Global Well-Posedness and Decay Rates for the Three-Dimensional Compressible Phan-Thein–Tanner Model

Abstract

In this paper, we are concerned with the global well-posedness and decay rates of strong solutions for the three-dimensional compressible Phan-Thein–Tanner model. We prove that this set of equations admits a unique global strong solution provided the initial data are close to the constant equilibrium state in $H^3$-framework. Moreover, if the initial data belong to $L^{1}$, the convergence rates of the higher-order spatial derivatives of the solution are obtained by combining the decay estimates for the linearized equations and the Fourier splitting method.

1 Introduction

The theory of Phan-Thein–Tanner model recently gained quite some attention, this model is derived from network theory for the polymeric fluid. This type of fluids is described by the following set of equations

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \rho _t+\mathrm{div}(\rho u)=0,\\ \rho (u_t+u\cdot \nabla u)-\mu (\triangle u+\nabla \mathrm{div}u)+\nabla p=\mu _{1}\mathrm{div}\tau ,\\ \tau _t+ u\cdot \nabla \tau +Q(\tau ,\nabla u)+(a+btr\tau )\tau =\mu _{2}D(u),\\ (\rho ,u,\tau )|_{t=0}=(\rho _{0},u_{0},\tau _{0}),(t,x)\in (0,\infty )\times \mathbb {R}^{3}. \end{array} \right. \end{aligned}$$

(1.1)

The unknown $\rho , u, \tau , p$ are the density, velocity, stress tensor and scalar pressure of fluid respectively. D(u) is the symmetric part of $\nabla u$, that is

$$\begin{aligned} D(u)=\frac{1}{2}(\nabla u+(\nabla u)^{T}). \end{aligned}$$

$Q(\tau ,\nabla u)$ is a given bilinear form

$$\begin{aligned} Q(\tau ,\nabla u)=\tau \Omega (u)-\Omega (u)\tau +\lambda (D(u)\tau +\tau D(u)). \end{aligned}$$

where $\Omega (u)$ is the skew-symmetric part of $\nabla u$, namely

$$\begin{aligned} \Omega (u)=\frac{1}{2}(\nabla u-(\nabla u)^{T}). \end{aligned}$$

$\mu >0$ is the viscosity coefficient and $\mu _{1}$ is the elastic coefficient. a and $\mu _{2}$ are associated to the Debroah number $De=\frac{\mu _{2}}{a}$, which indicates the relation between the characteristic flow time and elastic time [3]. $\lambda \in [-1,1]$ is a physical parameter, in particular, we call the system co-rotational case when $\lambda =0$. $b\ge 0$ is a constant relate to the rate of creation or destruction for the polymeric network junctions.

To complete the system (1.1), the initial data are given by

$$\begin{aligned} (\rho , u, \tau )|_{t=0}=(\rho _{0}, u_{0}, \tau _{0}), \ \ x\in \mathbb {R}^{3}, \end{aligned}$$

(1.2)

with the far field behavior:

$$\begin{aligned} (\rho ,u,\tau )(t,x)=(\bar{\rho },0,0)\ \text{ as } \ |x|\rightarrow \infty , t\ge 0. \end{aligned}$$

Let us review some previous works about the model (1.1) and the related models. If $b=0$, the system (PTT) reduces to the famous Oldroyd-B model (See [31]) which has been studied widely, most of the results on Oldroyd-B fluids in the literature are about the incompressible model. Guillop$\acute{e}$ and Saut [14, 15] proved the existence of local strong solutions and the global existence of one dimensional shear flows. Later, the smallness restriction on the coupling constant $\omega$ in [14] was removed by Molinet and Talhouk [28]. In [24], Lin et al. proved that if the initial data is a small perturbation around equilibrium, then the strong solution is global in time. The similar results were obtained in several papers by virtue of different methods, see Lei and Zhou [22], Lei et al. [21], Zhang and Fang [40], Zhu [43]. Fang et al. [8, 9] proved the global existence of strong solutions with a class of large data. On the other hand, there are relatively few results for the compressible model. Lei [20] proved the local and global existence of classical solutions for a compressible Oldroyd-B system in a torus with small initial data. He also studied the incompressible limit problem and showed that the compressible flows with well-prepared initial data converge to incompressible ones when the Mach number converges to zero. Later on, Guillop$\acute{e}$ et al. [13] obtained similar results on some bounded domain $\Omega \in \mathbb {R}^3$, but with small coupling constant . The case of ill prepared initial data was considered by Fang and Zi [11] in the whole space $\mathbb {R}^d$, $d\ge 2$. Recently, the smallness restriction on coupling constant was removed by Zi in [41]. On the other hand, in suitable Sobolev spaces, Fang and Zi [10] obtained the unique local strong solution to (1.1) with initial density vanishing from below and a blow-up criterion for this solution. Zhou et al. [42] proved the existence of global strong solution provided the initial data are close to the constant equilibrium state in $H^2$-framework and obtained the convergence rates of the solutions. For the compressible Oldroyd type model based on the deformation tensor, see the results [18, 23, 34, 39] and references therein.

When $\tau$ is zero, (1.1) reduces to the compressible Navier-Stokes(NS) equations. The convergence rates of solution for the compressible Navier-Stokes equations to the steady state has been investigated extensively since the first global existence of small solutions in $H^3$ was improved by Matsumura and Nishida [26, 27]. When the initial perturbation $(\rho _{0}-1, u_{0})\in L^{p}\cap H^{N}(N\ge 3)$ with $p\in [1, 2]$, the $L^2$ optimal decay rate of the solution to the NS system is

$$\begin{aligned} \Vert (\rho -1,u)(t)\Vert _{L^2}\le C(1+t)^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{2})}. \end{aligned}$$

For the small initial perturbation belonging to $H^3$ only, Matsumura [25] employed the weighted energy method to show the $L^2$ decay rates. Ponce [33] obtained the optimal $L^p$ convergence rate. In [36], Schonbek and Wiegner studied the large time behavior of solutions to the Navier-Stokes equation in $H^m(\mathbb {R}^{n})$ for all $n\le 5$. In order to establish optimal decay rates for higher order spatial derivatives of solutions, if the initial perturbation is bounded in $H^{-s}(s\in [0, \frac{3}{2}))$ norm instead of $L^1$-norm, Guo and Wang [16] developed the time convergence rates as follows by using a general energy method,

$$\begin{aligned} \Vert \nabla ^{l}(\rho -1,u)(t)\Vert _{H^{N-l}}\le C(1+t)^{-\frac{l+s}{2}}, \end{aligned}$$

for $0\le l\le N-1$. In addition, the decay rate of solutions to the NS system was investigated in [6, 7, 37] and the references therein.

In this paper, we focus on the PTT model ($b\ne 0$). To our knowledge, there are a lot of numerical results about the PTT model (See [2, 12, 29, 32]). Recently, Chen et al. [5] proved that the strong solution in critical Besov spaces exists globally when the initial data is a small perturbation over around the equilibrium. Chen et al. [4] proved that the strong solution will blow up in finite time and proved the global existence of strong solution with small initial data. However, there are few results to our knowledge on the compressible PTT model, especially the large-time behavior. Compared with the incompressible models, the compressible equations of PTT model are more difficult to deal with because of the strong nonlinearities and interactions among the physical quantities. The main purpose in this paper is to study the global existence and decay rates of smooth solutions for the compressible PTT model. We first establish the global solution of the system (1.1)–(1.2) in the whole space $\mathbb {R}^{3}$ near the constant equilibrium state under the assumption that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrarily large. Then we establish the large time behavior by combining the decay estimates for the linearized equations and the Fourier splitting method by assuming that the initial data belongs to $L^{1}(\mathbb {R}^{3})$ additionally.

Throughout of the paper, without loss of generality, we set $\mu =\mu _{1}=\mu _{2}=a=b=\bar{\rho }=1$ in the following. Before stating the main results, we explain the notations and conventions throughout this paper. $\partial _{j}$ stands for $\partial _{x_{j}}$, $\nabla ^{l}$ with an integer $l\ge 0$ stands for the usual any spatial derivatives of order l. When $l<0$ or l is not a positive integer, $\nabla ^{l}$ stands for $\Lambda ^{l}$ defined by $\Lambda ^{s}u=\mathscr {F}^{-1}(|\xi |^{s}\hat{u}(\xi ))$, where $\hat{u}$ is the Fourier transform of u and $\mathscr {F}^{-1}$ its inverse. We will employ the notation $A\lesssim B$ to mean that $A\le CB$ for a universal constant $C>0$ that only depends on the parameters coming from the problem. For the sake of conciseness, we write $\Vert (A,B)\Vert _{X}:=\Vert A\Vert _{X}+\Vert B\Vert _{X}$.

Now, we state our main result about the global existence and decay properties of solution to the system (1.1)–(1.2) in the following theorems.

Theorem 1.1

Let $N\ge 3$, assume that $(\rho _{0}-1, u_{0}, \tau _{0})\in H^{N}$, then there exists a constant $\delta _{0}>0$ such that if

$$\begin{aligned} \Vert (\rho _{0}-1, u_{0}, \tau _{0})\Vert _{H^{3}} \le \delta _{0}, \end{aligned}$$

(1.3)

then the problem (1.1)–(1.2) has a unique global solution $(\rho , u, \tau )$ satisfying that for all $t\ge 0$,

$$\begin{aligned} \Vert (\rho -1,u,\tau )(t)\Vert _{H^{N}}^{2}+\int _{0}^{t} (\Vert \nabla \rho (s)\Vert _{H^{N-1}}^{2}+\Vert \nabla u(s)\Vert _{H^{N}}^{2}+\Vert \tau (s)\Vert _{H^{N}}^{2})ds \le C\Vert (\rho _{0}-1, u_{0}, \tau _{0})\Vert _{H^{N}}^{2}. \end{aligned}$$

(1.4)

If further, $(\rho _{0}-1, u_{0}, \tau _{0})\in L^{1}(\mathbb {R}^3)$, then we have

$$\begin{aligned} \Vert \nabla ^{l}(\rho -1,u,\tau )(t)\Vert _{H^{N-l}}\le C(1+t)^{-\frac{3+2l}{4}}, \end{aligned}$$

(1.5)

where $l=0,1, \cdot \cdot \cdot , N-1$.

Based on the Sobolev interpolation of Gagliardo-Nirenberg inequality and the results in Theorem 1.1, we can deduce the time decay rates.

Theorem 1.2

Under all the assumptions in Theorem 1.1, the global smooth solution $(\rho , u, \tau )$ of the Cauchy problem (1.1)–(1.2) has the time decay rates

$$\begin{aligned} \Vert \nabla ^{l}(\rho -1,u,\tau )(t)\Vert _{L^{p}}\le C(1+t)^{-\frac{3}{2}(1-\frac{1}{p})-\frac{l}{2}}, \end{aligned}$$

(1.6)

for $l=0,1, \cdot \cdot \cdot , N-2$, where $2\le p\le \infty$. Especially,

$$\begin{aligned} \Vert \nabla ^{l}(\rho -1,u,\tau )(t)\Vert _{L^{\infty }}\le C(1+t)^{-\frac{3+l}{2}}. \end{aligned}$$

(1.7)

Moreover, we establish decay rates for the mixed space-time derivatives of solutions to the Cauchy problem (1.1)–(1.2).

Theorem 1.3

Under all the assumptions in Theorem 1.1, the global classical solution $(\rho , u, \tau )$ of Cauchy problem (1.1)-(1.2) has the time decay rates

$$\begin{aligned} \Vert \nabla ^{l}(\rho _t,u_t)(t)\Vert _{L^{2}}\le C(1+t)^{-\frac{5+2l}{4}},\end{aligned}$$

(1.8)

$$\begin{aligned} \Vert \nabla ^{l}\tau _t(t)\Vert _{L^{2}}\le C(1+t)^{-\frac{3+2l}{4}}, \end{aligned}$$

(1.9)

where $l=0,1, \cdot \cdot \cdot , N-2$.

2 The Global Existence of Solution

In this section, we are going to prove our main result. the proof of local well-posedness for PTT is similar to the Oldroyd-B model (See [10, 17]) and we omit the detail here. Global solutions will follow in a standard continuity argument after we establish (1.4) a priori. We first reformulate the system (1.1). We set $\varrho =\rho -1$, then the initial value problem (1.1)–(1.2) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \varrho _t+\mathrm{div}u=S_{1},\\ u_t+\gamma \nabla \varrho -(\triangle u+\nabla \mathrm{div}u)-\mathrm{div}\tau =S_{2},\\ \tau _t+\tau -D(u)=S_{3}, \end{array} \right. \end{aligned}$$

(2.1)

where the nonlinear terms $S_i(i=1, 2, 3)$ are defined as

$$\begin{aligned}&S_{1}=-\mathrm{div}(\varrho u),\\&S_{2}=-u\cdot \nabla u-f(\varrho )(\triangle u+\nabla \mathrm{div}u)-g(\varrho )\nabla \varrho -f(\varrho )\mathrm{div}\tau ,\\&S_{3}=-u\cdot \nabla \tau -Q(\tau ,\nabla u)-tr\tau \tau , \end{aligned}$$

with

$$\begin{aligned} (\varrho ,u,\tau )(x,0)=(\varrho _{0}, u_{0},\tau _{0})\rightarrow 0 \ \text{ as } \ |x|\rightarrow \infty , \end{aligned}$$

(2.2)

and here

$$\begin{aligned} \gamma =\frac{P'(1)}{1},\quad f(\varrho )=\frac{\varrho }{\varrho +1},\quad g(\varrho )=\frac{P'(\varrho +1)}{\varrho +1}-\frac{P'(1)}{1}. \end{aligned}$$

(2.3)

Then, we will derive the a priori nonlinear energy estimates for the system (2.1). Hence we assume a priori that for sufficiently small $\delta >0$,

$$\begin{aligned} \sqrt{\mathcal {E}^{3}_{0}(t)}=\Vert (\varrho ,u,\tau )(t) \Vert _{H^{3}}\le \delta . \end{aligned}$$

(2.4)

First of all, by (2.4) and Sobolev’s inequality, we obtain

$$\begin{aligned} \frac{1}{2}\le \varrho +1\le 2. \end{aligned}$$

Hence, we immediately have

$$\begin{aligned} |f(\varrho )|, |g(\varrho )|\le C|\varrho |, \ \ |f^{(k)}(\varrho )|, |g^{(k)}(\varrho )|\le C \ \ \text{ for } \text{ any } \ \ k\ge 1. \end{aligned}$$

(2.5)

Befor establishing the global existence of solution under the assumption of (2.4), we derive some energy estimates which are easy to establish just following the idea by Guo and Wang [16]. Hence, we only state the results here for the sake of brevity.

Lemma 2.1

If $\sqrt{\mathcal {E}^{3}_{0}(t)}\le \delta$, then for $k=0, \cdot \cdot \cdot , N-1$, we have

$$\begin{aligned}&\frac{d}{dt}\Vert \nabla ^{k}(\varrho ,u,\tau )\Vert _{L^{2}}^{2} +C(\Vert \nabla ^{k+1}u\Vert _{L^{2}}^{2}+\Vert \nabla ^{k}\tau \Vert _{L^{2}}^{2})\lesssim \delta (\Vert \nabla ^{k+1}(\varrho ,u)\Vert _{L^{2}}^{2}+\Vert \nabla ^{k}\tau \Vert _{L^{2}}^{2}). \end{aligned}$$

(2.6)

$$\begin{aligned}&\frac{d}{dt}\Vert \nabla ^{k+1}(\varrho ,u,\tau )\Vert _{L^{2}}^{2} +C(\Vert \nabla ^{k+2}u\Vert _{L^{2}}^{2}+\Vert \nabla ^{k+1}\tau \Vert _{L^{2}}^{2})\lesssim \delta (\Vert \nabla ^{k+1}(\varrho ,\tau )\Vert _{L^{2}}^{2}+\Vert \nabla ^{k+2}u\Vert _{L^{2}}^{2}). \end{aligned}$$

(2.7)

$$\begin{aligned}&\frac{d}{dt}\int _{\mathbb {R}^{3}} \nabla ^{k}u\cdot \nabla ^{k+1}\varrho dx+C\Vert \nabla ^{k+1}\varrho \Vert _{L^{2}}^{2}\lesssim \Vert \nabla ^{k+1}(u,\tau )\Vert _{L^{2}}^{2}+\Vert \nabla ^{k+2}u\Vert _{L^{2}}^{2}. \end{aligned}$$

(2.8)

Next, we will combine all the energy estimates that we have derived to prove (1.4) of Theorem 1.1.

Proof

We first close the energy estimates at each l-th level in our weaker sense. Let $N\ge 3$ and $0\le l\le m-1$ with $1\le m\le N$. Summing up the estimates (2.6) of Lemma 2.1 for from $k=l$ to $m-1$, we obtain

$$\begin{aligned} \frac{d}{dt}\sum _{l\le k\le m-1} \Vert \nabla ^{k}(\varrho ,u,\tau )\Vert _{L^{2}}^{2}+C\left( \sum _{l+1\le k\le m}\Vert \nabla ^{k}u\Vert _{L^{2}}^{2}+\sum _{l\le k\le m-1}\Vert \nabla ^{k}\tau \Vert _{L^{2}}^{2}\right) \lesssim \delta \sum _{l+1\le k\le m}\Vert \nabla ^{k}\varrho \Vert _{L^{2}}^{2}. \end{aligned}$$

(2.9)

Let $k=m-1$ in the estimates (2.7) of Lemma 2.1 , we have

$$\begin{aligned} \frac{d}{dt}\Vert \nabla ^{m}(\varrho ,u,\tau )\Vert _{L^{2}}^{2}+C(\Vert \nabla ^{m+1} u\Vert _{L^{2}}^{2}+\Vert \nabla ^{m}\tau \Vert _{L^{2}}^{2})\lesssim \delta \Vert \nabla ^{m}\varrho \Vert _{L^{2}}^{2}. \end{aligned}$$

(2.10)

Adding the inequality (2.10) with (2.9), we get

$$\begin{aligned} \frac{d}{dt}\sum _{l\le k\le m} \Vert \nabla ^{k}(\varrho ,u,\tau )\Vert _{L^{2}}^{2}+C_{1}\left( \sum _{l+1\le k\le m+1}\Vert \nabla ^{k}u\Vert _{L^{2}}^{2}+\sum _{l\le k\le m}\Vert \nabla ^{k}\tau \Vert _{L^{2}}^{2}\right) \le C_{2}\delta \sum _{l+1\le k\le m} \Vert \nabla ^{k}\varrho \Vert _{L^{2}}^{2}. \end{aligned}$$

(2.11)

Summing up the estimates (2.8) of Lemma 2.1 for from $k=l$ to $m-1$, we have

$$\begin{aligned} \frac{d}{dt}\sum _{l\le k\le m-1}\int _{\mathbb {R}^{3}} \nabla ^{k}u\cdot \nabla ^{k+1}\varrho dx+C_{3}\sum _{l+1\le k\le m}\Vert \nabla ^{k}\varrho \Vert _{L^{2}}^{2}\le C_{4}(\sum _{l+1\le k\le m+1} \Vert \nabla ^{k}u\Vert _{L^{2}}^{2}+\sum _{l+1\le k\le m}\Vert \nabla ^{k}\tau \Vert _{L^{2}}^{2}). \end{aligned}$$

(2.12)

Multiplying (2.12) by $\frac{2C_{2}\delta }{C_{3}}$, adding it with (2.11), since $\delta >0$ is small, we deduce that there exists a constant $C_{5}>0$ such that for $0\le l\le m-1$

$$\begin{aligned}&\frac{d}{dt}\left\{ \sum _{l\le k\le m} \Vert \nabla ^{k}(\varrho ,u,\tau )\Vert _{L^{2}}^{2}+\frac{2C_{2}\delta }{C_{3}}\sum _{l\le k\le m-1}\int _{\mathbb {R}^{3}}\nabla ^{k}u\cdot \nabla ^{k+1}\varrho dx\right\} \nonumber \\&\quad +C_{5}\left\{ \sum _{l+1\le k\le m}\Vert \nabla ^{k}\varrho \Vert _{L^{2}}^{2}+\sum _{l+1\le k\le m+1}\Vert \nabla ^{k}u\Vert _{L^{2}}^{2}+\sum _{l\le k\le m}\Vert \nabla ^{k}\tau \Vert _{L^{2}}^{2}\right\} \le 0. \end{aligned}$$

(2.13)

Next, we define $\mathcal {E}^{m}_{l}(t)$ to be $C_{5}^{-1}$ times the expression under the time derivative in (2.13). Observe that since $\delta$ is small, $\mathcal {E}^{m}_{l}(t)$ is equivalent to $\Vert \nabla ^{l}(\varrho ,u,\tau )(t)\Vert _{H^{m-l}}^{2}$, that is, there exists a constant $C_{6}>0$ such that for $0\le l\le m-1$

$$\begin{aligned} C_{6}^{-1}\Vert \nabla ^{l}(\varrho ,u,\tau )(t)\Vert _{H^{m-l}}^{2}\le \mathcal {E}^{m}_{l}(t) \le C_{6}\Vert \nabla ^{l}(\varrho ,u,\tau )(t)\Vert _{H^{m-l}}^{2}. \end{aligned}$$

(2.14)

Then we may write (2.13) as that for $0\le l\le m-1$

$$\begin{aligned} \frac{d}{dt}\mathcal {E}^{m}_{l}(t)+C(\Vert \nabla ^{l+1}\varrho \Vert _{H^{m-l-1}}^{2}+ \Vert \nabla ^{l+1}u\Vert _{H^{m-l}}^{2}+\Vert \nabla ^{l}\tau \Vert _{H^{m-l}}^{2}) \le 0. \end{aligned}$$

(2.15)

Taking $l=0$ and $m=3$ in (2.15), and then integrating directly in time, we get

$$\begin{aligned} \Vert (\varrho ,u,\tau )(t)\Vert _{H^{3}}^{2}\lesssim \mathcal {E}^{3}_{0}(t)\lesssim \mathcal {E}^{3}_{0}(0)\lesssim \Vert (\varrho _{0},u_{0},\tau _{0})\Vert _{H^{3}}^{2}. \end{aligned}$$

(2.16)

By a standard continuity argument, this closes the a priori estimates (2.4). This in turn allows us to take $l=0$ and $m=N$ in (2.15), and then integrate it directly in time to obtain

$$\begin{aligned} \Vert (\varrho ,u,\tau )(t)\Vert _{H^{N}}^{2}+C\int _{0}^{t} (\Vert \nabla \varrho (s)\Vert _{H^{N-1}}^{2}+\Vert \nabla u(s)\Vert _{H^{N}}^{2}+\Vert \tau (s)\Vert _{H^{N}}^{2}) ds \le C\Vert (\varrho _{0},u_{0},\tau _{0})\Vert _{H^{N}}^{2}. \end{aligned}$$

This proved (1.4). $\square$

3 Convergence Rate of the Solution

The aim of this section is to establish the decay rates of the solution stated in Theorem 1.1 under additional assumptions that the initial data belong to $L^{1}$. Firstly, we give the decay rates for the linearized PTT model. Then, we establish the decay rates for the flows (2.1) by the method of Fourier-splitting method and energy estimates. The Cauchy problem to the linearized PTT model is as follows:

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \varrho _t+\mathrm{div}u=0,\\ u_t+\gamma \nabla \varrho -(\triangle u+\nabla \mathrm{div}u)-\mathrm{div}\tau =0,\\ \tau _t+\tau -D(u)=0. \end{array} \right. \end{aligned}$$

(3.1)

Initial data of the system is given as

$$\begin{aligned} (\varrho ,u,\tau )(x,0)=(\varrho _{0}, u_{0},\tau _{0})\rightarrow 0 \ \text{ as } \ |x|\rightarrow \infty . \end{aligned}$$

(3.2)

Let us denote the matrix-valued differential operator associated with (3.1) by

$$\begin{aligned} B:=\left( \begin{array}{cccc} 0 &{} -\mathrm{div} &{}0\\ -\gamma \nabla &{} \triangle +\nabla \mathrm{div} &{} \mathrm{div}\\ 0 &{} \frac{1}{2}(\nabla +\nabla ^{T}) &{} -1 \end{array}\right) , \end{aligned}$$

where $\nabla ^{T}u :=(\nabla u)^T$ for any vector $u\in \mathbb {R}^3$. Then (3.1) can be rewritten as

$$\begin{aligned} U_{t}=BU, \end{aligned}$$

(3.3)

where we have used the notation $U:=(\varrho ,u,\tau )^T.$

Let K(t) be the semigroup defined by $K(t)=e^{tB}, t\ge 0$, then the solution of (3.3) takes the form

$$\begin{aligned} U(t)=K(t)U_{0}, \end{aligned}$$

with $U_{0}:=(\varrho _{0},u_{0},\tau _{0})^T.$ Applying the Fourier energy method to the Cauchy problem (3.1), [42] showed the following $L^{2}$ estimates of $U(t)=K(t)U_{0}$.

Lemma 3.1

Let $m\ge 0$ be an integer. Then for any $t\ge 0$, the solution $U(t)=K(t)U_{0}$ of system (3.1) satisfies

$$\begin{aligned} \Vert \nabla ^{m}K(t)U_{0}\Vert _{L^{2}}\le C(1+t)^{-\frac{3}{4}-\frac{m}{2}}\Vert U_{0}\Vert _{L^{1}\bigcap {H^{m}}}. \end{aligned}$$

(3.4)

Now, we turn to establish the time decay rates for the PTT model (2.1)-(2.2).

Lemma 3.2

Under the assumptions of Theorem 1.1, the global solution $(\varrho ,u,\tau )$ of problem (2.1)–(2.2) satisfies

$$\begin{aligned} \Vert \nabla ^{l}(\varrho ,u,\tau )(t)\Vert _{H^{N-l}}\le C(1+t)^{-\frac{3+2l}{4}}, \ \ \text{ for } \ \ l=0,1. \end{aligned}$$

(3.5)

Proof

Adding $\Vert \nabla ^{l}(\varrho ,u,\tau )(t)\Vert _{L^{2}}^{2}$ to both sides of (2.15) gives

$$\begin{aligned} \frac{d}{dt}\mathcal {E}^{m}_{l}(t)+C\mathcal {E}^{m}_{l}(t)\le \Vert \nabla ^{l}(\varrho ,u,\tau )(t)\Vert _{L^{2}}^{2}. \end{aligned}$$

(3.6)

Taking $l=1$ and $m=N$ in (3.6), we get

$$\begin{aligned} \frac{d}{dt}\mathcal {E}^{N}_{1}(t)+C\mathcal {E}^{N}_{1}(t)\le \Vert \nabla (\varrho ,u,\tau )(t)\Vert _{L^{2}}^{2}. \end{aligned}$$

(3.7)

It follows from Gronwall inequality that

$$\begin{aligned} \mathcal {E}^{N}_{1}(t)\le \mathcal {E}^{N}_{1}(0)e^{-Ct}+\int _{0}^{t} e^{-C(t-\tau )}\Vert \nabla (\varrho ,u,\tau )(\tau )\Vert _{L^{2}}^{2}d\tau . \end{aligned}$$

(3.8)

In order to derive the time decay rate for $\mathcal {E}^{N}_{1}(t)$, we need to control the term $\Vert \nabla (\varrho ,u,\tau )(t)\Vert _{L^{2}}^{2}$.

First, for the nonlinear terms of the model (2.1), employing the Hölder’s inequality, Lemma 5.1 and Lemma 5.2, we get

$$\begin{aligned} \left\{ \begin{array}{l}\displaystyle \Vert (S_{1},S_{2},S_{3})\Vert _{L^{1}\cap L^{2}}\lesssim \delta \Vert \nabla (\varrho ,u,\tau )\Vert _{H^{1}},\\ \Vert \nabla (S_{1},S_{2},S_{3})\Vert _{L^{2}}\lesssim \delta \Vert \nabla (\varrho ,u,\tau )\Vert _{H^{2}}. \end{array} \right. \end{aligned}$$

(3.9)

By Duhamels principle, it holds that

$$\begin{aligned} U(t)=K(t)U_{0}+\int _{0}^{t}K(t-s)G(s)ds,\ t\ge 0, \end{aligned}$$

(3.10)

where $G:=(S_{1},S_{2},S_{3})^{T}$. Then, Together with (3.10), Lemmas 3.1 and 5.4, we have

$$\begin{aligned} \Vert \nabla (\varrho ,u,\tau )\Vert _{L^{2}}^{2}&\le (1+t)^{-\frac{5}{2}}+C\int _{0}^{t} (1+t-s)^{-\frac{5}{2}}\Vert (S_{1},S_{2},S_{3})\Vert _{L^{1}\cap H^1}^{2}ds\nonumber \\&\le C(1+t)^{-\frac{5}{2}}+C\int _{0}^{t} \delta (1+t-s)^{-\frac{5}{2}}\Vert \nabla (\varrho ,u,\tau )\Vert _{H^{2}}^{2}ds\nonumber \\&\le C(1+t)^{-\frac{5}{2}}+C\delta M(t)\int _{0}^{t} (1+t-s)^{-\frac{5}{2}}(1+s)^{-\frac{5}{2}}ds\nonumber \\&\le C(1+t)^{-\frac{5}{2}}+C\delta M(t)(1+t)^{-\frac{5}{2}}\nonumber \\&\le C(1+t)^{-\frac{5}{2}}(1+\delta M(t)), \end{aligned}$$

(3.11)

where $M(t)=\sup \limits _{0\le s\le t}(1+s)^{\frac{5}{2}}\mathcal {E}^{N}_{1}(s).$

Inserting (3.11) into (3.8), it follows

$$\begin{aligned} \mathcal {E}^{N}_{1}(t)&\le \mathcal {E}^{N}_{1}(0)e^{-Ct}+C\int _{0}^{t} e^{-C(t-s)}(1+s)^{-\frac{5}{2}}(1+\delta M(s))ds\nonumber \\&\le \mathcal {E}^{N}_{1}(0)e^{-Ct}+C(1+\delta M(t))\int _{0}^{t} e^{-C(t-s)}(1+s)^{-\frac{5}{2}}ds\nonumber \\&\le \mathcal {E}^{N}_{1}(0)e^{-Ct}+C(1+\delta M(t))(1+t)^{-\frac{5}{2}}\nonumber \\&\le C(1+\delta M(t))(1+t)^{-\frac{5}{2}}, \end{aligned}$$

(3.12)

where we have used the fact:

$$\begin{aligned} \int _{0}^{t} e^{-C(t-s)}(1+s)^{-\frac{5}{2}}ds&=\int _{0}^{\frac{t}{2}} e^{-C(t-s)}(1+s)^{-\frac{5}{2}}ds+\int _{\frac{t}{2}}^{t} e^{-C(t-s)}(1+s)^{-\frac{5}{2}}ds\\&\le e^{-\frac{Ct}{2}}\int _{0}^{\frac{t}{2}} (1+s)^{-\frac{5}{2}}ds+(1+\frac{t}{2})^{-\frac{5}{2}}\int _{\frac{t}{2}}^{t} e^{-C(t-s)}ds\\&\le C(1+t)^{-\frac{5}{2}}. \end{aligned}$$

Noticing the definition of M(t), we get

$$\begin{aligned} M(t) \le C(1+\delta M(t)), \end{aligned}$$

which implies

$$\begin{aligned} M(t)\le C, \end{aligned}$$

(3.13)

since $\delta >0$ is sufficiently small.

Hence, we have the following decay rates

$$\begin{aligned} \Vert \nabla (\varrho ,u,\tau )(t)\Vert _{H^{N-1}}\le C(1+t)^{-\frac{5}{4}}. \end{aligned}$$

(3.14)

On the other hand, by (3.9), (3.10), Lemma 3.1 and Lemma 5.4, it is easy to deduce

$$\begin{aligned} \Vert (\varrho ,u,\tau )\Vert _{L^{2}}^{2}&\le (1+t)^{-\frac{3}{2}}+C\int _{0}^{t} (1+t-\tau )^{-\frac{3}{2}}\Vert (S_{1},S_{2},S_{3})\Vert _{L^{1}\cap L^{2}}^{2}d\tau \nonumber \\&\le C(1+t)^{-\frac{3}{2}}+C\int _{0}^{t} \delta (1+t-\tau )^{-\frac{3}{2}}\Vert \nabla (\varrho ,u,\tau )\Vert _{H^{1}}^{2}d\tau \nonumber \\&\le C(1+t)^{-\frac{3}{2}}+C\int _{0}^{t} (1+t-\tau )^{-\frac{3}{2}}(1+\tau )^{-\frac{5}{2}}d\tau \nonumber \\&\le C(1+t)^{-\frac{3}{2}}+C\int _{0}^{t} (1+t-\tau )^{-\frac{5}{2}}(1+\tau )^{-\frac{3}{2}}d\tau \nonumber \\&\le C(1+t)^{-\frac{3}{2}}, \end{aligned}$$

(3.15)

which together with (3.14) implies (3.5).

Lemma 3.3

Under the assumptions of Theorem 1.1, the global solution $(\varrho ,u,\tau )$ of problem (2.1)–(2.2) satisfies

$$\begin{aligned} \Vert \nabla ^{l}(\varrho ,u,\tau )(t)\Vert _{H^{N-l}}\le C(1+t)^{-\frac{3+2l}{4}}, \ \ \text{ for } \ \ l=0,1, \cdot \cdot \cdot , N-1. \end{aligned}$$

(3.16)

Proof

We are ready to prove (3.16) by induction. When $l=0, 1$, inequality (3.5) has been established in Lemma 3.2, suppose (3.16) holds for the case $l=k-1, \ \text{ and } \ \ k=2,3, \cdot \cdot \cdot , N-1$, that is

$$\begin{aligned} \Vert \nabla ^{k-1}(\varrho ,u,\tau )(t)\Vert _{H^{N-k+1}}\le C(1+t)^{-\frac{1+2k}{4}}. \end{aligned}$$

(3.17)

We need show (3.16) holds for $l=k$. Let $l=k$ and $m=N$ in the estimates (2.15), we have

$$\begin{aligned} \frac{d}{dt}\mathcal {E}^{N}_{k}(t)+C(\Vert \nabla ^{k+1}\varrho (t)\Vert _{H^{N-k-1}}^{2} +\Vert \nabla ^{k+1}(u,\tau )(t)\Vert _{H^{N-k}}^{2})\le 0. \end{aligned}$$

(3.18)

Adding $\Vert \nabla ^{k+1}\varrho (t)\Vert _{L^{2}}^{2}$ to both sides of (3.18) gives

$$\begin{aligned} \frac{d}{dt}\mathcal {E}^{N}_{k}(t)+C(\Vert \nabla ^{k+1}\varrho (t)\Vert _{L^{2}}^{2}+\Vert \nabla ^{k+1}\varrho (t)\Vert _{H^{N-k-1}}^{2} +\Vert \nabla ^{k+1}(u,\tau )(t)\Vert _{H^{N-k}}^{2})\le 0. \end{aligned}$$

(3.19)

As in [35], we define

$$\begin{aligned} S(t)=\{\xi \in \mathbb {R}^{3}:|\xi |\le (\frac{a}{1+t})^{\frac{1}{2}}\}, \end{aligned}$$

for a constant a that will be specified below. Then

$$\begin{aligned} \Vert \nabla ^{k+1}\varrho \Vert _{L^{2}}^{2}&=\int _{\mathbb {R}^{3}}|\xi |^{2(k+1)}|\hat{\varrho }|^{2} d\xi \ge \int _{\mathbb {R}^{3}/S}|\xi |^{2(k+1)}|\hat{\varrho }|^{2}d\xi \\&\ge \frac{a}{1+t}\int _{\mathbb {R}^{3}/S}|\xi |^{2k}|\hat{\varrho }|^{2}d\xi \\&\ge \frac{a}{1+t}\int _{\mathbb {R}^{3}}|\xi |^{2k}|\hat{\varrho }|^{2}d\xi -\frac{a^{2}}{(1+t)^{2}}\int _{S}|\xi |^{2(k-1)}|\hat{\varrho }|^{2}d\xi \\&\ge \frac{a}{1+t}\int _{\mathbb {R}^{3}}|\xi |^{2k}|\hat{\varrho }|^{2}d\xi -\frac{a^{2}}{(1+t)^{2}}\int _{\mathbb {R}^{3}}|\xi |^{2(k-1)}|\hat{\varrho }|^{2}d\xi . \end{aligned}$$

Thus, we have

$$\begin{aligned} \Vert \nabla ^{k+1}\varrho \Vert _{L^{2}}^{2}\ge \frac{a}{1+t}\Vert \nabla ^{k}\varrho \Vert _{L^{2}}^{2} -\frac{a^{2}}{(1+t)^{2}}\Vert \nabla ^{k-1}\varrho \Vert _{L^{2}}^{2}. \end{aligned}$$

(3.20)

Similarly, one has

$$\begin{aligned} \Vert \nabla ^{k+1}(u,\tau )\Vert _{L^{2}}^{2}\ge \frac{a}{1+t}\Vert \nabla ^{k}(u,\tau )\Vert _{L^{2}}^{2} -\frac{a^{2}}{(1+t)^{2}}\Vert \nabla ^{k-1}(u,\tau )\Vert _{L^{2}}^{2}. \end{aligned}$$

(3.21)

Summing up the estimates (3.21) for k from k to N, one has

$$\begin{aligned} \Vert \nabla ^{k+1}(u,\tau )\Vert _{H^{N-k}}^{2}\ge \frac{a}{1+t}\Vert \nabla ^{k}(u,\tau ) \Vert _{H^{N-k}}^{2}-\frac{a^{2}}{(1+t)^{2}}\Vert \nabla ^{k-1}(u,\tau )\Vert _{H^{N-k}}^{2}. \end{aligned}$$

(3.22)

Substituting the inequalities (3.20),(3.22) into (3.19), applying (3.17), it follows

$$\begin{aligned}&\quad \frac{d}{dt}\mathcal {E}^{N}_{k}(t)+\frac{Ca}{1+t}(\Vert \nabla ^{k}\varrho (t)\Vert _{L^{2}}^{2} +\Vert \nabla ^{k+1}\varrho (t)\Vert _{H^{N-k-1}}^{2} +\Vert \nabla ^{k}(u,\tau )\Vert _{H^{N-k}}^{2})\nonumber \\&\le \frac{Ca^{2}}{(1+t)^{2}}(\Vert \nabla ^{k-1}\varrho (t)\Vert _{L^{2}}^{2} +\Vert \nabla ^{k-1}(u,\tau )\Vert _{H^{N-k}}^{2})\nonumber \\&\le C(1+t)^{-\frac{5+2k}{2}}, \end{aligned}$$

where we have used

$$\begin{aligned} \frac{a}{1+t}\Vert \nabla ^{k+1}\varrho (t)\Vert _{H^{N-k-1}}^{2}\le \Vert \nabla ^{k+1}\varrho (t)\Vert _{H^{N-k-1}}^{2}, \end{aligned}$$

for some sufficiently large time $t\ge a-1$ , such that $\frac{a}{1+t}\le 1$.

This, together with the definition of $\mathcal {E}^{N}_{k}(t)$, implies that

$$\begin{aligned} \frac{d}{dt}\mathcal {E}^{N}_{k}(t)+\frac{Ca}{1+t}\mathcal {E}^{N}_{k}(t)\le C(1+t)^{-\frac{5+2k}{2}}. \end{aligned}$$

(3.23)

Choosing

$$\begin{aligned} a=\frac{k+2}{C}, \end{aligned}$$

and multiplying both sides of (3.23) by $(1+t)^{k+2}$, we get

$$\begin{aligned} \frac{d}{dt}[(1+t)^{k+2}\mathcal {E}^{N}_{k}(t)]\le C(1+t)^{-\frac{1}{2}}. \end{aligned}$$

(3.24)

Solving the inequality directly yields

$$\begin{aligned} \Vert \nabla ^{k}(\varrho ,u,\tau )(t)\Vert _{H^{N-k}}^{2}\le C(1+t)^{-\frac{3+2k}{2}}. \end{aligned}$$

Hence, we have verified that (3.16) holds on for the case $l=k$, this concludes the proof of the lemma.

Proof of 1.1

With the help of Lemmas 3.2 and 3.3, it is easy to obtain the conclusion (1.5). $\square$

Proof of Theorem 1.3

First of all, we shall estimate $\Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}$. For $l=0,1,\cdot \cdot \cdot ,N-2$, applying $\nabla ^{l}$ to (2.1)$_{1}$, multiplying the resulting identities by $\nabla ^{l}\varrho _{t}$ and integrating the resulting equation over $\mathbb {R}^{3}$, one gets

$$\begin{aligned} \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2}=-\int _{\mathbb {R}^{3}}\nabla ^{l}(\varrho \mathrm{div}u+u\cdot \nabla \varrho +\mathrm{div}u)\cdot \nabla ^{l}\varrho _{t} dx =K_{1}+K_{2}+K_{3}. \end{aligned}$$

(3.25)

We shall estimate each term in the right hand side of (3.25). First, for the term $K_{1}$, by Lemma 5.2 and Young’s inequality, we obtain

$$\begin{aligned} K_{1}&\lesssim (\Vert \nabla ^{l}\varrho \Vert _{L^{6}}\Vert \nabla u\Vert _{L^{3}}+ \Vert \varrho \Vert _{L^{\infty }}\Vert \nabla ^{l+1} u\Vert _{L^{2}})\Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}} \nonumber \\&\le \varepsilon \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2}+C\Vert \nabla ^{l} \varrho \Vert _{L^{6}}^{2}\Vert \nabla u\Vert _{L^{3}}^{2}+ C\Vert \varrho \Vert _{L^{\infty }}^{2}\Vert \nabla ^{l+1} u\Vert _{L^{2}}^{2}\nonumber \\&\le \varepsilon \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2}+C(1+t)^{-\frac{5+2l}{2}}(1+t)^{-3} +C(1+t)^{-3}(1+t)^{-\frac{5+2l}{2}}\nonumber \\&\le \varepsilon \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2}+C(1+t)^{-\frac{11+2l}{2}}, \end{aligned}$$

(3.26)

where $\varepsilon$ is small enough.

Similarly, we can bound

$$\begin{aligned} K_{2} \le \varepsilon \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2}+C(1+t)^{-\frac{11+2l}{2}}. \end{aligned}$$

(3.27)

Using Young’s inequality, $K_{3}$ can be estimated as follows:

$$\begin{aligned} K_{3} \le \varepsilon \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2}+C\Vert \nabla ^{l+1}u\Vert _{L^{2}}^{2} \le \varepsilon \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2}+C(1+t)^{-\frac{5+2l}{2}}. \end{aligned}$$

(3.28)

Combining (3.26)-(3.28), we deduce from (3.25) that

$$\begin{aligned} \Vert \nabla ^{l}\varrho _{t}\Vert _{L^{2}}^{2} \lesssim (1+t)^{-\frac{5+2l}{2}}. \end{aligned}$$

(3.29)

Then we shall estimate $\Vert \nabla ^{l}u_{t}\Vert _{L^{2}}$. For $l=0,1,\cdot \cdot \cdot ,N-2$, applying $\nabla ^{l}$ to (2.1)$_{2}$, multiplying the resulting identities by $\nabla ^{l}u_{t}$ and integrating the resulting equation over $\mathbb {R}^{3}$, and using Young’s inequality, one gets

$$\begin{aligned} \Vert \nabla ^{l}u_{t}\Vert _{L^{2}}^{2}&=\int _{\mathbb {R}^{3}}\nabla ^{l} [-u\cdot \nabla u+\frac{1}{\varrho +1}(\triangle u+\nabla \mathrm{div}u+\mathrm{div})-g(\varrho )\nabla \varrho -\gamma \nabla \varrho ]\cdot \nabla ^{l}u_{t} dx\nonumber \\&\lesssim \varepsilon \Vert \nabla ^{l}u_{t}\Vert _{L^{2}}^{2}+\Vert \nabla ^{l} [-u\cdot \nabla u+h(\varrho )(\triangle u+\nabla \mathrm{div}u+\mathrm{div})]\Vert _{L^{2}}^{2}+\Vert \nabla ^{l}[-g(\varrho )\nabla \varrho -\gamma \nabla \varrho ]\Vert _{L^{2}}^{2}\nonumber \\&=\varepsilon \Vert \nabla ^{l}u_{t}\Vert _{L^{2}}^{2}+X_{1}+X_{2}, \end{aligned}$$

(3.30)

where we have used the notation $h(\varrho ):=\frac{1}{\varrho +1}.$ Then from (2.4) and Sobolev’s inequality, we obtain

$$\begin{aligned} |h^{(l)}(\varrho )|\le C \ \ \text{ for } \text{ any } \ \ l\ge 0. \end{aligned}$$

(3.31)

This, together with Lemma 5.2, the second factor in the inequality (3.30) can be estimated as follows:

$$\begin{aligned} X_{1}&\lesssim \Vert \nabla ^{l}u\Vert _{L^{3}}^{2}\Vert \nabla u\Vert _{L^{6}}^{2}+ \Vert u\Vert _{L^{\infty }}^{2}\Vert \nabla ^{l+1} u\Vert _{L^{2}}^{2}+\Vert \nabla ^{l}h(\varrho )\Vert _{L^{6}}^{2}\Vert \nabla ^{2}u\Vert _{L^{3}}^{2} +\Vert h(\varrho )\Vert _{L^{\infty }}^{2}\Vert \nabla ^{l+2}u\Vert _{L^{2}}^{2}\nonumber \\&\lesssim (1+t)^{-2-l}(1+t)^{-\frac{7}{2}}+(1+t)^{-3}(1+t)^{-\frac{5+2l}{2}}+(1+t)^{-\frac{5+2l}{2}}(1+t)^{-4}+(1+t)^{-\frac{3+2(l+1)}{2}} \nonumber \\&\lesssim (1+t)^{-\frac{5+2l}{2}}. \end{aligned}$$

(3.32)

Similar to the estimate of the term $X_{1}$, for the term $X_{2}$, we have

$$\begin{aligned} X_{2}&\lesssim \Vert \nabla ^{l}g(\varrho )\Vert _{L^{6}}^{2}\Vert \nabla \varrho \Vert _{L^{3}}^{2}+ \Vert g(\varrho )\Vert _{L^{6}}^{2}\Vert \nabla ^{l+1} \varrho \Vert _{L^{3}}^{2} +\Vert \nabla ^{l+1}\varrho \Vert _{L^{2}}^{2}\nonumber \\&\lesssim (1+t)^{-\frac{5+2l}{2}}(1+t)^{-3}+(1+t)^{-\frac{5}{2}}(1+t)^{-3-l} +(1+t)^{-\frac{5+2l}{2}}\nonumber \\&\lesssim (1+t)^{-\frac{5+2l}{2}}. \end{aligned}$$

(3.33)

Combining (3.32)–(3.33), we deduce from (3.30) that

$$\begin{aligned} \Vert \nabla ^{l}u_{t}\Vert _{L^{2}}^{2} \lesssim (1+t)^{-\frac{5+2l}{2}}. \end{aligned}$$

(3.34)

Similar to the estimate of the term $\Vert \nabla ^{l}u_{t}\Vert _{L^{2}}$, For $l=0,1,\cdot \cdot \cdot ,N-2$, applying $\nabla ^{l}$ to (2.1)$_{3}$, multiplying the resulting identities by $\nabla ^{l}\tau _{t}$ and integrating the resulting equation over $\mathbb {R}^{3}$, and using Young’s inequality, one gets

$$\begin{aligned} \Vert \nabla ^{l}\tau _{t}\Vert _{L^{2}}^{2}&=\int _{\mathbb {R}^{3}}\nabla ^{l} [-u\cdot \nabla \tau +-Q(\tau ,\nabla u)-tr\tau \tau -\tau +D(u)]\cdot \nabla ^{l}\tau _{t} dx\nonumber \\&\le \varepsilon \Vert \nabla ^{l}\tau _{t}\Vert _{L^{2}}^{2}+C\Vert \nabla ^{l} [-u\cdot \nabla \tau +-Q(\tau ,\nabla u)-tr\tau \tau -\tau +D(u)]\Vert _{L^{2}}^{2}\nonumber \\&\le \varepsilon \Vert \nabla ^{l}\tau _{t}\Vert _{L^{2}}^{2}+C\Vert \nabla ^{l}u\Vert _{L^{3}}^{2}\Vert \nabla \tau \Vert _{L^{6}}^{2}+ C\Vert u\Vert _{L^{\infty }}^{2}\Vert \nabla ^{l+1} \tau \Vert _{L^{2}}^{2}+C\Vert \tau \Vert _{L^{\infty }}^{2}\Vert \nabla ^{l+1}u\Vert _{L^{2}}^{2} \nonumber \\&\quad +C\Vert \nabla ^{l}\tau \Vert _{L^{3}}^{2}\Vert \nabla u\Vert _{L^{6}}^{2} +C\Vert \nabla ^{l}tr\tau \Vert _{L^{3}}^{2}\Vert \tau \Vert _{L^{6}}^{2} +C\Vert tr\tau \Vert _{L^{\infty }}^{2}\Vert \nabla ^{l+1}\tau \Vert _{L^{2}}^{2} +C\Vert \nabla ^{l}\tau \Vert _{L^{2}}^{2}+C\Vert \nabla ^{l+1}u\Vert _{L^{2}}^{2}\nonumber \\&\le \varepsilon \Vert \nabla ^{l}\tau _{t}\Vert _{L^{2}}^{2}+C(1+t)^{-2-l} (1+t)^{-\frac{7}{2}}+C(1+t)^{-3}(1+t)^{-\frac{5+2l}{2}}\nonumber \\&\quad +C(1+t)^{-3}(1+t)^{-\frac{5+2l}{2}}+C(1+t)^{-2-l}(1+t)^{-\frac{7}{2}} +C(1+t)^{-2-l}(1+t)^{-\frac{5}{2}}\nonumber \\&\quad +C(1+t)^{-3}(1+t)^{-\frac{5+2l}{2}} +C(1+t)^{-\frac{3+2l}{2}}+C(1+t)^{-\frac{5+2l}{2}}\nonumber \\&\le \varepsilon \Vert \nabla ^{l}\tau _{t}\Vert _{L^{2}}^{2}+C(1+t)^{-\frac{3+2l}{2}}. \end{aligned}$$

(3.35)

Then, we have

$$\begin{aligned} \Vert \nabla ^{l}\tau _{t}\Vert _{L^{2}}^{2} \lesssim (1+t)^{-\frac{3+2l}{2}}. \end{aligned}$$

(3.36)

Thus, the proof of Theorem 1.3 is completed. $\square$

Appendix A. Analytic Tools

We will extensively use the Sobolev interpolation of the Gagliardo-Nirenberg inequality.

Lemma 5.1

Let $0\le m,\alpha \le l$, then we have

$$\begin{aligned} \Vert \nabla ^{\alpha }f\Vert _{L^{p}}\lesssim \Vert \nabla ^{m}f\Vert _{L^{q}}^{1-\theta }\Vert \nabla ^{l}f\Vert _{L^{r}}^{\theta }. \end{aligned}$$

(5.1)

where $0\le \theta \le 1$ and $\alpha$ satisfies

$$\begin{aligned} \frac{\alpha }{3}-\frac{1}{p}=\left( \frac{m}{3}-\frac{1}{q}\right) (1-\theta )+\left( \frac{l}{3}-\frac{1}{r}\right) \theta . \end{aligned}$$

(5.2)

Here when $p=\infty$ we require that $0<\theta <1$.

Proof

This can be found in [30, p. 125, Theorem]. $\square$

We recall the following commutator estimate:

Lemma 5.2

Let $m\ge 1$ be an integer and define the commutator

$$\begin{aligned}{}[\nabla ^{m},f]g=\nabla ^{m}(fg)-f\nabla ^{m}g, \end{aligned}$$

then we have

$$\begin{aligned} \Vert [\nabla ^{m},f]g\Vert _{L^{p}}\lesssim \Vert \nabla f\Vert _{L^{p_{1}}}\Vert \nabla ^{m-1}g\Vert _{L^{p_{2}}}+\Vert \nabla ^{m}f\Vert _{L^{p_{3}}}\Vert g\Vert _{L^{p_{4}}}. \end{aligned}$$

(5.3)

and for $m\ge 0$

$$\begin{aligned} \Vert \nabla ^{m}(fg)\Vert _{L^{p}}\lesssim \Vert f\Vert _{L^{p_{1}}}\Vert \nabla ^{m}g\Vert _{L^{p_{2}}}+\Vert \nabla ^{m}f\Vert _{L^{p_{3}}}\Vert g\Vert _{L^{p_{4}}}. \end{aligned}$$

(5.4)

where $p, p_{2}, p_{3}\in (1, \infty )$ and $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}.$

Proof

For $p=p_{2}=p_{3}=2$, it can be proved by using Lemma 5.1. For the general cases, one may refer to [19, Lemma 3.1]. $\square$

We should now recall the following elementary but useful inequality.

Lemma 5.3

Assume that $\Vert \varrho \Vert _{L^{\infty }}\le 1$ and $p>1$. Let $g(\varrho )$ be a smooth function of $\varrho$ with bounded derivatives of any order, then for any integer $m\ge 1$, we have

$$\begin{aligned} \Vert \nabla ^{m}g(\varrho )\Vert _{L^{p}}\lesssim \Vert \nabla ^{m}\varrho \Vert _{L^{p}}. \end{aligned}$$

(5.5)

Proof

The proof is similar to the proof of Lemma A.2 in [38] and is omitted here. $\square$

Lemma 5.4

[6] If $r_1>1$ and $r_2\in [0, r_1]$, then it holds that

$$\begin{aligned} \int _{0}^{t} (1+t-s)^{-r_1}(1+s)^{-r_2}ds\le C(r_1,r_2)(1+t)^{-r_2}. \end{aligned}$$

(5.6)

Lemma 5.5

[1] Let $f\in H^2(\mathbb {R}^{3})$. Then we have

$$\begin{aligned}&\Vert f\Vert _{L^{\infty }}\lesssim \Vert \nabla f\Vert _{L^{2}}^{\frac{1}{2}}\Vert \nabla f\Vert _{H^{1}}^{\frac{1}{2}} \lesssim \Vert \nabla f\Vert _{H^{1}},\\&\Vert f\Vert _{L^{6}}\lesssim \Vert \nabla f\Vert _{L^{2}},\\&\Vert f\Vert _{L^{q}}\lesssim \Vert f\Vert _{H^{1}},\ 2\le q\le 6. \end{aligned}$$