Asymptotic profiles and convergence rate of the compressible fluid models of Korteweg type

Abstract

In this paper, we consider the initial value problem for the compressible fluid models of Korteweg type in ${\mathbb {R}}^n(n\ge 3)$ and asymptotic profile of global solutions and the corresponding convergence rate are established. The structure of the nonlinear term plays a very important role in constructing asymptotic profile. The proof is based on the decay estimate of solutions operator, decay estimate and weighted decay estimate of global solutions.

1 Introduction

This paper concerns the initial value problem for the compressible fluid models of Korteweg type

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t\rho + \nabla \cdot m=0,\\ \partial _t m+\nabla \cdot ( \frac{m\otimes m}{\rho } )+\nabla P(\rho )- \mu \Delta \frac{m}{\rho }-(\mu +\lambda )\nabla \nabla \cdot (\frac{m}{\rho })=\alpha \rho \nabla \Delta \rho \end{array}\right. \end{aligned}$$

(1.1)

with the initial condition

$$\begin{aligned} t=0: \rho =\rho _0(x),\;\; m=m_0(x), \;\; x\in {\mathbb {R}}^n. \end{aligned}$$

(1.2)

The variables are the density $\rho$ and the momentum m. Furthermore, $p = p(\rho )$ is the pressure function satisfying $P'(\rho ) > 0$ for $\rho > 0$. The viscosity coefficients satisfy $\mu> 0, 2\mu +n\lambda > 0$, $\alpha >0$.

This compressible fluid model of Korteweg type describes the dynamics of a liquid–vapor mixture in the setting of the diffuse interface approach: between the two phases lies a thin region of continuous transition and the phase changes are described through the variations of the density, for example a Van der Waals pressure. The compressible fluid model of Korteweg type was derived rigorously by Dunn and Serrin [4] (see also [2, 5]).

Let $u=\frac{m}{\rho }$, then (1.1) may be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t\rho + \nabla \cdot (\rho u)=0,\\ \partial _t (\rho u)+\nabla \cdot ( \rho u\otimes u)+\nabla P(\rho )- \mu \Delta u-(\mu +\lambda )\nabla (\nabla \cdot u)=\alpha \rho \nabla \Delta \rho . \end{array}\right. \end{aligned}$$

(1.3)

There are numerous works dedicated to the study of the compressible fluid models of Korteweg type, and lots of important results were established. For well-posedness results, we refer to [1, 3, 6, 7, 9, 15]. Global existence of classical solutions in Sobolev space was established by [9]. Danchin and Desjardins [3] proved that global well-posedness in the critical Besov spaces for the initial data is close enough to stable equilibria. Moreover, local existence of solutions for initial densities bounded away from zero was also established. Bresch et al. [1] and Haspot [6] proved the global existence of weak solutions for the compressible Navier–Stokes–Korteweg system, respectively. Global strong solutions to the compressible Navier–Stokes–Korteweg system in two space dimensions have been proved in [7].

For decay estimate results of global solutions, we may refer to [12, 20,21,22, 27, 28]. Wang and Tan [27] established the $L^2$ and $L^p(p\ge 2)$ decay rates for the classical solutions by the detailed study of the linear decay estimates and nonlinear energy estimates. For the global existence and decay estimate of strong solutions, please refer to [20]. Tan and Zhang [22] obtained the faster decay estimate by assuming suitable condition on the initial value. For more decay estimates results, we may refer to [12, 21, 28]. The optimal decay estimates of global mild solutions in the critical nonhomogeneous Besov spaces to the problem (1.1), (1.2) were established in [25], provided that the initial perturbations of density and velocity are small in the space $B^{\frac{n}{2}}_{2, 1}\bigcap \dot{B}^{0}_{1, \infty }$ and $B^{\frac{n}{2}-1}_{2, 1}\bigcap \dot{B}^{0}_{1, \infty }$.

Our main purpose of this paper is to investigate asymptotic profile of global solutions obtained by Hattori and Li [9] to the problem (1.1), (1.2) and the corresponding convergence rate in the sprit of [14]. The nonlinear term plays a very important role in asymptotic profile of global solutions obtained in this paper. For the details, we refer to the following Theorem 1.3. The proof is based mainly on the decay estimate of solutions operator in the low-frequency region, the high-frequency region, the structure of the nonlinear term and global solutions obtained in [9] and decay estimate of global solutions established by Wang and Tan [27].

To state our asymptotic profile and the corresponding convergence rate results, we firstly state global solution and decay estimate results established by Hattori and Li [9] and Wang and Tan [27], respectively, as follows:

Theorem 1.1

( [9]) Let $n\ge 3$ and $s=[\frac{n}{2}]+1$ be an integer. Assume that $\rho _0-1\in H^{s+1}$, $m_0 \in H^s$. Put

$$\begin{aligned} E_0=\Vert \rho _0-1\Vert _{H^{s+1}}+\Vert m_0\Vert _{H^s}. \end{aligned}$$

Then there is a positive constant $\delta _0$ such that if $E_0\le \delta _0$, then the problem (1.3), (1.2) has a unique global solution $(\rho , u)$ satisfying

(1.4)

Theorem 1.2

( [27]) Let $n\ge 3$ and $s=[\frac{n}{2}]+2$ be an integer. Assume that $\rho _0-1\in H^{s+1}$, $m_0 \in H^s$ and $L^1$ norm of $(\rho _0-1, m)$ is finite. Let $(\rho , u)$ be the global solution to the problem (1.3), (1.2) obtained in Theorem 1.1. Then

$$\begin{aligned} \Vert (\rho -1, u)(t)\Vert _{L^p}\le & {} C(\Vert \rho _0-1\Vert _{H^{s+1}\bigcap L^1} \nonumber \\&+\Vert m_0\Vert _{H^{s}\bigcap L^1})(1+t)^{-\frac{n}{2}(1-\frac{1}{p})}, \;\; \forall 2\le p\le \infty . \end{aligned}$$

(1.5)

$$\begin{aligned} \Vert \partial _x(\rho -1, u)(t)\Vert _{L^p}\le & {} C(\Vert \rho _0-1\Vert _{H^{s+1}\bigcap L^1} \nonumber \\&+\Vert m_0\Vert _{H^{s}\bigcap L^1})(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{1}{2}}, \;\; \forall 2\le p\le 6. \end{aligned}$$

(1.6)

and

$$\begin{aligned} \Vert \partial _x(\rho -1)(t)\Vert _{H^s}+\Vert \partial _x u(t)\Vert _{H^{s-1}} \le C(\Vert \rho _0-1\Vert _{H^{s+1}\bigcap L^1}+\Vert m_0\Vert _{H^{s}\bigcap L^1})(1+t)^{-\frac{n}{4}-\frac{1}{2}}. \end{aligned}$$

(1.7)

Let $U=(\rho -1, m)^\tau =(\sigma , m)^\tau$. We state our asymptotic profile of global solutions obtained by Hattori and Li [9] and the corresponding convergence rate as follows:

Theorem 1.3

Let $n\ge 3$, $s=[n/2]+2$ and $\kappa = \frac{1}{2}$ when $n=3$ and $\kappa = 1$ when $n\ge 4$. Assume that $\rho _0-1\in H^{s+1}\cap L^1_\kappa \bigcap L^2_\kappa$, $m_0\in H^{s}\cap L^1_\kappa \bigcap L^2_\kappa$ and $\partial _x \rho _0 \in L^2_\kappa$. Put

$$\begin{aligned} E_1=\Vert \rho _0-1\Vert _{H^{s+1}\bigcap L^1}+\Vert m_0\Vert _{H^{s}\bigcap L^1}. \end{aligned}$$

Let $(\rho , m)$ be the global solution to the problem (1.1), (1.2) obtained in Theorem 1.1. If $E_1$ is suitable small, then it holds that

$$\begin{aligned}&{\displaystyle \Big \Vert U(t)-G(t)*U_0- \sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_0\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau \Big \Vert _{L^2}} \nonumber \\&\quad \le {\displaystyle C\left\{ \begin{array}{ll} (1+t)^{-\frac{n}{4}-\frac{1}{2}}, &{} n=3, 4 \\ (1+t)^{-\frac{n}{4}-1}, &{} n\ge 5 \\ \end{array}\right. } \end{aligned}$$

(1.8)

where

$$\begin{aligned} {\mathbb {F}}_i(U)=\left( \begin{array}{c} 0 \\ \overline{F}_{i}(U) \\ \end{array} \right) \end{aligned}$$

(1.9)

and $\overline{F}_{i}(U)$ is the i-th column vector of the matrix $\overline{F}_{ij}(U)$ defined by

$$\begin{aligned} \overline{F}_{ij}(U)=-\frac{m_im_j}{\sigma +1}-[P(1+\sigma )-P'(1)\sigma ]\delta _{ij}-\frac{\alpha }{2}|\nabla \sigma |^2\delta _{ij}-\alpha \partial _{x_i}\sigma \partial _{x_j}\sigma \end{aligned}$$

(1.10)

and $G_l(t)$ is defined by (2.8).

Remark 1.4

The result in Theorem 1.3 implies that the solutions to the problem (1.1), (1.2) are asymptotic to a new asymptotic profile, which is given by the nonlinear term in $\overline{F}_{ij}(U)$. In fact, the corresponding decay rate of the other nonlinear term in $F_{ij}$ (see (2.2)) is much faster.

The paper is organized as follows. We make the detail analysis for solution operators to (1.1), (1.2) in Sect. 2. In Sect. 3, weighted decay estimate of global solutions to the problem (1.3), (1.2) is established. Section 4 is devoted to derive the asymptotic profile of global solutions to the problem (1.1), (1.2) and the corresponding convergence rate.

Notations Let ${\mathcal {F}}[f]$ denote the Fourier transform of f defined by

$$\begin{aligned} \widehat{f}(\xi )={\mathcal {F}}[f](\xi ) :=\int \limits _{{\mathbb {R}}^n}e^{-i\xi \cdot x}f(x)dx. \end{aligned}$$

We denote its inverse transform by ${\mathcal {F}}^{-1}$. $L^p=L^p({\mathbb {R}}^n)(1\le p\le \infty )$ denotes the usual Lebesgue space with the norm $\Vert \cdot \Vert _{L^p}$. $L^p_\kappa =L^p_\kappa ({\mathbb {R}}^n)(1\le p< \infty )$ denotes the weighted Lebesgue space with the norm

$$\begin{aligned} \Vert f\Vert _{L^p_\kappa }=\left( \,\int \limits _{{\mathbb {R}}^n}|(1+|x|^\kappa )f(x)|^pdx\right) ^{\frac{1}{p}}<\infty . \end{aligned}$$

The usual Sobolev space of order s is defined by $H^{s}=(I-\partial ^2_x)^{-\frac{s}{2}}L^2$ with the norm $\Vert f\Vert _{H^{s}}=\Vert (I-\partial ^2_x)^{\frac{s}{2}}f\Vert _{L^2}$.

For a nonnegative integer k, $\partial _x^k$ denotes the totality of all the k-th order derivatives with respect to $x\in {\mathbb {R}}^n$. $\partial _t^l$ denotes the totality of all the l-th order derivatives with respect to $t \in {\mathbb {R}}_{+}$. Also, for an interval I and a Banach space X, $C^k(I;X)$ denotes the space of k-times continuously differential functions on I with values in X.

2 Decay properties of solution operators

This section is devoted to derive the solution operators to the compressible fluid models of Korteweg type (1.1), (1.2). To do so, let $\sigma =\rho -1$ and $P'(1) =1$. (1.1) may be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \sigma + \nabla \cdot m=0,\\ \partial _t m+\nabla \sigma - \mu \Delta m-(\mu +\lambda )\nabla (\nabla \cdot m)-\alpha \nabla \Delta \sigma =\nabla \cdot F,\\ \end{array}\right. \end{aligned}$$

(2.1)

where F is the $n \times n$ matrix $F_{ij}$ defined by

$$\begin{aligned} \begin{array}{lll} {\displaystyle F_{ij}} &{}=&{} {\displaystyle - \frac{m_i m_j}{\sigma +1}-\delta _{ij}\mu \nabla \cdot \left( \frac{\sigma m}{\sigma +1}\right) -(\mu +\lambda ) \partial _{x_j}\left( \frac{\sigma m_i}{\sigma +1} \right) -\delta _{ij}[P(\sigma +1)-P'(1) \sigma ]}\\ &{}&{}{\displaystyle +\frac{\alpha }{2}(\Delta \sigma ^2-|\nabla \sigma |^2)\delta _{ij}-\alpha \partial _{x_i}\sigma \partial _{x_j}\sigma } \end{array} \end{aligned}$$

(2.2)

with the i-th component of $\nabla \cdot F$ given by ${\displaystyle \sum ^n_{j=1}\partial _{x_j}F_{ij}}$.

Let $U=(\sigma , m)$, then (1.1) may be written as

$$\begin{aligned} \partial _t U-AU=\nabla \cdot {\mathfrak {F}}(U), \end{aligned}$$

(2.3)

where

$$\begin{aligned} A=\left( \begin{array}{cc} 0 &{} -\nabla \cdot \\ -\nabla +\alpha \nabla \Delta &{} \mu \Delta +(\mu +\lambda )\nabla \nabla \cdot \\ \end{array} \right) \end{aligned}$$

and ${\mathfrak {F}}(U)=(0, F(U))^{\tau }$.

Let $G(t)*$ be the solution operators to the compressible fluid models of Korteweg type (1.1). Then Fourier transform of G(t, x) is given by

$$\begin{aligned} \widehat{G}(t, \xi )= \left( \begin{array}{ll} \widehat{G}^{11} &{}\quad \widehat{G}^{12} \\ \widehat{G}^{21} &{}\quad \widehat{G}^{22} \\ \end{array} \right) \end{aligned}$$

(2.4)

with

$$\begin{aligned} \left\{ \begin{array}{l} \widehat{G}^{11}=\hat{{\mathcal {H}}}, \;\; \widehat{G}^{12}=-i\xi ^\tau \hat{{\mathcal {G}}}, \\ \widehat{G}^{21}=-i(1+|\xi |^2)\xi \widehat{{\mathcal {G}}}, \;\; \widehat{G}^{22}=(2\mu +\lambda )\xi \xi ^\tau \widehat{{\mathcal {G}}} +\widehat{{\mathcal {H}}}I_{n\times n}, \\ \widehat{{\mathcal {G}}}(\xi ,t)= \frac{e^{\lambda _{+}t}-e^{\lambda _{-}t}}{\lambda _{+}-\lambda _{-}}, \\ \widehat{{\mathcal {H}}}(\xi , t)=\frac{\lambda _{+}e^{\lambda _{-}t}-\lambda _{-}e^{\lambda _{+}t}}{\lambda _{+}-\lambda _{-}} \end{array}\right. \end{aligned}$$

(2.5)

and

$$\begin{aligned} \lambda _{\pm }(\xi )=\frac{ -(2\mu +\lambda )|\xi |^2\pm \sqrt{(2\mu +\lambda )^2|\xi |^4-4(|\xi |^2+\alpha |\xi |^4)}}{2}. \end{aligned}$$

(2.6)

Due to Duhamel principle, the solution to the problem (1.1), (1.2) may expressed as

$$\begin{aligned} U(t)=G(t)*\left( \begin{array}{c} \sigma _0\\ m_0\\ \end{array} \right) +\int \limits ^t_0G(t-\tau )*\nabla \cdot {\mathfrak {F}}(U)(\tau )d\tau . \end{aligned}$$

(2.7)

Let ${\mathcal {F}} \widehat{\phi } \in C^\infty ({\mathbb {R}}^n)$ such that

$$\begin{aligned} {\mathcal {F}} \widehat{\phi } = \left\{ \begin{array}{l} {\displaystyle 1, \quad |\xi |< \epsilon }\\ {\displaystyle 0, \quad |\xi |\ge 2\epsilon } \\ \end{array}\right. \end{aligned}$$

where $0< \epsilon < 1$ is a constant. Set

$$\begin{aligned} G_l=\phi *G, \;\; G_h=G-\phi *G=G-G_l. \end{aligned}$$

(2.8)

The above Fourier splitting frequency technique was early introduced in [13] and so on. Then (2.7) implies

$$\begin{aligned} U(t)=U_l(t)+U_h(t), \end{aligned}$$

(2.9)

where

$$\begin{aligned} U_l(t)=G_l(t)* U_0+\int \limits ^t_0 G_l(t-\tau )*\nabla \cdot {\mathfrak {F}}(U)(\tau )d\tau \end{aligned}$$

(2.10)

and

$$\begin{aligned} U_h(t)=G_h(t)* U_0+\int \limits ^t_0G_h(t-\tau )*\nabla \cdot {\mathfrak {F}}(U)(\tau )d\tau . \end{aligned}$$

(2.11)

$\widehat{{\mathcal {G}}}_l(t, \xi )$ and $\widehat{{\mathcal {H}}}_l(t, \xi )$ may be written

$$\begin{aligned} \widehat{{\mathcal {G}}}_l(t, \xi )=\frac{1}{|\xi |\vartheta (\xi )}e^{-\frac{2\mu +\lambda }{2}|\xi |^2t}\sin (|\xi |\vartheta (\xi ) t) \end{aligned}$$

(2.12)

and

$$\begin{aligned} \widehat{{\mathcal {H}}}_l(t, \xi ) = \frac{(2\mu +\lambda )|\xi |}{4 \vartheta (\xi )}e^{-\frac{ 2\mu +\lambda }{2}|\xi |^2t} \sin (|\xi |\vartheta (\xi ) t)+e^{-\frac{ 2\mu +\lambda }{2}|\xi |^2t} \cos (|\xi |\vartheta (\xi ) t), \end{aligned}$$

(2.13)

respectively, where

$$\begin{aligned} \vartheta (\xi )=\sqrt{1+(\alpha -\frac{(2\mu +\lambda )^2}{4})|\xi |^2}. \end{aligned}$$

Due to Taylor formula, it is not difficult to find that

$$\begin{aligned} \widehat{{\mathcal {G}}}_l(t, \xi )=\frac{(1+O(|\xi |^2))}{|\xi |}e^{-\frac{2\mu +\lambda }{2}|\xi |^2t}\sin (|\xi |+O(|\xi |^3))t \end{aligned}$$

(2.14)

and

$$\begin{aligned} \begin{array}{lll} {\displaystyle \widehat{{\mathcal {H}}}_l(t, \xi )} &{}=&{}{\displaystyle \frac{(2\mu +\lambda ) |\xi |}{4}(1+O(|\xi |^2)) e^{-\frac{2\mu +\lambda }{2}|\xi |^2t}\sin (|\xi |+O(|\xi |^3))t}\\ &{}&{}{\displaystyle +e^{-\frac{2\mu +\lambda }{2}|\xi |^2t}\cos (|\xi |+O(|\xi |^3))t.}\\ \end{array} \end{aligned}$$

(2.15)

We state the pointwise estimates for the solution operators $\hat{{\mathcal {G}}}$ and $\hat{{\mathcal {H}}}$ to the generalized Boussinesq equation (see [23] and [26]) as follows, which comes from [25].

Lemma 2.1

Let $\widehat{{\mathcal {G}}}$ and $\widehat{{\mathcal {H}}}$ be given by (2.5). Then we have the pointwise estimates

$$\begin{aligned} \begin{aligned}&|\widehat{{\mathcal {G}}}(t, \xi )|\le C|\xi |^{-1}(1+|\xi |^{2})^{-\frac{1}{2}}e^{-c|\xi |^2t}, \\&|\widehat{{\mathcal {H}}}(t, \xi )|\le Ce^{-c|\xi |^2t}, \\&|\widehat{\mathcal \partial _t{\mathcal {G}} }(t, \xi )|\le Ce^{-c|\xi |^2t},\\&|\widehat{\mathcal \partial _t{H}}(t, \xi )|\le C|\xi |(1+|\xi |^2)^{\frac{1}{2}}e^{-c|\xi |^2t} \end{aligned} \end{aligned}$$

(2.16)

for $\xi \in {\mathbb {R}}^n$ and $t\ge 0$.

From the above results in Lemma 2.1, it is not difficult to find that

$$\begin{aligned} |\nabla ^k_\xi \widehat{{\mathcal {G}}}_h|\le C|\xi |^{-k-2}{e^{-ct}}, \;\; |\nabla ^k_\xi \widehat{ {\mathcal {H}}}_h|\le C|\xi |^{-k}{e^{-ct}}. \end{aligned}$$

(2.17)

By Lemma 2.1 and the Plancherel theorem, the following decay estimates of solution operators can be established.

Lemma 2.2

Let $1\le q\le 2$, and let k, j and l be nonnegative integers. Then we have

$$\begin{aligned} \Vert \partial ^k_x G(t)*U_0\Vert _{L^2} \le C(1+t)^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{2}) -\frac{k-j}{2}}\Vert \partial _x^j U_0\Vert _{L^q} +Ce^{-ct}\Vert \partial ^{k+l}_xU_0\Vert _{L^2} \end{aligned}$$

(2.18)

and

$$\begin{aligned} \Vert \partial ^k_x \partial _tG(t)*U_0\Vert _{L^2} \le C(1+t)^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{2}) -\frac{k+1-j}{2}}\Vert \partial _x^j U_0\Vert _{L^q} +Ce^{-ct}\Vert \partial ^{k+l+1}_xU_0\Vert _{L^2}. \end{aligned}$$

(2.19)

Lemma 2.3

Let k, j be nonnegative integers. Then

• (i) If $2\le p\le \infty$, it holds that

  $$\begin{aligned} \Vert \partial ^j_t\partial ^k_x G_{l}(t)\Vert _{L^p}\le C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{k+j}{2}}. \end{aligned}$$

  (2.20)

• (ii) If $1\le p< 2$, it holds that

  $$\begin{aligned} \Vert \partial ^j_t\partial ^k_x G_{l}(t)\Vert _{L^p}\le C\left\{ \begin{array}{l} \displaystyle (1+t)^{ -\frac{n}{2}(1-\frac{1}{p})-\frac{n-1}{4}(1-\frac{2}{p})-\frac{k+j}{2}}, \quad n\ge 3 \text { and }n \text { is odd}\\ \displaystyle (1+t)^{ -\frac{n}{2}(1-\frac{1}{p})-\frac{n}{4}(1-\frac{2}{p})-\frac{k+j}{2}}, \quad \; \; n\ge 2 \text { and }n \text { is even} \\ \end{array}\right. \end{aligned}$$

  (2.21)

Proof

We may refer to [10, 11, 16,17,18] and [24] for the proof. The details are omitted. $\square$

From Lemma 2.2 and (2.8), (2.17), we get immediately the decay properties of $G_h(t, x)$.

Lemma 2.4

Let k be a nonnegative integer. Then

$$\begin{aligned} \Vert \partial ^k_x(G(t)*U_0-G_l(t)*U_0)\Vert _{L^2}\le Ce^{-ct}\Vert \partial ^k_xU_0\Vert _{L^2}. \end{aligned}$$

(2.22)

To establish the weighted decay estimate of global solutions, we need the following decay properties of solution operators in low-frequency parts.

Lemma 2.5

Let $\kappa =0, \frac{1}{2}, 1$ and let k be nonnegative integer. Then we have

$$\begin{aligned} \Vert |x|^\kappa \partial ^k_x G_{l}(t)*U_0\Vert _{L^2}\le C(1+t)^{-\frac{n}{4}-\frac{k}{2}+\kappa }\Vert U_0\Vert _{L^1}+(1+t)^{-\frac{n}{4}-\frac{k}{2}}\Vert |x|^\kappa U_0\Vert _{L^1}. \end{aligned}$$

(2.23)

Proof

When $\kappa =0$, by the Plancherel theorem, it is not difficult to see that (2.23) holds. Owing to the properties of Fourier transform and (2.14), (2.15), (2.5), we have

$$\begin{aligned} {\displaystyle \Vert |x| \partial ^k_x G_{l}(t) \Vert ^2_{L^2} }\le & {} {\displaystyle C\int \limits _{|\xi |\le 2r} \Big |\nabla _\xi \Big ((i\xi )^k \widehat{G}_{l} \Big )\Big |^2d\xi }\nonumber \\\le & {} {\displaystyle C\int \limits _{|\xi |\le 2r} |\xi |^{2(k-1)} e^{-c|\xi |^2t}d\xi + Ct^2\int \limits _{|\xi |\le 2r} |\xi |^{2k} e^{-c|\xi |^2t}d\xi }\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{2}-k+2}.} \end{aligned}$$

(2.24)

From Hölder inequality and (2.24), we have

$$\begin{aligned} {\displaystyle \Vert |x|^{\frac{1}{2}} \partial ^k_xG_{l}(t) \Vert _{L^2} }\le & {} {\displaystyle \Vert |x| \partial ^k_x G_{l}(t) \Vert ^{\frac{1}{2}}_{L^2}\Vert \partial ^k_x G_{l}(t) \Vert ^{\frac{1}{2}}_{L^2}}\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{k}{2}+\frac{1}{2}}.} \end{aligned}$$

(2.25)

When $\kappa =\frac{1}{2}, 1$, it follows from Young inequality and (2.24), (2.25) that

$$\begin{aligned} {\displaystyle \Vert |x|^\kappa \partial ^k_x G_{l}(t)* U_0 \Vert _{L^2} }= & {} {\displaystyle \Vert |x|^\kappa \partial ^k_x G_{l}(t)*U_0 \Vert _{L^2}}\\\le & {} {\displaystyle \Vert |x|^\kappa \partial ^k_x G_{l}(t) \Vert _{L^2}\Vert U_0\Vert _{L^1} +\Vert \partial ^k_x G_{l}(t) \Vert _{L^2}\Vert |x|^\kappa U_0\Vert _{L^1} }\\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{k}{2}+\kappa }\Vert U_0\Vert _{L^1}+(1+t)^{-\frac{n}{4}-\frac{k}{2}}\Vert |x|^\kappa U_0\Vert _{L^1}.} \end{aligned}$$

The lemma is proved. $\square$

Lemma 2.6

Assume that $U_0 \in L^1_2({\mathbb {R}}^n)$. Then we have

$$\begin{aligned}&{\displaystyle \Big \Vert G_{l}(t)* U_0-G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)dy +\sum _{|\alpha |=1}\partial ^\alpha _xG_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)y^\alpha dy \Big \Vert _{L^2}}\nonumber \\&\quad \le {\displaystyle C(1+t)^{-\frac{n}{4}-1}\Vert U_1\Vert _{L^1_2}}. \end{aligned}$$

(2.26)

Proof

The definition of convolution, Taylor formula, Young inequality and (2.20) entails that

$$\begin{aligned}&{\displaystyle \Big \Vert G_{l}(t)* U_0- G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)dy +\sum _{|\alpha |=1}\partial ^\alpha _x G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)y^\alpha dy\Big \Vert _{L^2}} \\&\quad ={\displaystyle \Big \Vert \int \limits _{{\mathbb {R}}^n} \Big ( G_{l}(t, x-y)- G_{l}(t, x) +\sum _{|\alpha |=1}\partial ^\alpha _x G_{l}(t, x)y^\alpha \Big ) U_0(y) dy\Big \Vert _{L^2}}\\&\quad ={\displaystyle \Big \Vert \int \limits _{{\mathbb {R}}^n} \sum _{|\alpha |=2}\partial ^\alpha _x G_{l}(t, x-\theta y)) U_0(y)y^\alpha dy\Big \Vert _{L^2}}\\&\qquad \le {\displaystyle \Vert \partial ^2_x G_{l}(t)\Vert _{L^2}\Vert |x|^2U_0\Vert _{L^1}}\\&\qquad \le {\displaystyle C(1+t)^{-\frac{n}{4}-1}\Vert U_0\Vert _{L^1_2},} \end{aligned}$$

where $\theta \in (0, 1)$. We complete the proof of Lemma 2.6. $\square$

To derive the weighted decay estimate of solution operators in high-frequency part, we need the following lemma, which comes from [14].

Lemma 2.7

Let $\partial ^k_\xi \widehat{ f}\in L^\infty$ for $k=0, 1$, and let $(1+|x|^{\frac{1}{2}})g \in L^2$. Then we have

$$\begin{aligned} \Vert |x|^{\frac{1}{2}}f*g\Vert _{L^2}\le C(\Vert \widehat{f}\Vert _{L^\infty }+\Vert \partial _\xi \widehat{f}\Vert _{L^\infty })(\Vert g\Vert _{L^2}+\Vert |x|^{\frac{1}{2}}g\Vert _{L^2}). \end{aligned}$$

(2.27)

Lemma 2.8

Let $\kappa =0, \frac{1}{2}, 1$ and let k be nonnegative integer. Then we have

$$\begin{aligned} \Vert |x|^\kappa \partial ^k_x G_{h}(t)* U_0\Vert _{L^2}\le Ce^{-ct}(\Vert \partial ^k_xU_0\Vert _{L^2}+\Vert |x|^\kappa \partial ^k_xU_0\Vert _{L^2}) \end{aligned}$$

(2.28)

and

$$\begin{aligned} \Vert |x|^\kappa \partial ^{k+1}_x G^{12}_{h}(t)* u_0\Vert _{L^2}\le Ce^{-ct}(\Vert \partial ^k_xu_0\Vert _{L^2}+\Vert |x|^\kappa \partial ^k_xu_0\Vert _{L^2}). \end{aligned}$$

(2.29)

Proof

When $\kappa =0, 1$, (2.28) immediately follows from the Plancherel theorem. When $\kappa =\frac{1}{2}$, making use of (2.27) and (2.5), (2.17), we deduce that

$$\begin{aligned} {\displaystyle \Big \Vert |x|^\kappa \partial ^k_x G_{h}(t)* U_0 \Big \Vert _{L^2} }= & {} {\displaystyle \Big \Vert |x|^\kappa G_{h}(t)*\partial ^k_xU_0 \Big \Vert _{L^2} }\\\le & {} {\displaystyle \left( \Big \Vert \partial _\xi \widehat{G}_{h}(t) \Big \Vert _{L^\infty }+\Big \Vert \widehat{G}_{h}(t) \Big \Vert _{L^\infty }\right) \left( \Vert \partial ^k_xU_0\Vert _{L^2} +\Vert |x|^\kappa \partial ^k_xU_0\Vert _{L^2} \right) }\\\le & {} {\displaystyle Ce^{-ct}\left( \Vert \partial ^k_xU_0\Vert _{L^2}+\Vert |x|^\kappa \partial ^k_xU_0\Vert _{L^2}\right) .} \end{aligned}$$

Similarly, we may prove (2.29). Then the proof of Lemma 2.8 is completed. $\square$

3 Weighted decay estimate of global solutions

The purpose of this section is to establish weighted decay estimate of global solutions obtained in [9]. To this end, let $\sigma =\rho -1$, $u=\frac{m}{\rho }$, $P'(1) =1$ and $V=(\sigma , u)^\tau$,$V_0=(\sigma _0, u_0)^\tau =(\sigma _0, \frac{m_0}{\sigma _0+1})^\tau$ . Then (1.3) may be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \sigma + \nabla \cdot u= \widetilde{F}_1(V),\\ \partial _t u+\nabla \sigma - \mu \Delta u-(\mu +\lambda )\nabla (\nabla \cdot u)-\alpha \nabla \Delta \sigma =\widetilde{F}_2(V),\\ \end{array}\right. \end{aligned}$$

(3.1)

The initial value becomes

$$\begin{aligned} t=0:\;\; V=V_0, \end{aligned}$$

(3.2)

where

$$\begin{aligned} \widetilde{F}_1(V)=-\nabla \cdot (\sigma u) \end{aligned}$$

(3.3)

and

$$\begin{aligned} \widetilde{F}_2(V)=-u\cdot \nabla u-\mu \frac{\sigma }{\sigma +1}\Delta u-(\mu +\lambda )\frac{\sigma }{\sigma +1} \nabla (\nabla \cdot u) -\frac{1}{\sigma +1}[\nabla (P(1+\sigma )-P'(1)\sigma )]. \end{aligned}$$

(3.4)

The problem (3.1), (3.2) may be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t V-AV= \widetilde{F}(V),\\ t=0:\;\; V=V_0,\\ \end{array}\right. \end{aligned}$$

(3.5)

where $\widetilde{F}(V)=(\widetilde{F}_1(V), \widetilde{F}_2(V))^\tau$.

Noting that (2.8), then the solution to the problem (3.5) is given by

$$\begin{aligned} {\displaystyle V(t)}= & {} {\displaystyle G(t)*V_0+\int \limits ^t_0G(t-\tau )*\widetilde{F}(V)(\tau )d\tau }\nonumber \\= & {} {\displaystyle G_l(t)*V_0+G_h(t)*V_0+\int \limits ^t_0G_l(t-\tau )*\widetilde{F}(V)(\tau )d\tau }\nonumber \\&+ {\displaystyle \int \limits ^t_0G_h(t-\tau )*\widetilde{F}(V)(\tau )d\tau .} \end{aligned}$$

(3.6)

We state the weighted decay estimate of global solutions obtained in [9] as follows.

Theorem 3.1

Let $n\ge 3$ and $\kappa = \frac{1}{2}$ when $n=3$ and $\kappa = 1$ when $n\ge 4$. Assume that the conditions of Theorem 1.2 hold. Moreover, assume that $V_0 \in L^1_\kappa \bigcap L^2_\kappa$ and $\partial _x\sigma _0 \in L^2_\kappa$. The solutions V to the problem (3.1), (3.2) satisfy

$$\begin{aligned} \Vert |x|^\kappa V(t)\Vert _{L^2}\le C(1+t)^{-\frac{n}{4}+\kappa } \end{aligned}$$

(3.7)

and

$$\begin{aligned} \Vert |x|^\kappa \partial _x\sigma (t)\Vert _{L^2}\le C(1+t)^{-\frac{n}{4}+\kappa }. \end{aligned}$$

(3.8)

Proof

Let

$$\begin{aligned} X(t)=\sup _{0 \le \tau \le t}\Big \{(1+\tau )^{\frac{n}{4}-\kappa } (\Vert |x|^\kappa V(\tau ) \Vert _{L^2}+ \Vert |x|^\kappa \partial _x\sigma (\tau ) \Vert _{L^2})\Big \}. \end{aligned}$$

(3.9)

Thanks to (3.6) and Minkowski inequality, we obtain

$$\begin{aligned} {\displaystyle \Vert |x|^\kappa V(\tau )\Vert _{L^2} }\le & {} {\displaystyle \Vert |x|^\kappa G_{l}(t)* V_0 \Vert _{L^2}+ \Vert |x|^\kappa G_{h}(t)*V_0 \Vert _{L^2}}\nonumber \\&{\displaystyle +\int \limits ^t_0 \Vert |x|^\kappa G_{1}(t-\tau )* \widetilde{F}(V)(\tau ) \Vert _{L^2}d\tau }\nonumber \\&{ \displaystyle +\int \limits ^t_0 \Vert |x|^\kappa G_{h}(t-\tau )*\widetilde{F}(V)(\tau ) \Vert _{L^2}d\tau }\nonumber \\=: & {} {\displaystyle I_1+I_2+I_3+I_4.} \end{aligned}$$

(3.10)

By virtue of (2.23), we can get

$$\begin{aligned} {\displaystyle I_1}\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}+\kappa }(\Vert V_0\Vert _{L^1}+\Vert |x|^\kappa V_0\Vert _{L^1})}\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}+\kappa }\Vert V_0\Vert _{L^1_\kappa }.} \end{aligned}$$

(3.11)

(2.28) entails that

$$\begin{aligned} {\displaystyle I_2}\le & {} {\displaystyle Ce^{-ct}(\Vert V_0\Vert _{L^2}+\Vert |x|^\kappa V_0\Vert _{L^2})}\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}+\kappa }\Vert V_0\Vert _{L^2_\kappa }.} \end{aligned}$$

(3.12)

It follows from (2.23) that

$$\begin{aligned} {\displaystyle I_3}\le & {} {\displaystyle C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}+\kappa }\Vert \widetilde{F}(V)(\tau )\Vert _{L^1} d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}+\kappa }\Vert |x|^\kappa \widetilde{F}(V)(\tau )\Vert _{L^1} d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}+\kappa }\Vert V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^2}+\Vert \partial ^2_x V(\tau )\Vert _{L^2}) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}+\kappa }\Vert |x|^\kappa V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^2}+\Vert \partial ^2_xV(\tau )\Vert _{L^2}) d\tau }\nonumber \\\le & {} {\displaystyle CE^2_1\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}+\kappa }(1+\tau )^{-\frac{n}{2}-\frac{1}{2}} d\tau }\nonumber \\&{\displaystyle +CE_1X(t)\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}+\kappa } (1+\tau )^{-\frac{n}{2}-\frac{1}{2}+\kappa } d\tau }\nonumber \\\le & {} {\displaystyle CE^2_1(1+t)^{-\frac{n}{4}+\kappa }+CE_1X(t)(1+t)^{-\frac{n}{4}+\kappa }.} \end{aligned}$$

(3.13)

Owing to (2.28), we arrive at

$$\begin{aligned} {\displaystyle I_4}\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Vert \widetilde{F}(V)(\tau )\Vert _{L^2} d\tau +C\int \limits ^t_0e^{-c(t-\tau )}\Vert |x|^\kappa \widetilde{F}(V)(\tau )\Vert _{L^2} d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Vert V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^\infty }+\Vert \partial ^2_x u(\tau )\Vert _{L^\infty }) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0e^{-c(t-\tau )}\Vert |x|^\kappa V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^\infty }+\Vert \partial ^2_x u(\tau )\Vert _{L^\infty }) d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Vert V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{H^{s-1}}+\Vert \partial _x u(\tau )\Vert _{H^{s}}) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0e^{-c(t-\tau )}\Vert |x|^\kappa V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{H^{s-1}}+\Vert \partial _x u(\tau )\Vert _{H^{s}}) d\tau }\nonumber \\\le & {} {\displaystyle C\left( \,\int \limits ^t_0e^{-2c(t-\tau )}(\Vert V(\tau )\Vert ^2_{L^2}+\Vert |x|^\kappa V(\tau )\Vert ^2_{L^2})d\tau \right) ^{\frac{1}{2}}}\nonumber \\&{\displaystyle \times \left( \,\int \limits ^t_0(\Vert \partial _x V(\tau )\Vert ^2_{H^{s-1}}+\Vert \partial _x u(\tau )\Vert ^2_{H^{s}})d\tau \right) ^{\frac{1}{2}} }\nonumber \\\le & {} {\displaystyle CE^2_1\left( \,\int \limits ^t_0e^{-2c(t-\tau )}(1+t)^{-\frac{n}{2}} d\tau \right) ^{\frac{1}{2}}}\nonumber \\&{\displaystyle + CE_1X(t)\left( \,\int \limits ^t_0e^{-2c(t-\tau )}(1+t)^{-\frac{n}{2}+2\kappa } d\tau \right) ^{\frac{1}{2}}}\nonumber \\\le & {} {\displaystyle CE^2_1(1+t)^{-\frac{n}{4}}+CE_1X(t)(1+t)^{-\frac{n}{4}+\kappa }.} \end{aligned}$$

(3.14)

Making use of (3.6) and Minkowski inequality, we obtain

$$\begin{aligned} {\displaystyle \Vert |x|^\kappa \partial _x \sigma (\tau )\Vert _{L^2} }\le & {} {\displaystyle \Vert |x|^\kappa \partial _xG^{11}_{l}(t)* \sigma _0 \Vert _{L^2}+ \Vert |x|^\kappa \partial _xG^{11}_{h}(t)*\sigma _0 \Vert _{L^2}}\nonumber \\\le & {} {\displaystyle \Vert |x|^\kappa \partial _xG^{12}_{l}(t)* u_0 \Vert _{L^2}+ \Vert |x|^\kappa \partial _xG^{12}_{h}(t)*u_0 \Vert _{L^2}}\nonumber \\&{\displaystyle +\int \limits ^t_0 \Vert |x|^\kappa \partial _xG^{11}_{1}(t-\tau )* \widetilde{F}_1(V)(\tau ) \Vert _{L^2}d\tau }\nonumber \\&{\displaystyle +\int \limits ^t_0 \Vert |x|^\kappa \partial _x G^{11}_{h}(t-\tau )* \widetilde{F}_1(V)(\tau ) \Vert _{L^2}d\tau }\nonumber \\&{ \displaystyle +\int \limits ^t_0 \Vert |x|^\kappa \partial _xG^{12}_{l}(t-\tau )*\widetilde{F}_2(V)(\tau ) \Vert _{L^2}d\tau }\nonumber \\&{ \displaystyle +\int \limits ^t_0 \Vert |x|^\kappa \partial _xG^{12}_{h}(t-\tau )*\widetilde{F}_2(V)(\tau ) \Vert _{L^2}d\tau }\nonumber \\=: & {} {\displaystyle J_1+J_2+J_3+J_4+J_5+J_6+J_7+J_8.} \end{aligned}$$

(3.15)

Thanks to (2.23), we arrive at

$$\begin{aligned} {\displaystyle J_1}\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }(\Vert \sigma _0\Vert _{L^1}+\Vert |x|^\kappa \sigma _0\Vert _{L^1})}\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert \sigma _0\Vert _{L^1_\kappa }.} \end{aligned}$$

(3.16)

Applying (2.28), this gives

$$\begin{aligned} {\displaystyle J_2}\le & {} {\displaystyle Ce^{-ct}(\Vert \partial _x\sigma _0\Vert _{L^2}+\Vert |x|^\kappa \partial _x\sigma _0\Vert _{L^2})}\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert \partial _x\sigma _0\Vert _{L^2_\kappa }.} \end{aligned}$$

(3.17)

By using (2.23), we get

$$\begin{aligned} {\displaystyle J_3}\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }(\Vert u_0\Vert _{L^1}+\Vert |x|^\kappa u_0\Vert _{L^1})}\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert u_0\Vert _{L^1_\kappa }.} \end{aligned}$$

(3.18)

(2.29) entails that

$$\begin{aligned} {\displaystyle J_4}\le & {} {\displaystyle Ce^{-ct}(\Vert u_0\Vert _{L^2}+\Vert |x|^\kappa u_0\Vert _{L^2})}\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert u_0\Vert _{L^2_\kappa }.} \end{aligned}$$

(3.19)

By a similar calculation to (3.13), it follows from (2.23) that

$$\begin{aligned} {\displaystyle J_5}\le & {} {\displaystyle C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert \widetilde{F}_1(V)(\tau )\Vert _{L^1} d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert |x|^\kappa \widetilde{F}_1(V)(\tau )\Vert _{L^1} d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert V(\tau )\Vert _{L^2} \Vert \partial _x V(\tau )\Vert _{L^2} d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert |x|^\kappa V(\tau )\Vert _{L^2}\Vert \partial _x V(\tau )\Vert _{L^2} d\tau }\nonumber \\\le & {} {\displaystyle CE^2_1\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }(1+\tau )^{-\frac{n}{2}-\frac{1}{2}} d\tau }\nonumber \\&{\displaystyle +CE_1X(t)\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa } (1+\tau )^{-\frac{n}{2}-\frac{1}{2}+\kappa } d\tau }\nonumber \\\le & {} {\displaystyle CE^2_1(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }+CE_1X(t)(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }.} \end{aligned}$$

(3.20)

Due to (2.28), it holds that

$$\begin{aligned} {\displaystyle J_6}\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Vert \widetilde{F}_1(V)(\tau )\Vert _{L^2} d\tau +C\int \limits ^t_0e^{-c(t-\tau )}\Vert |x|^\kappa \widetilde{F}_1(V)(\tau )\Vert _{L^2} d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Big (\Vert \sigma (\tau )\Vert _{L^2}\Vert \partial ^2_x u(\tau )\Vert _{L^\infty }+ \Vert \partial _x\sigma (\tau )\Vert _{L^2}\Vert \partial _x u(\tau )\Vert _{L^\infty }}\nonumber \\&{\displaystyle +\Vert u(\tau )\Vert _{L^2}\Vert \partial ^2_x \sigma (\tau )\Vert _{L^\infty }\Big ) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0e^{-c(t-\tau )}(\Vert |x|^\kappa \sigma (\tau )\Vert _{L^2}\Vert \partial ^2_x u(\tau )\Vert _{L^\infty }+ \Vert |x|^\kappa \partial _x\sigma (\tau )\Vert _{L^2} \Vert \partial _x u(\tau )\Vert _{L^\infty }}\nonumber \\&{\displaystyle +\Vert |x|^\kappa u(\tau )\Vert _{L^2}\Vert \partial ^2_x \sigma (\tau )\Vert _{L^\infty }) d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}(\Vert V(\tau )\Vert _{L^2}\Vert \partial _x V(\tau )\Vert _{H^{s}}+\Vert \partial _xV(\tau )\Vert _{L^2}\Vert \partial _x V(\tau )\Vert _{H^{s-1}}) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0e^{-c(t-\tau )}(\Vert |x|^\kappa V(\tau )\Vert _{L^2}\Vert \partial _x V(\tau )\Vert _{H^{s}}+ \Vert |x|^\kappa \partial _x \sigma (\tau )\Vert _{L^2} \Vert \partial _x V(\tau )\Vert _{H^{s-1}}) d\tau }\nonumber \\\le & {} {\displaystyle C\left( \int \limits ^t_0e^{-2c(t-\tau )}(\Vert V(\tau )\Vert ^2_{L^2}+\Vert \partial _xV(\tau )\Vert ^2_{L^2}+\Vert |x|^\kappa V(\tau )\Vert ^2_{L^2}+ \Vert |x|^\kappa \partial _x \sigma (\tau )\Vert ^2_{L^2} )d\tau \right) ^{\frac{1}{2}} }\nonumber \\&{\displaystyle \times \left( \int \limits ^t_0(\Vert \partial _x V(\tau )\Vert ^2_{H^{s-1}}+\Vert \partial _x V(\tau )\Vert ^2_{H^{s}})d\tau \right) ^{\frac{1}{2}} }\nonumber \\\le & {} {\displaystyle CE^2_1\left( \int \limits ^t_0e^{-2c(t-\tau )}(1+t)^{-\frac{n}{2}} d\tau \right) ^{\frac{1}{2}} }\nonumber \\&{\displaystyle +CE_1X(t)\left( \int \limits ^t_0e^{-2c(t-\tau )}(1+t)^{-\frac{n}{2}+2\kappa } d\tau \right) ^{\frac{1}{2}}}\nonumber \\\le & {} {\displaystyle CE^2_1(1+t)^{-\frac{n}{4}}+CE_1X(t)(1+t)^{-\frac{n}{4}+\kappa }.} \end{aligned}$$

(3.21)

We estimate $J_7$ as follows by (2.23)

$$\begin{aligned} {\displaystyle J_7}\le & {} {\displaystyle C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert \widetilde{F}_2(V)(\tau )\Vert _{L^1} d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert |x|^\kappa \widetilde{F}_2(V)(\tau )\Vert _{L^1} d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^2}+\Vert \partial ^2_x V(\tau )\Vert _{L^2}) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }\Vert |x|^\kappa V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^2}+\Vert \partial ^2_x V(\tau )\Vert _{L^2}) d\tau }\nonumber \\\le & {} {\displaystyle CE^2_1\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa }(1+\tau )^{-\frac{n}{2}-\frac{1}{2}} d\tau }\nonumber \\&{\displaystyle +CE_1X(t)\int \limits ^t_0(1+t-\tau )^{-\frac{n}{4}-\frac{1}{2}+\kappa } (1+\tau )^{-\frac{n}{2}-\frac{1}{2}+\kappa } d\tau }\nonumber \\\le & {} {\displaystyle CE^2_1(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }+CE_1X(t)(1+t)^{-\frac{n}{4}-\frac{1}{2}+\kappa }.} \end{aligned}$$

(3.22)

Making use of (2.29), we see that

$$\begin{aligned} {\displaystyle J_8}\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Vert \widetilde{F}_2(V)(\tau )\Vert _{L^2} d\tau +C\int \limits ^t_0e^{-c(t-\tau )}\Vert |x|^\kappa \widetilde{F}_2(V)(\tau )\Vert _{L^2} d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Vert V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^\infty }+\Vert \partial ^2_x u(\tau )\Vert _{L^\infty }) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0e^{-c(t-\tau )}\Vert |x|^\kappa V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{L^\infty }+\Vert \partial ^2_x u(\tau )\Vert _{L^\infty }) d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^t_0e^{-c(t-\tau )}\Vert V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{H^{s-1}}+\Vert \partial _x u(\tau )\Vert _{H^{s}}) d\tau }\nonumber \\&{\displaystyle +C\int \limits ^t_0e^{-c(t-\tau )}\Vert |x|^\kappa V(\tau )\Vert _{L^2}(\Vert \partial _x V(\tau )\Vert _{H^{s-1}}+\Vert \partial _x u(\tau )\Vert _{H^{s}}) d\tau }\nonumber \\\le & {} {\displaystyle C\left( \int \limits ^t_0e^{-2c(t-\tau )}(\Vert V(\tau )\Vert ^2_{L^2}+\Vert |x|^\kappa V(\tau )\Vert ^2_{L^2})d\tau \right) ^{\frac{1}{2}}}\nonumber \\&{\displaystyle \times \left( \int \limits ^t_0(\Vert \partial _x V(\tau )\Vert ^2_{H^{s-1}}+\Vert \partial _x u(\tau )\Vert ^2_{H^{s}})d\tau \right) ^{\frac{1}{2}} }\nonumber \\\le & {} {\displaystyle CE^2_1\left( \int \limits ^t_0e^{-2c(t-\tau )}(1+t)^{-\frac{n}{2}} d\tau \right) ^{\frac{1}{2}}}\nonumber \\&{\displaystyle + CE_1X(t)\left( \int \limits ^t_0e^{-2c(t-\tau )}(1+t)^{-\frac{n}{2}+2\kappa } d\tau \right) ^{\frac{1}{2}}}\nonumber \\\le & {} {\displaystyle CE^2_1(1+t)^{-\frac{n}{4}}+CE_1X(t)(1+t)^{-\frac{n}{4}+\kappa }.} \end{aligned}$$

(3.23)

Inserting (3.11)–(3.14) into (3.10) and (3.16)–(3.23) into (3.15) yields

$$\begin{aligned} X(t)\le C(\Vert V_0\Vert _{L^1_\kappa }+\Vert V_0\Vert _{L^2_\kappa }+\Vert \partial _x\sigma _0\Vert _{L^2_\kappa })+ CE^2_1+CE_1X(t), \end{aligned}$$

(3.24)

which implies

$$\begin{aligned} X(t)\le C(\Vert V_0\Vert _{L^1_\kappa }+\Vert V_0\Vert _{L^2_\kappa }+\Vert \partial _x\sigma _0\Vert _{L^2_\kappa })+ CE^2_1, \end{aligned}$$

provided that $E_1$ is suitably small. The proof of Theorem 3.1 is completed. $\square$

4 Asymptotic profile of global solutions

In this section, our main goal is to establish asymptotic profile of global solutions and the corresponding convergence rate. To this end, we state the $L^2$ decay rate of global solutions obtained in Theorem 1.1 in high-frequency part as follows.

Lemma 4.1

Under the conditions of Theorem 1.3, $U_h(t, x)$ satisfies

$$\begin{aligned} \Vert U_h(t)\Vert _{L^2}\le C(1+t)^{-\frac{n}{4}-1}. \end{aligned}$$

(4.1)

Proof

(4.1) immediately follows from Theorem 1.1, 1.2 and the decay property of $G_h(x,t)$. Here we omit the details. $\square$

In what follows, we give the proof of Theorem 1.3.

Proof

From (2.7), (2.9) and (2.10), we arrive at

$$\begin{aligned}&{\displaystyle U(t)-G(t)*U_0- \sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_0\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau }\nonumber \\&\quad ={\displaystyle U_h(t)+G_l(t)*U_0-G(t)*U_0+\int \limits ^t_0G_l(t-\tau )*\nabla \cdot {\mathfrak {F}}(U)(\tau )d\tau }\nonumber \\&\quad \quad {\displaystyle - \sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_0\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau }\nonumber \\&\quad ={\displaystyle U_h(t)+(G_l(t)*U_0-G(t)*U_0)}\nonumber \\&\quad \quad + {\displaystyle \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \Big (\partial _{x_i}G_l(t-\tau , x-y)-\partial _{x_i}G_l(t, x)\Big ){\mathbb {F}}_i(U)dyd\tau }\nonumber \\&\quad \quad + {\displaystyle \int \limits ^{\frac{t}{2}}_0 \sum ^n_{i=1} \partial _{x_i}G_l(t-\tau )* \Big ({\mathfrak {F}}_i(U)-{\mathbb {F}}_i(U)\Big )(\tau )d\tau }\nonumber \\&\quad \quad + {\displaystyle \int \limits ^t_{\frac{t}{2}} \sum ^n_{i=1} \partial _{x_i}G_l(t-\tau )* {\mathfrak {F}}_i(U)(\tau )d\tau }\nonumber \\&\quad \quad - {\displaystyle \sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_{\frac{t}{2}}\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau .} \end{aligned}$$

(4.2)

Then (4.2) and Minkowski inequality give

$$\begin{aligned}&\Big \Vert U(t)-G(t)*U_0- \sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_0\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau \Big \Vert _{L^2}\nonumber \\&\quad \le \Vert U_h(t)\Vert _{L^2}+\Big \Vert G_l(t)*U_0-G(t)*U_0\Big \Vert _{L^2}\nonumber \\&\quad \quad +\left\| \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \Big (\partial _{x_i}G_l(t-\tau , x-y)-\partial _{x_i}G_l(t, x)\Big ){\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}\nonumber \\&\quad \quad + \left\| \int \limits ^{\frac{t}{2}}_0 \sum ^n_{i=1} \partial _{x_i}G_l(t-\tau )* \Big ({\mathfrak {F}}_i(U)-{\mathbb {F}}_i(U)\Big )(\tau )d\tau \right\| _{L^2}\nonumber \\&\quad \quad + \left\| \int \limits ^t_{\frac{t}{2}} \sum ^n_{i=1} \partial _{x_i}G_l(t-\tau )* {\mathfrak {F}}_i(U)(\tau )d\tau \right\| _{L^2}\nonumber \\&\quad \quad +\left\| \sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_{\frac{t}{2}}\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}\nonumber \\&\quad =:K_1+K_2+K_3+K_4+K_5+K_6. \end{aligned}$$

(4.3)

By virtue of (4.1), it holds that

$$\begin{aligned} K_1\le C(1+t)^{-\frac{n}{4}-1}. \end{aligned}$$

(4.4)

Thanks to (2.22), we have

$$\begin{aligned} K_2\le Ce^{-ct}\Vert U_0\Vert _{L^2}. \end{aligned}$$

(4.5)

$K_3$ may be rewritten as

$$\begin{aligned} {\displaystyle K_3}= & {} {\displaystyle \left\| \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \Big (\partial _{x_i}G_l(t-\tau , x-y)-\partial _{x_i}G_l(t, x)\Big ){\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}}\\\le & {} {\displaystyle \left\| \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \Big (\partial _{x_i}G_l(t-\tau , x-y)-\partial _{x_i}G_l(t-\tau , x)\Big ){\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}}\\&{\displaystyle +\left\| \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \Big (\partial _{x_i}G_l(t-\tau , x)-\partial _{x_i}G_l(t, x)\Big ){\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}} \\=: & {} {\displaystyle K_{31}+K_{32}.} \end{aligned}$$

In what follows, we estimate $K_{31}$ for $n=3$ and $n\ge 4$, respectively. When $n=3$, mean value theorem, Young inequality, Hölder inequality and (2.20), (3.7), (3.8) entail that

$$\begin{aligned} {\displaystyle K_{31}}= & {} {\displaystyle \left\| \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \partial _x\partial _{x_i}G_l(t-\tau , x-\theta _1 y)y{\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}}\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{3}{4}-1}\Vert x{\mathbb {F}}_i(U)\Vert _{L^1}d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{3}{4}-1}\Big (\Vert |x|^{\frac{1}{2}}U(\tau )\Vert ^2_{L^2}+\Vert |x|^{\frac{1}{2}}\partial _x\sigma (\tau )\Vert ^2_{L^2}\Big )d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{3}{4}-1}(1+\tau )^{-\frac{1}{2}}d\tau }\nonumber \\\le & {} { (1+t)^{-\frac{5}{4}},} \end{aligned}$$

(4.6)

where $\theta _1 \in (0, 1).$

When $n\ge 4$, using mean value theorem, Young inequality, Hölder inequality and (2.20), (1.5), (1.6), (3.7), (3.8), the similar estimate of (4.6) leads to

$$\begin{aligned} {\displaystyle K_{31}}= & {} {\displaystyle \left\| \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \partial _x\partial _iG_l(t-\tau , x-\theta _2 y)y{\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}}\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}\Vert x{\mathbb {F}}_i(U)\Vert _{L^1}d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}\Big (\Vert |x|U(\tau )\Vert _{L^2}\Vert U(\tau )\Vert _{L^2}+\Vert |x|\partial _x\sigma (\tau )\Vert _{L^2}\Vert \partial _x\sigma (\tau )\Vert _{L^2}\Big )d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}(1+\tau )^{-\frac{n}{2}+1}d\tau }\nonumber \\\le & {} {\displaystyle C\left\{ \begin{array}{ll} (1+t)^{-\frac{n}{4}-1}\log (1+t), &{}\quad n= 4 \\ (1+t)^{-\frac{n}{4}-1}, &{} \quad n\ge 5 \\ \end{array}\right. } \end{aligned}$$

(4.7)

where $\theta _2 \in (0, 1).$

Making use of mean value theorem, Young inequality, Hölder inequality and (2.20), (1.5), (1.6), we have

$$\begin{aligned} {\displaystyle K_{32}}= & {} {\displaystyle \left\| \int \limits ^{\frac{t}{2}}_0\int \limits _{{\mathbb {R}}^n} \sum ^n_{i=1} \partial _t\partial _{x_i}G_l(t-\theta _3\tau , x)\tau {\mathbb {F}}_i(U)dyd\tau \right\| _{L^2}}\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}\tau \Vert {\mathbb {F}}_i(U)\Vert _{L^1}d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}\tau \Big (\Vert U(\tau )\Vert ^2_{L^2}+\Vert \partial _x\sigma (\tau )\Vert ^2_{L^2}\Big )d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}\tau (1+\tau )^{-\frac{n}{2}}d\tau }\nonumber \\\le & {} {\displaystyle C\left\{ \begin{array}{ll} (1+t)^{-\frac{n}{4}-\frac{1}{2}}, &{} \quad n= 3 \\ (1+t)^{-\frac{n}{4}-1}\log (1+t), &{} \quad n= 4 \\ (1+t)^{-\frac{n}{4}-1}, &{} \quad n\ge 5 \\ \end{array}\right. } \end{aligned}$$

(4.8)

where $\theta _3 \in (0, 1).$

It follows from (2.23) with $\kappa =0$ and (1.5) that

$$\begin{aligned} {\displaystyle K_{4}}\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}\Vert U(\tau )\Vert ^2_{L^2}d\tau +C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-\frac{3}{2}}\Vert \sigma (\tau )\Vert ^2_{L^2}d\tau }\nonumber \\\le & {} {\displaystyle C\int \limits ^{\frac{t}{2}}_0 (1+t-\tau )^{-\frac{n}{4}-1}(1+\tau )^{-\frac{n}{2}}d\tau }\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-1}.} \end{aligned}$$

(4.9)

Young inequality, (2.21), (1.5), (1.6) entail that

$$\begin{aligned} {K_{5}}\le & {} {\displaystyle C\int \limits ^t_{\frac{t}{2}}\Big (\Vert \partial _x {\mathscr {G}}_l(t-\tau )\Vert _{L^1}(\Vert |U|^2(\tau )\Vert _{L^2}+\Vert |\partial _x\sigma |^2(\tau )\Vert _{L^2})}\nonumber \\&+ {\displaystyle \Vert \partial ^2_x {\mathscr {G}}_l(t-\tau )\Vert _{L^1}\Vert |U|^2(\tau )\Vert _{L^2} + \Vert \partial ^3_x {\mathscr {G}}_l(t-\tau )\Vert _{L^1}\Vert \sigma (\tau )\Vert _{L^2}\Big )d\tau }\nonumber \\\le & {} {\displaystyle C\left\{ \begin{array}{ll} \displaystyle \int \limits ^t_{\frac{t}{2}}(1+t-\tau )^{\frac{n-1}{4}-\frac{1}{2}}(\Vert U(\tau )\Vert _{L^\infty }\Vert U\Vert _{L^2}+\Vert \partial _x\sigma (\tau )\Vert ^2_{L^4})d\tau , &{} n\ge 3\text { and }n \text { is odd} \\ \displaystyle \int \limits ^t_{\frac{t}{2}}(1+t-\tau )^{\frac{n}{4}-\frac{1}{2}}(\Vert U(\tau )\Vert _{L^\infty }\Vert U\Vert _{L^2}+\Vert \partial _x\sigma (\tau )\Vert ^2_{L^4})d\tau , &{} n\ge 4 \text { and }n \text { is even} \\ \end{array}\right. } \nonumber \\\le & {} {\displaystyle C\left\{ \begin{array}{ll} \displaystyle \int \limits ^t_{\frac{t}{2}}(1+t-\tau )^{\frac{n-1}{4}-\frac{1}{2}}(1+\tau )^{-\frac{3n}{4}}d\tau , &{} n\ge 3\text { and }n \text { is odd} \\ \displaystyle \int \limits ^t_{\frac{t}{2}}(1+t-\tau )^{\frac{n}{4}-\frac{1}{2}}(1+\tau )^{-\frac{3n}{4}}d\tau , &{} n\ge 4 \text { and }n \text { is even} \\ \end{array}\right. } \nonumber \\\le & {} {\displaystyle C\left\{ \begin{array}{ll} \displaystyle (1+t)^{-\frac{n}{4}-\frac{1}{2}}, &{} \quad n=3, 4 \\ \displaystyle (1+t)^{-\frac{n}{4}-1}, &{}\quad n\ge 5 \\ \end{array}\right. } \end{aligned}$$

(4.10)

Owing to (2.20) and (1.5), (1.6), we deduce, for $n\ge 3$

$$\begin{aligned} {\displaystyle K_{6}}\le & {} {\displaystyle \sum ^n_{i=1}\Vert \partial _{x_i}G_l(t)\Vert _{L^2}\int \limits ^\infty _{\frac{t}{2}}\int \limits _{{\mathbb {R}}^n} |{\mathbb {F}}_i(U) |dyd\tau }\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}}\int \limits ^\infty _{\frac{t}{2}} (\Vert U(\tau )\Vert ^2_{L^2}+\Vert \partial _x\sigma (\tau )\Vert ^2_{L^2})d\tau }\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-\frac{1}{2}}\int \limits ^\infty _{\frac{t}{2}} (1+\tau )^{-\frac{n}{2}}d\tau }\nonumber \\\le & {} {\displaystyle C(1+t)^{-\frac{n}{4}-1}.} \end{aligned}$$

(4.11)

Substituting (4.4)-(4.11) into (4.3) yields (1.8). This concludes Theorem 1.3.$\square$

Remark 4.2

Assume that the conditions of Theorem 1.3 hold. Furthermore, assume that $U_0 \in L^1_2$. Then from (2.22) and (2.26), we have

$$\begin{aligned}&{\displaystyle \Big \Vert G(t)* U_0- G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)dy +\sum _{|\alpha |=1}\partial ^\alpha _x G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)y^\alpha dy \Big \Vert _{L^2}}\nonumber \\&\quad \le {\displaystyle C(1+t)^{-\frac{n}{4}-1}(\Vert U_0\Vert _{L^1_2}+\Vert U_0\Vert _{L^2})}. \end{aligned}$$

(4.12)

Therefore, (1.8) and (4.12) and Minkowski inequality imply that

$$\begin{aligned}&{\displaystyle \Big \Vert u(t)- G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)dy +\sum _{|\alpha |=1}\partial ^\alpha _x G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)y^\alpha dy}\nonumber \\&\quad \quad {\displaystyle -\sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_0\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau \Big \Vert _{L^2} } \nonumber \\&\quad \le {\displaystyle \Big \Vert u(t) -G (t)* U_0-\sum ^n_{i=1} \partial _{x_i} G_l(t)\int \limits ^{\infty }_0\int \limits _{{\mathbb {R}}^n} {\mathbb {F}}_i(U)dyd\tau \Big \Vert _{L^2}}\nonumber \\&\quad \quad +{\displaystyle \Big \Vert G(t)* U_0 - G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)dy +\sum _{|\alpha |=1}\partial ^\alpha _x G_{l}(t)\int \limits _{{\mathbb {R}}^n} U_0(y)y^\alpha dy\Big \Vert _{L^2}}\nonumber \\&\quad \le {\displaystyle C\left\{ \begin{array}{ll} (1+t)^{-\frac{n}{4}-\frac{1}{2}}, &{}\quad n=3, 4 \\ (1+t)^{-\frac{n}{4}-1}, &{} \quad n\ge 5. \\ \end{array}\right. } \end{aligned}$$

(4.13)