 1 θ 2 H 2 . (3.80)
Using Lemma 2.1 and invoking the decompositions u = u + u we write K 22 as

K 22 := -g 0 ∂ 1 u 2 u 1 ∂ 1 ∂ 1 θ dx = g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx + g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx := K 221 + K 222 . (3.81)
By Lemmas 2.4, 2.5 and the divergence-free condition of u,

K 221 := g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx ≤ c ∂ 11 θ 1 2 L 2 ∂ 2 ∂ 11 θ L 2 u 1 1 2 L 2 ≤ ∂ 1 u 1 1 2 L 2 ∂ 1 u 1 1 2 L 2 = ∂ 2 u 2 1 2 L 2 ∂ 1 u 2 L 2 ≤ c u H 2 ∂ 2 u H 2 ∂ 1 θ H 2 ≤ c u H 2 ∂ 2 u 2 H 2 + ∂ 1 θ H 2 . (3.82)
To bound K 222 , we first use Lemma 2.1, Hölder's inequality and then Lemma 2.2 to obtain

K 222 := g 0 ∂ 11 θ u 1 ∂ 1 u 2 dx = g 0 R u 1 T ∂ 11 θ∂ 1 u 2 dx 1 dx 2 ≤ c R |u 1 | ∂ 11 θ L 2 x 1 ∂ 1 u 2 L 2 x 1 dx 2 ≤ c u 1 L ∞ x 2 ∂ 11 θ L 2 x 2 L 2 x 1 ∂ 1 u 2 L 2 x 2 L 2 x 1 ≤ c u 1 H 1 ∂ 11 θ L 2 ∂ 1 u 2 L 2 ≤ c u H 2 ∂ 1 u 2 2 L 2 + ∂ 1 θ 2 H 2 . ( 3
K 22 ≤ c u H 2 ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 + ∂ 1 θ 2 H 2 . (3.84) By ∂ 1 u 2 = ∂ 1 u 2 and θ = θ + θ, we rewrite K 23 as K 23 := -g 0 ∂ 1 u 2 ∂ 1 u 2 ∂ 2 θ dx = -g 0 ∂ 1 u 2 ∂ 1 u 2 ∂ 2 θ dx -g 0 ∂ 1 u 2 ∂ 1 u 2 ∂ 2 θ dx := K 231 + K 232 . (3.85)
To estimate K 231 , we make use of Lemma 2.1, Hölder's inequality and then Lemma 2.2 to get

K 231 := g 0 ∂ 1 u 2 ∂ 1 u 2 ∂ 2 θ dx = g 0 R ∂ 2 θ T ∂ 1 u 2 ∂ 1 u 2 dx 1 dx 2 ≤ c R |∂ 2 θ| ∂ 1 u 2 L 2 x 1 ∂ 1 u 2 L 2 x 1 dx 2 ≤ c ∂ 2 θ L ∞ x 2 ∂ 1 u 2 L 2 x 2 L 2 x 1 ∂ 1 u 2 L 2 x 2 L 2 x 1 ≤ c ∂ 2 θ H 1 ∂ 1 u 2 2 L 2 ≤ c θ H 2 ∂ 1 u 2 2 L 2 . (3.86)
Via Lemma 2.4,

K 232 := g 0 ∂ 1 u 2 ∂ 1 u 2 ∂ 2 θ dx ≤ c ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 u 2 1 2 L 2 ∂ 2 ∂ 1 u 2 1 2 L 2 ∂ 1 u 2 L 2 ≤ c θ 1 2 H 2 u 1 2 H 2 ∂ 1 θ 1 2 H 2 ∂ 1 u 2 3 2 L 2 ≤ c (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 . (3.87)
Inserting the bounds for K 231 and K 232 in (3.85), we find

K 23 ≤ c (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 .
(3.88)

The last term K 24 can also be bounded due to the fact that u 2 = 0, Lemmas 2.4 and 2.5

K 24 := -g 0 ∂ 1 u 2 u 2 ∂ 1 ∂ 2 θ dx = -g 0 ∂ 1 u 2 u 2 ∂ 1 ∂ 2 θ dx ≤ c u 2 1 2 L 2 ≤ ∂ 1 u 2 1 2 L 2 ∂ 1 u 2 1 2 L 2 ∂ 1 u 2 1 2 L 2 ∂ 2 ∂ 1 u 2 1 2 L 2 ∂ 1 ∂ 2 θ L 2 ≤ c θ 1 2 H 2 u 1 2 H 2 ∂ 1 θ 1 2 H 2 ∂ 1 u 2 3 2 L 2 ≤ c (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 . ( 3
K 2 ≤ c (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 1 u 2 2 L 2 + ∂ 2 u 2 H 2 . ( 3
1 2 g 0 ∂ 1 u 2 2 L 2 ≤ c ∂ 1 θ 2 H 2 -g 0 d dt ∂ 1 θ ∂ 1 u 2 dx + c (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 . (3.91)
Integrating (3.91) over the time interval [0, t], we find

t 0 g 0 ∂ 1 u 2 2 L 2 dτ ≤ c t 0 ∂ 1 θ 2 H 2 dτ -2g 0 ∂ 1 θ ∂ 1 u 2 dx + 2g 0 ∂ 1 θ 0 ∂ 1 u 02 dx + c t 0 (u, θ) H 2 ∂ 1 θ 2 H 2 + ∂ 2 u 2 H 2 + ∂ 1 u 2 2 L 2 dτ ≤ c t 0 ∂ 1 θ 2 H 2 dτ + c t 0 ∂ 2 u 2 H 2 dτ + c ( u 2 H 1 + θ 2 H 1 ) + c ( u 0 2 H 1 + θ 0 2 H 1 ) + c E(t) 3 2 . (3.92)
To conclude, we combine the H 1 -bound in (3.10), the homogeneous H 2 -bound in (3.60) and the bound for the extra regularization term in (3.92). When doing so, we need eliminate the quadratic terms on the right-hand side of (3.92) by the corresponding terms on the left-hand side, then it suffices to multiply both sides of (3.92) by a suitable small coefficient δ > 0. Taking (3.10) + (3.60) + δ (3.92), leads to

u(t) 2 H 2 + θ(t) 2 H 2 + 2ν t 0 ∂ 2 u 2 H 2 dτ + 2η t 0 ∂ 1 θ 2 H 2 dτ + δ t 0 g 0 ∂ 1 u 2 2 L 2 ≤ E(0) + c E(t) 3 2 + c δ ( u(t) 2 H 2 + θ(t) 2 H 2 ) + c δ ( u 0 2 H 2 + θ 0 2 H 2 ) + c δ t 0 ∂ 2 u 2 H 2 dτ + c δ t 0 ∂ 1 θ 2 H 2 dτ + c δ E(t) 3 2 . (3.93)
If δ > 0 is chosen to be sufficiently small, say

c δ ≤ 1 2 , c δ ≤ ν, c δ ≤ η, then (3.93) gives E(t) ≤ C 1 E(0) + C 2 E(t) 3 2 , (3.94)
where C 1 and C 2 are positive constants. The proof of the desired stability result, is then completed by applying the bootstrapping argument on (3.94). Indeed, if the initial data (u 0 , θ 0 ), is sufficiently small, say, and we then show that E(t) actually admits an even smaller bound by taking the initial H 2 -norm E(0) sufficiently small. In fact, Inserting (3.96) in (3.94) yields

E(0) = (u 0 , θ 0 ) 2 H 2 ≤ ε 2 := 1 16C 1 C 2 2 , ( 3
E(t) ≤ C 1 E(0) + C 2 E(t) 3 2 ≤ C 1 ε 2 + C 2 1 2C 2 E(t).
That is,

1 2 E(t) ≤ C 1 ε 2 or E(t) ≤ 2 C 1 1 16C 1 C 2 2 = 1 8C 2 2 = 2C 1 ǫ 2 , for all t ≤ T .
The bootstrapping argument then assesses that (3.96) holds for all time when E(0) satisfies (3.95). This establishes the global stability.

Finally, we establish the uniqueness of H 2 -solutions to (1.2). Assume that (u (1) , p (1) , θ (1) ) and (u (2) , p (2) , θ (2) ) are two solutions of (1.2) with one of them in the H 2 -regularity class say (u (1) , θ (1) ) ∈ L ∞ (0, T ; H 2 ). The difference between the two solutions (u * , p * , θ * ) with u * = u (2)u (1) , p * = p (2)p (1) and θ * = θ (2)θ (1) verifies

         ∂ t u * + u (2) • ∇u * + u * • ∇u (1) + ∇p * = ν∂ 22 u * + g 0 θ * e 2 , ∂ t θ * + u (2) • ∇θ * + u * • ∇θ (1) + g 0 u 2 * = η∂ 11 θ * , ∇ • u * = 0, u * (x, 0) = 0, θ * (x, 0) = 0.
(3.97)

We estimate the difference (u * , p * , θ * ) in L 2 (Ω). Taking the L 2 -inner product of (3.97) with (u * , θ * ) and applying the divergence-free condition, we get 1 2

d dt (u * , θ * ) 2 L 2 + ν ∂ 2 u * 2 L 2 + η ∂ 1 θ * 2 L 2 = -u * • ∇u (1) • u * dx -u * • ∇θ (1) • θ * dx = I 1 + I 2 . (3.98)
Due to Lemma 2.3 and the uniformly global bound for u (1) H 2 ,

I 1 := -u * • ∇u (1) • u * dx ≤ c ∇u (1) 1 2 L 2 ∇u (1) L 2 + ∂ 1 ∇u (1) L 2 1 2 ≤c u * 1 2 L 2 ∂ 2 u * 1 2 L 2 u * L 2 ≤ c u * 3 2 L 2 ∂ 2 u * 1 2 L 2 ≤ c u * 2 L 2 + ν 4 ∂ 2 u * 2 L 2 . (3.99)
Similarly, by Lemma 2.3 and the uniformly global bound for θ (1) H 2 ,

I 2 := -u * • ∇θ (1) • θ * dx ≤ c ∇θ (1) 1 2 L 2 ∇θ (1) L 2 + ∂ 1 ∇θ (1) L 2 1 2 ≤c u * 1 2 L 2 ∂ 2 u * 1 2 L 2 θ * L 2 ≤ c u * 1 2 L 2 ∂ 2 u * 1 2 L 2 θ * L 2 ≤ c θ * L 2 u * L 2 + ∂ 2 u * L 2 ≤ c θ * 2 L 2 + c u * 2 L 2 + ν 4 ∂ 2 u * 2 L 2 . ( 3
d dt (u * , θ * ) 2 L 2 + ν ∂ 2 u * 2 L 2 + η ∂ 1 θ * 2 L 2 ≤ c u * 2 L 2 + θ * 2 L 2 + ν 2 ∂ 2 u * 2 L 2 or d dt (u * , θ * ) 2 L 2 + ν ∂ 2 u * 2 L 2 + η ∂ 1 θ * 2 L 2 ≤ c (u * , θ * ) 2 L 2 . (3.101)
Grönwall's inequality then implies,

u * (t) L 2 = θ * (t) L 2 = 0.
In other words, these two solutions coincide. This finishes the proof of Theorem 1.1.

Decay Rates Result

This section is devoted to the proof the decay rates presented in Theorem 1.2.

Proof of Theorem 1.2. Taking the average of the system (1.2) and using the fact that u • ∇u = 0, we write the equations of (u, θ),

     ∂ t u + u • ∇ u + 0 ∂ 2 p = g 0 0 θ + ν∂ 2 2 u , ∂ t θ + u • ∇ θ = 0, (4.1)
where g 0 is a negative constant. By subtracting (4.1) from (1.2), we get

   ∂ t u + u • ∇ u + u 2 ∂ 2 u -ν∂ 2 2 u + ∇ p = g 0 θe 2 , ∂ t θ + u • ∇ θ + u 2 ∂ 2 θ -η∂ 2 1 θ + g 0 u 2 = 0. (4.2)
Taking the L 2 -inner product of (

d dt u 2 L 2 + θ 2 L 2 + ν ∂ 2 u 2 L 2 + η ∂ 1 θ 2 L 2 = -u • ∇ u • udx -u 2 ∂ 2 u • udx -u • ∇ θ • θdx -u 2 ∂ 2 θ • θdx := A 1 + A 2 + A 3 + A 4 . u, θ) with (4.2) yields, 1 2
Now, we estimate A 1 through A 4 . The first term A 1 is clearly zero due to ∇ • u = 0 and Lemma 2.1,

A 1 := -u • ∇ u • udx = -u • ∇ u • udx =0 + u • ∇ u • udx =0 = 0. (4.4)
Likewise,

A 3 := u • ∇ θ • θdx = 0. ( 4.5)
To bound A 2 we first write it as,

A 2 := -u 2 ∂ 2 u • udx := -u 2 ∂ 2 u 1 u 1 dx -u 2 ∂ 2 u 2 u 2 dx := A 21 + A 22 . (4.6)
Due to the fact that u 2 = 0 we have,

A 22 = -u 2 ∂ 2 u 2 u 2 dx = 0. (4.7)
Applying Lemmas 2.4 and 2.5, the divergence-free condition of u and then Young's inequality leads to

A 21 := -u 2 ∂ 2 u 1 u 1 dx ≤ c ∂ 2 u L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 u 1 1 2 L 2 ≤ ∂ 1 u 1 1 2 L 2 = ∂ 2 u 2 1 2 L 2 ∂ 1 u 1 1 2 L 2 = ∂ 2 u 2 1 2 L 2 ≤ c u H 2 u 2 1 2 L 2 ∂ 2 u 3 2 L 2 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 L 2 . (4.8)
Inserting (4.7) and (4.8) in (4.6) we get

A 2 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 L 2 . (4.9)
The last term A 4 can be bounded via Lemma 2.1, Hölder's inequality, and Lemmas 2.2 and 2.5,

A 4 := -u 2 ∂ 2 θ • θdx = - R ∂ 2 θ T θ u 2 dx 1 dx 2 ≤ c ∂ 2 θ L ∞ x 2 u 2 L 2 θ L 2 ≤ c ∂ 2 θ H 1 u 2 L 2 ∂ 1 θ L 2 ≤ c θ H 2 u 2 L 2 ∂ 1 θ L 2 ≤ c θ H 2 u 2 2 L 2 + ∂ 1 θ 2 L 2 . (4.10)
Combining the estimates of A 1 through A 4 , we get 1 2

d dt u 2 L 2 + θ 2 L 2 + ν ∂ 2 u 2 L 2 + η ∂ 1 θ 2 L 2 ≤ c (u, θ) H 2 u 2 2 L 2 + ∂ 2 u 2 L 2 + ∂ 1 θ 2 L 2 . (4.11)
Applying ∇ to (4.2), we write

   ∂ t ∇ u + ∇( u • ∇ u) + ∇( u 2 ∂ 2 u) -ν∂ 2 2 ∇ u + ∇∇ p = g 0 ∇( θe 2 ) , ∂ t ∇ θ + ∇( u • ∇ θ) + ∇( u 2 ∂ 2 θ) -η∂ 2 1 ∇ θ + g 0 ∇ u 2 = 0.
(4.12) Dotting (4.12) by (∇ u, ∇ θ), we get 1 2

d dt ∇ u(t) 2 L 2 + ∇ θ(t) 2 L 2 + ν ∂ 2 ∇ u 2 L 2 + η ∂ 1 ∇ θ 2 L 2 = -∇( u • ∇ u) • ∇ udx -∇( u 2 ∂ 2 u) • ∇ udx -∇( u • ∇ θ) • ∇ θdx -∇( u 2 ∂ 2 θ) • ∇ θdx := B 1 + B 2 + B 3 + B 4 . (4.13)
The terms B 1 through B 4 can be bounded as follows. We start with B 1 . Using Lemma 2.1, we write B 1 as,

B 1 := -∇( u • ∇ u) • ∇ udx = -∇(u • ∇ u) • ∇ u dx + ∇(u • ∇ u) • ∇ udx =0 = -∂ 1 u 1 ∂ 1 u • ∂ 1 udx -∂ 1 u 2 ∂ 2 u • ∂ 1 udx -∂ 2 u 1 ∂ 1 u • ∂ 2 udx -∂ 2 u 2 ∂ 2 u • ∂ 2 udx := B 11 + B 12 + B 13 + B 14 . (4.14)
Further, we divide the first term B 11 into the following two integrals,

B 11 := -∂ 1 u 1 ∂ 1 u • ∂ 1 udx = -∂ 1 u 1 ∂ 1 u 1 ∂ 1 u 1 dx -∂ 1 u 1 ∂ 1 u 2 ∂ 1 u 2 dx := B 111 + B 112 . (4.15)
By the divergence-free condition of u and Lemma 2.4

B 111 := -∂ 1 u 1 ∂ 1 u 1 ∂ 1 u 1 dx = ∂ 2 u 2 ∂ 2 u 2 ∂ 2 u 2 dx ≤ c ∂ 2 u 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ≤ c u H 2 ∂ 2 u 2 H 1 . (4.16)
Due to ∇ • u = 0, integration by parts and Lemma, 2.4

B 112 := -∂ 1 u 1 ∂ 1 u 2 ∂ 1 u 2 dx = ∂ 2 u 2 ∂ 1 u 2 ∂ 1 u 2 dx = 2 u 2 ∂ 2 ∂ 1 u 2 ∂ 1 u 2 dx ≤ c ∂ 2 ∂ 1 u 2 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 u 2 1 2 L 2 ∂ 1 ∂ 1 u 2 1 2 L 2 ≤ c u H 2 u 2 1 2 L 2 ∂ 2 u 3 2 H 1 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . (4.17)
Inserting the upper bound for B 111 and B 112 in (4.15) we get

B 11 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . (4.18)
To deal with B 12 , we write it first as

B 12 := -∂ 1 u 2 ∂ 2 u • ∂ 1 udx = -∂ 1 u 2 ∂ 2 u 1 ∂ 1 u 1 dx -∂ 1 u 2 ∂ 2 u 2 ∂ 1 u 2 dx := B 121 + B 122 . (4.19)
For B 121 , we use the divergence-free condition of u and Lemma 2.4

B 121 := -∂ 1 u 2 ∂ 2 u 1 ∂ 1 u 1 dx = ∂ 1 u 2 ∂ 2 u 1 ∂ 2 u 2 dx ≤ c ∂ 1 u 2 L 2 ∂ 2 u 1 1 2 L 2 ∂ 1 ∂ 2 u 1 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ≤ c u H 2 ∂ 2 u 2 H 1 . (4.20)
The second piece B 122 can be bounded using integrating by parts, Lemma 2.4 and then Young's inequality

B 122 := -∂ 1 u 2 ∂ 2 u 2 ∂ 1 u 2 dx = 2 ∂ 2 ∂ 1 u 2 u 2 ∂ 1 u 2 dx ≤ c ∂ 2 ∂ 1 u 2 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 u 2 1 2 L 2 ∂ 1 ∂ 1 u 2 1 2 L 2 ≤ c u H 2 u 2 1 2 L 2 ∂ 2 u 3 2 H 1 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 (4.21)
Combining (4.20) and (4.21) and inserting them in (4. [START_REF] Larios | Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion[END_REF]) we obtain

B 12 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . (4.22)
The term B 13 is naturally divided into two integrals,

B 13 := -∂ 2 u 1 ∂ 1 u • ∂ 2 udx = -∂ 2 u 1 ∂ 1 u 1 ∂ 2 u 1 dx -∂ 2 u 1 ∂ 1 u 2 ∂ 2 u 2 dx := B 131 + B 132 . (4.23)
Due to ∇ • u = 0 and Lemma 2.4,

B 131 := -∂ 2 u 1 ∂ 1 u 1 ∂ 2 u 1 dx = ∂ 2 u 1 ∂ 2 u 2 ∂ 2 u 1 dx ≤ c ∂ 2 u 1 L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 2 u 2 1 2 L 2 ∂ 2 u 1 1 2 L 2 ∂ 2 ∂ 2 u 1 1 2 L 2 ≤ c u H 2 ∂ 2 u 2 H 1 . (4.24)
Integrating by parts, making use of Lemma 2.4 and then Young's inequality

B 132 := -∂ 2 u 1 ∂ 1 u 2 ∂ 2 u 2 dx = ∂ 1 ∂ 2 u 1 u 2 ∂ 2 u 2 dx + ∂ 2 u 1 u 2 ∂ 1 ∂ 2 u 2 dx ≤ c ∂ 1 ∂ 2 u 1 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 2 u 2 1 2 L 2 + c ∂ 2 u 1 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 1 ∂ 2 u 2 1 2 L 2 ≤ c u H 2 u 2 1 2 L 2 ∂ 2 u 3 2 H 1 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . (4.25)
Inserting the estimates (4.24) and (4.25) in (4.23) we get

B 13 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . (4.26)
The last term B 14 can be bounded directly via Lemma 2.4,

B 14 := -∂ 2 u 2 ∂ 2 u • ∂ 2 udx ≤ c ∂ 2 u 2 L 2 ∂ 2 u 1 2 L 2 ∂ 1 ∂ 2 u 1 2 L 2 ∂ 2 u 1 2 L 2 ∂ 2 ∂ 2 u 1 2 L 2 ≤ c u H 2 ∂ 2 u 2 H 1 . ( 4
B 1 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . (4.28)
The next term B 2 is naturally split into four parts,

B 2 := -∇( u 2 ∂ 2 u) • ∇ udx = -∂ 1 u 2 ∂ 2 u • ∂ 1 udx -∂ 2 u 2 ∂ 2 u • ∂ 2 udx -u 2 ∂ 1 ∂ 2 u • ∂ 1 udx -u 2 ∂ 2 ∂ 2 u • ∂ 2 udx := B 21 + B 22 + B 23 + B 24 . (4.29)
We rewrite B 21 as,

B 21 := -∂ 1 u 2 ∂ 2 u • ∂ 1 udx = -∂ 1 u 2 ∂ 2 u 1 ∂ 1 u 1 dx -∂ 1 u 2 ∂ 2 u 2 ∂ 1 u 2 dx := B 211 + B 212 . (4.30)
Clearly, due to u 2 = 0,

B 212 := -∂ 1 u 2 ∂ 2 u 2 ∂ 1 u 2 dx = 0. (4.31)
By the divergence-free condition of u, integration by parts, Lemma 2.1, Hölder's inequality and then Lemma 2.2

B 211 := -∂ 1 u 2 ∂ 2 u 1 ∂ 1 u 1 dx = ∂ 1 u 2 ∂ 2 u 1 ∂ 2 u 2 dx = u 2 ∂ 2 u 1 ∂ 1 ∂ 2 u 2 dx = R ∂ 2 u 1 T u 2 ∂ 1 ∂ 2 u 2 dx 1 dx 2 ≤ c ∂ 2 u 1 L ∞ x 2 u 2 L 2 ∂ 1 ∂ 2 u 2 L 2 ≤ c ∂ 2 u 1 H 1 u 2 L 2 ∂ 1 ∂ 2 u 2 L 2 ≤ c u H 2 u 2 L 2 ∂ 2 u H 1 ≤ c θ H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . ( 4
B 22 := -∂ 2 u 2 ∂ 2 u • ∂ 2 udx ≤ c ∂ 2 u L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 2 u 2 1 2 L 2 ∂ 2 u 1 2 L 2 ∂ 2 ∂ 2 u 1 2 L 2 ≤ c u H 2 ∂ 2 u 2 H 1 . (4.34)
By definition of u,

B 23 := -u 2 ∂ 1 ∂ 2 u • ∂ 1 udx = 0. (4.35)
Due Lemma 2.4 and Young's inequality,

B 24 := -u 2 ∂ 2 ∂ 2 u • ∂ 2 udx ≤ c ∂ 2 ∂ 2 u L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 u 1 2 L 2 ∂ 1 ∂ 2 u 1 2 L 2 ≤ c u H 2 u 2 1 2 L 2 ∂ 2 u 3 2 H 1 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . ( 4
B 2 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 . (4.37)
To bound B 3 , we first write u = u + u and use Lemma 2.1

B 3 := -∇( u • ∇ θ) • ∇ θdx = -∇(u • ∇ θ) • ∇ θdx + ∇(u • ∇ θ) • ∇ θdx =0 = -∂ 1 θ∂ 1 u 1 ∂ 1 θdx -∂ 2 θ∂ 1 u 2 ∂ 1 θdx -∂ 1 θ∂ 2 u 1 ∂ 2 θdx -∂ 2 θ∂ 2 u 2 ∂ 2 θdx := B 31 + B 32 + B 33 + B 34 . (4.38)
All terms in (4.38) can be bounded suitably. In fact, by Lemma 2.4,

B 31 := -∂ 1 θ∂ 1 u 1 ∂ 1 θdx ≤ c ∂ 1 u 1 L 2 ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ∂ 2 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ∂ 1 θ 2 H 1 . (4.39)
For B 32 , B 33 and B 34 we use Lemmas 2.4 and 2.5,

B 32 := -∂ 2 θ∂ 1 u 2 ∂ 1 θdx ≤ c ∂ 1 u 2 L 2 ∂ 2 θ 1 2 L 2 ≤ ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ∂ 2 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ∂ 1 θ 2 H 1 , (4.40)
B 33 := -∂ 1 θ∂ 2 u 1 ∂ 2 θdx ≤ c ∂ 2 u 1 L 2 ∂ 2 θ 1 2 L 2 ≤ ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ∂ 2 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ∂ 1 θ 2 H 1 , (4.41)
and

B 34 := -∂ 2 θ∂ 2 u 2 ∂ 2 θdx ≤ c ∂ 2 θ L 2 ∂ 2 θ 1 2 L 2 ≤ ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ≤ c θ H 2 ∂ 2 u H 1 ∂ 1 θ H 1 ≤ c θ H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . ( 4
B 3 ≤ c (u, θ) H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . (4.43)
To deal with B 4 , we split it into four pieces,

B 4 := -∇( u 2 ∂ 2 θ) • ∇ θdx = -∂ 1 ( u 2 ∂ 2 θ) • ∂ 1 θdx -∂ 2 ( u 2 ∂ 2 θ) • ∂ 2 θdx = -∂ 1 u 2 ∂ 2 θ∂ 1 θdx -u 2 ∂ 1 ∂ 2 θ∂ 1 θdx -∂ 2 u 2 ∂ 2 θ∂ 2 θdx -u 2 ∂ 2 ∂ 2 θ∂ 2 θdx := B 41 + B 42 + B 43 + B 44 . (4.44)
The terms above can be bounded as follows. Due to the definition of the horizontal average θ,

B 42 := -u 2 ∂ 1 ∂ 2 θ∂ 1 θdx = 0. ( 4

.45)

For B 41 , we use integration by parts, Lemma 2.1, Hölder's inequality and then Lemma 2.2

B 41 := -∂ 1 u 2 ∂ 2 θ∂ 1 θdx = -u 2 ∂ 2 θ∂ 1 ∂ 1 θdx = R ∂ 2 θ T u 2 ∂ 1 ∂ 1 θdx 1 dx 2 ≤ c ∂ 2 θ L ∞ x 2 u 2 L 2 ∂ 1 ∂ 1 θ L 2 ≤ c ∂ 2 θ H 1 u 2 L 2 ∂ 1 ∂ 1 θ L 2 ≤ c θ H 2 u 2 L 2 ∂ 1 θ H 1 ≤ c θ H 2 u 2 2 L 2 + ∂ 1 θ 2 H 1 . (4.46)
The other two terms B 43 , B 44 can be bounded via Lemmas 2.4 and 2.5,

B 43 := -∂ 2 u 2 ∂ 2 θ∂ 2 θdx ≤ c ∂ 2 θ L 2 ∂ 2 θ 1 2 L 2 ≤ ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 2 u 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ≤ c θ H 2 ∂ 1 θ H 1 ∂ 2 u H 1 ≤ c θ H 2 ∂ 1 θ 2 H 1 + ∂ 2 u 2 H 1 , (4.47)
B 44 := -u 2 ∂ 2 ∂ 2 θ∂ 2 θdx ≤ c ∂ 2 ∂ 2 θ L 2 ∂ 2 θ 1 2 L 2 ≤ ∂ 1 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ≤ c θ H 2 u 2 1 2 L 2 ∂ 2 u 1 2 H 1 ∂ 1 θ H 1 ≤ c θ H 2 ∂ 2 u 2 H 1 + u 2 2 L 2 + ∂ 1 θ 2 H 1 . (4.48)
Inserting all the bounds obtained above for B 41 through B 44 in (4.44) leads to

B 4 ≤ c (u, θ) H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 + u 2 2 L 2 . ( 4
d dt ∇ u(t) 2 L 2 + ∇ θ(t) 2 L 2 + ν ∂ 2 ∇ u 2 L 2 + η ∂ 1 ∇ θ 2 L 2 ≤ c (u, θ) H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 + u 2 2 L 2 . ( 4
- d dt δ( u 2 , θ) = -δ(∂ t u 2 , θ) -δ( u 2 , ∂ t θ),
with δ > 0 is a small constant to be fixed at the end of the proof. Doing so, we generate an extra regularization term that helps bound u 2 L 2 . Note that, this stabilizing term comes from the interaction between u and θ. Due to Hölder's inequality, we have, for sufficiently small δ > 0, ( u, θ) 2 H 1δ( u 2 , θ) ≥ 0. Using the first equation of (4.2) and u 2 = 0, we write

∂ t u 2 + u • ∇ u 2 + u 2 ∂ 2 u 2 =0 -ν∂ 2 2 u 2 + ∂ 2 p = g 0 θ. ( 4

.51)

Applying ∇• to the first equation of (4.2), we obtain

∇ • ( u • ∇ u) + ∇ • ( u 2 ∂ 2 u) + ∆ p = g 0 ∂ 2 θ. (4.52)
Making use of (4.52), we have

p = -∆ -1 ∇ • ( u • ∇ u) -∆ -1 ∇ • ( u 2 ∂ 2 u) + g 0 ∆ -1 ∂ 2 θ.
Then,

∂ 2 p = -∂ 2 ∆ -1 ∇ • ( u • ∇ u) -∂ 2 ∆ -1 ∇ • ( u 2 ∂ 2 u) + g 0 ∂ 2 ∂ 2 ∆ -1 θ. (4.53)
By (4.51) and the second equation of (4.2), we write

-δ d dt ( u 2 , θ) = -δ(∂ t u 2 , θ) -δ( u 2 , ∂ t θ) = -δ(g 0 θ -∂ 2 p + ν∂ 2 2 u 2 -u • ∇ u 2 , θ) -δ( u 2 , -g 0 u 2 + η∂ 2 1 θ -u 2 ∂ 2 θ -u • ∇ θ) = -g 0 δ θ 2 L 2 + ∂ 2 p θdx -δν ∂ 2 2 u 2 θdx + δ u • ∇ u 2 θdx + g 0 δ u 2 2 L 2 -δη ∂ 2 1 θ u 2 dx + δ u 2 u 2 ∂ 2 θdx + δ u • ∇ θ u 2 dx := N 1 + • • • + N 8 .
(4.54)

The terms N 1 through N 8 obey the following bounds. For N 2 , we use (4.53) to rewrite it as,

N 2 := δ ∂ 2 p θdx = -δ ∂ 2 ∆ -1 ∇ • ( u • ∇ u) • θdx -δ ∂ 2 ∆ -1 ∇ • ( u 2 ∂ 2 u) • θdx + g 0 δ ∂ 2 ∂ 2 ∆ -1 θ • θdx := N 21 + N 22 + N 23 . (4.55)
By Lemma 2.1 and integration by parts we split N 21 into three pieces

N 21 := -δ ∂ 2 ∆ -1 ∇ • ( u • ∇ u) • θdx = -δ ∂ 2 ∆ -1 ∇ • (u • ∇ u) • θdx + δ ∂ 2 ∆ -1 ∇ • (u • ∇ u) • θdx =0 = -δ ∂ 2 ∆ -1 ∂ 1 (u 1 ∂ 1 u) • θdx -δ ∂ 2 ∆ -1 ∂ 2 (u 2 ∂ 2 u) • θdx = -δ (u 1 ∂ 1 u) • ∂ 2 ∆ -1 ∂ 1 θdx -δ (u 2 ∂ 2 u) • ∂ 2 ∆ -1 ∂ 2 θdx (4.56) = -δ ∂ 1 u 1 u • ∂ 2 ∆ -1 ∂ 1 θdx -δ u 1 u • ∂ 1 ∂ 2 ∆ -1 ∂ 1 θdx -δ (u 2 ∂ 2 u) • ∂ 2 ∆ -1 ∂ 2 θdx = N 211 + N 212 + N 213 . ( 4

.57)

Due to ∇ • u = 0, Lemma 2.4 and the boundedness of the Riesz transform,

N 211 = -δ ∂ 1 u 1 u • ∂ 2 ∆ -1 ∂ 1 θdx = δ ∂ 2 u 2 u • ∂ 2 ∆ -1 ∂ 1 θdx ≤ c u L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∆ -1 ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 2 ∆ -1 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ∂ 2 u H 1 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ∂ 2 u H 1 ∂ 1 θ H 1 ≤ c u H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . (4.58)
According to Lemma 2.4, the boundedness of the Riesz transform, Lemma 2.5 and ∇ • u = 0,

N 212 = -δ u 1 u • ∂ 1 ∂ 2 ∆ -1 ∂ 1 θdx ≤ c u 1 L 2 u 1 2 L 2 ∂ 2 u 1 2 L 2 ∂ 1 ∂ 2 ∆ -1 ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 ∂ 2 ∆ -1 ∂ 1 θ 1 2 L 2 ≤ c u H 2 u 1 2 L 2 ∂ 2 u 1 2 L 2 ∂ 1 θ 1 2 L 2 ∂ 1 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ( u 1 L 2 + u 2 L 2 ) 1 2 ∂ 2 u 1 2 H 1 ∂ 1 θ H 1 ≤ c u H 2 u 1 2 L 2 ≤ ∂ 1 u 1 2 L 2 = ∂ 2 u 2 2 L 2 + u 2 2 L 2 + ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . (4.59)
Applying Lemma 2.4, the boundedness of the Riesz transform and then Lemma 2.5,

N 213 = -δ (u 2 ∂ 2 u) • ∂ 2 ∆ -1 ∂ 2 θdx ≤ c u 2 L 2 ∂ 2 u 1 2 L 2 ∂ 2 ∂ 2 u 1 2 L 2 ∂ 2 ∆ -1 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 ∆ -1 ∂ 2 θ 1 2 L 2 ≤ c u H 2 ∂ 2 u H 1 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ∂ 2 u H 1 ∂ 1 θ H 1 ≤ c u H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . (4.60)
The bounds in (4.58), (4.59) and (4.60) lead to

N 21 ≤ c u H 2 u 2 2 L 2 + ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . (4.61)
Now we turn to the next term N 22 . Using Hölder's inequality, the boundedness of the Riesz transform and Lemmas 2.1, 2.2 and 2.5

N 22 := -δ ∂ 2 ∆ -1 ∇ • ( u 2 ∂ 2 u) • θdx ≤ cδ ∂ 2 ∆ -1 ∇ • ( u 2 ∂ 2 u) L 2 θ L 2 ≤ cδ u 2 ∂ 2 u L 2 θ L 2 ≤ cδ ∂ 2 u L ∞ x 2 u 2 L 2 θ L 2 ≤ cδ ∂ 2 u H 1 u 2 L 2 θ L 2 ≤ cδ u H 2 u 2 L 2 ∂ 1 θ L 2 ≤ cδ u H 2 u 2 2 L 2 + ∂ 1 θ 2 H 1 . (4.62)
To deal with N 23 , we integrate by parts, use Plancherel's theorem and then Lemma 2.5,

N 23 := g 0 δ ∂ 2 ∂ 2 ∆ -1 θ • θdx = g 0 δ ∂ 2 ∆ -1 2 θ • ∂ 2 ∆ -1 2 θdx = g 0 δ ∂ 2 Λ -1 θ 2 L 2 = g 0 δ k∈Z k =0 R ξ 2 2 k 2 + ξ 2 2 | θ(k, ξ 2 )| 2 dξ 2 ≤ cδ k∈Z k =0 R ξ 2 2 | θ(k, ξ 2 )| 2 dξ 2 = cδ ∂ 2 θ 2 L 2 ≤ cδ ∂ 1 ∂ 2 θ 2 L 2 ≤ cδ ∂ 1 θ 2 H 1 , (4.63)
where we denote Λ = (-∆)

1 2 and we have used the fact that the oscillation part θ(0, ξ 2 ) has the horizontal mode equal to 0, namely θ(0, ξ 2 ) = 0. Collecting (4.57), (4.62), (4.63) and (4.55), we find

N 2 ≤ cδ (u, θ) H 2 ∂ 2 u 2 H 1 + u 2 2 L 2 + ∂ 1 θ 2 H 1 + cδ ∂ 1 θ 2 H 1 . (4.64)
To deal with N 3 we use ∇ • u = 0, integration by parts, Hölder's inequality and Lemma 2.5,

N 3 := -δν ∂ 2 2 u 2 θdx = δν ∂ 2 ∂ 1 u 1 θdx = -δν u 1 ∂ 2 ∂ 1 θdx ≤ δν u 1 L 2 ∂ 2 ∂ 1 θ L 2 ≤ cδ u 1 2 L 2 ≤ ∂ 1 u 1 2 L 2 = ∂ 2 u 2 2 L 2 + ∂ 2 ∂ 1 θ 2 L 2 ≤ cδ ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . (4.65)
To estimate N 4 , we make use of Lemma 2.1 and integration by parts, to write it as

N 41 := -δ ∂ 1 u u 2 θdx ≤ c ∂ 1 u L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ≤ c u H 2 u 2 1 2 L 2 ∂ 2 u 1 2 L 2 ∂ 1 θ H 1 ≤ c u H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 + u 2 2 L 2 . (4.67)
Similarly, Clearly, the term N 5 can be bounded via Lemma 2.5,

N 42 = δ u∂ 2 u 2 θdx ≤ c u L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 ∂ 2 u 2 1 2 L 2 θ 1 2 L 2 ∂ 1 θ 1 2 L 2 ≤ c u H 2 ∂ 2 u H 1 ∂ 1 θ H 1 ≤ c u H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 . ( 4
N 5 := -g 0 δ θ 2 L 2 ≤ cδ ∂ 1 θ 2 L 2 ≤ cδ ∂ 1 θ 2 H 1 .
(4.70) Applying Hölder's inequality and Young's inequality,

N 6 := -δη ∂ 2 1 θ u 2 dx ≤ cδ ∂ 2 1 θ L 2 u 2 L 2 ≤ cδ ∂ 1 θ H 1 u 2 L 2 ≤ cδ ∂ 1 θ 2 H 1 -g 0 δ 4 u 2 2 L 2 . (4.71)
Using integration by parts and Lemma 2.4, we obtain

N 7 := δ u 2 u 2 ∂ 2 θdx = 2δ ∂ 2 u 2 u 2 θdx ≤ cδ ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 2 u 2 1 2 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 θ L 2 ≤ cδ ∂ 2 u 3 2 L 2 u 2 1 2 L 2 θ H 2 ≤ cδ θ H 2 ( ∂ 2 u 2 H 1 + u 2 2 L 2 ). ( 4
L 2 ∂ 1 u 1 1 2 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 1 θ L 2 ≤ cδ ∂ 2 u 3 2 L 2 u 2 1 2 L 2 ∂ 1 θ L 2 ≤ cδ θ H 2 ( ∂ 2 u 2 H 1 + u 2 2 L 2 ). ( 4
N 82 := δ u 1 ∂ 1 θ u 2 dx ≤ δ u 1 L ∞ x 2 ∂ 1 θ u 2 L 1 ≤ cδ u 1 L ∞ x 2 ∂ 1 θ L 2 u 2 L 2 ≤ cδ u H 1 ∂ 1 θ L 2 u 2 L 2 ≤ cδ u H 2 ∂ 1 θ 2 H 1 + u 2 2 L 2 . (4.75)
Due to u 2 = 0, integration by parts, Lemma 2.4 and Young's inequality

N 83 := δ u 2 ∂ 2 θ u 2 dx = δ u 2 ∂ 2 θ u 2 dx = 2δ ∂ 2 u 2 θ u 2 dx ≤ cδ ∂ 2 u 2 1 2 L 2 ∂ 1 ∂ 2 u 2 1 2 L 2 u 2 1 2 L 2 ∂ 2 u 2 1 2 L 2 ∂ 2 θ L 2 ≤ cδ ∂ 2 u 3 2
H 1 u 2

+ 2η ∂ 1 θ 2 H 1 ≤ c (u, θ) H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 + u 2 2 L 2 + g 0 3δ 4 u 2 2 L 2 + cδ (u, θ) H 2 ∂ 2 u 2 H 1 + u 2 2 L 2 + cδ ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 .
Using Theorem 1.1, if ε > 0 is sufficiently small and u 0 L 2 + θ 0 L 2 ≤ ε, then (u(t), θ(t)) H 2 ≤ cε and so,

d dt u 2 H 1 + θ 2 H 1 -δ( u 2 , θ) + 2ν ∂ 2 u 2 H 1 + 2η ∂ 1 θ 2 H 1 ≤ cǫ ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 + u 2 2 L 2 + g 0 3δ 4 u 2 2 L 2 + cδǫ ∂ 2 u 2 H 1 + u 2 2 L 2 + cδ ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 .
Choosing ǫ > 0 such that cǫ ≤ -g 0 min(

  .46) Thus, by (3.16), (3.27), (3.31), (3.46), and (3.12),

  .77) It then follows from (3.69), (3.70), (3.71), (3.72) and (3.77) that

  .83) Then (3.82), (3.83) and (3.81) together leads to

  .89) Inserting (3.80), (3.84), (3.88), (3.89), in (3.79) we obtain

  .90) Collecting the bounds obtained above for K 1 through K 3 in (3.64), (3.65), (3.78) and (3.90) and inserting them in (3.63), we get

  .100) Putting the estimates (3.99) and (3.100) in (3.98) leads to 1 2

N 4 :

 4 = δ u • ∇ u 2 θdx = δ u • ∇ u 2 θdxδ u • ∇ u 2 θdx =0 = δ u∂ 1 u 2 θdx + δ u∂ 2 u 2 θdx = -δ ∂ 1 u u 2 θdx + δ u∂ 2 u 2 θdx = N 41 + N 42 .(4.66) By Lemmas 2.4 and 2.5

≤ cδ u 1 1 2

 2 .72) It remain to bound the last term N 8 . Making use of Lemma 2.1, we divide it into three partsN 8 := δ u • ∇ θ u 2 dx = δ u • ∇ θ u 2 dxδ u • ∇ θ u 2 dx =0 = δ u 1 ∂ 1 θ u 2 dx + δ u 1 ∂ 1 θ u 2 dx + δ u 2 ∂ 2 θ u 2 dx := N 81 + N 82 + N 83 .(4.73)Due to Lemmas 2.4, 2.5 and divergence-free condition of u, we haveN 81 := δ u 1 ∂ 1 θ u 2 dx

  .68) Inserting (4.67) and (4.68) in (4.66) we findN 4 ≤ cδ u H 2 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 + u 2

	2 L 2 .	(4.69)

  .74) By Lemma 2.1, Hölder's inequality and Lemma 2.2,

  L 2 θ H 2 ≤ cδ θ H 2 ∂ 2 u 2 H 1 + u 2 L 2 + cδ (u, θ) H 2 ∂ 2 u 2 H 1 + u 2 + θ 2 H 1δ( u 2 , θ) + 2ν ∂ 2 u 2 H 1

						1
						2
						2 L 2 .	(4.76)
	Inserting (4.74), (4.75) and (4.76) in (4.73) leads to
		N 8 ≤ cδ (u, θ) H 2 ∂ 2 u 2 H 1 + u 2	2 L 2 + ∂ 1 θ 2 H 1 .	(4.77)
	Considering (4.54) and collecting (4.64), (4.65), (4.66), (4.70), (4.71), (4.72) and
	(4.77), we obtain
	-δ	d dt	( u 2 , θ) ≤ g 0 δ u 2	2	2 L 2
			-g 0	δ 4	u 2	2 L 2 + cδ ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 .	(4.78)
	It then follows from (4.11), (4.50) and (4.78) that
	d dt	u 2 H 1

  L 2 + cδ ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 .Choosing δ > 0 such that cδ ≤ min(ν, η, c 2 ), we getd dt u 2 H 1 + θ 2 H 1δ( u 2 , θ) + ν ∂ 2 u 2 H 1 + η ∂ 1 θ 2 H 1g 0Combining with the time integral bounds from Theorem 1.1,

									δ 4	u 2	2 L 2 ≤ 0. (4.79)
	Due to the above choice of δ, we obtain
				1 2	u 2 H 1 + θ 2 H 1 -δ( u 2 , θ) ≥ 0.
	or
	1 2	( u 2 H 1 + θ 2 H 1 ) ≤ u 2 H 1 + θ 2 H 1 -δ( u 2 , θ) ≤	3 2	( u 2 H 1 + θ 2 H 1 ).
	For any 0 ≤ s ≤ t, integrating (4.79) in time leads to
	1 2	( u(t) 2 H 1 + θ(t) 2 H 1 ) +	s	t	(ν ∂ 2 u 2 H 1 + η ∂ 1 θ 2 H 1 -g 0	δ 4	u 2	2 L 2 ) dτ
	≤	3 2	( u(s) 2 H 1 + θ(s) 2 H 1 ).
									H 1 )	(4.80)
	and
	∞ L ∞ 0 (ν ∂ 2 u 2 H 1 + η ∂ 1 θ 2 H 1 -g 0 δ 4 2 u 2 ∞	∞
	0		∂ 2 u 2 H 2 dt < ∞,	0		∂ 1 u 2	2 L 2 dt < ∞ and	0	∂ 1 θ 2 H 2 dt < ∞,
	we get
				∞
	1 4 , δ H 1 + θ(t) 2 ( u(t) 2 H 1 ) dt < ∞. 4 ), we obtain Finally, applying Lemma 2.6 to (4.80) and (4.81) leads to 0 d dt u 2 H 1 + θ 2 H 1 -δ( u 2 , θ) + 2ν ∂ 2 u 2 H 1 + 2η ∂ 1 θ 2 H 1 ≤ δ 4 ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1 -g 0 δ 4 u 2 u(t) 2 H 1 + θ(t) 2 H 1 ≤ c(1 + t) -1 , 2 L 2 + g 0 3δ 4 u 2 2 L 2 -g 0 and the asymptotic behavior, as t → ∞, δ 4 ∂ 2 u 2 H 1 + u 2 2 L 2 t ( u(t) 2 H 1 + θ(t) 2 H 1 ) → 0.	(4.81)
									+ cδ ∂ 2 u 2 H 1 + ∂ 1 θ 2 H 1
				≤ g 0	δ 4	u 2	2

Then, for any 0 ≤ s ≤ t, we have

u(t) 2 H 1 + θ(t) 2 H 1 ≤ 3( u(s) 2 H 1 + θ(s) 2 2 ) dτ ≤ C < ∞.

This completes the proof of Theorem 1.2.

Mathematics Subject Classification. 35Q35, 35Q86, 76D03, 76D50. Key words and phrases. Boussinesq equations; Hydrostatic balance; Partial dissipation; Stability.1

Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA Email address: obensai@ostatemail.okstate.edu

Laboratoire Analyse, Géométrie et Applications, Université Paris 13, 99 Avenue Jean Baptiste Clément 93430 Villetaneuse, FranceEmail address: bensaid@univ-paris13.fr