Two-phase flow refers to a flow system consisting of two substances in different phases (at least one of which is a fluid). The flow system may be in different combinations such as gas-liquid and liquid-solid. Two-phase flow is also widely present in nature and other engineering fields, such as the drift of rain, snow, clouds and fog [1] [2] , blood circulation in living organisms [3] [4] , sediment movement [5] [6] , high-speed aerated water flow in hydraulic engineering, and air pollution caused by smoke and dust in environmental engineering. With the passage of time and the development of science, the problem of two-phase flow has received increasing attention. Two-phase flow plays an important role in aerospace [7] [8] , biomedical engineering [9] - [13] , chemical engineering [14] - [16] and other fields. However, the analysis of two-phase flow problem is relatively complex. It involves knowledge from multiple fields such as fluid mechanics, thermodynamics, and heat transfer. People use different models to study two-phase flow problems. The most common is the compressible two-fluid model in [17] established using the Navier-Stokes equations:

$\{\begin{array}{l}{\partial }_t\left({\sigma }^{+}\rho \right)+\text{div}\left({\sigma }^{+}\rho w\right)=\mathrm{0,}\hfill \\ {\partial }_t\left({\sigma }^{+}\rho w\right)+\text{div}\left({\sigma }^{+}\rho w\otimes w\right)+{\sigma }^{+}\nabla P^{+}\left(\rho \right)=\text{div}\left({\sigma }^{+}{\tau }_1\right)\hfill \\ {\partial }_t\left({\sigma }^{-}\eta \right)+\text{div}\left({\sigma }^{-}\eta n\right)=\mathrm{0,}\hfill \\ {\partial }_t\left({\sigma }^{-}\eta n\right)+\text{div}\left({\sigma }^{-}\eta n\otimes n\right)+{\sigma }^{-}\nabla P^{-}\left(\eta \right)=\text{div}\left({\sigma }^{-}{\tau }_2\right)\hfill \\ P=P^{+}\left(\rho \right)=P^{-}\left(\eta \right)=a^{+}{\left(\rho \right)}^{{\bar{\gamma }}^{+}}=a^{-}{\left(\eta \right)}^{{\bar{\gamma }}^{-}}\hfill \end{array}$ (1.1)

in which $\rho$, $\eta$ are no less than 0, ${\bar{\gamma }}^{\pm }\ge 1$, ${\sigma }^{+}$ is the volume fraction of liquid, while ${\sigma }^{-}$ is that of gas, both ${\sigma }^{+}$ and ${\sigma }^{-}$ are positive constants. $w\left(x\mathrm{,}t\right)$, $n\left(x\mathrm{,}t\right)$ represent the velocities of each phase. Let $a^{+}=a^{-}=1$. Moreover, ${\tau }_1$, ${\tau }_2$ are the viscous stress tensors:

$\{\begin{array}{l}{\tau }_1\mathrm{<mo>\; :\; </mo>}={\mu }_1\left(\nabla w+{\nabla }^{t}w\right)+{\lambda }_1\text{div}w\text{Id}\mathrm{,}\hfill \\ {\tau }_2\mathrm{<mo>\; :\; </mo>}={\mu }_2\left(\nabla n+{\nabla }^{t}n\right)+{\lambda }_2\text{div}n\text{Id}\mathrm{,}\hfill \end{array}$ (1.2)

where ${\mu }_1$, ${\mu }_2$ and ${\lambda }_1$, ${\lambda }_2$ are shear and bulk viscosity coefficients: ${\mu }_1$, ${\mu }_2\mathrm{<mo>\; >\; </mo>0}$, $2{\mu }_1+3{\lambda }_1\ge 0$, $2{\mu }_2+3{\lambda }_2\ge 0$, then ${\mu }_1+{\lambda }_1\mathrm{<mo>\; >\; </mo>0}$, ${\mu }_2+{\lambda }_2>0$.

When ${\mu }_1$, ${\mu }_2$, ${\lambda }_1$ and ${\lambda }_2$ are constants, there are many studies on the solutions of (1.1). Bresch, Huang and Li obtained global weak solutions of a non-conservative viscous compressible two-phase model in one space dimension in [18] . For the multi-dimensional case, Yao, Zhang and Zhu obtained the existence and asymptotic behavior of global weak solutions for a two-phase model of two-dimensional viscous liquid gas flow by using the idea of studying single-phase Navier-Stokes equations in [19] . Vasseur, Wen and Yu in [20] got the existence of global weak solutions for the model like (1.1), where the pressure contingent upon two variables. Cui, Wang, Yao and Zhu in [21] studied the time-decay rate of a non conservative compressible model by analysing systems and estimating energy. When $P^{+}=P^{-}$, Wu and Yao obtained the properties of the classical solutions to the three dimensional space in [22] .

With the intensive study of the single phase Navier-Stokes equations, researchers have obtained some important conclusions, one of which is particularly noteworthy: The dynamic viscosity coefficient and volumetric viscosity coefficient both depend on density, see [23] - [27] and the references therein. For the two-fluid model, we make the following assumptions:

$\{\begin{array}{l}{\tau }_1\mathrm{<mo>\; :\; </mo>}={\rho }^{{\mu }_1}\left(\nabla w+{\nabla }^{t}w\right)+{\rho }^{{\lambda }_1}\text{div}w\text{Id}\mathrm{,}\hfill \\ {\tau }_2\mathrm{<mo>\; :\; </mo>}={\eta }^{{\mu }_2}\left(\nabla n+{\nabla }^{t}n\right)+{\eta }^{{\lambda }_2}\text{div}n\text{Id}\mathrm{,.}\hfill \end{array}$ (1.3)

where ${\mu }_1$, ${\lambda }_1\mathrm{,}$ ${\mu }_2$, ${\lambda }_2\ge 1$. The biggest difference between this article and previous research is the viscosity term, in which both $\mu$ and $\lambda$ are functions of density.

When the velocities of the two fluids are the same, Zodji in [28] proved the existence of a unique local-time solution when density and dynamic viscosity coefficients have an infimum. Li, Wang and Zhang in [29] researched a model analogous to (1.1), where the dynamic viscosity coefficient and volumetric viscosity coefficient are functions of density. There are no finite-energy classical solutions to this system for any small time. The assumption made by Bresch in [30] is that the viscosity coefficients of two-phase motion are ${\mu }_1\rho$ and ${\mu }_2\eta$ respectively, and the volume viscosity coefficients are both 0, Bresch obtained the global weak solutions, while $1<{\gamma }^{\pm }<6$. Evje, Wang and Wen in [31] supposed:

$P^{+}\left(\rho \right)-P^{-}\left(\eta \right)={\rho }^{{\bar{\gamma }}^{+}}-{\eta }^{{\bar{\gamma }}^{-}}=f\left({\sigma }^{-}\eta \right),$ (1.4)

where $f$ represents capillary pressure, $f\in C^3\left(\left[\mathrm{0,}\infty \right)\right)$. If f satisfies

$-\frac{r_{-}^2\left(1,1\right)}{{\sigma }^{-}\left(1,1\right)}<f^{\prime }\left(1\right)<0,$ (1.5)

then $f$ decreases strictly monotonically in the vicinity of the equilibrium point.

Where $r_{+}^2\mathrm{<mo>\; :\; </mo>}=\frac{\text{d}{P}^{+}}{\text{d}\rho }\left(\rho \right)={\bar{\gamma }}^{+}\frac{P^{+}\left(\rho \right)}{\rho }$, $r_{-}^2\mathrm{<mo>\; :\; </mo>}=\frac{\text{d}{P}^{-}}{\text{d}\eta }\left(\eta \right)={\bar{\gamma }}^{-}\frac{P^{-}\left(\eta \right)}{\eta }$ represent the

sound speed respectively. They achieved global existence and decay rates of the solutions in [31] .

There are currently few research results on viscous stress tensors (1.3). Therefore, we study the properties of solutions when both $\mu$ and $\lambda$ are functions of density.

The research methods and techniques used in this article are derived from [22] , which obtains the global goodness and large time behavior of the classical solution of a three-dimensional compressible two-phase flow model with common pressure through energy estimation and continuity induction.

We observed (1.1)₅ and obtained the following relationship

$\text{d}P=r_{+}^2\text{d}\rho =r_{-}^2\text{d}\eta \mathrm{,}$ (1.6)

We introduce fractional density like in [30]

$M^{+}={\sigma }^{+}\rho \mathrm{,}{M}^{-}={\sigma }^{-}\eta \mathrm{,}$ (1.7)

Combined with ${\sigma }^{+}+{\sigma }^{-}=1$, it gives

$\text{d}\rho =\frac{1}{{\sigma }^{+}}\left(\text{d}{M}^{+}-\rho \text{d}{\sigma }^{+}\right)\mathrm{,}\text{d}\eta =\frac{1}{{\sigma }^{-}}\left(\text{d}{M}^{-}+\eta \text{d}{\sigma }^{-}\right)\mathrm{.}$ (1.8)

From (1.6) and (1.7), we get

$\text{d}{\sigma }^{+}=\frac{{\sigma }^{-}{r}_{+}^2}{{\sigma }^{-}\rho r_{+}^2+{\sigma }^{+}\eta r_{-}^2}\text{d}{M}^{+}-\frac{{\sigma }^{+}{r}_{-}^2}{{\sigma }^{-}\rho r_{+}^2+{\sigma }^{+}\eta r_{-}^2}\text{d}{M}^{-}\mathrm{.}$ (1.9)

Putting (1.9) into (1.8), we derive

$\text{d}\rho =\frac{\rho \eta r_{-}^2}{M^{-}{\left(\rho \right)}^2{r}_{+}^2+M^{+}{\left(\eta \right)}^2{r}_{-}^2}\left(\eta \text{d}{M}^{+}+\rho \text{d}{M}^{-}\right)\mathrm{,}$

and

$\text{d}\eta =\frac{\rho \eta r_{+}^2}{M^{-}{\left(\rho \right)}^2{r}_{+}^2+M^{+}{\left(\eta \right)}^2{r}_{-}^2}\left(\eta \text{d}{M}^{+}+\rho \text{d}{M}^{-}\right)\mathrm{,}$

We get $\text{d}P$:

$\text{d}P=ℒ\left(\eta \text{d}{M}^{+}+\rho \text{d}{M}^{-}\right)$

where

$ℒ\mathrm{<mo>\; :\; </mo>\; <mo>\; =\; </mo>}\frac{r_{-}^2{r}_{+}^2}{{\sigma }^{-}\rho r_{+}^2+{\sigma }^{+}\eta r_{-}^2}\mathrm{.}$

From ${\sigma }^{+}+{\sigma }^{-}=1$, we get

$\frac{M^{+}}{\rho }+\frac{M^{-}}{\eta }=\mathrm{1,\; <mi>\; </mi>\; <mi>\; </mi>}\text{thus}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{M}^{-}=\frac{\eta \left(\rho -M^{+}\right)}{\rho }=\frac{P^{1/{\gamma }^{-}}\left(P^{1/{\gamma }^{+}}-M^{+}\right)}{P^{1/{\gamma }^{+}}}\mathrm{.}$ (1.10)

With (1.1)₅, (1.7) and (1.10), we define ${\sigma }^{\pm }$

$\{\begin{array}{l}{\sigma }^{+}\left(P\mathrm{,}{M}^{+}\right)=\frac{M^{+}}{P^{1/{\gamma }^{+}}}\mathrm{,}\hfill \\ {\sigma }^{-}\left(P\mathrm{,}{M}^{+}\right)=1-\frac{M^{+}}{P^{1/{\gamma }^{+}}}\mathrm{.}\hfill \end{array}$ (1.11)

Rewrite (1.1) using $\left(M^{\pm }\mathrm{,}P\mathrm{,}u\mathrm{,}v\right)$:

$\{\begin{array}{l}{\partial }_t{M}^{+}+\text{div}\left(M^{+}w\right)=\mathrm{0,}\\ {\partial }_t{M}^{-}+\text{div}\left(M^{-}n\right)=\mathrm{0,}\\ {\partial }_{t}P+ℒ\eta \text{div}\left(M^{+}w\right)+ℒ\rho \text{div}\left(M^{-}n\right)=\mathrm{0,}\\ {\partial }_t\left(M^{+}w\right)+\text{div}\left(M^{+}w\otimes w\right)+{\sigma }^{+}\nabla P=\text{div}\left\{{\sigma }^{+}\left[{\rho }^{{\mu }_1}\left(\nabla w+{\nabla }^{t}w\right)+{\rho }^{{\lambda }_1}\text{div}w\text{Id}\right]\right\}\mathrm{,}\\ {\partial }_t\left(M^{-}n\right)+\text{div}\left(M^{-}n\otimes n\right)+{\sigma }^{-}\nabla P=\text{div}\left\{{\sigma }^{-}\left[{\eta }^{{\mu }_2}\left(\nabla n+{\nabla }^{t}n\right)+{\eta }^{{\lambda }_2}\text{div}n\text{Id}\right]\right\}\mathrm{.}\end{array}$(1.12)

This paper considers (1.12) in 3D space with initial data.

$\left(M^{\pm },P,u,v\right)\left(x,0\right)=\left(M_0^{\pm },P_0,u_0,v_0\right)\left(x\right)\to \left({\bar{M}}^{\pm },\bar{P},\stackrel{\to }{0},\stackrel{\to }{0}\right)\text{as}\left|x\right|\to \infty \in {ℝ}^3,$ (1.13)

${\bar{M}}^{\pm }$ and $\bar{P}$ are constants for the background doping profile. Let ${\bar{M}}^{\pm }=1$, thus $\bar{P}$ depends on $M^{-}\left(\bar{P}\mathrm{,1}\right)-1=0$.

Moreover, setting

$Z=P-\bar{P},m^{\pm }=M^{\pm }-1,$

We restate (1.12)-(1.13) based on the above equation:

$\{\begin{array}{l}{\partial }_t{m}^{+}+\text{div}w=-\text{div}\left(m^{+}w\right)\mathrm{,}\hfill \\ {\partial }_t{m}^{-}+\text{div}n=-\text{div}\left(m^{-}n\right)\mathrm{,}\hfill \\ {\partial }_{t}Z+{\beta }_1\text{div}w+{\beta }_2\text{div}n=G_1\mathrm{,}\hfill \\ {\partial }_{t}w+{\beta }_3\nabla Z-{\kappa }_1\Delta w-{\kappa }_2\nabla \text{div}w=H_1+R_1\mathrm{,}\hfill \\ {\partial }_{t}n+Z_4\nabla Z-{\varkappa }_1\Delta n-{\varkappa }_2\nabla \text{div}n=H_2+R_2\mathrm{,}\hfill \end{array}$ (1.14)

with initial data

$\left(m^{\pm }\mathrm{,}Z\mathrm{,}w\mathrm{,}n\right)\left(x\mathrm{,0}\right)=\left(m_0^{\pm }\mathrm{,}{Z}_0\mathrm{,}{u}_0\mathrm{,}{v}_0\right)\left(x\right)\to \left(\mathrm{0,0,}\stackrel{\to }{0}\mathrm{,}\stackrel{\to }{0}\right)\mathrm{,}\text{as}\left|x\right|\to +\infty \mathrm{.}$ (1.15)

where ${\beta }_1=ℒ\left(\mathrm{1,}\bar{P}\right)\eta \left(\bar{P}\right)$, ${\beta }_2=ℒ\left(\mathrm{1,}\bar{P}\right)\rho \left(\bar{P}\right)$, ${\beta }_3=\frac{1}{\rho \left(\bar{P}\right)}$, ${\beta }_4=\frac{1}{\eta \left(\bar{P}\right)}$, ${\kappa }_1=\frac{\rho {\left(\bar{P}\right)}^{{\mu }_1}}{\rho \left(\bar{P}\right)}$, ${\varkappa }_1=\frac{\eta {\left(\bar{P}\right)}^{{\mu }_2}}{\eta \left(\bar{P}\right)}$, ${\kappa }_2=\frac{\rho {\left(\bar{P}\right)}^{{\mu }_1}+\rho {\left(\bar{P}\right)}^{{\lambda }_1}}{\rho \left(\bar{P}\right)}$, ${\varkappa }_2=\frac{\eta {\left(\bar{P}\right)}^{{\mu }_2}+\eta {\left(\bar{P}\right)}^{{\lambda }_2}}{\eta \left(\bar{P}\right)}$ and the nonlinear terms are given by

$\begin{array}{c}{G}_1=-h_1\left(m^{+}\mathrm{,}Z\right)\text{div}w-h_2\left(m^{+}\mathrm{,}Z\right)\text{div}n-ℒ\left(m^{+}+\mathrm{1,}Z+\bar{P}\right)\eta u\cdot \nabla m^{+}\\ -ℒ\left(m^{+}+\mathrm{1,}Z+\bar{P}\right)\rho n\cdot \nabla m^{-}\mathrm{,}\end{array}$ (1.16)

$\begin{array}{c}{H}_1=-w\cdot \nabla w-\left(\frac{1}{\rho }-{\beta }_3\right)\nabla Z+\left(\frac{{\rho }^{{\mu }_1}}{\rho }-{\kappa }_1\right)\Delta w\\ +\left(\frac{{\rho }^{{\mu }_1}+{\rho }^{{\lambda }_1}}{\rho }-{\kappa }_2\right)\nabla \text{div}w\mathrm{,}\end{array}$ (1.17)

and

$\begin{array}{c}{R}_1=\frac{{\rho }^{{\mu }_1}\left(\nabla w+{\nabla }^{t}w\right)\nabla {\sigma }^{+}}{M^{+}}+\frac{{\rho }^{{\lambda }_1}\text{div}w\nabla {\sigma }^{+}}{M^{+}}\\ +\frac{\nabla \left({\rho }^{{\mu }_1}\right)\left(\nabla w+{\nabla }^{t}w\right)}{\rho }+\frac{\nabla \left({\rho }^{{\lambda }_1}\right)\text{div}w}{\rho }\mathrm{,}\end{array}$ (1.18)

$\begin{array}{c}{H}_2=-n\cdot \nabla n-\left(\frac{1}{\eta }-{\beta }_4\right)\nabla Z+\left(\frac{{\eta }^{{\mu }_2}}{\eta }-{\varkappa }_1\right)\Delta n\\ +\left(\frac{{\eta }^{{\mu }_2}+{\eta }^{{\lambda }_2}}{\eta }-{\varkappa }_2\right)\nabla \text{div}n\mathrm{,}\end{array}$ (1.19)

$\begin{array}{c}{R}_2=\frac{{\eta }^{{\mu }_2}\left(\nabla n+{\nabla }^{t}n\right)\nabla {\sigma }^{-}}{M^{-}}+\frac{{\eta }^{{\lambda }_2}\text{div}n\nabla {\sigma }^{-}}{M^{-}}\\ +\frac{\nabla \left({\eta }^{{\mu }_2}\right)\left(\nabla n+{\nabla }^{t}n\right)}{\eta }+\frac{\nabla \left({\eta }^{{\lambda }_2}\right)\text{div}n}{\eta }\mathrm{,}\end{array}$ (1.20)

where $h_i\left(i=\mathrm{1,2}\right)$ are given by

$\{\begin{array}{l}{h}_1\left(m^{+}\mathrm{,}Z\right)=ℒ\left(m^{+}+\mathrm{1,}Z+\bar{P}\right)\eta M^{+}-{\beta }_1\mathrm{,}\\ h_2\left(m^{+}\mathrm{,}Z\right)=ℒ\left(m^{+}+\mathrm{1,}Z+\bar{P}\right)\rho M^{-}-{\beta }_2\mathrm{.}\end{array}$ (1.21)

Now, we will show the difficulties in the proof and how to fix them.

First of all, we rewrite (1.12) as (1.14) by taking $m^{\pm }=M^{\pm }-1$, $Z=P-\bar{P}$. We obtain the corresponding linear system (1.26) for (1.14). To study time-decay estimates of the linear system (1.26), we must study the properties of the semigroup in detail. We use the Hodge decomposition technique [32] to decompose the linear system into three equations. We can get linear $L^2$ estimates using the Fourier transform.

Second, similar to [22] , we multiply (1.14)₃, (1.14)₄, (1.14)₅ by $\frac{{\beta }_3}{{\beta }_1}{\nabla }^{k}Z$, ${\nabla }^{k}w$,

and ${\nabla }^{k}n$ $\left(k=\mathrm{0,1,2}\right)$, respectively, and then integrate them. The difference between this paper and [22] is that when estimating the term on the right, we need to estimate the norms of ${\rho }^{{\mu }_1}$, ${\rho }^{{\lambda }_1}$, ${\eta }^{{\mu }_2}$ and ${\eta }^{{\lambda }_2}$. We noticed that $\rho$ and $\eta$ are related to $P$, making $Z=P-\bar{P}$. Therefore, norm estimates related to ${\rho }^{{\mu }_1}$, ${\rho }^{{\lambda }_1}$, ${\eta }^{{\mu }_2}$ and ${\eta }^{{\lambda }_2}$ can all be calculated using norm estimates related to $Z$. We deduce the energy of $\left(Z\mathrm{,}w\mathrm{,}n\right)$ and its derivatives.

In the last, we need handle $ℒ\eta w\cdot \nabla m^{+}$ and $ℒ\rho n\cdot \nabla m^{-}$ in (1.16). because we have chosen $P$ as our research variable. Dealing with items like ${\displaystyle {\int }_{{ℝ}^3}}ℒ\eta w\cdot \nabla {\nabla }^3{m}^{+}{\nabla }^{3}Z\text{d}x$ and ${\displaystyle {\int }_{{ℝ}^3}}ℒ\rho n\cdot \nabla {\nabla }^3{m}^{-}{\nabla }^{3}Z\text{d}x$ can be more difficult for us, so we have rewritten it as (5.11). At the same time, applying related inequalities, We deduced the time decay rates for $\left(Z\mathrm{,}n\mathrm{,}w\right)$. We have proven Theorem 1.2.

Main results.

Theorem 1.1. If $M_0^{\pm }-1\in H^3\left({ℝ}^3\right)$, $P_0-\bar{P}\mathrm{,}{w}_0\mathrm{,}{n}_0\in H^3\left({ℝ}^3\right)\cap L^1\left({ℝ}^3\right)$, then there is a constant ${\delta }_0$ such that if

$N_0\mathrm{<mo>\; :\; </mo>}={\Vert \left(M_0^{\pm }-1\right)\Vert }_{H^3}+{\Vert \left(P_0-\bar{P}\mathrm{,}{w}_0\mathrm{,}{n}_0\right)\Vert }_{H^3\cap L^1}\le {\delta }_0\mathrm{,}$ (1.22)

then the Cauchy problem (1.9)-(1.10) admits a unique global solution $\left(M^{\pm }\mathrm{,}P\mathrm{,}w\mathrm{,}n\right)$ in time, satisfying

$\begin{array}{l}{M}^{\pm }-\mathrm{1,}P-\bar{P}\in C^0\left(\left[\mathrm{0,}\infty \right)\mathrm{<mo>\; ;\; </mo>}{H}^3\left({ℝ}^3\right)\right)\cap C^1\left(\left[\mathrm{0,}\infty \right)\mathrm{<mo>\; ;\; </mo>}{H}^2\left({ℝ}^3\right)\right)\mathrm{,}\\ w\mathrm{,}n\in C^0\left(\left[\mathrm{0,}\infty \right)\mathrm{<mo>\; ;\; </mo>}{H}^3\left({ℝ}^3\right)\right)\cap C^1\left(\left[\mathrm{0,}\infty \right)\mathrm{<mo>\; ;\; </mo>}{H}^1\left({ℝ}^3\right)\right)\mathrm{.}\end{array}$

After obtaining the global fitness of the model solutions, we further investigate the properties of the solution and study the time decay rate of the solutions.

Theorem 1.2. If $\left(M^{\pm }\mathrm{,}P\mathrm{,}w\mathrm{,}n\right)$ satisfies the conditions in Theorem 1.1, for any $t\ge 0$, constant $C_0>0$ exists and is independent of $t$, we can get estimates of the solution $\left(M^{\pm }\mathrm{,}P\mathrm{,}w\mathrm{,}n\right)$:

${\Vert \left(P-\bar{P}\mathrm{,}w\mathrm{,}n\right)\left(t\right)\Vert }_{L^2}\le C_0{N}_0{\left(1+t\right)}_{}^{-\frac{3}{4}}\mathrm{,}$ (1.23)

${\Vert \nabla P\left(t\right)\Vert }_{H^1}+{\Vert \nabla \left(w\mathrm{,}n\right)\left(t\right)\Vert }_{H^2}\le C_0{N}_0{\left(1+t\right)}_{}^{-\frac{5}{4}}\mathrm{.}$ (1.24)

and

${\Vert {\nabla }^{3}P\left(t\right)\Vert }_{L^2}+{\Vert \left(M^{+}-\mathrm{1,}{M}^{-}-1\right)\left(t\right)\Vert }_{H^3}\le C_0{N}_0\mathrm{.}$ (1.25)

In this paper, ${\Vert \cdot \Vert }_{H^k}$ is expressed as the norm in Sobolev space $H^k\left({ℝ}^3\right)$. $L^p\left({ℝ}^3\right)$ is expressed as the norm in $L^p$ space, $1\le p\le \infty$. The norm ${\Vert \left(g\mathrm{,}h\right)\Vert }_X$ is short for ${\Vert g\Vert }_X+{\Vert h\Vert }_X$. $x\lesssim y$ means that $x\le Ky$, where positive constant $K>0$. We represent $\nabla ={\partial }_x=\left({\partial }_1\mathrm{,}{\partial }_2\mathrm{,}{\partial }_3\right)$ like [33] , let ${\partial }_x^{\ell }g={\nabla }^{\ell }g=\nabla \left({\nabla }^{\ell -1}g\right)$. ${\Lambda }^r$ is a pseudo differential operator, defined as

${\Lambda }^{r}g={\mathfrak{G}}^{-1}\left({\left|\xi \right|}^r\hat{g}\right)\mathrm{,\; <mi>\; </mi>}\text{for}\mathrm{<mi>\; </mi>}r\in ℝ\mathrm{,}$

where $\hat{g}$ and $\mathfrak{G}\left(g\right)$ are the Fourier transform of $g$. ${\Vert g\Vert }_{{\stackrel{\dot{}}{H}}^r}\triangleq {\Vert {\Lambda }^{r}g\Vert }_{L^2}$ represents the norm of homogenous Sobolev space ${\stackrel{\dot{}}{H}}^r\left({ℝ}^3\right)$. The radial function $\Psi$ satisfies

$\{\begin{array}{l}\Psi \left(\varsigma \right)=\mathrm{1,}\left|\varsigma \right|\le \frac{\zeta }{2}\hfill \\ \Psi \left(\varsigma \right)=\mathrm{0,}\left|\varsigma \right|\ge \zeta \hfill \end{array}$

see Lemma 3.1 for the definition of $\zeta$.

We define

$g^l={\mathfrak{G}}^{-1}\left[\Psi \left(\varsigma \right)\hat{g}\right],g^h={\mathfrak{G}}^{-1}\left[\left(1-\Psi \left(\varsigma \right)\right)\hat{g}\right]\mathrm{.}$

where $g^l$ and $g^h$ respectively represent the high-frequency and low-frequency parts of $g$. If the Fourier transform of $g$ exists, then $g=g^l+g^h$.

Moreover, for the convenience of the reader, here is a summary of the Gagliardo-Nirenberg inequality.

Lemma 2.1. If $0\le l\mathrm{,}k\le s$, then

${\Vert {\nabla }^{l}g\Vert }_{L^{\alpha }}\lesssim {\Vert {\nabla }^{k}g\Vert }_{L^{\theta }}^{1-b}{\Vert {\nabla }^{s}g\Vert }_{L^r}^b$

where $b$ belongs to $\left[\frac{l}{s}\mathrm{,1}\right]$ and satisfies

$\frac{l}{3}-\frac{1}{\alpha }=\left(\frac{k}{3}-\frac{1}{\theta }\right)\left(1-b\right)+\left(\frac{s}{3}-\frac{1}{r}\right)b\mathrm{.}$

Particularly, if $p$, $q$ and $r$ are all equal to 2, we get

${\Vert {\nabla }^{l}g\Vert }_{L^2}\lesssim {\Vert {\nabla }^{k}g\Vert }_{L^2}^{\frac{s-l}{s-k}}{\Vert {\nabla }^{k}g\Vert }_{L^2}^{\frac{l-k}{s-k}}\mathrm{.}$

Proof. The proof is shown on page 125 of [34] . □

Lemma 2.2. For $s\ge 1$, we have

${\Vert {\nabla }^s\left(gh\right)\Vert }_{L^r}\lesssim {\Vert g\Vert }_{L^{r_1}}{\Vert {\nabla }^{s}h\Vert }_{L^{r_2}}+{\Vert {\nabla }^{s}g\Vert }_{L^{r_3}}{\Vert h\Vert }_{L^{r_4}}\mathrm{,}$

and

${\Vert {\nabla }^s\left(gh\right)-f{\nabla }^{s}h\Vert }_{L^r}\lesssim {\Vert \nabla g\Vert }_{L^{r_1}}{\Vert {\nabla }^{s-1}h\Vert }_{L^{r_2}}+{\Vert {\nabla }^{s}g\Vert }_{L^{r_3}}{\Vert h\Vert }_{L^{r_4}}\mathrm{,}$

where $r\mathrm{,}{r}_1\mathrm{,}{r}_2\mathrm{,}{r}_3\mathrm{,}{r}_4\in \left[\mathrm{1,}\infty \right]$ and

$\frac{1}{r}=\frac{1}{r_1}+\frac{1}{r_2}=\frac{1}{r_3}+\frac{1}{r_4}\mathrm{.}$

Proof. The lemma has been proved in [35] and [36] . □