On stability of Boussinesq equations without thermal conduction

Abstract

We investigate the stability of a specific stationary solution to Boussinesq equations without thermal conduction in the flat strip $\Omega = \mathbb {T}\times (0,1)$. Explicit decay rates of the vorticity/velocity are given as well as the limit state of the temperature. Our method is based on time-weighted energy estimates.

1 Introduction

1.1 Presentation of the problem

In the mathematical study of fluid dynamics, the Boussinesq approximation is one of important models among various simplified ones to the Navier–Stokes–Fourier system. In such an approximation, both temperature and density of the flow are assumed to vary small so that the variation of temperature is (inversely) proportional to that of density. Hence, the fluid is assumed to be divergence free and only the action of gravity is considered. In a general 3D setting, the Boussinesq equations read as

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\mathbf {v}+\mathbf {v}\cdot \nabla \mathbf {v}-\nu \Delta \mathbf {v} +\nabla p = g \vartheta \mathbf {e}_{3},\\&\nabla \cdot \mathbf {v} = 0, \\&\partial _{t}\vartheta +\mathbf {v}\cdot \nabla \vartheta - \kappa \Delta \vartheta = 0. \end{aligned} \right. \end{aligned}$$

(1.1)

Here $\mathbf {v},p$ and $\vartheta$ are the velocity, pressure and temperature of the fluid, respectively, while $\nu$ is the viscosity, $\kappa$ is the thermal conductivity, g is the constant of gravity and $\mathbf {e}_3$ is the inverse direction of the gravity. Note that the term $g\vartheta \mathbf {e}_3$ on the right-hand side of momentum Eq. (1.1)$_1$ is nothing but the effect of gravity/buoyancy. In case of $\nu ,\kappa >0$, the Boussinesq equations is an elliptic–parabolic coupled nonlinear PDE system which plays a key role in the study of hydrodynamic instability problems, especially the Rayleigh-Bénard convection, see [3, 10] among others. When the thermal conductivity is neglected, the temperature variation $\vartheta$ can be considered as the inverse of density variation. This explains the physical significance of the transport equation in (1.1), which now becomes an elliptic–parabolic–hyperbolic coupled system.

The present paper is concerned with Boussinesq equations

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\mathbf {v} +\mathbf {v}\cdot \nabla \mathbf {v}-\Delta \mathbf {v} +\nabla p = \vartheta \mathbf {e}_{2},\\&\nabla \cdot \mathbf {v} = 0,\\&\partial _{t}\vartheta +\mathbf {v}\cdot \nabla \vartheta = 0 \end{aligned} \right. \end{aligned}$$

(1.2)

in the two-dimensional domain $\Omega =\mathbb {T}\times (0,1)\subset \mathbb {R}^{2}$. Here the vector field $\mathbf {v}=(v_{1}(\mathbf {x},t), v_{2}(\mathbf {x},t))$ is the velocity describing the motion of viscous fluid under the action of buoyancy force, while the temperature $\vartheta$ is transported by the fluid motion. The fluid viscosity $\nu$ and the constant of gravity g are assumed to be 1, which are irrelevant in the following analysis. We use $\mathbf {x}$ to denote the space variable $(x,y) \in \mathbb {R}^{2}$ and $\mathbf {e}_{2} :=\nabla {y} = (0,1)$ the direction of buoyancy force.

We supplement system (1.2) with the following initial and (slip) boundary conditions.

$$\begin{aligned} \left\{ \begin{aligned}&\mathbf {v}(\mathbf {x},0) = \mathbf {v}_{0}(\mathbf {x}),\, \vartheta (\mathbf {x},0)=\vartheta _{0}(\mathbf {x})\text { in }\Omega ,\\&\left( \mathbf {v}\cdot \mathbf {n}\right) (\mathbf {x},t) = 0,\,\left( \mathbb {S}(\nabla \mathbf {v})\mathbf {n}\cdot \mathbf {\tau }\right) (\mathbf {x},t) =0 \text { on }\partial \Omega ,\,t>0, \end{aligned}\right. \end{aligned}$$

(1.3)

where $\mathbf {n}$ ($\tau$) is the outward unit normal (unit tangential direction) to $\partial \Omega =\mathbb {T}\times \{y=0,1\}$ and the stress tensor $\mathbb {S}(\nabla \mathbf {v})$ is the symmetric part of $\nabla \mathbf {v}$. Since the boundary $\partial \Omega$ is flat, (1.3)$_2$ is equivalent to

$$\begin{aligned} v_{2}(\mathbf {x},t)=0,\,\partial _{2}v_{1}(\mathbf {x},t)= 0\text { on }\partial \Omega ,\,t>0. \end{aligned}$$

(1.4)

By setting $\mathbf {v}=0$ in (1.2), we immediately obtain that

$$\begin{aligned} \mathbf {v}_s =0, \, p_s = p_s(y), \,\vartheta _s = p'_s(y),\, y\in [0,1] \end{aligned}$$

(1.5)

is a stationary solution (hydrostatic equilibrium) to (1.2), where $p_s(\cdot )$ is an arbitrary smooth function. It is well known that when $\vartheta '_s(y_0)<0$ for some $y_0\in [0,1]$, such a stationary solution is unstable—the Rayleigh–Taylor instability happens. In the present work, we are interested in the opposite case $\vartheta '_s(y)>0$ for all $y\in [0,1]$, which implies that fluid with lower temperature (higher density) lies below the fluid with higher temperature (lower density). Specifically we choose $\vartheta _s(y)=y$ and succeed to show that this steady solution is stable in the sense of Lyapunov, meaning that the solution to (1.2)-(1.3) starting from initial data close to this stationary solution is close to it for all time $t>0$ in a suitable sense. Moreover, we obtain that the velocity $\mathbf {v}$ and $\partial _1\vartheta$ converge to zero in $H^2$ with explicit decay rates $(1+t)^{-1}$ as $t\rightarrow \infty$, see Theorem 1.1.

1.2 Related results on the Boussinesq equations

Boussinesq equations have rich physical background and mathematical features. In particular, the two-dimensional model keeps some key features of the 3D Navier–Stokes/Euler equations, see [20, 23]. Over the past years, there have been many works devoted to Boussinesq equations. For the reason of brevity, we only review some related results on the two-dimensional Boussinesq system (1.2).

First result on global well-posedness of Cauchy problem to (1.2) has been obtained in [6] and [15] with arbitrary large initial data. Since then, other kinds of initial (boundary) value problems to (1.2) have been investigated by many authors under different settings, see [1, 7, 13, 14, 16, 19], among others. It is then natural to study the large-time behavior of solutions to (1.2). In [17], N. Ju has obtained the global-in-time uniform boundedness of $\mathbf {v}$ in $H^{2}$ and the exponential growth $e^{c t^2}$ of $\Vert \nabla \vartheta (t)\Vert _{L^{2}}$. It is then improved by I. Kukavica and W. Wang in [18] that the growth of $\Vert \nabla \vartheta (t)\Vert _{L^{2}}$ is at most $e^{ct}$. In [8], for general initial data it is shown that $\mathbf {v}$ converges in $H^{1}$ to zero and $\Vert \nabla ^{2}\mathbf {v}(t)\Vert _{L^2}$ is uniformly bounded. Very recently, by using spectral analysis, the authors in [24] have investigated the global asymptotic stability of the specific hydrostatic equilibrium (1.6). Besides analysis for the linearized system, they also give explicit decay rate $(1+t)^{-1/2}$ of $\Vert \mathbf {v}(t)\Vert _{L^{2}}$ under certain assumptions on the solution $(\mathbf {v},\vartheta )$ to (1.2) on periodic domain $\mathbb {T}^2$. In the most relevant work [26], R. Wan has obtained asymptotic behavior and explicit decay rates for solutions to the perturbed system (1.7) in $\mathbb {R}^2$ by using spectral analysis. Finally we refer the reader to [4, 5, 25] for works on the global existence and stability of the 2D Boussinesq equations with a velocity damping term (without viscosity) near the hydrostatic equilibrium (1.6).

1.3 Main result

Choosing $\vartheta _{s}(y) = y$, we infer from (1.5) that

$$\begin{aligned} \mathbf {v}_s =0, \, p_s = \frac{1}{2}y^{2}, \,\vartheta _s = y,\, y\in [0,1]. \end{aligned}$$

(1.6)

Perturbing this specific stationary solution, we have

$$\begin{aligned} \mathbf {v}=\mathbf {u},\,p=q +\frac{1}{2}y^{2},\,\vartheta = y +\theta . \end{aligned}$$

Then $(\mathbf {u},q,\theta )$ satisfies the perturbed equations as follows.

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\mathbf {u}+\mathbf {u}\cdot \nabla \mathbf {u}-\Delta \mathbf {u} +\nabla q = \theta \mathbf {e}_{2}, \\&\partial _{t}\theta + \mathbf {u}\cdot \nabla \theta = -u_{2},\\&\nabla \cdot \mathbf {u} = 0 \end{aligned} \right. \end{aligned}$$

(1.7)

with initial and boundary conditions

$$\begin{aligned} \left\{ \begin{aligned}&(\mathbf {u}, \theta )(\mathbf {x},0) = (\mathbf {u}_{0}, \theta _{0})(\mathbf {x})\text { in } \Omega ,\\&u_{2}(\mathbf {x},t) = 0,~\partial _{2} u_{1}(\mathbf {x},t) = 0\text { on }\partial \Omega ,\,t>0. \end{aligned} \right. \end{aligned}$$

(1.8)

Since $\nabla \cdot \mathbf {u} = 0$, there exists a stream function $\psi$ such that $\mathbf {u} =\nabla ^{\perp }\psi = (\partial _{2}\psi , -\partial _{1}\psi )$. Here we choose $\psi$ satisfying

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta \psi = \omega , \\&\psi |_{\partial \Omega } = 0, \end{aligned} \right. \end{aligned}$$

(1.9)

where $\omega =\partial _{1}u_{2} - \partial _{2}u_{1}$. By denoting $\psi = (-\Delta )^{-1}\omega$,

$$\begin{aligned} \mathbf {u} =\nabla ^{\perp }(-\Delta )^{-1}\omega . \end{aligned}$$

Hence, the equations for $(\omega , \theta )$ are written as

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\omega +\mathbf {u}\cdot \nabla \omega -\Delta \omega = \partial _{1}\theta , \\&\partial _{t}\theta +\mathbf {u}\cdot \nabla \theta = -u_{2},\\&\mathbf {u} = \nabla ^{\bot }(-\Delta )^{-1}\omega , \\ \end{aligned} \right. \end{aligned}$$

(1.10)

together with initial and boundary conditions

$$\begin{aligned} \left\{ \begin{aligned}&(\omega , \theta )(\mathbf {x},0) = (\omega _{0}, \theta _{0})(\mathbf {x})\text { in }\Omega ,\\&\omega (\mathbf {x},t)=0\text { on }\partial \Omega ,\,t>0. \end{aligned} \right. \end{aligned}$$

(1.11)

In the following we focus on the analysis of (1.10)-(1.11) instead of considering (1.2)-(1.3) directly. Assume that

$$\begin{aligned}&\omega _{0} \in H^{m}(\Omega )~\text {and}~\partial _{2}^{n}\omega _{0}= 0 \text { on }\partial \Omega ,~\text {for}~n= 0,2,\cdots ,2[(m-1)/2], \end{aligned}$$

(1.12)

$$\begin{aligned}&\theta _{0} \in H^{m+1}(\Omega )~\text {and}~\partial _{2}^{n}\theta _{0}= 0 \text { on }\partial \Omega ,~\text {for}~n= 0,2,\cdots ,2[m/2], \end{aligned}$$

(1.13)

where $[m/2]=k$ if $m=2k$ or $2k+1$, $k=0,1,2,\cdots$. We remark that assumptions (1.12) and (1.13) on initial data are motivated by [5]. According to (1.10)$_{2}$ and $u_{2}(t)|_{\partial \Omega }=0$, we get the transport equation

$$\begin{aligned} \partial _{t}\theta (t)|_{\partial \Omega } + u_{1}(t)\partial _{1}\theta (t)|_{\partial \Omega } = 0. \end{aligned}$$

Then $\theta _{0}|_{\partial \Omega }=0$ implies $\theta (t)|_{\partial \Omega }= 0$. Taking $\partial _{2}$ on $\nabla \cdot \mathbf {u} = 0$ and using $\partial _{2}u_{1}(t)|_{\partial \Omega }=0$ yield

$$\begin{aligned} \partial _{2}^{2}u_{2}(t)|_{\partial \Omega } = -\partial _{2}\partial _{1}u_{1}(t)|_{\partial \Omega } = 0. \end{aligned}$$

Now operating $\partial _{2}^{2}$ on (1.10)$_{2}$ and restricting to the boundary, one gets

$$\begin{aligned} \partial _{t}\partial _{2}^{2}\theta (t)|_{\partial \Omega } + u_{1}(t)\partial _{1}\partial _{2}^{2}\theta (t)|_{\partial \Omega } + 2\partial _{2}u_{2}(t)\partial _{2}^{2}\theta (t)|_{\partial \Omega } = 0. \end{aligned}$$

This implies that $\partial _{2}^{2}\theta _{0}|_{\partial \Omega } = 0$ is preserved in time. Furthermore, from the evolution Eq. (1.10)$_{1}$, it follows that

$$\begin{aligned} -\Delta \omega (t)|_{\partial \Omega } = \partial _{1}\theta (t)|_{\partial \Omega } - \partial _{t}\omega (t)|_{\partial \Omega } - u_{1}(t)\partial _{1}\omega (t)|_{\partial \Omega } - u_{2}(t)\partial _{2}\omega (t)|_{\partial \Omega }. \end{aligned}$$

Then $\partial _{2}^{2}\omega (t)|_{\partial \Omega } = 0$. By using this fact and the incompressibility condition for $\mathbf {u}$,

$$\begin{aligned} \partial _{2}^{3}u_{1}(t)|_{\partial \Omega } = 0,\,\partial _{2}^{4}u_{2}(t)|_{\partial \Omega } = 0. \end{aligned}$$

Similarly, from $\partial _{2}^{n}\theta _{0}|_{\partial \Omega } =0$ for $n= 2,4\cdots$, we conclude that the sufficiently smooth function $(\omega ,\theta ,\mathbf {u})$ satisfies

$$\begin{aligned} \partial _{2}^{n}\theta (t)|_{\partial \Omega }=0,\,\partial _{2}^{n+2}\omega (t)|_{\partial \Omega }= 0,\,\partial _{2}^{n+3}u_{1}(t)|_{\partial \Omega }= 0,\,\partial _{2}^{n+4}u_{2}(t)|_{\partial \Omega }= 0. \end{aligned}$$

(1.14)

Hence, we assume that the initial data $(\omega _0,\theta _{0})$ satisfies the special setting (1.12)-(1.13).

We now state the main result of this paper as follows.

Theorem 1.1

Let $m\ge 5$ be a fixed integer and $(\omega _{0}, \theta _{0})$ satisfy (1.12)-(1.13). There exists a constant $\epsilon _{0}>0$ such that if

$$\begin{aligned} \Vert \omega _{0}\Vert _{H^{m}}^{2} +\Vert \theta _{0}\Vert _{H^{m + 1}}^{2}\le \epsilon _{0}^{2}, \end{aligned}$$

(1.15)

then (1.10)-(1.11) admits a unique global smooth solution

$$\begin{aligned} (\omega ,\theta )\in C([0,\infty );H^m)\times C([0,\infty ),H^{m+1}) \end{aligned}$$

satisfying

$$\begin{aligned} \Vert \omega (t)\Vert _{H^{m}}^{2}+\Vert \theta (t)\Vert _{H^{m + 1}}^{2}\lesssim \epsilon _{0}^{2}, \text { for all }t >0. \end{aligned}$$

Moreover,

$$\begin{aligned} \Vert \omega (t)\Vert _{H^{1}}+\Vert \partial _1\omega (t)\Vert _{H^{1}}\lesssim (1+t)^{-1},\,\Vert \partial _{1}\theta (t)\Vert _{H^{2}}\lesssim (1+t)^{-1}. \end{aligned}$$

Remark 1.1

It follows from Theorem 1.1 that the solution to (1.2)-(1.3) satisfies

$$\begin{aligned} \Vert \mathbf {v}(t)\Vert _{L^{\infty }}\lesssim \Vert \mathbf {v}(t)\Vert _{H^{2}}\lesssim (1+t)^{-1}, \end{aligned}$$

$$\begin{aligned} \Vert \vartheta (x,y,t)-\overline{\vartheta }(y,t)\Vert _{L^{\infty }}\lesssim \Vert \vartheta (x,y,t)-\overline{\vartheta }(y,t)\Vert _{H^{2}}\lesssim \Vert \partial _1\vartheta (x,y,t)\Vert _{H^{2}}\lesssim (1+t)^{-1}. \end{aligned}$$

This means that $\overline{\vartheta }(y,t)$ determines the large-time asymptotics of $\vartheta (x,y,t)$. Furthermore, we deduce from (1.2)$_{3}$ that

$$\begin{aligned} \overline{\vartheta }(y,t) = \overline{\vartheta }_{0}(y) + \partial _{2}\int _{0}^{t}\overline{v_{2}\vartheta }(y,\tau )\mathrm{d} \tau , \end{aligned}$$

which has been proved in [24]. Here the bar denotes the horizontal average, that is,

$$\begin{aligned} \overline{f}(y,t) = \int _{\mathbb {T}}f(x,y,t)\mathrm{d} x. \end{aligned}$$

The method of the proof of Theorem 1.1 is motivated by [22]. To prove Theorem 1.1, it is necessary to show the global uniform estimates of the solution $(\omega , \theta )$ to (1.10)-(1.11). However, we have to face the difficulty arising from the absence of thermal conduction. By making full use of the structure of (1.10), we obtain the only partial dissipation of $\theta$. This partial dissipation implies that it is difficult to control the growth of $\mathbf {u}\cdot \nabla \theta$. In the estimate process of $\mathcal {E}(T)$ defined in (3.4), we find that the key point is to obtain $L^{1}$ estimate of $\Vert \partial \mathbf {u}(t)\Vert _{L^{\infty }}$ in time and $L^{1}$ estimate of $\Vert \partial _{2}^{2}u_{2}(t)\Vert _{L^{\infty }}$ in time. Using a carefully designed time-weighted energy $\mathcal {F}(T)$ defined in (3.5) and applying Poincaré inequality in the x-direction, we overcome these above challenges and prove uniform estimates of $\mathcal {E}(T)$ and $\mathcal {F}(T)$.

The structure of this paper is as follows. In Sect. 2, we give some notations and lemmas which are used in the sequel. Section 3 is devoted to the proof of Theorem 1.1. In Sect. 4, we present some remarks.

2 Preliminaries

The following notations and results will be used in the paper. For simplicity, we set $\partial _{1} = \partial _{x}$, $\partial _{2} = \partial _{y}$ and use $\langle \cdot ,\cdot \rangle$ as the inner product in $L^{2}$. Here $A \lesssim B$ means that $A\le CB$, where C is a generic constant. Throughout this paper, $\nabla =(\partial _1,\partial _2)$, $\Delta =\nabla \cdot \nabla =\partial _{11} + \partial _{22}$ is the 2D Laplacian operator. For $m\in \mathbb {N}$, the inhomogeneous Sobolev space with derivatives up to order m in $L^{2}(\Omega )$ is denoted by $H^{m}(\Omega )$. Moreover, we use $\dot{H}^{m}(\Omega )$ to denote the homogeneous Sobolev space with the mth-order derivatives in $L^{2}(\Omega )$. For $f \in H^m(\Omega )$, $\Vert f\Vert _{H^{m}(\Omega )} = \Vert f\Vert _{L^{2}(\Omega )} + \Vert f\Vert _{\dot{H}^{m}(\Omega )}$. In this paper, we may use $L^{p}$, $\dot{H}^{m}$ and $H^{m}$ to stand for $L^{p}(\Omega )$, $\dot{H}^{m}(\Omega )$ and $H^{m}(\Omega )$, respectively, in some places.

In this section, we give some necessary lemmas.

Lemma 2.1

([2]) Let $0\le m_{1}\le m \le m_{2}$. If $f\in H^{m_{2}}(\Omega )$, then

$$\begin{aligned} \Vert f\Vert _{H^{m}(\Omega )}\lesssim \Vert f\Vert _{H^{m_{1}}(\Omega )}^{s}\Vert f\Vert _{H^{m_{2}}(\Omega )}^{1-s},\text { with } m= s m_{1} +(1-s)m_{2},\,0\le s\le 1. \end{aligned}$$

(2.1)

The following estimates are classical, see [11, 21], among others.

Lemma 2.2

For $m \in \mathbb {N}$, we have

• If $f, g \in H^{m}(\Omega ) \cap L^{\infty }(\Omega )$, then

  $$\begin{aligned} \Vert fg\Vert _{H^{m}(\Omega )}\lesssim \Vert f\Vert _{H^{m}(\Omega )}\Vert g\Vert _{L^{\infty }(\Omega )} + \Vert f\Vert _{L^{\infty }(\Omega )}\Vert g\Vert _{H^{m}(\Omega )}. \end{aligned}$$

  (2.2)

• If $f \in H^{m}(\Omega ) \cap W^{1,\infty }(\Omega )$ and $g \in H^{m-1}(\Omega ) \cap L^{\infty }(\Omega )$, then for $|\alpha |\lesssim m$,

  $$\begin{aligned} \Vert \partial ^{\alpha }(fg) - f\partial ^{\alpha }g\Vert _{L^{2}(\Omega )}\lesssim \Vert f\Vert _{W^{1,\infty }(\Omega )}\Vert g\Vert _{H^{m-1}(\Omega )} + \Vert f\Vert _{H^{m}(\Omega )}\Vert g\Vert _{L^{\infty }(\Omega )}. \end{aligned}$$

  (2.3)

The following lemma related to Poincaré inequality in the x-direction is important. The proof of this lemma is standard, see [2, 22], among others.

Lemma 2.3

Let $m \in \mathbb {N}$. If $f \in H ^{1+m}(\Omega$),

$$\begin{aligned} \left\| f(x,y) - \int _{\mathbb {T}}f(x,y)\mathrm{d} x\right\| _{H^{m}(\Omega )} \lesssim \Vert \partial _{1}f\Vert _{H^{m}(\Omega )}. \end{aligned}$$

(2.4)

Furthermore, if f satisfies $\int _{\mathbb {T}}f(x,y)\mathrm{d} x=0$, then

$$\begin{aligned} \Vert f\Vert _{H ^{m}(\Omega )} \lesssim \Vert \partial _{1}f\Vert _{H ^{m}(\Omega )}. \end{aligned}$$

(2.5)

The following result is well known, see [12], among others.

Lemma 2.4

Let $m \in \mathbb {N}$. Consider the elliptic equations

$$\begin{aligned} \left\{ \begin{aligned}&- \Delta U = f, \\&U|_{\partial \Omega } = 0. \end{aligned} \right. \end{aligned}$$

If $f \in H^{m}(\Omega )$, then

$$\begin{aligned} \Vert U\Vert _{H^{m+2}(\Omega )} \lesssim \Vert f\Vert _{H^{m}(\Omega )}. \end{aligned}$$

From (1.9) and Lemma 2.4, we obtain

Corollary 2.1

Let $m \in \mathbb {N}$. Assume that $\mathbf {v} \in H^{m}(\Omega )$, $\nabla \cdot \mathbf {v} =0$ in $\Omega$ and $\mathbf {v}\cdot \mathbf {n} = 0$ on $\partial \Omega$. Then

$$\begin{aligned} \Vert \mathbf {v}\Vert _{H^{m}(\Omega )}\lesssim \Vert \nabla \times \mathbf {v}\Vert _{H^{m-1}(\Omega )}. \end{aligned}$$

3 Proof of the main result

The local existence of solution to (1.10)-(1.11) for general smooth initial data in a bounded domain $\Omega$ can be proved by the classical methods, see [5, 21] and references therein. We omit the proof of the following result for brevity.

Proposition 3.1

Let $m \ge 2$ be an integer and $(\omega _{0}, \theta _{0})$ satisfy (1.12)-(1.13). Then there exists $T_{0}>0$ such that (1.10)-(1.11) admits a unique solution

$$\begin{aligned} (\omega ,\theta )\in C([0,T_{0}];H^{m}(\Omega ))\times C([0,T_{0}];H^{m+1}(\Omega )) \end{aligned}$$

satisfying

$$\begin{aligned} \partial _{2}^{s}\omega |_{\partial \Omega }= & {} 0,~\text {for}~s= 0,2,...,2[(m-1)/2], \end{aligned}$$

(3.1)

$$\begin{aligned} \partial _{2}^{n}\theta |_{\partial \Omega }= & {} 0,~\text {for}~n= 0,2,...,2[m/2]. \end{aligned}$$

(3.2)

Moreover, if $T^{*}$ is the lifespan to the solution $( \omega ,\theta )$ and $T^{*}<\infty$, then

$$\begin{aligned} \int _{0}^{T^{*}}\left( \Vert \nabla \mathbf {u}(t)\Vert _{L^{\infty }}+\Vert \nabla \theta (t)\Vert _{L^{\infty }}\right) \mathrm{d} t = \infty . \end{aligned}$$

(3.3)

Based on Proposition 3.1, it is enough to show global a priori bounds for the smooth solution $(\omega ,\theta )$. To this end, we define for $m \in \mathbb {N}$,

$$\begin{aligned} \mathcal {E}(T)= & {} \sup _{0\le t\le T}\left( \Vert \theta (t)\Vert _{H^{m+1}}^{2} + \Vert \omega (t)\Vert _{H^{m}}^{2}\right) + \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{H^{m-1}}^{2}\mathrm{d} t + \int _{0}^{T}\Vert \omega (t)\Vert _{H^{m+1}}^{2}\mathrm{d} t ,\end{aligned}$$

(3.4)

$$\begin{aligned} \mathcal {F}(T)= & {} \sup _{0\le t\le T} (1+t)^{2}\left( \Vert \partial _{1}\theta (t)\Vert _{H^{2}}^{2} +\Vert \partial _{1}\omega (t)\Vert _{H^{1}}^{2}+ \Vert \omega (t)\Vert _{H^{1}}^{2} \right) \nonumber \\&+\int _{0}^{T}(1+t)^{2}\left( \Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2} + \Vert \partial _{1}\omega (t)\Vert _{H^{2}}^{2} + \Vert \omega (t)\Vert _{H^{2}}^{2}\right) \mathrm{d} t . \end{aligned}$$

(3.5)

The estimates of $\mathcal {E}(T)$ and $\mathcal {F}(T)$ will be given, respectively, in the following two sections.

3.1 A priori estimate of $\mathcal {E}(T)$

To begin, we set

$$\begin{aligned} \mathcal {E}_{1}(T)= & {} \sup _{0\le t\le T}\left( \Vert \theta (t)\Vert _{H^{m+1}}^{2} + \Vert \omega (t)\Vert _{H^{m}}^{2}\right) + \int _{0}^{T}\Vert \omega (t)\Vert _{H^{m+1}}^{2}\mathrm{d} t , \end{aligned}$$

(3.6)

$$\begin{aligned} \mathcal {E}_{2}(T)= & {} \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{H^{m-1}}^{2}\mathrm{d} t . \end{aligned}$$

(3.7)

Then $\mathcal {E}(T) = \mathcal {E}_{1}(T) +\mathcal {E}_{2}(T)$.

We start with controlling the bound of $\mathcal {E}_{1}(T)$ by the combination of energies defined in (3.4)-(3.5).

Lemma 3.1

Let $m \ge 2$. Then

$$\begin{aligned} \mathcal {E}_{1}(T) \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.8)

Proof

Multiplying (1.10)$_{1}$ by $\omega$ and testing (1.10)$_{2}$ by $-\Delta \theta$,

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t }\left( \Vert \nabla \theta \Vert _{L^{2}}^{2} + \Vert \omega \Vert _{L^{2}}^{2}\right) + \Vert \nabla \omega \Vert _{L^{2}}^{2} = -\langle \nabla \mathbf {u}\cdot \nabla \theta ,\nabla \theta \rangle + \langle -\nabla u_{2}, \nabla \theta \rangle + \langle \partial _{1}\theta , \omega \rangle . \end{aligned}$$

(3.9)

Due to $u_2=-\partial _1(-\Delta )^{-1}\omega$, we use integrations by parts to obtain

$$\begin{aligned} \langle -\nabla u_{2}, \nabla \theta \rangle + \langle \partial _{1}\theta , \omega \rangle = \langle \nabla \partial _1(-\Delta )^{-1}\omega , \nabla \theta \rangle + \langle \partial _{1}\theta , \omega \rangle =0. \end{aligned}$$

(3.10)

By integrating (3.9) in time from 0 to T and using (3.10),

$$\begin{aligned}&\sup _{0\le t\le T}\left( \Vert \nabla \theta (t)\Vert _{L^{2}}^{2} + \Vert \omega (t)\Vert _{L^{2}}^{2}\right) + \int _{0}^{T}\Vert \nabla \omega (t)\Vert _{L^{2}}^{2}\mathrm{d} t \nonumber \\&\quad \lesssim \Vert \nabla \theta _{0}\Vert _{L^{2}}^{2} + \Vert \omega _{0}\Vert _{L^{2}}^{2} +\int _{0}^{T}\Vert \nabla \mathbf {u}(t)\Vert _{L^{\infty }}\mathrm{d} t \sup _{0\le t\le T}\Vert \nabla \theta (t)\Vert _{L^{2}}^{2}\nonumber \\&\quad \lesssim \Vert \nabla \theta _{0}\Vert _{L^{2}}^{2} + \Vert \omega _{0}\Vert _{L^{2}}^{2}+\left( \int _{0}^{T}(1+t)^{2}\Vert \omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}} \left( \int _{0}^{T}(1+t)^{-2}\mathrm{d} t \right) ^{\frac{1}{2}}\sup _{0\le t \le T}\Vert \theta (t)\Vert _{H^{1}}^{2}\nonumber \\&\quad \lesssim \mathcal {E}(0) +\mathcal {F}^{\frac{1}{2}}(T)\mathcal {E}(T)\lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T), \end{aligned}$$

(3.11)

where Young’s inequality is used in the last inequality. Moreover, we get for $m\ge 1$,

$$\begin{aligned}&\langle \partial _{t}\partial ^{m}\omega , \partial ^{m}\omega \rangle + \langle \partial ^{m}(\mathbf {u}\cdot \nabla \omega ), \partial ^{m}\omega \rangle - \langle \Delta \partial ^{m}\omega , \partial ^{m}\omega \rangle = \langle \partial ^{m}\partial _{1}\theta , \partial ^{m}\omega \rangle , \end{aligned}$$

(3.12)

$$\begin{aligned}&\langle \partial _{t}\partial ^{m}\nabla \theta , \partial ^{m}\nabla \theta \rangle + \langle \partial ^{m}\nabla (\mathbf {u}\cdot \nabla \theta ), \partial ^{m}\nabla \theta \rangle = \langle -\partial ^{m}\nabla u_{2}, \partial ^{m}\nabla \theta \rangle . \end{aligned}$$

(3.13)

Adding (3.12) to (3.13) yields

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t }\left( \Vert \theta \Vert _{\dot{H}^{m + 1}}^{2} + \Vert \omega \Vert _{\dot{H}^{m}}^{2}\right) + \Vert \omega \Vert _{\dot{H}^{m + 1}}^{2} = I_{1} + I_{2} + I_{3} \end{aligned}$$

(3.14)

with

$$\begin{aligned} I_{1}= & {} - \langle \partial ^{m}(\mathbf {u}\cdot \nabla \omega ), \partial ^{m}\omega \rangle ,\\ I_{2}= & {} -\langle \partial ^{m}\nabla (\mathbf {u}\cdot \nabla \theta ), \partial ^{m}\nabla \theta \rangle ,\\ I_{3}= & {} \langle -\partial ^{m}\nabla u_{2}, \partial ^{m}\nabla \theta \rangle + \langle \partial ^{m}\partial _{1}\theta , \partial ^{m}\omega \rangle . \end{aligned}$$

According to (2.3) in Lemma 2.2, together with Corollary 2.1 and the Sobolev imbedding theorem, we have

$$\begin{aligned} I_{1}= & {} -\left\langle \partial ^{m}(\mathbf {u}\cdot \nabla \omega )-\mathbf {u}\cdot \nabla \partial ^{m}\omega , \partial ^{m}\omega \right\rangle = -\left\langle \partial ^{m}\text {div}(\mathbf {u}\omega )-\mathbf {u}\cdot \nabla \partial ^{m}\omega , \partial ^{m}\omega \right\rangle \\\lesssim & {} \Vert \mathbf {u}\Vert _{W^{1,\infty }}\Vert \omega \Vert _{H^{m}}^{2} + \Vert \omega \Vert _{L^{\infty }}\Vert \mathbf {u}\Vert _{H^{m+1}}\Vert \omega \Vert _{H^{m}} \lesssim \Vert \omega \Vert _{H^{2}}\Vert \omega \Vert _{H^{m}}^{2}, \end{aligned}$$

where we use the fact that $\langle \mathbf {u}\cdot \nabla \partial ^{m}\omega , \partial ^{m}\omega \rangle = 0$. Hence, for $m\ge 2$,

$$\begin{aligned} \int _{0}^{T}|I_{1}(t)|\mathrm{d} t \lesssim \sup _{0\le t \le T}\Vert \omega (t)\Vert _{H^{m}}\int _{0}^{T}\Vert \omega (t)\Vert _{H^{m}}^{2}\mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T). \end{aligned}$$

(3.15)

Note that

$$\begin{aligned} I_{2}&= -\int _{\Omega }\partial ^{m}\partial _{1}(\mathbf {u}\cdot \nabla \theta )\partial ^{m}\partial _{1}\theta \mathrm{d} x\mathrm{d} y-\int _{\Omega }\partial ^{m}\partial _{2}(\mathbf {u}\cdot \nabla \theta )\partial ^{m}\partial _{2}\theta \mathrm{d} x\mathrm{d} y\\&=: I_{21} + I_{22}. \end{aligned}$$

For $I_{21}$, using the fact that $\left\langle \mathbf {u}\cdot \nabla \partial ^{m}\partial _{1}\theta , \partial ^{m}\partial _{1}\theta \right\rangle = 0$ and applying integration by parts together with the boundary conditions (1.14),

$$\begin{aligned} I_{21}= & {} -\sum _{1\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial ^{\alpha }\mathbf {u}\cdot \nabla \partial ^{m+1-\alpha }\theta \partial ^{m}\partial _{1}\theta \mathrm{d} x\mathrm{d} y\\= & {} \sum _{2\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial ^{\alpha +1}\mathbf {u}\cdot \nabla \partial ^{m+1-\alpha }\theta \partial ^{m-1}\partial _{1}\theta \mathrm{d} x\mathrm{d} y\\&+ \sum _{2\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial ^{\alpha }\mathbf {u}\cdot \nabla \partial ^{m+2-\alpha }\theta \partial ^{m-1}\partial _{1}\theta \mathrm{d} x\mathrm{d} y\\&-(m+1)\int _{\Omega }\partial \mathbf {u}\cdot \nabla \partial ^{m}\theta \partial ^{m}\partial _{1}\theta \mathrm{d} x\mathrm{d} y, \end{aligned}$$

where $\partial ^{m}\partial _{1}$ is denoted by $\partial ^{m+1}$ which has at least one derivative on x. Then

$$\begin{aligned} I_{21}\lesssim & {} \Vert \mathbf {u}\Vert _{H^{m+2}} \Vert \theta \Vert _{H^{m+1}}\Vert \partial _{1}\theta \Vert _{H^{m-1}}\\&+ \Vert \mathbf {u}\Vert _{H^{m+1}} \Vert \theta \Vert _{H^{m+1}}\Vert \partial _{1}\theta \Vert _{H^{m-1}} + \Vert \partial \mathbf {u}\Vert _{L^{\infty }}\Vert \theta \Vert _{H^{m+1}}^{2}\\\lesssim & {} \Vert \omega \Vert _{H^{m+1}}\Vert \theta \Vert _{H^{m+1}}\Vert \partial _{1}\theta \Vert _{H^{m-1}} + \Vert \omega \Vert _{H^{2}}\Vert \theta \Vert _{H^{m+1}}^{2}. \end{aligned}$$

With the help of Hölder inequality and Young’s inequality,

$$\begin{aligned} \int _{0}^{T}|I_{21}(t)| \mathrm{d} t\lesssim & {} \left( \int _{0}^{T}\Vert \omega (t)\Vert _{H^{m+1}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\left( \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{H^{m-1}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}} \sup _{0\le t \le T}\Vert \theta (t)\Vert _{H^{m+1}}\nonumber \\&+ \left( \int _{0}^{T}(1+t)^{2}\Vert \omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}} \left( \int _{0}^{T}(1+t)^{-2}\mathrm{d} t \right) ^{\frac{1}{2}}\sup _{0\le t \le T}\Vert \theta (t)\Vert _{H^{m+1}}^{2}\nonumber \\\lesssim & {} \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{1}{2}}(T)\mathcal {E}(T) \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.16)

For $I_{22}$, we analyze the case, namely $\partial ^{m}$ has only derivative on y. Other cases can be estimated by the method as employed in the proof of $I_{21}$. Since $\left\langle \mathbf {u}\cdot \nabla \partial ^{m}\partial _{2}\theta , \partial ^{m}\partial _{2}\theta \right\rangle = 0$,

$$\begin{aligned} I_{22}= & {} -\sum _{1\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial _{2}^{\alpha }\mathbf {u}\cdot \nabla \partial _{2}^{m+1-\alpha }\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y\\= & {} I_{221} + I_{222} \end{aligned}$$

with

$$\begin{aligned} I_{221}= & {} -\sum _{1\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial _{2}^{\alpha }u_{1}\partial _{1}\partial _{2}^{m+1-\alpha }\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y,\\ I_{222}= & {} -\sum _{1\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial _{2}^{\alpha }u_{2}\partial _{2}^{m+2-\alpha }\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y. \end{aligned}$$

Note that

$$\begin{aligned} I_{221}= & {} -\sum _{2\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial _{2}^{\alpha }u_{1}\partial _{2}^{m+1-\alpha }\partial _{1}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y\\&-(m+1)\int _{\Omega }\partial _{2}u_{1}\partial _{2}^{m}\partial _{1}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y. \end{aligned}$$

Similarly,

$$\begin{aligned} I_{221}\lesssim & {} \Vert \mathbf {u}\Vert _{H^{m+1}}\Vert \partial _{1}\theta \Vert _{H^{m-1}}\Vert \theta \Vert _{H^{m+1}} + \Vert \partial _{2}u_{1}\Vert _{L^{\infty }}\Vert \theta \Vert _{H^{m+1}}^{2}\\\lesssim & {} \Vert \omega \Vert _{H^{m}}\Vert \partial _{1}\theta \Vert _{H^{m-1}}\Vert \theta \Vert _{H^{m+1}} + \Vert \omega \Vert _{H^{2}}\Vert \theta \Vert _{H^{m+1}}^{2}. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{T}|I_{221}(t)| \mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{1}{2}}(T)\mathcal {E}(T) \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.17)

Using the fact that $\partial _{2}u_{2} = - \partial _{1}u_{1}$ and integrating by parts yield

$$\begin{aligned} I_{222}= & {} -\sum _{3\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial _{2}^{\alpha -1}u_{1}\partial _{2}^{m+2-\alpha }\partial _{1}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y\\&- \sum _{3\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial _{2}^{\alpha -1}u_{1}\partial _{2}^{m+2-\alpha }\theta \partial _{1}\partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y\\&- (m+1)\int _{\Omega }\partial _{2}u_{2}\partial _{2}^{m+1}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y- C_{m+1}^{2}\int _{\Omega }\partial _{2}^{2}u_{2} \partial _{2}^{m}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y\\= & {} -\sum _{3\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\partial _{2}^{\alpha -1}u_{1}\partial _{2}^{m+2-\alpha }\partial _{1}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y\\&-\sum _{3\le \alpha \le m+1} C_{m+1}^{\alpha }\int _{\Omega }\sum _{0\le \beta \le 2}C_{2}^{\beta }\partial _{2}^{\alpha -1+\beta }u_{1}\partial _{2}^{m+2-\alpha +2-\beta }\theta \partial _{1}\partial _{2}^{m-1}\theta \mathrm{d} x\mathrm{d} y\\&+ (m+1)\int _{\Omega }\partial _{1}u_{1}\partial _{2}^{m+1}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y+ C_{m+1}^{2}\int _{\Omega }\partial _{2}\partial _{1}u_{1} \partial _{2}^{m}\theta \partial _{2}^{m+1}\theta \mathrm{d} x\mathrm{d} y. \end{aligned}$$

Furthermore, one gets

$$\begin{aligned} I_{222}\lesssim & {} \Vert \mathbf {u}\Vert _{H^{m}}\Vert \partial _{1}\theta \Vert _{H^{m-1}}\Vert \theta \Vert _{H^{m+1}} + \Vert \mathbf {u}\Vert _{H^{m+2}}\Vert \partial _{1}\theta \Vert _{H^{m-1}}\Vert \theta \Vert _{H^{m+1}}\\&+ \Vert \partial _{1}u_{1}\Vert _{L^{\infty }}\Vert \theta \Vert _{H^{m+1}}^{2} + \Vert \partial _{2}\partial _{1}u_{1}\Vert _{L^{\infty }}\Vert \theta \Vert _{H^{m+1}}^{2}\\\lesssim & {} \Vert \omega \Vert _{H^{m+1}}\Vert \partial _{1}\theta \Vert _{H^{m-1}}\Vert \theta \Vert _{H^{m+1}} + \Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \theta \Vert _{H^{m+1}}^{2}. \end{aligned}$$

Thus,

$$\begin{aligned} \int _{0}^{T}|I_{222}(t)| \mathrm{d} t\lesssim & {} \mathcal {E}^{\frac{3}{2}}(T)\nonumber \\&+ \left( \int _{0}^{T}(1+t)^{2}\Vert \partial _1\omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}} \left( \int _{0}^{T}(1+t)^{-2}\mathrm{d} t \right) ^{\frac{1}{2}}\sup _{0\le t \le T}\Vert \theta (t)\Vert _{H^{m+1}}^{2}\nonumber \\\lesssim & {} \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{1}{2}}(T)\mathcal {E}(T) \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.18)

Putting (3.17) and (3.18) together,

$$\begin{aligned} \int _{0}^{T}|I_{22}(t)| \mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.19)

Moreover, the preceding estimates (3.16) and (3.19) show that

$$\begin{aligned} \int _{0}^{T}|I_{2}(t)| \mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.20)

To handle the last term, substituting $u_{2} = -\partial _{1}(-\Delta )^{-1}\omega$ into $I_{3}$ and then integrating by parts give

$$\begin{aligned} I_{3} = \langle \partial ^{m}\nabla \partial _{1}(-\Delta )^{-1}\omega , \partial ^{m}\nabla \theta \rangle + \langle \partial ^{m}\partial _{1}\theta , \partial ^{m}\omega \rangle = 0. \end{aligned}$$

(3.21)

Integrating (3.14) in time from 0 to T and summing up (3.15), (3.20), one has

$$\begin{aligned} \sup _{0\le t\le T}\left( \Vert \theta (t)\Vert _{\dot{H}^{m+1}}^{2} + \Vert \omega (t)\Vert _{\dot{H}^{m}}^{2}\right) + \int _{0}^{T}\Vert \omega (t)\Vert _{\dot{H}^{m+1}}^{2}\mathrm{d} t \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.22)

Finally adding (3.11) to (3.22), we get by Poincaré inequality that

$$\begin{aligned} \mathcal {E}_{1}(T)\lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.23)

$\square$

In the following lemma, we proceed to deal with the estimate of $\mathcal {E}_{2}(T)$.

Lemma 3.2

Let $m \ge 2$. Then

$$\begin{aligned} \mathcal {E}_{2}(T) \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.24)

Proof

Applying $\partial ^{m-1}$ on (1.10)$_{1}$ and taking inner product with $\partial _{1}\partial ^{m-1}\theta$, one gets

$$\begin{aligned} \Vert \partial _{1}\theta \Vert _{\dot{H}^{m-1}}^{2}&= \langle \partial ^{m-1}\partial _{t}\omega , \partial _{1}\partial ^{m-1}\theta \rangle + \langle \partial ^{m-1}(\mathbf {u}\cdot \nabla \omega ), \partial _{1}\partial ^{m-1}\theta \rangle - \langle \partial ^{m-1}\Delta \omega , \partial _{1}\partial ^{m-1}\theta \rangle \nonumber \\&= \frac{\mathrm{d}}{\mathrm{d} t }\langle \partial ^{m-1}\omega , \partial _{1}\partial ^{m-1}\theta \rangle - \langle \partial ^{m-1}\omega , \partial _{1}\partial ^{m-1}\partial _{t}\theta \rangle \nonumber \\&\quad + \langle \partial ^{m-1}(\mathbf {u}\cdot \nabla \omega ), \partial _{1}\partial ^{m-1}\theta \rangle - \langle \partial ^{m-1}\Delta \omega , \partial _{1}\partial ^{m-1}\theta \rangle \nonumber \\&=: N_{1} + N_{2} + N_{3} + N_{4}. \end{aligned}$$

(3.25)

We need estimate $N_{j}, j =1,2,3,4$, respectively. For $N_{1}$, by Hölder inequality,

$$\begin{aligned} \int _{0}^{T}|N_{1}(t)|\mathrm{d} t \lesssim \sup _{0\le t \le T}\Vert \theta (t)\Vert _{H^{m}}\sup _{0\le t \le T}\Vert \omega (t)\Vert _{H^{m-1}}\lesssim \mathcal {E}_{1}(T). \end{aligned}$$

(3.26)

Substituting $\partial _{t}\theta = -\mathbf {u}\cdot \nabla \theta -u_{2}$ into $N_{2}$ and using (2.2) in Lemma 2.2 give

$$\begin{aligned} N_{2}= & {} \langle \partial ^{m-1}\omega , \partial _{1}\partial ^{m-1}(\mathbf {u}\cdot \nabla \theta )\rangle + \langle \partial ^{m-1}\omega , \partial _{1}\partial ^{m-1}u_{2}\rangle \\\lesssim & {} \Vert \omega \Vert _{H^{m+1}}^{2}\Vert \theta \Vert _{H^{m+1}} + \Vert \omega \Vert _{H^{m+1}}^{2}. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{T}|N_{2}(t)|\mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \int _{0}^{T}\Vert \omega (t)\Vert _{H^{m+1}}^{2}\mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {E}_{1}(T). \end{aligned}$$

(3.27)

Similarly for $N_3$,

$$\begin{aligned} \int _{0}^{T}|N_{3}(t)|\mathrm{d} t \lesssim \int _{0}^{T}\Vert \omega (t)\Vert _{H^{m+1}}^{2}\Vert \partial _{1}\theta (t)\Vert _{H^{m-1}}\mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T). \end{aligned}$$

(3.28)

By Hölder inequality and Young’s inequality,

$$\begin{aligned} N_{4} \le C_{\varepsilon }\Vert \omega \Vert _{H^{m+1}}^{2} + \varepsilon \Vert \partial _{1}\theta \Vert _{\dot{H}^{m-1}}^{2}, \end{aligned}$$

where $C_{\varepsilon }$ is a constant depending on $\varepsilon$. Thus,

$$\begin{aligned} \int _{0}^{T}|N_{4}(t)|\mathrm{d} t\le & {} C_{\varepsilon }\int _{0}^{T}\Vert \omega (t)\Vert _{H^{m+1}}^{2}\mathrm{d} t + \varepsilon \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{\dot{H}^{m-1}}^{2}\mathrm{d} t \nonumber \\\le & {} C_{\varepsilon }\mathcal {E}_{1}(T) + \varepsilon \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{\dot{H}^{m-1}}^{2}\mathrm{d} t . \end{aligned}$$

(3.29)

Integrating (3.25) in time from 0 to T, summing up (3.26)-(3.29) and then taking $\varepsilon$ small enough, we infer from Lemma 3.1 that

$$\begin{aligned} \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{\dot{H}^{m-1}}^{2}\mathrm{d} t \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {E}_{1}(T) \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.30)

Similarly,

$$\begin{aligned} \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.31)

Finally, the desired estimate (3.24) follows from (3.30) and (3.31). $\square$

From Lemma 3.1-3.2, we obtain

Lemma 3.3

Let $m \ge 2$. Then

$$\begin{aligned} \mathcal {E}(T) \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.32)

3.2 A priori estimate of $\mathcal {F}(T)$

Here we set

$$\begin{aligned} \mathcal {F}_{1}(T)= & {} \sup _{0\le t\le T} (1+t)^{2}\left( \Vert \partial _{1}\theta (t)\Vert _{H^{2}}^{2} +\Vert \partial _{1}\omega (t)\Vert _{H^{1}}^{2} \right) + \int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t , \end{aligned}$$

(3.33)

$$\begin{aligned} \mathcal {F}_{2}(T)= & {} \int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t , \end{aligned}$$

(3.34)

$$\begin{aligned} \mathcal {F}_{3}(T)= & {} \sup _{0\le t\le T} (1+t)^{2}\Vert \omega (t)\Vert _{H^{1}}^{2} + \int _{0}^{T}(1+t)^{2}\Vert \omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t . \end{aligned}$$

(3.35)

To obtain a priori estimate of $\mathcal {F}(T)$, we shall estimate $\mathcal {F}_{1}(T)$, $\mathcal {F}_{2}(T)$ and $\mathcal {F}_{3}(T)$, respectively.

Lemma 3.4

Let $m \ge 5$. Then

$$\begin{aligned} \mathcal {F}_{1}(T) \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.36)

Proof

Taking $\partial \partial _{1}$ on (1.10)$_{1}$ and testing by $\partial \partial _{1}\omega$ yield

$$\begin{aligned} \langle \partial _{t}\partial \partial _{1}\omega , \partial \partial _{1}\omega \rangle + \langle \partial \partial _{1}(\mathbf {u}\cdot \nabla \omega ), \partial \partial _{1}\omega \rangle - \langle \Delta \partial \partial _{1}\omega , \partial \partial _{1}\omega \rangle = \langle \partial \partial _{11}\theta , \partial \partial _{1}\omega \rangle . \end{aligned}$$

(3.37)

Applying $\partial \partial _{1}\nabla$ to (1.10)$_{2}$ and taking inner product with $\partial \partial _{1}\nabla \theta$ give

$$\begin{aligned} \langle \partial _{t}\partial \partial _{1}\nabla \theta , \partial \partial _{1}\nabla \theta \rangle + \langle \partial \partial _{1}\nabla (\mathbf {u}\cdot \nabla \theta ), \partial \partial _{1}\nabla \theta \rangle = \langle -\partial \partial _{1}\nabla u_{2}, \partial \partial _{1}\nabla \theta \rangle . \end{aligned}$$

(3.38)

By adding (3.37) to (3.38) and multiplying the time weight $(1+t)^{2}$,

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t }(1+t)^{2} \left( \Vert \partial _{1}\theta \Vert _{\dot{H}^{2}}^{2} + \Vert \partial _{1}\omega \Vert _{\dot{H}^{1}}^{2}\right) + (1+t)^{2}\Vert \partial _{1}\omega \Vert _{\dot{H}^{2}}^{2} = J_{1} + J_{2} +J_{3} + J_{4}, \end{aligned}$$

(3.39)

where

$$\begin{aligned} J_{1}= & {} -(1+t)^{2}\langle \partial \partial _{1}(\mathbf {u}\cdot \nabla \omega ), \partial \partial _{1}\omega \rangle ,\\ J_{2}= & {} -(1+t)^{2}\langle \partial \partial _{1}\nabla (\mathbf {u}\cdot \nabla \theta ), \partial \partial _{1}\nabla \theta \rangle ,\\ J_{3}= & {} -(1+t)\left( \Vert \partial _{1}\theta \Vert _{\dot{H}^{2}}^{2} + \Vert \partial _{1}\omega \Vert _{\dot{H}^{1}}^{2}\right) ,\\ J_{4}= & {} -(1+t)^{2}\left( \langle \partial \partial _{11}\theta , \partial \partial _{1}\omega \rangle + \langle -\partial \partial _{1}\nabla u_{2}, \partial \partial _{1}\nabla \theta \rangle \right) . \end{aligned}$$

$\mathbf {Estimate}~\mathbf {of}~J_{1}$. Note that

$$\begin{aligned} J_{1}= & {} -(1+t)^{2}\langle \partial \partial _{1}\mathbf {u}\cdot \nabla \omega , \partial \partial _{1}\omega \rangle -(1+t)^{2}\langle \partial \mathbf {u}\cdot \nabla \partial _{1}\omega , \partial \partial _{1}\omega \rangle \\&-(1+t)^{2}\langle \partial _{1}\mathbf {u}\cdot \nabla \partial \omega , \partial \partial _{1}\omega \rangle -(1+t)^{2}\langle \mathbf {u}\cdot \nabla \partial \partial _{1}\omega , \partial \partial _{1}\omega \rangle . \end{aligned}$$

By the incompressibility condition for $\mathbf {u}$,

$$\begin{aligned} \langle \mathbf {u}\cdot \nabla \partial \partial _{1}\omega , \partial \partial _{1}\omega \rangle = 0. \end{aligned}$$

Then the Sobolev imbedding theorem implies that

$$\begin{aligned} J_{1}\lesssim & {} (1+t)^{2}\Vert \partial \partial _{1}\mathbf {u}\Vert _{L^{2}}\Vert \nabla \omega \Vert _{L^{\infty }}\Vert \partial _{1}\omega \Vert _{H^{1}}\\&+ (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{1}}^{2}\Vert \partial \mathbf {u}\Vert _{L^{\infty }} + (1+t)^{2}\Vert \partial _{1}\mathbf {u}\Vert _{L^{\infty }}\Vert \omega \Vert _{H^{2}}\Vert \partial _{1}\omega \Vert _{H^{1}}\\\lesssim & {} (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{1}}^{2}\Vert \omega \Vert _{H^{m}}. \end{aligned}$$

Thus, by Young’s inequality,

$$\begin{aligned} \int _{0}^{T}|J_{1}(t)|\mathrm{d} t\lesssim & {} \int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \sup _{0\le t \le T}\Vert \omega (t)\Vert _{H^{m}}\nonumber \\\lesssim & {} \mathcal {F}(T)\mathcal {E}^{\frac{1}{2}}(T) \lesssim \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{3}{2}}(T). \end{aligned}$$

(3.40)

$\mathbf {Estimate}~\mathbf {of}~J_{2}$. We divide $J_{2}$ into six parts as follows.

$$\begin{aligned} J_{2} = -(1+t)^{2}\langle \partial ^{2}\partial _{1}(\mathbf {u}\cdot \nabla \theta ), \partial ^{2}\partial _{1}\theta \rangle = J_{21} + J_{22} + J_{23} + J_{24} + J_{25} + J_{26} \end{aligned}$$

with

$$\begin{aligned} J_{21}= & {} -(1+t)^{2}\langle \partial ^{2}\partial _{1}\mathbf {u}\cdot \nabla \theta , \partial ^{2}\partial _{1}\theta \rangle ,~ J_{22} =-(1+t)^{2}\langle \partial ^{2}\mathbf {u}\cdot \nabla \partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle ,\\ J_{23}= & {} -2(1+t)^{2}\langle \partial \partial _{1}\mathbf {u}\cdot \nabla \partial \theta , \partial ^{2}\partial _{1}\theta \rangle ,~ J_{24} =-(1+t)^{2}\langle \partial _{1}\mathbf {u}\cdot \nabla \partial ^{2}\theta , \partial ^{2}\partial _{1}\theta \rangle ,\\ J_{25}= & {} -2(1+t)^{2}\langle \partial \mathbf {u}\cdot \nabla \partial \partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle ,~ J_{26} =-(1+t)^{2}\langle \mathbf {u}\cdot \nabla \partial ^{2}\partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle . \end{aligned}$$

For $J_{21}$, we calculate that

$$\begin{aligned} J_{21}&= -(1+t)^{2}\langle \partial ^{2}\partial _{1}u_{1}\partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle -(1+t)^{2}\langle \partial ^{2}\partial _{1}u_{2}\partial _{2}\theta , \partial ^{2}\partial _{1}\theta \rangle \\&= -(1+t)^{2}\langle \partial ^{2}\partial _{1}u_{1}\partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle + (1+t)^{2}\langle \partial ^{2}u_{2}\partial _{2}\partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle + (1+t)^{2}\langle \partial ^{2}u_{2}\partial _{2}\theta , \partial ^{2}\partial _{11}\theta \rangle \\&=: J_{211} +J_{212} + J_{213}. \end{aligned}$$

From Lemma 2.3, we find that

$$\begin{aligned} J_{211} \lesssim (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{1}}\Vert \partial _{1}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}} \lesssim (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{1}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}. \end{aligned}$$

(3.41)

Integrating by parts and using the fact that $\mathbf {u} = \nabla ^{\perp }\psi =\nabla ^{\perp }(-\Delta )^{-1}\omega$ yield

$$\begin{aligned} J_{212}= & {} (1+t)^{2}\int _{\Omega }\sum _{0\le \beta \le 2}\partial ^{2 + \beta }u_{2}\partial ^{2- \beta }\partial _{2}\partial _{1}\theta \partial _{1}\theta \mathrm{d} x\mathrm{d} y\\= & {} -(1+t)^{2}\int _{\Omega }\sum _{0\le \beta \le 2}\partial ^{2 + \beta }\partial _{1}\psi \partial ^{2- \beta }\partial _{2}\partial _{1}\theta \partial _{1}\theta \mathrm{d} x\mathrm{d} y. \end{aligned}$$

From Lemma 2.3-2.4, it follows that

$$\begin{aligned} J_{212} \lesssim (1+t)^{2}\Vert \partial _{1}\psi \Vert _{H^{4}}\Vert \partial _{1}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}} \lesssim (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}. \end{aligned}$$

(3.42)

In a similar way,

$$\begin{aligned} J_{213}= & {} (1+t)^{2}\int _{\Omega }\sum _{0\le \beta \le 2}\partial ^{2 + \beta }u_{2}\partial ^{2- \beta }\partial _{2}\theta \partial _{11}\theta \mathrm{d} x\mathrm{d} y\\= & {} -(1+t)^{2}\int _{\Omega }\sum _{0\le \beta \le 2}\partial ^{2 + \beta }\partial _{1}\psi \partial ^{2- \beta }\partial _{2}\theta \partial _{11}\theta \mathrm{d} x\mathrm{d} y. \end{aligned}$$

Then

$$\begin{aligned} J_{213} \lesssim (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}. \end{aligned}$$

(3.43)

Summing up (3.41)-(3.43),

$$\begin{aligned} J_{21} \lesssim (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}. \end{aligned}$$

(3.44)

Note that

$$\begin{aligned} J_{22}= & {} -(1+t)^{2}\langle \partial ^{2}u_{1}\partial _{11}\theta , \partial ^{2}\partial _{1}\theta \rangle - (1+t)^{2}\langle \partial ^{2}u_{2}\partial _{2}\partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle \\= & {} -(1+t)^{2}\langle \partial ^{2}u_{1}\partial _{11}\theta , \partial ^{2}\partial _{1}\theta \rangle + (1+t)^{2}\langle \partial _{2}\partial ^{2}u_{2}\partial _{1}\theta , \partial ^{2}\partial _{1}\theta \rangle + (1+t)^{2}\langle \partial ^{2}u_{2}\partial _{1}\theta , \partial ^{2}\partial _{2}\partial _{1}\theta \rangle . \end{aligned}$$

This implies that

$$\begin{aligned} J_{22} \lesssim (1+t)^{2}\Vert \omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}. \end{aligned}$$

(3.45)

Similarly, we obtain

$$\begin{aligned} J_{23}\lesssim & {} (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}, \end{aligned}$$

(3.46)

$$\begin{aligned} J_{24}\lesssim & {} (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}, \end{aligned}$$

(3.47)

and

$$\begin{aligned} J_{25} \lesssim (1+t)^{2}\Vert \omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}. \end{aligned}$$

(3.48)

Moreover, the incompressibility condition for $\mathbf {u}$ implies that

$$\begin{aligned} J_{26} = 0. \end{aligned}$$

(3.49)

We eventually deduce from (3.44)-(3.49) that

$$\begin{aligned} J_{2} \lesssim (1+t)^{2}\Vert \omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}} + (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \theta \Vert _{H^{5}}. \end{aligned}$$

Consequently, by Hölder inequality and Young’s inequality,

$$\begin{aligned} \int _{0}^{T} |J_{2}(t)|\mathrm{d} t\lesssim & {} \left( \int _{0}^{T}(1+t)^{2}\Vert \omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\left( \int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}} \sup _{0\le t \le T}\Vert \theta (t)\Vert _{H^{m}}\nonumber \\&+ \left( \int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\left( \int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\sup _{0\le t \le T}\Vert \theta (t)\Vert _{H^{m}}\nonumber \\\lesssim & {} \mathcal {F}(T)\mathcal {E}^{\frac{1}{2}}(T) \lesssim \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{3}{2}}(T). \end{aligned}$$

(3.50)

$\mathbf {Estimate}~\mathbf {of}~J_{3}$. By the interpolation inequality (2.1) in Lemma 2.1,

$$\begin{aligned} J_{3}\lesssim & {} (1+t)\Vert \partial _{1}\theta \Vert _{H^{2}}^{2} + (1+t)\Vert \partial _{1}\omega \Vert _{H^{2}}^{2}\\\lesssim & {} (1+t)\Vert \partial _{1}\theta \Vert _{L^{2}}\Vert \partial _{1}\theta \Vert _{H^{4}} + (1+t)\Vert \partial _{1}\omega \Vert _{H^{2}}^{2}. \end{aligned}$$

Hence, for $m\ge 5$,

$$\begin{aligned} \int _{0}^{T} |J_{3}(t)|\mathrm{d} t\lesssim & {} \left( \int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\left( \int _{0}^{T}\Vert \partial _{1}\theta (t)\Vert _{H^{m-1}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\nonumber \\&+\left( \int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\left( \int _{0}^{T}\Vert \omega (t)\Vert _{H^{m+1}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\nonumber \\\lesssim & {} \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.51)

$\mathbf {Estimate}~\mathbf {of}~J_{4}$. Substituting $u_{2}= -\partial _{1}(-\Delta )^{-1}\omega$ into $J_{4}$ and integrating by parts give

$$\begin{aligned} J_{4} = 0. \end{aligned}$$

(3.52)

Integrating (3.39) in time from 0 to T and putting (3.40), (3.50), (3.51), (3.52) together, we obtain

$$\begin{aligned}&\sup _{0\le t\le T} (1+t)^{2}\left( \Vert \partial _{1}\theta (t)\Vert _{\dot{H}^{2}}^{2} +\Vert \partial _{1}\omega (t)\Vert _{\dot{H}^{1}}^{2} \right) + \int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\omega (t)\Vert _{\dot{H}^{2}}^{2}\mathrm{d} t \nonumber \\&\quad \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.53)

Similarly,

$$\begin{aligned}&\sup _{0\le t\le T} (1+t)^{2}\left( \Vert \nabla \partial _{1}\theta (t)\Vert _{L^{2}}^{2} +\Vert \partial _{1}\omega (t)\Vert _{L^{2}}^{2} \right) + \int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\nabla \omega (t)\Vert _{L^{2}}^{2}\mathrm{d} t \nonumber \\&\quad \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.54)

Hence, summing up (3.53)-(3.54) and using Poincaré inequality give

$$\begin{aligned} \mathcal {F}_{1}(T) \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

This completes the proof of Lemma 3.4. $\square$

Lemma 3.5

Let $m \ge 5$. Then

$$\begin{aligned} \mathcal {F}_{2}(T) \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.55)

Proof

Taking $\partial _{1}$ on (1.10)$_{1}$ and testing by $\partial _{11}\theta$ yield

$$\begin{aligned} \Vert \partial _{11}\theta \Vert _{L^{2}}^{2}= & {} \langle \partial _{t}\partial _{1}\omega , \partial _{11}\theta \rangle + \langle \partial _{1}(\mathbf {u}\cdot \nabla \omega ), \partial _{11}\theta \rangle - \langle \Delta \partial _{1}\omega , \partial _{11}\theta \rangle \nonumber \\= & {} \frac{\mathrm{d}}{\mathrm{d} t }\langle \partial _{1}\omega , \partial _{11}\theta \rangle - \langle \partial _{1}\omega , \partial _{11}\partial _{t}\theta \rangle + \langle \partial _{1}(\mathbf {u}\cdot \nabla \omega ), \partial _{11}\theta \rangle - \langle \Delta \partial _{1}\omega , \partial _{11}\theta \rangle . \end{aligned}$$

(3.56)

Multiplying (3.56) by $(1+t)^{2}$, we have

$$\begin{aligned} (1+t)^{2}\Vert \partial _{11}\theta \Vert _{L^{2}}^{2} =&\frac{\mathrm{d}}{\mathrm{d} t }(1+t)^{2}\langle \partial _{1}\omega , \partial _{11}\theta \rangle -2(1+t)\langle \partial _{1}\omega , \partial _{11}\theta \rangle - (1+t)^{2}\langle \partial _{1}\omega , \partial _{11}\partial _{t}\theta \rangle \nonumber \\&+ (1+t)^{2}\langle \partial _{1}(\mathbf {u}\cdot \nabla \omega ), \partial _{11}\theta \rangle - (1+t)^{2} \langle \Delta \partial _{1}\omega , \partial _{11}\theta \rangle \nonumber \\ =:&K_{1} + K_{2} + K_{3} + K_{4} + K_{5}. \end{aligned}$$

(3.57)

We now estimate $K_1, K_2,\cdots , K_5$ one by one. For $K_{1}$, by Hölder inequality,

$$\begin{aligned} \int _{0}^{T} |K_{1}(t)|\mathrm{d} t \lesssim \sup _{0\le t\le T} (1+t)\Vert \partial _{1}\theta (t)\Vert _{H^{2}}\sup _{0\le t\le T}(1+t)\Vert \partial _{1}\omega (t)\Vert _{H^{1}}\lesssim \mathcal {F}_{1}(T). \end{aligned}$$

(3.58)

For $K_{2}$, one gets

$$\begin{aligned} \int _{0}^{T} |K_{2}(t)|\mathrm{d} t \lesssim \left( \int _{0}^{T}\Vert \partial _{1}\omega (t)\Vert _{L^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\left( \int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}} \lesssim \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.59)

By substituting $\partial _{t}\theta = -\mathbf {u}\cdot \nabla \theta - u_{2}$ into $K_{3}$,

$$\begin{aligned} K_{3}= & {} (1+t)^{2}\langle \partial _{1}\omega , \partial _{11}(\mathbf {u}\cdot \nabla \theta )\rangle + (1+t)^{2}\langle \partial _{1}\omega , \partial _{11}u_{2}\rangle \\\lesssim & {} (1+t)^{2}\Vert \omega \Vert _{H^{2}}^{2}\Vert \theta \Vert _{H^{3}} + (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}^{2}. \end{aligned}$$

Thus,

$$\begin{aligned} \int _{0}^{T} |K_{3}(t)|\mathrm{d} t\lesssim & {} \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}(T) + \int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \nonumber \\\lesssim & {} \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {F}_{1}(T). \end{aligned}$$

(3.60)

Note that

$$\begin{aligned} K_{4} \lesssim (1+t)^{2}\Vert \partial _{11}\theta \Vert _{L^{2}}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \omega \Vert _{H^{1}}. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{T} |K_{4}(t)|\mathrm{d} t \lesssim \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}(T) \lesssim \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.61)

For the last term, by Young’s inequality,

$$\begin{aligned} K_{5}\lesssim & {} (1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}\Vert \partial _{11}\theta \Vert _{L^{2}}\\\le & {} C_{\varepsilon }(1+t)^{2}\Vert \partial _{1}\omega \Vert _{H^{2}}^{2} + \varepsilon (1+t)^{2}\Vert \partial _{11}\theta \Vert _{L^{2}}^{2}. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{T} |K_{5}(t)|\mathrm{d} t\le & {} C_{\varepsilon }\int _{0}^{T}(1+t)^{2}\Vert \partial _{1}\omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t + \varepsilon \int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t \nonumber \\\le & {} C_{\varepsilon }\mathcal {F}_{1}(T) + \varepsilon \int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t . \end{aligned}$$

(3.62)

Finally, integrating (3.57) in time from 0 to T and summing up (3.58)-(3.62), we take $\varepsilon$ small enough to obtain

$$\begin{aligned} \mathcal {F}_{2}(T)\lesssim & {} \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {F}_{1}(T)+ \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T)\\\lesssim & {} \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T), \end{aligned}$$

where we apply (3.36) in Lemma 3.4. $\square$

Lemma 3.6

Let $m \ge 5$. Then

$$\begin{aligned} \mathcal {F}_{3}(T) \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T)+ \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.63)

Proof

Testing (1.10)$_{1}$ by $-\Delta \omega$ yields

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t }\Vert \omega \Vert _{\dot{H}^{1}}^{2} + \Vert \Delta \omega \Vert _{L^{2}}^{2} = -\langle \mathbf {u}\cdot \nabla \omega , \Delta \omega \rangle - \langle \partial _{1}\theta , \Delta \omega \rangle . \end{aligned}$$

(3.64)

Furthermore, multiplying (3.64) by $(1+t)^{2}$ gives

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t } (1+t)^{2}\Vert \omega \Vert _{\dot{H}^{1}}^{2} + (1+t)^{2}\Vert \Delta \omega \Vert _{L^{2}}^{2} = R_{1} + R_{2} + R_{3}, \end{aligned}$$

(3.65)

where

$$\begin{aligned} R_{1}= & {} -(1+t)^{2}\langle \mathbf {u}\cdot \nabla \omega , \Delta \omega \rangle ,\\ R_{2}= & {} - (1+t)^{2}\langle \partial _{1}\theta , \Delta \omega \rangle ,\, R_{3} = (1+t)\Vert \omega \Vert _{\dot{H}^{1}}^{2}. \end{aligned}$$

By Hölder inequality and Young’s inequality,

$$\begin{aligned} \int _{0}^{T} |R_{1}(t)|\mathrm{d} t\lesssim & {} \int _{0}^{T}(1+t)^{2}\Vert \omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \sup _{0\le t \le T}\Vert \omega (t)\Vert _{H^{m}}\nonumber \\\lesssim & {} \mathcal {F}(T)\mathcal {E}^{\frac{1}{2}}(T) \lesssim \mathcal {F}^{\frac{3}{2}}(T)+\mathcal {E}^{\frac{3}{2}}(T). \end{aligned}$$

(3.66)

Similarly for $R_{2}$, we deduce from Lemma 2.3 that

$$\begin{aligned} \int _{0}^{T} |R_{2}(t)|\mathrm{d} t\le & {} C_{\varepsilon }\int _{0}^{T}(1+t)^{2}\Vert \partial _{11}\theta (t)\Vert _{L^{2}}^{2}\mathrm{d} t + \varepsilon \int _{0}^{T}(1+t)^{2}\Vert \Delta \omega (t)\Vert _{L^{2}}^{2}\mathrm{d} t \nonumber \\\le & {} C_{\varepsilon }\mathcal {F}_{2}(T) + \varepsilon \int _{0}^{T}(1+t)^{2}\Vert \Delta \omega (t)\Vert _{L^{2}}^{2}\mathrm{d} t . \end{aligned}$$

(3.67)

For $R_{3}$, we find that

$$\begin{aligned} \int _{0}^{T} |R_{3}(t)|\mathrm{d} t\lesssim & {} \left( \int _{0}^{T}(1+t)^{2}\Vert \omega (t)\Vert _{H^{2}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\left( \int _{0}^{T}\Vert \omega (t)\Vert _{H^{1}}^{2}\mathrm{d} t \right) ^{\frac{1}{2}}\nonumber \\\lesssim & {} \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

(3.68)

Integrating (3.65) in time from 0 to T and summing up (3.66)-(3.68), we take $\varepsilon$ small enough to yield

$$\begin{aligned}&\sup _{0\le t\le T} (1+t)^{2}\Vert \omega (t)\Vert _{\dot{H}^{1}}^{2} + \int _{0}^{T}(1+t)^{2}\Vert \Delta \omega (t)\Vert _{L^{2}}^{2}\mathrm{d} t \\&\quad \lesssim \mathcal {F}(0) + \mathcal {F}_{2}(T)+ \mathcal {F}^{\frac{3}{2}}(T)+\mathcal {E}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

Finally, we use Poincaré inequality, Lemma 2.4 and (3.55) in Lemma 3.5 to obtain

$$\begin{aligned} \mathcal {F}_{3}(T)\lesssim & {} \mathcal {F}(0) + \mathcal {F}_{2}(T) + \mathcal {F}^{\frac{3}{2}}(T)+\mathcal {E}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T)\\\lesssim & {} \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T)+ \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

$\square$

The next lemma gives estimate of $\mathcal {F}(T)$.

Lemma 3.7

Let $m \ge 5$. Then

$$\begin{aligned} \mathcal {F}(T) \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T). \end{aligned}$$

(3.69)

Proof

From Lemma 3.4-3.6, it follows that

$$\begin{aligned} \mathcal {F}(T) \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}^{\frac{1}{2}}(T)\mathcal {F}^{\frac{1}{2}}(T). \end{aligned}$$

By virtue of Young’s inequality,

$$\begin{aligned} \mathcal {F}(T) \lesssim \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T)+ \mathcal {F}^{\frac{3}{2}}(T) + C_{\varepsilon }\mathcal {E}(T) + \varepsilon \mathcal {F}(T). \end{aligned}$$

Then taking $\varepsilon$ small enough and using (3.32) in Lemma 3.3 give

$$\begin{aligned} \mathcal {F}(T) \lesssim \mathcal {E}(0) + \mathcal {F}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T) + \mathcal {E}(T) \lesssim \mathcal {E}(0) + \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T), \end{aligned}$$

where we use the fact $\mathcal {F}(0)\lesssim \mathcal {E}(0)$. Hence, the proof of Lemma 3.7 is completed. $\square$

3.3 Proof of theorem 1.1

From Lemma 3.3 and Lemma 3.7, there exists a constant $C_0>0$ such that

$$\begin{aligned} \mathcal {E}(T) + \mathcal {F}(T) \le C_{0}\mathcal {E}(0)+ C_{0}\left( \mathcal {E}^{\frac{3}{2}}(T) + \mathcal {F}^{\frac{3}{2}}(T)\right) . \end{aligned}$$

(3.70)

By denoting $\mathcal {G}(T)=\mathcal {E}(T) + \mathcal {F}(T)$, one deduces from (3.70) that

$$\begin{aligned} \mathcal {G}(T) \le C_{0}\mathcal {E}(0)+ C_{0}\mathcal {G}^{\frac{3}{2}}(T). \end{aligned}$$

(3.71)

Assume that

$$\begin{aligned} \Vert \omega _{0}\Vert _{H^{m}}^{2}+\Vert \theta _{0}\Vert _{H^{m + 1}}^{2}\le \epsilon _{0}^{2}, \end{aligned}$$

(3.72)

with $\epsilon _0\in (0,1)$ to be determined later. Then there exists a constant $C_1>0$ such that

$$\begin{aligned} C_{0}\mathcal {E}(0) + \mathcal {G}(0)= & {} C_{0}\mathcal {E}(0) + \mathcal {E}(0) + \mathcal {F}(0)\nonumber \\\le & {} C_{0}\epsilon _{0}^{2} + 3\epsilon _{0}^{2} \le C_1\epsilon _{0}^{2}. \end{aligned}$$

(3.73)

According to Proposition 3.1, there exists a positive time $T_{0}<T^{*}$ such that

$$\begin{aligned} \mathcal {G}(T) \le 4C_1\epsilon _{0}^{2}, \text { for all } T\in [0,T_{0}]. \end{aligned}$$

(3.74)

Since $T^{*}$ is the life span to the solution $(\omega , \theta )$, we only need to show $T^{*}=\infty$ while completing the proof of Theorem 1.1. Otherwise, if $T^{*}<\infty$, the solution $(\omega , \theta )$ satisfies (3.3). Then we define

$$\begin{aligned} \tilde{T} \triangleq \sup \{T<T^{*}:\mathcal {G}(T)\le 4C_1\epsilon _{0}^{2}\}. \end{aligned}$$

(3.75)

Moreover, we take $\epsilon _{0}$ small enough to yield $8C_{0} C_1^{\frac{1}{2}}\epsilon _{0}<1$. From (3.71) and (3.73), it follows that

$$\begin{aligned} \mathcal {G}(T)\le C_{0}\epsilon _{0}^{2} + 8C_{0}C_{1}^{\frac{3}{2}}\epsilon _{0}^{3} \le C_1\epsilon _{0}^{2} + 8C_{0}C_{1}^{\frac{3}{2}}\epsilon _{0}^{3} < 2C_1\epsilon _{0}^{2}, \text { for all }t\in [0,\tilde{T}). \end{aligned}$$

By using a continuity argument,

$$\begin{aligned} \mathcal {G}(T)\le 2C_1\epsilon _{0}^{2}, \text { for all }t\in [0,T^{*}), \end{aligned}$$

which gives a contradiction with (3.75) if $\tilde{T}<T^{*}<\infty$. This in turn implies that $\tilde{T} =T^{*}=\infty$. Thus, we finish the proof of Theorem 1.1.

4 Concluding remarks

In this article, we show stability of the specific stationary solution $\omega _s=0$, $\vartheta _s=y$ to Boussinesq equations without thermal conduction in the two-dimensional domain $\mathbb {T}\times (0,1)$. For the vorticity/velocity, we obtain asymptotic stability and explicit decay rate while for the temperature only the stability in the sense of Lyapunov is given. One may expect the temperature $\vartheta$ converges to $\vartheta _s$ as time goes to infinity. However, this is not true in general, once we realize that the stationary solution

$$\begin{aligned} \tilde{\omega }_s=0,\, \tilde{\vartheta }_s = y+\epsilon \sin 2\pi y \end{aligned}$$

is a small perturbation of $(\omega _s,\vartheta _s)$ if $\epsilon$ is small. Note that the existence of such a small perturbation is due to the choice of our underlying domain $\mathbb {T}\times (0,1)$, which is periodic in the horizontal direction. In a future paper [9], we will consider the underlying domain $\Omega =\mathbb {R}\times (0,1)$, in which case it seems possible to give the asymptotic stability of the temperature under suitable setting.