Optimal decay rates of higher–order derivatives of solutions for the compressible nematic liquid crystal flows in $\mathbb R^3$

1.   Introduction

In this paper, we investigate the upper and lower bounds of decay rates for global solution to compressible nematic liquid crystal flows in three–dimensional whole space:

where $t \geq 0$ is time and $x \in \mathbb{ R }^{3}$ is the spatial coordinate. Here, the symbol $\otimes$ is the Kronecker tensor product. We denote the unknown functions $\rho(x, t)$ denotes the fluid density, $u = (u_1, u_2, u_3)$ is the fluid velocity, $d$ is the macroscopic average of the nematic liquid crystal orientation field. The pressure $P = P (\rho) = a\rho^\gamma(a > 0, \gamma \ge 1)$ is a smooth function in a neighborhood of $1$ with $P'(1) = 1$. $\mu$ and $\nu$ are shear viscosity and the bulk viscosity coefficients of the fluid, respectively, which satisfy the physical assumptions:

The positive constants $\gamma$ and $\theta$ represent the competition between the kinetic energy and the potential energy, and the microscopic elastic relaxation time for the molecular orientation field, respectively. For the sake of simplicity, we set the constants $\gamma$ and $\theta$ to be $1$. We consider the Cauchy problem of the system (1.1) subject to the initial conditions:

where $\omega_{0}$ is a unit constant vector.

1.1.   History of the problem

Liquid crystal has important physical and chemical properties such as photoelectric effect, thermal effect, photochemical effect and so on. After nearly a century of research, liquid crystals have been widely used in production, life, and scientific research. Particularly, liquid crystal displays have been widely used in LED technology, airplanes, medical equipment, bioengineering, machinery manufacturing, and other fields. Next, let us present some explanations about the above model. The nematic liquid crystal flows are a coupling between the compressible Navier–Stokes equations and the transported flow harmonic maps. Ericksen ^[4] and Leslie ^[20] first established the continuum theory of liquid crystals in the $1960$s. Since then, due to the physical importance and mathematical challenges, the study of the full Ericksen–Leslie model has attracted many physicists and mathematicians. Considering the compressible liquid crystal flows, Ding–Lin–Wang ^[2] gained both existence and uniqueness of global strong solution in one–dimensional space. For the case of multi-dimensional space, Jiang–Jiang–Wang in ^[16,17] proved the global existence of weak solutions to the initial–boundary problem with large initial energy. Huang–Wang–Wen in ^[13] obtained the blow up criterion of strong solutions. Hu–Wu ^[11] showed the existence and uniqueness of global strong solution in critical Besov spaces. Under the assumption that the initial energy is suitably small, Wu–Tan in ^[32] proved the global existence of small energy weak solution. Li–Xu–Zhang in ^[22] established the global existence of classical solution with smooth initial data which are of small energy but possibly large oscillations. Recently, Gao–Tao–Yao ^[8] obtained the global well–posedness of classical solution under the condition that the initial data is a small perturbation of the constant equilibrium state in the $H^{N}(\mathbb R^{3}) (N\geq 3)$–framework. Furthermore, if the initial perturbation data belongs to $L^{1}$ additionally, they obtained the optimal decay rate of k–th $(k\leq N-1)$ order spatial derivative of solution in ^[8] as follow:

Here, the positive constant $C$ is independent of time.

When the director is a constant vector, then the compressible nematic liquid crystal flow (1.1) becomes the compressible Navier–Stokes equation. There are a lot of basic results about the global existence, unique and time decay rates of the solutions to the compressible Navier–Stokes equations, cf.^{[3,6,7,10,14,23,24,28,29]} and references cited therein. We also mention that the problem of time-asymptotic behavior for the solutions of hydrodynamic equations is a hot topic, see ^{[5,12,15,18,30,31]} and references cited therein.

By observing the results of ^[8], decay rate (1.3) shows that the highest–order spatial derivative of density and velocity converge to zero at $L^2$–rate $(1+t)^{-\frac{ 1}{4}-\frac{N }{2 }}$, which are slower than the $L^2$–rate $(1+t)^{-\frac{ 3}{4}-\frac{N }{2 }}$ for the heat equation, and thus are not optimal in this sense. Thus, we are caught up in the upper and lower optimal decay rates of higher–order spatial derivatives of the solutions to the $3$D compressible liquid crystal flows (1.1). More precisely, we focus on the following three problems:

(ⅰ) Can we show that the highest order spatial derivative of the density and velocity converge to zero at the same $L^2$–rate $(1+t)^{- \frac{3}{4}- \frac{N}{2}}$ as that of the heat equation?

(ⅱ) Can we provide some lower bounds of decay rate for the solution as well as its all–order spatial derivatives?

(ⅲ) Can we provide some information on the upper and lower bound of decay rate for the time derivatives of solution?

The main purpose of this article is to give a clear answer to the above three problems.

1.2.   Main results

In this article, we use $H^k(\mathbb R^3)$ to denote the usual Sobolev spaces with norm $\|\cdot\|_{H^k}$. Generally, we use $L^p$, $1\leq p\leq \infty$ to denote the usual $L^p(\mathbb R^3)$ spaces with norm $\|\cdot\|_{L^p}$. The notation $a\lesssim b$ means that $a\leq Cb$ for a universal positive constant which is independent of time $t$. Let $\Lambda^{s}$ be the pseudo–differential operator defined by

where $\widehat{f}$ and $\mathfrak{F}$ are the Fourier transform of $f$. For a radial function $\phi\in C_0^\infty(\mathbb R^3_{{\bf \xi}})$ such that $\phi({\bf \xi}) = 1$ when $|{\bf \xi}|\leq 1$ and $\phi({\bf \xi}) = 0$ when $|{\bf \xi}|\geq 2$, we define the low–frequency part and the high–frequency part of $f$ as follows

Before stating our main results, let us recall the result of ^[8] in the following.

Theorem 1.1.(See ^[8]) Assume that the initial data $(\rho_{0}-1, u_{0}, \nabla d_{0})\in H^{N}(\mathbb R^{3})\cap L^{1}(\mathbb R^{3})$ for any integer $N\geq 3$, $d_0 (x) = 1$ in $\mathbb R^{3}$ and there exists a small constant $\delta_0 > 0$ such that

then the Cauchy problem (1.1)–(1.2) admits a unique globally classical solution $(\rho, u, d)$ such that for any $t\in [0, \infty)$,

If the initial data $\| d_{0}- \omega_{0} \|_{ L^2 }$ and $\| (\rho_{0}-1, u_{0}, \nabla d_{0}) \|_{ L^1}$ are finite additionally, the global solution $(\rho, u, d)$ of problem (1.1)–(1.2) satisfies for all $t\geq0$

Here, the positive constant $C$ is independent of time.

Now, we are in a position to state our main results, which are stated in the following four theorems. First, we show that the highest–order spatial derivative of the density and velocity converge to zero at the same $L^2$–rate as ones of the heat equation.

Theorem 1.2. Under the assumptions of Theorem 1.1, for all $t \geq 0$, it holds that

Here, the positive constant $C$ is independent of time.

Next, we address the lower bound of decay rate for the solution as well as its all–order spatial derivatives.

Theorem 1.3. Assume that all the hypotheses of Theorem 1.1 are in force, denote $m_{0} = \rho_{0}u_{0}$, $b_{0} = \Lambda d_0$, and the Fourier transform $(\widehat{\rho_0 -1}, \widehat{m}_{0}, \widehat{b_{0} })$ satisfies

where $c_0$ is a positive constant. Then, the global solution $(\rho, u, d)$ has the decay rates for all $t\geq t_{*}$

Here, $t_{*}$ is a positive large time, the two positive constants c$_1$ and c$_2$ are independent of time.

Next, we will establish the upper bounds of decay rates for the time derivatives of the solution to the $3$D compressible liquid crystal flows (1.1).

Theorem 1.4. Under the assumptions of Theorem 1.1 and Theorem 1.2, the global solution $(\rho, u, d)$ of Cauchy problem (1.1)–(1.2) satisfies for all $t\geq0$

Here, the positive constant $C$ is independent of time.

Finally, we will establish the lower bounds of decay rates for the time derivatives of the solution to the $3$D compressible liquid crystal flows (1.1).

Theorem 1.5. Under all the assumptions in Theorem 1.1, and the condition (1.9) holds, then the global solution $(\rho, u, d)$ of Cauchy problem (1.1)–(1.2) satisfies for all $t\geq t_{*}$

Furthermore, if there exists a small constant $\delta_{1}$ such that $\| u_{0} \|_{L^{1}}\leq \delta_{1}$, it holds that for all $t\geq t_{*}$

Here, $t_{*}$ is a positive large time, and the positive constant c$_3$ is independent of time.

Remark 1.6. Compared to Theorem 1.1 of ^[8], the main innovation of Theorem 1.2–1.5 lies in the following three aspects: First, by observing the decay rates in (1.8), we prove that the highest–order spatial derivative of the density and velocity converge to their corresponding equilibrium states at the $L^2$–rate $(1+t)^{-\frac{3}{4} - \frac{ N }{ 2}}$, which is the same as one of the heat equation and particularly faster than the rates $(1+t)^{-\frac{1}{4} - \frac{ N }{ 2}}$ in ^[8]. Second, for well–chosen initial data, Theorem 1.3 also gives the lower bounds on solution as well as its all–order spatial derivatives. Thus, our time decay rates are really optimal in this sense. Third, we also gives the lower bound of decay rates for the time derivatives of density, velocity and macroscopic average for the $3D$ compressible liquid crystal flows, which converge to zero at the $L^2$–rates $(1+t)^{- \frac{5}{4}}$, $(1+t)^{- \frac{5}{4}}$ and $(1+t)^{- \frac{5}{4}}$, respectively.

Now, let us sketch the main strategy of proving Theorem 1.2–Theorem 1.5 and explain some main difficulties and techniques involved in the process. Roughly speaking, we will make full use of the benefit of the low–frequency and high–frequency decomposition $f = f^{l} + f^{h}$, where $f^l$ and $f^h$ stand for the low–frequency part and high–frequency part of $f$, respectively.

For the proof of Theorem 1.2, motivated by the work in ^[27], we will establish the highest–order spatial derivatives of the density and velocity to compressible nematic liquid crystal flows (1.1). There are four steps to achieve this. First, we rewrite the Cauchy problem (1.1)–(1.2) into the system (2.1). We notice that the low-frequency part $\nabla^{k}(\varrho, u) (0\leq k\leq N)$ of the corresponding linear system to (2.1) has been obtained by ^[21]. Then, by employing Duhamel's principle, the key linear decay estimates in Lemma 3.1 and nonlinear energy estimates, we can get the optimal decay rate of $\|\nabla^N (\varrho^l, u^l)(t)\|_{L^2}$ (see the proof of Lemma 3.3 for details). Second, we want to obtain the optimal decay rate of $\|\nabla^N (\varrho^h, u^h)(t)\|_{L^2}$. Through high–frequency and low–frequency decomposition and precise energy estimates, we establish the energy inequality as follows:

Third, note that the energy equality (1.14) only gives the dissipation estimate for $u^h$. In order to explore the dissipation estimates for $\varrho^h$, we will construct the new interactive energy functionals between $u^h$ and $\varrho^h$. Therefore, we have

Fourth, we choose two sufficiently large positive constant $D_0$ and $T_0$, then define the temporal energy functional as

Notice that ${\bf{E}}(t)$ is equivalent to $\|\nabla^{N}(\varrho^{h}, u^{h})\|_{L^{2}}^{2}$. Multiplying (1.14) by $D_{0}$, adding the resulting inequality with (1.15), and then for all $t\geq T_0$, we obtain the Lyapunov-type energy inequality as follow:

By virtue of Lemma 3.3 and Gronwall's inequality, we obtain the optimal $L^2$ time decay rates of $\|\nabla^{N}(\varrho^{h}, u^{h})\|_{L^{2}}^{2}$. In addition, together with the $N$–order low–frequency decay rates of (3.7), we obtain the decay rates of (1.8) immediately. Thus, we have completed the proof of Theorem 1.2.

For Theorem 1.3, we show the lower bounds on the decay rates of solutions. Compared to the proof of Theorem 1.2, the new difficulty we encounter is that the system (2.1) is not a conservative form, which implies that it seems impossible to obtain the lower optimal decay rates as in (1.10). To this end, let's break it down into two parts. For the part of Navier–stokes system, the key idea here is that, instead of using the system (2.1)$_1$–(2.1)$_2$, we will employ the system (4.1) of $(\varrho, m)$ with $m = (\varrho +1)u$, which can be rewritten in the conservative form. As a result, one can shift the derivative onto the solution semigroup to obtain the desired lower optimal decay rates (See the proof of (4.4) for details). For the part of macroscopic average, we notice that the linearized system of (4.6) is mere heat equation on $\nabla n$. When the initial data satisfy (1.9), we will employ Plancherel theorem and careful analysis on the solution semigroup to obtain an optimal lower bound estimate for the linear part. We find a similar structure between (4.6) and the system (4.1), so we can prove the lower bound on the convergence rates in $L^{2}$–norm for the macroscopic average(see the proof of Lemma 4.2 for details). Thus, we complete the proof of Theorem 1.3.

For Theorem 1.4 and Theorem 1.5, we establish the upper and lower bound of decay rate for the time derivative of solution in $L^2$–norm. It is worth mentioning that the lower decay rate estimate for the time derivative of solution can be obtained which is inspired by the work of Gao–Lyu–yao in ^[9]. However, the lower bound of decay estimates in ^[9] are established for the compressible fluid model of Korteweg type.

2.   Reformulation

In this section, we will reformulate the problem firstly. Set $\varrho = \rho-1$ and $n = d-\omega_0$, the Cauchy problem (1.1)–(1.2) can be reformulated into:

Here $S_{i}(i = 1, 2, 3)$ are defined as

where the three nonlinear functions of $\varrho$ are defined by

Assume there exists a small positive constant $\delta$ satisfying following estimate

for all $t \in [0, T]$. By virtue of (2.4) and Sobolev inequality, it is easy to get

Hence, we immediately have

And, for the nonlinear terms of the model (2.1), employing the Hölder's inequality, we obtain

3.   Proof of Theorem 1.2

Let us to consider the following linearized compressible nematic liquid crystal system:

with the initial conditions:

Since the system (3.1) is a decoupled system of the classical linearized Navier–Stokes equations and heat equations. If we set

Then the solution to (3.1)$_{1}$–(3.1)$_{2}$ can be written as

where $\mathcal{A}$ is a matrix–valued differential operators of the form

The solution semigroup $e^{-\mathcal{A}t}$ has the following lemma on the decay in time.

Lemma 3.1. Let $k \geq -\frac{ 3 }{ 2 }$ and $2\leq r \leq \infty$, then for any $t \geq 0$, it holds that

Moreover, if initial data satisfy $(1.9)$, then there exists a positive large time $t_{*}$such that for all $t\geq t_{*}$, we have

for $k = 0, 1$.Here, the positive constant c$_4$ is independent of time.

Proof. The proof can be seen in ^[21].

To treat the macroscopic average, we notice that the Eq (3.1)$_{3}$ is a mere heat equation on $\nabla n$. We state the large-time behavior of solutions to the heat equation as the following lemma which can be obtained by direct calculation or more can refer to ^[26,33].

Lemma 3.2. Let $k \geq -\frac{ 3 }{ 2 }$ and $2\leq r \leq \infty$, then for any $t \geq 0$, it holds that

Moreover, if initial data satisfy $(1.9)$, then there exists a positive large time $t_{*}$, such that for all $t\geq t_{*}$, we have

Here, the positive constant c$_4$ is independent of time.

Second, we state the $L^{2}$–time decay rates on the low–frequency part of the solution in the nonlinear system (2.1).

Lemma 3.3. Under the assumptions of Theorem 1.1, then solution $(\varrho, u)$ of the nonlinear system $(2.1)$ satisfiesthe following decay rates:

Proof. We write the nonlinear terms $S: = (\ S_{1}, \ S_{2})^{T}$. By virtue of the Eq (2.1)$_1$–(2.1)$_2$, Lemma 3.1, Duhamel's principle, Plancherel theorem and Hausdorff–Young's inequality, we have

On the other hand, by employing decay rate (1.7), Lemma 7.1, Lemma 7.3 and Hölder's inequality, we can bound the third term on the right–hand of (3.8) as follows

Here, for the term $\| \nabla n \cdot \Delta n \|_{L^{2}}$, we have

For the term $\|\nabla^{N-2} (\nabla n \cdot \Delta n) \|_{L^{2}}$, we get

Substituting (3.9) and (2.6) into (3.8), we can get the follow estimates

Hence, we complete the proof of this lemma.

Third, we will give the energy estimates which contains the dissipation of $\nabla^{N}u^{h}$.

Lemma 3.4. Under the assumptions of Theorem 1.1, then we have

for any $t\in [0, \infty)$.

Proof. Taking

and using integration by parts, we can obtain

The right-hand side of (3.15) can be estimated one by one. For the term $I_1$, it holds that

The first term $I_{11}$ can be rewritten as follows

For the term $I_{111}$, due to Lemma 7.4, Hölder's inequality, Young's inequality and Sobolev interpolation theorem, we arrive at

Here we have defined the commutator:

For the term $I_{112}$, we can obtain

Similarly, it is easy to see that

Substituting (3.18)–(3.20) into (3.17), we can conclude that

For the term $I_{12}$, by using the Lemma 7.3, Hölder's inequality, Young's inequality, Sobolev interpolation theorem, we have

For the term $I_{2}$, it holds that

For the term $I_{21}$, making use of integration by parts, we have

For the term $I_{22}$, we obtain

Similarly, it is easy to see that

For the term $I_{24}$, similar to the proof of (3.24), we have

For the term $I_{25}$, by virtue of Lemma 7.3, we have

Substituting (3.24)–(3.28) into (3.23), we have

Substituting (3.21)–(3.22) and (3.29) into (3.15), we obtain the estimate (3.13).

Next, we will establish the dissipation estimates for $\nabla^{N}\varrho^{h}$.

Lemma 3.5. Suppose that the assumptions of Theorem 1.1 are in force, then we have

for any $t\in [0, \infty)$.

Proof. Taking $\langle\mathfrak{F}^{-1}(1-\phi({\bf \xi})) \nabla^{N-1} (2.1)_2, \nabla^{N} \varrho^h \rangle$, and making use of the integration by parts, it hold that

In order to deal with the term $\int_{\mathbb{R}^3} \nabla^{N-1} u_{t}^{h} \nabla^{N} \varrho^{h} \mathrm{d}x$, we apply the transport equation (2.1)$_{1}$. More precisely, we have

Substituting (3.33) into (3.32), and employing Young's inequality, it is easy to deduce

Here, we write

For the term $\Pi_{1}$, with the help of Hölder's inequality and Lemma 7.3, it holds that

For the term $\Pi_{2}$, similarly to the proof of (3.16), we have

For the term $\Pi_{21}$, we have

For the term $\Pi_{22}$, it easy to deduce

For the term $\Pi_{23}$, we can get

For the term $\Pi_{24}$, similarly to the proof of (3.34), we have

For the term $\Pi_{25}$, by virtue of Lemma 7.3, it is easy to obtain

Substituting (3.36)–(3.41) into (3.35), yields directly

Substituting (3.34) and (3.41) into (3.33), we complete the proof of the Lemma 3.5.

Finally, we will close the estimate by combining them which have been already proved along the proof of Lemma (3.3)–(3.5). To achieve this, we choose sufficiently large time $T_{0}$ and positive constant $D_{0}$, and then define the temporary energy functional

for $t\geq T_{0}$, it is noticed that ${\bf{E}}(t)$ is equivalent to $\|\nabla^{N}(\varrho^{h}, u^{h})\|_{L^{2}}^{2}$. Substituting (3.13) and (3.30) into

which together with the smallness of $\delta$, for $t \geq T_0$, it holds that

where we have used the fact that $T_{0}$ is large enough. On other hand, it is clear that

Hence, by virtue of estimate (3.7) and Gronwall's inequality, we can arrive at

So, the proof of Theorem 1.2 has been completed.

4.   Proof of Theorem 1.3

In this section, we will establish the lower optimal decay rates of the solution as well as its all–order spatial derivatives in Theorem 1.3. At first, we rewrite (1.1)$_1$–(1.1)$_2$ with $m = (\varrho +1) u$ in the perturbation form as follow:

where the functions $\mathcal{F} = \mathcal{F} (\varrho, u, n)$ is defined as

Here $\nabla n \odot\nabla n = (\partial_{i} n, \partial_j n)_{1\leq i, j \leq 3}$. It is easy to check

with the initial conditions:

Next, we will establish the lower bound decay rate of $\nabla^{k} (\varrho, u)$ to the system (2.1).

Lemma 4.1. Under all the assumptions of Theorem 1.3, then the global solution $(\varrho, u)$ has the decay rates for all $t\geq t_{*}$ and $0\leq k \leq N$

Here, $t_{*}$ is a positive large time, the positive constant c$_1$ is independent of time.

Proof. By virtue of Duhamel's principle, system (4.1), Lemma 3.1, Theorem 1.1 for $k = 0, 1$, we have

Then, for $2\leq k \leq N$, by applying (4.3) and Lemma 7.2, we have from Sobolev interpolation that

Therefore, we have completed the proof of this lemma.

In order to obtain the decay rate of $\| \nabla^{k+1}n \|_{L^2}$, employing $\nabla$ operator to the equation (2.3)$_3$, we have

Then, we can deduce the lower bound decay rate of $\nabla^{k+1}n$ which is given in the following lemma.

Lemma 4.2. Under all the assumptions of Theorem 1.3, then there is a positive constant c$_2$ independent of time, such that for all $t\geq t_{*}$ and $0\leq k \leq N$,

where $t_{*}$ is a positive large time.

Proof. If $t_{*}$ is large enough, by using Lemma 3.2 and Lemma 7.5, for $k = 0, 1$, we have

Next, we employ interpolation inequality to establish the lower bound of decay rate for the higher–order spatial derivative of solution. The decay rates (4.8) together with the interpolations $(k \ge 2)$

we have

for large time $t_{*}$. Thus the proof of Lemma 4.2 is completed.

Therefore, we have completed the proof of Theorem 1.3.

5.   Proof of Theorem 1.4

In this section, we will establish the decay rate for the mixed space–time derivatives of solution to the Cauchy problem (1.1)–(1.2).

Lemma 5.1. Suppose that the assumptions of Theorem 1.1 and Theorem 1.2 are in force, the global solution $(\varrho, u, n)$ hasthe time decay rate:

Here, the positive constant $C$ is independent of time.

Proof. First of all, we shall estimate $\|\nabla^k \partial_{t}\varrho \|_{L^{2}}$. For $l = 0, 1, 2, ..., N-1$, taking $l$–th spatial derivative to (2.1)$_{1}$, then multiplying the resulting identities by $\nabla^l \partial_{t} \varrho$ and integrating over $\mathbb{R}^3$, we have

We will estimate the right hand side of (5.2) term by term. First, for the term $K_{1}$, by virtue of Lemma 7.3, and Young's inequality, we obtain

where $\delta$ is small enough. Similarly, we can get

For the term $K_{3}$, using Young's inequality, it is easy to see that

Combining (5.3), (5.4) with (5.5), we obtain

Then we shall estimate $\|\nabla^k \partial_{t} u \|_{L^{2}}$. For $k = 0, 1, 2, ..., N-2$, taking $k$–th spatial derivative to (2.1)$_{2}$, multiplying the resulting identities by $\nabla^k \partial_{t} u$ and integrating over $\mathbb{R}^3$, we have

By applying Lemma 7.3 again, we estimate the second term in the right hand of (5.7).

Similarly, for the term $L_{2}$, we can bound

For the term $L_{3}$, it is easy to get

By employing the Leibniz formula and Lemma 7.3, we estimate the last term of (5.7)

Substituting estimates (5.8)–(5.11) into (5.7), we have

Similar to the estimate of the term $\|\nabla^l \partial_{t} \varrho \|_{L^{2}}$. For $l = 0, 1, 2, ..., N-1$, taking $l$–th spatial derivative to (2.1)$_{3}$, then multiplying the resulting identities by $\nabla^l \partial_{t} n$ and integrating over $\mathbb{R}^3$, we have

By using Lemma 7.3 again, we estimate the second term of (5.13)

By employing the Leibniz formula and Lemma 7.3, we estimate the last term of (5.13)

Here, we have the basic fact that $|n(x, t) +\omega_0| = |d(x, t)| = 1$, and

Combining (5.14) with (5.15), we arrive at

Combining estimates (5.6), (5.12) with (5.17), then we complete the proof of this lemma.

6.   Proof of Theorem 1.5

In this section, we will establish the lower bound of decay rate for the time derivative of solution of the system (2.1).

Lemma 6.1. Under all the assumptions of Theorem 1.3, then the global solution $(\varrho, u, n)$has the time decay rate for all $t\geq t_{*}$

Here, $t_{*}$ is a positive large constant, the positive constant c$_3$ is independent of time.

Proof. At first, we establish the lower bound time decay rate for $\partial_{t} n$ in the $L^2$–norm. With the help of the Eq (2.1)$_{3}$, we have

By applying the Sobolev inequality and decay rate (1.7), it is easy to get

And hence, we have

for all some large time $t_{*}$.

Next, we establish lower bound time decay rate for $\partial_{t} u$ in the $L^2$–norm. Using the Eq (2.1)$_{2}$, we can obtain

And hence, we have

By virtue of the Sobolev inequality and time decay rate (1.7), it is easy to deduce

Together with estimates (6.6) and (6.7), yields directly

Finally, we establish lower bound time decay rate for $\partial_{t} \varrho$ in the $L^2$–norm. To accomplish this objective, we use the Eq (2.1)$_{1}$ to obtain

By virtue of the Sobolev inequality and decay rate (1.7), we can get

and hence, we arrive at

Now, we need to establish the lower bound decay rate for $\| \mathrm{div} u \|_{L^{2}}$. Notice the relation differential relation $\Delta = \nabla \mathrm{div} - \nabla \times \nabla \times$, we get

And hence, we have

which implies that we need to establish upper bound decay rate for $\| \nabla \times u \|_{L^{2}}$. To this end, we take the $\nabla \times$ operator the the velocity equation (2.1)$_{2}$ to get

With the help of the Sobolev inequality, uniform bound (1.6), decay rate (1.7) and calculus identity (4.2), we have

By virtue of the Duhamel principle formula, we obtain

which, together with estimate (6.13) and smallness of $\delta_{i}(i = 0, 1)$, yields directly

This and the estimate (6.11) yields

which implies directly

for some large time $t_{*}$. Therefore, we complete the proof of this lemma.

7.   Analytic tools

For ease of use and clear reference, some Sobolev inequalities are listed as follows:

Lemma 7.1. (i) If $u(x) \in H^1 (\mathbb{R}^3)$, then the following inequalities hold:

(ii) Assume $u(x) \in H^2 (\mathbb{R}^3)$, then

Proof. One can found them in ^[1].

Now, we will introduce the Gagliardo-Nirenberg inequality that is frequently used in this paper.

Lemma 7.2. Let $0 \leq i, j \leq k$; then we have

where $\theta$ satisfies

Especially, while $p = q = r = 2$, we have

Proof. The proof can be seen in ^[25].

Next, to estimate the $L^2$-norm of the spatial derivatives of the product of two functions, we shall use the following formula:

Lemma 7.3. Let $k\ge 1$ be an integer; then one have

where $p, p_2, p_3\in[1, +\infty]$ and

Proof. For $p = p_{2} = p_{3} = 2$, it can be proved by using Lemma 7.2. For the general case, one may to refer to ^[19].

From Lemma 7.3, we can deduce the following commutator estimate:

Lemma 7.4. Let $f$ and $g$ be smooth functions belonging to $H^k \cap L^\infty$ for any integer $k\geq 1$ and define the commutator

Then we have

Here $p_{i} (i = 1, 2, 3, 4)$ are defined in Lemma 7.3.

Finally, we introduce the estimate for the low–frequency part and the high–frequency part of $f$.

Lemma 7.5. If $2\leq p \leq \infty$, $f\in L^{p}(\mathbb{R}^3)$, then we have

Proof. For $2\leq p \leq \infty$, by Young's inequality for convolutions, for the low-frequency, we have

and hence

Acknowledgments

This work is partially supported by Guangxi Natural Science Foundation #2019JJG110003, #2019AC20214, and #2019JJA110071.

Conflict of interest

The authors declare that they have no conflict of interest.