The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation

1.   Introduction

The Boussinesq equations model buoyancy-driven flows such as geophysical fluids and various Rayleigh-Bénard convection (see, e.g., ^[1,2,3,4]). The Boussinesq equations are mathematically significant ^[3]. This paper concerns a special anisotropic 2D incompressible Boussinesq equation with only vertical dissipations.

where $U$ represents the velocity field of the fluid, $P$ the pressure, $\varTheta$ the temperature, and $e_2 = (0, 1)$ is the unit vector in the vertical direction. Here $\mu > 0$ is the kinematic viscosity, $\eta > 0$ is the thermal diffusivity and the spatial domain $\Omega$ is given by

with $\mathbb{T} = [-\frac{1}{2}, \frac{1}{2}]$ being a 1D periodic box.

This paper attempts to achieve two main goals. The first is to understand the stability and large-time behavior of perturbations near hydrostatic fluid equilibrium given by

It is easy to verify that hydrostatic fluid equilibrium($U_{he}, \varTheta _{he}, P_{he}$) is a steady state solution of (1.1). We consider the perturbation $(u, \theta)$ with

Then $(u, \theta)$ satisfies

The second is to help better reveal the smoothing and stabilization effect of the temperature by considering the system (1.2) with the vertical dissipation in only the second component of the velocity.

The standard incompressible Boussinesq equations with full dissipation read as

The physical background and mathematical features of (1.3) make the model a rich area for mathematical investigations. Over the past decades, the Boussinesq equations have attracted considerable interest from mathematical scholars. Major concerns are oriented around the global well-posedness and finite-time blow up of large-data classical solutions and global regularity for the Boussinesq equations with full partial dissipation, i.e., $\mu = 0$ or $\eta = 0$, or the mixed partial dissipation case (see, e.g., ^{[5,6,7,8,9,10,11,12,13]})

In recent years, the problems of stability and large-time behavior of its solutions has garnered a lot attention, and significant progress have been made. For the 2D case, Doering, Wu, Zhao, and Zheng ^[14] rigorously proved the global asymptotic stability near a special type of hydrostatic equilibrium without buoyancy diffusion on a bounded domain subject to stress-free boundary conditions. Later, Tao and Wu ^[15] resolved some of the problems left open in ^[14]. They studied the stability problem for perturbations near hydrostatic equilibrium of the 2D Boussinesq equations without thermal diffusion in the periodic domain ${\mathbb{T}}^2$. Ben Said, Pandey, and Wu ^[16] solved the stability problem for a 2D Boussinesq system with only vertical dissipation and horizontal thermal diffusion in $\mathbb{R}^2$. Furthermore, when the dissipation is the opposite of that in ^[16], i.e. the horizontal dissipation and the vertical thermal diffusion, ^[17] established the stability in the Sobolev space $H^2$ and obtained algebraic decay rates for the oscillation parts in the $H^1$-norm when the spatial domain $\Omega$ is $\mathbb{T}\times\mathbb{R}$. More results with partial dissipation in two dimensions can be found in ^{[15,18,19,20,21,22,23,24,25]}. For the 3D case, there are also some developments on the stability of solutions (see, e.g., ^{[26,27,28,29,30,31,32]}). Here we recall a recent result obtained by Wu and Zhang in ^[32]. They considered a 3D anisotropic Boussinesq system in the periodic domain $\Omega = \mathbb{R}^2 \times \mathbb{T}$. The stability and large-time behavior problem on perturbations near the hydrostatic balance were established.

Our paper here focuses on the 2D Boussinesq equations with only vertical dissipations. In order to better understand relevant progress and our difficulties, let's review some related results, which means the system with partial dissipation only in one direction. Cao and Wu ^[33] established the global-in-time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation in ${\mathbb{R}}^2$ and solved the global regularity problem. The stability of the 2D Boussinesq equations with only horizontal or vertical dissipation remains an open problem. Some recent works are devoted to this system in the periodic domain. Dong, Wu, Xu, and Zhu ^[34] investigated the stability and exponential decay of the 2D Boussinesq equations with horizontal dissipation in the domain $\mathbb{T} \times \mathbb{R}$. Also, ^[35] proved the nonlinear stability of Couette flow in a uniform magnetic field with only vertical dissipation in the same domain as ^[34]. Now whether the solution of the 2D system remains stable in a periodic domain if the velocity has horizontal or vertical dissipation only in one component equation, say, $u_1$ or $u_2$ equation.

Motivated by the above works related to only one-direction dissipation, we examine the 2D Boussinesq (1.2) in ${\mathbb{T}}^2$ and establish the stability result and the exponential decay rates of the solution. Before stating our results, we first assume that $u_0$ and $\theta_0$ satisfy the symmetry as follows:

Theorem 1.1. Consider the 2D Boussinesq equation (1.2) with the initial data $(u_0, \theta_0)\in H^2(\Omega)$ satisfying $\nabla\cdot u_0 = 0$ and the symmetry condition (1.4). Then there exists $\delta > 0$ such that, if

then (1.1) possesses a unique global solution satisfying, for any $t > 0$,

for some universal constant $C > 0$.

Remark 1.2. The symmetry property (1.4) for the solution $(u, {\theta})$ at $t = 0$ can persist for any time $t > 0$, namely,

A similar proof can be found in ^[32] and ^[36].

Remark 1.3. If we consider the 2D Boussinesq equation with horizontal dissipation

the stability result in Theorem 1.1 still holds provided that the symmetry condition (1.4) is replaced by

Theorem 1.1 assesses the global-in-time existence and stability of small solutions to (1.2). Due to the lack of the horizontal dissipation, the proof of Theorem 1.1 is nontrivial. Especially, the velocity equation involves only vertical dissipation of $u_2$; it is extremely challenging to control the growth of the Navier-Stokes nonlinear term, i.e., $u\cdot{\nabla} u$. In fact, when Navier-Stokes possesses the dissipation in one direction, namely

the global existence in time of solutions in the whole space ${\mathbb{R}}^2$ remains an open problem. Here we consider the periodic domain ${\mathbb{T}}^2$, which will greatly help solve this problem. More precisely, our proof will take advantage of the domain and explore many significant properties. Based on these properties, several key anisotropic inequalities will then be introduced. There are two important observations. The first is that by separating a physical quantity into its average, including both horizontal and vertical directions and the corresponding oscillations, we are able to establish the strong Poincaré-type inequalities, which are very powerful tools and also play a crucial role in the proof. The second observation is that if $(u_0, \theta_0)$ satisfies the symmetry given in (1.4), then $(u, \theta)$ maintains the same symmetries, namely,

This can be achieved via the uniqueness of the solution. Specifically, define

It easily verifies that $U = (U_1, U_2)$, $P$, and $\varTheta$ are still the solution of (1.2). Then the uniqueness implies the symmetries (1.8). Based on the symmetric property, another strong version of the Poincaré inequality can be obtained. With these properties and inequalities at our disposal, we can resolve all the difficult items.

Let us briefly outline the sketch of the proof. The framework in the proof of Theorem 1.1 is the bootstrapping argument. We first introduce some notations. For a sufficiently smooth function $f = f(x_1, x_2)$, we define its horizontal average $\overline{f}^{(1)}$ and vertical average $\overline{f}^{(2)}$ by

and the corresponding oscillation part

This decomposition is extremely useful due to some of the related properties (see Lemma 2.1). We remark that the most important property is $\overline{\widetilde{f}^{(i)}}^{(i)} = 0$ for $i = 1, 2$, which allows us to establish a strong Poincaré-type inequality,

Meanwhile, from the symmetries (1.1) we can also obtain another strong Poincaré-type inequality

Furthermore, to deal with the triple products that stem from the nonlinear terms, the anisotropic inequality involving triple products associated with $\widetilde{f}^{(2)}$ is provided,

To obtain the global existence of the solutions in the Sobolev setting $H^2$, we now introduce the $H^2$-energy $E(t)$ defined by

As aforementioned, the most difficult term is the integral involving the nonlinear term in the velocity, i.e.

However, with the help of the strong Poincaré-type inequality and anisotropic inequalities, we are able to settle the difficulty. Take one term for instance, by integrations by parts, (1.11) and (1.13), the following nonlinear integral can be bounded as

Therefore, through a series of subtle bounds, we can control the growth of all nonlinear terms and establish the closed priori estimate:

Then applying a bootstrapping argument to (1.15) implies the uniform upper bound (1.5) for the initial data is small enough.

Next, we show the second theorem assessing the large-time behavior of the solutions of (1.2). More precisely, the exponential decay rates for the oscillation part of the solution are established.

Theorem 1.4. Assume the initial data $(u_0, \theta_0)\in H^2(\Omega)$ with $\nabla\cdot u_0 = 0$ satisfying the symmetry condition (1.4) and

for some $\delta > 0$ small enough. Let $(u, \theta)$ be the corresponding solution of (1.2). Then the oscillation part $(\widetilde u^{(2)}, \widetilde \theta^{(2)})$ decays exponentially in time,

for all $t \ge 0$ and some constant $C > 0$.

Remark 1.5. Following the decay results from Theorem 1.4, the solution $(u, \theta)$ of (1.2) is asymptotically close to the vertical average $(u, \theta)$ in $H^1(\Omega)$ satisfying ($\overline{u}^{(2)}$, $\overline{\theta}^{(2)}$) satisfies

We explain the main idea in the proof of Theorem 1.4. Due to the degeneracy in the viscous dissipation and the heat diffusion, especially, the very weak dissipation for the velocity, it is impossible to establish the large-time behavior for $(u, {\theta})$. We remark that classical approaches such as Schonbek's Fourier splitting method ^[37,38] that solve the fully dissipated system in whole space no longer apply. Therefore, we have to develop some new techniques. Based on one key observation, i.e., the strong Poincaré-type inequality

we are content to investigate the decay of $(\widetilde{u}^{(2)}, \widetilde{{\theta}}^{(2)})$ of the Boussinesq system (1.2) with $\widetilde{u}^{(2)}$ and $\widetilde{{\theta}}^{(2)}$ obeying

Our goal is to derive a differential inequality of the form

which implies the exponential decay rates $X(t)\le C e^{-ct}$. The proof of Theorem 1.4 is divided into two stages. The first stage proves the exponential decay rate for $\|(\widetilde{u}^{(2)}, \widetilde{\theta}^{(2)})\|_{H^1}$ while the second is to estimate $\lVert (\partial _2\nabla \widetilde{u}^{(2)}_2, \partial _2\nabla \widetilde{\theta}^{(2)}) \rVert _{L^2}$. The estimates are more complicated than that of the stability. Besides the inequalities (1.11), (1.12), and (1.13), we need to introduce two additional anisotropic inequalities associated with the $L^4$-norm and $L^{\infty}$-norm (see Lemma 2.4 for details), which will be frequently used in the proof of the decay rates. After a long and delicate estimate, it is obtained that

and

Using (1.11) again yields the desired exponential decay (1.16) and (1.17) in Theorem 1.4. More technical details can be found in the proof of Theorem 1.4 in Section 4.

The rest of this paper is organized as follows. Section 2 presents four tool lemmas to be used in the proof of Theorems 1.1 and 1.4. Section 3 is devoted to the proof of Theorem 1.1. Section 4 proves the exponential decay estimate of Theorem 1.4. At the end, we claim that $C$ may be different for each line in this article.

2.   Decomposition and anisotropic inequalities

To prepare for the proofs in the subsequent sections, we provide some preliminary lemmas. The first presents some properties on $\widetilde{f}^{(i)}$, $\overline{f}^{(i)}$ for $i = 1, 2$, and their derivatives. The second contains three Poincaré-type inequalities, which provide the powerful tools for proving our theorems. The third proposes an anisotropic upper bound for triple products, whereas the last states anisotropic inequalities related to the $L^4$-norm and $L^{\infty}$-norm that serve the proof of the large-time behavior.

We start with the properties of the composition for $f$, which can be derived via the definitions (1.9) and (1.10).

Lemma 2.1. Let $\widetilde{f}^{ (i)}$ and $\overline{f}^{ (i)}$ for $i = 1, 2$ be defined as in (1.9) and (1.10). Then we have

$(1)$ The average operator and the oscillation operator can commute with the derivatives, i.e.

In particular, if $\nabla\cdot f = 0$, then

$(2)$ The corresponding average of the oscillation $\widetilde{f}^{(i)}$ is zero, for $i = 1, 2$

$(3)$ For any $k\geq0$, $\widetilde{f}^{ (i)}$ and $\overline{f}^{ (i)}$ are orthogonal in Sobolev space $H^k$.

The second lemma provides the strong Poincaré-type inequalities associated with the oscillation $\widetilde{f}^{(i)}$ for $i = 1, 2$.

Lemma 2.2. Assume $f \in H^1(\Omega)$. Then for $i = 1, 2$ it holds,

where $C > 0$ is a pure constant. In addition, if we further assume $\overline{f}^{(1)} = 0$, then

Proof.. Without loss of generality, we prove (2.1) for the case $i = 2$. Thanks to the fact that the vertical average of $\widetilde{f}^{(2)}$ is zero, the proof for (2.1) is easy. In fact, by the integral mean value theorem, for any $x_2 \in \mathbb{T}$, there exists $y \in \mathbb{T}$ such that

Using Leibniz's formula yields

By Hölder's inequality,

Then integrating in space $\Omega$, we obtain

(2.2) follows from $\overline{\widetilde{f}^{(2)}}^{(1)} = 0$. By the definition of $\overline{\widetilde{f}^{(2)}}^{(1)}$, $f = \overline{f}^{(2)}+\widetilde{f}^{(2)}$ and $\overline{f}^{(1)} = 0$, we obtain

Then a similar argument to (2.1) yields the desired strong Poincaré-type inequality (2.2) in $x_1$-direction. This concludes the proof of Lemma 2.2. □

The third lemma assesses an anisotropic upper bound for triple products, which will be used frequently in both Theorem 1.1 and 1.4. Similar anisotropic inequalities in ${\mathbb{R}}^2$ are also available (see e.g. ^[39]). We are able to use a similar proof to that in ^[39] together with the Poincaré-type inequality (2.1) to obtain the anisotropic inequality in periodic domain $\Omega$.

Lemma 2.3. For any functions $f, g, h, \partial_2g, \partial_1f \in L^2(\Omega)$, then

Proof.. By Hölder's inequality, Minkowski's inequality, (2.1), (2.7), and (2.8), we have

□

We now state the last lemma, which provides two anisotropic upper bounds on $L^4$-norm and $L^{\infty}$-norm of $\widetilde{f}^{(2)}$. It can be achieved via 1D inequalities of $L^\infty$-norm and the strong Poincaré-type inequality in Lemma 2.2.

Lemma 2.4. Assume $\partial_1 f \in L^2(\Omega)$ and $\partial_2 f \in H^1(\Omega)$. Then the following inequalities holds,

where $C > 0$ are some pure constants.

Proof.. To prove inequality (2.4), we need the 1D inequalities of $L^\infty$-norm,

which can be obtained through a slight modification of the proof for (2.1).

Applying Hölder's inequality in one component and Minkowski's inequality, combining (2.1) and (2.8), we have

which, combining with $\|\widetilde{f}^{(2)}\|\le \| \partial_2\widetilde{f}^{(2)}\|$, derives (2.4).

(2.6) is the direct consequence of Hölder's inequality and Poincaré inequality (2.1).

This completes the proof of Lemma 2.4. □

3.   Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1, which claims the global existence and stability of solutions of (1.2). To obtain this result, we need to establish a global priori estimate of the energy $E(t)$, as shown in Proposition 3.1. With the energy inequality at our disposal, we are then able to prove Theorem 1 by using the bootstrapping argument (see [40, p.21]).

Proposition 3.1. Assume the initial data $(u_0, \theta_0)$ satisfies the conditions in (1.4). Let $E(t)$ be an energy functional defined by

Then there exist two constants $C_0$ and $C_1$, depending on $\mu$ and $\eta$ such that, for $0 < t < T$,

Proof of proposition 3.1. First, we have the $L^2$-bound

Note that the norm $\|(u(t), \theta(t))\|_{H^{2}}$ is equivalent to $\|(u(t), \theta(t))\|_{L^2}+\|(u(t), \theta(t))\|_{\dot{H}^{2}}$. Thus it suffices to bound $\|(u(t), \theta(t))\|_{\dot{H}^{2}}$. Applying $\partial _{i}^{2} (i = 1, 2)$ to (1.2), taking the $L^2$-inner product of the resulted equations with $(\partial_{i}^{2}u, \partial_{i}^{2}\theta)$, and using divergence-free condition for $u$, we obtain

where we have used

To make full use of the anisotropic dissipation, by integrations by parts and ${\nabla}\cdot u = 0$, we first split $I_1$ into four parts.

We now bound $I_{11}$ through $I_{14}$ one by one. For $I_{11}$, we further decompose it as follows:

Thanks to the dissipation of $u_2$ in the $x_2$-direction, direct applications of Hölder's inequality, Sobolev's inequality, and $\nabla \cdot u = 0$ can show that

According to Lemma 2.1, we obtain $\partial _1\partial _2u_2 = \partial _1\partial _2\widetilde{u}_2^{(2)}$. Then by integration by parts, Hölder's inequality, Sobolev's inequality, (2.1), and Lemma 2.3, $I_{11, 2}$ can be bounded as

Similarly,

For $I_{11, 3}$, with the help of the symmetry $\overline{u}_1^{(1)} = 0$ together with (2.2), it is easy to obtain

Combining all estimates above for $I_{11, 1}$ through $I_{11, 4}$ yields

$I_{12}$ can be handled similarly to $I_{11}$. We first rewrite it as follows:

Then invoking (3.4), (3.5), and (3.6) and applying Hölder's inequality, Sobolev's inequality to the third term yields

We proceed to bound $I_{13}$. As $I_{11}$, $I_{13}$ is first divided into four parts.

Lemma 2.1, integration by parts, Hölder's inequality together with (2.1) lead to

Similarly, $I_{13, 3}$ can be estimated as

By (2.3) and $\|u_1\|_{L^2}\le C\| \partial_1u_1\|_{L^2}$, we obtain

Also, we have

Thus, we obtain

With a nearly same argument with $I_{13}$, we derive

In summary, we obtain the upper bound for $I_1$

Next, we turn to deal with $I_2$. It remains to be divided into four parts:

We first split $I_{21}$ into two terms, then use integration by parts and combine with Hölder's inequality, Sobolev's inequality, and (2.1) to obtain

where we have used $\partial_1 \partial_2 u_2 = \partial_1 \partial_2\widetilde{u_2}^{(2)}$ and $\partial_2 {\theta} = \partial_2\widetilde{{\theta}}^{(2)}$ by Lemma 2.1. Similarly, $I_{22}$ can be estimated as follows:

To bound $I_{23}$, we need to resort to the fact that $\overline{u}_1^{(1)} = 0$, i.e., $u_1 = \widetilde{u}_1^{(1)}$. Then by (2.3) and (2.1), we find

For the last term $I_{24}$, it is clear that

As a result of the above estimates, we obtain

Inserting (3.7), (3.8) into (3.3) and integrating it in time, we conclude

which together with (3.2) implies the desired estimates (3.1). The proof of Proposition 3.1 is thus completed. □

We now prove Theorem 1.1.

Proof of Theorem1.1. We now have established a priori estimate on the $H^2$-norm of $(u, \theta)$, namely,

The bootstrapping argument then allows us to prove the stability of the solution, provided that the initial data is sufficiently small, i.e.

To apply the bootstrapping argument, we start with the ansatz that

Then (3.9) together with the small assumption (3.10) implies

or

Thus, the bootstrapping argument asserts for any $t\ge0$,

which means the perturbed solution of (1.2) exists globally for all time. We complete the proof of Theorem 1.1. □

4.   Proof of Theorem 1.4

This section is committed to proving Theorem 1.4. As aforementioned in the introduction, to obtain the exponential decay, we will derive the following differential inequality:

In the proof, we will make extensive use of the anisotropic inequalities and Poincaré inequality presented in Section 2, which play a crucial role in establishing this type of inequality (4.1).

Proof of Theorem1.4. We first construct the equations of $(\widetilde{u}^{(2)}, \widetilde{{\theta}}^{(2)})$. By taking the vertical average of (1.2), it is easy to verify that ($\overline{u}^{(2)}$, $\overline{\theta}^{(2)}$) satisfies

Taking the difference between (1.2) and (4.2), we obtain

Step 1. Decay for $\lVert (\widetilde{u}^{(2)}, \widetilde{{\theta}}^{(2)})\rVert _{H^1}$

Dotting the system (4.3) by $(\widetilde{u}^{(2)}, \widetilde{\theta}^{(2)})$ yields, we obtain

where

Before bounding $J_1$ through $J_6$, we first make the following decompositions by $u = \overline{u}^{(2)}+\widetilde{u}^{(2)}$.

Subsisting (4.5) in $J_1$ and using (2.2) and $\lVert \widetilde{f}^{(2)} \rVert _{L^2}\le \lVert f \rVert _{L^2}$ yields

where we have used, due to $\partial _1\overline{u}_{1}^{(2)} = -\partial _2\overline{u}_{2}^{(2)} = 0$,

Using a similar argument and replacing to apply the Poincaré inequality (2.2) by (2.1), $J_4$ can be estimated as

Invoking $\partial _2\widetilde{f}^{(2)} = \partial _2f$ and $\partial _2\overline{f}^{(2)} = 0$, and noticing the following facts

we have

Then Hölder's inequality, (2.2) and (2.4) lead to

Similarly, $J_3$ is first rewritten as three parts and employing $\|\widetilde{u}_2^{(2)}\|\le C\| \partial_2\widetilde{u}_2^{(2)}\|$ and $\|\widetilde{u}_1^{(2)}\|\le C\| \partial_1\widetilde{u}_1^{(2)}\|$ yields

Next, we estimate $J_5$. By means of Hölder's inequality, Lemma 2.2, Lemma 2.3, and $\|\widetilde {f}^{(2)}\|_{L^2}\le \|f\|_{L^2}$, $J_5$ is bounded by

where we have used

Now to start estimating the last term $J_6$, we use the above Lemma 2.2 and Hölder's inequality, we get

Collecting all estimates above yields

In what follows, we show the differential inequality of $\lVert(\nabla \widetilde{u}^{(2)}, \nabla \widetilde{\theta}^{(2)}) \rVert _{L^2}$. Taking the gradient of (4.3) and multiplying the resulting equations by $(\nabla\widetilde{u}^{(2)}, \nabla \widetilde{\theta}^{(2)})$, we have

By integration by parts and (4.5), $K_1$ is divided into two parts.

Due to Hölder's inequality and Lemma 2.2, we obtain

Also,

which together with the estimates for $K_{11}$ gives

Going through a similar process as in the derivation of (4.10), we have

The estimates of $K_2$ are similar to those of $J_2$. We divide $K_2$ into three parts.

We first bound $K_{21}$, $K_{22}$. At first glance, it seems there are eight terms that need to be estimated. However, due to the fact that $\partial_2\overline{u}^{(2)} = 0$, the decomposition for $K_{21}$ and $K_{22}$ is reduced to three items. Then applying Hölder's inequality, Lemma 2.2, and Lemma 2.3, we obtain

Here we have used

which can be proved by integration by parts and $\partial_2\overline{u_2}^{(2)} = 0$. For $K_{23}$, invoking Poincaré inequality (2.1), (2.2), and the anisotropic inequalities (2.3) and (2.4) yields

We now deal with $K_3$. By (4.7), it naturally divides $K_3$ into three parts.

Again by the good property $\partial _2\overline{f}^{(2)} = 0$, $K_{31}$ and $K_{32}$ can be reformulated as

where we use

Then making full use of (2.1), (2.2), and (2.3), we infer

For $K_{33}$, noticing that by (2.4) and Lemma 2.2

Then it can be estimated as follows:

Now we focus on estimating the term $K_5$, we are able to establish the upper bound in a similar way as in $J_5$. Since $K_5$ has more terms, according to (4.8), this can be divided into three terms,

We proceed to estimate each of these three items separately, owing to ${\nabla} \cdot \overline{u}^{(2)} = 0$ and $\partial_2\overline{f}^{(2)} = 0$, $K_{51}$ passes through the decomposition with only one term,

Applying (2.1) and (2.3) yields

By Hölder's inequality and (4.12), we obtain

Finally, we estimate the last term, this can be decomposed into four terms by a similar method as in $J_6$ to obtain

Combining the estimates for $K_1$ through $K_6$, we obtain

which together with (4.9) derives, for a pure constant $C_1$

Recalling the stability result in Theorem 1.1, we can select $\delta > 0$ in (1.5) to be sufficiently small such that

Moreover, using Poincaré-type inequalities in Lemma 2.2 yields

Then we have

which implies the exponential decay (1.16) in Theorem1.4.

Step 2. Decay for $\lVert (\partial _2\nabla \widetilde{u}^{(2)}, \partial _2\nabla \widetilde{{\theta}}^{(2) })\rVert _{L^2}$

The routine and the procedure of the proof are similar to $\lVert (\widetilde{u}^{(2)}, \widetilde{{\theta}}^{(2)}) \rVert _{H^1}^{2}$. Applying $\partial_2{\nabla}$ operator to (4.3), multiplying the resulting equations by $(\partial_2{\nabla} \widetilde{u}^{(2)}, \partial_2{\nabla} \widetilde{{\theta}}^{(2)})$ and then integrating over $\Omega$ yields

The estimates for $L_1$ and $L_4$ are simple. Invoking (4.5) along with $\partial _2\overline{f}^{(2)} = 0$, applying integration by parts, $u_1 = \widetilde{u}_{1}^{(2)}+\overline{u}_{1}^{(2)}$ and $\nabla u_1 = \nabla\widetilde{u}_{1}^{(2)}+\nabla\overline{u}_{1}^{(2)}$, we obtain

Similarly,

To estimate $L_2$, we first divide it into two terms according to $i = 1$ and $i = 2$.

Again, based on the fact $\partial _2\overline{f}^{(2)} = 0$, $L_{21}$ is further decomposed as follows:

Then Poincaré inequality (2.2), (2.6) along with $\|\widetilde {f}^{(2)}\|_{L^2}, \|\overline {f}^{(2)}\|_{L^2}\le \|f\|_{L^2}$ leads to

Similarly, by Lemma 2.2 and Lemma 2.3 we obtain

where we have used, by integration by parts,

Thus,

The bound for $L_3$ is subtle. We first rewrite it as

Observe that ${\nabla} \overline{u}_1^{(2)} = 0$ and

which can be verified by integration by parts. It is easy to see

Also, by Lemma 2.2 and (2.4), we infer

where we have used

For $L_{33}$, according to Lemma 2.1, $\partial _1\nabla \widetilde{f}^{\left(2 \right)} = \widetilde{\partial _1\nabla f}^{\left(2 \right)}$, and $\partial_1\nabla(\widetilde{u}_1^{(2)} \widetilde{u}_2^{(2)})$ can be decomposed into these four terms:

Then, invoking $\lVert \widetilde{f}^{(2)} \rVert _{L^2}\le \lVert f \rVert _{L^2}$, (2.1), (2.5), and (4.12), $L_{33}$ can be bounded,

Therefore,

With reference to $K_5$, $L_5$ can be first shown as follows:

Applying the equality $\partial _2\overline{f}^{(2)} = 0$ and $\partial_1\overline{u}_1^{(2)} = 0$, using integration by parts, we then obtain

Using (2.3) again together with (2.6) and (2.2), $L_{52}$ can be bounded as

Similarly to $K_{53}$, $L_{53}$ can be obtained as follows:

Combining all estimates above for $L_{51}, L_{52}$, and $L_{53}$, we obtain

After integration by parts, $L_6$ is split into four terms

Similarly,

As a consequence of (4.13), (2.6), (4.15), (4.16), (4.17), and (4.18), we conclude that there exist two constants $C_3$ and $C_4$ such that

Then (4.19) along with the stability result of Theorem 1.1 implies

for some positive constants $C_3$, $C_4$, provided that the initial data is suitable to satisfy

We thus complete the proof of Theorem 1.4.

□

Author contributions

All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. Hongxia Lin, Sabana, and Qing Sun made for mathematical analysis and the derivation of the proof. Qing Sun and Ruiqi You prepared the original manuscript with contributions from all co-authors. Sabana Checked all English editing and grammar. Xiaochuan Guo performed the review and revision.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Hongxia Lin was partially supported by the Natural Science Foundation of Sichuan Province under (Grant No. 2023NSFSC0056), the National Natural Science Foundation of China NSFC (Grant No. 11701049), and Creative Research Groups of the Natural Science Foundation of Sichuan (Grant No. 2023NSFSC1984).

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.