Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain

Abstract

We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based on spectral analysis combined with energy estimates. To the best of our knowledge, our results provide first sharp decay rates for the temperature fluctuation and the vertical velocity to the linearly stratified Boussinesq equations in all intermediate norms.

1 Introduction

We consider the Boussinesq equations for buoyant fluids

$$\begin{aligned} \left\{ \begin{array}{ll} v_t +\nu (-\Delta )^\alpha v +(v \cdot \nabla )v = -\nabla p +\rho e_d,\\ \rho _t +\kappa (-\Delta )^\beta \rho +(v \cdot \nabla )\rho =0,\\ {\text {div}}\,\, v =0,\\ v(x, 0)=v_0(x),\,\, \rho (x, 0)=\rho _0(x), \end{array} \right. \end{aligned}$$

(1.1)

where v, p, and $\rho$ denote the fluid velocity field, scalar pressure and density (or temperature) respectively. The parameter $\alpha \ge 0$ and $\beta \ge 0$ represent the strength of dissipation and thermal diffusion, while the parameters $\nu \ge 0$ and $\kappa \ge 0$ stand for the nonnegative constant fluid viscosity and thermal diffusivity, respectively. The d-dimensional vector $e_d$ stands for $(0, \cdots , 0, 1)^T$.

The Boussinesq equations (1.1) arise in geophysical fluid dynamics to model and study atmospheric and oceanographic flows [36, 39] and describe interesting physical phenomena such as Rayleigh-Bénard convection [18, 22] and turbulence [10]. From a mathematical point of view, the Boussinesq equations are intimately tied to the Euler and Navier-Stokes equations and they share important features such as the vortex stretching. In fact, the two-dimensional inviscid Boussinesq equations can be viewed as the three-dimensional axisymmetric Euler equations for swirling flows [37]. Due to its physical and mathematical relevance, there have been a lot of works and progress made on the Boussinesq system in the past decades: for instance, see [1, 2, 4, 6, 7, 9, 11, 12, 20, 23, 24, 26, 27, 29,30,31, 33,34,35, 41,42,45] and references therein on the local, global well-posedness and regularity problem.

On the other hand, it is well-known that the system (1.1) has the exact solutions, called hydrostatic equilibrium, with the balance equation

$$\begin{aligned} v = 0, \quad \frac{\partial }{\partial x_{d}}p(x_d)= \rho (x_d). \end{aligned}$$

In recent years, the stability around the linearly stratified state $(v_s, \rho _s, p_s):=(0, \cdots , 0, x_d, x_d^2/2)$ has been a subject of active research in the presence of dissipation where damping is understood as a limit of fractional diffusion. For $d=2$, there exist many stability results (see [2, 3] and references therein), while less works are available for other space dimension. Among others, asymptotic stability with velocity damping was studied in ${\mathbb {R}}^3$ [15], and the stability result has been extended to ${\mathbb {R}}^d$ with more general initial data in [28].

In this paper, we focus on the domain with boundary, in particular $\Omega =\mathbb T^{d-1} \times [-1,1]$. This type of domain with $\rho = 1$ and $\rho = -1$ fixed on the bottom boundary and top boundary has been used to demonstrate the Rayleigh-Bénard convection [18, 22], which leads to the instability of the solution by a continuously heated bottom fluid. On the contrary, the opposite case where $\rho = -1$ and $\rho = 1$ on the bottom and top boundary respectively stabilizes the system. We will show stabilizing aspects of the latter by analyzing the dynamics near linearly stratified hydrostatic equilibrium $(v_s, \rho _s, p_s)=(0, \cdots , 0, x_d, x_d^2/2)$. We consider two cases: $\alpha = 0$ (velocity damping) and $\alpha =1$ (velocity diffusion) without thermal diffusion $(\kappa = 0)$. When $\alpha =0$, we take the no-penetration boundary condition $v \cdot n = 0$ and when $\alpha = 1$, we impose the stress free boundary condition, also known as the Lions boundary condition $v \cdot n = 0$ and ${\text {curl}}v \times n = 0$, where the temperature is fixed at $\rho _s = -1$ and $\rho _s = 1$ on the each boundary. Here, n denotes the outward unit normal vector to $\partial \Omega$. Let us set

$$\begin{aligned} \rho (x,t)= x_d+\theta (x,t),\qquad p(x,t)=x_d^2/2 +P(x,t). \end{aligned}$$

Then, the perturbed system is given by

$$\begin{aligned} \left\{ \begin{aligned}&v_t + (-\Delta )^{\alpha }v + (v \cdot \nabla )v = -\nabla P + \theta e_d, \qquad \textrm{div}\, v = 0, \\&\theta _t + (v \cdot \nabla )\theta = -v_d, \\&v(x, 0)=v_0(x),\,\, \theta (x, 0)=\theta _0(x), \end{aligned} \right. \end{aligned}$$

(1.2)

where the boundary conditions of the velocity field are preserved and $\theta$ vanishes on $\partial \Omega$ in each case $\alpha = 0$ and $\alpha = 1$ with $\theta _0|_{\partial \Omega } =0$.

We now discuss some relevant prior works regarding (1.2) starting with the case $\alpha = 0$. Castro, Córdoba, and Lear [5] showed the asymptotic stability of (1.2) for $d=2$. In particular, the authors showed that high order compatibility conditions are satisfied for well-prepared data, and introduced proper solution spaces $X^m(\Omega )$, $Y^m(\Omega ) \subset H^m(\Omega )$ with orthonormal bases (see Sect. 2.2 for the definitions). For their main result, for $m \in \mathbb N$ with $m \ge 17$, the small data global existence with temporal decay estimate $(1+t)^{\frac{m-7}{8}} (\Vert v(t) \Vert _{H^4} + \Vert \bar{\theta } (t) \Vert _{H^4}) \le C$ was obtained, where $\bar{\theta }(t) := \theta (t) - \int _{\mathbb T} \theta (t,x) \,\textrm{d}x_1$. It is worth pointing out that the temporal decay rates of $H^4$-norm increase as m gets larger, namely the solutions are more regular. Next we consider the case $\alpha = 1$. In $d=2$, long time behavior was first considered by Doering et al [13] for $v\in H^2$ and $\theta \in H^1$ and explicit decay rates were given in ${\mathbb {T}}^2$ by Tao et al [40] using the spectral analysis. Recently, Dong and Sun considered the asymptotic stability problem on the infinite flat strip $\mathbb R^{d-1} \times (0,1)$ for $d = 2$ and 3 in [16, 17] respectively, and Dong [14] obtained the stability result on $\mathbb T\times (0,1)$.

In the aforementioned works, some explicit decay rates were obtained with high regularity index m or global existence (2D) with more general initial data was obtained without explicit decay rates. However, the convergence of the temperature fluctuation and its optimal equilibration rate has remained elusive. The goal of this paper is to establish the global existence in $H^m$, $m>2+\alpha + \frac{d}{2}$ satisfying high order compatibility conditions, the convergence of $\theta$ to the asymptotic profile $\sigma$, and sharp decay rates of $(v, \theta -\sigma )$ in $H^s$ norms for all $s\in [0,m]$. We now state the main results:

Theorem 1.1

Let $d \in \mathbb {N}$ with $d \ge 2$ and let $m \in \mathbb {N}$ satisfying $m > 3+\frac{d}{2}$. Then there exists a constant $\delta > 0$ such that if initial data $(v_0,\,\theta _0) \in \mathbb X^m \times X^m(\Omega )$ with ${\text {div}} v_0 = 0$, $\int _{\Omega } v_0 \,\textrm{d}x = 0$, and $\Vert (v_0, \theta _0) \Vert _{H^m}^2 < \delta ^2$, then (1.2) with $\alpha = 1$ possesses a unique global classical solution $(v,\,\theta )$ satisfying

$$\begin{aligned} v \in C([0,\infty ); \mathbb X^m(\Omega )) \cap L^2([0,\infty ); \mathbb X^{m+1}(\Omega )), \qquad \theta \in C([0,\infty ); X^m(\Omega )) \end{aligned}$$

with

$$\begin{aligned} \sup _{t \in [0,\infty )} \Vert (v,\theta )(t) \Vert _{H^m}^2 + \int _0^{\infty } \Vert \nabla v(t) \Vert _{H^m}^2 \,\textrm{d}t+ \int _0^{\infty } \Vert \nabla _h \theta (t) \Vert _{H^{m-2}}^2 \,\textrm{d}t \le 4 \Vert (v_0,\theta _0) \Vert _{H^m}^2.\nonumber \\ \end{aligned}$$

(1.3)

Moreover, there exists a function

$$\begin{aligned} \sigma (x_d) := \int _{\mathbb T^{d-1}} \theta _0\,\textrm{d}x_h - \int _{\mathbb T^{d-1}} \int _0^\infty \left( (v \cdot \nabla )\theta + v_d \right) \,\textrm{d}t \textrm{d}x_h \end{aligned}$$

(1.4)

such that

$$\begin{aligned} (1+t)^{\frac{m-s}{4}} \Vert \theta (t) - \sigma (x_d) \Vert _{H^s} + (1+t)^{\frac{1}{2} + \frac{m-s}{4}} \Vert v(t)\Vert _{H^s} + (1+t)^{1+\frac{m-s}{4}} \Vert v_d (t) \Vert _{H^s} \le C\nonumber \\ \end{aligned}$$

(1.5)

for any $s \in [0,m]$.

Remark 1.2

The assumption $\int _{\Omega } v_{0}\,\textrm{d}x = 0$ is essential for the velocity field v decaying in t (see Lemma 2.1).

Remark 1.3

Indeed for any $\epsilon >0$, there exists a constant $C>0$ such that

$$\begin{aligned} t^{\frac{3}{4} + \frac{m-s}{4}} \Vert v (t) \Vert _{H^{s-\epsilon }} \le C \end{aligned}$$

for any $s \in [0,m+1]$. See Proposition 6.6.

Theorem 1.4

Let $d \in \mathbb {N}$ with $d \ge 2$ and let $m \in \mathbb {N}$ satisfying $m > 2+\frac{d}{2}$. Then there exists a constant $\delta > 0$ such that if initial data $(v_0,\,\theta _0) \in \mathbb X^m \times X^m(\Omega )$ with ${\text {div}} v_0 = 0$ and $\Vert (v_0, \theta _0) \Vert _{H^m}^2 < \delta ^2$, then (1.2) with $\alpha = 0$ possesses a unique global classical solution $(v,\,\theta )$ satisfying

$$\begin{aligned} v \in C([0,\infty ); \mathbb X^m(\Omega )) \cap L^2([0,\infty ); \mathbb X^{m}(\Omega )), \qquad \theta \in C([0,\infty ); X^m(\Omega )) \end{aligned}$$

with

$$\begin{aligned} \sup _{t \in [0,\infty )} \Vert (v,\theta )(t) \Vert _{H^m}^2 + \int _0^{\infty } \Vert v(t) \Vert _{H^m}^2 \,\textrm{d}t+ \int _0^{\infty } \Vert \nabla _h \theta (t) \Vert _{H^{m-1}}^2 \,\textrm{d}t \le 4 \Vert (v_0,\theta _0) \Vert _{H^m}^2.\nonumber \\ \end{aligned}$$

(1.6)

Moreover, there exists a function $\sigma (x_d)$ defined by (1.4) such that

$$\begin{aligned} (1+t)^{\frac{m-s}{2}} \Vert \theta (t) - \sigma (x_d) \Vert _{H^s} + (1+t)^{\frac{1}{2} + \frac{m-s}{2}} \Vert v (t) \Vert _{H^s} + (1+t)^{1+\frac{m-s}{2}} \Vert v_d (t) \Vert _{H^s} \le C\nonumber \\ \end{aligned}$$

(1.7)

for any $s \in [0,m]$.

Remark 1.5

The decay rates for $\theta$ and $v_d$ in Theorem 1.1 and 1.4 are sharp (see Sect. 7).

To the best of our knowledge, our results provide the first sharp decay rates for the temperature fluctuation and the vertical velocity in all intermediate norms. In particular, they show the enhanced $L^2$ decay rate for higher order initial data, while $H^m$ decay rate doesn’t change for both velocity damping and velocity diffusion. This is in contrast to parabolic equations for which higher norms enjoy faster decay rates. The regularity index m required in our analysis is higher than the one required for the local existence, but it is still significantly smaller than the ones required in the previous results. Also our results demonstrate that the velocity damping leads to faster decay than the velocity diffusion in the presence of the slip boundary, despite having the Poincaré inequality for the velocity field in hand. This is because of coupling structure between the velocity field and the temperature fluctuation of Boussinesq equations: it causes the temperature to decay much slower than the velocity field and the velocity diffusion weakens the temperature damping in high frequency. Moreover, the method developed in this paper is robust and applicable to the periodic box ${\mathbb {T}}^d$, and to various partially dissipative PDEs including non-resistive MHD and IPM (cf. [25]).

The main difficulty comes from the non-decaying $\theta$ and weak damping in $\nabla _h\theta$, which makes the standard energy estimates alone hard to bootstrap the local theory to global theory and to capture precise decay rates. To establish the results, we employ the spectral analysis using the orthonormal basis associated to our domain with the slip boundary together with energy estimates, first to obtain the global existence and then to prove the decay rates by relying on the already established uniform bounds of the solutions. The relaxed condition for m comes from estimating the key quantities $\int \Vert \nabla v(t) \Vert _{L^{\infty }} \,\textrm{d}t$ and $\int \Vert \partial _d v_d(t) \Vert _{L^{\infty }} \,\textrm{d}t$ which appear in the energy estimates. The previous works on the stability problem of (1.2) ($d=2$) were devoted to obtaining the temporal decay estimate for $\Vert u(t) \Vert _{H^4}$ or $\Vert \partial _1 {\text {curl}} v(t) \Vert _{H^2}$, which obviously require stronger condition for m (see [5, 14]). Getting decay rates in bounded domains turns out to be more subtle than in the whole space, since $\theta$ does not decay, while it decays in the whole space. To prove the sharp decay rates of $(v, \theta -\sigma )$ in $H^s$ norms for all $s\in [0,m]$ in our domain, we adapt Elgindi’s the splitting scheme of the density first used for the linearly stratified IPM equation in $\mathbb T^2$ [19]. In particular, splitting the density into decaying part and non-decay part and using the boundedness of high norms obtained from the global existence part, the decay of low norms can be obtained through optimizing splitting scale of frequency in spirit of [19]. We refer to Lemma 3.1 for a clear view of the sharp decay estimates for the linearized system of (1.2), and Sect. 6 for controlling the nonlinear terms in (1.2) with the splitting scheme.

The rest of this paper proceeds as follows. In Sect. 2, we give some preliminary results used for the paper and introduce key function spaces $X^m(\Omega ), Y^{m}(\Omega ), \mathbb X^m(\Omega )$ and their orthonormal bases. Section 3 is devoted to spectral analysis of (1.2) in frequency variables and the proof of linear decay estimates. In Sect. 4, we present the energy-dissipation inequalities for (1.2). In Sect. 5, we extend the local existence to global-in-time result by combining the energy estimates with spectral analysis to estimate key quantities (5.1) appearing in the energy estimates. Section 6 is devoted to the proof of temporal decay estimates based on the spectral analysis and the splitting scheme. In Sect. 7, we argue that the decay rates are sharp by showing that the linear decay rates can’t be algebraically improved.

2 Preliminaries

We first introduce some notations that will be used throughout this paper. Let $\langle \cdot ,\cdot \rangle$ be the standard inner product on $\mathbb C^d$ for any $d \ge 2$. We use $\gamma$ as a multi-index, and let $v_h := (v_1, \cdots , v_{d-1})^T$, $x_h := (x_1, \cdots , x_{d-1})^T$, and $\nabla _h := (\partial _1, \cdots , \partial _{d-1})^T$. For any smooth function $f:\Omega \rightarrow \mathbb R$, we use the notation

$$\begin{aligned} \bar{f} := f - \int _{\mathbb T^{d-1}} f(x) \,\textrm{d}x_h. \end{aligned}$$

Next we investigate the average of the solution $(v,\theta )$ over time.

Lemma 2.1

Let $(v,\theta )$ be a smooth solution to (1.2) with $\alpha \in \{0, 1\}$. Then, there hold

$$\begin{aligned} \int _{\mathbb T^{d-1}} v_d(t,x) \,\textrm{d}x_h = 0, \qquad x_d \in [-1,1] \end{aligned}$$

(2.1)

and

$$\begin{aligned} \int _{\Omega } \theta (t,x) \,\textrm{d}x = \int _{\Omega } \theta _0(x) \,\textrm{d}x \end{aligned}$$

(2.2)

for all $t \ge 0$. Moreover, if $\alpha = 1$, then

$$\begin{aligned} \int _{\Omega } v_h(t,x) \,\textrm{d}x = \int _{\Omega } v_h(0,x) \,\textrm{d}x, \end{aligned}$$

(2.3)

and if $\alpha = 0$,

$$\begin{aligned} \int _{\Omega } v_h(t,x) \,\textrm{d}x = e^{-t}\int _{\Omega } v_h(0,x) \,\textrm{d}x. \end{aligned}$$

(2.4)

Proof

By the divergence-free condition and the boundary condition $v_d(x_h,-1) = 0$, we have

$$\begin{aligned} 0 = -\int _{\mathbb T^{d-1} \times [-1,x_d]} \nabla _h \cdot v_h \,\textrm{d}x = \int _{\mathbb T^{d-1} \times [-1,x_d]} \partial _d v_d \,\textrm{d}x = \int _{\mathbb T^{d-1}} v_d(x_h,x_d) \,\textrm{d}x_h \end{aligned}$$

for all $x_d \in [-1,1]$. From the $v_h$ equation in (1.2), we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_h \,\textrm{d}x + \int _{\Omega } (-\Delta )^{\alpha } v_h \,\textrm{d}x + \int _{\Omega } (v \cdot \nabla ) v_h \,\textrm{d}x = -\int _{\Omega } \nabla _h P \,\textrm{d}x. \end{aligned}$$

Integration by parts and the boundary condition for $v_d$ yield

$$\begin{aligned} \int _{\Omega } (v \cdot \nabla ) v_h \,\textrm{d}x = \int _{\Omega } \nabla _h P \,\textrm{d}x = 0, \end{aligned}$$

thus,

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_h \,\textrm{d}x + \int _{\Omega } (-\Delta )^{\alpha } v_h \,\textrm{d}x = 0. \end{aligned}$$

This gives (2.4) when $\alpha = 0$. In the case of $\alpha = 1$, $\partial _d v_h = 0$ on $\partial \Omega$ implies (2.3). Similarly, we can obtain (2.2) by the use of (2.1). This completes the proof. $\square$

2.1 Boundary conditions

In the section, we briefly show in both cases $\alpha = 0$ and $\alpha =1$ the high compatibility conditions, whose statement is as follows: Let $(v,\theta )$ be a global-in-time smooth solution to (1.2) and suppose that there exists $n \in \mathbb N$ such that $\partial _d^{2k} \theta _0 = 0$ holds on the boundary for all $0 \le k \le n$. Then, we have

$$\begin{aligned} \partial _d^{2k} v_d = \partial _d^{2k-1+2\alpha } v_h = \partial _d^{2k} \theta = \partial _d^{2k-1} P = 0 \end{aligned}$$

(2.5)

for any $1 \le k \le n$.

When $d=2$, Castro, Córdoba, and Lear [5] and Dong [14] showed (2.5) for $\alpha =0$ and $\alpha = 1$ respectively. It is not hard to extend it to the $d \ge 3$ case. Here, we only give details for the case $\alpha = 1$.

From our boundary conditions, we see that

$$\begin{aligned} v_d(x) = \theta (x) = 0 \qquad \text{ and } \qquad \partial _d v_h(x) = 0, \qquad x \in \partial \Omega . \end{aligned}$$

(2.6)

By (2.6) and the incompressibility, it holds

$$\begin{aligned} \partial _d^2 v_d(x) = -\nabla _h \cdot \partial _dv_h(x) = 0, \qquad x \in \partial \Omega . \end{aligned}$$

Then from the $v_d$ equation in (1.2), we can see

$$\begin{aligned} -\partial _d P = \partial _t v_d -\Delta v_d + (v \cdot \nabla )v_d - \theta = 0 \end{aligned}$$

on the boundary. Next, we apply $\partial _d$ to the $v_h$ equation in (1.2) and have

$$\begin{aligned} \partial _t \partial _d v_h - \Delta \partial _d v_h + \partial _d (v \cdot \nabla ) v_h = -\nabla _h \partial _d P. \end{aligned}$$

The previous results imply that $\partial _d^3 v_h = 0$ on the boundary. From the $\theta$ equation in (1.2), we can see

$$\begin{aligned} \partial _t \partial _d^2 \theta + \partial _d^2 (v \cdot \nabla ) \theta = -\partial _d^2 v_d, \end{aligned}$$

hence,

$$\begin{aligned} \partial _t \partial _d^2 \theta + \partial _d v_d \partial _d^2 \theta + (v_h \cdot \nabla _h) \partial _d^2 \theta = 0, \qquad x \in \partial \Omega . \end{aligned}$$

Consider the flow map $\Phi (t,x)$ with $\partial _t \Phi (t,x) = (v_h(t,\Phi (t,x)),0)$. Then, it holds

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \partial _d^2 \theta (t,\Phi (t,x)) + \partial _dv_d(t,\Phi (t,x)) \partial _d^2 \theta (t,\Phi (t,x))= 0. \end{aligned}$$

By the use of Grönwall’s inequality, we have

$$\begin{aligned} \partial _d^2 \theta (t,\Phi (t,x)) = \partial _d^2 \theta _0(x)\exp \left( \int _0^t \partial _dv_d(\tau ,\Phi (\tau ,x)) \,\textrm{d}\tau \right) . \end{aligned}$$

Thus, $\partial _d^2 \theta _0 = 0$ is conserved over time on the boundary, whenever $\partial _d v_d \in L^1_t$. Thus, (2.5) with $k=1$ is obtained. It is clear that

$$\begin{aligned} \partial _d^4 v_d(x) = -\nabla _h \cdot \partial _d^3v_h(x) = 0, \qquad x \in \partial \Omega . \end{aligned}$$

Repeating the above processes, we can deduce (2.5) for all $1 \le k \le n$.

2.2 Functional spaces and orthonormal bases

To introduce our solution spaces, we define orthonormal sets $\{ b_q \}_{q \in \mathbb N}$ and $\{ c_q \}_{q \in \mathbb N\cup \{0\}}$ by

$$\begin{aligned} b_{q} (x_d)&= {\left\{ \begin{array}{ll} \displaystyle \sin (\frac{\pi }{2} q x_d) &{}\quad q : \text{ even } \\ \displaystyle \cos (\frac{\pi }{2} q x_d) &{}\quad q : \text{ odd } \end{array}\right. } \qquad \text{ with } x_d \in [-1,1],\\ c_{q} (x_d)&= {\left\{ \begin{array}{ll} \displaystyle -\sin (\frac{\pi }{2} q x_d) &{}\quad q : \text{ odd } \\ \displaystyle \cos (\frac{\pi }{2} q x_d) &{}\quad q : \text{ even } \end{array}\right. } \qquad \text{ with } x_d \in [-1,1]. \end{aligned}$$

Note that each set is orthonormal basis for $L^2([-1,1])$. Let

$$\begin{aligned}{} & {} \mathscr {B}_{n,q}(x) := e^{2\pi i n \cdot x_h} b_{q}(x_d), \qquad (n,q) \in \mathbb Z^{d-1} \times \mathbb N,\\{} & {} \mathscr {C}_{n,q}(x) := e^{2 \pi i n \cdot x_h} c_{q}(x_d), \qquad (n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}. \end{aligned}$$

Then we have the following relations

$$\begin{aligned} \nabla _h \mathscr {B}_{n,q} = 2\pi i n \mathscr {B}_{n,q}, \quad \nabla _h \mathscr {C}_{n,q} = 2\pi i n \mathscr {C}_{n,q}, \quad \partial _d \mathscr {B}_{n,q} = \frac{\pi }{2}q \mathscr {C}_{n,q}, \quad \partial _d \mathscr {C}_{n,q} = -\frac{\pi }{2} q \mathscr {B}_{n,q}. \end{aligned}$$

Now, we consider the function spaces

$$\begin{aligned} X^m(\Omega )&:= \{ f \in H^m(\Omega ) ; \partial _d^k f |_{\partial \Omega } = 0, \quad k = 0,2,4, \cdots , m^*\}, \\ Y^m(\Omega )&:= \{ f \in H^m(\Omega ) ; \partial _d^k f |_{\partial \Omega } = 0, \quad k = 1,3,5, \cdots , m_*\}, \end{aligned}$$

where

$$\begin{aligned} m^* := {\left\{ \begin{array}{ll} \displaystyle m-2, &{}\quad m : \text{ even } \\ \displaystyle m-1, &{}\quad m : \text{ odd } \end{array}\right. } \qquad and \qquad m_* := {\left\{ \begin{array}{ll} \displaystyle m-1, &{}\quad m : \text{ even } \\ \displaystyle m-2, &{}\quad m : \text{ odd } . \end{array}\right. } \end{aligned}$$

Then, $\{ \mathscr {B}_{n,q} \}_{(n,q) \in \mathbb Z^{d-1} \times \mathbb N}$ and $\{ \mathscr {C}_{n,q} \}_{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{0\}}$ become orthonormal bases of $X^m(\Omega )$ and $Y^m(\Omega )$ respectively. For the velocity field, we define a d-dimensional vector space $\mathbb X^m(\Omega )$ by

$$\begin{aligned} \mathbb X^m(\Omega ) := \{ v \in H^m(\Omega ) ; v = (v_h,v_d) \in Y^m(\Omega ) \times X^m(\Omega )\}. \end{aligned}$$

We introduce series expansions of the elements in $X^m(\Omega )$ and $Y^m(\Omega )$. Let

$$\begin{aligned} \mathscr {F}_b f(n,q) := \int _{\Omega } f(x) \overline{\mathscr {B}_{n,q}(x)} \,\textrm{d}x, \qquad \mathscr {F}_c f(n,q) := \int _{\Omega } f(x) \overline{\mathscr {C}_{n,q}(x)} \,\textrm{d}x \end{aligned}$$

for each $(n,q) \in \mathbb Z^{d-1} \times \mathbb N$ and $(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}$ respectively. Then for any $f \in X^m(\Omega )$ and $g \in Y^m(\Omega )$, we can write

$$\begin{aligned} f(x) = \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} \mathscr {F}_bf(n,q) \mathscr {B}_{n,q}(x), \qquad g(x) = \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}} \mathscr {F}_cg(n,q) \mathscr {C}_{n,q}(x). \end{aligned}$$

We refer to [5, Lemma 3.1] for details.

We give two simple lemmas. The first one implies $fg \in X^m$ when $f \in X^m$ and $g \in Y^m$, and the second one implies $fg \in Y^m$ when $f,g \in X^m$ or $f,g \in Y^m$ for any given $m \in \mathbb N$ with $m > d/2$. Since the proofs are elementary, we omit them.

Lemma 2.2

Let $q_e$ and $q_o$ be even and odd number respectively. Then, there hold

$$\begin{aligned} \begin{aligned} -\sin (\frac{\pi }{2}q_e x_d) \sin (\frac{\pi }{2} q_o x_d)&= \frac{1}{2} \left( \cos (\frac{\pi }{2}(q_e + q_o)x_d) - \cos (\frac{\pi }{2}(q_e - q_o)x_d) \right) , \\ \sin (\frac{\pi }{2}q_e x_d) \cos (\frac{\pi }{2}q'_e x_d)&= \frac{1}{2} \left( \sin (\frac{\pi }{2}(q_e+q_e')x_d) + \sin (\frac{\pi }{2}(q_e-q_e')x_d) \right) , \\ -\cos (\frac{\pi }{2} q_o x_d) \sin (\frac{\pi }{2} q_o' x_d)&= -\frac{1}{2} \left( \sin (\frac{\pi }{2}(q'_o+q_o)x_d) + \sin (\frac{\pi }{2}(q_o'-q_o)x_d) \right) , \\ \cos (\frac{\pi }{2}q_o x_d) \cos (\frac{\pi }{2} q_e x_d)&= \frac{1}{2} \left( \cos (\frac{\pi }{2}(q_o+q_e)x_d) + \cos (\frac{\pi }{2}(q_o - q_e)x_d) \right) . \end{aligned} \end{aligned}$$

Lemma 2.3

Let $q-q'$ and $q-q''$ be odd and even number respectively. Then, there hold

$$\begin{aligned} \begin{aligned} -\sin (\frac{\pi }{2}q x_d) \sin (\frac{\pi }{2} q'' x_d)&= \frac{1}{2} \left( \cos (\frac{\pi }{2}(q + q'')x_d) - \cos (\frac{\pi }{2}(q - q'')x_d) \right) , \\ \sin (\frac{\pi }{2}q x_d) \cos (\frac{\pi }{2}q' x_d)&= \frac{1}{2} \left( \sin (\frac{\pi }{2}(q+q')x_d) + \sin (\frac{\pi }{2}(q-q')x_d) \right) , \\ -\cos (\frac{\pi }{2} q x_d) \sin (\frac{\pi }{2} q' x_d)&=- \frac{1}{2} \left( \sin (\frac{\pi }{2}(q'+q)x_d) + \sin (\frac{\pi }{2}(q'-q)x_d) \right) , \\ \cos (\frac{\pi }{2}q x_d) \cos (\frac{\pi }{2} q'' x_d)&= \frac{1}{2} \left( \cos (\frac{\pi }{2}(q+q'')x_d) + \cos (\frac{\pi }{2}(q - q'')x_d) \right) . \end{aligned} \end{aligned}$$

The next proposition provides convolution estimates similar to the Fourier expansion.

Proposition 2.4

Let $f,f' \in X^m$ and $g,g' \in Y^m$ for some $m \in \mathbb N$ with $m > \frac{d}{2}$. Then, there hold

$$\begin{aligned} \begin{aligned} \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} |\mathscr {F}_b[fg](n,q)|&\le \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} |\mathscr {F}_b f(n,q)| \right) \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}} |\mathscr {F}_c g(n,q)| \right) , \\ \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}} |\mathscr {F}_c[ff'](n,q)|&\le \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} |\mathscr {F}_b f(n,q)| \right) \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} |\mathscr {F}_b f'(n,q)| \right) , \\ \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{0\}} |\mathscr {F}_c[gg'](n,q)|&\le \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}} |\mathscr {F}_c g(n,q)| \right) \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}} |\mathscr {F}_c g'(n,q)| \right) . \end{aligned} \end{aligned}$$

Proof

We only show the first inequality because the others can be proved similarly. By the series expansions of $f \in X^m(\Omega )$ and $g \in Y^m(\Omega )$, it holds

$$\begin{aligned} fg(x)&= \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} \mathscr {F}_bf(n,q) \mathscr {B}_{n,q}(x) \right) \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}} \mathscr {F}_cg(n,q) \mathscr {C}_{n,q}(x) \right) . \end{aligned}$$

By the use of Lemma 2.2, we can see for each $(n,q) \in \mathbb Z^{d-1} \times \mathbb N$ that

$$\begin{aligned}{} & {} |\mathscr {F}_b[fg](n,q)| \\{} & {} \quad \le \sum _{n'+n''=n} \left( \frac{1}{2} \sum _{q'+q''=q} |\mathscr {F}_bf(n',q')| |\mathscr {F}_cg(n'',q'')| + \frac{1}{2} \sum _{|q'-q''|=q} |\mathscr {F}_bf(n',q')| |\mathscr {F}_cg(n'',q'')| \right) . \end{aligned}$$

This estimate infers

$$\begin{aligned} \begin{aligned} \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} |\mathscr {F}_b[fg]|&\le \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N} |\mathscr {F}_b f| \right) \left( \sum _{(n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{ 0 \}} |\mathscr {F}_c g| \right) . \end{aligned} \end{aligned}$$

This finishes the proof. $\square$

3 Spectral analysis

In this section, we give a different form of (1.2) via spectral analysis. Then, we provide temporal decay estimates for the linear operator of (1.2). From now on, we use the notations $\tilde{n} := 2\pi n$ and $\tilde{q} := \frac{\pi }{2} q$ for each $n \in \mathbb Z^{d-1}$ and $q \in \mathbb N\cup \{0\}$. We define two sets

$$\begin{aligned} I:= \{\eta = (\tilde{n},\tilde{q}) ; (n,q) \in \mathbb Z^{d-1} \times \mathbb N\cup \{0\} \}, \qquad J:= \{\eta = (\tilde{n},\tilde{q}) ; (n,q) \in \mathbb Z^{d-1} \times \mathbb N\}. \end{aligned}$$

We estimate the pressure term first. From the v equation in (1.2), we can see

$$\begin{aligned} \textrm{div}\, (v \cdot \nabla ) v = -\Delta P + \partial _d \theta . \end{aligned}$$

Using the basis $\mathscr {C}_{n,q}(x) = \mathscr {C}_{\eta }(x)$, we have for each $\eta \in I \setminus \{0\}$ that

$$\begin{aligned} \begin{aligned} \mathscr {F}_c P(\eta )&= \frac{1}{|\eta |^2} \mathscr {F}_c [\textrm{div}\, (v \cdot \nabla ) v] (\eta ) - \frac{1}{|\eta |^2} \mathscr {F}_c \partial _d \theta (\eta ). \end{aligned} \end{aligned}$$

Since $\nabla _h \mathscr {C}_{\eta } = i\tilde{n} \mathscr {C}_{\eta }$ and $\partial _d \mathscr {C}_{\eta } = -\tilde{q} \mathscr {B}_{\eta }$, we can see

$$\begin{aligned} \mathscr {F}_c [\textrm{div}\, (v \cdot \nabla ) v] (\eta )&= \int _{\Omega } \nabla _h \cdot (v \cdot \nabla )v_h(x) \overline{\mathscr {C}_{\eta }(x)}\,\textrm{d}x + \int _{\Omega } \partial _d (v \cdot \nabla )v_d (x)\overline{\mathscr {C}_{\eta }(x)} \,\textrm{d}x \\&= i\tilde{n} \cdot \int _{\Omega } (v \cdot \nabla )v_h(x) \overline{\mathscr {C}_{\eta }(x)}\,\textrm{d}x + \tilde{q} \int _{\Omega } (v \cdot \nabla )v_d(x) \overline{\mathscr {B}_{\eta }(x)}\,\textrm{d}x \\&= i\tilde{n} \cdot \mathscr {F}_c [(v \cdot \nabla )v_h] (\eta ) + \tilde{q} \mathscr {F}_b [(v \cdot \nabla )v_d] (\eta ) \end{aligned}$$

and

$$\begin{aligned} \mathscr {F}_c \partial _d \theta (\eta ) = \tilde{q} \mathscr {F}_b \theta (\eta ). \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \nabla P(x) = \left( \sum _{\eta \in I} \mathscr {F}_c \nabla _h P(\eta ) \mathscr {C}_{\eta }(x), \sum _{\eta \in J} \mathscr {F}_b \partial _d P(\eta ) \mathscr {B}_{\eta }(x) \right) ^{T}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \mathscr {F}_c \nabla _h P(\eta )&= -\frac{\tilde{n} \otimes \tilde{n}}{|\eta |^2} \mathscr {F}_c [(v \cdot \nabla )v_h] (\eta ) + i\frac{\tilde{q}\tilde{n}}{|\eta |^2} \mathscr {F}_b [(v \cdot \nabla )v_d] (\eta ) - i \frac{\tilde{q}\tilde{n}}{|\eta |^2} \mathscr {F}_b \theta (\eta ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \mathscr {F}_b \partial _d P(\eta )&= -i\frac{\tilde{q}\tilde{n}}{|\eta |^2} \cdot \mathscr {F}_c [(v \cdot \nabla )v_h] (\eta ) - \frac{\tilde{q}^2}{|\eta |^2} \mathscr {F}_b [(v \cdot \nabla )v_d] (\eta ) + \frac{\tilde{q}^2}{|\eta |^2} \mathscr {F}_b \theta (\eta ). \end{aligned} \end{aligned}$$

From these formulas, we have

$$\begin{aligned} \begin{aligned} \partial _t \mathscr {F}_c v_h&+ |\eta |^{2\alpha } \mathscr {F}_c v_h + \left( I - \frac{\tilde{n} \otimes \tilde{n}}{|\eta |^2} \right) \mathscr {F}_c \left[ (v \cdot \nabla )v_h\right] + i\frac{\tilde{q}\tilde{n}}{|\eta |^2} \mathscr {F}_b \left[ (v \cdot \nabla )v_d\right] \\&\quad - i\frac{\tilde{q} \tilde{n}}{|\eta |^2} \mathscr {F}_b \theta = 0 \end{aligned} \end{aligned}$$

(3.1)

for $\eta \in I \setminus \{0\}$, and

$$\begin{aligned} \partial _t \mathscr {F}_b v_d+ & {} |\eta |^{2\alpha } \mathscr {F}_b v_d + \left( 1- \frac{\tilde{q}^2}{|\eta |^2} \right) \mathscr {F}_b \left[ (v \cdot \nabla )v_d\right] \nonumber \\{} & {} \quad - i\frac{\tilde{q} \tilde{n}}{|\eta |^2} \cdot \mathscr {F}_c [(v \cdot \nabla )v_h] - \frac{|\tilde{n}|^2}{|\eta |^2} \mathscr {F}_b \theta = 0, \end{aligned}$$

(3.2)

$$\begin{aligned} \partial _t \mathscr {F}_b \theta+ & {} \mathscr {F}_b \left[ (v \cdot \nabla )\theta \right] + \mathscr {F}_b v_d = 0 \end{aligned}$$

(3.3)

for $\eta \in J$. Due to the linear structure of (3.2) and (3.3), we can observe a partially dissipative nature by writing the two equaitons at once with $\textbf{u} := (v_d,\theta )^{T}$.

Let us define an operator $\mathscr {F} : (L^2)^{d-1} \times L^2 \rightarrow \mathbb C^{d-1} \times \mathbb C$ by $\mathscr {F} := (\mathscr {F}_c,\mathscr {F}_b)$. Then, it follows

$$\begin{aligned} \partial _t \mathscr {F}_b\textbf{u} + M \mathscr {F}_b \textbf{u} + \langle \mathbb {P}\, \mathscr {F}(v \cdot \nabla )v, e_{d} \rangle e_1 + \mathscr {F}_b [(v \cdot \nabla )\theta ] e_2 = 0, \end{aligned}$$

(3.4)

where

$$\begin{aligned} \mathbb {P} := I - \frac{1}{|\eta |^2} \left( \begin{array}{c|c} \tilde{n} \otimes \tilde{n} &{} -i \tilde{q}\tilde{n} \\ \hline i\tilde{q} \tilde{n} &{} \tilde{q}^2 \end{array} \right) , \qquad M := \begin{pmatrix} |\eta |^{2\alpha } &{} -\frac{|\tilde{n}|^2}{|\eta |^2} \\ 1 &{} 0 \end{pmatrix}. \end{aligned}$$

For simplicity, we use the notation

$$\begin{aligned} N(v,\theta ) := \langle \mathbb {P}\, \mathscr {F}(v \cdot \nabla )v, e_{d} \rangle e_1 + \mathscr {F}_b [(v \cdot \nabla )\theta ] e_2. \end{aligned}$$

Since the characteristic equation of $M^{T}$ is given by

$$\begin{aligned} {\text {det}} \big ( M^{T} - \lambda I \big ) = \lambda ^2 - |\eta |^{2\alpha } \lambda + \frac{|\tilde{n}|^2}{|\eta |^2}, \end{aligned}$$

the two pair of eigenvalue and eigenvector $(\lambda _\pm (\eta ), \overline{\textbf{a}_\pm (\eta )})$ satisfy

$$\begin{aligned} \lambda _{\pm }(\eta ) = \frac{|\eta |^{2\alpha } \pm \sqrt{|\eta |^{4\alpha } - {4 |\tilde{n}|^2}/{|\eta |^2}}}{2}, \qquad \overline{\textbf{a}_{\pm } (\eta )} = \begin{pmatrix} \lambda _{\pm } \\ -\frac{|\tilde{n}|^2}{|\eta |^2} \end{pmatrix} , \end{aligned}$$

where $M^{T} \overline{\textbf{a}_\pm (\eta )} = \lambda (\eta )_\pm \overline{\textbf{a}_\pm (\eta )}$ holds. We note that there is no pair $\eta \in J$ satisfying $|\eta |^{4\alpha } - {4 |\tilde{n}|^2}/{|\eta |^2} = 0$. Since

$$\begin{aligned} A:= (\overline{\textbf{a}_+}\,\, \overline{\textbf{a}_-}) \qquad \text{ and } \qquad B := \frac{1}{\lambda _{+}-\lambda _{-}} \begin{pmatrix} 1 &{} \frac{| \eta |^2}{|\tilde{n}|^2} \lambda _{-} \\ -1 &{} -\frac{| \eta |^2}{|\tilde{n}|^2} \lambda _{+} \end{pmatrix} = \begin{pmatrix} \textbf{b}_+ \\ \textbf{b}_- \end{pmatrix} \end{aligned}$$

satisfy $BA = I$, it follows by Duhamel’s principle

$$\begin{aligned} \langle \mathscr {F}_b\textbf{u}(t),\textbf{a}_\pm \rangle \textbf{b}_\pm = e^{-\lambda _\pm t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_\pm \rangle \textbf{b}_\pm - \int _0^t e^{-\lambda _\pm (t - \tau )} \langle N(v,\theta )(\tau ),\textbf{a}_\pm \rangle \textbf{b}_\pm \,\textrm{d}\tau . \end{aligned}$$

(3.5)

However, using this formula directly can be problematic because of the unboundedness of $|\textbf{b}_{\pm }|$ around the set $\{ |\eta |^{4\alpha } = 4|\tilde{n}|^2/|\eta |^2 \}$. For this reason, we employ

$$\begin{aligned} \begin{aligned} \mathscr {F}_b \theta (t)&= \sum _{j \in \pm } e^{-\lambda _j t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_j \rangle \langle \textbf{b}_j,e_{2} \rangle - \sum _{j \in \pm } \int _0^t e^{-\lambda _j (t - \tau )} \langle N(v,\theta )(\tau ),\textbf{a}_j \rangle \langle \textbf{b}_j,e_{2} \rangle \,\textrm{d}\tau \\&= (e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_{2} \rangle + e^{-\lambda _+ t} \mathscr {F}_b \theta _0\\&\quad - \int _0^t (e^{-\lambda _- (t - \tau )} - e^{-\lambda _+ (t - \tau )}) \langle N(v,\theta )(\tau ),\textbf{a}_- \rangle \langle \textbf{b}_-,e_{2} \rangle \,\textrm{d}\tau \\&\quad - \int _0^t e^{-\lambda _+ (t - \tau )} \mathscr {F}_b [(v \cdot \nabla )\theta ] \,\textrm{d}\tau , \end{aligned} \end{aligned}$$

(3.6)

which allows us to get rid of the singularity of $\textbf{b}_{\pm }$. Here, we note some useful calculations when using (3.6). From the definition of $\lambda _{\pm }$, $\textbf{a}_{\pm }$, and $\textbf{b}_{\pm }$, we have

$$\begin{aligned} |e^{-\lambda _{+}(\eta )t}|\le & {} e^{-|\eta |^{2\alpha } \frac{t}{2}}, \qquad |e^{\lambda _{-}(\eta )t}| \le {\left\{ \begin{array}{ll} e^{-|\eta |^{2\alpha } \frac{t}{2}}, &{}\qquad |\eta |^{2+4\alpha } - 4|\tilde{n}|^2 \le 0, \\ e^{-\frac{|\tilde{n}|^2}{|\eta |^{2+2\alpha }}t}, &{}\qquad |\eta |^{2+4\alpha } - 4|\tilde{n}|^2 \ge 0, \end{array}\right. }\\ |\textbf{a}_{-}|^2= & {} |\lambda _-|^2 + \frac{|\tilde{n}|^4}{|\eta |^4}, \qquad |\langle \textbf{b}_-,e_2 \rangle |^2 = \frac{|\eta |^4|\lambda _{+}|^2}{|\tilde{n}|^4|\lambda _+- \lambda _-|^2}.\nonumber \end{aligned}$$

(3.7)

Thus, it follows

$$\begin{aligned} |\textbf{a}_-||\langle \textbf{b}_-,e_2 \rangle | \le {\left\{ \begin{array}{ll} \displaystyle \frac{C}{|\lambda _+ - \lambda _-|}, &{}\qquad |\eta |^{2+4\alpha } - 4|\tilde{n}|^2 \le 0, \\ \displaystyle \frac{C|\eta |^{2\alpha }}{|\lambda _+ - \lambda _-|}, &{}\qquad |\eta |^{2+4\alpha } - 4|\tilde{n}|^2 \ge 0. \end{array}\right. } \end{aligned}$$

Let us consider the three sets

$$\begin{aligned} \begin{aligned} D_1&:= \{ \eta \in J ; |\eta |^{4\alpha } - \frac{4 |\tilde{n}|^2}{|\eta |^2} \le 0\}, \\ D_2&:= \{ \eta \in J ; 0 \le |\eta |^{4\alpha } - \frac{4 |\tilde{n}|^2}{|\eta |^2} \le \frac{1}{4} |\eta |^{4\alpha }\}, \\ D_3&= \{ \eta \in J ; |\eta |^{4\alpha } - \frac{4 |\tilde{n}|^2}{|\eta |^2} \ge \frac{1}{4} |\eta |^{4\alpha }\}, \end{aligned} \end{aligned}$$

with $J = D_1 \cup D_2 \cup D_3$. Then, for any $\textbf{f} \in \mathbb C^2$, there exists a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \textbf{f},\textbf{a}_- \rangle \langle \textbf{b}_-,e_{2} \rangle |&\le C e^{-|\eta |^{2\alpha } \frac{t}{4}} |\textbf{f}|,&\quad \eta \in D_1, \\ |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \textbf{f},\textbf{a}_- \rangle \langle \textbf{b}_-,e_{2} \rangle |&\le Ce^{-|\eta |^{2\alpha } \frac{t}{4}} |\textbf{f}|,&\quad \eta \in D_2, \\ |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \textbf{f},\textbf{a}_- \rangle \langle \textbf{b}_-,e_{2} \rangle |&\le Ce^{-\frac{|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\textbf{f}|,&\quad \eta \in D_3. \end{aligned} \end{aligned}$$

(3.8)

For the first and second inequalities, we apply the mean value theorem so that

$$\begin{aligned} |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \textbf{f},\textbf{a}_- \rangle \langle \textbf{b}_-,e_{2} \rangle | \le C \frac{|e^{-\lambda _- t} - e^{-\lambda _+ t}|}{|\lambda _+ - \lambda _-|} |\textbf{f}| \le Cte^{-|\eta |^{2\alpha } \frac{t}{2}} |\textbf{f}| \end{aligned}$$

for any $\eta \in D_1$ and

$$\begin{aligned} |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \textbf{f},\textbf{a}_- \rangle \langle \textbf{b}_-,e_{2} \rangle | \le C|\eta |^{2\alpha } \frac{|e^{-\lambda _- t} - e^{-\lambda _+ t}|}{|\lambda _+ - \lambda _-|} |\textbf{f}| \le C|\eta |^{2\alpha }te^{-|\eta |^{2\alpha } \tau } |\textbf{f}|, \quad \tau \in (\frac{t}{4} , \frac{3t}{4}) \end{aligned}$$

for any $\eta \in D_2$. One can easily obtain the last inequality in (3.8) by the use of $|\lambda _+ - \lambda _-| \ge \frac{1}{2}|\eta |^{2\alpha }$.

Now, we are ready to show temporal decay estimates of solutions to the linearized system of (3.4):

$$\begin{aligned} \partial _t \mathscr {F}_b\textbf{u} + M \mathscr {F}_b \textbf{u} = 0. \end{aligned}$$

(3.9)

Lemma 3.1

Let $d \in \mathbb {N}$ with $d \ge 2$ and $m \in \mathbb {N}$. Let $\textbf{u}_0 \in X^m(\Omega )$. Then, there exists a unique smooth global smooth global solution $\textbf{u}= (v_d,\theta )$ to (3.9) such that

$$\begin{aligned} \Vert v_d(t) \Vert _{{\dot{H}}^s} \le Ce^{-\frac{t}{4}} \Vert \textbf{u}_0 \Vert _{{\dot{H}}^s} + C(1+t)^{-(1+\frac{m-s}{2(1+\alpha )})} \Vert \textbf{u}_0 \Vert _{{\dot{H}}^m} \end{aligned}$$

(3.10)

and

$$\begin{aligned} \Vert \bar{\theta }(t) \Vert _{{\dot{H}}^s} \le Ce^{-\frac{t}{4}} \Vert \textbf{u}_0 \Vert _{{\dot{H}}^s} + C(1+t)^{-\frac{m-s}{2(1+\alpha )}} \Vert \textbf{u}_0 \Vert _{{\dot{H}}^m} \end{aligned}$$

(3.11)

for all $s \in [0,m]$.

Proof

We recall

$$\begin{aligned} \mathscr {F}_b\textbf{u}= (e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \textbf{b}_- + e^{-\lambda _+ t} \mathscr {F}_b \textbf{u}_0 \end{aligned}$$

and prove (3.11) first. We can see

$$\begin{aligned} \Vert \bar{\theta } \Vert _{{\dot{H}}^s}\le & {} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_2 \rangle |^2 \right) ^{\frac{1}{2}}\\{} & {} + \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |e^{-\lambda _+ t} \langle \mathscr {F}_b \textbf{u}_0,e_2 \rangle |^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

From (3.7) it is clear that

$$\begin{aligned} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |e^{-\lambda _+ t} \langle \mathscr {F}_b \textbf{u}_0,e_2 \rangle |^2 \right) ^{\frac{1}{2}} \le e^{-\frac{t}{2}} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

On the other hand, (3.8) gives

$$\begin{aligned}{} & {} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_2 \rangle |^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \le Ce^{-\frac{t}{4}} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}} + C\left( \sum _{\{ \tilde{n} \ne 0 \} \cap D_3} e^{-\frac{2|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

Since

$$\begin{aligned} e^{-\frac{2|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2&\le \left( \frac{|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t \right) ^{-\frac{m-s}{1+\alpha }} \left( \frac{|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t \right) ^{\frac{m-s}{1+\alpha }} e^{-\frac{2|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \\&\le Ct^{-\frac{m-s}{1+\alpha }} |\eta |^{2m} |\mathscr {F}_b \textbf{u}_0|^2, \end{aligned}$$

and

$$\begin{aligned} e^{-\frac{2|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \le |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2, \end{aligned}$$

it follows

$$\begin{aligned} \left( \sum _{\{ \tilde{n} \ne 0 \} \cap D_3} e^{-\frac{|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}} \le C(1+t)^{-\frac{m-s}{2(1+\alpha )}} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2m} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

(3.12)

Collecting the above estimates, we deduce (3.11).

It remains to show (3.10). We can see

$$\begin{aligned} \Vert v_d \Vert _{{\dot{H}}^s} \le \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_1 \rangle |^2 \right) ^{\frac{1}{2}} + e^{-\frac{t}{2}} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

Since $\langle \textbf{b}_{-},e_1 \rangle = \frac{|\tilde{n}|^2}{|\lambda _+||\eta |^2}\langle \textbf{b}_{-},e_2 \rangle$, using $\frac{|\tilde{n}|^2}{|\lambda _+||\eta |^2} \le C$ for $\eta \in D_1 \cup D_2$ and $\frac{|\tilde{n}|^2}{|\lambda _+||\eta |^2} \le 2 \frac{|\tilde{n}|^2}{|\eta |^{2+2\alpha }}$ for $\eta \in D_3$, we have

$$\begin{aligned}{} & {} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |(e^{-\lambda _- t} - e^{-\lambda _+ t}) \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_1 \rangle |^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \le Ce^{-\frac{t}{4}} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}} + C\left( \sum _{\{ \tilde{n} \ne 0 \} \cap D_3} \frac{|\tilde{n}|^4}{|\eta |^{4+4\alpha }} e^{-\frac{2|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

Estimating as in (3.12), we deduce

$$\begin{aligned} \left( \sum _{\{ \tilde{n} \ne 0 \} \cap D_3} \frac{|\tilde{n}|^4}{|\eta |^{4+4\alpha }} e^{-\frac{2|\tilde{n}|^2}{|\eta |^{2+2\alpha }} t} |\eta |^{2s} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}} \le C(1+t)^{-(1+\frac{m-s}{2(1+\alpha )})} \left( \sum _{\tilde{n} \ne 0} |\eta |^{2m} |\mathscr {F}_b \textbf{u}_0|^2 \right) ^{\frac{1}{2}} \end{aligned}$$

from which we obtain (3.10). This completes the proof. $\square$

4 Energy estimates

In this section, we provide the energy estimates which specify the quantities that should be computed via the spectral analysis. We start with the following standard local existence result. For the proof, we refer to [5, 37].

Proposition 4.1

(Local well-posedness) Let $d \in \mathbb {N}$ with $d \ge 2$ and $\alpha \in \{0,1\}$. Let $m \in \mathbb N$ with $m > 1+\frac{d}{2}-\alpha$ and an initial data $\theta _0 \in X^m$ and $v_0 \in \mathbb X^m$. Then there exists a $T>0$ such that there exists a unique classical solution $(v, \theta )$ to the stratified Boussinesq equations (1.2) satisfying

$$\begin{aligned} v \in L^\infty (0,T ; \mathbb X^m(\Omega )), \qquad \theta \in L^\infty (0,T ; X^m(\Omega )). \end{aligned}$$

Let $T^* \in (0,\infty ]$ be the maximal time of existence. Moreover, if $T^* < \infty$, then it holds

$$\begin{aligned} \lim _{t \nearrow T^*} \big ( \Vert v( t) \Vert _{H^m}^2 + \Vert \theta ( t) \Vert _{H^m}^2 \big ) = \infty . \end{aligned}$$

We will frequently use the following result on the product estimates

(see [21] for the proof).

Lemma 4.2

Let $m \in \mathbb N$. Then for any subset $D \subset \{ \gamma ; |\gamma | = m \}$, there exists a constant $C=C(m)>0$ such that

$$\begin{aligned} \left\| \sum _{\gamma \in D} \partial ^{\gamma } (fg) \right\| _{L^2} \le C \left( \Vert f \Vert _{H^m} \Vert g \Vert _{L^{\infty }} + \Vert f \Vert _{L^{\infty }} \Vert g \Vert _{H^m} \right) \end{aligned}$$

for all $f, g \in H^m(\Omega ) \cap L^{\infty }(\Omega )$. Moreover, if $m > \frac{d}{2}$, then it holds

$$\begin{aligned} \left\| \sum _{\gamma \in D} \partial ^{\gamma } (fg) \right\| _{L^2} \le C \Vert f \Vert _{H^m} \Vert g \Vert _{H^m}. \end{aligned}$$

Let us use the notations for $k \in \mathbb N$

$$\begin{aligned} E_k(t):= & {} \left( \Vert v(t) \Vert _{H^k}^2 + \Vert \theta (t) \Vert _{H^k}^2 \right) ^{\frac{1}{2}},\\ A_k(t):= & {} \sum _{|\gamma |=1}^{k} \int _{\Omega } \partial ^\gamma v_d(t) \partial ^\gamma \theta (t) \,\textrm{d}x. \end{aligned}$$

Note that Young’s inequality implies

$$\begin{aligned} |A_{k}(t)| \le \frac{1}{2} E_{k}(t)^2 . \end{aligned}$$

(4.1)

Proposition 4.3

Let $d \in \mathbb {N}$ with $d \ge 2$ and $m \in \mathbb {N}$ with $m \ge 2+\frac{d}{2}$. Assume that $(v, \theta )$ is a smooth global solution to (1.2) with $\alpha = 1$. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}( E_m(t)^2 - A_{m-1}(t) ) + \frac{1}{2} \Vert \nabla v(t) \Vert _{H^m}^2 + \frac{1}{2} \Vert \nabla _h \theta (t) \Vert _{H^{m-2}}^2 \\&\quad \le C \Vert \theta (t) \Vert _{H^m} \big (\Vert \nabla v(t) \Vert _{H^m}^2 + \Vert \nabla _h \theta (t) \Vert _{H^{m-2}}^2\big ) + C\big (\Vert \theta (t) \Vert _{H^m}^2 + \Vert v(t) \Vert _{H^m}^2 \big ) \Vert \nabla v(t) \Vert _{L^{\infty }} \end{aligned} \end{aligned}$$

(4.2)

for all $t>0$.

Proof

From the system (1.2), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t}\big ( \Vert v(t) \Vert _{H^m}^2 + \Vert \theta (t) \Vert _{H^m}^2 \big ) + \Vert \nabla v \Vert _{H^m}^2 \\&\quad \le - \sum _{1 \le |\gamma | \le m} \int _{\Omega } \partial ^\gamma (v \cdot \nabla ) v \partial ^\gamma v \, \textrm{d}x - \sum _{1 \le |\gamma | \le m} \int _{\Omega } \partial ^\gamma (v \cdot \nabla ) \theta \partial ^\gamma \theta \, \textrm{d}x. \end{aligned} \end{aligned}$$

We only consider the $|\gamma | = m$ case because the others can be treated similarly. It is clear by the divergence-free condition and the boundary condition that

$$\begin{aligned} \int _{\Omega } (v \cdot \nabla \partial ^\gamma v) \partial ^\gamma v \,\textrm{d}x = \int _{\Omega } (v \cdot \nabla \partial ^\gamma \theta ) \partial ^\gamma \theta \,\textrm{d}x = 0, \qquad |\gamma | = m. \end{aligned}$$

Thus, Lemma 4.2 implies

$$\begin{aligned}{} & {} \left| - \sum _{|\gamma | = m} \int _{\Omega } \partial ^\gamma (v \cdot \nabla ) v \partial ^\gamma v \, \textrm{d}x -\sum _{|\gamma |=m-1} \int _{\Omega } (\nabla v \cdot \partial ^\gamma \nabla \theta ) \cdot \partial ^\gamma \nabla \theta \,\textrm{d}x \right| \\{} & {} \quad \le C\Vert \nabla v \Vert _{L^{\infty }} (\Vert v \Vert _{H^m}^2+\Vert \theta \Vert _{H^m}^2). \end{aligned}$$

For estimating the remainder term

$$\begin{aligned} \left| \sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma (\Delta v \cdot \nabla \theta ) \partial ^\gamma \Delta \theta \,\textrm{d}x \right| , \end{aligned}$$

we use a simple formula $\Delta v \cdot \nabla \theta = \Delta v_h \cdot \nabla _h \theta + \Delta v_d \partial _d \theta$. We can infer from Hölder’s inequality with Sobolev embeddings that

$$\begin{aligned}&\left| \sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma (\Delta v_h \cdot \nabla _h \theta ) \partial ^\gamma \Delta \theta \,\textrm{d}x \right| \\&\quad \le \Vert \Delta v_h \cdot \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^{m}} \\&\quad \le C\Vert \nabla v \Vert _{H^m} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^m} + \Vert \Delta v \Vert _{L^{\infty }} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^m}. \end{aligned}$$

Since $m \ge 2 + d/2$ implies

$$\begin{aligned} \Vert \Delta v \Vert _{L^{\infty }} \le C\Vert \nabla v \Vert _{L^{\infty }} + C\Vert \nabla v \Vert _{H^m}, \end{aligned}$$

it follows

$$\begin{aligned} \left| \sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma (\Delta v_h \cdot \nabla _h \theta ) \partial ^\gamma \Delta \theta \,\textrm{d}x \right|&\le C\Vert \nabla v \Vert _{H^m} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^m} + C\Vert \nabla v \Vert _{L^{\infty }} \Vert \theta \Vert _{H^m}^2. \end{aligned}$$

Otherwise, we have

$$\begin{aligned} \begin{aligned}&\left| \sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma (\Delta v_d \partial _d \theta ) \partial ^\gamma \Delta \theta \,\textrm{d}x \right| \\&\quad \le \left| \sum _{|\gamma | = m-3} \int _{\Omega } \partial ^\gamma \nabla _h(\Delta v_d \partial _d \theta ) \cdot \partial ^\gamma \Delta \nabla _h \theta \,\textrm{d}x \right| + \left| \int _{\Omega } \partial _d^{m-2}(\partial _d^2 v_d \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right| . \end{aligned} \end{aligned}$$

(4.3)

Here, we need to estimate carefully with the boundary conditions. The first integral on the right-hand side is bounded by

$$\begin{aligned} \left| \sum _{|\gamma | = m-3} \int _{\partial \Omega } \partial ^\gamma \nabla _h(\Delta v_d \partial _d \theta ) \cdot \partial _d \partial ^\gamma \nabla _h \theta \,\textrm{d}x_h \right| + \left| \sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma \nabla _h(\Delta v_d \partial _d \theta ) \cdot \partial ^\gamma \nabla _h \theta \,\textrm{d}x \right| . \end{aligned}$$

Since $v_d \in X^{m+1}(\Omega )$ and $\theta \in X^m(\Omega )$ for a.e. $t>0$, the boundary term vanishes. Lemma 4.2 implies

$$\begin{aligned} \left| \sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma \nabla _h(\Delta v_d \partial _d \theta ) \cdot \partial ^\gamma \nabla _h \theta \,\textrm{d}x \right| \le C\Vert \nabla v \Vert _{H^m} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^m}. \end{aligned}$$

On the other hand, we write the second integral in (4.3) as

$$\begin{aligned} \left| \int _{\Omega } \partial _d^{m-2}(\partial _d^2 v_d \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right|&\le \left| \int _{\Omega } \partial _d^{m-2}\nabla _h \cdot (\partial _d v_h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right| \\&\quad + \left| \int _{\Omega } \partial _d^{m-2}(\partial _d v_h \cdot \nabla _h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right| . \end{aligned}$$

It can be shown by Hölder’s inequalities and Sobolev embeddings that

$$\begin{aligned} \left| \int _{\Omega } \partial _d^{m-2}(\partial _d v_h \cdot \nabla _h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right|&\le (\Vert \partial _d^{m-3}(\partial _d^2 v_h \cdot \nabla _h \partial _d \theta ) \Vert _{L^2} + \Vert \partial _d v_h \cdot \nabla _h \partial _d^{m-1} \theta \Vert _{L^2}) \Vert \partial _d^m \theta \Vert _{L^2} \\&\le C \Vert \nabla v \Vert _{H^m} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^{m}} + C \Vert \nabla v \Vert _{L^{\infty }} \Vert \theta \Vert _{H^{m}}^2. \end{aligned}$$

Since $\partial _d v_h \partial _d \theta \in X^{m-1}$ and $\theta \in X^m$, it holds

$$\begin{aligned} \begin{aligned} \left| \int _{\Omega } \partial _d^{m-2} \nabla _h \cdot (\partial _d v_h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right|&= \left| \sum _{\eta \in J} \tilde{q}^{m-2} i\tilde{n} \cdot \mathscr {F}_b (\partial _d v_h \partial _d \theta )(\eta ) \overline{\tilde{q}^{m} \mathscr {F}_b \theta (\eta )} \right| \\&= \left| \sum _{\eta \in J} \tilde{q}^{m-1} \mathscr {F}_b (\partial _d v_h \partial _d \theta )(\eta ) \cdot \overline{i\tilde{n} \tilde{q}^{m-1} \mathscr {F}_b \theta (\eta )} \right| \\&= \left| \int _{\Omega } \partial _d^{m-1} (\partial _d v_h \partial _d \theta ) \cdot \nabla _h \partial _d^{m-1} \theta \,\textrm{d}x \right| . \end{aligned} \end{aligned}$$

(4.4)

Then, we have

$$\begin{aligned}&\left| \int _{\Omega } \partial _d^{m-1}(\partial _d v_h \partial _d \theta ) \cdot \nabla _h \partial _d^{m-1} \theta \,\textrm{d}x \right| \\&\quad \le \left| \int _{\Omega } (\partial _d v_h \partial _d^m \theta ) \cdot \nabla _h \partial _d^{m-1} \theta \,\textrm{d}x \right| + \left| \int _{\Omega } \partial _d^{m-2}(\partial _d^2 v_h \partial _d \theta ) \cdot \nabla _h \partial _d^{m-1} \theta \,\textrm{d}x \right| \\&\quad = \left| \int _{\Omega } (\partial _d v_h \partial _d^m \theta ) \cdot \nabla _h \partial _d^{m-1} \theta \,\textrm{d}x \right| + \left| \int _{\Omega } \partial _d^{m-1}(\partial _d^2 v_h \partial _d \theta ) \cdot \nabla _h \partial _d^{m-2} \theta \,\textrm{d}x \right| \\&\quad \le C \Vert \nabla v \Vert _{L^{\infty }} \Vert \theta \Vert _{H^{m}}^2 + C \Vert \nabla v \Vert _{H^m} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^{m}}. \end{aligned}$$

Combining the above estimates, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t}\big ( \Vert v(t) \Vert _{H^m}^2 + \Vert \theta (t) \Vert _{H^m}^2 \big ) + \Vert \nabla v \Vert _{H^m}^2 \\&\quad \le C \Vert \nabla v \Vert _{L^{\infty }} (\Vert v \Vert _{H^m}^2 + \Vert \theta \Vert _{H^{m}}^2) + C \Vert \nabla v \Vert _{H^m} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^{m}}. \end{aligned} \end{aligned}$$

(4.5)

Now we claim that

$$\begin{aligned} \frac{3}{2} \Vert \nabla v \Vert _{H^m}^2 \ge \frac{1}{2} \Vert \nabla _h \theta \Vert _{H^{m-2}}^2 - \frac{\textrm{d}}{\textrm{d}t}A_{m-1}(t) - C E_m(t) \Vert \nabla v \Vert _{H^{m-2}}^2. \end{aligned}$$

(4.6)

We recall the $v_d$ equation in (1.2)

$$\begin{aligned} \partial _t v_d -\Delta v_d + \langle \mathbb {P}\,(v \cdot \nabla )v , e_d \rangle = \langle \mathbb {P} \,\theta e_d,e_d \rangle . \end{aligned}$$

(4.7)

We first take $-\Delta$ on the both sides of (4.7). Since the definition of $\mathbb {P}$ implies

$$\begin{aligned} -\Delta \langle \mathbb {P}\,(v \cdot \nabla )v , e_d \rangle = -\Delta (v\cdot \nabla ) v_d + \partial _d \nabla \cdot (v\cdot \nabla ) v \end{aligned}$$

and

$$\begin{aligned} -\Delta \langle \mathbb {P} \,\theta e_d,e_d \rangle = - \Delta _h \theta , \end{aligned}$$

it follows

$$\begin{aligned} \partial _t (-\Delta )v_d + (-\Delta )^2 v_d -\Delta (v\cdot \nabla ) v_d + \partial _d \nabla \cdot (v\cdot \nabla ) v = -\Delta _h \theta . \end{aligned}$$

Then, we have for $|\gamma | = m-2$ that

$$\begin{aligned}{} & {} \int _{\Omega } \partial _t \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x + \int _{\Omega } \nabla (-\Delta ) \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x + \int _{\Omega } \nabla \partial ^{\gamma } (v \cdot \nabla ) v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x \\{} & {} - \int _{\Omega } \nabla \cdot \partial ^{\gamma } (v \cdot \nabla )v \partial _d \partial ^{\gamma } \theta \,\textrm{d}x = \int _{\Omega } | \nabla _h \partial ^{\gamma } \theta |^2 \,\textrm{d}x. \end{aligned}$$

On the other hand, we have from the $\theta$ equation in (1.2)

$$\begin{aligned} \int _{\Omega } \partial _t \nabla \partial ^{\gamma } \theta \cdot \nabla \partial ^{\gamma } v_d \,\textrm{d}x + \int _{\Omega } \nabla \partial ^{\gamma } (v\cdot \nabla ) \theta \cdot \nabla \partial ^{\gamma } v_d \,\textrm{d}x + \int _{\Omega } |\nabla \partial ^{\gamma } v_d|^2 \,\textrm{d}x = 0. \end{aligned}$$

Adding these two equalities gives

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \, \textrm{d}x + \int _{\Omega } \nabla (-\Delta ) \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x - \int _{\Omega } \nabla \cdot \partial ^{\gamma } (v \cdot \nabla )v \partial _d \partial ^{\gamma } \theta \,\textrm{d}x \\{} & {} \quad + \int _{\Omega } \big ( \nabla \partial ^{\gamma } (v \cdot \nabla ) v_d \cdot \nabla \partial ^{\gamma } \theta + \nabla \partial ^{\gamma } (v\cdot \nabla ) \theta \cdot \nabla \partial ^{\gamma } v_d\big ) \, \textrm{d}x\\{} & {} \quad + \int _{\Omega } |\nabla \partial ^{\gamma } v_d|^2 \,\textrm{d}x = \int _{\Omega } | \nabla _h \partial ^{\gamma } \theta |^2 \,\textrm{d}x. \end{aligned}$$

We have by Lemma 4.2 and the divergence-free condition that

$$\begin{aligned} \bigg | -\int _{\Omega } \nabla \cdot \partial ^{\gamma } (v \cdot \nabla )v \partial _d \partial ^{\gamma } \theta \,\textrm{d}x \bigg | \le C \Vert v \Vert _{H^{m}}^2 \Vert \theta \Vert _{H^{m}}. \end{aligned}$$

From $\partial _dv_d = -\nabla _h \cdot v_h$ and the cancellation property, we deduce

$$\begin{aligned} \left| \int _{\Omega } \nabla (-\Delta ) \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x \right| \le \Vert \Delta \partial ^{\gamma } \nabla v \Vert _{L^2} \Vert \partial ^{\gamma } \nabla _h \theta \Vert _{L^2} \le \frac{1}{2} \Vert \Delta \partial ^{\gamma } \nabla v \Vert _{L^2}^2 + \frac{1}{2} \Vert \partial ^{\gamma } \nabla _h \theta \Vert _{L^2}^2 \end{aligned}$$

and

$$\begin{aligned} \bigg | \int _{\Omega } \big ( \nabla \partial ^{\gamma } (v \cdot \nabla ) v_d \cdot \nabla \partial ^{\gamma } \theta + \nabla \partial ^{\gamma } (v\cdot \nabla ) \theta \cdot \nabla \partial ^{\gamma } v_d\big ) \, \textrm{d}x \bigg | \le C \Vert v \Vert _{H^{m}}^2 \Vert \theta \Vert _{H^{m}} \end{aligned}$$

respectively. The above estimates yield

$$\begin{aligned}{} & {} \sum _{|\gamma |=m-2} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \, \textrm{d}x + \frac{1}{2} \Vert \nabla v \Vert _{{\dot{H}}^m}^2 + \Vert \nabla v_d \Vert _{{\dot{H}}^{m-2}}^2 + C \Vert v \Vert _{H^{m}}^2 \Vert \theta \Vert _{H^{m}}\\{} & {} \quad \ge \frac{1}{2} \Vert \nabla _h \theta \Vert _{{\dot{H}}^{m-2}}^2. \end{aligned}$$

Similarly, we can repeat the above procedure for the lower order derivatives. Then, (4.6) is obtained.

We multiply (4.5) by 2

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\big ( \Vert v(t) \Vert _{H^m}^2 + \Vert \theta (t) \Vert _{H^m}^2 \big ) + 2\Vert \nabla v \Vert _{H^m}^2 \\ \le C \Vert \nabla v \Vert _{L^{\infty }} (\Vert v \Vert _{H^m}^2 + \Vert \theta \Vert _{H^{m}}^2) + C \Vert \nabla v \Vert _{H^m} \Vert \nabla _h \theta \Vert _{H^{m-2}} \Vert \theta \Vert _{H^{m}}, \end{aligned} \end{aligned}$$

and recall (4.6)

$$\begin{aligned} -\frac{3}{2} \Vert \nabla v \Vert _{H^m}^2 + \frac{1}{2} \Vert \nabla _h \theta \Vert _{H^{m-2}}^2 - \frac{\textrm{d}}{\textrm{d}t}A_{m-1}(t) \le C E_m(t) \Vert \nabla v \Vert _{H^{m-2}}^2. \end{aligned}$$

Adding these two inequality, we obtain (4.2). This completes the proof. $\square$

Proposition 4.4

Let $d \in \mathbb {N}$ with $d \ge 2$ and $m \in \mathbb {N}$ with $m > 1+\frac{d}{2}$. Assume that $(v, \theta )$ is a smooth global solution to (1.2) with $\alpha = 0$. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}( E_m(t)^2 - A_m(t) ) + \frac{1}{2} \Vert v(t) \Vert _{H^m}^2 + \frac{1}{2} \Vert \nabla _h \theta (t) \Vert _{H^{m-1}}^2 \\&\quad \le C \Vert \theta (t) \Vert _{H^m}\big (\Vert v(t) \Vert _{H^m}^2 + \Vert \nabla _h \theta (t) \Vert _{H^{m-1}}^2\big ) \\&\qquad + C\Vert v(t) \Vert _{H^m}^3 + C\big (\Vert \theta (t) \Vert _{H^m}^2 + \Vert v(t) \Vert _{H^m}^2 \big ) \Vert \partial _d v_d(t) \Vert _{L^{\infty }} \end{aligned} \end{aligned}$$

(4.8)

for all $t>0$.

Proof

From (1.2), we can have

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t}\big ( \Vert v(t) \Vert _{H^m}^2 + \Vert \theta (t) \Vert _{H^m}^2 \big ) + \Vert v \Vert _{H^m}^2 \\&\quad \le - \sum _{1 \le |\gamma | \le m} \int _{\Omega } \partial ^{\gamma } (v \cdot \nabla ) v \partial ^{\gamma } v \,\textrm{d}x - \sum _{1 \le |\gamma | \le m} \int _{\Omega } \partial ^\gamma (v \cdot \nabla ) \theta \partial ^\gamma \theta \, \textrm{d}x. \end{aligned} \end{aligned}$$

We only estimate $|\gamma | = m$ case because the others can be treated similarly. The first integral on the right-hand side can be estimated by lemma 4.2 and the divergence-free condition that

$$\begin{aligned} \left| \int _{\Omega } \partial ^{\gamma } (v \cdot \nabla ) v \partial ^{\gamma } v \,\textrm{d}x \right| \le C \Vert v \Vert _{H^m}^3. \end{aligned}$$

To estimate the remainder term, we consider the case $\partial ^{\gamma } \ne \partial _d^m$ first. As estimating the first one, we can see

$$\begin{aligned} \left| \int _{\Omega } \partial ^\gamma (v \cdot \nabla ) \theta \partial ^\gamma \theta \,\textrm{d}x \right| \le C \Vert v \Vert _{H^m} \Vert R_h \theta \Vert _{H^m} \Vert \theta \Vert _{H^m}. \end{aligned}$$

In the case of $\partial ^{\gamma } = \partial _d^m$, we have

$$\begin{aligned} \int _{\Omega } \partial _d^m (v \cdot \nabla ) \theta \partial _d^m \theta \,\textrm{d}x&= \int _{\Omega } \partial _d^{m-1} (\partial _d v_h \cdot \nabla _h \theta ) \partial _d^m \theta \,\textrm{d}x + \int _{\Omega } \partial _d^{m-1} (\partial _d v_d \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \\&\quad \le C \Vert v \Vert _{H^m} \Vert R_h \theta \Vert _{H^m} \Vert \theta \Vert _{H^m} + C\Vert \partial _d v_d \Vert _{L^{\infty }} \Vert \theta \Vert _{H^m}^2 \\&\qquad + \int _{\Omega } \partial _d^{m-2} (\partial _d^2 v_d \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x. \end{aligned}$$

We note that

$$\begin{aligned} \int _{\Omega } \partial _d^{m-2} (\partial _d^2 v_d \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x&= -\int _{\Omega } \partial _d^{m-2} \nabla _h \cdot (\partial _d v_h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x\\&\quad + \int _{\Omega } \partial _d^{m-2} (\partial _d v_h \cdot \nabla _h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x\\&\le - \int _{\Omega } \partial _d^{m-2} \nabla _h \cdot (\partial _d v_h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x\\&\quad + C \Vert v \Vert _{H^m} \Vert R_h \theta \Vert _{H^m} \Vert \theta \Vert _{H^m}. \end{aligned}$$

By (4.4), we can deduce

$$\begin{aligned} \begin{aligned} \left| \int _{\Omega } \partial _d^{m-2} \nabla _h \cdot (\partial _d v_h \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right| \le C \Vert v \Vert _{H^m} \Vert R_h \theta \Vert _{H^m} \Vert \theta \Vert _{H^m}, \end{aligned} \end{aligned}$$

thus,

$$\begin{aligned} \left| \int _{\Omega } \partial _d^{m-2} (\partial _d^2 v_d \partial _d \theta ) \partial _d^m \theta \,\textrm{d}x \right| \le C \Vert v \Vert _{H^m} \Vert R_h \theta \Vert _{H^m} \Vert \theta \Vert _{H^m}. \end{aligned}$$

Collecting the above estimates, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t}\big ( \Vert v(t) \Vert _{H^m}^2 + \Vert \theta (t) \Vert _{H^m}^2 \big ) + \Vert v \Vert _{H^m}^2 \le C \Vert v \Vert _{H^m}^3 + C \Vert v \Vert _{H^m} \Vert R_h \theta \Vert _{H^m} \Vert \theta \Vert _{H^m} \\&\quad + C\Vert \partial _d v_d \Vert _{L^{\infty }} \Vert \theta \Vert _{H^m}^2. \end{aligned} \end{aligned}$$

(4.9)

As estimating (4.6), we can show

$$\begin{aligned} \frac{3}{2} \Vert v \Vert _{H^m}^2 \ge \frac{1}{2} \Vert \nabla _h \theta \Vert _{H^{m-1}}^2 - \frac{\textrm{d}}{\textrm{d}t}A_{m}(t) - C E_m(t) \Vert v \Vert _{H^{m}}^2. \end{aligned}$$

(4.10)

We only consider the highest derivative case. We can see from the $v_d$ equation in (1.2)

$$\begin{aligned} \partial _t (-\Delta )v_d + (-\Delta ) v_d -\Delta (v\cdot \nabla v_d) + \partial _d \nabla \cdot (v\cdot \nabla ) v = -\Delta _h \theta . \end{aligned}$$

Thus, we have for $|\gamma | = m-1$ that

$$\begin{aligned}{} & {} \int _{\Omega } \partial _t \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x + \int _{\Omega } \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x + \int _{\Omega } \nabla \partial ^{\gamma } (v \cdot \nabla v_d) \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x \\{} & {} \quad - \int _{\Omega } \nabla \cdot \partial ^{\gamma } (v \cdot \nabla )v \partial _d \partial ^{\gamma } \theta \,\textrm{d}x = \int _{\Omega } | \nabla _h \partial ^{\gamma } \theta |^2 \,\textrm{d}x. \end{aligned}$$

From the $\theta$ equation in (1.2), it follows

$$\begin{aligned} \int _{\Omega } \partial _t \nabla \partial ^{\gamma } \theta \cdot \nabla \partial ^{\gamma } v_d \,\textrm{d}x + \int _{\Omega } \nabla \partial ^{\gamma } (v\cdot \nabla \theta ) \cdot \nabla \partial ^{\gamma } v_d \,\textrm{d}x + \int _{\Omega } |\nabla \partial ^{\gamma } v_d|^2 \,\textrm{d}x = 0. \end{aligned}$$

Combining the two above gives

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \, \textrm{d}x + \int _{\Omega } \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x - \int _{\Omega } \nabla \cdot \partial ^{\gamma } (v \cdot \nabla )v \partial _d \partial ^{\gamma } \theta \,\textrm{d}x \\{} & {} \qquad + \int _{\Omega } \big ( \nabla \partial ^{\gamma } (v \cdot \nabla v_d) \cdot \nabla \partial ^{\gamma } \theta + \nabla \partial ^{\gamma } (v\cdot \nabla \theta ) \cdot \nabla \partial ^{\gamma } v_d\big ) \, \textrm{d}x + \int _{\Omega } |\nabla \partial ^{\gamma } v_d|^2 \,\textrm{d}x\\{} & {} \quad = \int _{\Omega } | \nabla _h \partial ^{\gamma } \theta |^2 \,\textrm{d}x. \end{aligned}$$

The divergence-free condition and Lemma 4.2 imply

$$\begin{aligned} \bigg | \int _{\Omega } \nabla \cdot \partial ^{\gamma } (v \cdot \nabla )v \partial _d \partial ^{\gamma } \theta \,\textrm{d}x \bigg | \le C \Vert v \Vert _{H^{m}}^2 \Vert \theta \Vert _{H^{m}}. \end{aligned}$$

We also have with the divergence-free condition that

$$\begin{aligned} \left| \int _{\Omega } \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \,\textrm{d}x \right| \le \Vert \partial ^{\gamma } \nabla v \Vert _{L^2} \Vert \partial ^{\gamma } \nabla _h \theta \Vert _{L^2} \le \frac{1}{2} \Vert \partial ^{\gamma } \nabla v \Vert _{L^2}^2 + \frac{1}{2} \Vert \partial ^{\gamma } \nabla _h \theta \Vert _{L^2}^2. \end{aligned}$$

The cancellation property yields

$$\begin{aligned} \bigg | \int _{\Omega } \big ( \nabla \partial ^{\gamma } (v \cdot \nabla v_d) \cdot \nabla \partial ^{\gamma } \theta + \nabla \partial ^{\gamma } (v\cdot \nabla \theta ) \cdot \nabla \partial ^{\gamma } v_d\big ) \, \textrm{d}x \bigg | \le C \Vert v \Vert _{H^{m}}^2 \Vert \theta \Vert _{H^{m}}. \end{aligned}$$

Therefore, we deduce that

$$\begin{aligned} \sum _{|\gamma |=m-1} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } \nabla \partial ^{\gamma } v_d \cdot \nabla \partial ^{\gamma } \theta \, \textrm{d}x + \frac{3}{2} \Vert v \Vert _{{\dot{H}}^m}^2 + C \Vert v \Vert _{H^{m}}^2 \Vert \theta \Vert _{H^{m}} \ge \frac{1}{2} \Vert \nabla _h \theta \Vert _{{\dot{H}}^{m-1}}^2, \end{aligned}$$

which implies (4.10).

Multiplying (4.9) by 2 and adding (4.10) gives (4.8). This completes the proof. $\square$

5 Global-in-time existence

In this section, we prove the global existence part of Theorem 1.1 and 1.4. It remains to estimate the key quantities in Proposition 4.3 and Proposition 4.4, namely,

$$\begin{aligned} \int \Vert \nabla v(t) \Vert _{L^{\infty }} \,\textrm{d}t \qquad \text{ and } \qquad \int \Vert \partial _d v_d(t) \Vert _{L^{\infty }} \,\textrm{d}t \end{aligned}$$

(5.1)

respectively. For this purpose, we recall the notations introduced in Sect. 3. From now on, we use the notations

$$\begin{aligned} R_h f = \nabla _h \Lambda ^{-1} f = \sum _{|\tilde{n}| \ne 0} \frac{i\tilde{n}}{|\eta |} \mathscr {F}_bf(\eta ) \mathscr {B}_\eta (x), \qquad R_h g = \nabla _h \Lambda ^{-1} g = \sum _{|\tilde{n}| \ne 0} \frac{i\tilde{n}}{|\eta |} \mathscr {F}_cg(\eta ) \mathscr {C}_{\eta }(x), \end{aligned}$$

for $f \in X^m(\Omega )$ and $g \in Y^m(\Omega )$.

5.1 Proof of Theorem 1.1:Global-in-time existence part

Here, we fix $\alpha = 1$. We show the two propositions that provide proper upper-bound of the key quantity. Then combining with Proposition 4.3, we finish the proof.

Proposition 5.1

Let $d \in \mathbb {N}$ with $d \ge 2$ and $m \in \mathbb {N}$ satisfying $m > 1+\frac{d}{2}$. Assume that $(v, \theta )$ is a smooth global solution to (1.2) with $\alpha = 1$. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned}&\sum _{\eta \in I} \int _0^T |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \le C \Vert v_0 \Vert _{H^m} \\&\quad + C \sup _{t \in [0,T]} \Vert v(t) \Vert _{H^m} \left( \sum _{\eta \in I} \int _0^T |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) \\&\quad + \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} |\mathscr {F}_b \theta (t)| \,\textrm{d}t \end{aligned} \end{aligned}$$

(5.2)

for all $T > 0$.

Proof

We first note that the divergence-free condition implies

$$\begin{aligned} |\eta ||\mathscr {F}_b v_d(\eta )| \le C |\tilde{n}| |\mathscr {F}_cv_h(\eta )| + C |\tilde{n}| |\mathscr {F}_bv_d(\eta )|. \end{aligned}$$

(5.3)

Thus, it suffices to show

$$\begin{aligned}{} & {} \sum _{\eta \in J} \int _0^T |\tilde{n}||\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\tilde{n}||\eta | |\mathscr {F}_bv_d(\eta )| \,\textrm{d}t \le C \Vert v_0 \Vert _{H^m} \\{} & {} \quad + C \sup _{t \in [0,T]} \Vert v(t) \Vert _{H^m} \left( \sum _{\eta \in I} \int _0^T |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) \\{} & {} \quad + \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} |\mathscr {F}_b \theta (t)| \,\textrm{d}t. \end{aligned}$$

We only estimate the first term on the left-hand side because the other can be treated similarly. Applying Duhamel’s principle to (3.1), we can have

$$\begin{aligned} \sum _{\eta \in J} \int _0^T |\tilde{n}||\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t \le I_1 + I_2 + I_3 + I_4, \end{aligned}$$

where

$$\begin{aligned} I_1&:= \sum _{\eta \in J} \int _0^T |\tilde{n}||\eta | e^{-|\eta |^2t} |\mathscr {F} v_0| \,\textrm{d}t, \\ I_2&:= \sum _{\eta \in J} \int _0^T \int _0^t |\tilde{n}||\eta |e^{-|\eta |^2(t-\tau )} |\mathscr {F}_c[(v \cdot \nabla )v_h](\tau )| \,\textrm{d}\tau \textrm{d}t, \\ I_3&:= \sum _{\eta \in J} \int _0^T \int _0^t |\tilde{n}||\eta |e^{-|\eta |^2(t-\tau )} |\mathscr {F}_b[(v \cdot \nabla )v_d](\tau )| \,\textrm{d}\tau \textrm{d}t, \\ I_4&:= \sum _{\eta \in J} \int _0^T \int _0^t |\tilde{n}||\eta | e^{-|\eta |^2(t - \tau )} \frac{|\tilde{n}|}{|\eta |} |\mathscr {F}_b \theta (\tau )| \,\textrm{d}\tau \textrm{d}t. \end{aligned}$$

We have used that $|\mathscr {F} \mathbb {P} f| \le |\mathscr {F} f|$ for $I_2$ and $I_3$. It is clear that

$$\begin{aligned} I_1 \le \sum _{\eta \in J} \int _0^T |\eta |^2 e^{-|\eta |^2t} |\mathscr {F}v_0| \,\textrm{d}t \le \sum _{\eta \in I} |\mathscr {F}v_0| \le C\Vert v_0 \Vert _{H^m}. \end{aligned}$$

We can see by Fubini’s theorem that

$$\begin{aligned} I_4 \le \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} |\mathscr {F}_b \theta (t)| \, \textrm{d}t. \end{aligned}$$

Similarly,

$$\begin{aligned} I_2 \le C \sum _{\eta \in I} \int _0^T |\mathscr {F}_c[(v \cdot \nabla )v_h](t)| \,\textrm{d}t. \end{aligned}$$

Using $(v \cdot \nabla )v_h = (v_h \cdot \nabla _h)v_h + v_d \partial _d v_h$ and Proposition 2.4, we have

$$\begin{aligned}{} & {} I_2 \le \int _0^T \left( \sum _{\eta \in I} |\mathscr {F}_c v_h(t)| \right) \left( \sum _{\eta \in I} | \mathscr {F}_c \nabla _h v_h(t)| \right) \,\textrm{d}t \\{} & {} \qquad + \int _0^T \left( \sum _{\eta \in J} |\mathscr {F}_b v_d(t)| \right) \left( \sum _{\eta \in J} | \mathscr {F}_b \partial _d v_h(t)| \right) \,\textrm{d}t \\{} & {} \quad \le C \sup _{t \in [0,T]} \Vert v(t) \Vert _{H^m} \left( \sum _{\eta \in I} \int _0^T |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) . \end{aligned}$$

In a similar way with the above, we can have the same upper-bound for $I_3$. Collecting the estimates for $I_1$, $I_2$, $I_3$, and $I_4$, we obtain the claim. This completes the proof. $\square$

Proposition 5.2

Let $d \in \mathbb {N}$ with $d \ge 2$ and $m \in \mathbb {N}$ satisfying $m > 3+\frac{d}{2}$. Assume that $(v, \theta )$ is a smooth global solution to (1.2) with $\alpha = 1$, and $\int _{\Omega } v_0 \,\textrm{d}x$ be satisfied. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} |\mathscr {F}_b \theta (t)| \,\textrm{d}t \le C \Vert (v_0,\theta _0) \Vert _{H^{m}} + C\int _0^T \Vert \nabla v(t) \Vert _{H^{m}}^2 \,\textrm{d}t + C\int _0^T \Vert \nabla _h \theta (t) \Vert _{H^{m-2}}^2 \,\textrm{d}t \\ + C \sup _{t \in [0,T]} (\Vert v(t) \Vert _{H^m} + \Vert \theta (t) \Vert _{H^m}) \left( \int _0^T \sum _{\eta \in I} |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \int _0^T \sum _{\eta \in J} |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) . \end{aligned}\nonumber \\ \end{aligned}$$

(5.4)

for all $T>0$.

Proof

We recall (3.6) and have

$$\begin{aligned} \sum _{\eta \in J}\int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} |\mathscr {F}_b \theta (t)| \,\textrm{d}t \le I_5 + I_6 + I_7 + I_8, \end{aligned}$$

where

$$\begin{aligned} I_5&:= \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} (e^{-\lambda _- t} - e^{-\lambda _+ t}) |\langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}t, \\ I_6&:= \sum _{\eta \in J} \int _0^T e^{-\lambda _+ t} |\mathscr {F}_b \theta _0| \, \textrm{d}t, \\ I_7&:= \sum _{\eta \in J} \int _0^T \int _0^t \frac{|\tilde{n}|^2}{|\eta |^2} (e^{-\lambda _- (t - \tau )} - e^{-\lambda _+ (t - \tau )}) |\langle N(v,\theta )(\tau ),\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}\tau \textrm{d}t,\\ I_8&:= \sum _{\eta \in J} \int _0^T \int _0^t e^{-\lambda _+ (t - \tau )} |\mathscr {F}_b [(v \cdot \nabla )\theta ](\tau )| \,\textrm{d}\tau \textrm{d}t. \end{aligned}$$

We estimate $I_6$ and $I_8$ first. By (3.7) we have

$$\begin{aligned} I_6 \le \sum _{\eta \in J} \int _0^T |\eta |^2 e^{-|\eta |^2 \frac{t}{2}} |\mathscr {F}_b \theta _0| \, \textrm{d}t \le C\Vert \theta _0 \Vert _{H^m}. \end{aligned}$$

With Fubini’s theorem, we also have

$$\begin{aligned} I_8&\le C \sum _{\eta \in J} \int _0^T |\mathscr {F}_b [(v \cdot \nabla )\theta ](t)| \,\textrm{d}t. \end{aligned}$$

Due to (2.1) and (2.3), Poincaré inequality implies

$$\begin{aligned} I_8&\le C\int _0^T \left( \sum _{\eta \in I} |\mathscr {F} (v-\int _{\Omega } v \,\textrm{d}x )(t)| \right) \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_b \theta (t)| \right) \,\textrm{d}t \\&\le C \sup _{t \in [0,T]} \Vert \theta (t) \Vert _{H^m} \left( \int _0^T \sum _{\eta \in I} |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \int _0^T \sum _{\eta \in J} |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) . \end{aligned}$$

Now, we estimate $I_5$ and $I_7$. To apply (3.8), we consider $\eta \in D_2$ and $\eta \in D_3$ separately. Note that $D_1 = \emptyset$ when $\alpha = 1$. In the former case, as the previous estimates, we have

$$\begin{aligned}&\sum _{\eta \in D_2} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} (e^{-\lambda _- t} - e^{-\lambda _+ t}) |\langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}t\\&\quad \le C\sum _{\eta \in J} \int _0^T e^{-|\eta |^2 \frac{t}{4}} |\mathscr {F}_b \textbf{u}_0| \, \textrm{d}t \le C\Vert \textbf{u}_0 \Vert _{H^m} \end{aligned}$$

and

$$\begin{aligned}&\sum _{\eta \in D_2}\int _0^T \int _0^t \frac{|\tilde{n}|^2}{|\eta |^2} (e^{-\lambda _- (t - \tau )} - e^{-\lambda _+ (t - \tau )}) |\langle N(v,\theta )(\tau ),\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}\tau \textrm{d}t \\&\quad \le C\sum _{\eta \in J} \int _0^T |N(v,\theta )(t)| \, \textrm{d}t \\&\quad \le C\sum _{\eta \in J} \int _0^T (|\mathscr {F}[(v \cdot \nabla )v](t)| + |\mathscr {F}_b[((v-\int _{\Omega } v \,\textrm{d}x) \cdot \nabla )\theta ](t)|) \, \textrm{d}t \\&\quad \le C\int _0^T \left\{ \left( \sum _{\eta \in I} |\mathscr {F}v(t)| \right) \left( \sum _{\eta \in I} |\eta | |\mathscr {F}v(t)| \right) + \left( \sum _{\eta \in I} |\eta | |\mathscr {F}v(t)| \right) \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_b \theta (t)|\right) \right\} \, \textrm{d}t \\&\quad \le C \sup _{t \in [0,T]} (\Vert v(t) \Vert _{H^m} + \Vert \theta (t) \Vert _{H^m}) \left( \int _0^T \sum _{\eta \in I} |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \int _0^T \sum _{\eta \in J} |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) . \end{aligned}$$

In the latter case,

$$\begin{aligned} \sum _{\eta \in D_3} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} (e^{-\lambda _- t} - e^{-\lambda _+ t}) |\langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}t&\le C\sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |^2} e^{-\frac{|\tilde{n}|^2}{|\eta |^4} t} |\mathscr {F}_b \textbf{u}_0| \, \textrm{d}t \\&\quad \le C\sum _{\eta \in J} |\eta |^2 |\mathscr {F}_b \textbf{u}_0| \\&\quad \le C \Vert \textbf{u}_0 \Vert _{H^m}. \end{aligned}$$

On the other hand, we similarly have

$$\begin{aligned}&\sum _{\eta \in D_3}\int _0^T \int _0^t \frac{|\tilde{n}|^2}{|\eta |^2} (e^{-\lambda _- (t - \tau )} - e^{-\lambda _+ (t - \tau )}) |\langle N(v,\theta )(\tau ),\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}\tau \textrm{d}t \\&\quad \le \sum _{\eta \in J} \int _0^T |\eta |^2 |N(v,\theta )(t)| \, \textrm{d}t \\&\quad \le \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}(v \cdot \nabla )v(t)| \, \textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_b(v \cdot \nabla )\theta (t)| \, \textrm{d}t. \end{aligned}$$

Let $\tilde{n}' + \tilde{n}'' = \tilde{n}$ and $\tilde{q}' + \tilde{q}'' = \tilde{q}$. Then, it holds

$$\begin{aligned} |\eta |^2 = |\tilde{n}|^2 + |\tilde{q}|^2 \le 2|\tilde{n}'|^2 + 2|\tilde{q}'|^2 + 2|\tilde{n}''|^2 + 2|\tilde{q}''|^2 = 2|\eta '|^2 + 2|\eta ''|^2. \end{aligned}$$

Similarly, for $\tilde{n}' + \tilde{n}'' = \tilde{n}$ and $|\tilde{q}' - \tilde{q}''| = \tilde{q}$, we can see

$$\begin{aligned} |\eta |^2 \le 2|\eta '|^2 + 2|\eta ''|^2. \end{aligned}$$

They imply

$$\begin{aligned}{} & {} \sum _{\eta \in J} \int _0^T |\eta |^2 (|\mathscr {F}(v \cdot \nabla )v(t)| \, \textrm{d}t \\{} & {} \quad \le C\int _0^T \left\{ \left( \sum _{\eta \in I} |\eta |^2 |\mathscr {F}v(t)| \right) \left( \sum _{\eta \in I} |\eta | |\mathscr {F}v(t)| \right) + \left( \sum _{\eta \in I} |\mathscr {F}v(t)| \right) \left( \sum _{\eta \in I} |\eta |^3 |\mathscr {F}v(t)| \right) \right\} \, \textrm{d}t \\{} & {} \quad \le C \int _0^T \Vert \nabla v \Vert _{H^m}^2 \,\textrm{d}t. \end{aligned}$$

On the other hand, with $\mathscr {F}_b(v \cdot \nabla )\theta = \mathscr {F}_b(v_h \cdot \nabla _h)\theta + \mathscr {F}_b[v_d \partial _d \theta ]$ it follows for $m > 3+d/2$

$$\begin{aligned}{} & {} \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_b(v \cdot \nabla )\theta (t)| \, \textrm{d}t \\{} & {} \quad \le \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_b(v_h \cdot \nabla _h)\theta (t)| \, \textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_b[v_d \partial _d \theta ](t)| \, \textrm{d}t \\{} & {} \quad \le C\int _0^T \left\{ \left( \sum _{\eta \in I} |\eta |^2 |\mathscr {F}_cv_h(t)| \right) \left( \sum _{\eta \in J} |\tilde{n}| |\mathscr {F}_b\theta (t)| \right) \right. \\{} & {} \quad \left. + \left( \sum _{\eta \in I} |\mathscr {F}_cv_h(t)| \right) \left( \sum _{\eta \in J} |\eta |^2 |\tilde{n}| |\mathscr {F}_b \theta (t)|\right) \right\} \, \textrm{d}t\\{} & {} \quad + C\int _0^T \left\{ \left( \sum _{\eta \in J} |\eta |^2 |\mathscr {F}_bv_d(t)| \right) \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_b\theta (t)| \right) + \left( \sum _{\eta \in J} |\mathscr {F}_bv_d(t)| \right) \left( \sum _{\eta \in J} |\eta |^3 |\mathscr {F}_b\theta (t)| \right) \right\} \, \textrm{d}t \\{} & {} \quad \le C\int _0^T \left( \sum _{\eta \in I} |\eta |^2 |\mathscr {F}_cv_h(t)| \right) \left( \sum _{\eta \in J} |\tilde{n}| |\mathscr {F}_b\theta (t)| \right) \, \textrm{d}t \\{} & {} \qquad +C \sup _{t \in [0,T]} \Vert \theta (t) \Vert _{H^m} \left( \int _0^T \sum _{\eta \in I} |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \int _0^T \sum _{\eta \in J} |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) . \end{aligned}$$

Since

$$\begin{aligned} \sum _{\eta \in J} |\tilde{n}| |\mathscr {F}_b\theta (t)| = \sum _{\eta \in J} |\mathscr {F}_b \nabla _h \theta (t)| \le C \Vert \nabla _h \theta (t) \Vert _{H^{m-2}} \end{aligned}$$

and

$$\begin{aligned} \sum _{\eta \in I} |\eta |^2 |\mathscr {F}_cv_h(t)| \le C \Vert \nabla v (t) \Vert _{H^{m}}, \end{aligned}$$

we have

$$\begin{aligned}{} & {} C\int _0^T \left( \sum _{\eta \in I} |\eta |^2 |\mathscr {F}_cv_h(t)| \right) \left( \sum _{\eta \in J} |\tilde{n}| |\mathscr {F}_b\theta (t)| \right) \, \textrm{d}t \le C\int _0^T \Vert \nabla v(t) \Vert _{H^{m}}^2 \,\textrm{d}t \\{} & {} \quad + C\int _0^T \Vert \nabla _h \theta (t) \Vert _{H^{m-2}}^2 \,\textrm{d}t. \end{aligned}$$

Collecting the estimates for $I_5$, $I_6$, $I_7$, and $I_8$, we complete the proof. $\square$

Now, we are ready to prove the global existence part of Theorem 1.1. Let $T^*>0$ and $(v,\theta )$ be the maximal time of existence and the local solution given in Proposition 4.1 respectively. We define

$$\begin{aligned} B_m(T) := \left( \sup _{t \in [0,T]} E_m(t)^2 + \int _0^T \Vert \Lambda ^{\alpha } v(t) \Vert _{H^m}^2 \,\textrm{d}t+ \int _0^T \Vert \nabla _h \theta (t) \Vert _{H^{m-1-\alpha }}^2 \,\textrm{d}t \right) ^{\frac{1}{2}}. \end{aligned}$$

(5.5)

Then, from (5.2) and (5.4), we have

$$\begin{aligned}{} & {} \sum _{\eta \in I} \int _0^T |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \le C \Vert (v_0,\theta _0) \Vert _{H^m}\\{} & {} \quad + C_1B_m(T) \left( \sum _{\eta \in I} \int _0^T |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \right) + C_1B_m(T)^2 \end{aligned}$$

for some $C_1>0$. For a while, we assume that $C_1B_m(T) \le \frac{1}{2}$ for all $T \in (0,T^*)$. Then, we have

$$\begin{aligned} \sum _{\eta \in I} \int _0^T |\eta | |\mathscr {F}_cv_h(t) | \,\textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta |^2 |\mathscr {F}_bv_d(t) | \,\textrm{d}t \le C \Vert (v_0,\theta _0) \Vert _{H^m} + B_m(T). \end{aligned}$$

On the other hand, we recall (4.2) and integrate it on the interval [0, T]. Then by (4.1), we have

$$\begin{aligned} \frac{1}{2} B_m(T)^2 \le \frac{3}{2} \Vert (v_0,\theta _0) \Vert _{H^m}^2 + CB_m(T)^3 + CB_m(T)^2 \int _0^T \Vert \nabla v(t) \Vert _{L^{\infty }} \,\textrm{d}t. \end{aligned}$$

Since

$$\begin{aligned} \Vert \nabla v \Vert _{L^{\infty }} \le \sum _{\eta \in I} |\eta ||\mathscr {F}_c v_h| + \sum _{\eta \in J} |\eta |^2|\mathscr {F}_bv_d|, \end{aligned}$$

it holds

$$\begin{aligned} \frac{1}{2} B_m(T)^2&\le \frac{3}{2}\Vert (v_0,\theta _0) \Vert _{H^m}^2 + CB_m(T)^3 + CB_m(T)^2 \left( \Vert (v_0,\theta _0) \Vert _{H^m} + B_m(T) \right) \\&\le \frac{3}{2} \Vert (v_0,\theta _0) \Vert _{H^m}^2 + C_2 \Vert (v_0,\theta _0) \Vert _{H^m} B_m(T)^2 + C_2B_m(T)^3 \end{aligned}$$

for some $C_2>0$. If we assume $C_2 \Vert (v_0,\theta _0) \Vert _{H^m} \le \frac{1}{16}$ and $C_2 B_m(T) \le \frac{1}{16}$, then

$$\begin{aligned} B_m(T)^2 \le 4 \Vert (v_0,\theta _0) \Vert _{H^m}^2 \le 4\delta ^2. \end{aligned}$$

(5.6)

Here, we take $\delta >0$ such that $C_1 (2\delta ) < \frac{1}{2}$ and $C_2 (2\delta ) < \frac{1}{16}$. By the above estimates, we can deduce that (5.6) holds for all $T \in (0,T^*)$, hence, $T^* = \infty$. Thus, (1.3) is obtained. This completes the proof.

5.2 Proof of Theorem 1.4:Global-in-time existence part

In this subsection, we fix $\alpha = 0$. We only provide two propositions counterparts of Proposition 5.1 and 5.2, because the rest of the proof is similar with that of theorem 1.1.

Proposition 5.3

Let $d \in \mathbb {N}$ with $d \ge 2$ and $m \in \mathbb {N}$ satisfying $m > 2+\frac{d}{2}$. Assume that $(v, \theta )$ is a smooth global solution to (1.2) with $\alpha = 0$. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_b v_d(t) | \,\textrm{d}t \le C \Vert v_0 \Vert _{H^m} + C \int _0^T \Vert v(t) \Vert _{H^m}^2 \,\textrm{d}t + \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |} |\mathscr {F}_b \theta (t)| \,\textrm{d}t \end{aligned}\nonumber \\ \end{aligned}$$

(5.7)

for all $T > 0$.

Proof

From (3.2) and Duhamel’s principle, we can have

$$\begin{aligned} \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_bv_d(t) | \,\textrm{d}t \le J_1 + J_2 + J_3 + J_4, \end{aligned}$$

where

$$\begin{aligned} J_1&:= \sum _{\eta \in J} \int _0^T |\eta | e^{-t} |\mathscr {F}_bv_0| \,\textrm{d}t, \\ J_2&:= \sum _{\eta \in J} \int _0^T \int _0^t |\eta |e^{-(t-\tau )} |\mathscr {F}_b[(v \cdot \nabla )v_d](\tau )| \,\textrm{d}\tau \textrm{d}t,\\ J_3&:= \sum _{\eta \in J} \int _0^T \int _0^t |\eta |e^{-(t-\tau )} |\mathscr {F}_c[(v \cdot \nabla )v_h](\tau )| \,\textrm{d}\tau \textrm{d}t, \\ J_4&:= \sum _{\eta \in J} \int _0^T \int _0^t e^{-(t - \tau )} \frac{|\tilde{n}|^2}{|\eta |} |\mathscr {F}_b \theta (\tau )| \,\textrm{d}\tau \textrm{d}t. \end{aligned}$$

We can easily show

$$\begin{aligned} J_1 \le \sum _{\eta \in J} |\eta | |\mathscr {F}_bv_0| \le C \Vert v_0 \Vert _{H^m} \end{aligned}$$

and

$$\begin{aligned} J_4 \le \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |} |\mathscr {F}_b \theta (t)| \, \textrm{d}t. \end{aligned}$$

Fubini’s theorem and Propostion 2.4 gives

$$\begin{aligned} J_2 \le \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_b[(v \cdot \nabla )v_d](t)| \,\textrm{d}t \le C\int _0^T \Vert v(t) \Vert _{H^m}^2 \,\textrm{d}t. \end{aligned}$$

Similarly, we can estimate $J_3$ and have

$$\begin{aligned} J_3 \le C\int _0^T \Vert v(t) \Vert _{H^m}^2 \,\textrm{d}t. \end{aligned}$$

From the estimates for $J_1$, $J_2$, $J_3$, and $J_4$, we deduce (5.7). This completes the proof. $\square$

Proposition 5.4

Let $d \in \mathbb {N}$ with $d \ge 2$ and $m \in \mathbb {N}$ satisfying $m > 2+\frac{d}{2}$. Assume that $(v, \theta )$ is a smooth global solution to (1.2) with $\alpha = 1$. Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |} |\mathscr {F}_b \theta (t)| \,\textrm{d}t&\le C \Vert \textbf{u}_0 \Vert _{H^{m}} + C \int _0^T \Vert v(t) \Vert _{H^m}^2 \,\textrm{d}t + C \int _0^T \Vert \nabla _h \theta (t) \Vert _{H^{m-1}}^2 \,\textrm{d}t \\&\quad + C \sup _{t \in [0,T]} \Vert \theta (t) \Vert _{H^m} \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_bv_d(t) | \,\textrm{d}t . \end{aligned} \end{aligned}$$

(5.8)

for all $T>0$.

Proof

We recall (3.6) and have

$$\begin{aligned} \sum _{\eta \in J}\int _0^T \frac{|\tilde{n}|^2}{|\eta |} |\mathscr {F}_b \theta (t)| \,\textrm{d}t \le J_5 + J_6 + J_7 + J_8, \end{aligned}$$

where

$$\begin{aligned} J_5&:= \sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |} (e^{-\lambda _- t} - e^{-\lambda _+ t}) |\langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}t, \\ J_6&:= \sum _{\eta \in J} \int _0^T |\tilde{n}| e^{-\lambda _+ t} |\mathscr {F}_b \theta _0| \, \textrm{d}t, \\ J_7&:= \sum _{\eta \in J} \int _0^T \int _0^t \frac{|\tilde{n}|^2}{|\eta |} (e^{-\lambda _- (t - \tau )} - e^{-\lambda _+ (t - \tau )}) |\langle N(v,\theta )(\tau ),\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}\tau \textrm{d}t,\\ J_8&:= \sum _{\eta \in J} \int _0^T \int _0^t |\tilde{n}| e^{-\lambda _+ (t - \tau )} |\mathscr {F}_b [(v \cdot \nabla )\theta ](\tau )| \,\textrm{d}\tau \textrm{d}t. \end{aligned}$$

It is clear by (3.7)

$$\begin{aligned} J_6 \le \sum _{\eta \in J} \int _0^T e^{-\frac{t}{2}} |\mathscr {F}_b \nabla _h \theta _0| \, \textrm{d}t \le C\Vert \theta _0 \Vert _{H^m}. \end{aligned}$$

Similarly, we have with Proposition 2.4 that

$$\begin{aligned} J_8&\le C \sum _{\eta \in J} \int _0^T |\mathscr {F}_b [\nabla _h (v \cdot \nabla )\theta ]| \,\textrm{d}t \\&\le C \int _0^T \left\{ \left( \sum _{\eta \in I} |\eta | |\mathscr {F}_c v_h| \right) \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_b \nabla _h \theta | \right) + \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_b v_d| \right) \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_b \partial _d \theta | \right) \right\} \,\textrm{d}t \\&\le C \int _0^T \Vert v \Vert _{H^m}^2 \,\textrm{d}t + C \int _0^T \Vert \nabla _h \theta \Vert _{H^{m-1}}^2 \,\textrm{d}t + C \sup _{t \in [0,T]} \Vert \theta (t) \Vert _{H^m} \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_bv_d | \,\textrm{d}t . \end{aligned}$$

To estimate $J_5$ and $J_7$ with (3.8), we consider $\eta \in D_1 \cup D_2$ and $\eta \in D_3$ separately. We note that

$$\begin{aligned}&\sum _{\eta \in D_1 \cup D_2} \int _0^T \frac{|\tilde{n}|^2}{|\eta |} (e^{-\lambda _- t} - e^{-\lambda _+ t}) |\langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}t \\&\quad \le C\sum _{\eta \in J} \int _0^T |\tilde{n}| e^{- \frac{t}{4}} |\mathscr {F}_b \textbf{u}_0| \, \textrm{d}t \le C\Vert \textbf{u}_0 \Vert _{H^m} \end{aligned}$$

and

$$\begin{aligned}&\sum _{\eta \in D_1 \cup D_2}\int _0^T \int _0^t \frac{|\tilde{n}|^2}{|\eta |} (e^{-\lambda _- (t - \tau )} - e^{-\lambda _+ (t - \tau )}) |\langle N(v,\theta )(\tau ),\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}\tau \textrm{d}t \\&\quad \le C\sum _{\eta \in J} \int _0^T |\tilde{n}| |N(v,\theta )(t)| \, \textrm{d}t \\&\quad \le C\sum _{\eta \in J} \int _0^T (|\mathscr {F}[\nabla _h(v \cdot \nabla )v](t)| + |\mathscr {F}_b[\nabla _h(v \cdot \nabla )\theta ](t)|) \, \textrm{d}t \\&\quad \le C \int _0^T \Vert v \Vert _{H^m}^2 \,\textrm{d}t + C \int _0^T \Vert \nabla _h \theta \Vert _{H^{m-1}}^2 \,\textrm{d}t + C \sup _{t \in [0,T]} \Vert \theta (t) \Vert _{H^m} \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_bv_d | \,\textrm{d}t . \end{aligned}$$

On the other hand,

$$\begin{aligned} \sum _{\eta \in D_3} \int _0^T \frac{|\tilde{n}|^2}{|\eta |} (e^{-\lambda _- t} - e^{-\lambda _+ t}) |\langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}t&\le C\sum _{\eta \in J} \int _0^T \frac{|\tilde{n}|^2}{|\eta |} e^{-\frac{|\tilde{n}|^2}{|\eta |^2} t} |\mathscr {F}_b \textbf{u}_0| \, \textrm{d}t \\&\le C\sum _{\eta \in J} |\eta | |\mathscr {F}_b \textbf{u}_0| \\&\le C \Vert (v_0,\theta _0) \Vert _{H^m}. \end{aligned}$$

We can see

$$\begin{aligned}&\sum _{\eta \in D_3}\int _0^T \int _0^t \frac{|\tilde{n}|^2}{|\eta |} (e^{-\lambda _- (t - \tau )} - e^{-\lambda _+ (t - \tau )}) |\langle N(v,\theta )(\tau ),\textbf{a}_- \rangle | |\langle \textbf{b}_-,e_{2} \rangle | \,\textrm{d}\tau \textrm{d}t \\&\quad \le \sum _{\eta \in J} \int _0^T |\eta | |N(v,\theta )(t)| \, \textrm{d}t \\&\quad \le \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}(v \cdot \nabla )v(t)| \, \textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_b(v \cdot \nabla )\theta (t)| \, \textrm{d}t. \end{aligned}$$

As estimating $I_7$ on the set $D_3$, we can deduce

$$\begin{aligned}{} & {} \sum _{\eta \in J} \int _0^T |\eta | (|\mathscr {F}(v \cdot \nabla )v(t)| \, \textrm{d}t \\{} & {} \quad \le C\int _0^T \left\{ \left( \sum _{\eta \in I} |\eta | |\mathscr {F}v(t)| \right) ^2 + \left( \sum _{\eta \in I} |\mathscr {F}v(t)| \right) \left( \sum _{\eta \in I} |\eta |^2 |\mathscr {F}v(t)| \right) \right\} \, \textrm{d}t \\{} & {} \quad \le C \int _0^T \Vert v(t) \Vert _{H^m}^2 \,\textrm{d}t \end{aligned}$$

and

$$\begin{aligned}{} & {} \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_b(v \cdot \nabla )\theta (t)| \, \textrm{d}t \\{} & {} \le \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_b(v_h \cdot \nabla _h)\theta (t)| \, \textrm{d}t + \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_b[v_d \partial _d \theta ](t)| \, \textrm{d}t \\{} & {} \le C\int _0^T \left\{ \left( \sum _{\eta \in I} |\eta | |\mathscr {F}_cv_h(t)| \right) \left( \sum _{\eta \in J} |\tilde{n}| |\mathscr {F}_b\theta (t)| \right) + \left( \sum _{\eta \in I} |\mathscr {F}_cv_h(t)| \right) \left( \sum _{\eta \in J} |\eta | |\tilde{n}| |\mathscr {F}_b \theta (t)|\right) \right\} \, \textrm{d}t \\{} & {} \quad + C\int _0^T \left\{ \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_bv_d(t)| \right) \left( \sum _{\eta \in J} |\eta | |\mathscr {F}_b\theta (t)| \right) + \left( \sum _{\eta \in J} |\mathscr {F}_bv_d(t)| \right) \left( \sum _{\eta \in J} |\eta |^2 |\mathscr {F}_b\theta (t)| \right) \right\} \, \textrm{d}t \\{} & {} \le C \int _0^T \Vert v \Vert _{H^m}^2 \,\textrm{d}t + C \int _0^T \Vert \nabla _h \theta \Vert _{H^{m-1}}^2 \,\textrm{d}t + C \sup _{t \in [0,T]} \Vert \theta (t) \Vert _{H^m} \sum _{\eta \in J} \int _0^T |\eta | |\mathscr {F}_bv_d | \,\textrm{d}t \end{aligned}$$

for $m > 2+d/2$. Collecting the estimates for $J_5$, $J_6$, $J_7$, and $J_8$, we obtain (5.8). This completes the proof. $\square$

6 Proof of temporal decay estimates

In this section, let $(v,\theta )$ be a smooth global-in-time solution to (1.2). In addition, we assume that (1.6) or (1.3) holds in each case with

$$\begin{aligned} \Vert (v_0,\theta _0) \Vert _{H^m} \le \delta \end{aligned}$$

(6.1)

for sufficiently small $\delta >0$. The next three propositions are for the temporal decay estimates of $\Vert \bar{\theta }(t) \Vert _{L^2}$, $\Vert v(t) \Vert _{L^2}$, and $\Vert v_d(t) \Vert _{L^2}$ in both cases $\alpha =0$ and $\alpha = 1$. After that, we prove (1.7) and (1.5) combining with the temporal decay estimates for $\Vert v(t) \Vert _{{\dot{H}}^m}$ and $\Vert v_d(t) \Vert _{{\dot{H}}^m}$.

Proposition 6.1

Let $d \in \mathbb {N}$ with $d \ge 2$ and $\alpha \in \{0,1\}$. Let $m \in \mathbb {N}$ with $m > 1+\frac{d}{2} +\alpha$ and $(v, \theta )$ be a smooth global solution to (1.2) with (1.3) or (1.6). Suppose that (6.1) be satisfied. Then, there exists a constant $C > 0$ such that

$$\begin{aligned} \Vert \bar{\theta }(t) \Vert _{L^2}^2 \le C (1+t)^{-\frac{m}{1+\alpha }}. \end{aligned}$$

(6.2)

Proof

From the v equations in (1.2), we have

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } |v|^2 \,\textrm{d}x&= -\Vert \Lambda ^{\alpha } v\Vert _{L^2}^2 + \int _{\Omega } v_d \theta \,\textrm{d}x. \end{aligned}$$

On the other hand, we have from (2.1) and the $\theta$ equation in (1.2) that

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } |\bar{\theta }|^2 \,\textrm{d}x&= - \int _{\Omega } (v \cdot \nabla ) (\widetilde{\theta } +\bar{\theta }) \cdot \bar{\theta } \,\textrm{d}x -\int _{\Omega } v_d \bar{\theta } \,\textrm{d}x \\&\quad = - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot \bar{\theta } \,\textrm{d}x -\int _{\Omega } v_d \theta \,\textrm{d}x, \end{aligned}$$

where

$$\begin{aligned} \widetilde{\theta } := \int _{\mathbb T^{d-1}} \theta (x) \,\textrm{d}x_h. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} (\Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2) = -\Vert \Lambda ^{\alpha } v \Vert _{L^2}^2 - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot \bar{\theta } \,\textrm{d}x. \end{aligned}$$

We can deduce from (3.2), (3.3), and (2.1) that

$$\begin{aligned} -\frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_d \Lambda ^{-2\alpha }\theta \,\textrm{d}x= & {} -\int _{\Omega } \partial _tv_d \Lambda ^{-2\alpha } \bar{\theta } \,\textrm{d}x - \int _{\Omega } \partial _t\theta \Lambda ^{-2\alpha }v_d \,\textrm{d}x \\\le & {} \Vert (v \cdot \nabla )v\Vert _{L^2} \Vert \Lambda ^{-2\alpha } \bar{\theta } \Vert _{L^2} + \Vert (v \cdot \nabla )\theta \Vert _{L^2} \Vert \Lambda ^{-2\alpha } v_d \Vert _{L^2} \\{} & {} - \frac{1}{2} \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 + \frac{3}{2} \Vert \Lambda ^{\alpha } v_d \Vert _{L^2}^2 . \end{aligned}$$

Combining the above, we have

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 - \int _{\Omega } v_d \Lambda ^{-2\alpha }\theta \,\textrm{d}x \right) \le -\frac{1}{4} (\Vert \Lambda ^{\alpha } v \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2) \\{} & {} \quad + C\Vert v\Vert _{L^2} \Vert \nabla \theta \Vert _{L^{\infty }} \Vert \Lambda ^{-2\alpha } v_d \Vert _{L^2} + C \Vert v \Vert _{L^2} \Vert v \Vert _{{\dot{H}}^m} \Vert \bar{\theta } \Vert _{L^2} - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot \bar{\theta } \,\textrm{d}x. \end{aligned}$$

To estimate the integral on the right-hand side, we note

$$\begin{aligned} \left| - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot \bar{\theta } \,\textrm{d}x \right| \le \left| - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot -\Delta _h (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| + \left| - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot -\partial _d^2 (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| . \end{aligned}$$

We consider $\alpha = 0$ case first. The right-hand side is bounded by

$$\begin{aligned}{} & {} \Vert v_d \Vert _{L^2} \Vert \partial _d \theta \Vert _{L^{\infty }} \Vert R_h^2 \theta \Vert _{L^2} + \left| -\int _{\Omega } (\nabla _h \cdot v_h) \partial _d \widetilde{\theta } \cdot \partial _d (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x - \int _{\Omega } v_d \partial _d^2 \widetilde{\theta } \cdot \partial _d (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| \\{} & {} \le \Vert v_d \Vert _{L^2} \Vert \partial _d \theta \Vert _{L^{\infty }} \Vert R_h^2 \theta \Vert _{L^2} + \Vert v_h \Vert _{L^2} \Vert \partial _d \widetilde{\theta } \Vert _{L^{\infty }} \Vert R_h \theta \Vert _{L^2} + \left| - \int _{\Omega } v_d \partial _d^2 \widetilde{\theta } \cdot \partial _d (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| . \end{aligned}$$

Using $v_d = -(\Delta _h)(-\Delta )^{-1} v_d + \partial _d \nabla _h (-\Delta )^{-1} v_h$, we have

$$\begin{aligned}{} & {} \left| - \int _{\Omega } v_d \partial _d^2 \widetilde{\theta } \cdot \partial _d (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| \\{} & {} \quad \le \left| - \int _{\Omega } \nabla _h (-\Delta )^{-1} v_d \partial _d^2 \widetilde{\theta } \cdot \nabla _h \partial _d (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| + \left| \int _{\Omega } \partial _d (-\Delta )^{-1}v_h \partial _d^2 \widetilde{\theta } \cdot \nabla _h \partial _d (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| \\{} & {} \quad \le \Vert \nabla (-\Delta )^{-1} v \Vert _{L^2_{x_h}L^{\infty }_{x_d}} \Vert \partial _d^2 \widetilde{\theta } \Vert _{L^{2}} \Vert R_h \theta \Vert _{L^2} \\{} & {} \quad \le \Vert v \Vert _{L^2} \Vert \theta \Vert _{H^m} \Vert R_h \theta \Vert _{L^2}. \end{aligned}$$

For $\alpha = 1$, we can see

$$\begin{aligned} \left| - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot -\Delta _h (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right|= & {} \left| - \int _{\Omega } \nabla _h v_d \partial _d \widetilde{\theta } \cdot \nabla _h (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| \\\le & {} \Vert \nabla _h v_d \Vert _{L^2} \Vert \partial _d \theta \Vert _{L^{\infty }} \Vert \Lambda ^{-1} R_h \theta \Vert _{L^2} \end{aligned}$$

and

$$\begin{aligned}{} & {} \left| - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot -\partial _d^2 (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| \\{} & {} \quad \le \left| \int _{\Omega } (\nabla _h \cdot \partial _d v_h) \partial _d \widetilde{\theta } \cdot (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| + 2\left| \int _{\Omega } (\nabla _h \cdot v_h) \partial _d^2 \widetilde{\theta } \cdot (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| \\{} & {} \qquad + \left| \int _{\Omega } v_d \partial _d^3 \widetilde{\theta } \cdot (-\Delta )^{-1} \bar{\theta } \,\textrm{d}x \right| \\{} & {} \quad \le \Vert \partial _d v_h \Vert _{L^2} \Vert \partial _d \theta \Vert _{L^{\infty }} \Vert \Lambda ^{-1} R_h \theta \Vert _{L^2} + 2\Vert v_h \Vert _{L^2} \Vert \partial _d^2 \theta \Vert _{L^{\infty }} \Vert \Lambda ^{-1} R_h \theta \Vert _{L^2} \\{} & {} \qquad + \Vert \nabla (-\Delta )^{-1} v \Vert _{L^2_{x_h}L^{\infty }_{x_d}} \Vert \partial _d^3 \widetilde{\theta } \Vert _{L^{2}} \Vert (-\Delta )^{-1} \theta \Vert _{L^2}, \end{aligned}$$

where $v_d = -(\Delta _h)(-\Delta )^{-1} v_d + \partial _d \nabla _h (-\Delta )^{-1} v_h$ also used here. Hence, we can deduce

$$\begin{aligned} \left| - \int _{\Omega } v_d \partial _d \widetilde{\theta } \cdot \bar{\theta } \,\textrm{d}x \right| \le C \Vert \Lambda ^{\alpha } v \Vert _{L^2} \Vert \theta \Vert _{H^m} \Vert \Lambda ^{-\alpha } R_h \theta \Vert _{L^2} \end{aligned}$$

in both cases. Combining the above and using (6.1), we can have

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 - \int _{\Omega } v_d \Lambda ^{-2\alpha }\theta \,\textrm{d}x \right) \\{} & {} \quad \le -(\frac{1}{4} - C(\Vert v \Vert _{H^m}^2 + \Vert \theta \Vert _{H^m}^2)) (\Vert \Lambda ^{\alpha } v \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2) + C \Vert v \Vert _{L^2} \Vert v \Vert _{{\dot{H}}^m} \Vert \bar{\theta } \Vert _{L^2} \\{} & {} \quad \le -\frac{1}{8} (\Vert v \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2) + C \Vert v \Vert _{{\dot{H}}^m}^2 \Vert \bar{\theta } \Vert _{L^2}^2. \end{aligned}$$

Let $M\ge 1$ which will be specified later. Since

$$\begin{aligned} \begin{aligned} \frac{1}{M} \Vert \bar{\theta } \Vert _{L^2}^2 - \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2&= \sum _{|\tilde{n}| \ne 0} \left( \frac{1}{M} - \frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \right) |\mathscr {F} \theta (\eta )|^2 \\&\le \frac{1}{M} \sum _{\frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \le \frac{1}{M}, |\tilde{n}| \ne 0} |\mathscr {F} \theta (\eta )|^2 \\&\le \frac{1}{M^{1+\frac{m-1-\alpha }{1+\alpha }}} \Vert \bar{\theta } \Vert _{{\dot{H}}^{m-1-\alpha }}^2 \\&\le \frac{1}{M^{\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \left| \int _{\Omega } v_d \Lambda ^{-2\alpha } \theta \,\textrm{d}x \right| \le \Vert v_d \Vert _{L^2} \Vert \Lambda ^{-2\alpha } \bar{\theta } \Vert _{L^2} \le \frac{1}{2} \Vert v \Vert _{L^2}^2 + \frac{1}{2} \Vert \bar{\theta } \Vert _{L^2}^2, \end{aligned}$$

(6.3)

it holds

$$\begin{aligned} \begin{aligned}&-\frac{1}{8} (\Vert v \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2) \le -\frac{1}{8M} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 -\frac{1}{2} \int _{\Omega } v_d \Lambda ^{-2\alpha } \theta \,\textrm{d}x\right) \\&\qquad + \frac{1}{16M} \int _{\Omega } v_d \Lambda ^{-2\alpha } \theta \,\textrm{d}x + \frac{1}{8} \left( \frac{1}{M} \Vert \bar{\theta } \Vert _{L^2}^2 - \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 \right) \\&\quad \le -\frac{1}{16M} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 -\int _{\Omega } v_d \Lambda ^{-2\alpha } \theta \,\textrm{d}x \right) + \frac{1}{8M^{\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2. \end{aligned} \end{aligned}$$

(6.4)

Thus,

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 - \int _{\Omega } v_d \Lambda ^{-2\alpha }\theta \,\textrm{d}x \right) \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 -\int _{\Omega } v_d \Lambda ^{-2\alpha } \theta \,\textrm{d}x \right) + \frac{1}{8M^{\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 + C \Vert v \Vert _{{\dot{H}}^m}^2 \Vert \bar{\theta } \Vert _{L^2}^2. \end{aligned}$$

Taking $M = 1+\frac{t}{8\frac{m}{1+\alpha }}$ and multiplying both terms by $2M^{\frac{m}{1+\alpha }}$, we obtain by (6.3) that

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \left( (1+\frac{t}{8\frac{m}{1+\alpha }})^{\frac{m}{1+\alpha }} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 - \int _{\Omega } v_d \Lambda ^{-2\alpha }\theta \,\textrm{d}x \right) \right) \\{} & {} \le C \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 + C \Vert v \Vert _{{\dot{H}}^m}^2 (1+\frac{t}{8\frac{m}{1+\alpha }})^{\frac{m}{1+\alpha }} \left( \Vert v \Vert _{L^2}^2 + \Vert \bar{\theta } \Vert _{L^2}^2 - \int _{\Omega } v_d \Lambda ^{-2\alpha }\theta \,\textrm{d}x \right) . \end{aligned}$$

Using Grönwall’s inequality, we obtain (6.2). This completes the proof. $\square$

Proposition 6.2

Let $d \in \mathbb {N}$ with $d \ge 2$ and $\alpha \in \{0,1\}$. Let $m \in \mathbb {N}$ with $m > 1+\frac{d}{2} +\alpha$ and $(v, \theta )$ be a smooth global solution to (1.2) with (1.3) or (1.6). Suppose that (6.1) be satisfied. Then, there exists a constant $C > 0$ such that

$$\begin{aligned} \Vert v(t) \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta (t) \Vert _{L^2}^2 \le C (1+t)^{-(1+\frac{m}{1+\alpha })}. \end{aligned}$$

(6.5)

Proof

From the v equations in (1.2), we have

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } |v|^2 \,\textrm{d}x&\le -\Vert \Lambda ^{\alpha } v\Vert _{L^2}^2 + C\left( \sum _{|\tilde{n}| \ne 0} \frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} |\mathscr {F}_b \theta (\eta )|^{2} \right) ^{\frac{1}{2}} \left( \sum _{|\tilde{n}| \ne 0} |\eta |^{2\alpha } |\mathscr {F} v(\eta )|^2 \right) ^{\frac{1}{2}} \\&\le -\frac{1}{2} \Vert v \Vert _{L^2}^2 + C \sum _{|\tilde{n}| \ne 0} \frac{|\tilde{n}|^{2}}{|\eta |^{2(1+\alpha )}} |\mathscr {F}_b \theta (\eta )|^2. \end{aligned}$$

Since Duhamel’s principle implies

$$\begin{aligned} \Vert v(t) \Vert _{L^2}^2&\le e^{-t} \Vert v_0 \Vert _{L^2}^2 + C\int _0^t e^{-(t-\tau )} \sum _{|\tilde{n}| \ne 0} \frac{|\tilde{n}|^{2}}{|\eta |^{2(1+\alpha )}} |\mathscr {F}_b \theta (\eta )|^2 \,\textrm{d}\tau \\&\quad \le e^{-t} \Vert v_0 \Vert _{L^2}^2 + C\sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{1+\alpha }} \Vert R_h\Lambda ^{-\alpha } \theta (\tau ) \Vert _{L^2}^2 \int _0^t e^{-(t-\tau )} (1+\tau )^{-(1+\frac{m}{1+\alpha })} \,\textrm{d}\tau , \end{aligned}$$

we have

$$\begin{aligned} \sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{1+\alpha }} \Vert v(\tau ) \Vert _{L^2}^2 \le C\left( \Vert v_0 \Vert _{L^2}^2 + \sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{1+\alpha }} \Vert R_h\Lambda ^{-\alpha } \theta (\tau ) \Vert _{L^2}^2 \right) .\nonumber \\ \end{aligned}$$

(6.6)

On the other hand, from (3.2) and (3.3), we have

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \sum _{\eta \in J} |\mathscr {F}_b \Lambda ^{-\alpha } v_d(\eta )|^2\le & {} -\Vert v_d\Vert _{L^2}^2 + \Vert (v \cdot \nabla )v\Vert _{L^2} \Vert \Lambda ^{-2\alpha } v_d \Vert _{L^2}\\{} & {} + \sum _{\eta \in J} \frac{|\tilde{n}|^{2}}{|\eta |^{2(1+\alpha )}} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \sum _{\eta \in J} |\mathscr {F}_b R_h \Lambda ^{-\alpha } \theta (\eta )|^2 \le -\int _{\Omega } (v \cdot \nabla )\theta R_h^{2} \Lambda ^{-2\alpha }\theta \,\textrm{d}x - \sum _{\eta \in J} \frac{|\tilde{n}|^{2}}{|\eta |^{2(1+\alpha )}} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ) \end{aligned}$$

respectively. Thus,

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} (\Vert \Lambda ^{-\alpha } v_d\Vert _{L^2}^2 + \Vert R_h\Lambda ^{-\alpha } \theta \Vert _{L^2}^2)\le & {} -\Vert v_d\Vert _{L^2}^2 + \Vert (v \cdot \nabla )v\Vert _{L^2} \Vert v_d \Vert _{L^2}\\{} & {} -\int _{\Omega } (v \cdot \nabla )\theta R_h^{2} \Lambda ^{-2\alpha } \theta \,\textrm{d}x. \end{aligned}$$

Moreover, we can deduce from (3.2) and (3.3) that

$$\begin{aligned} -\frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_d R_h^{2} \Lambda ^{-4\alpha }\theta \,\textrm{d}x= & {} -\int _{\Omega } \partial _tv_d R_h^{2} \Lambda ^{-4\alpha }\theta \,\textrm{d}x - \int _{\Omega } \partial _t\theta R_h^{2} \Lambda ^{-4\alpha }v_d \,\textrm{d}x \\\le & {} \Vert (v \cdot \nabla )v\Vert _{L^2} \Vert R_h^{2} \Lambda ^{-4\alpha } \theta \Vert _{L^2} + \int _{\Omega } (v \cdot \nabla )\theta R_h^{2} \Lambda ^{-4\alpha } v_d \,\textrm{d}x \\{} & {} - \frac{1}{2} \Vert R_h^{2} \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 + \frac{3}{2} \Vert v_d \Vert _{L^2}^2 . \end{aligned}$$

Combining the above, we have

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert \Lambda ^{-\alpha }v_d\Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{2} \Lambda ^{-4\alpha }\theta \,\textrm{d}x \right) \le -\frac{1}{4} (\Vert v_d\Vert _{L^2}^2 + \Vert R_h^{2}\Lambda ^{-2\alpha } \theta \Vert _{L^2}^2) \\{} & {} \quad + C\Vert v\Vert _{L^2} \Vert \nabla v \Vert _{L^\infty } (\Vert v_d \Vert _{L^2} + \Vert R_h^{2} \Lambda ^{-2\alpha }\theta \Vert _{L^2}) -\int _{\Omega } (v \cdot \nabla )\theta (R_h^{2} \Lambda ^{-2\alpha }\theta - \frac{1}{2} R_h^{2} \Lambda ^{-2\alpha }v_d) \,\textrm{d}x. \end{aligned}$$

By $(v \cdot \nabla )\theta = (v_h \cdot \nabla _h) \theta + v_d \partial _d \theta$, we deduce

$$\begin{aligned}{} & {} \left| -\int _{\Omega } (v \cdot \nabla )\theta (R_h^{2} \Lambda ^{-2\alpha }\theta - \frac{1}{2} R_h^{2} \Lambda ^{-2\alpha }v_d) \,\textrm{d}x \right| \\{} & {} \le \Vert v_h \Vert _{L^2} \Vert \nabla _h \theta \Vert _{L^{\infty }} (\Vert v_d \Vert _{L^2} + \Vert R_h^{2} \Lambda ^{-2\alpha } \theta \Vert _{L^2}) + C(\Vert v_d \Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2) \Vert \partial _d \theta \Vert _{L^{\infty }}. \end{aligned}$$

Thus, by $W^{1,\infty }(\Omega ) \hookrightarrow H^{m-\alpha }(\Omega )$, (6.1), and Young’s inequality, we have

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert \Lambda ^{-\alpha }v_d\Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{2} \Lambda ^{-4\alpha }\theta \,\textrm{d}x \right) \\{} & {} \quad \le -(\frac{1}{4} - C \Vert \theta \Vert _{H^m}) ( \Vert R_h^{2}\Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 + \Vert v_d \Vert _{L^2}^2) + C\Vert v\Vert _{L^2} (\Vert \nabla v \Vert _{L^\infty } \\{} & {} \qquad + \Vert \nabla _h \theta \Vert _{L^{\infty }}) (\Vert v_d \Vert _{L^2} + \Vert R_h^{2} \Lambda ^{-2\alpha }\theta \Vert _{L^2}) \\{} & {} \quad \le -\frac{1}{8} ( \Vert R_h^{2} \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 + \Vert v_d \Vert _{L^2}^2)+ C \Vert v \Vert _{L^2}^2 (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2). \end{aligned}$$

Let $M\ge 1$ which will be specified later. Since

$$\begin{aligned} \begin{aligned} \frac{1}{M} \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 - \Vert R_h^{2} \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2&= \sum _{\eta \in J} \left( \frac{1}{M} - \frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \right) |\mathscr {F} R_h \Lambda ^{-\alpha } \theta (\eta )|^2 \\&\le \frac{1}{M} \sum _{\frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \le \frac{1}{M}, |\tilde{n}| \ne 0} |\mathscr {F} R_h \Lambda ^{-\alpha } \theta (\eta )|^2 \\&\le \frac{1}{M^{1+\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2, \end{aligned} \end{aligned}$$

together with

$$\begin{aligned} \left| -\int _{\Omega } v_d R_h^2 \Lambda ^{-4\alpha } \theta \,\textrm{d}x \right| \le \Vert \Lambda ^{-2\alpha } v_d \Vert _{L^2} \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2} \le \frac{1}{2} \Vert v_d \Vert _{L^2}^2 + \frac{1}{2} \Vert R_h \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2,\nonumber \\ \end{aligned}$$

(6.7)

we can have as estimating (6.4) that

$$\begin{aligned}{} & {} -\frac{1}{8} \left( \Vert R_h^{2} \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 + \Vert v_d \Vert _{L^2}^2 \right) \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 + \Vert v_d \Vert _{L^2}^2 -\int _{\Omega } v_d R_h^2 \Lambda ^{-4\alpha } \theta \,\textrm{d}x \right) + \frac{1}{8M^{1+\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2. \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert \Lambda ^{-\alpha }v_d\Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{2} \Lambda ^{-4\alpha }\theta \,\textrm{d}x \right) \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 + \Vert \Lambda ^{-\alpha } v_d \Vert _{L^2}^2 -\int _{\Omega } v_d R_h^2 \Lambda ^{-4\alpha } \theta \,\textrm{d}x \right) \\{} & {} \qquad + \frac{1}{8M^{1+\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 + C \Vert v \Vert _{L^2}^2 (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2). \end{aligned}$$

We take $M = 1+\frac{t}{8(1+\frac{m}{1+\alpha })}$. Then, we can have with (6.6) that

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \left( (1+\frac{t}{8(1+\frac{m}{1+\alpha })})^{1+\frac{m}{1+\alpha }} \left( \Vert \Lambda ^{-\alpha } v_d\Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^2 \Lambda ^{-4\alpha } \theta \,\textrm{d}x \right) \right) \\{} & {} \quad \le C \sup _{\tau \in [0,t]} (1+\frac{\tau }{8(1+\frac{m}{1+\alpha })})^{1+\frac{m}{1+\alpha }} \left( \Vert \Lambda ^{-\alpha } v_d(\tau ) \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta (\tau ) \Vert _{L^2}^2\right) (\Vert v\Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2) \\{} & {} \qquad + C (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2). \end{aligned}$$

We integrate it over time and use (6.7) with

$$\begin{aligned} \int _0^{\infty } (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2) \,\textrm{d}t \le C. \end{aligned}$$

Then, for

$$\begin{aligned} f(t) := \sup _{\tau \in [0,t]} (1+\frac{\tau }{8(1+\frac{m}{1+\alpha })})^{1+\frac{m}{1+\alpha }} \left( \Vert \Lambda ^{-\alpha } v_d(\tau ) \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta (\tau ) \Vert _{L^2}^2\right) , \end{aligned}$$

it holds

$$\begin{aligned} f(t) \le C + \int _0^t f(\tau ) (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}) \,\textrm{d}\tau . \end{aligned}$$

Applying Grönwall’s ineqaulity, we obtain

$$\begin{aligned} \sup _{\tau \in [0,t]} (1+\frac{\tau }{8(1+\frac{m}{1+\alpha })})^{1+\frac{m}{1+\alpha }} \left( \Vert \Lambda ^{-\alpha } v_d(\tau ) \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } \theta (\tau ) \Vert _{L^2}^2\right) \le C. \end{aligned}$$

With (6.6), we deduce (6.5). This completes the proof. $\square$

Proposition 6.3

Let $d \in \mathbb {N}$ with $d \ge 2$ and $\alpha \in \{0,1\}$. Let $m \in \mathbb {N}$ with $m > 2+\frac{d}{2} +\alpha$ and $(v, \theta )$ be a smooth global solution to (1.2) with (1.3) or (1.6). Suppose that (6.1) be satisfied. Then, there exists a constant $C > 0$ such that

$$\begin{aligned} \Vert v_d(t) \Vert _{L^2}^2 + \Vert R_h^{2} \Lambda ^{-2\alpha } \theta (t) \Vert _{L^2}^2 \le C (1+t)^{-(2 + \frac{m}{1+\alpha })}. \end{aligned}$$

(6.8)

Proof

Recalling the definition of $\textbf{b}_{\pm }$, we can verify that

$$\begin{aligned} \mathscr {F}_b v_d = \frac{1}{\lambda _{+}} \frac{|\tilde{n}|^2}{|\eta |^2} \mathscr {F}_b \theta + \frac{|\tilde{n}|^2}{|\eta |^2} \left( \frac{1}{\lambda _{-}} -\frac{1}{\lambda _{+}} \right) \langle \mathscr {F}_b \textbf{u}, \textbf{a}_{+} \rangle \langle \textbf{b}_{+}, e_2 \rangle , \qquad \eta \in J. \end{aligned}$$

We note that

$$\begin{aligned} \frac{1}{|\lambda _{+}|} \le \frac{|\eta |}{|\tilde{n}|} \le \frac{2}{|\eta |^{2\alpha }}, \qquad \eta \in D_1 \qquad \text{ and } \qquad \frac{1}{|\lambda _{+}|} \le \frac{2}{|\eta |^{2\alpha }}, \qquad \eta \not \in D_1. \end{aligned}$$

Together with

$$\begin{aligned} \frac{|\tilde{n}|^2}{|\eta |^2} \left( \frac{1}{\lambda _{-}} -\frac{1}{\lambda _{+}} \right) \langle \mathscr {F}_b \textbf{u}, \textbf{a}_{+} \rangle \langle \textbf{b}_{+}, e_2 \rangle = \frac{1}{\lambda _+} \langle \mathscr {F}_b \textbf{u}, \textbf{a}_{+} \rangle , \end{aligned}$$

we have

$$\begin{aligned} \Vert v_d (t) \Vert _{L^2} \le C \Vert R_h^2 \Lambda ^{-2\alpha } \theta (t) \Vert _{L^2} + C \left( \sum _{\eta \in J} \frac{1}{|\eta |^{4\alpha }} | \langle \mathscr {F}_b \textbf{u},\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

(6.9)

We show

$$\begin{aligned}{} & {} \left( \sum _{\eta \in J} \frac{1}{|\eta |^{4\alpha }} | \langle \mathscr {F}_b \textbf{u},\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}}\nonumber \\{} & {} \quad \le (1+t)^{-(1+\frac{m}{2(1+\alpha )})} (C + C \delta \sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{2(1+\alpha )}} \Vert v_d(\tau ) \Vert _{L^2}), \end{aligned}$$

(6.10)

where $\delta$ is a small constant in (6.1). Then by taking $\delta$ small enough, we obtain

$$\begin{aligned} \sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{2(1+\alpha )}} \Vert v_d(\tau ) \Vert _{L^2} \le C + C \sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{2(1+\alpha )}} \Vert R_h^2 \Lambda ^{-2\alpha } \theta (\tau ) \Vert _{L^2}.\nonumber \\ \end{aligned}$$

(6.11)

We recall (3.5) and have

$$\begin{aligned} \langle \mathscr {F}_b\textbf{u}(t),\textbf{a}_+ \rangle = e^{-\lambda _+ t} \langle \mathscr {F}_b \textbf{u}_0,\textbf{a}_+ \rangle - \int _0^t e^{-\lambda _+ (t - \tau )} \langle N(v,\theta )(\tau ),\textbf{a}_+ \rangle \,\textrm{d}\tau . \end{aligned}$$

Since $|e^{-\lambda _+ t}| \le e^{-|\eta |^{2\alpha } \frac{t}{2}}$ for $\eta \in J$, it follows by the Minkowski inequality

$$\begin{aligned}{} & {} \left( \sum _{\eta \in J} \frac{1}{|\eta |^{4\alpha }} | \langle \mathscr {F}_b \textbf{u},\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \le \left( \sum _{\eta \in J} \frac{1}{|\eta |^{4\alpha }} e^{-|\eta |^{2\alpha }t} | \langle \mathscr {F}_b \textbf{u}_0,\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}} + \int _0^t \left( \sum _{\eta \in J} \frac{1}{|\eta |^{4\alpha }} e^{-|\eta |^{2\alpha }(t - \tau )} |\langle N(v,\theta )(\tau ),\textbf{a}_+ \rangle |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau . \end{aligned}$$

From the simple fact $|\textbf{a}_{+}|^2 = |\lambda _{+}|^2 + \frac{|\tilde{n}|^4}{|\eta |^4} \le C|\eta |^{4\alpha }$ with (6.1), we have

$$\begin{aligned} \left( \sum _{\eta \in J} \frac{1}{|\eta |^{4\alpha }} e^{-|\eta |^{2\alpha }t} | \langle \mathscr {F}_b \textbf{u}_0,\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}} \le Ce^{-t} \Vert \textbf{u}_0 \Vert _{L^2} \le C (1+t)^{-(1+\frac{m}{2(1+\alpha )})}. \end{aligned}$$

We note that

$$\begin{aligned} |\langle N(v,\theta )(\tau ),\textbf{a}_+ \rangle | \le |\mathscr {F} (v\cdot \nabla )v||\lambda _{+}| + |\mathscr {F}_b (v \cdot \nabla )\theta | \frac{|\tilde{n}|^2}{|\eta |^2}. \end{aligned}$$

(6.12)

Thus, it holds

$$\begin{aligned}{} & {} \int _0^t \left( \sum _{\eta \in J} \frac{1}{|\eta |^{4\alpha }} e^{-|\eta |^{2}(t - \tau )} |\langle N(v,\theta )(\tau ),\textbf{a}_+ \rangle |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau \\{} & {} \quad \le \int _0^t \left( \sum _{\eta \in J} e^{-|\eta |^{2}(t - \tau )} | \mathscr {F}(v \cdot \nabla )v |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau + \int _0^t \left( \sum _{\eta \in J} e^{-|\eta |^{2}(t - \tau )} \frac{1}{|\eta |^{4\alpha }} | \mathscr {F}_b(v \cdot \nabla )\theta |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau \\{} & {} \quad \le \int _0^t e^{-(t - \tau )} (\Vert (v \cdot \nabla )v (\tau ) \Vert _{L^2} + \Vert (v_h \cdot \nabla _h)\bar{\theta } (\tau ) \Vert _{L^2} + \Vert v_d \partial _d \theta (\tau ) \Vert _{L^2}) \,\textrm{d}\tau . \end{aligned}$$

We have used

$$\begin{aligned} (v \cdot \nabla )\theta = (v_h \cdot \nabla _h)\bar{\theta } + v_d \partial _d \theta \end{aligned}$$

in the last inequality. We note by $H^{m-2-\alpha } \hookrightarrow L^{\infty }$

$$\begin{aligned} \Vert (v \cdot \nabla )v \Vert _{L^2}\le & {} \Vert v \Vert _{L^2} \Vert \nabla v \Vert _{L^{\infty }} \le C \Vert v \Vert _{L^2}^{1+\frac{1+\alpha }{m}} \Vert v \Vert _{{\dot{H}}^m}^{1-\frac{1+\alpha }{m}},\\ \Vert (v_h \cdot \nabla _h) \bar{\theta } \Vert _{L^2}\le & {} \Vert v \Vert _{L^2} \Vert \nabla _h \bar{\theta } \Vert _{L^{\infty }} \le C \Vert v \Vert _{L^2} \Vert \bar{\theta } \Vert _{L^2}^{\frac{1+\alpha }{m}} \Vert \bar{\theta } \Vert _{{\dot{H}}^{m}}^{1-\frac{1+\alpha }{m}}, \end{aligned}$$

and

$$\begin{aligned} \Vert v_d \partial _d \theta \Vert _{L^2} \le \Vert v_d \Vert _{L^2} \Vert \partial _d \theta \Vert _{L^{\infty }} \le C \Vert v_d \Vert _{L^2} \Vert \theta \Vert _{H^m}. \end{aligned}$$

Combining (6.5), (6.2) and our assumptions, we can see

$$\begin{aligned}{} & {} (1+\tau )^{1+\frac{m}{2(1+\alpha )}} (\Vert (v \cdot \nabla )v (\tau ) \Vert _{L^2} + \Vert (v_h \cdot \nabla _h)\bar{\theta } (\tau ) \Vert _{L^2} + \Vert v_d \partial _d \theta \Vert _{L^2}) \\{} & {} \quad \le C + C\delta \sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{2(1+\alpha )}} \Vert v_d(\tau ) \Vert _{L^2}, \end{aligned}$$

where $\delta$ is a small constant in (6.1). Hence,

$$\begin{aligned}{} & {} \int _0^t e^{-(t - \tau )} (\Vert (v \cdot \nabla )v (\tau ) \Vert _{L^2} + \Vert (v_h \cdot \nabla _h)\bar{\theta } (\tau ) \Vert _{L^2} + \Vert v_d \partial _d \theta \Vert _{L^2}) \,\textrm{d}\tau \\{} & {} \quad \le C (1+t)^{-(1+\frac{m}{2(1+\alpha )})} (C + C\delta \sup _{\tau \in [0,t]} (1+\tau )^{1+\frac{m}{2(1+\alpha )}} \Vert v_d(\tau ) \Vert _{L^2}). \end{aligned}$$

Collecting the above estimates, we obtain (6.10) and (6.11).

Now, we show

$$\begin{aligned} \Vert R_h^2 \Lambda ^{-2\alpha } \theta (t) \Vert _{L^2} \le C(1+t)^{-(1+\frac{m}{2(1+\alpha )})}. \end{aligned}$$

(6.13)

Since we have from (3.2) and (3.3),

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \sum _{\eta \in J} |\mathscr {F} R_h \Lambda ^{-2\alpha } v_d(\eta )|^2 \\{} & {} \quad \le -\Vert R_h \Lambda ^{-\alpha }v_d\Vert _{L^2}^2 + \Vert (v \cdot \nabla )v\Vert _{L^2} \Vert R_h^2\Lambda ^{-4\alpha } v_d \Vert _{L^2} + \sum _{\eta \in J} \frac{|\tilde{n}|^{4}}{|\eta |^{4(1+\alpha )}} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \sum _{\eta \in J} |\mathscr {F} R_h^2 \Lambda ^{-2\alpha } \theta (\eta )|^2 \le -\int _{\Omega } (v \cdot \nabla )\theta R_h^{4} \Lambda ^{-4\alpha }\theta \,\textrm{d}x - \sum _{\eta \in J} \frac{|\tilde{n}|^{4}}{|\eta |^{4(1+\alpha )}} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ) \end{aligned}$$

respectively, it holds

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} (\Vert R_h \Lambda ^{-2\alpha } v_d\Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2) \\{} & {} \le -\Vert R_h \Lambda ^{-\alpha } v_d\Vert _{L^2}^2 + \Vert (v \cdot \nabla )v\Vert _{L^2} \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2} -\int _{\Omega } (v \cdot \nabla )\theta R_h^{4} \Lambda ^{-4\alpha } \theta \,\textrm{d}x. \end{aligned}$$

We can infer from (3.2) and (3.3) that

$$\begin{aligned} -\frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_d R_h^{4} \Lambda ^{-6\alpha }\theta \,\textrm{d}x= & {} -\int _{\Omega } \partial _tv_d R_h^{4} \Lambda ^{-6\alpha }\theta \,\textrm{d}x - \int _{\Omega } \partial _t\theta R_h^{4} \Lambda ^{-6\alpha }v_d \,\textrm{d}x \\\le & {} \Vert (v \cdot \nabla )v\Vert _{L^2} \Vert R_h^{4} \Lambda ^{-6\alpha } \theta \Vert _{L^2} + \int _{\Omega } (v \cdot \nabla )\theta R_h^{4} \Lambda ^{-6\alpha } v_d \,\textrm{d}x \\{} & {} - \frac{1}{2} \Vert R_h^{3} \Lambda ^{-3\alpha } \theta \Vert _{L^2}^2 + \frac{3}{2} \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}^2. \end{aligned}$$

Combining the above, we have

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h \Lambda ^{-2\alpha }v_d\Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{4} \Lambda ^{-6\alpha }\theta \,\textrm{d}x \right) \\{} & {} \quad \le -\frac{1}{4} (\Vert R_h \Lambda ^{-\alpha } v_d\Vert _{L^2}^2 + \Vert R_h^{3}\Lambda ^{-3\alpha } \theta \Vert _{L^2}^2) \\{} & {} \qquad + \Vert v \Vert _{L^2} \Vert \nabla v \Vert _{L^{\infty }} (\Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2} + \Vert R_h^{3} \Lambda ^{-3\alpha }\theta \Vert _{L^2})\\{} & {} \qquad -\int _{\Omega } (v \cdot \nabla )\theta (R_h^{4} \Lambda ^{-4\alpha }\theta - \frac{1}{2} R_h^{4} \Lambda ^{-6\alpha }v_d) \,\textrm{d}x. \end{aligned}$$

We estimate the last integral with $(v \cdot \nabla )\theta = (v_h \cdot \nabla _h) \theta + v_d \partial _d \theta$. Hölder’s inequality implies

$$\begin{aligned}{} & {} \left| -\int _{\Omega } (v_h \cdot \nabla _h)\theta (R_h^{4} \Lambda ^{-4\alpha }\theta - \frac{1}{2} R_h^{4} \Lambda ^{-6\alpha }v_d) \,\textrm{d}x\right| \\{} & {} \le C\Vert v \Vert _{L^2} \Vert \nabla _h \theta \Vert _{L^p}(\Vert R_h^4 \Lambda ^{-6\alpha } v_d \Vert _{L^q} + \Vert R_h^{3} \Lambda ^{-4\alpha }\theta \Vert _{L^q}), \end{aligned}$$

where $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$. We take $\frac{1}{q} = \frac{1}{2} - \frac{\alpha }{d}$. Then for $\epsilon \in (0,1)$ with $m>2+ \frac{d}{2} + \alpha +2\epsilon$, we can see

$$\begin{aligned} \Vert \nabla _h \theta \Vert _{L^p} \le C\Vert R_h \theta \Vert _{{\dot{H}}^{1 + \frac{d}{2} + \epsilon }} \le C\Vert R_h \theta \Vert _{{\dot{H}}^{m-1-\epsilon }}, \qquad \alpha = 0, \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla _h \theta \Vert _{L^p} \le C \Vert R_h \theta \Vert _{{\dot{H}}^{\frac{d}{2} + \epsilon }} \le C\Vert R_h \theta \Vert _{{\dot{H}}^{m-3-\epsilon }}, \qquad \alpha =1, \end{aligned}$$

together with $\Vert R_h^4 \Lambda ^{-6\alpha } v_d \Vert _{L^q} + \Vert R_h^{3} \Lambda ^{-4\alpha }\theta \Vert _{L^q} \le C\Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2} + C\Vert R_h^{3} \Lambda ^{-3\alpha }\theta \Vert _{L^2}$, we have

$$\begin{aligned}{} & {} \left| -\int _{\Omega } (v_h \cdot \nabla _h)\theta (R_h^{4} \Lambda ^{-4\alpha }\theta - \frac{1}{2} R_h^{4} \Lambda ^{-6\alpha }v_d) \,\textrm{d}x\right| \\{} & {} \le C\Vert v \Vert _{L^2} \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}(\Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2} + \Vert R_h^{3} \Lambda ^{-3\alpha }\theta \Vert _{L^2}). \end{aligned}$$

On the other hand, it holds

$$\begin{aligned}{} & {} -\int _{\Omega } v_d \partial _d\theta (R_h^{4} \Lambda ^{-4\alpha }\theta - \frac{1}{2} R_h^{4} \Lambda ^{-6\alpha }v_d) \,\textrm{d}x = -\int _{\Omega } \nabla _h (v_d \partial _d \theta ) \cdot \nabla _h(-\Delta )^{-1} (R_h^2 \Lambda ^{-4\alpha } \theta \\{} & {} \qquad - \frac{1}{2} R_h^2 \Lambda ^{-6\alpha } v_d) \,\textrm{d}x \\{} & {} \quad = -\int _{\Omega } (\nabla _hv_d \partial _d \theta ) \cdot \nabla _h(-\Delta )^{-1} (R_h^2 \Lambda ^{-4\alpha } \theta - \frac{1}{2} R_h^2 \Lambda ^{-6\alpha } v_d) \,\textrm{d}x \\{} & {} \qquad - \int _{\Omega } (v_d \partial _d \nabla _h\theta ) \cdot \nabla _h (-\Delta )^{-1} (R_h^2 \Lambda ^{-4\alpha } \theta - \frac{1}{2} R_h^2 \Lambda ^{-6\alpha } v_d) \,\textrm{d}x. \end{aligned}$$

The second integral on the right-hand side is bounded by

$$\begin{aligned}{} & {} \left| - \int _{\Omega } (v_d \partial _d \nabla _h\theta ) \cdot \nabla _h (-\Delta )^{-1} (R_h^2 \Lambda ^{-4\alpha } \theta - \frac{1}{2} R_h^2 \Lambda ^{-6\alpha } v_d) \,\textrm{d}x \right| \\{} & {} \quad \le C \Vert v_d \Vert _{L^2} \Vert \partial _d \nabla _h \theta \Vert _{L^{\infty }} ( \Vert R_h^3\Lambda ^{-3\alpha } \theta \Vert _{L^2} + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}). \end{aligned}$$

We note

$$\begin{aligned}{} & {} \left| -\int _{\Omega } (\nabla _hv_d \partial _d \theta ) \cdot \nabla _h(-\Delta )^{-1} (R_h^2 \Lambda ^{-4\alpha } \theta - \frac{1}{2} R_h^2 \Lambda ^{-6\alpha } v_d) \,\textrm{d}x \right| \\{} & {} \quad = \left| \int _{\Omega } (\nabla _h \Delta (-\Delta )^{-1} v_d \partial _d \theta ) \cdot \nabla _h(-\Delta )^{-1} (R_h^2 \Lambda ^{-4\alpha } \theta - \frac{1}{2} R_h^2 \Lambda ^{-6\alpha } v_d) \,\textrm{d}x \right| . \end{aligned}$$

When $\alpha = 0$, with the integration by parts, it holds

$$\begin{aligned}{} & {} \left| \int _{\Omega } (\nabla _h \Delta (-\Delta )^{-1} v_d \partial _d \theta ) \cdot \nabla _h(-\Delta )^{-1} (R_h^2 \theta - \frac{1}{2} R_h^2 v_d) \,\textrm{d}x \right| \\{} & {} \quad \le \Vert R_h v_d \Vert _{L^2} (\Vert \partial _d \nabla \theta \Vert _{L^{\infty }} + \Vert \partial _d \theta \Vert _{L^{\infty }}) ( \Vert R_h^3 \theta \Vert _{L^2} + \Vert R_h v_d \Vert _{L^2}) \\{} & {} \quad \le C \Vert \theta \Vert _{H^m} ( \Vert R_h^3 \Lambda ^{-3\alpha } \theta \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}^2). \end{aligned}$$

For $\alpha = 1$, we have

$$\begin{aligned}{} & {} \left| \int _{\Omega } (\nabla _h \Delta (-\Delta )^{-1} v_d \partial _d \theta ) \cdot \nabla _h(-\Delta )^{-1} (R_h^2 \Lambda ^{-4} \theta - \frac{1}{2} R_h^2 \Lambda ^{-6} v_d) \,\textrm{d}x \right| \\{} & {} \quad \le C \Vert R_h \Lambda ^{-1} v_d \Vert _{L^2} (\Vert \Delta \partial _d \theta \Vert _{L^{\infty }} + \Vert \nabla \partial _d \theta \Vert _{L^{\infty }}+ \Vert \partial _d \theta \Vert _{L^{\infty }}) ( \Vert R_h^3 \Lambda ^{-3} \theta \Vert _{L^2} + \Vert R_h \Lambda ^{-1} v_d \Vert _{L^2}) \\{} & {} \quad \le C \Vert \theta \Vert _{H^m} ( \Vert R_h^3 \Lambda ^{-3\alpha } \theta \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}^2). \end{aligned}$$

Therefore,

$$\begin{aligned}{} & {} \left| -\int _{\Omega } (v \cdot \nabla )\theta (R_h^{4} \Lambda ^{-4\alpha }\theta - \frac{1}{2} R_h^{4} \Lambda ^{-6\alpha }v_d) \,\textrm{d}x \right| \\{} & {} \quad \le C\Vert v \Vert _{L^2} \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}(\Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2} + \Vert R_h^{3} \Lambda ^{-3\alpha }\theta \Vert _{L^2}) \\{} & {} \qquad + C \Vert v_d \Vert _{L^2} \Vert \partial _d \nabla _h \theta \Vert _{L^{\infty }} ( \Vert R_h^3\Lambda ^{-3\alpha } \theta \Vert _{L^2} + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}) \\{} & {} \qquad + C \Vert \theta \Vert _{H^m} ( \Vert R_h^3 \Lambda ^{-3\alpha } \theta \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}^2). \end{aligned}$$

With Young’s inequality and (6.1) we can have

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h \Lambda ^{-2\alpha }v_d\Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{4} \Lambda ^{-6\alpha }\theta \,\textrm{d}x \right) \\{} & {} \quad \le -(\frac{1}{4} - C\Vert \theta \Vert _{H^m}) (\Vert R_h \Lambda ^{-\alpha } v_d\Vert _{L^2}^2 + \Vert R_h^{3}\Lambda ^{-3\alpha } \theta \Vert _{L^2}^2) \\{} & {} \qquad + C\Vert v \Vert _{L^2} (\Vert v \Vert _{{\dot{H}}^{m-1-\alpha -\epsilon }} + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}) (\Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2} + \Vert R_h^{3} \Lambda ^{-3\alpha }\theta \Vert _{L^2}) \\{} & {} \qquad + C \Vert v_d \Vert _{L^2} \Vert \partial _d \nabla _h \theta \Vert _{L^{\infty }} ( \Vert R_h^3\Lambda ^{-3\alpha } \theta \Vert _{L^2} + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}) \\{} & {} \quad \le -\frac{1}{8} (\Vert R_h \Lambda ^{-\alpha } v_d\Vert _{L^2}^2 + \Vert R_h^{3}\Lambda ^{-3\alpha } \theta \Vert _{L^2}^2) + C\Vert v \Vert _{L^2}^2 (\Vert v \Vert _{{\dot{H}}^{m-1-\alpha -\epsilon }}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}^2) \\{} & {} \qquad + C \Vert v_d \Vert _{L^2}^2 \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2. \end{aligned}$$

Let $M\ge 1$ which will be specified later. Since

$$\begin{aligned} \begin{aligned} \frac{1}{M} \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 - \Vert R_h^{3} \Lambda ^{-3\alpha } \theta \Vert _{L^2}^2&= \sum _{|\tilde{n}| \ne 0} \left( \frac{1}{M} - \frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \right) |\mathscr {F} R_h^2 \Lambda ^{-2\alpha } \theta (\eta )|^2 \\&\le \frac{1}{M} \sum _{\frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \le \frac{1}{M}, |\tilde{n}| \ne 0} |\mathscr {F} R_h^2 \Lambda ^{-2\alpha } \theta (\eta )|^2 \\&\le \frac{1}{M^{2+\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \left| -\int _{\Omega } v_d R_h^4 \Lambda ^{-6\alpha } \theta \,\textrm{d}x \right|\le & {} \Vert R_h\Lambda ^{-3\alpha } v_d \Vert _{L^2} \Vert R_h^3 \Lambda ^{-3\alpha } \theta \Vert _{L^2} \nonumber \\\le & {} \frac{1}{2} \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}^2 + \frac{1}{2} \Vert R_h^3 \Lambda ^{-3\alpha } \theta \Vert _{L^2}^2, \end{aligned}$$

(6.14)

we have

$$\begin{aligned}{} & {} -\frac{1}{8} \left( \Vert R_h^{3} \Lambda ^{-3\alpha } \theta \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}^2 \right) \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-\alpha } v_d \Vert _{L^2}^2 -\int _{\Omega } v_d R_h^4 \Lambda ^{-6\alpha } \theta \,\textrm{d}x \right) \\{} & {} \qquad + \frac{1}{8M^{2+\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2. \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h \Lambda ^{-2\alpha }v_d\Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{4} \Lambda ^{-6\alpha }\theta \,\textrm{d}x \right) \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 + \Vert R_h \Lambda ^{-2\alpha } v_d \Vert _{L^2}^2 -\int _{\Omega } v_d R_h^4 \Lambda ^{-6\alpha } \theta \,\textrm{d}x \right) \\{} & {} \qquad + \frac{1}{8M^{2+\frac{m}{1+\alpha }}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2.\\{} & {} \qquad + C\Vert v \Vert _{L^2}^2 (\Vert v \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}^2) + C \Vert v_d \Vert _{L^2}^2 \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2. \end{aligned}$$

We take $M = 1+\frac{t}{8(2+\frac{m}{1+\alpha })}$ and multiply $2M^{2+\frac{m}{1+\alpha }}$ both sides. Then,

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \left( (1+\frac{t}{8(2+\frac{m}{1+\alpha })})^{2+\frac{m}{1+\alpha }} (\Vert R_h \Lambda ^{-2\alpha }v_d\Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{4} \Lambda ^{-6\alpha }\theta \,\textrm{d}x) \right) \\{} & {} \quad \le C \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 \\{} & {} \qquad + C(1+\frac{t}{8(2+\frac{m}{1+\alpha })})^{1+\frac{m}{1+\alpha }} \Vert v \Vert _{L^2}^2 (1+\frac{t}{8(2+\frac{m}{1+\alpha })}) (\Vert v \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}^2) \\{} & {} \qquad + C (1+\frac{t}{8(2+\frac{m}{1+\alpha })})^{2+\frac{m}{1+\alpha }} \Vert v_d \Vert _{L^2}^2 \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2. \end{aligned}$$

Since the interpolation inequality implies

$$\begin{aligned} \Vert v \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }} + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}\le & {} \Vert v \Vert _{L^2}^{\frac{1+\alpha +\epsilon }{m}} \Vert v \Vert _{{\dot{H}}^m}^{1-\frac{1+\alpha +\epsilon }{m}}\\{} & {} + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{L^2}^{\frac{1+\alpha +\epsilon }{m}} \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^m}^{1-\frac{1+\alpha +\epsilon }{m}}, \end{aligned}$$

we have from (6.5)

$$\begin{aligned}{} & {} C(1+\frac{t}{8(2+\frac{m}{1+\alpha })})^{1+\frac{m}{1+\alpha }} \Vert v \Vert _{L^2}^2 (1+\frac{t}{8(2+\frac{m}{1+\alpha })}) (\Vert v \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-1-\alpha - \epsilon }}^2) \\{} & {} \quad \le C (1+t)^{1-\frac{1+\alpha +\epsilon }{m} (1+\frac{m}{1+\alpha })} (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-\alpha }}^2)^{1-\frac{1+\alpha +\epsilon }{m}} \\{} & {} \quad \le C(1+t)^{-\frac{1+\alpha +\epsilon }{m} - \frac{\epsilon }{1+\alpha }} (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-\alpha }}^2)^{1-\frac{1+\alpha +\epsilon }{m}} . \end{aligned}$$

By (6.11), it holds

$$\begin{aligned} (1+\frac{t}{8(2+\frac{m}{1+\alpha })})^{2+\frac{m}{1+\alpha }} \Vert v_d \Vert _{L^2}^2 \le C + C \sup _{\tau \in [0,t]} (1+\frac{\tau }{8(2+\frac{m}{1+\alpha })})^{2+\frac{m}{1+\alpha }} \Vert R_h^2 \Lambda ^{-2\alpha } \theta (\tau ) \Vert _{L^2}^2. \end{aligned}$$

Then, we deduce that

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \left( (1+\frac{t}{8(2+\frac{m}{1+\alpha })})^{2+\frac{m}{1+\alpha }} (\Vert R_h \Lambda ^{-2\alpha }v_d\Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2 - \int _{\Omega } v_d R_h^{4} \Lambda ^{-6\alpha }\theta \,\textrm{d}x) \right) \\{} & {} \quad \le C \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 + C(1+t)^{-\frac{1+\alpha +\epsilon }{m} - \frac{\epsilon }{1+\alpha }} (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-\alpha }}^2)^{1-\frac{1+\alpha +\epsilon }{m}} \\{} & {} \qquad +C\left( \sup _{\tau \in [0,t]} (1+\frac{\tau }{8(2+\frac{m}{1+\alpha })})^{2+\frac{m}{1+\alpha }} (\Vert R_h \Lambda ^{-2\alpha }v_d\Vert _{L^2}^2 + \Vert R_h^2 \Lambda ^{-2\alpha } \theta \Vert _{L^2}^2) \right) \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2. \end{aligned}$$

Since we can verify

$$\begin{aligned} \int _0^\infty \left( C \Vert R_h \theta \Vert _{{\dot{H}}^{m-\alpha }}^2 + C(1+t)^{-\frac{1+\alpha +\epsilon }{m} - \frac{\epsilon }{1+\alpha }} (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \Lambda ^{-\alpha } \theta \Vert _{{\dot{H}}^{m-\alpha }}^2)^{1-\frac{1+\alpha +\epsilon }{m}} \right) \,\textrm{d}t \le C, \end{aligned}$$

using Grönwall’s inequality and (6.14), (6.13) is obtained. Then, (6.8) follows from (6.11). This completes the proof. $\square$

6.1 Proof of Theorem 1.4:Temporal decay part

In this section, we completes the proof of Theorem 1.4. For this purpose, we suppose (6.15) and (6.16) hold true, which will be proved in the following proposition. From (6.5) and (6.15), we obtain

$$\begin{aligned} (1+t)^{1+m-s} \Vert v(t) \Vert _{H^s}^2 \le C. \end{aligned}$$

On the other hand, (6.8) and (6.16) imply

$$\begin{aligned} (1+t)^{2+m-s} \Vert v_d(t) \Vert _{H^s}^2 \le C. \end{aligned}$$

Hence, it suffices to prove

$$\begin{aligned} (1+t)^{m} \Vert \theta (t) - \sigma \Vert _{L^2}^2 \le C \end{aligned}$$

because of (1.6). We recall (1.4) and use (6.2) to have

$$\begin{aligned} \Vert \theta - \sigma \Vert _{L^2}&\le \Vert \bar{\theta } \Vert _{L^2} + \left\| \int _{\mathbb T^{d-1}} \int _t^\infty \left( (v \cdot \nabla )\theta + v_d \right) \,\textrm{d}\tau \textrm{d}x_h \right\| _{L^2} \\&\le C(1+t)^{-\frac{m}{2}} +\int _t^\infty \Vert (v \cdot \nabla )\theta + v_d \Vert _{L^2} \,\textrm{d}\tau . \end{aligned}$$

Since

$$\begin{aligned} \Vert (v \cdot \nabla )\theta + v_d \Vert _{L^2}&\le \Vert (v_h \cdot \nabla _h)\theta \Vert _{L^2} + \Vert v_d \partial _d \theta \Vert _{L^2} + \Vert v_d \Vert _{L^2} \\&\le C \Vert v \Vert _{L^2} \Vert R_h \theta \Vert _{H^{m}} + C\Vert v_d \Vert _{L^2} \Vert \theta \Vert _{H^m} + \Vert v_d \Vert _{L^2} \\&\le C (1+\tau )^{-(\frac{1}{2} +\frac{m}{2})} \Vert R_h \theta \Vert _{H^{m}} + C (1+\tau )^{-(1+\frac{m}{2})} \end{aligned}$$

by (6.5) and (6.8), we can deduce

$$\begin{aligned} \int _t^\infty \Vert (v \cdot \nabla )\theta + v_d \Vert _{L^2} \,\textrm{d}\tau \le C \left\| (1+\tau )^{-(\frac{1}{2} +\frac{m}{2})} \right\| _{L^2(t,\infty )} + C (1+t)^{-\frac{m}{2}} \le C (1+t)^{-\frac{m}{2}}. \end{aligned}$$

This completes the proof.

Proposition 6.4

Let $d \in \mathbb {N}$ with $d \ge 2$ and $\alpha = 0$. Let $m \in \mathbb {N}$ with $m > 2+\frac{d}{2}$ and $(v, \theta )$ be a smooth global solution to (1.2) with (1.6). Suppose that (6.1) be satisfied. Then, there exists a constant $C > 0$ such that

$$\begin{aligned} \Vert v(t) \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \theta (t) \Vert _{{\dot{H}}^{m}}^2 \le C (1+t)^{-1} \end{aligned}$$

(6.15)

and

$$\begin{aligned} \Vert v_d(t) \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h v(t) \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h^2 \theta (t) \Vert _{{\dot{H}}^{m}}^2 \le C (1+t)^{-2}. \end{aligned}$$

(6.16)

Proof

From the v equations in (1.2), it follows

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert v \Vert _{{\dot{H}}^m}^2 + \Vert v \Vert _{{\dot{H}}^m}^2 \le C\Vert \nabla v \Vert _{L^\infty } \Vert v \Vert _{{\dot{H}}^m}^2 - \sum _{\eta \in J} |\eta |^{2m} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ). \end{aligned}$$

Using (5.3) gives

$$\begin{aligned} \left| - \sum _{\eta \in J} |\eta |^{2m} \mathscr {F} \theta (\eta )\mathscr {F} v_d(\eta ) \right|&\le C\left( \sum _{\eta \in J} |\eta |^{2m} |\mathscr {F}_b R_h \theta (\eta )|^2 \right) ^{\frac{1}{2}} \left( \sum _{\eta \in J} |\eta |^{2m} |\mathscr {F} v(\eta )|^2 \right) ^{\frac{1}{2}} \\&\le \frac{1}{4} \Vert v \Vert _{{\dot{H}}^m}^2 + C \Vert R_h \theta \Vert _{{\dot{H}}^{m}}^2. \end{aligned}$$

By (6.1) with (1.6), we have

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert v \Vert _{{\dot{H}}^m}^2&\le - (\frac{3}{4} -C \Vert (v_0,\theta _0) \Vert _{H^m}^2) \Vert v \Vert _{{\dot{H}}^{m}}^2 + C\Vert R_h \theta \Vert _{{\dot{H}}^{m}}^2 \le -\frac{1}{2} \Vert v \Vert _{{\dot{H}}^m}^2 + C\Vert R_h \theta \Vert _{{\dot{H}}^{m}}^2. \end{aligned}$$

Then, applying Duhamel’s principle shows

$$\begin{aligned} \Vert v(t) \Vert _{{\dot{H}}^m}^2 \le e^{-t} \Vert v_0 \Vert _{{\dot{H}}^m}^2 + C \int _0^t e^{-(t-\tau )} \Vert R_h \theta \Vert _{{\dot{H}}^{m}}^2 \,\textrm{d}\tau , \end{aligned}$$

thus,

$$\begin{aligned} \Vert v(t) \Vert _{{\dot{H}}^m}^2 \le C(1+t)^{-1} \left( \Vert v_0 \Vert _{{\dot{H}}^m}^2 + \sup _{\tau \in [0,t]} (1+\tau ) \Vert R_h \theta (\tau ) \Vert _{{\dot{H}}^{m}}^2 \right) . \end{aligned}$$

(6.17)

Since (5.3) implies $\Vert v_d (t) \Vert _{{\dot{H}}^m}^2 \le C\Vert R_h v(t) \Vert _{{\dot{H}}^m}^2$, we can similarly obtain

$$\begin{aligned} \Vert v_d (t) \Vert _{{\dot{H}}^m}^2 + \Vert R_h v(t) \Vert _{{\dot{H}}^m}^2 \le C(1+t)^{-2} \left( \Vert v_0 \Vert _{{\dot{H}}^m}^2 + \sup _{\tau \in [0,t]} (1+\tau )^2 \Vert R_h^2 \theta (\tau ) \Vert _{{\dot{H}}^{m}}^2 \right) .\nonumber \\ \end{aligned}$$

(6.18)

We omit the details.

Now, we show that

$$\begin{aligned} \Vert R_h^2 \theta (t) \Vert _{{\dot{H}}^{m}}^2 \le C(1+t)^{-2}. \end{aligned}$$

(6.19)

Since this implies

$$\begin{aligned} \Vert R_h \theta (t) \Vert _{{\dot{H}}^{m}}^2 \le \Vert R_h^2 \theta (t) \Vert _{{\dot{H}}^{m}} \Vert \theta (t) \Vert _{{\dot{H}}^{m}} \le C(1+t)^{-1}, \end{aligned}$$

(6.15) and (6.16) follow by (6.17) and (6.18) respectively. Since $H^m(\Omega )$ is a Banach algebra, we can show from (3.2) that

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 + \Vert R_h v_d \Vert _{{\dot{H}}^m}^2\le & {} C\Vert R_h v \Vert _{{\dot{H}}^m} \Vert v \Vert _{{\dot{H}}^m} \Vert R_h v_d \Vert _{{\dot{H}}^m}\\{} & {} + \sum _{|\tilde{n}| \ne 0} |\tilde{n}|^4|\eta |^{2m-4} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ). \end{aligned}$$

From (3.3),

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 \le -\sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma \partial _h^2 (v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x - \sum _{|\tilde{n}| \ne 0} |\tilde{n}|^4|\eta |^{2m-4} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ). \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 \right) + \Vert R_h v_d \Vert _{{\dot{H}}^{m}}^2 \\{} & {} \quad \le C\Vert R_h v \Vert _{{\dot{H}}^m} \Vert v \Vert _{{\dot{H}}^m} \Vert R_h v_d \Vert _{{\dot{H}}^m} -\sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma \partial _h^2 (v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x . \end{aligned}$$

We note that

$$\begin{aligned} \sum _{|\gamma | = m-2} \int _{\Omega } \partial ^\gamma \partial _h^2 (v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x = \sum _{|\gamma | = m-2} \left( K_1 + K_2 + K_3 \right) , \end{aligned}$$

where

$$\begin{aligned} K_1&= \int _{\Omega } \partial ^\gamma (\partial _h^2 v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x, \\ K_2&= \int _{\Omega } \partial ^\gamma (\partial _h v \cdot \nabla ) \partial _h \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x, \\ K_3&:= \int _{\Omega } \partial ^\gamma (v \cdot \nabla ) \partial _h^2 \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x . \end{aligned}$$

The integration by parts and $(v \cdot \nabla ) \theta = (v_h \cdot \nabla _h) \theta + v_d \partial _d \theta$ show

$$\begin{aligned} K_1 + K_2 = - \int _{\Omega } \partial ^\gamma (\partial _h v_h \cdot \nabla _h) \theta \cdot \partial ^\gamma \partial _h^3 \theta \,\textrm{d}x - \int _{\Omega } \partial ^\gamma (\partial _h v_d \partial _d \theta ) \cdot \partial ^\gamma \partial _h^3 \theta \,\textrm{d}x. \end{aligned}$$

Again using the integration by parts with the continuous embedding $L^{\infty }(\Omega ) \hookrightarrow H^{m-1}(\Omega )$, we obtain

$$\begin{aligned} |K_1 + K_2| \le C\Vert R_h v_h \Vert _{{\dot{H}}^m} \Vert R_h \theta \Vert _{{\dot{H}}^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} + C\Vert R_h v_d \Vert _{{\dot{H}}^m} \Vert \theta \Vert _{H^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}. \end{aligned}$$

On the other hand, due to the cancellation property, we can have

$$\begin{aligned} |K_3|\le & {} C\Vert v_h \Vert _{{\dot{H}}^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m} + C \Vert \nabla v_d \Vert _{L^{\infty }} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 \\{} & {} + \int _{\Omega } \partial _d^{m-4} (\partial _d^2 v_d \partial _d \partial _h^2 \theta ) \cdot \partial _d^{m-2} \partial _h^2 \theta \,\textrm{d}x. \end{aligned}$$

The divergence-free condition and the integration by parts imply

$$\begin{aligned} \left| \int _{\Omega } \partial _d^{m-4} (\partial _d^2 v_d \partial _d \partial _h^2 \theta ) \cdot \partial _d^{m-2} \partial _h^2 \theta \,\textrm{d}x \right|&= \left| \int _{\Omega } \partial _d^{m-4} (\partial _d \nabla _h \cdot v_h \partial _d \partial _h^2 \theta ) \cdot \partial _d^{m-2} \partial _h^2 \theta \,\textrm{d}x \right| \\&\le C\Vert v \Vert _{{\dot{H}}^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}. \end{aligned}$$

Thus,

$$\begin{aligned} |K_3| \le C\Vert v \Vert _{{\dot{H}}^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m} + C \Vert \nabla v_d \Vert _{L^{\infty }} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2. \end{aligned}$$

From the above estimates, we deduce

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 \right) + \Vert R_h v_d \Vert _{{\dot{H}}^{m}}^2 \\{} & {} \quad \le C(\Vert R_h v \Vert _{{\dot{H}}^m} + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m})(\Vert v \Vert _{{\dot{H}}^m} + \Vert R_h \theta \Vert _{{\dot{H}}^m}) (\Vert R_h v_d \Vert _{{\dot{H}}^m} + \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}) \\{} & {} \qquad + C\Vert R_h v_d \Vert _{{\dot{H}}^m} \Vert \theta \Vert _{H^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} + C \Vert \nabla v_d \Vert _{L^{\infty }} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2. \end{aligned}$$

On the other hand, using (3.2) and

$$\begin{aligned} -\Delta \langle \mathbb {P} (v \cdot \nabla )v , e_d \rangle = -\Delta (v \cdot \nabla )v_d + \partial _d \nabla \cdot (v \cdot \nabla )v, \end{aligned}$$

(6.20)

we have

$$\begin{aligned} -\int _{\Omega } \partial _tv_d (-\Delta )^{m} R_h^4 \theta \,\textrm{d}x&= \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^{m} R_h^4 \theta \,\textrm{d}x - \int _{\Omega } \partial _d \nabla \cdot ((v \cdot \nabla )v) (-\Delta )^{m-1} R_h^4 \theta \,\textrm{d}x \\&\quad - \int _{\Omega } v_d (-\Delta )^{m} R_h^4 \theta \,\textrm{d}x - \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m}}^2 \\&\le \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x + C\Vert R_h v \Vert _{{\dot{H}}^m} \Vert v \Vert _{{\dot{H}}^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} \\&\quad + \frac{1}{2} \Vert R_hv_d \Vert _{{\dot{H}}^{m}}^2 - \frac{1}{2}\Vert R_h^3 \theta \Vert _{{\dot{H}}^{m}}^2. \end{aligned}$$

Since (3.3) yields

$$\begin{aligned} -\int _{\Omega } \partial _t\theta (-\Delta )^{m} R_h^4 v_d \,\textrm{d}x \le \int _{\Omega } (v \cdot \nabla )\theta (-\Delta )^{m} R_h^4 v_d \,\textrm{d}x + \Vert R_h^2 v_d \Vert _{{\dot{H}}^{m}}^2, \end{aligned}$$

we have

$$\begin{aligned}{} & {} -\frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_d (-\Delta )^{m} R_h^4 \theta \,\textrm{d}x \le \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^{m} R_h^4 \theta \,\textrm{d}x + \int _{\Omega } (v \cdot \nabla )\theta (-\Delta )^{m} R_h^4 v_d \,\textrm{d}x \\{} & {} \quad + C\Vert R_h v \Vert _{{\dot{H}}^m} \Vert v \Vert _{{\dot{H}}^s} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} - \frac{1}{2} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m}}^2+ \frac{3}{2} \Vert R_h v_d \Vert _{{\dot{H}}^{m}}^2. \end{aligned}$$

We note that

$$\begin{aligned} \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^{m} R_h^4 \theta \,\textrm{d}x + \int _{\Omega } (v \cdot \nabla )\theta (-\Delta )^{m} R_h^4 v_d \,\textrm{d}x = \sum _{|\gamma | = m-2} \left( K_4 + K_5 + K_6 \right) , \end{aligned}$$

where

$$\begin{aligned} K_4&:= \int _{\Omega } \partial ^{\gamma }(\partial _h^2 v \cdot \nabla )v_d \partial ^{\gamma } \partial _h^2 \theta \,\textrm{d}x + \int _{\Omega } \partial ^{\gamma }( \partial _h^2 v \cdot \nabla )\theta \partial ^{\gamma } \partial _h^2 v_d \,\textrm{d}x, \\ K_5&:= \int _{\Omega } \partial ^{\gamma }(\partial _h v \cdot \nabla ) \partial _h v_d \partial ^{\gamma } \partial _h^2 \theta \,\textrm{d}x + \int _{\Omega } \partial ^{\gamma }( \partial _h v \cdot \nabla )\partial _h \theta \partial ^{\gamma } \partial _h^2 v_d \,\textrm{d}x,\\ K_6&:= \int _{\Omega } \partial ^{\gamma }(v \cdot \nabla ) \partial _h^2 v_d \partial ^{\gamma } \partial _h^2 \theta \,\textrm{d}x + \int _{\Omega } \partial ^{\gamma }(v \cdot \nabla )\partial _h^2 \theta \partial ^{\gamma } \partial _h^2 v_d \,\textrm{d}x. \end{aligned}$$

We can see

$$\begin{aligned}{} & {} |K_4| \le C\Vert R_h^2 v_h \Vert _{{\dot{H}}^m} \Vert R_h v_d \Vert _{{\dot{H}}^m} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m} + C\Vert R_h^2 v_d \Vert _{{\dot{H}}^m} \Vert v_d \Vert _{{\dot{H}}^m} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m} \\{} & {} \quad + C \Vert R_h^2 v_h \Vert _{{\dot{H}}^m} \Vert R_h \theta \Vert _{{\dot{H}}^m} \Vert R_h^2 v_d \Vert _{{\dot{H}}^m} + C \Vert R_h^2 v_d \Vert _{{\dot{H}}^m}^2 \Vert \theta \Vert _{H^m} \end{aligned}$$

and

$$\begin{aligned} |K_5| \le C\Vert R_h v \Vert _{{\dot{H}}^m} \Vert R_hv_d \Vert _{{\dot{H}}^m} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m} + C \Vert R_h v \Vert _{{\dot{H}}^m} \Vert R_h \theta \Vert _{{\dot{H}}^m} \Vert R_h^2 v_d \Vert _{{\dot{H}}^m}. \end{aligned}$$

Due to the cancellation property, we have

$$\begin{aligned} |K_6| \le C\Vert v \Vert _{{\dot{H}}^m} \Vert R_h^2 v_d \Vert _{{\dot{H}}^m} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}. \end{aligned}$$

Collecting the above estimates gives

$$\begin{aligned}{} & {} -\frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \le - \frac{1}{2} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}^2 + \frac{3}{2} \Vert R_h v_d \Vert _{{\dot{H}}^{m}}^2 \\{} & {} \quad + C(\Vert R_h v \Vert _{{\dot{H}}^m} + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m})(\Vert v \Vert _{{\dot{H}}^m} + \Vert R_h \theta \Vert _{{\dot{H}}^m})(\Vert R_h v_d \Vert _{{\dot{H}}^m} \\{} & {} \quad + \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}) + C \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 \Vert \theta \Vert _{H^m}. \end{aligned}$$

Thus, we arrived at

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 - \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \right) \\{} & {} \quad \le -(\frac{1}{4} - C\Vert \theta \Vert _{H^m}) \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}^2 \right) \\{} & {} \qquad + C(\Vert R_h v \Vert _{{\dot{H}}^m} + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m})(\Vert v \Vert _{{\dot{H}}^m} + \Vert R_h \theta \Vert _{{\dot{H}}^m}) (\Vert R_h v_d \Vert _{{\dot{H}}^m} \\{} & {} \qquad + \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}) + C \Vert \nabla v_d \Vert _{L^{\infty }} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2. \end{aligned}$$

By Young’s inequality and (6.1), it follows

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 - \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \right) \le -\frac{1}{8} \left( \Vert R_h v_d\Vert _{{\dot{H}}^m}^2 + \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}^2 \right) \\{} & {} \quad + C(\Vert R_h v \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2)(\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}). \end{aligned}$$

We consider $M \ge 1$ which will be specified later. Since

$$\begin{aligned} \begin{aligned} \frac{1}{M} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 - \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}^2&= \sum _{|\tilde{n}| \ne 0} \left( \frac{1}{M} - \frac{|\tilde{n}|^2}{|\eta |^2} \right) |\eta |^{2m} |\mathscr {F} R_h^2 \theta (\eta )|^2 \\&\le \frac{1}{M} \sum _{\frac{|\tilde{n}|^2}{|\eta |^2} \le \frac{1}{M}, |\tilde{n}| \ne 0} |\eta |^{2m} |\mathscr {F} R_h^2 \theta (\eta )|^2 \\&\le \frac{1}{M^2} \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \left| \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \right| \le \Vert R_h v_d \Vert _{{\dot{H}}^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^m} \le \frac{1}{2} \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 + \frac{1}{2} \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2, \end{aligned}$$

(6.21)

it holds

$$\begin{aligned}{} & {} -\frac{1}{8} \left( \Vert R_h v_d\Vert _{{\dot{H}}^m}^2 + \Vert R_h^3 \theta \Vert _{{\dot{H}}^m}^2 \right) \le -\frac{1}{8M} \left( \Vert R_h v_d\Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 \right) \\{} & {} \qquad + \frac{1}{16M} \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \\{} & {} \qquad + \frac{1}{8M^2} \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 - \frac{1}{16M} \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h v_d\Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 - \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \right) + \frac{1}{8M^2} \Vert R_h \theta \Vert _{{\dot{H}}^m}^2. \end{aligned}$$

Hence,

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 - \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \right) \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h v_d\Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 - \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x \right) + \frac{1}{8M^2} \Vert R_h \theta \Vert _{{\dot{H}}^m}^2\\{} & {} \qquad + C (\Vert R_h v \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2) (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}). \end{aligned}$$

Here, we take $M = 1+\frac{t}{16}$. Then multiplying the both sides by $2M^2$ and using (6.18), we have

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \left( (1+\frac{t}{16})^2 (\Vert R_h v_d\Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^m}^2 - \int _{\Omega } v_d (-\Delta )^m R_h^4 \theta \,\textrm{d}x) \right) \\{} & {} \quad \le C\Vert R_h \theta \Vert _{{\dot{H}}^m}^2 + C \Vert v_0 \Vert _{{\dot{H}}^m}^2 (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}) \\{} & {} \qquad + C \sup _{\tau \in [0,t]} (1+\frac{\tau }{16})^2 \left( \Vert R_h v_d(\tau ) \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta (\tau ) \Vert _{{\dot{H}}^m}^2 \right) (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}). \end{aligned}$$

We integrate it over time and use (6.21) and

$$\begin{aligned} \int _0^{\infty } (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}) \,\textrm{d}t \le C. \end{aligned}$$

Then, for

$$\begin{aligned} f(t) := \sup _{\tau \in [0,t]} (1+\frac{\tau }{16})^2 \left( \Vert R_h v_d(\tau ) \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \theta (\tau ) \Vert _{{\dot{H}}^m}^2 \right) , \end{aligned}$$

it holds

$$\begin{aligned} f(t) \le C + \int _0^t f(\tau ) (\Vert v \Vert _{{\dot{H}}^m}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^m}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}) \,\textrm{d}\tau . \end{aligned}$$

By Grönwall’s ineqaulity, we obtain (6.19). This completes the proof. $\square$

6.2 Proof of Theorem 1.1:Temporal decay part

Now, we finish the proof of Theorem 1.1 assuming (6.22) and (6.23), which are given in Proposition 6.5. We also provide Proposition 6.6 for improved temporal estimates. From (6.5) and (6.22), we obtain

$$\begin{aligned} (1+t)^{1+\frac{m-s}{2}} \Vert v(t) \Vert _{H^s}^2 \le C. \end{aligned}$$

On the other hand, (6.8) and (6.23) imply

$$\begin{aligned} (1+t)^{2+\frac{m-s}{2}} \Vert v_d(t) \Vert _{H^s}^2 \le C. \end{aligned}$$

It suffices to prove

$$\begin{aligned} (1+t)^{\frac{m}{2}} \Vert \theta (t) - \sigma \Vert _{L^2}^2 \le C \end{aligned}$$

due to (1.3). Recalling (1.4), we can estimate from (6.2) that

$$\begin{aligned} \Vert \theta - \sigma \Vert _{L^2}&\le \Vert \bar{\theta } \Vert _{L^2} + \left\| \int _{\mathbb T^{d-1}} \int _t^\infty \left( (v \cdot \nabla )\theta + v_d \right) \,\textrm{d}\tau \textrm{d}x_h \right\| _{L^2} \\&\le C(1+t)^{-\frac{m}{2}} +\int _t^\infty \Vert (v \cdot \nabla )\theta + v_d \Vert _{L^2} \,\textrm{d}\tau . \end{aligned}$$

Since

$$\begin{aligned} \Vert (v \cdot \nabla )\theta + v_d \Vert _{L^2}&\le \Vert (v_h \cdot \nabla _h)\theta \Vert _{L^2} + \Vert v_d \partial _d \theta \Vert _{L^2} + \Vert v_d \Vert _{L^2} \\&\le C \Vert v \Vert _{L^2} \Vert R_h \theta \Vert _{H^{m-1}} + C\Vert v_d \Vert _{L^2} \Vert \theta \Vert _{H^m} + \Vert v_d \Vert _{L^2} \\&\le C (1+\tau )^{-(\frac{1}{2} +\frac{m}{4})} \Vert R_h \theta \Vert _{H^{m-1}} + C (1+\tau )^{-(1+\frac{m}{4})} \end{aligned}$$

by (6.5) and (6.8), it holds

$$\begin{aligned} \int _t^\infty \Vert (v \cdot \nabla )\theta + v_d \Vert _{L^2} \,\textrm{d}\tau \le C \left\| (1+\tau )^{-(\frac{1}{2} +\frac{m}{4})} \right\| _{L^2(t,\infty )} + C (1+t)^{-\frac{m}{4}} \le C (1+t)^{-\frac{m}{4}}. \end{aligned}$$

This completes the proof.

Proposition 6.5

Let $d \in \mathbb {N}$ with $d \ge 2$ and $\alpha =1$. Let $m \in \mathbb {N}$ with $m > 3+\frac{d}{2}$ and $(v, \theta )$ be a smooth global solution to (1.2) with (1.3). Suppose that (6.1) be satisfied. Then, there exists a constant $C > 0$ such that

$$\begin{aligned} \Vert v(t) \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \Lambda ^{-1} \theta (t) \Vert _{{\dot{H}}^{m}}^2 \le C (1+t)^{-1}, \end{aligned}$$

(6.22)

and

$$\begin{aligned} \Vert v_d(t) \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \Lambda ^{-1} v(t) \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h^2 \Lambda ^{-2}\theta (t) \Vert _{{\dot{H}}^{m}}^2 \le C (1+t)^{-2}. \end{aligned}$$

(6.23)

Proof

From the v equations in (1.2), it follows

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert v \Vert _{{\dot{H}}^m}^2 + \Vert v \Vert _{{\dot{H}}^{m+1}}^2 \le C\Vert \nabla v \Vert _{L^\infty } \Vert v \Vert _{{\dot{H}}^m}^2 - \sum _{|\tilde{n}| \ne 0} |\eta |^{2m} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ). \end{aligned}$$

Using (5.3) gives

$$\begin{aligned}&\left| - \sum _{|\tilde{n}| \ne 0} |\eta |^{2m} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ) \right| \\&\quad \le \left( \sum _{|\tilde{n}| \ne 0} |\eta |^{2(m-1)} |\mathscr {F}_b R_h \theta (\eta )|^2 \right) ^{\frac{1}{2}} \left( \sum _{|\tilde{n}| \ne 0} |\eta |^{2(m+1)} |\mathscr {F} v(\eta )|^2 \right) ^{\frac{1}{2}} \\&\quad \le \frac{1}{4} \Vert v \Vert _{{\dot{H}}^{m+1}}^2 + \Vert R_h \Lambda ^{-1} \theta \Vert _{{\dot{H}}^{m}}^2. \end{aligned}$$

By (6.1) and (1.3), we have

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert v \Vert _{{\dot{H}}^m}^2&\le - (\frac{3}{4} -C \Vert (v_0,\theta _0) \Vert _{H^m}^2) \Vert v \Vert _{{\dot{H}}^{m+1}}^2 + C\Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2 \\&\le -\frac{1}{2} \Vert v \Vert _{{\dot{H}}^m}^2 + C\Vert R_h \Lambda ^{-1} \theta \Vert _{{\dot{H}}^{m}}^2. \end{aligned}$$

Then, applying Duhamel’s principle shows

$$\begin{aligned} \Vert v(t) \Vert _{{\dot{H}}^m}^2 \le e^{-t} \Vert v_0 \Vert _{{\dot{H}}^m}^2 + C \int _0^t e^{-(t-\tau )} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2 \,\textrm{d}\tau . \end{aligned}$$

Thus, from

$$\begin{aligned} (1+\tau ) \Vert R_h \Lambda ^{-1} \theta (\tau ) \Vert _{{\dot{H}}^{m}}^2 \le (1+\tau ) \Vert R_h^2 \Lambda ^{-2} \theta (\tau ) \Vert _{{\dot{H}}^{m}} \sup _{\tau \in [0,\infty )} \Vert \theta (\tau ) \Vert _{{\dot{H}}^{m}}, \end{aligned}$$

it follows

$$\begin{aligned} \Vert v(t) \Vert _{{\dot{H}}^m}^2 \le C(1+t)^{-1} \left( C + \sup _{\tau \in [0,t]} (1+\tau )^2 \Vert R_h^2 \Lambda ^{-2} \theta (\tau ) \Vert _{{\dot{H}}^{m}}^2 \right) . \end{aligned}$$

(6.24)

Recalling from the v equations in (1.2) that

$$\begin{aligned} \partial _t R_h v + (-\Delta ) R_h v + R_h(v \cdot \nabla )v = R_h \mathbb {P} \theta e_d. \end{aligned}$$

and using (6.1), we can see

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert R_h v \Vert _{{\dot{H}}^{m-1}}^2&\le - \Vert R_h v \Vert _{{\dot{H}}^m}^2 + C \Vert v \Vert _{{\dot{H}}^{m}} \Vert R_h v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h^2 \Lambda ^{-2} \theta \Vert _{{\dot{H}}^{m}} \Vert R_h v \Vert _{{\dot{H}}^{m}} \\&\le -\frac{1}{2} \Vert R_h v \Vert _{{\dot{H}}^m}^2 + \Vert R_h^2 \Lambda ^{-2} \theta \Vert _{{\dot{H}}^{m}}^2. \end{aligned}$$

By Duhamel’s principle,

$$\begin{aligned} \Vert R_hv(t) \Vert _{{\dot{H}}^{m-1}}^2 \le \left( C + \sup _{\tau \in [0,t]} (1+\tau )^{2} \Vert R_h^2 \Lambda ^{-2} \theta (\tau ) \Vert _{{\dot{H}}^{m}}^2 \right) . \end{aligned}$$

(6.25)

Now, we show that

$$\begin{aligned} \Vert R_h^2 \Lambda ^{-2} \theta (t) \Vert _{{\dot{H}}^{m}}^2 \le C(1+t)^{-2}. \end{aligned}$$

(6.26)

We can show from (3.2) that

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert R_h \Lambda ^{-2} v_d \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \Lambda ^{-1} v_d \Vert _{{\dot{H}}^{m}}^2\\{} & {} \quad = - \sum _{|\gamma | = m-3} \int _{\Omega } \nabla _h \partial ^{\gamma } \langle \mathbb {P} (v \cdot \nabla )v,e_d \rangle \cdot \nabla _h \partial ^{\gamma } v_d \,\textrm{d}x + \sum _{|\tilde{n}| \ne 0} |\tilde{n}|^4|\eta |^{2(m-4)} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ). \end{aligned}$$

From (3.3),

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \Vert R_h^2 \Lambda ^{-2} \theta \Vert _{{\dot{H}}^{m}}^2= & {} -\sum _{|\gamma | = m-4} \int _{\Omega } \partial ^\gamma \partial _h^2 (v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x \\{} & {} - \sum _{|\tilde{n}| \ne 0} |\tilde{n}|^4|\eta |^{2(m-4)} \mathscr {F}_b \theta (\eta )\mathscr {F}_b v_d(\eta ). \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h \Lambda ^{-2} v_d \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h^2 \Lambda ^{-2} \theta \Vert _{{\dot{H}}^{m}}^2 \right) + \Vert R_h \Lambda ^{-1} v_d \Vert _{{\dot{H}}^{m}}^2 \\{} & {} \quad \le - \sum _{|\gamma | = m-4} \int _{\Omega } \nabla _h \partial ^{\gamma } \langle \mathbb {P} (v \cdot \nabla )v,e_d \rangle \cdot \nabla _h \partial ^{\gamma } (-\Delta ) v_d \,\textrm{d}x\\{} & {} \qquad -\sum _{|\gamma | = m-4} \int _{\Omega } \partial ^\gamma \partial _h^2 (v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x. \end{aligned}$$

Using integration by parts and Hölder’s inequality, we have

$$\begin{aligned}{} & {} \left| - \sum _{|\gamma | = m-4} \int _{\Omega } \nabla _h \partial ^{\gamma } \langle \mathbb {P} (v \cdot \nabla )v,e_d \rangle \cdot \nabla _h \partial ^{\gamma } (-\Delta ) v_d \,\textrm{d}x \right| \\{} & {} \quad \le C(\Vert (v_h \cdot \nabla _h) v \Vert _{{\dot{H}}^{m-3}} + \Vert \nabla _h (v_d \partial _d v) \Vert _{{\dot{H}}^{m-4}}) \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} \\{} & {} \quad \le C\Vert v \Vert _{{\dot{H}}^{m}} \Vert R_h v \Vert _{{\dot{H}}^{m-1}} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} + C\Vert v \Vert _{H^m} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2. \end{aligned}$$

We note that

$$\begin{aligned} \sum _{|\gamma | = m-4} \int _{\Omega } \partial ^\gamma \partial _h^2 (v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x = \sum _{|\gamma | = m-4} \left( K_7 + K_8 + K_9 \right) , \end{aligned}$$

where

$$\begin{aligned} K_7&= \int _{\Omega } \partial ^\gamma (\partial _h^2 v \cdot \nabla ) \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x, \\ K_8&= \int _{\Omega } \partial ^\gamma (\partial _h v \cdot \nabla ) \partial _h \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x, \\ K_9&:= \int _{\Omega } \partial ^\gamma (v \cdot \nabla ) \partial _h^2 \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x . \end{aligned}$$

The integration by parts and $(v \cdot \nabla ) \theta = (v_h \cdot \nabla _h) \theta + v_d \partial _d \theta$ show

$$\begin{aligned} K_7 + K_8 = - \int _{\Omega } \partial ^\gamma (\partial _h v_h \cdot \nabla _h) \theta \cdot \partial ^\gamma \partial _h R_h^2 (-\Delta ) \theta \,\textrm{d}x - \int _{\Omega } \partial ^\gamma (\partial _h v_d \partial _d \theta ) \cdot \partial ^\gamma \partial _h R_h^2 (-\Delta ) \theta \,\textrm{d}x. \end{aligned}$$

By the use of the integration by parts, we can estimate the second integral on the right-hand side as

$$\begin{aligned} \left| - \int _{\Omega } \partial ^\gamma (\partial _h v_d \partial _d \theta ) \cdot \partial ^\gamma \partial _h R_h^2 (-\Delta ) \theta \,\textrm{d}x \right|&= \left| - \int _{\Omega } (-\Delta ) \partial ^{\gamma } (\partial _h v_d \partial _d \theta ) \cdot \partial ^{\gamma } \partial _h R_h^2 \theta \,\textrm{d}x \right| \\&\le C\Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} \Vert \theta \Vert _{H^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}. \end{aligned}$$

Similarly, the first one is bounded by

$$\begin{aligned} \left| \int _{\Omega } (-\Delta ) \partial ^{\gamma } (\partial _h v_h \cdot \nabla _h) \theta \cdot \partial ^{\gamma } \partial _h R_h^2 \theta \,\textrm{d}x \right| \le C \Vert R_h v \Vert _{{\dot{H}}^{m-1}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}. \end{aligned}$$

Thus,

$$\begin{aligned} |K_7 + K_8| \le C\Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} \Vert \theta \Vert _{H^m} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} + C \Vert R_h v \Vert _{{\dot{H}}^{m-1}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}. \end{aligned}$$

Due to the cancellation property, we have for $|\gamma '|=1$ that

$$\begin{aligned} K_9&= \int _{\Omega } \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} v_h \cdot \nabla _h) \partial _h^2 \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x + \int _{\Omega } \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} v_d \partial _d \partial _h^2 \theta ) \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x. \end{aligned}$$

By integration by parts and the calculus inequality, it can be shown that

$$\begin{aligned}{} & {} \left| \int _{\Omega } \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} v_h \cdot \nabla _h) \partial _h^2 \theta \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x \right| \\{} & {} \quad \le \left| \int _{\Omega } \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} \nabla _h \cdot v_h \partial _h^2 \theta ) \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x \right| + \left| \int _{\Omega } \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} v_h \partial _h^2 \theta ) \cdot \nabla _h \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x \right| . \end{aligned}$$

The first term on the right-hand side is bounded by

$$\begin{aligned} \left| \int _{\Omega } \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} \partial _d v_d \partial _h^2 \theta ) \cdot \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x \right|&\le C(\Vert v_d \Vert _{{\dot{H}}^{m-3}} \Vert \partial _h^2 \theta \Vert _{L^{\infty }} \\&\quad + \Vert \nabla \partial _d v_d \Vert _{L^p} \Vert \partial ^{\gamma -\gamma '} \partial _h^2 \theta \Vert _{L^q}) \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}} \\&\le C \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}, \end{aligned}$$

where $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$ and $\frac{1}{p} = \frac{2}{d} + (\frac{1}{2} - \frac{m-2}{d})\frac{2}{m-2}$.On the other hand, the integration by parts yields

$$\begin{aligned}{} & {} \left| \int _{\Omega } \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} v_h \partial _h^2 \theta ) \cdot \nabla _h \partial ^\gamma \partial _h^2 \theta \,\textrm{d}x \right| \\{} & {} \quad = \left| \int _{\Omega } (-\Delta ) \partial ^{\gamma -\gamma '} (\partial ^{\gamma '} v_h \partial _h^2 \theta ) \cdot \partial ^\gamma \partial _h R_h^2 \theta \,\textrm{d}x \right| \\{} & {} \quad \le C ( \Vert \nabla v_h \Vert _{{\dot{H}}^{m-3}} \Vert \partial _h^2 \theta \Vert _{L^{\infty }} + \Vert \nabla v_h \Vert _{L^{\infty }} \Vert \partial _h^2 \theta \Vert _{{\dot{H}}^{m-3}}) \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} \\{} & {} \quad \le C \Vert v \Vert _{{\dot{H}}^{m-2}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}. \end{aligned}$$

Hence,

$$\begin{aligned} |K_9| \le C \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}} + C \Vert v \Vert _{{\dot{H}}^{m-2}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}. \end{aligned}$$

By the above estimates, we deduce

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 \right) + \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2 \\{} & {} \quad \le C(\Vert v \Vert _{H^m} + \Vert \theta \Vert _{H^m}) (\Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}) \\{} & {} \qquad + C (\Vert R_h v \Vert _{{\dot{H}}^{m-1}} + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}} + \Vert v \Vert _{{\dot{H}}^{m-2}}) (\Vert v \Vert _{{\dot{H}}^{m}} \\{} & {} \qquad + \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}) ( \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} + \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}). \end{aligned}$$

On the other hand, using (3.2) and (6.20), we have

$$\begin{aligned} -\int _{\Omega } \partial _tv_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x&= \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x\\&\quad - \int _{\Omega } \partial _d \nabla \cdot ((v \cdot \nabla )v) (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \\&\quad - \int _{\Omega } (-\Delta ) v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x - \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2 \\&\le \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x\\&\quad + C \Vert v \Vert _{{\dot{H}}^{m-2}}\Vert v \Vert _{{\dot{H}}^{m-1}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} \\&\quad + \frac{1}{2} \Vert R_hv_d \Vert _{{\dot{H}}^{m-1}}^2 - \frac{1}{2}\Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2. \end{aligned}$$

Since (3.3) yields

$$\begin{aligned} -\int _{\Omega } \partial _t\theta (-\Delta )^{m-3} R_h^4 v_d \,\textrm{d}x \le \int _{\Omega } (v \cdot \nabla )\theta (-\Delta )^{m-3} R_h^4 v_d \,\textrm{d}x + \Vert R_h^2 v_d \Vert _{{\dot{H}}^{m-3}}^2, \end{aligned}$$

we have

$$\begin{aligned}{} & {} -\frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \le \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x\\{} & {} \quad + \int _{\Omega } (v \cdot \nabla )\theta (-\Delta )^{m-3} R_h^4 v_d \,\textrm{d}x \\{} & {} \quad + C\Vert v \Vert _{{\dot{H}}^{m-2}} \Vert v \Vert _{{\dot{H}}^{m}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} - \frac{1}{2} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2+ \frac{3}{2} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2. \end{aligned}$$

It is clear

$$\begin{aligned} \left| \int _{\Omega } (v \cdot \nabla )v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \right|\le & {} C \Vert v \Vert _{{\dot{H}}^{m-3}} \Vert v_d \Vert _{{\dot{H}}^{m-2}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} \\{} & {} \le C \Vert v \Vert _{{\dot{H}}^{m}} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}. \end{aligned}$$

Since $H^{m-3}(\Omega )$ is banach algebra, we can see

$$\begin{aligned}{} & {} \left| \int _{\Omega } (v \cdot \nabla )\theta (-\Delta )^{m-3} R_h^4 v_d \,\textrm{d}x \right| \\{} & {} \quad \le \Vert (v_h \cdot \nabla _h)\theta \Vert _{{\dot{H}}^{m-3}} \Vert R_h^4 v_d \Vert _{{\dot{H}}^{m-3}} + C\Vert v_d \partial _d \theta \Vert _{{\dot{H}}^{m-3}} \Vert R_h^4 v_d \Vert _{{\dot{H}}^{m-3}} \\{} & {} \quad \le C\Vert v \Vert _{{\dot{H}}^{m-3}} \Vert \nabla _h \theta \Vert _{{\dot{H}}^{m-3}} \Vert R_h^4 v_d \Vert _{{\dot{H}}^{m-3}} + C\Vert v_d \Vert _{{\dot{H}}^{m-3}} \Vert \partial _d \theta \Vert _{{\dot{H}}^{m-3}} \Vert R_h^4 v_d \Vert _{{\dot{H}}^{m-3}} \\{} & {} \quad \le C\Vert v \Vert _{{\dot{H}}^{m-2}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} + C\Vert \theta \Vert _{{\dot{H}}^{m}} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2. \end{aligned}$$

Collecting the above estimates gives

$$\begin{aligned}{} & {} -\frac{\textrm{d}}{\textrm{d}t} \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \le - \frac{1}{2} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2 + \frac{3}{2} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2\\{} & {} \quad + C\Vert v \Vert _{{\dot{H}}^{m-2}} \Vert v \Vert _{{\dot{H}}^{m}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} \\{} & {} \quad + C \Vert v \Vert _{{\dot{H}}^{m}} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} +C\Vert v \Vert _{{\dot{H}}^{m-2}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}\\{} & {} \quad + C\Vert \theta \Vert _{{\dot{H}}^{m}} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2. \end{aligned}$$

Now, we arrived at

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 - \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \right) \\{} & {} \quad \le -(\frac{1}{4} - C(\Vert v \Vert _{H^m} + \Vert \theta \Vert _{H^m})) \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2 \right) \\{} & {} \quad + C (\Vert R_h v \Vert _{{\dot{H}}^{m-1}} + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}} + \Vert v \Vert _{{\dot{H}}^{m-2}}) (\Vert v \Vert _{{\dot{H}}^{m}} + \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}) ( \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}\\{} & {} \qquad + \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}) \\{} & {} \quad \le -\frac{1}{8} \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2 \right) \\{} & {} \qquad + C (\Vert R_h v \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 + \Vert v \Vert _{{\dot{H}}^{m-2}}^2) (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2). \end{aligned}$$

We have used Young’s inequality and (6.1) in the last inequality. We consider $M \ge 1$ which will be specified later. Since

$$\begin{aligned} \begin{aligned} \frac{1}{M} \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 - \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2&= \sum _{|\tilde{n}| \ne 0} \left( \frac{1}{M} - \frac{|\tilde{n}|^2}{|\eta |^4} \right) |\eta |^{2(m-2)} |\mathscr {F} R_h^2 \theta (\eta )|^2 \\&\le \frac{1}{M} \sum _{\frac{|\tilde{n}|^2}{|\eta |^4} \le \frac{1}{M}, |\tilde{n}| \ne 0} |\eta |^{2(m-2)} |\mathscr {F} R_h^2 \theta (\eta )|^2 \\&\le \frac{1}{M^2} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2 \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \left| \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \right| \le \Vert R_h v_d \Vert _{{\dot{H}}^{m-3}} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}} \le \frac{1}{2} \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2 + \frac{1}{2} \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2,\nonumber \\ \end{aligned}$$

(6.27)

it holds

$$\begin{aligned}{} & {} -\frac{1}{8} \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^3 \theta \Vert _{{\dot{H}}^{m-3}}^2 \right) \le -\frac{1}{8M} \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 \right) \\{} & {} \qquad + \frac{1}{16M} \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \\{} & {} \qquad + \frac{1}{8M^2} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2 - \frac{1}{16M} \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h v_d\Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 - \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \right) + \frac{1}{8M^2} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2. \end{aligned}$$

Hence,

$$\begin{aligned}{} & {} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} \left( \Vert R_h v_d \Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 - \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \right) \\{} & {} \quad \le -\frac{1}{16M} \left( \Vert R_h v_d\Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 - \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x \right) + \frac{1}{8M^2} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2 \\{} & {} \qquad + C (\Vert R_h v \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 + \Vert v \Vert _{{\dot{H}}^{m-2}}^2) (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2). \end{aligned}$$

Here, we take $M = 1+\frac{t}{16}$. Then multiplying the both sides by $2M^2$ and using (6.18), we have

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \left( (1+\frac{t}{16})^2 (\Vert R_h v_d \Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 - \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x) \right) \le C \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2 \\{} & {} \quad + C (1+\frac{t}{16})^2 (\Vert R_h v \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 + \Vert v \Vert _{{\dot{H}}^{m-2}}^2) (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2). \end{aligned}$$

From (6.25), it is clear

$$\begin{aligned} C (1+\frac{t}{16})^2 (\Vert R_h v \Vert _{{\dot{H}}^{m-1}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2) \le C + C\sup _{\tau \in [0,t]} (1+\tau )^{2} \Vert R_h^2 \theta (\tau ) \Vert _{{\dot{H}}^{m-2}}^2. \end{aligned}$$

Using (6.5), (6.24) and the interpolation inequality gives

$$\begin{aligned} (1+\frac{t}{16})^2 \Vert v \Vert _{{\dot{H}}^{m-2}}^2&\le (1+\frac{t}{16})^2 \Vert v \Vert _{L^2}^{\frac{4}{m}} \Vert v \Vert _{{\dot{H}}^{m}}^{2-\frac{4}{m}} \\&\quad \le C(1+t)^{1-\frac{2}{m}} \Vert v \Vert _{{\dot{H}}^{m}}^{2-\frac{4}{m}} \\&\quad \le C + C\sup _{\tau \in [0,t]} (1+\tau )^2 \Vert R_h^2 \theta (\tau ) \Vert _{{\dot{H}}^{m-2}}^2. \end{aligned}$$

Therefore,

$$\begin{aligned}{} & {} \frac{\textrm{d}}{\textrm{d}t} \left( (1+\frac{t}{16})^2 (\Vert R_h v_d \Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta \Vert _{{\dot{H}}^{m-2}}^2 - \int _{\Omega } v_d (-\Delta )^{m-3} R_h^4 \theta \,\textrm{d}x) \right) \\{} & {} \quad \le \left( C + C\sup _{\tau \in [0,t]} (1+\frac{\tau }{16})^2 (\Vert R_h v_d(\tau ) \Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta (\tau ) \Vert _{{\dot{H}}^{m-2}}^2) \right) (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2). \end{aligned}$$

We integrate it over time and use (6.27) with

$$\begin{aligned} \int _0^{\infty } (\Vert v \Vert _{{\dot{H}}^{m}}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}}^2) \,\textrm{d}t \le C. \end{aligned}$$

Then, for

$$\begin{aligned} f(t) := \sup _{\tau \in [0,t]} (1+\frac{\tau }{16})^2 \left( \Vert R_h v_d(\tau ) \Vert _{{\dot{H}}^{m-2}}^2 + \Vert R_h^2 \theta (\tau ) \Vert _{{\dot{H}}^{m-2}}^2 \right) , \end{aligned}$$

it holds

$$\begin{aligned} f(t) \le C + \int _0^t f(\tau ) (\Vert v \Vert _{{\dot{H}}^s}^2 + \Vert R_h \theta \Vert _{{\dot{H}}^s}^2 + \Vert \nabla v_d \Vert _{L^{\infty }}) \,\textrm{d}\tau . \end{aligned}$$

By Grönwall’s ineqaulity, we obtain (6.26).

Now, we prove that

$$\begin{aligned} \Vert v_d \Vert _{{\dot{H}}^m} \le C(1+t)^{-1}. \end{aligned}$$

(6.28)

As showing (6.9), we can have

$$\begin{aligned} \Vert v_d (t) \Vert _{{\dot{H}}^m} \le C \Vert R_h^2 \Lambda ^{-2} \theta (t) \Vert _{{\dot{H}}^{m}} + C \left( \sum _{\eta \in J} |\eta |^{2(m-2)} | \langle \mathscr {F} \textbf{u},\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

Thus, it suffices to show

$$\begin{aligned} \left( \sum _{\eta \in J} |\eta |^{2(m-2)} | \langle \mathscr {F} \textbf{u},\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}} \le C (1+t)^{-1}. \end{aligned}$$

(6.29)

We can see from (3.5) that

$$\begin{aligned} \langle \mathscr {F}_b\textbf{u}(t),\textbf{a}_+ \rangle = e^{-\lambda _+ t} \langle \mathscr {F}\textbf{u}_0,\textbf{a}_+ \rangle - \int _0^t e^{-\lambda _+ (t - \tau )} \langle N(v,\theta )(\tau ),\textbf{a}_+ \rangle \,\textrm{d}\tau . \end{aligned}$$

Due to $|e^{-\lambda _+ t}| \le e^{-\frac{|\eta |^{2}}{2} t}$ for $\eta \in J$, it follows by the Minkowski inequality

$$\begin{aligned}{} & {} \left( \sum _{\eta \in J} |\eta |^{2(m-2)} | \langle \mathscr {F} \textbf{u},\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \le \left( \sum _{\eta \in J} |\eta |^{2(m-2)} e^{-|\eta |^{2}t} | \langle \mathscr {F} \textbf{u}_0,\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}} \\{} & {} \quad + \int _0^t \left( \sum _{\eta \in J} |\eta |^{2(m-2)} e^{-|\eta |^{2}(t - \tau )} |\langle N(v,\theta )(\tau ),\textbf{a}_+ \rangle |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau . \end{aligned}$$

From the simple fact $|\textbf{a}_{+}|^2 = |\lambda _{+}|^2 + \frac{|\tilde{n}|^4}{|\eta |^4} \le C|\eta |^{4}$ with (6.1), we have

$$\begin{aligned} \left( \sum _{\eta \in J} |\eta |^{2(m-2)} e^{-|\eta |^{2\alpha }t} | \langle \mathscr {F} \textbf{u}_0,\textbf{a}_{+} \rangle |^2 \right) ^{\frac{1}{2}} \le Ce^{-t} \Vert \textbf{u}_0 \Vert _{H^m}. \end{aligned}$$

By (6.12) we have

$$\begin{aligned}{} & {} \int _0^t \left( \sum _{\eta \in J} |\eta |^{2(m-2)} e^{-|\eta |^{2}(t - \tau )} |\langle N(v,\theta )(\tau ),\textbf{a}_+ \rangle |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau \\{} & {} \quad \le \int _0^t \left( \sum _{\eta \in J} e^{-|\eta |^{2}(t - \tau )} |\eta |^{2m} | \mathscr {F}(v \cdot \nabla )v |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau \\{} & {} \quad + \int _0^t \left( \sum _{\eta \in J} e^{-|\eta |^{2}(t - \tau )} |\eta |^{2(m-2)} | \mathscr {F}_b(v \cdot \nabla )\theta |^2 \right) ^{\frac{1}{2}} \,\textrm{d}\tau \\{} & {} \quad \le \int _0^t e^{-(t - \tau )} (\Vert (v \cdot \nabla )v (\tau ) \Vert _{{\dot{H}}^{m}} + \Vert (v \cdot \nabla )\theta (\tau ) \Vert _{{\dot{H}}^{m-2}}) \,\textrm{d}\tau . \end{aligned}$$

We note

$$\begin{aligned} \Vert (v \cdot \nabla )v \Vert _{{\dot{H}}^m} \le C \Vert v \Vert _{{\dot{H}}^{m+1}} \Vert v \Vert _{{\dot{H}}^{m-2}} \le C \Vert v \Vert _{{\dot{H}}^{m+1}} \Vert v \Vert _{{\dot{H}}^m}^{1-\frac{2}{m}} \Vert v \Vert _{L^2}^{\frac{2}{m}} \end{aligned}$$

and

$$\begin{aligned} \Vert (v \cdot \nabla )\theta \Vert _{{\dot{H}}^{m-2}}&\le \Vert v \Vert _{{\dot{H}}^{m-2}} \Vert \theta \Vert _{{\dot{H}}^{m-1}} \le \Vert v \Vert _{{\dot{H}}^m}^{1-\frac{2}{m}} \Vert v \Vert _{L^2}^{\frac{2}{m}} \Vert \theta \Vert _{H^m}. \end{aligned}$$

Combining (6.5) and (6.22), we can see

$$\begin{aligned} (1+\tau ) (\Vert (v \cdot \nabla )v(\tau ) \Vert _{{\dot{H}}^m} + \Vert (v \cdot \nabla )\theta (\tau ) \Vert _{{\dot{H}}^{m-2}}) \le C (\Vert v \Vert _{{\dot{H}}^{m+1}} + \Vert \theta \Vert _{H^m}). \end{aligned}$$

Thus,

$$\begin{aligned} \int _0^t e^{-(t - \tau )} (\Vert (v \cdot \nabla )v (\tau ) \Vert _{{\dot{H}}^{m}} + \Vert (v \cdot \nabla )\theta (\tau ) \Vert _{{\dot{H}}^{m-2}}) \,\textrm{d}\tau \le C (1+t)^{-1}. \end{aligned}$$

Collecting the above estimates, we obtain (6.29), which implies (6.28). This completes the proof. $\square$

Proposition 6.6

Let $d \in \mathbb {N}$ with $d \ge 2$ and $\alpha =1$. Let $m \in \mathbb {N}$ with $m > 3+\frac{d}{2}$ and $(v, \theta )$ be a smooth global solution to (1.2) with (1.3). Suppose that (6.1) be satisfied. Then, for any $\epsilon \in (0,1)$, there exists a constant $C > 0$ such that

$$\begin{aligned} \Vert \Lambda ^{-\epsilon } v(t) \Vert _{L^2} \le C (1+t)^{-(\frac{3}{4} + \frac{m}{4})} \end{aligned}$$

(6.30)

and

$$\begin{aligned} \Vert \Lambda ^{-\epsilon } v(t) \Vert _{{\dot{H}}^{m+1}} \le C t^{-\frac{1}{2}}. \end{aligned}$$

(6.31)

Proof

Since (6.8) implies $\Vert \Lambda ^{-\epsilon } v_d(t) \Vert _{L^2} \le C t^{-(\frac{3}{4} + \frac{m}{4})}$, it suffices to show that

$$\begin{aligned} \Vert \Lambda ^{-\epsilon } v_h(t) \Vert _{L^2} \le C t^{-(\frac{3}{4} + \frac{m}{4})}. \end{aligned}$$

Applying Duhamel’s principle to (3.1) with (2.3), we obtain

$$\begin{aligned} \begin{aligned}&\left( \sum _{\eta \in I} |\eta |^{-2\epsilon } |\mathscr {F}_c v_h|^2 \right) ^{\frac{1}{2}} \\&\quad \le e^{- t} \Vert v_0 \Vert _{L^2} + \int _0^t e^{-(t-\tau )} \Vert (v \cdot \nabla )v \Vert _{L^2} \,\textrm{d}\tau + \left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} \frac{|\tilde{n}|}{|\eta |^{1+\epsilon }} |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

for any $\epsilon \in (0,1)$. We clearly have by (6.5) and (6.22) that

$$\begin{aligned} \int _0^t e^{-(t-\tau )} \Vert (v \cdot \nabla )v \Vert _{L^2}^2 \,\textrm{d}\tau \le \int _0^t e^{-(t-\tau )} \Vert v \Vert _{L^2} \Vert v \Vert _{H^m}^2 \,\textrm{d}\tau \le C(1+t)^{-(2+\frac{m}{2})}. \end{aligned}$$

On the other hand, we can see

$$\begin{aligned}{} & {} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} \frac{|\tilde{n}|}{|\eta |^{1+\epsilon }} |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \\{} & {} \quad \le C \left| e^{-\frac{t}{2}} \int _0^{\frac{t}{2}} |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 + C \left| \int _{\frac{t}{2}} ^t (t-\tau )^{-(1-\frac{\epsilon }{4})} \frac{|\tilde{n}|}{|\eta |^{3+\frac{\epsilon }{2}}} |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2. \end{aligned}$$

Thus, we have

$$\begin{aligned}{} & {} \left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} \frac{|\tilde{n}|}{|\eta |^{1+\epsilon }} |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \le C \left( \sum _{\eta \in I} \left| e^{-\frac{t}{2}} \int _0^{\frac{t}{2}} |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}}\\{} & {} \qquad + C \left( \sum _{\eta \in I} \left| \int _{\frac{t}{2}} ^t (t-\tau )^{-(1-\frac{\epsilon }{4})} \frac{|\tilde{n}|}{|\eta |^{3+\frac{\epsilon }{2}}} |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \le C t e^{-\frac{t}{2}} + C \int _{\frac{t}{2}}^t (t-\tau )^{-(1-\frac{\epsilon }{4})} \Vert R_h \Lambda ^{-(2+\frac{\epsilon }{2})} \theta \Vert _{L^2} \,\textrm{d}\tau . \end{aligned}$$

We can infer from (6.5) and (6.8) that

$$\begin{aligned} \Vert R_h \Lambda ^{-(2+\frac{\epsilon }{2})} \theta \Vert _{L^2} \le \Vert R_h^{\frac{3}{2} + \frac{\epsilon }{4}} \Lambda ^{-(\frac{3}{2} + \frac{\epsilon }{4})} \theta \Vert _{L^2} \le C(1+t)^{-(\frac{3}{2} + \frac{\epsilon }{4} +\frac{m}{2})}. \end{aligned}$$

Since this implies

$$\begin{aligned} \int _{\frac{t}{2}}^t (t-\tau )^{-(1-\frac{\epsilon }{4})} \Vert R_h \Lambda ^{-(2+\frac{\epsilon }{2})} \theta \Vert _{L^2} \,\textrm{d}\tau \le C(1+t)^{-(\frac{3}{2} + \frac{m}{2})}, \end{aligned}$$

combining the above estimates gives (6.30).

Using (2.3) and (2.1), we can infer from (3.1) and (3.2) that

$$\begin{aligned} \begin{aligned} \left( \sum _{\eta \in I} |\eta |^{2(m+1-\epsilon )} |\mathscr {F} v|^2 \right) ^{\frac{1}{2}}&\le \left( \sum _{\eta \in I} e^{-|\eta |^2t} |\eta |^{2(m+1-\epsilon )} |\mathscr {F} v_0|^2 \right) ^{\frac{1}{2}} \\&\quad + \left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} |\eta |^{m+1-\epsilon } | \mathscr {F} (v \cdot \nabla )v | \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}}\\&\quad + \left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} |\tilde{n}| |\eta |^{m-\epsilon } |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}} \end{aligned} \end{aligned}$$

for any $\epsilon \in (0,1)$. We can see

$$\begin{aligned} \left( \sum _{\eta \in I} e^{-|\eta |^2t} |\eta |^{2(m+1-\epsilon )} |\mathscr {F} v_0|^2 \right) ^{\frac{1}{2}} \le Ct^{-\frac{1-\epsilon }{2}} e^{-\frac{t}{2}} \Vert v_0 \Vert _{H^m}. \end{aligned}$$

Since

$$\begin{aligned}&\left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} |\eta |^{m+1-\epsilon } | \mathscr {F} (v \cdot \nabla )v | \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}}\\&\quad \le C\int _0^t (t-\tau )^{-\frac{2-\epsilon }{2}} e^{- \frac{t-\tau }{2}} \Vert (v \cdot \nabla )v \Vert _{{\dot{H}}^{m-1}} \,\textrm{d}\tau \\&\quad \le C \int _0^t (t-\tau )^{-\frac{2-\epsilon }{2}} e^{- \frac{t-\tau }{2}} \Vert v \Vert _{{\dot{H}}^m}^2 \,\textrm{d}\tau \end{aligned}$$

and

$$\begin{aligned} \left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} |\tilde{n}| |\eta |^{m-\epsilon } |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}}&\le C\int _0^t (t-\tau )^{-\frac{2-\epsilon }{2}} e^{- \frac{t-\tau }{2}} \Vert R_h \theta \Vert _{{\dot{H}}^{m-1}} \,\textrm{d}\tau , \end{aligned}$$

we have by (6.22) that

$$\begin{aligned}{} & {} \left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} |\eta |^{m+1-\epsilon } | \mathscr {F} (v \cdot \nabla )v | \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}} \\{} & {} \qquad + \left( \sum _{\eta \in I} \left| \int _0^t e^{-|\eta |^2 (t-\tau )} |\tilde{n}| |\eta |^{m-\epsilon } |\mathscr {F}_b \theta (\eta )| \,\textrm{d}\tau \right| ^2 \right) ^{\frac{1}{2}}\\{} & {} \quad \le Ct^{-\frac{1}{2}}. \end{aligned}$$

Collecting the above estimates gives (6.31). This completes the proof. $\square$

7 Sharpness of decay rates

In this section, we prove that the decay rates in Theorem 1.1 and 1.4 are sharp in the following sense. We recall the linearized system of (3.4):

$$\begin{aligned} \partial _t \mathscr {F}_b\textbf{u} + M \mathscr {F}_b \textbf{u} = 0, \qquad M := \begin{pmatrix} |\eta |^{2\alpha } &{} -\frac{|\tilde{n}|^2}{|\eta |^2} \\ 1 &{} 0 \end{pmatrix}, \end{aligned}$$

(7.1)

where $\textbf{u} = (v_d,\theta )^{T}$. The eigenvalues and eigenvectors of the linear operator are previously given by

$$\begin{aligned} \lambda _{\pm }(\eta ) = \frac{|\eta |^{2\alpha } \pm \sqrt{|\eta |^{4\alpha } - {4 |\tilde{n}|^2}/{|\eta |^2}}}{2}, \qquad \overline{\textbf{a}_{\pm } (\eta )} = \begin{pmatrix} \lambda _{\pm } \\ -\frac{|\tilde{n}|^2}{|\eta |^2} \end{pmatrix} , \end{aligned}$$

and it holds

$$\begin{aligned} \mathscr {F}_b \textbf{u} = \sum _{j = \pm } \langle \mathscr {F}_b\textbf{u}(t),\textbf{a}_j \rangle \textbf{b}_j = \sum _{j=\pm } e^{-\lambda _j t} \langle \mathscr {F}\textbf{u}_0,\textbf{a}_j \rangle \textbf{b}_j, \end{aligned}$$

(7.2)

where

$$\begin{aligned} \begin{pmatrix} \textbf{b}_+ \\ \textbf{b}_- \end{pmatrix} = \frac{1}{\lambda _{+}-\lambda _{-}} \begin{pmatrix} 1 &{} \frac{| \eta |^2}{|\tilde{n}|^2} \lambda _{-} \\ -1 &{} -\frac{| \eta |^2}{|\tilde{n}|^2} \lambda _{+} \end{pmatrix}. \end{aligned}$$

Note that If we consider $\textbf{u}_0$ such that $\mathscr {F}_b \textbf{u}_0 = 0$ for $\eta \not \in D_3$, then $|\textbf{a}_{\pm }||\textbf{b}_{\pm }| \le C$ for some $C>0$ not depending on $\eta$. Now, we are ready to provide the sharpness of the decay rates.

Proposition 7.1

Let $m \in \mathbb N$. Then for any $\epsilon > 0$, there exists an initial data $\textbf{u}_0 \in X^m(\Omega )$ such that the solution $\textbf{u}=(v_d,\theta )$ to (7.1) satisfies

$$\begin{aligned} \Vert \bar{\theta }(t) \Vert _{H^s} \ge Ct^{-\frac{m-s}{2(1+\alpha )} - \epsilon } \end{aligned}$$

(7.3)

and

$$\begin{aligned} \Vert v_d(t) \Vert _{H^s} \ge Ct^{-1-\frac{m-s}{2(1+\alpha )} - \epsilon } \end{aligned}$$

(7.4)

for any $s \in [0,m]$ and $t \ge C$.

Proof

Let $\epsilon > 0$ and

$$\begin{aligned} \textbf{u}_0 := (0, \sum _{\eta \in J} \mathscr {F}_b \theta _0(\eta ) \mathscr {B}_{\eta }(x))^{T}, \end{aligned}$$

where $\mathscr {F}_b \theta _0(\eta ) := |q|^{-(m+\frac{1}{2} + \epsilon )}$ for $\eta \in D_3 \cap \{|n| = 1\}$, $\mathscr {F}_b \theta _0(\eta ) = 0$ for $\eta \not \in D_3 \cap \{|n| = 1\}$. For simplicity, we use the notation $A:= D_3 \cap \{|n| = 1\}$. We show (7.3) first. From (7.2), we can see

$$\begin{aligned} \Vert \bar{\theta } \Vert _{H^s} \ge \Vert e^{-\lambda _{-}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_2 \rangle \Vert _{H^s} - \Vert e^{-\lambda _{+}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_+ \rangle \langle \textbf{b}_+, e_2 \rangle \Vert _{H^s}. \end{aligned}$$

Due to $|e^{-\lambda _{+}t}| \le e^{-|\eta |^{2\alpha } \frac{t}{2}} \le e^{-\frac{t}{2}}$ it is clear that

$$\begin{aligned} \Vert e^{-\lambda _{+}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_+ \rangle \langle \textbf{b}_+, e_2 \rangle \Vert _{H^s} \le Ce^{-\frac{t}{2}}. \end{aligned}$$

(7.5)

On the other hand, we have

$$\begin{aligned} |e^{-\lambda _{-}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_2 \rangle | = C|e^{-\lambda _{-}t} \mathscr {F}_b \theta | \ge Ce^{-\frac{2|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} t} |q|^{-(m+\frac{1}{2} + \epsilon )}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert e^{-\lambda _{-}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_2 \rangle \Vert _{H^s}&\ge C\left( \sum _{J \in A} e^{-\frac{4|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}}t} |q|^{-(2(m-s)+ 1 + 2\epsilon )} \right) ^{\frac{1}{2}} \\&\ge C\left( \sum _{|q| \ge C_1} e^{-\frac{Ct}{q^{2(1+\alpha )}}} |q|^{-(2(m-s)+ 1 + 2\epsilon )} \right) ^{\frac{1}{2}} \\&\ge Ct^{-\frac{m-s}{2(1+\alpha )}-\frac{1+2\epsilon }{4(1+\alpha )}} \left( \sum _{|q| \ge C_1} e^{-\frac{Ct}{q^{2(1+\alpha )}}} (\frac{Ct}{|q|^{2(1+\alpha )}})^{\frac{m-s}{1+\alpha }+\frac{1+2\epsilon }{2(1+\alpha )}} \right) ^{\frac{1}{2}} , \end{aligned}$$

for some $C_1>0$. Let

$$\begin{aligned} f(\tau ) := e^{-\frac{Ct}{|\tau |^{2(1+\alpha )}}} (\frac{Ct}{|\tau |^{2(1+\alpha )}})^{\frac{m-s}{1+\alpha }+\frac{1+2\epsilon }{2(1+\alpha )}}. \end{aligned}$$

Then, we can verify that there exists $C_2 \ge C_1$ not depending on t such that $f(\tau )$ is decreasing on the interval $(C_2 t^{\frac{1}{2(1+\alpha )}},\infty )$. Thus, it holds for $t \ge 1$ that

$$\begin{aligned} \sum _{|q| \ge C_1} e^{-\frac{Ct}{|q|^{2(1+\alpha )}}} (\frac{Ct}{|q|^{2(1+\alpha )}})^{\frac{m-s}{1+\alpha }+\frac{1+2\epsilon }{2(1+\alpha )}} \ge \int _{|\tau | \ge C_2 t^{\frac{1}{2(1+\alpha )}}} f(\tau ) \,\textrm{d}\tau . \end{aligned}$$

By the change of variable $\tilde{\tau } = \tau t^{-\frac{1}{2(1+\alpha )}}$, we can see

$$\begin{aligned} \int _{|\tau | \ge C_2 t^{\frac{1}{2(1+\alpha )}}} f(\tau ) \,\textrm{d}\tau = t^{\frac{1}{2(1+\alpha )}} \int _{|\tilde{\tau }| \ge C_2} e^{-\frac{1}{|\tilde{\tau }|^{2(1+\alpha )}}} |\tilde{\tau }|^{-(2(m-s)+1+2\epsilon )} \,\textrm{d}\tilde{\tau } \ge C \end{aligned}$$

for some $C>0$. Combining the above yields

$$\begin{aligned} \Vert e^{-\lambda _{-}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_2 \rangle \Vert _{H^s} \ge Ct^{-\frac{m-s+\epsilon }{2(1+\alpha )}}. \end{aligned}$$

Therefore, (7.3) is obtained.

The proof of (7.4) is similar with the previous one. By (7.2) and (7.5), it holds

$$\begin{aligned} \Vert v_d \Vert _{H^s} \ge \Vert e^{-\lambda _{-}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_1 \rangle \Vert _{H^s} - Ce^{-\frac{t}{2}}. \end{aligned}$$

Note that

$$\begin{aligned} |e^{-\lambda _{-}t} \langle \mathscr {F}_b\textbf{u}_0,\textbf{a}_- \rangle \langle \textbf{b}_-,e_1 \rangle | = \frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} |e^{-\lambda _{-}t} \mathscr {F}_b \theta | \ge \frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}}e^{-\frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \frac{t}{2}} |q|^{-(m+\frac{1}{2} + \epsilon )}. \end{aligned}$$

Using that $\frac{|\tilde{n}|^2}{|\eta |^{2(1+\alpha )}} \ge \frac{C}{|q|^{2(1+\alpha )}}$ for all $\eta \in A$, repeating the above procedures, we obtain (7.4). This completes the proof. $\square$