On the stability of stationary compressible Navier–Stokes flows in 3D

Abstract

This paper studies the stability of the stationary solution of the compressible Navier–Stokes equation in the 3D whole space with an external force which decays at spatial infinity. We obtain the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the small stationary solution. We also derive the time decay rates of the perturbations under the smallness assumption on the initial perturbations, and show the optimality of the time decay rates. The proofs are based on the combination of the spectral analysis and energy method in the framework of Besov spaces. The time-space integral estimate for the linearized semigroup around the constant state in some Besov spaces plays a crucial role in the proofs.

1 Introduction

We consider the Cauchy problem for the compressible Navier–Stokes equation:

$$\begin{aligned} \left\{ \, \begin{aligned}&\partial _{t}\rho + \textrm{div}(\rho v) = 0, \\&\partial _{t}(\rho v) + \textrm{div}(\rho v \otimes v) = \mu \Delta v + (\mu +\mu ')\nabla \textrm{div}\,v -\nabla P(\rho ) + \rho F(x), \end{aligned} \right. \end{aligned}$$

(1)

with initial data $(\rho ,v)|_{t=0}=(\rho _0,v_0)$ and the boundary condition $(\rho ,v)(t,x)\rightarrow (\rho _\infty ,0)$ as $|x|\rightarrow \infty .$ Here $t\ge 0,$ $x\in {\mathbb {R}}^3,$ $v=(v_1,v_2,v_3)$ is the fluid velocity, $\rho$ is the fluid density, $\rho _\infty$ is a given positive constant, P is a given pressure, $\mu$ and $\mu '$ are given viscosity coefficients and $F=(F_1,F_2,F_3)$ is a given stationary external force. We assume that $\mu$ and $\mu '$ are constants that satisfy $\mu >0$ and $2\mu /3+\mu '\ge 0,$ and P is a smooth function of $\rho$ in a neighborhood of $\rho _\infty$ with $P'(\rho _{\infty })>0.$ As F is a stationary external force, we introduce the corresponding stationary problem:

$$\begin{aligned} \left\{ \, \begin{aligned}&\textrm{div}(\rho ^{*} v^{*}) = 0, \\&\textrm{div}(\rho ^{*} v^{*} \otimes v^{*}) = \mu \Delta v^{*} + (\mu +\mu ')\nabla \textrm{div}\,v^{*} -\nabla P(\rho ^{*}) + \rho ^* F(x), \end{aligned} \right. \end{aligned}$$

(2)

with the boundary condition $(\rho ^*,v^*)(x)\rightarrow (\rho _\infty ,0)$ as $|x|\rightarrow \infty .$

It was shown by Shibata and Tanaka [17] that if the stationary external force F has the form $F=\textrm{div}F_1+F_2$ where $F_1$ is a $3\times 3$-matrix valued function and $F_2$ is a 3-vector valued function and F is small in the quantity:

$$\begin{aligned} {\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| F \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }&= \sum _{k=0}^{3} \Vert (1+|x|)^{k+1}\nabla ^{k}F\Vert _{L^2} + \Vert (1+|x|)^{3}F\Vert _{L^{\infty }} \\&\quad + \Vert (1+|x|)^2 F_{1}\Vert _{L^\infty } + \Vert F_2\Vert _{L^1}, \end{aligned}$$

then there exists a unique stationary solution $(\rho ^*,v^*)=(\sigma ^*+\rho _\infty ,v^*)$ which satisfies the decay properties

$$\begin{aligned} |v^{*}(x)|\lesssim \frac{\delta }{|x|},\quad |\nabla v^{*}(x)|\lesssim \frac{\delta }{|x|^{2}},\quad |\sigma ^*(x)|\lesssim \frac{\delta }{|x|^2}, \end{aligned}$$

(3)

as $|x|\rightarrow \infty$ with $\delta ={\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| F \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| },$ and the smoothness property

$$\begin{aligned} \Vert (1+|x|)^{j}\nabla ^{j}\sigma ^*\Vert _{L^2} + \Vert (1+|x|)^{j}\nabla ^{j+1}v^*\Vert _{L^2} \lesssim \delta \end{aligned}$$

for any $0\le j \le 4.$ In addition, Shibata and Tanaka [17] proved that if the initial perturbation $(\rho _0-\rho ^*, v_0-v^*)$ belongs to $H^3$ and is small in $H^3$ norm, then there exists a unique solution $(\rho ,v)=(\sigma +\rho ^{*},w+v^*)$ of (1) such that $\sigma \in C^{0}([0,\infty );H^3)\cap C^{1}([0,\infty );H^2),$ $w \in C^{0}([0,\infty );H^3)\cap C^{1}([0,\infty );H^1)$ and

$$\begin{aligned}&\sup _{0\le t<\infty }\Vert (\sigma ,w)(t)\Vert _{H^3}^2 +\int _{0}^t\Vert \nabla \sigma (\tau )\Vert _{{H}^{2}}^2 + \Vert \nabla w(\tau )\Vert _{{H}^{3}}^2 + \Vert \partial _t w(\tau )\Vert _{{H}^{2}}^2 d\tau \\&\quad \lesssim \Vert (\rho _{0}-\rho ^*,v_0-v^*)\Vert _{H^3}^2. \end{aligned}$$

In [18], Shibata and Tanaka then derived the decay rates of the perturbations

$$\begin{aligned} \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{H}^1}\lesssim _{\epsilon } (1+t)^{-\frac{1}{2}+\epsilon }\Vert (\rho _0-\rho ^*,v_0-v^*)\Vert _{L^{\frac{6}{5}}\cap H^3}, \end{aligned}$$

(4)

where $\epsilon >0$ is an arbitrary constant. Here, $\dot{H}^{s}$ denotes the homogeneous Sobolev space whose definition is given in Sect. 2 below.

On the other hand, when the external force $F=0,$ the corresponding stationary solution is the motionless state $(\rho _\infty ,0),$ and the time-decay estimates of its perturbation are given by

$$\begin{aligned} \Vert (\rho -\rho _\infty ,v)(t)\Vert _{\dot{H}^s}\lesssim (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}- \frac{1}{2}\right) }\Vert (\rho _0-\rho _\infty ,v_0)\Vert _{L^{p}\cap H^3} \end{aligned}$$

(5)

for $0\le s\le 2$ and $1\le p\le 2.$ (Cf. [11,12,13, 15].) Note that the decay rates in (4) are slower than in (5) with $s=1,$ $p=6/5.$ Since the stationary solution $(\rho ^*,v^*)$ is close to the motionless state $(\rho _\infty ,0),$ one could expect that the decay estimate (5) would also hold for the perturbation of $(\rho ^*,v^*).$ However, it is not straightforward to see this since the spatial decay of the stationary velocity field $v^*$ is slow as is written in (3).

The aim of this paper is the following twofold. The first is to derive the optimal decay rate of the perturbations of the stationary solution $(\rho ^*, v^*)$ under the smallness assumptions on the initial perturbation of $(\rho ^*, v^*).$ We prove that there holds the decay estimate

$$\begin{aligned} \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{H}^{s}} \lesssim _{s,p} (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}- \frac{1}{2}\right) }\Vert (\rho _0-\rho ^*,v_0-v^*)\Vert _{L^p\cap H^3}, \end{aligned}$$

(6)

where $-3/2<s<3/2$ and $1\le p \le 2$ with $s/2+3/2(1/p-1/2)>0$ or $s=0$ and $p=2,$ which especially shows that the decay estimate (5) with $0\le s<3/2$ and $1\le p\le 2$ hold for the perturbation of $(\rho ^*,v^*).$ In fact, we will derive the decay estimate (6) for the stationary solutions $(\rho ^*,v^*)$ in a larger class than that studied in [17, 18]. The second is to establish the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the stationary solution $(\rho ^*, v^*),$ without assuming that the initial perturbation $(\rho _0-\rho ^*,v_0-v^*)$ belongs to $L^2.$ In fact, we shall construct the global solution when the initial perturbation $(\rho _0-\rho ^*, v_0-v^*)$ belongs to $\dot{B}^{1/2}_{2,\infty } \cap \dot{H}^3$ with small norm. Here, $\dot{B}^{s}_{p,r}$ denotes the homogeneous Besov space whose definition is given in Sect. 2 below. Note that the velocity $v^*$ of the stationary solution is not necessarily in $L^2$ but in $\dot{B}^{1/2}_{2,\infty } \cap \dot{H}^3;$ and so our result claims the global existence of solutions of problem (1) for a class of initial data which contains not only the stationary solution obtained in [17] but also the one constructed in this paper. We shall consider the stationary solutions obtained in the following theorem.

Theorem 1.1

There exists a constant $\delta _0>0$ such that if $F\in \dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^{3}$ and

$$\begin{aligned} \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3} \le \delta _0, \end{aligned}$$

then the stationary problem (2) has a unique solution $(\rho ^*,v^*)=(\sigma ^*+\rho _\infty ,v^*)$ satisfying

$$\begin{aligned} \Vert \sigma ^*\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }\cap \dot{H}^{4}} + \Vert v^*\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{5}} \lesssim \delta _0. \end{aligned}$$

(7)

In the proof of the existence of stationary solutions, there is a difficulty in that the convection term $\textrm{div}(\sigma ^{*}v^*)$ causes a loss of derivative. This difficulty is overcome by rewriting the equation by using the Helmholtz decomposition as in Lemma 3.2 below and regularizing the convection term $\textrm{div}(\sigma ^{*}v^*).$ Theorem 1.1 shows the existence of stationary solutions for data F in a larger class than that of [17].

Let us now state our main theorem, which derives the global existence of (1) and the decay rates of the perturbations.

Theorem 1.2

Let $(\rho ^*,v^*)$ be the stationary solution satisfying (7) with $\Vert F\Vert _{\dot{B}^{-3/2}_{2,\infty }\cap \dot{H}^3}$ sufficiently small. Then,  there exists a constant $\delta >0$ such that if the initial perturbation $(\rho _0-\rho ^*,v_0-v^*)\in \dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3$ and

$$\begin{aligned} \Vert (\rho _0-\rho ^*,v_0-v^*)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} \le \delta , \end{aligned}$$

(8)

then the compressible Navier–Stokes equation (1) with initial data $(\rho _0,v_0)$ has a unique global solution $(\rho ,v)$ satisfying $(\rho -\rho ^*,v-v^*)\in C^0([0,\infty );\dot{B}^{1/2}_{2,\infty }\cap \dot{H}^3)$ and

$$\begin{aligned} \sup _{0\le t<\infty } \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}\lesssim \delta . \end{aligned}$$

(9)

In addition,  if $(\rho _0-\rho ^*,v_0-v^*)\in \dot{B}^{s_0}_{2,\infty }$ for some $-3/2\le s_0 \le 1/2,$ then the decay estimate

$$\begin{aligned} \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{s} (1+t)^{-\frac{s-s_0}{2}} \Vert (\rho _0-\rho ^*,v_0-v^*)\Vert _{\dot{B}^{s_0}_{2,\infty }\cap \dot{H}^3} \end{aligned}$$

(10)

holds for $-3/2<s<3/2$ with $s_0\le s$ and $t\ge 0.$

Remark 1.3

The time decay estimate of the perturbation when the initial perturbation belongs to $\dot{B}^{s_0}_{2,\infty }$ with some negative $s_0$ has already been studied by Danchin and Xu [9], Xu [20] for the case $(\rho ^*,v^*)=(\rho _\infty ,0).$ They derived the decay estimate of the solution constructed in the $L^p$ critical regularity framework.

Remark 1.4

The time decay estimate in Theorem 1.2 can be derived without an additional smallness assumption of the initial perturbation in $\dot{B}^{s_0}_{2,\infty }.$ In fact, we will show the following type estimate. (See Propositions 5.2 and 5.5 below.) For any $-3/2\le s_0\le 1/2,$ $\epsilon >0$ and $T>0,$ let

$$\begin{aligned} {\mathcal {D}}_{\epsilon ,s_0}(T) \equiv \sup _{\begin{array}{c} -3/2+\epsilon \le \eta \le 3/2-\epsilon , \\ s_0\le \eta \end{array}}\ \sup _{ 0 \le t \le T} (1+t)^{\frac{\eta -s_0}{2}}\Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{B}^{\eta }_{2,\infty }}. \end{aligned}$$

We then have

$$\begin{aligned} {\mathcal {D}}_{\epsilon ,s_0}(T)&\lesssim _{\epsilon } \Vert (\rho _0-\rho ^*,v_0-v^*)\Vert _{\dot{B}^{s_0}_{2,\infty }}\\&\quad +\Big ( \sup _{0\le t<\infty } \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}+\Vert F\Vert _{\dot{B}^{-3/2}_{2,\infty }\cap \dot{H}^3}\Big ){\mathcal {D}}_{\epsilon ,s_0}(T), \end{aligned}$$

and hence, we arrive at

$$\begin{aligned} \sup _{T>0} {\mathcal {D}}_{\epsilon ,s_0}(T)<\infty \end{aligned}$$

if $\Vert F\Vert _{\dot{B}^{-3/2}_{2,\infty }\cap \dot{H}^3}$ and $\Vert (\rho _0-\rho ^*,v_0-v^*)\Vert _{\dot{B}^{1/2}_{2,\infty }\cap \dot{H}^3}$ are small enough.

We also have the following estimate for the perturbation.

Theorem 1.5

Let $(\rho ^*,v^*),$ $(\rho ,v)$ be as in Theorem 1.2, with

$$\begin{aligned} \Vert (\rho _0-\rho ^*,v_0-v^*)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}+\Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3} \end{aligned}$$

sufficiently small. If the initial perturbation $(\rho _0-\rho ^*,v_0-v^*)\in L^p$ for some $1\le p\le 2,$ then the decay estimate

$$\begin{aligned} \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{H}^s}\lesssim _{s} (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) }\Vert (\rho _0-\rho ^*, v_{0}-v^*)\Vert _{L^{p}\cap H^{3}}\nonumber \\ \end{aligned}$$

(11)

holds for $-3/2<s<3/2$ with $s/2+3/2(1/p-1/2)>0$ or $s=0$ and $p=2.$

Here, we mention that in the case where the external force $F=0,$ the time decay of the perturbation was derived by Xu [20] when the initial perturbation belongs to Besov space with some negative exponents. We also mention that the smallness of the initial perturbation in $\dot{B}^{s_0}_{2,\infty }$ or $L^p$ are not needed in Theorems 1.2 and 1.5, since by using Lemma 5.3 below, the nonlinear estimates in the proof of decay estimate can be done under the smallness assumption in (9).

We obtain the following results regarding the optimality of the estimates in Theorem 1.5.

Theorem 1.6

Let $(\rho ^*,v^*)$ be the stationary solution satisfying (7) with $\Vert F\Vert _{\dot{B}^{-3/2}_{2,\infty }\cap \dot{H}^3}$ sufficiently small. Then,  the following hold : 

• (i) There exists an initial perturbation $(\rho _0-\rho ^*, v_{0}-v^*)\in L^{1}\cap H^3$ with sufficiently small $\Vert (\rho _0-\rho ^*, v_{0}-v^*)\Vert _{\dot{B}^{1/2}_{2,\infty } \cap \dot{H} ^{3}}$ such that the corresponding global solution $(\rho ,v)$ satisfies

  $$\begin{aligned} \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{H}^{s}} \sim _{s} (1+t)^{-\frac{s}{2}-\frac{3}{4}}\Vert (\rho _0-\rho ^*, v_{0}-v^*)\Vert _{L^{1}\cap H^{3}}, \end{aligned}$$

  where $-3/2<s<3/2$ and $t\gg 1.$

• (ii) Under the same assumption as in Theorem 1.5, if $(\rho _0-\rho ^*,v_0-v^*)\in L^{p,\infty }$ for some $1< p < 2,$ then the following estimate

  $$\begin{aligned} \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{H}^{s}} \lesssim _{s,p} (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) }\Vert (\rho _0-\rho ^*, v_{0}-v^*)\Vert _{L^{p,\infty }\cap H^{3}} \end{aligned}$$

  (12)

  holds for $-3/2<s<3/2$ with $s/2+3/2(1/p-1/2)>0$ and $t\ge 0.$ In addition,  there exists an initial perturbation $(\rho _0-\rho ^*, v_{0}-v^*)\in L^{p,\infty }\cap H^3$ with sufficiently small $\Vert (\rho _0-\rho ^*, v_{0}-v^*)\Vert _{\dot{B}^{1/2}_{2,\infty }\cap H^{3}}$ such that the corresponding global solution $(\rho ,v)$ satisfies

  $$\begin{aligned} \Vert (\rho -\rho ^*,v-v^*)(t)\Vert _{\dot{H}^{s}} \sim _{s,p} (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) }\Vert (\rho _0-\rho ^*, v_{0}-v^*)\Vert _{L^{p,\infty }\cap H^{3}}, \end{aligned}$$

  where $-3/2<s<3/2$ with $s/2+3/2(1/p-1/2)>0$ and $t\gg 1.$

The difficulty in deriving the time-decay estimate of the perturbation $(\rho -\rho ^*,v-v^*)$ arises from the slow spatial decay of the stationary solution. The spatial decay order of $\nabla v^*$ is expected to be at most $O(1/|x|^2),$ so the linear term $w\cdot \nabla v^{*}$ appearing in the perturbation equation may need to be treated as the inverse-square potential term. Indeed, Davies and Simon [10] showed that the inverse-square type potential term can affect the asymptotic behavior of the solution of the heat equation. For this reason, it is not straightforward that the linear term $w\cdot \nabla v^*$ is considered as a simple perturbation of the linearized operator around the motionless state $(\rho _{\infty },0).$ The same difficulty arises in the analysis of the linear term $v^*\cdot \nabla w,$ since the spatial decay order of $v^*$ is O(1/|x|).

To overcome the difficulty arising from the linear terms $v^*\cdot \nabla w$ and $w\cdot \nabla v^{*},$ we shall formulate the decay problem in a framework of weak-type Besov spaces and prove the estimate (10). The proof of decay estimate (10) is performed by decomposing the perturbation into low- and high-frequency parts. The analysis of the low frequency part is carried out by using the momentum formulation, while the analysis of the high-frequency part is carried out by using the velocity formulation in order to avoid the derivative loss. (Cf. [1, 19].) The low frequency part is estimated by spectral analysis around the motionless state $( \rho _{\infty },0).$ Here, a crucial role is played by the time-space integral estimate established by Danchin [6]. (See Lemma 4.1 below.) A similar analysis is found in Yamazaki [21] where the time-space integral estimate in the Lorentz spaces is effectively employed to study the incompressible Navier–Stokes equation under time-dependent external force. In this direction, we also mention that the work by Chemin [4], where the time-space integral estimate in Besov spaces was established for incompressible Navier–Stokes equation. The estimate of the high-frequency part is established by the energy method in Besov spaces developed by Danchin [6]. The proof of optimality in Theorem 1.6 is inspired by the argument in Kawashima, Matsumura and Nishida [12]. We finally note that, in the case of $F=0$ , the decay rate of the perturbation of the motionless state $(\rho _\infty ,0)$ was studied in critical spaces [5, 9, 14, 20]. It is also an interesting issue to consider the decay rate of the perturbation of the stationary solution $(\rho ^*,v^*)$ in critical spaces.

Organization of the paper. In Sect. 2, we present the notation used throughout this paper and the basic facts of the homogeneous Besov spaces. Section 3 is devoted to the proof of existence of solutions to the stationary problem (2). In Sect. 4, we construct a global solution under the smallness assumptions on an initial value and an external force. In Sect. 5, we derive the decay rate of the perturbation under the smallness assumptions on the initial perturbation. We also show the result regarding the optimality of the estimates in Theorem 1.5.

2 Preliminary

The notation $A\lesssim _{\alpha } B$ means that there exists a constant C depending on $\alpha$ such that $A\le CB.$ The notation $A\sim _\alpha B$ means that $A\lesssim _\alpha B$ and $B\lesssim _\alpha A.$ We denote a commutator by $[X,Y]\equiv XY-YX.$ We write ${\mathcal {S}}$ for the set of all Schwartz functions on ${\mathbb {R}}^{3},$ and we write ${\mathcal {S}}'$ for the set of all tempered distributions on ${\mathbb {R}}^{3}.$ The notations ${\hat{\cdot }},$ ${\mathcal {F}}$ stand for the Fourier transform

$$\begin{aligned} \hat{u}(\xi ) = {\mathcal {F}}(u)(\xi )\equiv \int _{{\mathbb {R}}^{3}} e^{-i x \cdot \xi } u(x)dx, \end{aligned}$$

and the notation ${\mathcal {F}}^{-1}$ denotes the inverse Fourier transform. The symbol ${\mathbb {P}}$ denotes the Helmholtz projection: ${\mathbb {P}}u \equiv u- \Delta ^{-1}\nabla \textrm{div}\,u,$ $u\in {\mathcal {S}}'.$ We denote the $L^{2}({\mathbb {R}}^3)$ inner product by $\langle u,v\rangle \equiv \int _{{\mathbb {R}}^3} u vdx.$ Let $s\in {\mathbb {R}}.$ The homogeneous Sobolev space $\dot{H}^{s}=\dot{H}^{s}({\mathbb {R}}^3)$ is the set of tempered distributions u on ${\mathbb {R}}^{3}$ such that $\hat{u}\in L^1_{loc},$ $\Vert u\Vert _{\dot{H}^{s}}\equiv \Vert |\cdot |^{s}\hat{u}\Vert _{L^2}<\infty .$ The inhomogeneous Sobolev space $H^{s}=H^{s}({\mathbb {R}}^3)$ is the set of tempered distributions u on ${\mathbb {R}}^{3}$ such that $\Vert u\Vert _{H^{s}}\equiv \Vert (1+|\cdot |)^{s}\hat{u}\Vert _{L^2}<\infty .$ Let $1\le p\le \infty .$ The weak $L^p$ space $L^{p,\infty }=L^{p,\infty }({\mathbb {R}}^3)$ is the set of measurable functions on ${\mathbb {R}}^{3}$ such that

$$\begin{aligned} \Vert u\Vert _{L^{p,\infty }}\equiv \sup _{t>0}t\, m(\{x\mid |u(x)|>t\})^{\frac{1}{p}}<\infty , \end{aligned}$$

where m is the Lebesgue measure on ${\mathbb {R}}^{3}.$ Let I be an interval in ${\mathbb {R}}$ and let X be a Banach space. The Bochner space $L^{p}(I;X)$ is the set of strongly measurable functions $u:I\rightarrow X$ such that

$$\begin{aligned} \Vert u\Vert _{L^{p}(I;X)} \equiv \left( \int _{I} \Vert u(t)\Vert _{X}^{p}dt \right) ^{\frac{1}{p}}<\infty . \end{aligned}$$

The rest of this section is devoted to introducing the homogeneous Besov spaces and presenting some basic facts. These will be applied effectively throughout this paper. To apply our analysis, we employ the squared dyadic partition of unity. Choose $\phi \in C^{\infty }({\mathbb {R}}^3)$ supported in the annulus ${\mathcal {C}}=\{\xi \in {\mathbb {R}}^3 \mid 3/4 \le |\xi | \le 8/3\}$ such that

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}}\phi ^2(2^{-j}\xi )=1\quad {\textrm{for}}\ \xi \ne 0. \end{aligned}$$

Define the dyadic blocks $(\dot{\Delta }_{j})_{j\in {\mathbb {Z}}}$ by the Fourier multiplier

$$\begin{aligned} \dot{\Delta }_{j}u \equiv {\mathcal {F}}^{-1} [\phi ^{2}(2^{-j}\cdot ) \hat{u}], \end{aligned}$$

and the square-rooted dyadic blocks by $\dot{\Delta }^{1/2}_{j}u\equiv {\mathcal {F}}^{-1}[ \phi (2^{-j}\cdot )\hat{u}].$ The homogeneous low frequency cutoff operator is denoted by

$$\begin{aligned} \dot{S}_{j}u \equiv \sum _{j' < j}\dot{\Delta }_{j'}u,\quad j\in {\mathbb {Z}}. \end{aligned}$$

(13)

At least formally, the decomposition

$$\begin{aligned} u=\sum _{j\in {\mathbb {Z}}}\dot{\Delta }_{j}u \end{aligned}$$

can be considered, which is called a Littlewood–Paley decomposition. We fix $\phi _0\in C_{0}^{\infty }({\mathbb {R}}^3)$ satisfying $\phi _0(0)\ne 0.$

Let $s\in {\mathbb {R}},$ $1\le p, r \le \infty .$ Then, the homogeneous Besov space $\dot{B}^{s}_{p,r}=\dot{B}^{s}_{p,r}({\mathbb {R}}^{3})$ is given by

$$\begin{aligned}&\dot{B}^{s}_{p,r}\equiv \big \{ u\in {\mathcal {S}}' \ \mid \ \lim _{j\rightarrow -\infty } \Vert {\mathcal {F}}^{-1} [\phi _{0}(2^{-j}\cdot ) \hat{u}]\Vert _{L^\infty }=0,\ \ \Vert u\Vert _{\dot{B}^{s}_{p,r}} < \infty \big \},\\&\Vert u\Vert _{\dot{B}^{s}_{p,r}}\equiv \left\| (2^{js}\Vert \dot{\Delta }_{j}u\Vert _{L^p})_{j\in {\mathbb {Z}}}\right\| _{\ell ^{r}}. \end{aligned}$$

We state some basic facts on homogeneous Besov spaces, which are frequently used in this paper.

Proposition 2.1

Let $s,\tilde{s}\in {\mathbb {R}},$ $1\le p,\tilde{p}, r,\tilde{r} \le \infty$ and $u,v\in {\mathcal {S}}'.$

• (i) (Derivative) For any $k\ge 0,$ $\Vert \nabla ^{k}u\Vert _{\dot{B}_{p,r}^{s}}\sim \Vert u\Vert _{\dot{B}_{p,r}^{s+k}}.$

• (ii) (Duality) Let $p'$ be a conjugate exponent of p and let $r'$ be a conjugate exponent of r. Let $v\in {\mathcal {S}}.$ Then,  we have the following duality estimates : 

  $$\begin{aligned} \langle u,v \rangle \lesssim \Vert u\Vert _{\dot{B}^{s}_{p,r}}\Vert u\Vert _{\dot{B}^{-s}_{p',r'}}\quad {\textrm{and}}\quad \Vert u\Vert _{\dot{B}^{s}_{p,r}}\lesssim \sup _{\psi }\langle u,\psi \rangle , \end{aligned}$$

  where the supremum is taken over the Schwartz functions $\psi$ with $\Vert \psi \Vert _{\dot{B}^{-s}_{p',r'}} \le 1$ and $0\notin {\text {supp}}{\mathcal {F}}\psi .$

• (iii) (Interpolation) Let $s_1 < s_2$ satisfy $s=(1-\theta )s_1 + \theta s_2$ for some $0<\theta <1.$ Then,  the interpolation inequality

  $$\begin{aligned} \Vert u\Vert _{\dot{B}^{s}_{p,r}} \lesssim _{\theta , s_1,s_2} \Vert u\Vert ^{1-\theta }_{\dot{B}^{s_1}_{p,\infty }}\Vert u\Vert ^{\theta }_{\dot{B}^{s_2}_{p,\infty }} \end{aligned}$$

  (14)

  holds.

• (iv) (Fatou property) Assume $s<3/p$ or $s=3/p,$ $r=1.$ If $\{u_n\}_n$ is a bounded sequence in $\dot{B}_{p,r}^{s}\cap \dot{B}_{\tilde{p},\tilde{r}}^{\tilde{s}},$ then there exists a subsequence of $\{u_n\}_n$ (without relabeling) and $u\in \dot{B}^{s}_{p,r}\cap \dot{B}_{\tilde{p},\tilde{r}}^{\tilde{s}}$ such that

  $$\begin{aligned} \lim _{n\rightarrow \infty } u_{n} = u\ {\textrm{in}}\ {\mathcal {S}}'\quad {\textrm{and}}\quad \Vert u\Vert _{\dot{B}^{s}_{p,r}\cap \dot{B}_{\tilde{p},\tilde{r}}^{\tilde{s}}} \lesssim \liminf _{n\rightarrow \infty } \Vert u_n\Vert _{\dot{B}^{s}_{p,r}\cap \dot{B}_{\tilde{p},\tilde{r}}^{\tilde{s}}}. \end{aligned}$$

• (v) (Paralinearization) Let $\Phi \in C^{\infty }({\mathbb {R}}^{3})$ and $u,v\in \dot{B}^{s}_{2,r}\cap \dot{B}^{3/2}_{2,1}$ with $-3/2\le s<3/2$ or $s=3/2,$ $r=1.$ Then,  we have

  $$\begin{aligned} \Vert \Phi (u)-\Phi (v)\Vert _{\dot{B}^{s}_{2,r}} \lesssim _{\Phi } (1+\Vert (u,v)\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}}) \Vert u-v\Vert _{\dot{B}^{s}_{2,r}}. \end{aligned}$$

• (vi) (Bilinear estimate) Let $s_1,s_2\in {\mathbb {R}}$ satisfy $s_1,s_2<3/2$ and $s_1+s_2>0.$ Let $1\le r_1,r_2 \le \infty$ satisfy $1/r_1+1/r_2=1/r.$ Then,  we have

  $$\begin{aligned} \Vert uv\Vert _{\dot{B}_{2,r}^{s_1+s_2-\frac{3}{2}}} \lesssim _{s_1,s_2} \Vert u\Vert _{\dot{B}_{2,r_1}^{s_1}}\Vert v\Vert _{\dot{B}_{2,r_2}^{s_2}}. \end{aligned}$$

  In the cases $s_1\le 3/2,$ $s_2 <3/2$ with $s_1+s_2\ge 0,$ we have

  $$\begin{aligned} \Vert uv\Vert _{\dot{B}_{2,\infty }^{s_1+s_2-\frac{3}{2}}} \lesssim \Vert u\Vert _{\dot{B}_{2,1}^{s_1}}\Vert v\Vert _{\dot{B}_{2,\infty }^{s_2}}. \end{aligned}$$

As for the proofs other than $({\textrm{v}}),$ see [2, Lemma 2.1, Proposition 2.22, 2.29, Theorem 2.25, 2.47, 2.52 and Corollary 2.91] for example. The proof of Proposition 5 (v) is same as in the proof of [6, Lemma 1.6 ii)].

Generalized Young’s inequality (see [16, pp. 31-32] for example) implies that, for any $j\in {\mathbb {Z}}$ and any $u\in L^{1}+L^{\infty },$

$$\begin{aligned}&2^{-3j\left( \frac{1}{p_1}-\frac{1}{p_2}\right) }\Vert \dot{\Delta }_{j}u\Vert _{L^{p_2}}\lesssim \Vert u\Vert _{L^{p_1}}\quad {\textrm{if}}\ 1\le p_1\le p_2\le \infty ,\\&2^{-3j\left( \frac{1}{p_1}-\frac{1}{p_2}\right) }\Vert \dot{\Delta }_{j}u\Vert _{L^{p_2}}\lesssim \Vert u\Vert _{L^{p_1,\infty }}\quad {\textrm{if}}\ 1<p_1<p_2<\infty . \end{aligned}$$

This shows the following proposition.

Proposition 2.2

Let $1\le p_1\le p_2 \le \infty .$ Then,  the space $L^{p_1}$ is continuously embedded in the space $\dot{B}_{p_2,\infty }^{-3(1/p_{1}-1/p_{2})}.$ In addition,  if $1<p_1<p_2<\infty ,$ then the space $L^{p_1,\infty }$ is continuously embedded in the space $\dot{B}_{p_2,\infty }^{-3(1/p_{1}-1/p_{2})}.$

We need the following commutator estimate. The proof is the same as that in [2, Lemma 2.100].

Lemma 2.3

Let $-3/2<s<5/2,$ $1\le r\le \infty$ and $\phi _0\in C^{\infty }_{0}({\mathbb {R}}^3)$ with ${\text {supp}}\phi _0\subset {\mathcal {C}}'$ for some annulus ${\mathcal {C}}'$ centered at the origin. Let us denote $\chi _{j}v\equiv {\mathcal {F}}^{-1}\left[ \phi _0(2^{-j}\cdot )\hat{v} \right]$ for any $v\in {\mathcal {S}}',$ $j\in {\mathbb {Z}}.$ Then,  we have

$$\begin{aligned} \left\| \left( 2^{js}\Vert [\chi _j, h\partial _k]u\Vert _{L^2}\right) _{j\in {\mathbb {Z}}}\right\| _{\ell ^{r}({\mathbb {Z}})} \lesssim _{s,\phi _0}\Vert \nabla h\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}} \Vert u\Vert _{\dot{B}^{s}_{2,r}}, \end{aligned}$$

where $1\le k \le 3$ and u,  h are scalar functions.

3 Existence of stationary solutions

This section is devoted to proving the existence of the stationary solutions of (2). Let $k \in {\mathbb {Z}}_{\ge 1}.$ Set the function spaces X,  Y and Z as

$$\begin{aligned} X=L^2 \times \dot{H}^{1},\quad Y=Y_0\cap Y_1\quad {\textrm{and}}\quad Z=\dot{B}^{-3/2}_{2,\infty }\cap \dot{H}^k, \end{aligned}$$

where

$$\begin{aligned} Y_0=\dot{B}^{-\frac{1}{2}}_{2,\infty }\times \dot{B}^{\frac{1}{2}}_{2,\infty },\quad Y_1=\dot{H}^{k+1}\times \dot{H}^{k+2}. \end{aligned}$$

Then, we have the following results.

Theorem 3.1

If $\Vert F\Vert _{Z}$ is sufficiently small,  then there exists a unique stationary solution $(\rho ,v)=(\sigma +\rho _{\infty },v)$ of (2) such that $(\sigma ,v)\in Y$ and $\Vert (\sigma ,v)\Vert _{Y} \lesssim \Vert F\Vert _{Z}.$

To prove Theorem 3.1, we first reformulate the stationary problem (2). By rewriting the stationary problem (2) using the Helmholtz decomposition, we obtain the following lemma.

Lemma 3.2

A pair of function $(\rho ,v)=(\sigma +\rho _\infty ,v),$ $(\sigma ,v)\in Y$ is the solution to the problem (2) if and only if $(\rho ,v)=(\sigma +\rho _\infty ,v),$ $(\sigma ,v)\in Y$ satisfies the following equations : 

$$\begin{aligned} \left\{ \, \begin{aligned}&\sigma + \alpha \textrm{div}\,(\sigma v) = \gamma ^{-2}\Delta ^{-1} \textrm{div}\,g , \\&v -\beta \Delta ^{-1}\nabla \sigma = -\mu _0^{-1}\Delta ^{-2}\nabla \textrm{div}\, g -\mu ^{-1}\Delta ^{-1} {\mathbb {P}}g, \\ \end{aligned} \right. \end{aligned}$$

(15)

where $\mu _0 = 2\mu + \mu ',$ $\alpha =\mu _0 / (P'(\rho _\infty )\rho _\infty ),$ $\beta =P'(\rho _\infty )/\mu _0,$ $\gamma =P'(\rho _\infty )^{1/2}$ and

$$\begin{aligned} g(\sigma ,v)= -\textrm{div}(\rho v\otimes v) -(P'(\rho )-P'(\rho _\infty ))\nabla \sigma + \rho F. \end{aligned}$$

Proof

Let $(\rho ,v)=(\sigma +\rho _\infty ,v)$ be a solution of the stationary problem (2). By letting the Helmholtz projection ${\mathbb {P}}$ and $\textrm{div}$ act on the second equation of (2), respectively, we obtain the following system of equations.

$$\begin{aligned} \left\{ \, \begin{aligned}&\Delta \sigma -\alpha \rho _\infty \Delta \textrm{div}\,v = \gamma ^{-2}\textrm{div}\,g,\\&\mu \Delta {\mathbb {P}} v = -{\mathbb {P}} g,\\&\rho _\infty \textrm{div}\,v + \textrm{div}(\sigma v) = 0. \end{aligned} \right. \end{aligned}$$

Therefore, $(\sigma ,v)$ satisfies (15). The rest can be shown in a similar way. $\square$

We introduce the linear operator

$$\begin{aligned} {\mathcal {L}}_{\tilde{v}}(\sigma ,v) \equiv \begin{bmatrix} \sigma + \alpha \textrm{div}\,(\sigma \tilde{v}) \\ v - \beta \Delta ^{-1} \nabla \sigma \end{bmatrix}. \end{aligned}$$

Then, the Eq. (15) is written as

$$\begin{aligned} {\mathcal {L}}_{{v}}(\sigma ,v) = N(\sigma ,v), \end{aligned}$$

where

$$\begin{aligned} N(\sigma ,{v}) = \begin{bmatrix} N_1(\sigma ,v)\\ N_2(\sigma ,v) \end{bmatrix} = \begin{bmatrix} \gamma ^{-2}\Delta ^{-1} \textrm{div}\,g(\sigma ,v)\\ -\left( \mu _0^{-1}\Delta ^{-2}\nabla \textrm{div}+ \mu ^{-1}\Delta ^{-1}{\mathbb {P}}\right) g(\sigma ,v) \end{bmatrix}. \end{aligned}$$

To solve (15), we first consider the following approximate problems:

$$\begin{aligned} {\mathcal {L}}_{\tilde{v},j}(\sigma ,v)=N(\tilde{\sigma },\tilde{v}),\ j\in {\mathbb {Z}}. \end{aligned}$$

Here, ${\mathcal {L}}_{\tilde{v},j}$ is the approximation operator

$$\begin{aligned} {\mathcal {L}}_{\tilde{v},j}(\sigma ,v) \equiv \begin{bmatrix} \sigma + \alpha \dot{S}_{j} \textrm{div}\,(\sigma \tilde{v}) \\ v - \beta \Delta ^{-1} \nabla \sigma \end{bmatrix},\quad j\in {\mathbb {Z}}, \end{aligned}$$

where $\dot{S}_{j}$ is the low frequency cut-off operator defined in (13).

Lemma 3.3

Let $\tilde{v}\in L^\infty \cap \dot{B}^{5/2}_{2,1}.$ If $\Vert \tilde{v}\Vert _{\dot{B}^{5/2}_{2,1}}$ is small,  then for any $j\in {\mathbb {Z}},$ the map

$$\begin{aligned} {\mathcal {L}}_{\tilde{v},j} : X \rightarrow X \end{aligned}$$

is bijective.

Proof

Using Young’s inequality, we have $\Vert \dot{S}_{j}\textrm{div}(\sigma \tilde{v})\Vert _{L^{2}}\lesssim 2^{j}\Vert \tilde{v}\Vert _{L^\infty }\Vert \sigma \Vert _{L^2}.$ Thus, ${\mathcal {L}}_{\tilde{v},j}(X)\subset X.$ Fix small $d>0.$ We define the inner product of X by

$$\begin{aligned} \left( (\sigma _1,v_1),(\sigma _2,v_2)\right) _{X} \equiv \langle \sigma _1, \sigma _2 \rangle + d\langle \nabla v_1, \nabla v_2\rangle , \end{aligned}$$

where $(\sigma _1,v_1),(\sigma _2,v_2)\in X.$

By Young’s inequality and $\Vert \textrm{div}\tilde{v}\Vert _{L^{\infty }}\lesssim \Vert \tilde{v}\Vert _{\dot{B}^{5/2}_{2,1}},$ we have

$$\begin{aligned} |\langle \dot{S}_{j}((\textrm{div}\tilde{v})\sigma ),\sigma \rangle | \lesssim \Vert \dot{S}_{j}((\textrm{div}\tilde{v})\sigma )\Vert _{L^2}\Vert \sigma \Vert _{L^2}\lesssim \Vert \tilde{v}\Vert _{\dot{B}^{\frac{5}{2}}_{2,1}} \Vert \sigma \Vert _{L^2}^2. \end{aligned}$$

By using Proposition 2.1(vi), Lemma 2.3 with $s=0$ and the identity $\langle \tilde{v}\cdot \nabla \sigma _{j'},\sigma _{j'} \rangle = -1/2\langle \textrm{div}\,\tilde{v},\sigma _{j'}^2\rangle$ with $\sigma _{j'} = \dot{\Delta }_{j'}^{1/2} \sigma ,$ we obtain

$$\begin{aligned} |\langle \dot{S}_{j} (\sigma \cdot \nabla \tilde{v}),\sigma \rangle |&\lesssim \sum _{j'\le j-1} |\langle \dot{\Delta }_{j'} (v\cdot \nabla \sigma ), \sigma \rangle | \nonumber \\&\lesssim \sum _{j'\in {\mathbb {Z}}} (|\langle \tilde{v}\cdot \nabla \sigma _{j'},\sigma _{j'}\rangle | +|\langle [\dot{\Delta }_{j'}^{\frac{1}{2}},\tilde{v}\cdot \nabla ]\sigma ,\sigma _{j'}\rangle |)\nonumber \\&\lesssim \sum _{j'\in {\mathbb {Z}}} (\Vert \textrm{div}\tilde{v}\Vert _{L^\infty }\Vert \sigma _{j'}\Vert _{L^2}^2+\Vert [\dot{\Delta }_{j'}^{\frac{1}{2}},\tilde{v}\cdot \nabla ]\sigma \Vert _{L^2}\Vert \sigma _{j'}\Vert _{L^2})\nonumber \\&\lesssim \Vert \tilde{v}\Vert _{\dot{B}^{\frac{5}{2}}_{2,1}} \Vert \sigma \Vert _{L^2}^2. \end{aligned}$$

(16)

Thus, we have

$$\begin{aligned} |\langle \dot{S}_{j}\textrm{div}(\tilde{v}\sigma ),\sigma \rangle | \lesssim \Vert \tilde{v}\Vert _{\dot{B}^{\frac{5}{2}}_{2,1}} \Vert \sigma \Vert _{L^2}^2. \end{aligned}$$

Hence, if d and $\Vert \tilde{v}\Vert _{\dot{B}^{5/2}_{2,1}}$ are sufficiently small, then

$$\begin{aligned} |( {\mathcal {L}}_{\tilde{v},j}(\sigma ,v), (\sigma , v))_{X}| \gtrsim \left\| (\sigma ,v)\right\| _{X}^{2} \end{aligned}$$

for any $(\sigma ,v)\in X,$ where $\Vert (\sigma ,v)\Vert _{X}\equiv ((\sigma ,v),(\sigma ,v))_{X}^{1/2}.$ The Lax–Milgram theorem completes the proof. $\square$

We show the nonlinear estimate.

Lemma 3.4

Let

$$\begin{aligned} (\tilde{\sigma },\tilde{v}),(\tilde{\sigma }_{1},\tilde{v}_{1}), (\tilde{\sigma }_{2},\tilde{v}_{2})\in Y_{\delta } \equiv \big \{(\sigma ,v)\in Y\mid \Vert (\sigma ,v)\Vert _{Y} \le \delta \big \}. \end{aligned}$$

Then,  for any $0<\delta \le 1,$ we have

$$\begin{aligned} \Vert g(\tilde{\sigma },\tilde{v})\Vert _{Z}&\lesssim \delta ^{2} + \Vert F\Vert _{Z}, \\ \Vert g(\tilde{\sigma }_1,\tilde{v}_1)-g(\tilde{\sigma }_2,\tilde{v}_2)\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}&\lesssim (\delta +\Vert F\Vert _Z) \Vert (\tilde{\sigma }_1-\tilde{\sigma }_2, \tilde{v}_{1}-\tilde{v}_{2})\Vert _{Y_0} . \end{aligned}$$

Proof

By Proposition 2.1(v), (vi),

$$\begin{aligned} \Vert g(\tilde{\sigma },\tilde{v})\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}&\lesssim \Vert \tilde{v}\Vert _{ \dot{B}^{\frac{1}{2}}_{2,\infty }}^{2} + \Vert \tilde{\sigma }\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}\cap \dot{B}^{-\frac{1}{2}}_{2,\infty }}^{2} + \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}\\&\lesssim \delta ^2 + \Vert F\Vert _Z,\\ \Vert g(\tilde{\sigma }_1,\tilde{v}_1)-g(\tilde{\sigma }_2,\tilde{v}_2)\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}&\lesssim \delta \Vert \tilde{v}_1-\tilde{v}_2\Vert _{ \dot{B}^{\frac{1}{2}}_{2,\infty }} + \delta \Vert \tilde{\sigma }_1-\tilde{\sigma }_2\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }} \\&\quad +\Vert \tilde{\sigma }_1-\tilde{\sigma }_2\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }}\Vert F\Vert _{\dot{B}^{\frac{1}{2}}_{2,1}}\\&\lesssim (\delta +\Vert F\Vert _Z) \Vert (\tilde{\sigma }_1-\tilde{\sigma }_2,\tilde{v}_1-\tilde{v}_2)\Vert _{Y_0}. \end{aligned}$$

Using Sobolev’s inequality and Proposition 2.1(vi), we have

$$\begin{aligned}&\Vert \textrm{div}\,(\tilde{\rho } \tilde{v}\otimes \tilde{v})\Vert _{\dot{H}^{k}} \lesssim \Vert \nabla ^{k+1}(\tilde{\rho } \tilde{v}\otimes \tilde{v})\Vert _{L^2} \lesssim \big (1+\Vert \tilde{\sigma }\Vert _{H^{k+1}}\big )\Vert \nabla \tilde{v}\Vert _{H^{k}}^{2},\\&\Vert (P'(\tilde{\rho })-P'(\rho _\infty ))\nabla \tilde{\sigma }\Vert _{\dot{H}^{k}} = \Vert Q(\tilde{\sigma })\tilde{\sigma }\nabla \tilde{\sigma }\Vert _{\dot{H}^{k}},\\&Q(\tilde{\sigma })\equiv \int ^{1}_0 P''(\rho _\infty +t\tilde{\sigma })dt, \lesssim \big (\Vert Q(\tilde{\sigma })\Vert _{L^\infty } + \Vert \nabla Q(\tilde{\sigma })\Vert _{H^{k-1}}\big )\Vert \tilde{\sigma }\Vert _{H^{k+1}}^{2}, \\&\Vert \tilde{\rho }F\Vert _{\dot{H}^{k}} \lesssim (1+\Vert \tilde{\sigma }\Vert _{H^{k}}) \Vert F\Vert _{H^k}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \Vert g(\tilde{\sigma },\tilde{v})\Vert _{Z} \lesssim \delta ^{2} + \Vert F\Vert _{Z}. \end{aligned}$$

$\square$

By virtue of Lemmas 3.3 and 3.4, we can define the maps

$$\begin{aligned} \Phi _j(\tilde{\sigma },\tilde{v}) \equiv {\mathcal {L}}_{\tilde{v},j}^{-1}N(\tilde{\sigma },\tilde{v}),\quad (\tilde{\sigma },\tilde{v})\in Y_{\delta },\ j\in {\mathbb {Z}}. \end{aligned}$$

Lemma 3.5

If $\delta >0$ and $\Vert F\Vert _Z$ are sufficiently small,  then,  for any $j\ge 0,$ the map $\Phi _j:Y_\delta \rightarrow Y_\delta$ satisfies $\Vert \Phi _j(\tilde{\sigma },\tilde{v})\Vert _Y\lesssim \delta ^2+\Vert F\Vert _{Z}$ for any $(\tilde{\sigma },\tilde{v})\in Y_\delta .$ Furthermore,  there exists a constant $c>0$ such that the maps $\Phi _j,$ $j\ge 0$ are contraction mappings in $Y_0$ norm with uniform Lipschitz constant $c\delta ,$ that is, 

$$\begin{aligned} \sup _{j\ge 0}\Vert \Phi _j(\tilde{\sigma }_1,\tilde{v}_1)-\Phi _j(\tilde{\sigma }_2,\tilde{v}_2)\Vert _{Y_0} \le c\delta \Vert (\tilde{\sigma }_1-\tilde{\sigma }_2,\tilde{v}_1-\tilde{v}_2)\Vert _{Y_0} \end{aligned}$$

for any $(\tilde{\sigma }_1,\tilde{v}_1), (\tilde{\sigma }_2,\tilde{v}_2)\in Y_\delta .$

Proof

Let $(\sigma ,v)=\Phi _j (\tilde{\sigma },\tilde{v}),$ $(\tilde{\sigma },\tilde{v})\in Y_\delta .$ Then, since

$$\begin{aligned} \sigma = -\dot{S}_{j} \textrm{div}(\sigma \tilde{v}) + \gamma ^{-2} \Delta ^{-1}\textrm{div}\, g(\tilde{\sigma },\tilde{v}), \end{aligned}$$

we have the estimate

$$\begin{aligned} \Vert \sigma \Vert _{H^{k+1}}&\lesssim \Vert \dot{S}_{j} \textrm{div}(\sigma \tilde{v})\Vert _{H^{k+1}} + \Vert g(\tilde{\sigma },\tilde{v})\Vert _{H^k}\\&\lesssim _{j} \Vert \tilde{v}\Vert _{L^{\infty }}\Vert \sigma \Vert _{L^2} + \delta ^2 + \Vert F\Vert _Z, \end{aligned}$$

where the last inequality is due to the fact that $\Vert \nabla ^{n}\dot{S}_j(\sigma \tilde{v})\Vert _{L^2}\lesssim 2^{nj}\Vert \sigma \tilde{v}\Vert _{L^2}$ for any $n\in {\mathbb {Z}}_{\ge 0}$ and Lemma 3.4. Thus, we have $\sigma \in H^{k+1}.$ We apply the argument for (16) again, with $\sigma$ replaced by $\partial _x^\alpha \sigma ,$ $|\alpha |= k+1,$ to obtain

$$\begin{aligned} |\langle \dot{S}_{j}\textrm{div}( \partial ^{\alpha }_{x}\sigma \,\tilde{v}),\partial _{x}^{\alpha } \sigma \rangle | \lesssim \Vert \nabla \tilde{v}\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}} \Vert \partial ^{\alpha }_x \sigma \Vert _{L^{2}}^2. \end{aligned}$$

This gives

$$\begin{aligned} \Vert \sigma \Vert ^{2}_{\dot{H}^{k+1}}&=\sum _{|\alpha |=k+1}\langle \partial _{x}^{\alpha }\sigma ,\partial _{x}^{\alpha }\sigma \rangle \lesssim \sum _{{|\alpha |=k+1,\,\beta \le \alpha }}|\langle \dot{S}_{j}\textrm{div}(\partial _{x}^{\beta }\sigma \,\partial _{x}^{\alpha -\beta }\tilde{v}) , \partial _{x}^{\alpha }\sigma \rangle | \\&\quad + \Vert \Delta ^{-1}\textrm{div}\,g(\tilde{\sigma },\tilde{v})\Vert _{\dot{H}^{k+1}} \Vert \sigma \Vert _{\dot{H}^{k+1}}\\&\lesssim \Vert \nabla \tilde{v}\Vert _{H^{k+1}} \Vert \sigma \Vert _{H^{k+1}}^2 + \Vert g(\tilde{\sigma },\tilde{v})\Vert _{\dot{H}^k}\Vert \sigma \Vert _{\dot{H}^{k+1}}. \end{aligned}$$

Thus, Lemma 3.4 shows $\Vert \sigma \Vert _{\dot{H}^{k+1}}\lesssim \delta \Vert (\sigma ,v)\Vert _{Y} +\delta ^2+\Vert F\Vert _{Z}.$ By directly estimating $(\sigma ,v)^{\textsf{T}}=(\sigma ,v)^{\textsf{T}}-{\mathcal {L}}_{\tilde{v}, j}(\sigma ,v)+N(\tilde{\sigma },\tilde{v}),$ we obtain

$$\begin{aligned} \Vert \sigma \Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }}&\lesssim \Vert \sigma \tilde{v}\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} + \Vert g(\tilde{\sigma },\tilde{v})\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }} \\&\lesssim \Vert \sigma \Vert _{\dot{B}^{\frac{3}{2}}_{2,1}}\Vert \tilde{v}\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} + \delta ^2 +\Vert F\Vert _Z \lesssim \delta \Vert (\sigma ,v)\Vert _{Y}+\delta ^2+\Vert F\Vert _Z, \\ \Vert v\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{k+2}}&\lesssim \Vert \sigma \Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }\cap \dot{H}^{k+1}} + \Vert g(\tilde{\sigma },\tilde{v})\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^{k}} \lesssim \delta \Vert (\sigma ,v)\Vert _{Y} +\delta ^2+\Vert F\Vert _Z. \end{aligned}$$

Then, we have $\Vert (\sigma ,v)\Vert _Y \lesssim \delta ^2 + \Vert F\Vert _Z$ for small $\delta >0.$ Let $(\sigma _i,v_i)=\Phi _j(\tilde{\sigma }_i,\tilde{v}_i),$ $(\tilde{\sigma }_i,\tilde{v}_i)\in Y_\delta ,$ $i=1,2.$ Applying Lemma 3.4 for estimating

$$\begin{aligned} v_1-v_2 = \beta \Delta ^{-1} \nabla (\sigma _1-\sigma _2)+N_2(\tilde{\sigma }_1,\tilde{v}_1)-N_2(\tilde{\sigma }_2,\tilde{v}_2) \end{aligned}$$

we obtain

$$\begin{aligned} \Vert v_1-v_2\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}&\lesssim \Vert \sigma _1-\sigma _2\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }} + \Vert g(\tilde{\sigma }_1,\tilde{v}_1)-g(\tilde{\sigma }_2,\tilde{v}_2)\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}\\&\lesssim \Vert \sigma _1-\sigma _2\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }}+\delta \Vert (\tilde{\sigma }_1-\tilde{\sigma }_2,\tilde{v}_1-\tilde{v}_2)\Vert _{Y_0}. \end{aligned}$$

The rest is to estimate $\Vert \sigma _1-\sigma _2\Vert _{\dot{B}^{-1/2}_{2,\infty }}.$ For any $n\in {\mathbb {Z}},$ there holds

$$\begin{aligned} \Vert \dot{\Delta }_{n}(\sigma _1-\sigma _2)\Vert _{L^2}^2&=-\langle \dot{\Delta }_{n}\alpha \dot{S}_{j}\textrm{div}(\sigma _1 \tilde{v_1}-\sigma _2\tilde{v}_2), \dot{\Delta }_{n}(\sigma _1-\sigma _2)\rangle \\&\quad + \langle \dot{\Delta }_{n}\gamma ^{-2}\Delta ^{-1}\textrm{div}(g(\tilde{\sigma }_1,\tilde{v}_1)-g(\tilde{\sigma }_2,\tilde{v}_2)), \dot{\Delta }_{n}(\sigma _1-\sigma _2)\rangle . \end{aligned}$$

Let $\chi _{n,j'}=\dot{\Delta }_{n}\dot{\Delta }_{j'}^{1/2},$ $\omega =\sigma _1-\sigma _2.$ Then, by using Proposition 2.1(vi), Lemma 2.3 and the identity

$$\begin{aligned} \langle \tilde{v}_1 \cdot \nabla \chi _{n,j'}\omega , \chi _{n,j'}\omega \rangle = -\frac{1}{2}\langle \textrm{div}\tilde{v}_1\,\chi _{n,j'}\omega , \chi _{n,j'}\omega \rangle , \end{aligned}$$

we have the estimate

$$\begin{aligned}&|\langle \dot{\Delta }_{n}\dot{S}_{j}\textrm{div}(\sigma _1 \tilde{v}_1-\sigma _2\tilde{v}_2), \dot{\Delta }_{n}\omega \rangle | \\&\quad \lesssim \sum _{j'<j,\ |n-j'|\le 1} \left( \Vert [\chi _{n,j'},\tilde{v}_{1}\cdot \nabla ]\omega \Vert _{L^2} \Vert \chi _{n,j'}\omega \Vert _{L^2}+ |\langle \tilde{v}_1 \cdot \nabla \chi _{n,j'}\omega , \chi _{n,j'}\omega \rangle |\right) \\&\qquad +\left( \Vert \dot{\Delta }_{n}(\textrm{div}\tilde{v}_1\, \omega )\Vert _{L^2} + \Vert \dot{\Delta }_{n}\textrm{div}((\tilde{v}_1-\tilde{v}_2)\sigma _2)\Vert _{L^2}\right) \Vert \dot{\Delta }_{n}\omega \Vert _{L^{2}}\\&\quad \lesssim 2^{\frac{1}{2}n}\left( \Vert \tilde{v}_1\Vert _{\dot{B}^{\frac{5}{2}}_{2,1}}\Vert \omega \Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }} + \Vert \sigma _2\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}}\Vert \tilde{v}_1-\tilde{v}_2\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \right) \Vert \dot{\Delta }_{n}\omega \Vert _{L^{2}}\\&\quad \lesssim 2^{\frac{1}{2}n}\delta \left( \Vert \omega \Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }}+ \Vert \tilde{v}_1-\tilde{v}_2\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}\right) \Vert \dot{\Delta }_{n}\omega \Vert _{L^{2}}. \end{aligned}$$

Thus, if $\delta >0$ is sufficiently small, then we have

$$\begin{aligned} \Vert \sigma _1-\sigma _2\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }}&\lesssim \delta \Vert \tilde{v}_1-\tilde{v}_2\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} + \Vert g(\tilde{\sigma }_1,\tilde{v}_1)-g(\tilde{\sigma }_2,\tilde{v}_2)\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}\\&\lesssim \delta \Vert (\tilde{\sigma }_1-\tilde{\sigma }_2,\tilde{v}_1-\tilde{v}_2)\Vert _{Y_0}. \end{aligned}$$

$\square$

Let $(\tilde{\sigma },\tilde{v})\in Y_\delta$ and let $\delta >0$ be small enough. By Proposition 2.1(iv), the bounded sequence $\{\Phi _j(\tilde{\sigma },\tilde{v})\}_{j\ge 0}$ has a convergence subsequence in ${\mathcal {S}}'.$ Let $(\sigma ,v)\in {\mathcal {S}}'$ be one of its convergent limits, and we write $\Phi (\tilde{\sigma },\tilde{v})=(\sigma ,v).$ Then, the following proposition holds.

Proposition 3.6

If $\delta >0$ and $\Vert F\Vert _Z$ are sufficiently small,  then the map $\Phi :Y_\delta \rightarrow Y_\delta$ is well-defined and is a contraction in the $Y_0$ norm.

Proof

Let us show the well-definedness of $\Phi .$ Let $(\tilde{\sigma },\tilde{v})\in Y_\delta ,$ and let $(\sigma _1,v_1),$ $(\sigma _2,v_2)$ be convergent limits in ${\mathcal {S}}'$ of subsequences $\{\Phi _{\psi _1(j)}(\tilde{\sigma },\tilde{v})\}_{j\ge 0},$ $\{\Phi _{\psi _2(j)}(\tilde{\sigma },\tilde{v})\}_{j\ge 0},$ respectively. By Proposition 2.1(iv) and Lemma 3.5, there exist subsequences of $\{\Phi _{\psi _i(j)}(\tilde{\sigma },\tilde{v})\}_{j\ge 0},$ $i=1,2$ (without relabeling) such that

$$\begin{aligned} \Vert (\sigma _i,v_i)\Vert _{Y} \lesssim \liminf _{j\rightarrow \infty } \Vert \Phi _{\psi _{i}(j)}(\tilde{\sigma },\tilde{v})\Vert _Y\lesssim \delta ^2 + \Vert F\Vert _{Z},\quad i=1,2. \end{aligned}$$

Thus, we have $(\sigma _i,v_i)\in Y_{\delta },$ $i=1,2$ for small $\delta >0$ and $\Vert F\Vert _Z.$ By subtracting ${\mathcal {L}}_{\tilde{v},\psi _{i}(j)}\Phi _{\psi _{i}(j)}(\tilde{\sigma },\tilde{v})=N(\tilde{\sigma },\tilde{v}),$ $i=1,2,$ we have

$$\begin{aligned} \left\{ \, \begin{aligned}&\sigma _{1,j}-\sigma _{2,j} + \alpha \dot{S}_{\psi _{1}(j)}\textrm{div}\,((\sigma _{1,j}-\sigma _{2,j})\tilde{v})-\alpha (\dot{S}_{\psi _{2}(j)}-\dot{S}_{\psi _{1}(j)})\textrm{div}\,(\sigma _{2,j}\tilde{v}) = 0, \\&v_{1,j}-v_{2,j} -\beta \Delta ^{-1}\nabla (\sigma _{1,j}-\sigma _{2,j}) =0, \\ \end{aligned} \right. \end{aligned}$$

where $(\sigma _{i,j},v_{i,j})\equiv \Phi _{\psi _{i}(j)}(\tilde{\sigma },\tilde{v}),$ $i=1,2,$ $j\ge 0.$ Letting $j\rightarrow \infty$ in ${\mathcal {S}}',$ we obtain the equalities

$$\begin{aligned} \left\{ \, \begin{aligned}&\sigma _{1}-\sigma _{2} + \alpha \textrm{div}\,((\sigma _{1}-\sigma _{2})\tilde{v}) = 0, \\&v_{1}-v_{2} -\beta \Delta ^{-1}\nabla (\sigma _{1}-\sigma _{2}) =0. \\ \end{aligned} \right. \end{aligned}$$

Then, letting $\eta =\sigma _1-\sigma _2,$ we have

$$\begin{aligned} \Vert \eta \Vert ^{2}_{L^2} = - \alpha \langle \textrm{div}(\eta \tilde{v}),\eta \rangle = -\frac{\alpha }{2}\langle \textrm{div}\,\tilde{v}, \eta ^2 \rangle \lesssim \delta \Vert \eta \Vert _{L^2}^2. \end{aligned}$$

Thus, if $\delta >0$ is small, then we have $\sigma _1=\sigma _2$ and $v_1=v_2.$ Hence, the map $\Phi :Y_\delta \rightarrow Y_\delta$ is well-defined.

Next, we show the map $\Phi$ is a contraction in the $Y_0$ norm. Let $(\sigma _i,v_i)=\Phi (\tilde{\sigma }_i,\tilde{v}_i),$ $(\tilde{\sigma }_i,\tilde{v}_i)\in Y_\delta ,$ $i=1,2.$ Since the definition of $\Phi$ does not depend on taking a weakly convergent subsequence, there exits a subsequence $\{\Phi _{\psi (j)}\}_{j\ge 0}$ such that $\Phi _{\psi (j)}(\tilde{\sigma }_i,\tilde{v}_i)\rightarrow (\sigma _i,v_i)$ in ${\mathcal {S}}'$ as $j\rightarrow \infty ,$ $i=1,2.$ By Proposition 2.1(iv) and Lemma 3.5, there exists a subsequence of $\{\Phi _{\psi (j)}(\tilde{\sigma }_i,\tilde{v}_i)\}_{j\ge 0},$ $i=1,2$ (without relabeling) such that

$$\begin{aligned} \Vert (\sigma _i,v_i)\Vert _{Y} \lesssim \liminf _{j\rightarrow \infty } \Vert \Phi _{\psi (j)}(\tilde{\sigma }_i,\tilde{v}_i)\Vert _Y\lesssim \delta ^2+\Vert F\Vert _{Z},\quad i=1,2, \end{aligned}$$

and

$$\begin{aligned} \Vert (\sigma _1-\sigma _2,v_1-v_2)\Vert _{Y_0}&\lesssim \liminf _{j\rightarrow \infty } \Vert \Phi _{\psi (j)}(\tilde{\sigma }_1,\tilde{v}_1)-\Phi _{\psi (j)}(\tilde{\sigma }_2,\tilde{v}_2)\Vert _{Y_0}\\&\le c\delta \Vert (\tilde{\sigma }_1-\tilde{\sigma }_2,\tilde{v}_1-\tilde{v}_2)\Vert _{Y_0}. \end{aligned}$$

Therefore, if $\delta >0$ and $\Vert F\Vert _Z$ are sufficiently small, then the map $\Phi :Y_\delta \rightarrow Y_\delta$ is well-defined and is a contraction in the $Y_0$ norm. $\square$

Let us now establish the proof of Theorem 3.1.

Proof of Theorem 3.1

Let $\delta$ and $\Vert F\Vert _Z$ be small. Then, by Proposition 3.6, the map $\Phi :Y_\delta \rightarrow Y_\delta$ is well-defined and is a contraction in the $Y_0$ norm. The contraction mapping principle and Proposition 2.1(iv) show that there exists a unique $(\sigma ,v)\in Y_{\delta }$ such that $\Phi (\sigma ,v)=(\sigma ,v).$ This implies ${\mathcal {L}}_{v}(\sigma ,v)=N(\sigma ,v),$ since $\dot{S}_{j}\rightarrow 1$ in ${\mathcal {S}}'$ as $j\rightarrow \infty .$ Hence, Lemma 3.2 shows that $(\sigma ,v)$ solves the stationary problem (2). $\square$

4 Non-stationary problem

This section is devoted to proving the existence of the solutions of (1). Let us consider the equations satisfied by the perturbation of the stationary solution. Let $(\rho ,v)$ be a solution of (1) and let $(\rho ^*,v^*)=(\sigma ^* +\rho _\infty , v^*)$ be a stationary solution of (2). After the rescaling

$$\begin{aligned}&(\rho (t, x), v(t,x)) \rightarrow (\rho (\lambda ^2 t,\lambda x),\lambda v(\lambda ^2 t,\lambda x)), \\&(\rho ^*(x), v^*(x)) \rightarrow (\rho ^*(\lambda x),\lambda v^*(\lambda x)), \end{aligned}$$

with $\lambda =\rho _\infty / P'(\rho _\infty )^{1/2},$ we assume without loss of generality that $\rho _\infty / P'(\rho _\infty )^{1/2} = 1.$ Then, the perturbation $(\sigma ,w)=(\rho -\rho ^*,v-v^*)$ satisfies the following system of equations:

$$\begin{aligned} \left\{ \, \begin{aligned}&\partial _{t}\sigma + \gamma _{0} \textrm{div}\,w = f(\sigma ,w), \\&\partial _{t}w - {\mathcal {A}}_{0} w + \gamma _{0} \nabla \sigma = g(\sigma ,w), \\&(\sigma ,w)|_{t=0} = (\sigma _0, w_0), \end{aligned} \right. \end{aligned}$$

(17)

where $\gamma _0=P'(\rho _\infty )^{1/2},$ $\nu _0=\mu /\rho _\infty ,$ $\nu _0'=\mu '/\rho _\infty ,$ ${\mathcal {A}}_{0}\equiv \nu _0 \Delta + (\nu _0+\nu _0')\nabla \textrm{div},$ $(\sigma _0,w_0)\equiv (\rho _{0} - \rho ^*, v_{0} - v^*);$ f and g are defined by the following:

$$\begin{aligned} f(\sigma ,w) = -\gamma _0\textrm{div}\left\{ (v^{*}+w)\sigma +\sigma ^*w \right\} ,\ g(\sigma ,w) = \sum _{i=1}^{4}g^{i} \end{aligned}$$

with

$$\begin{aligned}&g^1 = -v^* \cdot \nabla w - w \cdot \nabla v^* - w\cdot \nabla w,\\&g^{2} = -(\Phi (\sigma ^* + \sigma )-\Phi (\sigma ^{*})) \nabla \sigma ^* - (\Phi (\sigma ^* + \sigma )-\Phi (0))\nabla \sigma , \\&g^{3} = (\Psi (\sigma ^* + \sigma )-\Psi (\sigma ^*)){\mathcal {A}}_0 (v^* + w),\quad g^{4} = (\Psi (\sigma ^*)-\Psi (0)) {\mathcal {A}}_0 w,\\&\Phi (\zeta )=\frac{P'(\zeta +\rho _{\infty })}{\zeta +\rho _{\infty }},\ \ \Psi (\zeta )=\frac{1}{\zeta +\rho _{\infty }}. \end{aligned}$$

Next, we present some estimates for the solution to the linearized compressible Navier–Stokes equation around the constant state $(\rho _{\infty },0)$:

$$\begin{aligned} \left\{ \, \begin{aligned}&\partial _{t}b + \gamma \textrm{div}\,u= 0, \\&\partial _{t}u - {\mathcal {A}}u + \gamma \nabla b = 0, \end{aligned} \right. \end{aligned}$$

(18)

where $\gamma >0$ and ${\mathcal {A}}=\nu \Delta + (\nu +\nu ')\nabla \textrm{div}$ with $\nu >0,$ $2\nu /3+\nu '\ge 0;$ a is a scaler function and u is a 3-vector valued function. Let $e^{tA}$ be the semigroup associated with the linear equation (18):

$$\begin{aligned} e^{tA}U_0 = {\mathcal {F}}^{-1} \left[ e^{t\hat{A}(\xi )} \widehat{U_0} \right] ,\quad U_0=(U_{0,1},\ldots ,U_{0,4})^{\textsf{T}}\in {\mathcal {S}}'({\mathbb {R}}^3)^{4}, \end{aligned}$$

(19)

where $\hat{A}(\xi )$ is the matrix of the form:

$$\begin{aligned} \hat{A}(\xi ) = \begin{bmatrix} 0 &{}\quad \quad \qquad -i\gamma \xi ^{\textsf{T}}\\ -i\gamma \xi &{}\quad \quad -\nu |\xi |^{2}{\textrm{I}}_{3} \quad \ - (\nu +\nu ')\xi \otimes \xi \end{bmatrix}. \end{aligned}$$

(20)

Here $\xi = (\xi _1, \xi _2, \xi _3)^{\textsf{T}} \in {\mathbb {R}}^{3},$ $\xi \otimes \xi =\xi \xi ^{\textsf{T}}$ and ${\textrm{I}}_{3}$ is the $3\times 3$ identity matrix. By direct calculation, the eigenvalues of $\hat{A}(\xi )$ are given by

$$\begin{aligned} \lambda _{\pm }(\xi ) = -\frac{2\nu +\nu '}{2}|\xi |^{2} \pm \frac{\sqrt{(2\nu + \nu ')^{2}|\xi |^{4}-4\gamma ^2 |\xi |^{2}}}{2},\quad \lambda _{0}(\xi )=-\nu |\xi |^{2}. \end{aligned}$$

(21)

We set $P_{\pm }(\xi )$:

$$\begin{aligned} P_{\pm }(\xi ) = \frac{V_{\pm }\otimes V_{\pm }}{V_{\pm }\cdot V_{\pm }} \quad {\textrm{with}}\ V_{\pm }=\begin{bmatrix} -i\lambda _{\pm }^{-1}\gamma |\xi |^2\\ \xi \end{bmatrix} \end{aligned}$$

for $|\xi |\ne 0,\eta _0,$ where $\eta _0=\gamma /(\nu +\nu '/2),$ $V_{\pm } \cdot V_{\pm } \equiv V_{\pm }^{\textsf{T}}V_{\pm },$ and set the eigenprojection $P_0(\xi )$:

$$\begin{aligned} P_{0}(\xi ) = \begin{bmatrix} 0 &{}\quad 0 \\ 0 &{} \quad {\textrm{I}}_3 - \frac{\xi \otimes \xi }{|\xi |^2} \end{bmatrix}. \end{aligned}$$

Since $P_+(\xi )+P_-(\xi )+P_0(\xi )={\text {I}}_4 ,$ we have the spectral resolution

$$\begin{aligned} e^{t\hat{A}(\xi )} = e^{\lambda _{+}t}P_{+}(\xi ) + e^{\lambda _{-}t}P_{-}(\xi ) + e^{\lambda _0t}P_0(\xi )\quad {\textrm{for}}\ |\xi |\ne 0,\eta _0. \end{aligned}$$

(22)

If $|\xi |=\eta _0,$ then we have

$$\begin{aligned} e^{t\hat{A}(\xi )} = e^{-\nu _0 |\xi |^2 t} \begin{bmatrix} 1-\nu _0 |\xi |^2 t &{}\quad -i \gamma \xi ^{\textsf{T}} t \\ -i \gamma \xi t &{}\quad (1-\nu _0 |\xi |^2 t) \frac{\xi \otimes \xi }{|\xi |^2} \end{bmatrix} + e^{-\nu |\xi |^2t}P_{0}(\xi ), \end{aligned}$$

(23)

where $\nu _0=\nu +\nu '/2.$ This spectral resolution will be used in the proof of Theorem 1.6 in Sect. 5.3.

The following lemma shows some smoothing estimate for the low frequency part of the semigroup $e^{tA}$ and its adjoint $e^{tA^*}.$ This lemma has been proved in [2, Proposition 10.22]. (Cf. [4, 6, 21].)

Lemma 4.1

Let $j_{0}\in {\mathbb {Z}},$ $s\in {\mathbb {R}}.$ Set $e^{tA}_{L} \equiv \dot{S}_{j_{0}}e^{tA},$ $e^{tA^{*}}_{L} \equiv \dot{S}_{j_{0}}e^{tA^*},$ where $\dot{S}_{j_0}$ is the low frequency cut-off operator defined in (13).

• (i) For any $U_0\in \dot{B}_{2,r}^{s}$ and $\alpha \ge 0,$ we have

  $$\begin{aligned} \Vert e^{tA}_{L}U_0\Vert _{\dot{B}_{2,r}^{s+\alpha }},\ \Vert e^{tA^{*}}_{L}U_0\Vert _{\dot{B}_{2,r}^{s+\alpha }} \lesssim _{\alpha ,j_0} (1+t)^{-\frac{\alpha }{2}}\Vert U_0\Vert _{\dot{B}_{2,r}^{s}} \end{aligned}$$

  (24)

  for any $1\le r\le \infty .$

• (ii) The following time-space integral estimate holds : 

  $$\begin{aligned} \int ^{\infty }_{0} \Vert e^{tA}_{L} U_0 \Vert _{\dot{B}_{2,1}^{s+2}}dt,\ \int ^{\infty }_{0} \Vert e^{tA^{*}}_{L} U_0 \Vert _{\dot{B}_{2,1}^{s+2}}dt \lesssim _{j_0} \Vert U_0\Vert _{\dot{B}_{2,1}^{s}} \end{aligned}$$

  (25)

  for any $U_0 \in \dot{B}^{s}_{2,1}.$

4.1 Existence of non-stationary solutions

Let us prove the global existence result in Theorem 1.2. We shall prove the following theorem.

Theorem 4.2

Let $(\rho ^*,v^*)$ be a stationary solution of (2) satisfying (7) with $\Vert F\Vert _{\dot{B}^{-3/2}_{2,\infty }\cap \dot{H}^3}$ sufficiently small. Then,  there exists a constant $\delta >0$ such that if $(\sigma _0,w_0)$ satisfy

$$\begin{aligned} \Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}} \le \delta , \end{aligned}$$

then the Eq. (17) with initial data $(\sigma _0,w_0)$ has a global solution $(\sigma ,w)$ satisfying $(\sigma ,w)\in C^{0}([0,\infty );\dot{B}^{1/2}_{2,\infty }\cap \dot{H}^3)$ and

$$\begin{aligned} \sup _{t>0}\Vert (\sigma ,w)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}} \lesssim \delta . \end{aligned}$$

First, we show the a priori estimate for the perturbation.

Proposition 4.3

Let $(\rho ^*,v^*)$ be a stationary solution of (2) satisfying (7) with $\Vert F\Vert _{\dot{B}^{-3/2}_{2,\infty }\cap \dot{H}^3}$ sufficiently small. Let $(\sigma ,w) \in C^{0}([0,T);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3),$ $w\in L^{2}_{loc}([0,T);\dot{H}^4)$ be a solution of (17) with initial value $(\sigma _0,w_0)$ for some T,  $0<T\le \infty .$ Then,  there exist constants $\delta _1>0$ and $C_1>0$ such that if

$$\begin{aligned} \sup _{0\le t<T} \Vert (\sigma ,w)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} \le \delta _1, \end{aligned}$$

then we have

$$\begin{aligned} \sup _{0\le t<T}\Vert (\sigma ,w)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}} \le C_1 \Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}. \end{aligned}$$

(26)

Proof

Let us denote $U=(\sigma ,w)$ and $U_0=(\sigma _0,w_0).$ Fix $j_0\in {\mathbb {Z}}$ and decompose

$$\begin{aligned} U=U_L+U_H = (\sigma _L , w_L) + (\sigma _H , w_H), \end{aligned}$$

(27)

where $U_L=(\sigma _L, w_L)\equiv (\dot{S}_{j_0}\sigma ,\dot{S}_{j_0} w).$ In order to estimate the low frequency part $U_L,$ we shall rewrite the perturbation equation (17) in the momentum formulation. Let

$$\begin{aligned} m\equiv \rho v,\quad m^*\equiv \rho ^*v^*\quad {\textrm{and}}\quad n\equiv \frac{m-m^*}{\rho _{\infty }}. \end{aligned}$$

Then, the pair of functions $V=(\sigma ,n)=(\rho -\rho ^*,n)$ satisfies the system of equations:

$$\begin{aligned} \left\{ \, \begin{aligned}&\partial _{t}\sigma + \gamma _{1}\textrm{div}\,n= 0, \\&\partial _{t}n - {\mathcal {A}}_1 n + \gamma _{1} \nabla \sigma =h + \gamma _{1}^{-1}\sigma F(x), \end{aligned} \right. \end{aligned}$$

(28)

where $\gamma _1=P'(\rho _\infty )^{1/2}=\rho _{\infty },$ ${\mathcal {A}}_1 = \mu \Delta + (\mu +\mu ')\nabla \textrm{div};$ h is defined by

$$\begin{aligned} h=\sum _{i=1}^{4}h_i \end{aligned}$$

(29)

with

$$\begin{aligned} h_1&=-\textrm{div}\left( \frac{n\otimes m}{\rho } +\frac{m^*\otimes n}{\rho }+\gamma _1^{-1}\left( \Psi (\sigma ^*+\sigma )-\Psi (\sigma ^*)\right) m^*\otimes m\right) ,\\ h_2&=-\nabla \left( \Pi (\sigma ^*,\sigma )\sigma \right) ,\quad h_3={\mathcal {A}}_1\left( \left( \Psi (\sigma ^*+\sigma )-\Psi (0)\right) n\right) , \\ h_4&=\gamma _1^{-1}{\mathcal {A}}_1 \left( (\Psi (\sigma ^*+\sigma )-\Psi (\sigma ^*))m^* \right) ,\\ \Pi (\zeta _1,\zeta _2)&= \int _{0}^{1} \left( P'(\zeta _1+\theta \zeta _2+\rho _{\infty })-P'(\rho _\infty )\right) d\theta ,\quad \Psi (\zeta )=\frac{1}{\zeta +\rho _\infty }. \end{aligned}$$

Let $e^{tA}$ be the semigroup defined in (19) with $\gamma =\gamma _1$ and ${\mathcal {A}}={\mathcal {A}}_1.$ Then, the Duhamel principle gives

$$\begin{aligned} V_L(t) = e^{tA}_{L} V_0 + \int ^{t}_{0} e^{(t-\tau )A}_{L} \begin{bmatrix} 0 \\ h + \gamma _1^{-1} \sigma F(x) \end{bmatrix}(\tau )\,d\tau , \end{aligned}$$

(30)

where $V_L \equiv \dot{S}_{j_0} V,$ $e^{tA}_L\equiv \dot{S}_{j_0}e^{tA}$ and $V_0= V(0).$ Thanks to Lemma 4.1, for any $\psi =(\psi _1,\ldots ,\psi _4)^{\textsf{T}} \in {\mathcal {S}}^{4},$ we have

$$\begin{aligned} \langle V_L(t), \psi \rangle&= \langle e^{tA}_{L} V_0,\psi \rangle + \int _{0}^{t}\left\langle \begin{bmatrix} 0 \\ h + \gamma _1^{-1} \sigma F \end{bmatrix}(\tau ) , e^{(t-\tau )A^{*}}_{L}\psi \right\rangle d\tau \\&\lesssim \Vert V_0\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \Vert \psi \Vert _{\dot{B}^{-\frac{1}{2}}_{2,1}}+\sup _{0\le t<T}\Vert h + \gamma _1^{-1} \sigma F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }} \int _{0}^{\infty } \Vert e^{\tau A^{*}}_{L}\psi \Vert _{\dot{B}^{\frac{3}{2}}_{2,1}}d\tau \\&\lesssim \Big (\Vert V_0\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}+\sup _{0\le t<T} \Vert h + \gamma _1^{-1} \sigma F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}\Big ) \Vert \psi \Vert _{\dot{B}^{-\frac{1}{2}}_{2,1}}, \end{aligned}$$

where $0\le t < T.$ Then, Proposition 2.1(ii) implies that

$$\begin{aligned} \sup _{0\le t<T}\Vert V_L(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \lesssim \Vert V_0\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} + \sup _{0\le t< T} \Vert h + \gamma _1^{-1} \sigma F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}. \end{aligned}$$

Applying the bilinear estimate in Proposition 2.1(vi), we have

$$\begin{aligned} \Vert h + \gamma _1^{-1} \sigma F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }} \lesssim \Big (\Vert (\sigma ^*,v^*, \sigma , w)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{B}^{\frac{3}{2}}_{2,1}}+\Vert F\Vert _{\dot{B}^{-\frac{1}{2}}_{2,1}}\Big ) \Vert U\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}. \end{aligned}$$

Thus, if $\delta _1>0$ is small enough, then

$$\begin{aligned} \sup _{0\le t<T}\Vert U_L(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \lesssim \Vert U_0\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}+\delta _1 \sup _{0\le t<T}\Vert U(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}. \end{aligned}$$

Next, we estimate the high-frequency part $U_H.$ For any $u\in {\mathcal {S}}',$ we denote $u_L\equiv \dot{S}_{j_0}u$ and $u_H\equiv u-\dot{S}_{j_0}u.$ Since $U=(\sigma ,w)$ satisfies the Eq. (17), for any multi-index $\alpha _1,$ $\alpha _2,$ we have the following identities:

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\Vert \partial _{x}^{\alpha _1}U_H\Vert _{L^2}^{2} + \nu \Vert \partial _{x}^{\alpha _1}\nabla w_H\Vert _{L^2}^2 + (\nu +\nu ')\Vert \partial _{x}^{\alpha _1}\textrm{div}\,w_H\Vert _{L^2}^2 \\&\quad = \langle \partial _{x}^{\alpha _1} f_H, \partial _{x}^{\alpha _1} \sigma _H\rangle + \langle \partial _{x}^{\alpha _1} g_H, \partial _{x}^{\alpha _1} w_H \rangle , \\&\frac{d}{dt}\langle \partial _{x}^{\alpha _2}\nabla \sigma _H, \partial _{x}^{\alpha _2} w_H \rangle + \gamma _0\Vert \partial _{x}^{\alpha _2}\nabla \sigma _H\Vert _{L^2}^2 \\&\quad =\gamma _0\Vert \partial _{x}^{\alpha _2}\textrm{div}\, w_H\Vert _{L^2}^2+ \langle \partial _{x}^{\alpha _2}{\mathcal {A}} w_H,\partial _{x}^{\alpha _2}\nabla \sigma _H \rangle \\&\qquad + \langle \partial _{x}^{\alpha _2}\nabla f_H,\partial _{x}^{\alpha _2} w_H \rangle + \langle \partial _{x}^{\alpha _2}g_H, \partial _{x}^{\alpha _2}\nabla \sigma _H \rangle . \end{aligned}$$

Let $\kappa >0$ be small enough. We set

$$\begin{aligned} {\mathcal {E}}(t) = \frac{1}{2}\sum _{|\alpha _1| = 3} \Vert \partial _{x}^{\alpha _1}U_H(t)\Vert _{L^{2}}^2 + \sum _{|\alpha _2|= 2}\kappa \langle \partial _{x}^{\alpha _2}\nabla \sigma _H(t), \partial _{x}^{\alpha _2} w_H(t) \rangle . \end{aligned}$$

Then, we have the following inequality

$$\begin{aligned} \frac{d}{dt}{\mathcal {E}}(t)+ \widetilde{{\mathcal {E}}}(t) \lesssim&\sum _{|\alpha _1|= 3} \langle \partial _{x}^{\alpha _1} f_H, \partial _{x}^{\alpha _1} \sigma _H\rangle + \langle \partial _{x}^{\alpha _1} g_H, \partial _{x}^{\alpha _1} w_H \rangle \nonumber \\&+\sum _{|\alpha _2|= 2}\kappa \langle \partial _{x}^{\alpha _2}\nabla f_H,\partial _{x}^{\alpha _2} w_H \rangle + \langle \partial _{x}^{\alpha _2}g_H, \partial _{x}^{\alpha _2}\nabla \sigma _H \rangle , \end{aligned}$$

(31)

where $0<t<T$ and

$$\begin{aligned} \widetilde{{\mathcal {E}}}(t)&\equiv \sum _{|\alpha _1|= 3} \left( \nu \Vert \partial _{x}^{\alpha _1}\nabla w_H(t)\Vert _{L^2}^2 + (\nu + \nu ')\Vert \partial _{x}^{\alpha _1} \textrm{div}\,w_H(t)\Vert _{L^2}^2\right) \\&\quad + \sum _{|\alpha _2|= 2} \kappa \left( \Vert \partial _{x}^{\alpha _2}\nabla \sigma _H(t)\Vert _{L^2}^2 +\Vert \partial _{x}^{\alpha _2}\textrm{div}\,w_H(t)\Vert _{L^2}^2\right) . \end{aligned}$$

We also have

$$\begin{aligned} {\mathcal {E}}(t) \sim \Vert (\sigma _H,w_H)(t)\Vert _{\dot{H}^{3}}^{2},\quad \widetilde{{\mathcal {E}}}(t) \sim _{\kappa ,j_0} \Vert \sigma _H(t)\Vert _{\dot{H}^{3}}^2 + \Vert w_H(t)\Vert _{\dot{H}^4}^2 \end{aligned}$$

for $0\le t<T.$ Let us estimate the right-hand of (31). For any $\alpha _1$ with $|\alpha _1|=3,$ by Proposition 2.1(vi), we have the following estimate

$$\begin{aligned} \langle (v\cdot \nabla \partial _{x}^{\alpha _1}\sigma )_H,\partial _{x}^{\alpha _1}\sigma _H \rangle&\lesssim |\langle v\cdot \nabla \partial _x^{\alpha _1}\sigma _H,\partial _x^{\alpha _1}\sigma _H\rangle |+\big (\Vert v\cdot \nabla \partial _x^{\alpha _1}\sigma _L\Vert _{L^2}\\&\quad +\Vert (v\cdot \nabla \partial _x^{\alpha _1}\sigma )_L\Vert _{L^2}\big ) \Vert \partial _x^{\alpha _1}\sigma _H\Vert _{L^2} \\&\lesssim _{j_0} \Vert \textrm{div}\,v\Vert _{L^\infty }\Vert \partial _x^{\alpha _1}\sigma _H\Vert _{L^2}^2 +\Vert v\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}}\Vert \partial _x^{\alpha _1}\sigma \Vert _{L^2}\Vert \partial _x^{\alpha _1}\sigma _H\Vert _{L^2}, \end{aligned}$$

where $v=v^*+w.$ For any $\alpha _1$ with $|\alpha _1|=3,$ the above estimate leads us to

$$\begin{aligned}&\langle \partial _{x}^{\alpha _1} f_H, \partial _{x}^{\alpha _1} \sigma _H\rangle \\&\quad = \langle (v \cdot \nabla \partial _{x}^{\alpha _1} \sigma )_H, \partial _{x}^{\alpha _1} \sigma _H \rangle +\langle ((\textrm{div}\,v)\partial _{x}^{\alpha _1} \sigma )_H, \partial _{x}^{\alpha _1} \sigma _H \rangle \nonumber \\&\qquad +\sum _{0< \beta \le \alpha _1} \langle \textrm{div}(\partial ^{\beta }_{x}v\, \partial _{x}^{\alpha _1-\beta } \sigma )_H, \partial _{x}^{\alpha _1} \sigma _H \rangle + \langle \partial _x^{\alpha _1}\textrm{div}(\sigma ^*w)_H, \partial _{x}^{\alpha _1} \sigma _H \rangle \nonumber \\&\quad \lesssim \Vert \textrm{div}v\Vert _{L^\infty } \Vert \partial _{x}^{\alpha _1}\sigma _H\Vert _{L^{2}}^2 + \Vert v\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}\cap \dot{B}^{\frac{5}{2}}_{2,1}}\Vert \partial _x^{\alpha _1}\sigma \Vert _{L^2}\Vert \partial _x^{\alpha _1}\sigma _H\Vert _{L^2}\nonumber \\&\qquad + \Vert \nabla v\Vert _{H^{3}} \Vert \nabla \sigma \Vert _{H^{2}}\Vert \partial _x^{\alpha _1}\sigma _H\Vert _{L^{2}} + \Vert \nabla \sigma ^*\Vert _{H^{3}} \Vert \nabla w\Vert _{H^{3}}\Vert \partial _x^{\alpha _1}\sigma _H\Vert _{L^{2}} \nonumber \\&\quad \lesssim \delta _1 \big ( \Vert U\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}} + \Vert w\Vert _{\dot{H}^{4}} \big ) \Vert \partial _x^{\alpha _1}\sigma _H\Vert _{L^{2}} \lesssim \delta \Vert U\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}}^{2} +\delta _1 \widetilde{{\mathcal {E}}}. \end{aligned}$$

We also have

$$\begin{aligned} \langle \partial _{x}^{\alpha _2}\nabla f_H, \partial _x^{\alpha _2}w_H \rangle \lesssim \delta _1 \Vert U\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}}^{2}+\delta _1 \widetilde{{\mathcal {E}}}\quad {\textrm{for}}\ |\alpha _2|=2. \end{aligned}$$

By Proposition 2.1(i), (v) and (vi), we have

$$\begin{aligned} \sum _{|\alpha _1|= 3} \langle \partial _{x}^{\alpha _1} g_H, \partial _{x}^{\alpha _1} w_H \rangle +\sum _{|\alpha _2|=2}\kappa \langle \partial _{x}^{\alpha _2}g_H, \partial _{x}^{\alpha _2}\nabla \sigma _H \rangle \lesssim \delta _1 \Vert U\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}}^{2}+\delta _1 \widetilde{{\mathcal {E}}}. \end{aligned}$$

Thus, if $\delta _1>0$ is sufficiently small, then there exits a constant $c_0>0$ such that

$$\begin{aligned} \frac{d}{dt}{\mathcal {E}}(t) + c_0 \widetilde{{\mathcal {E}}}(t) \lesssim \delta _1 \Vert U(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}}^{2}\quad {\textrm{for}}\ 0< t<T. \end{aligned}$$

Since ${\mathcal {E}} \lesssim _{j_0} \widetilde{{\mathcal {E}}},$ we have the estimate

$$\begin{aligned} \frac{d}{dt}{\mathcal {E}}(t) + c_0 {\mathcal {E}}(t) \lesssim \delta _1 \Vert U(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}}^{2}\quad {\textrm{for}}\ 0< t<T. \end{aligned}$$

Then, Grönwall’s inequality shows that

$$\begin{aligned} \Vert U_H(t)\Vert _{\dot{H}^{3}}^2&\lesssim e^{-c_0 t}\Vert U_H(0)\Vert _{\dot{H}^3}^2 +\delta _1 \int _{0}^{t} e^{-c_0(t-\tau )}\Vert U(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}}^{2}d\tau \\&\lesssim \Vert U_0\Vert _{\dot{H}^3}^2 + \delta _1 \sup _{0\le \tau <T} \Vert U(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}^{2} \end{aligned}$$

for $0\le t<T.$ Thus, if $\delta _1>0$ is small enough, then the estimate (26) holds. $\square$

Next, we show the following local existence result. The proof of Proposition 4.4 below is inspired by the argument in Danchin [8]. Since $(\rho ^*-\rho _\infty ,v^*)$ is in $\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3},$ it suffices to show the following proposition to prove the local existence of (17).

Proposition 4.4

There exist constants $T_0>0,$ $\delta _{2,0}>0$ and $C_2>0$ such that if an external force F(x) and an initial value $(\rho _0,v_0)=(b_0+\rho _\infty ,v_0)$ satisfy

$$\begin{aligned} \Vert F\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }\cap \dot{H}^{2}} + \Vert (b_0,v_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3}} \le \delta _2 \end{aligned}$$

(32)

with $\delta _2\le \delta _{2,0},$ then there exists a unique solution $(\rho ,v)=(b+\rho _\infty ,v)$ of (1) on $[0,T_0)\times {\mathbb {R}}^3$ satisfying $(b,v)|_{t=0}=(b_0,v_0),$ $(b,v) \in C^{0}([0,T_0);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3),$ $v\in L^{2}_{t}((0,T_0);\dot{H}^4)$ and

$$\begin{aligned} \sup _{0\le t<T_0} \Vert (b,v)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}\le C_2 \delta _2. \end{aligned}$$

Proof

Let $0<T_0\le 1,$ and let $0<\delta _2\le \delta _{2,0}\le 1$ be satisfying (32). We denote the function spaces $X_{T_0}$ and $Y_{T_0}$ by

$$\begin{aligned} X_{T_0}&\equiv \left\{ ({b},{v}) \in C^{0}([0,T_0);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3) \,\Bigg |\, {v}\in L^{2}_{t}((0,T_0);\dot{H}^{4}) \right\} ,\\ Y_{T_0}&\equiv \left\{ ({b},{v}) \in C^{0}([0,T_0);\dot{B}^{\frac{1}{2}}_{2,\infty })\,\Bigg |\, {v}\in C^{0}([0,T_0);\dot{B}^{\frac{3}{2}}_{2,\infty }) \right\} \end{aligned}$$

with norms

$$\begin{aligned} \Vert ({b},{v})\Vert _{X_{T_0}}&\equiv \Vert ({b},{v})\Vert _{C^{0}([0,T_0);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3)} + \Vert {v}\Vert _{L^{2}_{t}((0,T_0);\dot{H}^{4})},\\ \Vert ({b},{v})\Vert _{Y_{T_0}}&\equiv \Vert ({b},{v})\Vert _{C^{0}([0,T_0);\dot{B}^{\frac{1}{2}}_{2,\infty })} + \Vert {v}\Vert _{C^{0}([0,T_0);\dot{B}^{\frac{3}{2}}_{2,\infty })}. \end{aligned}$$

We denote the norm of the Chemin–Lerner space $\tilde{L}_{t}^{\infty }((0,T_0);\dot{H}^3)$ by

$$\begin{aligned} \Vert u\Vert _{\tilde{L}_{t}^{\infty }((0,T_0);\dot{H}^3)}\equiv \left\| \{2^{3j}\Vert \dot{\Delta }_{j}u\Vert _{L^{\infty }_{t}((0,T_0);L^2)}\}_{j\in {\mathbb {Z}}} \right\| _{\ell ^{2}({\mathbb {Z}})}. \end{aligned}$$

(Cf. [3].) The Minkowski inequality implies the estimate

$$\begin{aligned} \Vert u\Vert _{{L}_{t}^{\infty }((0,T_0);\dot{H}^3)}\le \Vert u\Vert _{\tilde{L}_{t}^{\infty }((0,T_0);\dot{H}^3)}. \end{aligned}$$

Let us construct the approximation sequence

$$\begin{aligned} \{U_n\}_{n\ge -1} \equiv \{(b_n,v_n)^{\textsf{T}}\}_{n\ge -1}\subset X_{T_0} \end{aligned}$$

as follows: The sequence $\{U_n\}_{n\ge -1}$ defined by $(b_{-1},v_{-1})=(0,0)$ and

$$\begin{aligned} \left\{ \, \begin{aligned}&\partial _{t}b_{n+1} + v_{n}\cdot \nabla b_{n+1} = f_{0}(b_n,v_n), \\&\partial _{t}v_{n+1} - {\mathcal {A}}_{0} v_{n+1} = g_0(b_n,v_n), \\&(b_{n+1},v_{n+1})|_{t=0} = (b_0, v_0), \end{aligned} \right. \end{aligned}$$

(33)

where $\nu _0=\mu /\rho _\infty ,$ $\nu _0'=\mu '/\rho _\infty$ and ${\mathcal {A}}_{0}\equiv \nu _0 \Delta + (\nu _0+\nu _0')\nabla \textrm{div};$ $f_{0}$ and $g_{0}$ are defined by the following:

$$\begin{aligned} f_0(b,v)&= \rho _\infty \textrm{div}\,v+(\textrm{div}\,v)b,\quad g_0(b,v)= g_{0,1}(b,v)+g_{0,2}(b,v),\\ g_{0,1}(b,v)&= -\Phi (b) \nabla b-v\cdot \nabla v,\quad g_{0,2}(b,v) = (\Psi (b)-\Psi (0)){\mathcal {A}}_{0} v + F,\\ \Phi (\zeta )&=\frac{P'(\zeta +\rho _{\infty })}{\zeta +\rho _{\infty }},\quad \Psi (\zeta )=\frac{1}{\zeta +\rho _{\infty }}. \end{aligned}$$

Here, the solutions of the first equation of (33) is given by

$$\begin{aligned} b_{n+1}(t,x) = b_0(\psi _{n}^{-1}(t,x)) + \int _{0}^{t} f_{0}(b_n,v_n)(\tau , \psi _{n}(\tau ,\psi _{n}^{-1}(t,x))) d\tau , \end{aligned}$$

where $\psi _{n}$ is the solution of the ordinal differential equation:

$$\begin{aligned} \left\{ \, \begin{aligned}&\partial _{t}\psi _{n}(t,x)=v_{n}(t,\psi _{n}(t,x)), \\&\psi _{n}(0,x)=x. \end{aligned} \right. \end{aligned}$$

We shall prove by induction that if $T_0>0$ and $\delta _{2,0}>0$ are small enough, then for any $n\ge 1,$

$$\begin{aligned}&\Vert U_{n}\Vert _{X_{T_0}} + \Vert U_n\Vert _{\tilde{L}_{t}^{\infty }((0,T_0);\dot{H}^3)}\lesssim \delta _2, \end{aligned}$$

(34)

$$\begin{aligned}&\Vert U_n-U_{n-1}\Vert _{Y_{T_0}} \le \frac{1}{2}\Vert U_{n-1}-U_{n-2}\Vert _{Y_{T_0}}. \end{aligned}$$

(35)

Let $n\ge 1$ and assume that the inequalities (34), (35) hold for $1\le k\le n.$

Since $U_{n+1}=(b_{n+1},v_{n+1})^{\textsf{T}}$ is a solution of (33), for any $j\in {\mathbb {Z}},$ we have the energy estimates

$$\begin{aligned} \frac{d}{dt}\Vert \dot{\Delta }_{j}v_{n+1}\Vert _{L^2}^2 + c_0 \Vert \nabla \dot{\Delta }_{j}v_{n+1}\Vert _{L^2}^2&\lesssim \langle \dot{\Delta }_{j}g_{0}(b_n,v_n),\dot{\Delta }_{j}v_{n+1}\rangle \\&\lesssim 2^{-j}\Vert \dot{\Delta }_{j}g_0(b_n,v_n)\Vert _{L^{2}} \Vert \nabla \dot{\Delta }_{j}v_{n+1}\Vert _{L^2}, \end{aligned}$$

where $c_0>0$ is a constant, and

$$\begin{aligned}&\frac{d}{dt}\Vert \nabla \dot{\Delta }_{j}b_{n+1}\Vert _{L^2}^2\\&\quad \lesssim \langle -\nabla (\dot{\Delta }_{j}(v_n\cdot \nabla b_{n+1}))+\nabla \dot{\Delta }_{j}f_0(b_n,v_n), \nabla \dot{\Delta }_{j}b_{n+1}\rangle \\&\quad \lesssim |\langle \nabla (\dot{\Delta }_{j}(v_{n}\cdot \nabla b_{n+1})), \nabla \dot{\Delta }_{j}b_{n+1}\rangle | + \Vert \nabla \dot{\Delta }_{j}f_0(b_n,v_n)\Vert _{L^2} \Vert \nabla \dot{\Delta }_{j}b_{n+1}\Vert _{L^2}. \end{aligned}$$

For any $0\le t<T_0,$ by Proposition 2.1(v), (vi) and Lemma 2.3, there exists a sequence $\{d_j(t)\}_{j\in {\mathbb {Z}}}$ with $\Vert \{d_j(t)\}\Vert _{\ell ^2({\mathbb {Z}})}\le 1$ such that

$$\begin{aligned} \Vert \dot{\Delta }_{j}g_{0,1}(b_n,v_n)(t)\Vert _{L^{2}}&\lesssim 2^{-2j} d_j(t)\Vert g_{0,1}(b_n,v_n)(t)\Vert _{\dot{H^2}}\lesssim 2^{-2j} d_j(t) \delta _2^2,\\ \Vert \dot{\Delta }_{j}g_{0,2}(b_n,v_n)(t)\Vert _{L^2}&\lesssim 2^{-2j}d_j(t)\Vert g_{0,2}(b_n,v_n)(t)\Vert _{\dot{H}^2}\\&\lesssim 2^{-2j}d_j(t)\big (\Vert b_{n}(t)\Vert _{\dot{H}^1\cap \dot{H}^3}\Vert v_n(t)\Vert _{\dot{H}^{2}\cap \dot{H}^4}+ \Vert F\Vert _{\dot{H}^2}\big )\\&\lesssim 2^{-2j}d_{j}(t) \delta _2\Vert v_n(t)\Vert _{\dot{H}^4} + 2^{-2j}d_{j}(t) \delta _2^2 + 2^{-2j}d_j(t) \delta _2, \\ \Vert \nabla \dot{\Delta }_{j}f_0(b_n,v_n)(t)\Vert _{L^2}&\lesssim 2^{-2j}d_j(t) \Vert v_n(t)\Vert _{\dot{H}^{1}\cap \dot{H}^4}\Vert b_n(t)\Vert _{\dot{H}^1\cap \dot{H}^3}\\&\lesssim 2^{-2j}d_{j}(t) \delta _2\Vert v_n(t)\Vert _{\dot{H}^4}+ 2^{-2j}d_{j}(t) \delta _2^2 \end{aligned}$$

and

$$\begin{aligned}&|\langle \nabla (\dot{\Delta }_{j}(v_n\cdot \nabla b_{n+1})), \nabla \dot{\Delta }_{j}b_{n+1}\rangle (t)|\\&\quad \lesssim \Vert [\dot{\Delta }_{j},v_{n}\cdot \nabla ]\nabla b_{n+1}(t)\Vert _{L^2} \Vert \nabla \dot{\Delta }_{j}b_{n+1}(t)\Vert _{L^2} + \Vert \textrm{div}v_n(t)\Vert _{L^\infty } \Vert \nabla \dot{\Delta }_{j}b_{n+1}(t)\Vert _{L^2}^2 \\&\qquad +2^{-2j}d_j(t)\Vert U_n(t)\Vert _{\dot{H}^1\cap \dot{H}^3}^{2}\Vert \nabla \dot{\Delta }_{j}b_{n+1}(t)\Vert _{L^2}\\&\quad \lesssim 2^{-2j}d_{j}(t)\delta _2 \big (\Vert b_{n+1}(t)\Vert _{\dot{H}^{3}}+\delta _2\big ) \Vert \nabla \dot{\Delta }_{j}b_{n+1}(t)\Vert _{L^2} + \delta _2\Vert \nabla \dot{\Delta }_{j}b_{n+1}(t)\Vert _{L^2}^2. \end{aligned}$$

Here, we use the identities:

$$\begin{aligned} \langle v_n \cdot \nabla \partial _k \dot{\Delta }_{j}b_{n+1}, \partial _k \dot{\Delta }_{j}b_{n+1} \rangle =-\frac{1}{2}\langle \textrm{div}\,v_n \,\partial _k \dot{\Delta }_{j}b_{n+1}, \partial _k \dot{\Delta }_{j}b_{n+1} \rangle \end{aligned}$$

for $1\le k\le 3.$ Thus, if $T_0>0$ is small enough, we have the estimate

$$\begin{aligned} \Vert U_{n+1}\Vert _{\tilde{L}_t^{\infty }((0,T_0);\dot{H}^3)} + \Vert v_{n+1}\Vert _{L^2_{t}((0,T_0);\dot{H}^4)}\lesssim \delta _2 + \delta _2^{\frac{1}{2}} \Vert U_{n+1}\Vert _{X_{T_0}}. \end{aligned}$$

By the Duhamel principle, we may rewrite the second equation of (33) by

$$\begin{aligned} v_{n+1}(t) = e^{t{\mathcal {A}}_0}v_0 +\int _{0}^{t}e^{(t-\tau ){\mathcal {A}}_{0}}g_0(b_n,v_n)(\tau )d\tau . \end{aligned}$$

We use the following estimate for the semigroup $e^{t{\mathcal {A}}_{0}}.$ $\square$

Lemma 4.5

Let $s\in {\mathbb {R}},$ $1\le r\le \infty$ and $\alpha \ge 0.$ For any $\psi _1\in \dot{B}^{s}_{2,r},$ $\psi _2\in \dot{B}^{s}_{2,1},$ we have

$$\begin{aligned} \Vert e^{t{\mathcal {A}}_0}\psi _1\Vert _{\dot{B}^{s+\alpha }_{2,r}}\lesssim t^{-\frac{\alpha }{2}}\Vert \psi _1\Vert _{\dot{B}^{s}_{2,r}},\quad \int _{0}^{t} \Vert e^{\tau {\mathcal {A}}_0}\psi _2\Vert _{\dot{B}^{s+2}_{2,1}} d\tau \lesssim \Vert \psi _2\Vert _{\dot{B}^{s}_{2,1}}\quad {\textrm{for}}\ t> 0. \end{aligned}$$

As for the proof, see [4, Proposition 2.1], [7, Corollary 2.7] for example.

By Proposition 2.1(v), (vi) and Lemma 4.5, for any $0\le t<T_0,$ we have

$$\begin{aligned} \Vert e^{t{\mathcal {A}}_0}v_0\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}&\lesssim \Vert v_0\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }},\\ \left\| \int _{0}^{t}e^{(t-\tau ){\mathcal {A}}_0}g_{0}(b_n,v_n)(\tau )d\tau \right\| _{\dot{B}^{\frac{1}{2}}_{2,\infty } }&\lesssim \int _{0}^{t} \tau ^{-\frac{1}{2}} \Vert g_{0}(b_n,v_n)(\tau )\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }} d\tau \\&\lesssim {T_0}^{\frac{1}{2}} \Big ( \Vert U_n\Vert _{X_{T_0}}+ \Vert U_n\Vert _{X_{T_0}}^{2}+\Vert F\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }} \Big ). \end{aligned}$$

Let us write $g_{0}(b_n,v_n)-g_{0}(b_{n-1},v_{n-1})=\tilde{g}_1+\tilde{g}_2+\tilde{g}_3$ with

$$\begin{aligned} \tilde{g}_1&= -(\Psi (b_{n})-\Psi (b_{n-1}))\nabla b_{n} -(v_n-v_{n-1})\cdot \nabla v_n,\\ \tilde{g}_2&= -\Psi (b_{n-1}) \nabla (b_{n}-b_{n-1})- v_{n-1}\cdot \nabla (v_{n}-v_{n-1}), \\ \tilde{g}_3&= g_{0,2}(b_n,v_n) -g_{0,2}(b_{n-1},v_{n-1}). \end{aligned}$$

By Proposition 2.1(v), (vi) and Lemma 4.5, for any $0\le t<T_0,$ we have

$$\begin{aligned}&\left\| \int _{0}^{t}e^{(t-\tau ){\mathcal {A}}_0} \tilde{g}_1(\tau ) d\tau \right\| _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{B}^{\frac{3}{2}}_{2,\infty }}\lesssim \int _{0}^{t} \tau ^{-\frac{1}{2}} \Vert \tilde{g}_1(\tau )\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }\cap \dot{B}^{\frac{1}{2}}_{2,\infty }} d\tau \\&\quad \lesssim T_{0}^{\frac{1}{2}} (1+\Vert (U_{n},U_{n-1})\Vert _{X_{T_0}})\sup _{0\le \tau <T_0}\Vert (U_{n}-U_{n+1})(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \end{aligned}$$

and

$$\begin{aligned}&\left\| \int _{0}^{t}e^{(t-\tau ){\mathcal {A}}_0} \tilde{g}_2(\tau ) d\tau \right\| _{\dot{B}^{\frac{1}{2}}_{2,\infty }}\lesssim \int _{0}^{t} \tau ^{-\frac{1}{2}} \Vert \tilde{g}_2(\tau )\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }} d\tau \\&\quad \lesssim T_{0}^{\frac{1}{2}} (1+\Vert (U_{n},U_{n-1})\Vert _{X_{T_0}})\sup _{0\le \tau <T_0}\Vert (U_{n}-U_{n+1})(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}. \end{aligned}$$

For any $\psi \in {\mathcal {S}}$ and $0\le t <T_0,$ by Proposition 2.1(ii), (v), (vi) and Lemma 4.5, we have

$$\begin{aligned}&\left\langle \int _{0}^{t} e^{(t-\tau ){\mathcal {A}}_{0}}\tilde{g}_2(\tau ) d\tau ,\psi \right\rangle \lesssim \sup _{0\le \tau<T_0}\Vert \tilde{g}_2(\tau )\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }}\int _{0}^{\infty } \Vert e^{\tau {\mathcal {A}}_0}\psi \Vert _{\dot{B}^{\frac{1}{2}}_{2,1}} d\tau \\&\quad \lesssim \Vert (U_n,U_{n-1})\Vert _{X_{T_0}}\sup _{0\le \tau <T_0}\Vert (U_n-U_{n-1})(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}\Vert \psi \Vert _{\dot{B}^{-\frac{3}{2}}_{2,1}} \end{aligned}$$

and

$$\begin{aligned}&\left\langle \int _{0}^{t} e^{(t-\tau ){\mathcal {A}}_{0}}\tilde{g}_3(\tau ) d\tau ,\psi \right\rangle \\&\quad \lesssim \sup _{0\le \tau<T_0}\Vert \tilde{g}_3(\tau )\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{B}^{-\frac{1}{2}}_{2,\infty }}\int _{0}^{\infty } \Vert e^{\tau {\mathcal {A}}_0}\psi \Vert _{\dot{B}^{\frac{3}{2}}_{2,1}+\dot{B}^{\frac{1}{2}}_{2,1}} d\tau \\&\quad \lesssim \Vert (U_n,U_{n-1})\Vert _{X_{T_0}}\Big (\sup _{0\le \tau<T_0}\Vert (U_n-U_{n-1})(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \\&\qquad +\sup _{0\le \tau <T_0}\Vert (v_{n}-v_{n+1})(\tau )\Vert _{\dot{B}^{\frac{3}{2}}_{2,\infty }}\Big )\Vert \psi \Vert _{\dot{B}^{-\frac{1}{2}}_{2,1}+\dot{B}^{-\frac{3}{2}}_{2,1}}. \end{aligned}$$

Hence, by Proposition 2.1(ii) and the induction hypothesis,

$$\begin{aligned} \sup _{0\le t<T_0}\Vert v_{n+1}(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}&\lesssim \delta _2,\\ \sup _{0\le t<T_0}\Vert (v_{n+1}-v_{n})(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{B}^{\frac{3}{2}}_{2,\infty }}&\lesssim (T_{0}^{\frac{1}{2}}+\delta _2)\Vert U_{n}-U_{n+1}\Vert _{Y_{T_0}}. \end{aligned}$$

To estimate $\sup _{0\le t<T_0}\Vert b_{n+1}(t)\Vert _{\dot{B}^{1/2}_{2,\infty }}$ and $\sup _{0\le t<T_0}\Vert (b_{n+1}-b_{n})(t)\Vert _{\dot{B}^{1/2}_{2,\infty }},$ we use the following lemma.

Lemma 4.6

Let $\tilde{v},\tilde{f} \in X_{T_0},$ and let $\tilde{b}$ be a solution of the linear transport equation

$$\begin{aligned} \partial _t \tilde{b} + \tilde{v}\cdot \nabla \tilde{b} = \tilde{f}\quad {\textrm{on}}\ [0,T_0)\times {\mathbb {R}}^{3}. \end{aligned}$$

Then,  there exists a constant $C>0$ such that

$$\begin{aligned} \Vert \tilde{b}(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \lesssim e^{CE_{\tilde{v}}(t)} \left( \Vert \tilde{b}(0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} + \int _{0}^{t}e^{-CE_{\tilde{v}}(\tau )}\Vert \tilde{f}(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}d\tau \right) , \end{aligned}$$

for any $0\le t<T_0,$ where $E_{\tilde{v}}(t)\equiv \int _{0}^{t}\Vert \nabla \tilde{v}(\tau )\Vert _{\dot{B}^{3/2}_{2,1}}d\tau .$

This lemma is a special case of [7, Theorem 2.5]. Hence, we omit the proof.

Since $E_{v_n}(T_0)\lesssim 1,$ by Lemma 4.6, we have

$$\begin{aligned} \sup _{0<t\le T_0} \Vert b_{n+1}(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \lesssim \Vert b_0\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} +T_0\sup _{0<t\le T_0}\Vert f_0(b_n,v_n)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \end{aligned}$$

and

$$\begin{aligned} \sup _{0<t\le T_0} \Vert (b_{n+1}-b_{n})(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}&\lesssim T_0\sup _{0<t\le T_0}\Big ( \Vert ((v_n-v_{n-1})\cdot \nabla b_n)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}\\&\quad +\Vert (f_0(b_n,v_n)-f_0(b_{n-1},v_{n-1}))(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \Big ). \end{aligned}$$

Then, Proposition 2.1(vi) and the induction hypothesis show that

$$\begin{aligned}&\sup _{0<t\le T_0} \Vert b_{n+1}(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \lesssim \delta _2,\\&\sup _{0<t\le T_0} \Vert (b_{n+1}-b_{n})(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}\lesssim T_0 \Vert U_n-U_{n-1}\Vert _{Y_{T_0}}. \end{aligned}$$

If $T_0>0$ and $\delta _{2,0}>0$ are small enough, then we have

$$\begin{aligned}&\Vert U_{n+1}\Vert _{X_{T_0}} +\Vert U_{n+1}\Vert _{\tilde{L}_{t}^{\infty }((0,T_0);\dot{H}^3)}\lesssim \delta _2,\\&\Vert U_{n+1}-U_{n}\Vert _{Y_{T_0}}\le \frac{1}{2}\Vert U_{n}-U_{n-1}\Vert _{Y_{T_0}}. \end{aligned}$$

Therefore, by induction, we have the estimates (34) and (35) for any $n\ge 1.$ The estimate (35) implies that $U_n$ converges to some $U\in Y_{T_0}$ as $n\rightarrow \infty .$ Then, for any $j\in {\mathbb {Z}}$ and $\psi \in C^{\infty }_{0}((0,T_0)\times {\mathbb {R}}^3),$

$$\begin{aligned} \langle \dot{\Delta }_{j}U_n, \psi \rangle \rightarrow \langle \dot{\Delta }_{j}U, \psi \rangle \quad {\textrm{as}}\ n\rightarrow \infty . \end{aligned}$$

For any $j\in {\mathbb {Z}},$ there exists a sequence $\{\psi _{m}^{(j)}\}_{m\in {\mathbb {Z}}_{\ge 0}}\subset C^{\infty }_{0}((0,T_0)\times {\mathbb {R}}^3)$ such that $\sup _{m}\Vert \psi _{m}^{(j)}\Vert _{L^1_t((0,T_0);L^2)}\le 1$ and

$$\begin{aligned} \langle \dot{\Delta }_{j}U, \psi _{m}^{(j)}\rangle \rightarrow \Vert \dot{\Delta }_{j}U\Vert _{L^{\infty }_{t}((0,T_0);L^2)} \quad {\textrm{as}}\ m\rightarrow \infty . \end{aligned}$$

Thus, by Fatou’s lemma, we obtain

$$\begin{aligned} \Vert U\Vert _{\tilde{L}_t^{\infty }((0,T_0);\dot{H}^3)}&\le \liminf _{m\rightarrow \infty }\liminf _{n\rightarrow \infty } \left\| \{2^{3j} \langle \dot{\Delta }_{j}U_n,\psi _{m}^{(j)}\rangle \}_{j\in {\mathbb {Z}}}\right\| _{\ell ^{2}({\mathbb {Z}})}\\&\le \liminf _{n\rightarrow \infty }\Vert U_n\Vert _{\tilde{L}_t^{\infty }((0,T_0);\dot{H}^3)}\lesssim \delta _2. \end{aligned}$$

Since $U\in C^{0}((0,T_0);\dot{B}^{1/2}_{2,\infty }),$ for any $j\in {\mathbb {Z}},$ the low frequency part $\dot{S}_{j}U$ belongs to $C^{0}((0,T_0);\dot{B}^{1/2}_{2,\infty }\cap \dot{H}^3).$ By $U\in \tilde{L}_t^{\infty }((0,T_0);\dot{H}^3),$ we have

$$\begin{aligned} \Vert U-\dot{S}_{j}U\Vert _{L^{\infty }_{t}((0,T_0);\dot{H}^3)} \rightarrow 0\quad {\textrm{as}}\ j\rightarrow \infty . \end{aligned}$$

Then, we have $U\in C^{0}([0,T_0);\dot{H}^3).$ Since $U_n$ satisfy the relation (33), $(\rho ,v)\equiv U+(\rho _{\infty },0)$ is a solution of (1) such that $U|_{t=0}=(b_0,v_0),$ $U\in X_{T_0},$ $v\in L^{2}_{t}((0,T_0);\dot{H}^1\cap \dot{H}^4)$ and

$$\begin{aligned} \Vert U\Vert _{C^{0}([0,T_0);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3})} + \Vert v\Vert _{L^{2}_{t}((0,T_0);\dot{H}^{1}\cap \dot{H}^4)} \le c_1 \delta _2, \end{aligned}$$

where $c_1>0$ is a constant. Let us show the uniqueness. Let $\tilde{U}=(\tilde{b},\tilde{v})$ be a solution of (1) satisfying

$$\begin{aligned} \Vert \tilde{U}\Vert _{C^{0}([0,T_0);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{3})} + \Vert \tilde{v}\Vert _{L^{2}_{t}((0,T_0);\dot{H}^{1}\cap \dot{H}^4)} \le c_1 \delta _2. \end{aligned}$$

Then, by the proof of (34), the estimate

$$\begin{aligned} \Vert U-\tilde{U}\Vert _{Y_{T_0}} \le \frac{1}{2} \Vert U-\tilde{U}\Vert _{Y_{T_0}} \end{aligned}$$

holds. Hence, $U=\tilde{U}.$

The rest of this section is devoted to proving Theorem 4.2.

Proof of Theorem 4.2

Let $(\rho ^*,v^*)$ be the stationary solution satisfying (7). By Proposition 4.4 and (7), there exist $\delta _{2,0}>0,$ $T_0>0$ and $C_2 \ge 1$ such that if an initial perturbation $(\sigma _0,w_0)=(\rho _0-\rho ^*,v_0-v^*)$ and F(x) satisfy

$$\begin{aligned} \Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} + \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3} \le \delta _{2,0}, \end{aligned}$$

then there exists a unique solution $(\sigma ,w)=(\rho -\rho ^*,v-v^*)$ of (17) on $[0,2T_0)$ such that $(\sigma ,w)(0)=(\sigma _0,w_0),$ $(\sigma ,w)\in C^{0}([0,2T_0);\dot{H}^3)$ and

$$\begin{aligned} \sup _{0\le t < 2T_0} \Vert (\sigma ,w)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} \le C_2 \big (\Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} + \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3}\big ). \end{aligned}$$

Let $C_1\ge 1$ and $\delta _1>0$ be constants appearing in Proposition 4.3. We take an initial perturbation $(\sigma _0,w_0)$ and F(x) such that

$$\begin{aligned} C_1\Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} + \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3} \le \min \left\{ \delta _{2,0}, \frac{\delta _{1}}{C_2}\right\} . \end{aligned}$$

Let $N\ge 1,$ and let $(\sigma ,w)=(\rho -\rho ^*,v-v^*)\in C^{0}([0,NT_0);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3)$ be a solution of (17) on $[0,NT_0)$ with initial value $(\sigma _0,w_0)$ satisfying

$$\begin{aligned} \sup _{0\le t< NT_0} \Vert (\sigma ,w)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} \le C_1 \Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}. \end{aligned}$$

Since

$$\begin{aligned} \Vert (\sigma ,w)((N-1)T_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} + \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3} \le \delta _{2,0}, \end{aligned}$$

by Proposition 4.4, there exists a unique $(\tilde{\sigma },\tilde{w})\in C^{0}([0,2T_0);\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3)$ such that $(\tilde{\sigma },\tilde{w})$ is a solution of (17) with $(\tilde{\sigma },\tilde{w})(0)=(\sigma ,w)((N-1)T_0)$ satisfying

$$\begin{aligned} \sup _{0\le t < 2T_0} \Vert (\tilde{\sigma },\tilde{w})(t) \Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}&\le C_2 \left( \Vert (\sigma ,w)((N-1)T_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} + \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3}\right) \\&\le C_2 \left( C_1\Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} + \Vert F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }\cap \dot{H}^3}\right) \le \delta _1. \end{aligned}$$

Let $(\sigma ,w)(t)\equiv (\tilde{\sigma },\tilde{w})(t-NT_0),$ $t\in [NT_0,(N+1)T_0).$ Then, $(\sigma ,w)$ is a solution of (17) on $[0,(N+1)T_0)$ and, by Proposition 4.3, we have

$$\begin{aligned} \sup _{0\le t< (N+1)T_0} \Vert (\sigma ,w)(t)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3} \le C_1 \Vert (\sigma _0,w_0)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^3}. \end{aligned}$$

Hence, the proof is completed by induction on N. $\square$

5 The proof of decay estimates

Throughout this section, we fix the stationary solution $(\rho ^{*},v^{*})=(\sigma ^{*}+\rho _{\infty },v^{*})$ obtained in Theorem 1.1 and the perturbation $(\sigma ,w)=(\rho -\rho ^{*},v-v^*)$ obtained in Theorem 4.2. From now on, we denote the real numbers $\delta _1,$ $\delta _2$ and $\delta$ by

$$\begin{aligned} \delta _1\equiv \Vert \sigma ^*\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }\cap \dot{H}^{4}}+\Vert v^{*}\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{5}},\ \delta _2\equiv \sup _{t>0}\Vert (\sigma ,w)(t)\Vert _{\dot{B}^{1/2}_{2,\infty }\cap \dot{H}^3} \end{aligned}$$

and $\delta \equiv \delta _1+\delta _2.$ In order to prove the time decay estimates (10), (11) and (12), we show the following weak-type decay estimate.

Theorem 5.1

Let $-3/2\le s_0 \le 1/2.$ If $\delta =\delta _1+\delta _2$ is small enough,  then,  for any $-3/2<s<3/2$ with $s_0\le s,$ we have

$$\begin{aligned} \Vert (\sigma , w)(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{s}\, (1+t)^{-\frac{s-s_0}{2}}\Vert (\sigma _0, w_0)\Vert _{\dot{B}^{s_0}_{2,\infty }\cap \dot{H}^3}. \end{aligned}$$

(36)

The proof of Theorem 5.1 is performed by decomposing the perturbation into low- and high-frequency parts with respect to the Fourier space. We decompose the perturbation $(\sigma , w)$ into low- and high-frequency components for fixed $j_0\in {\mathbb {Z}}$:

$$\begin{aligned} (\sigma , w)= (\sigma _L , w_L) + (\sigma _H , w_H), \end{aligned}$$

(37)

where $(\sigma _L, w_L)\equiv (\dot{S}_{j_0}\sigma ,\dot{S}_{j_0} w).$

5.1 Estimate for the low frequency part

Let us establish the time decay estimate for the low frequency part of the perturbation $(\sigma _L, w_L).$

Proposition 5.2

Let $-3/2\le s_0\le 1/2,$ and let $\epsilon >0$ be a small number. If $\delta =\delta _1+\delta _2$ is sufficiently small,  then,  for any $T>0,$ we have

$$\begin{aligned} \sup _{0 \le t \le T} (1+t)^{\frac{s-s_0}{2}} \Vert (\sigma _L, w_L)(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{\epsilon ,j_0} \Vert (\sigma _0, w_0)\Vert _{\dot{B}^{s_0}_{2,\infty }}+ \delta {\mathcal {D}}_{\epsilon ,s_0}(T), \end{aligned}$$

(38)

where $-3/2+\epsilon \le s \le 3/2-\epsilon$ with $s_0\le s.$ Here,  the quantity ${\mathcal {D}}_{\epsilon ,s_0}(T)$ is defined by

$$\begin{aligned} {\mathcal {D}}_{\epsilon ,s_0}(T) \equiv \sup _{\begin{array}{c} -3/2+\epsilon \le \eta \le 3/2-\epsilon , \\ s_0\le \eta \end{array}}\ \sup _{ 0 \le t \le T} (1+t)^{\frac{\eta -s_0}{2}}\Vert (\sigma , w)(t)\Vert _{\dot{B}^{\eta }_{2,\infty }}. \end{aligned}$$

(39)

Proof

Let $n=(m-m^*)/\rho _\infty$ with $m=\rho v,$ $m^*=\rho ^* v^*.$ Then, $(\sigma ,n)$ satisfies the Eq. (28). By the definition of n,  we have $n=\gamma _{1}^{-1}w+\gamma _{1}(\sigma v-\sigma ^*w).$ Let $n_L=\dot{S}_{j_0}n.$ Then, for any $-3/2<s<3/2,$ Proposition 2.1(vi) shows

$$\begin{aligned} \Vert w_L\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _s \Vert n_L\Vert _{\dot{B}^{s}_{2,\infty }} + \delta \Vert (\sigma ,w)\Vert _{\dot{B}^{s}_{2,\infty }}. \end{aligned}$$

Thus, to prove Proposition 5.2, it is sufficient to show the inequalities

$$\begin{aligned} \sup _{0 \le t \le T} (1+t)^{\frac{s-s_0}{2}} \Vert (\sigma _L, n_L)(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{\epsilon ,p,j_0} \Vert (\sigma _0, w_0)\Vert _{\dot{B}^{s_0}_{2,\infty }}+ \delta {\mathcal {D}}_{\epsilon ,s_0}(T), \end{aligned}$$

where $-3/2+\epsilon \le s\le 3/2-\epsilon$ with $s_0\le s.$ Let $e^{tA}$ be the semigroup associated with the left-hand side of (28). Then, the Duhamel principle gives

$$\begin{aligned} \begin{bmatrix} \sigma _{L} \\ n_{L} \end{bmatrix}(t) = e^{tA}_{L} \begin{bmatrix} \sigma _0 \\ n_0 \end{bmatrix} + \int ^{t}_{0} e^{(t-\tau )A}_{L} \begin{bmatrix} 0 \\ h+\gamma _1^{-1}\sigma F(x) \end{bmatrix}(\tau )\,d\tau , \end{aligned}$$

(40)

where $e^{tA}_L\equiv \dot{S}_{j_0}e^{tA}$ and the function h is defined in (29). Let us denote $V_0 =(\sigma _0,n_0)^{\textsf{T}},$ $V=(\sigma ,n)^{\textsf{T}}.$ Then, according to Proposition 2.1(vi), Lemma 4.1(i), we have the following estimate for the first term in the right-hand side of (40):

$$\begin{aligned} \left\| e^{tA}_{L}V_0\right\| _{\dot{B}_{2,\infty }^{s}} \lesssim _{j_0} (1+t)^{-\frac{s-s_0}{2}}\Vert V_0\Vert _{\dot{B}^{s_0}_{2,\infty }} \lesssim (1+t)^{-\frac{s-s_0}{2}}\Vert (\sigma _0, w_0)\Vert _{\dot{B}^{s_0}_{2,\infty }}. \end{aligned}$$

(41)

To estimate the second term in (40), we use the following lemma. $\square$

Lemma 5.3

Let $-5/2<\beta <-1/2.$ Then,  we have

$$\begin{aligned} \Vert h\Vert _{\dot{B}_{2,\infty }^{\beta }} \lesssim _{\beta ,\eta }\, \delta _1 \Vert V\Vert _{\dot{B}_{2,\infty }^{\beta +2}} + \Vert V\Vert _{\dot{B}^{\frac{1}{2}+\eta }_{2,\infty }} \Vert V\Vert _{\dot{B}_{2,\infty }^{\beta +2-\eta }} + \Vert V\Vert _{ \dot{B}^{\frac{3}{2}-\eta }_{2,1}} \Vert V\Vert _{\dot{B}^{\beta +2+\eta }_{2,\infty }} \end{aligned}$$

(42)

for any $0\le \eta <\min \{\beta +5/2, -\beta -1/2, 1\}.$

Admitting Lemma 5.3 for a moment, we continue the proof of Proposition 5.2. Let us denote the second term in (40) by

$$\begin{aligned} N_L(t)&= \int ^{t}_{0} e^{(t-\tau )A}_{L} \begin{bmatrix} 0 \\ h \end{bmatrix}(\tau )\,d\tau +\int ^{t}_{0} e^{(t-\tau )A}_{L} \begin{bmatrix} 0 \\ \gamma _1^{-1}\sigma F(x)\end{bmatrix}(\tau )\,d\tau \nonumber \\&\equiv N_{L}^1(t)+N_{L}^{2}(t). \end{aligned}$$

(43)

We estimate the term $N_L^{1}(t).$ We first treat the case $-1/2<s\le 3/2-\epsilon$ with $s_0\le s.$ In this case, we estimate $N_{L}^1(t)$ by using the duality argument. Let $\psi =(\psi _1,\ldots ,\psi _4)^{\textsf{T}} \in {\mathcal {S}}^4.$ Fix a real number $\alpha _0$ satisfying $s-s_0<\alpha _0<s-s_0+2$ and $s+1/2+\epsilon \le \alpha _0 \le s+5/2-\epsilon .$ This $\alpha _0$ can be taken if $\epsilon <1.$ Proposition 2.1(ii) then yields

$$\begin{aligned} \left\langle N_L^{1}(t), \psi \right\rangle&\lesssim \int ^{t}_{\frac{t}{2}} \Vert h(\tau )\Vert _{\dot{B}^{s-2 }_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\quad +\int ^{\frac{t}{2}}_{0} \Vert h(\tau )\Vert _{\dot{B}^{s-\alpha _0}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{\alpha _0-s}_{2,1}}\,d\tau . \end{aligned}$$

By Proposition 2.1(vi),

$$\begin{aligned} \Vert V(\tau )\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim (1+\Vert U(\tau )\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}}) \Vert U(\tau )\Vert _{\dot{B}^{s}_{2,\infty }}\lesssim \Vert U(\tau )\Vert _{\dot{B}^{s}_{2,\infty }}, \end{aligned}$$

(44)

where $U=(\sigma ,w)^{\textsf{T}}.$ Then, Lemma 5.3 with $\beta =s-2$ and $\eta =0$ shows

$$\begin{aligned} \Vert h(\tau )\Vert _{\dot{B}^{s-2}_{2,\infty }} \lesssim (\delta _1 +\Vert U(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{B}^{\frac{3}{2}}_{2,1}} )\Vert U(\tau )\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim \delta \Vert U(\tau )\Vert _{\dot{B}^{s}_{2,\infty }}. \end{aligned}$$

By using the time-space integral estimate

$$\begin{aligned} \int ^{\infty }_{0} \Vert e^{\tau A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \lesssim \Vert \psi \Vert _{\dot{B}_{2,1}^{-s}} \end{aligned}$$

which follows from Lemma 4.1(ii), we obtain

$$\begin{aligned} \int ^{t}_{\frac{t}{2}} \Vert h(\tau )\Vert _{\dot{B}^{s-2}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau&\lesssim _{s} \delta \int ^{t}_{\frac{t}{2}} \Vert U(\tau )\Vert _{\dot{B}^{s}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\lesssim \delta {\mathcal {D}}_{\epsilon ,s_0}(T) (1+t)^{-\frac{s-s_0}{2}}\int ^{\infty }_{0} \Vert e^{\tau A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\lesssim _{ j_0} \delta {\mathcal {D}}_{\epsilon ,s_0}(T) (1+t)^{-\frac{s-s_0}{2}}\Vert \psi \Vert _{\dot{B}_{2,1}^{-s}}. \end{aligned}$$

Lemma 5.3 with $\beta =s-\alpha _0,$ $\eta =0$ shows

$$\begin{aligned} \Vert h(\tau )\Vert _{\dot{B}^{s-\alpha _0}_{2,\infty }} \lesssim \left( \delta _1+\Vert U(\tau )\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{B}^{\frac{3}{2}}_{2,1}} \right) \Vert U(\tau )\Vert _{\dot{B}^{s-\alpha _0+2}_{2,\infty }} \lesssim \delta \Vert U(\tau )\Vert _{\dot{B}^{s-\alpha _0+2}_{2,\infty }}. \end{aligned}$$

By using the time decay estimate

$$\begin{aligned} \Vert e^{(t-\tau ) A}_{L} \psi \Vert _{\dot{B}^{\alpha _0-s}_{2,\infty }} \lesssim (1+t-\tau )^{-\frac{\alpha _0}{2}} \Vert \psi \Vert _{\dot{B}^{-s}_{2,\infty }}, \end{aligned}$$

which follows from Lemma 4.1(i), we obtain

$$\begin{aligned}&\int ^{\frac{t}{2}}_{0} \Vert h(\tau )\Vert _{\dot{B}^{s-\alpha _0}_{2,\infty }}\Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{\alpha _0-s}_{2,1}}\,d\tau \\&\quad \lesssim _{s} \delta \int ^{\frac{t}{2}}_0 \Vert U(\tau )\Vert _{\dot{B}^{s-\alpha _0+2}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{\alpha _0-s}_{2,1}}\,d\tau \\&\quad \lesssim \delta {\mathcal {D}}_{\epsilon ,s_0}(T) \Vert \psi \Vert _{\dot{B}_{2,1}^{-s}} \int _{0}^{\frac{t}{2}}(1+\tau )^{-\frac{s-s_0+2-\alpha _0}{2}}(1+t-\tau )^{-\frac{\alpha _0}{2}} d\tau \\&\quad \lesssim _{\epsilon , j_0}\delta {\mathcal {D}}_{\epsilon ,s_0}(T) (1+t)^{-\frac{s-s_0}{2}}\Vert \psi \Vert _{\dot{B}_{2,1}^{-s}}. \end{aligned}$$

As $\psi$ is arbitrary, applying Proposition 2.1(ii), we obtain the inequality

$$\begin{aligned} \Vert N_L^{1}(t)\Vert _{\dot{B}^{s}_{2,\infty }}\lesssim _{\epsilon ,j_0}\delta {\mathcal {D}}_{\epsilon ,s_0}(T)\,(1+t)^{-\frac{s-s_{0}}{2}} \end{aligned}$$

(45)

for $-1/2<s\le 3/2-\epsilon ,$ $-3/2\le s_0\le 1/2$ with $s_0\le s.$

Next, we show the inequalities (45) for $-3/2+\epsilon \le s \le -1/2,$ $-3/2\le s_0\le 3/2$ with $s_0\le s.$ We take $\epsilon _1>0$ which satisfies $\epsilon _1 < 1/2.$ Then, using Lemma 4.1(i) with $\alpha =2+s-\epsilon _1$ and Lemma 5.3 with $\beta =-2+\epsilon _1$ and $\eta =0,$ we have

$$\begin{aligned} \Vert N_L^1(t)\Vert _{\dot{B}^{s}_{2,\infty }}&\lesssim _{j_0} \int ^{t}_{0} (1+t-\tau )^{-\frac{1}{2}\left( 2+s-\epsilon _1\right) }\Vert h(\tau )\Vert _{\dot{B}^{-2+\epsilon _1}_{2,\infty }}d\tau \\&\lesssim \delta \int ^{t}_{0} (1+t-\tau )^{-\frac{1}{2}\left( 2+s-\epsilon _1\right) }\Vert U(\tau )\Vert _{\dot{B}^{\epsilon _1}_{2,\infty }}d\tau \\&\lesssim _{\epsilon _1,p}\delta {\mathcal {D}}_{\epsilon ,s_0}(T)\,(1+t)^{-\frac{s-s_{0}}{2}}. \end{aligned}$$

Let us estimate $N_L^{2}(t).$ By Proposition 2.1(vi), Lemma 4.1(i) with $\alpha =s+3/2,$ we have

$$\begin{aligned} \Vert N_L^2(t)\Vert _{\dot{B}^{s}_{2,\infty }}&\lesssim _{j_0} \int ^{t}_{0} (1+t-\tau )^{-\frac{s}{2}-\frac{3}{4}}\Vert \sigma (\tau )F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}d\tau \\&\lesssim _{\epsilon } \Vert F\Vert _{\dot{B}^{-\frac{3}{2}+\epsilon }_{2,1}} \int ^{t}_{0} (1+t-\tau )^{-\frac{s}{2}-\frac{3}{4}}\Vert \sigma (\tau )\Vert _{\dot{B}^{\frac{3}{2}-\epsilon }_{2,\infty }}d\tau \\&\lesssim _{\epsilon }\delta {\mathcal {D}}_{\epsilon ,s_0}(T)\,(1+t)^{-\frac{s-s_{0}}{2}}, \end{aligned}$$

where $-3/2+\epsilon \le s\le 3/2-\epsilon ,$ $-3/2\le s_0\le 3/2$ with $s_0\le s.$ Hence, we obtain

$$\begin{aligned} \left\| N_L(t) \right\| _{\dot{B}^{s}_{2,\infty }}\lesssim _{\epsilon ,j_0}\delta {\mathcal {D}}_{\epsilon ,s_0}(T)\,(1+t)^{-\frac{s-s_{0}}{2}} \end{aligned}$$

(46)

for $-3/2+\epsilon \le s \le -1/2,$ $-3/2\le s_0\le 3/2$ with $s_0\le s.$

It remains to prove Lemma 5.3.

Proof of Lemma 5.3

Let $\Psi (\zeta )= 1/(\zeta +\rho _\infty ).$ By using Proposition 2.1(i), (vi), we obtain

$$\begin{aligned} \Vert h_1\Vert _{\dot{B}^{\beta }_{2,\infty }}&\lesssim \left\| \frac{n\otimes m}{\rho } +\frac{m^*\otimes n}{\rho }+\gamma _1^{-1}\left( \Psi (\sigma ^*+\sigma )-\Psi (\sigma ^*)\right) m^*\otimes m \right\| _{\dot{B}^{\beta +1}_{2,\infty }} \\&\lesssim _{\beta ,\eta } \Vert (\sigma ^{*},v^{*})\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}\Vert U\Vert _{\dot{B}^{\beta +2}_{2,\infty }} + \Vert U\Vert _{\dot{B}^{\frac{1}{2}+\eta }_{2,\infty }} \Vert U\Vert _{\dot{B}^{\beta +2-\eta }_{2,\infty }}, \\ \end{aligned}$$

since $0<\eta <1.$ We also have bounds for $h_2$ as

$$\begin{aligned} \Vert h_2\Vert _{\dot{B}^{\beta }_{2,\infty }} \lesssim \Vert \Pi (\sigma ^*,\sigma )\sigma \Vert _{\dot{B}^{\beta +1}_{2,\infty }}\lesssim \Vert \sigma ^*\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }}\Vert \sigma \Vert _{\dot{B}^{\beta +2}_{2,\infty }}+\Vert \sigma \Vert _{\dot{B}^{\frac{1}{2}+\eta }_{2,\infty }}\Vert \sigma \Vert _{\dot{B}^{\beta +2-\eta }_{2,\infty }}. \end{aligned}$$

By Proposition 2.1(i), (v), (vi), we obtain bounds for $g^3,$ $g^4$ as

$$\begin{aligned} \Vert h_3\Vert _{\dot{B}^{\beta }_{2,\infty }}&\lesssim \left\| \left( \Psi (\sigma ^*+\sigma )-\Psi (0) \right) \nabla n\right\| _{\dot{B}^{\beta +1}_{2,\infty }} \\&\quad +\left\| \nabla \left( \Psi (\sigma ^*+\sigma )-\Psi (0)\right) n\right\| _{\dot{B}^{\beta +1}_{2,\infty }}\\&\lesssim _{\beta ,\eta } \Vert \sigma ^*\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}} \Vert n\Vert _{\dot{B}^{\beta +2}_{2,\infty }} + \Vert U\Vert _{\dot{B}^{\frac{3}{2}-\eta }_{2,\infty }}\Vert U\Vert _{\dot{B}^{\beta +2+\eta }_{2,\infty }}, \\ \Vert h_{4}\Vert _{\dot{B}^{\beta }_{2,\infty }}&\lesssim _{\beta ,\eta } \left\| \left( \Psi (\sigma ^*+\sigma )-\Psi (\sigma ^*) \right) \nabla m^*\right\| _{\dot{B}^{\beta +1}_{2,\infty }} \\&\quad +\left\| \nabla \left( \Psi (\sigma ^*+\sigma )-\Psi (\sigma ^*)\right) m^*\right\| _{\dot{B}^{\beta +1}_{2,\infty }} \\&\lesssim _{\beta ,\eta } \Vert m^*\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}} \Vert \sigma \Vert _{\dot{B}^{\beta +2}_{2,\infty }}, \end{aligned}$$

since $-3/2<\beta ,\,\beta \pm \eta <-1/2.$ Therefore, we have the desired estimate (42). $\square$

Proposition 2.2 implies the continuous inclusions

$$\begin{aligned} L^{p,\infty } \hookrightarrow \dot{B}^{-3\left( \frac{1}{p}-\frac{1}{2}\right) }_{2,\infty }\quad {\textrm{for}}\ 1<p<2,\quad L^1\hookrightarrow \dot{B}^{-\frac{3}{2}}_{2,\infty }\quad {\textrm{and}}\quad L^2\hookrightarrow \dot{B}^{0}_{2,\infty }. \end{aligned}$$

Then, by Lemma 4.1(i), we have the estimate for the first term in the right-hand side of (40),

$$\begin{aligned} \left\| e^{tA}_{L}V_0\right\| _{\dot{B}_{2,\infty }^{s}}&\lesssim _{j_0} (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) }\Vert V_0\Vert _{L^{p,\infty }} \quad {\textrm{if}}\ 1<p<2, \end{aligned}$$

(47)

$$\begin{aligned} \left\| e^{tA}_{L}V_0\right\| _{\dot{B}_{2,\infty }^{s}}&\lesssim _{j_0} (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) }\Vert V_0\Vert _{L^{p}} \quad {\textrm{if}}\ p=1,2, \end{aligned}$$

(48)

where $s\in {\mathbb {R}}$ and $1\le p\le 2$ with $s/2+3/2(1/p-1/2)\ge 0.$ Thus, by the estimate (46), we obtain the following proposition.

Proposition 5.4

Let $\epsilon >0$ be a small number. If $\delta =\delta _1+\delta _2$ is sufficiently small,  then,  for any $T>0,$ we have

$$\begin{aligned} \sup _{0 \le t \le T} (1+t)^{\frac{s}{2}+\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) } \Vert (\sigma _L, w_L)(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{\epsilon ,p,j_0} \Vert (\sigma _0, w_0)\Vert _{\Lambda _p}+ \delta \widetilde{{\mathcal {D}}}_{\epsilon ,\,p}(T), \end{aligned}$$

(49)

where $-3/2+\epsilon \le s \le 3/2-\epsilon ,$ $1\le p\le 2$ with $s/2+3/2(1/p-1/2)\ge 0.$ Here,  the quantity $\widetilde{{\mathcal {D}}}_{\epsilon ,p}(T)$ is defined by

$$\begin{aligned} \widetilde{{\mathcal {D}}}_{\epsilon ,\,p}(T) \equiv \sup _{\begin{array}{c} -3/2+\epsilon \le \eta \le 3/2-\epsilon , \\ \eta /2+3/2(1/p-1/2)\ge 0 \end{array}}\ \sup _{ 0 \le t \le T} (1+t)^{\frac{\eta }{2}+\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) }\Vert (\sigma , w)(t)\Vert _{\dot{B}^{\eta }_{2,\infty }}, \end{aligned}$$

and the function space $\Lambda _p$ is defined by

$$\begin{aligned} \Lambda _p= {\left\{ \begin{array}{ll} L^{p,\infty }({\mathbb {R}}^3) &{} \text {if } 1<p<2, \\ L^p({\mathbb {R}}^3) &{} \text {if } p=1,2. \end{array}\right. } \end{aligned}$$

5.2 Estimate for the high frequency part

The estimate of the high frequency part of the perturbation $(\sigma _H,w_H)$ is proved as follows.

Proposition 5.5

If $\delta =\delta _1+\delta _2$ is small enough,  then we have the following estimate for all $T>0$ and small $\epsilon >0,$

$$\begin{aligned} \sup _{0 \le t \le T} (1+t)^{\frac{s-s_0}{2}} \Vert (\sigma _H, w_H)(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{j_0} \Vert (\sigma _0, w_0)\Vert _{\dot{B}^{s}_{2,\infty }} + \delta {\mathcal {D}}_{\epsilon ,s_0}(T), \end{aligned}$$

(50)

where $\delta =\delta _1+\delta _2,$ $-3/2+\epsilon \le s \le 3/2-\epsilon ,$ $-3/2\le s_0\le 1/2$ with $s_{0} \le s.$ Here,  ${\mathcal {D}}_{\epsilon ,s_0}(T)$ is the quantity defined in (39).

Proof

Since $\Vert (\sigma _H, w_H)(t)\Vert _{\dot{B}^{s_1}_{2,\infty }}\lesssim _{j_0} \Vert (\sigma _H, w_H)(t)\Vert _{\dot{B}^{s_2}_{2,\infty }}$ for $s_1\le s_2,$ it is sufficient to show that the estimate (50) with the highest order case $s=3/2-\epsilon .$ Since $(\sigma ,w)$ satisfies Eq. (17), we have the following identities for $(\sigma _j,w_j,f_j,g_j)\equiv \dot{\Delta }_{j}(\sigma ,w,f, g),$ where $\dot{\Delta }_{j}$ are the dyadic blocks,

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt} \Vert \nabla (\sigma _j, w_j)\Vert _{L^2}^2 + \nu \Vert \nabla ^{2}w_j\Vert _{L^2}^2 + (\nu + \nu ')\Vert \nabla \textrm{div}\,w_j\Vert _{L^2}^2 \\&\quad = \langle \nabla f_j, \nabla \sigma _j\rangle + \langle \nabla g_j, \nabla w_j \rangle ,\\&\frac{d}{dt}\langle \nabla \sigma _j, w_j \rangle + \gamma \Vert \nabla \sigma _j\Vert _{L^2}^2 =\gamma \Vert \textrm{div}\,w_j\Vert _{L^2}^2 + \langle {\mathcal {A}} w_j,\nabla \sigma _j \rangle {+} \langle \nabla f_j, w_j \rangle {+} \langle g_j, \nabla \sigma _j \rangle . \end{aligned}$$

Hence, there exists a constant $c_0>0$ such that for sufficiently small $\kappa >0,$

$$\begin{aligned}&\frac{d}{dt}( \Vert \nabla (\sigma _j,w_j)\Vert _{L^2}^2 + \kappa \langle \nabla \sigma _j, w_j \rangle ) + c_0\left( \kappa \Vert \nabla \sigma _j\Vert _{L^2}^2 + \Vert \nabla ^2 w_{j}\Vert _{L^2}^2\right) \\&\quad \lesssim \langle \nabla f_j,\nabla \sigma _j \rangle + \kappa \langle \nabla f_j, w_j \rangle + \sum _{i=1}^{4}(\kappa \langle g^{i}_j, \nabla \sigma _j \rangle + \langle \nabla g^{i}_j, \nabla w_j \rangle ), \end{aligned}$$

where $g_{j}^{i}=\dot{\Delta }_{j}g^{i}.$ By using Proposition 2.1(i), (vi) and Lemma 2.3, for any $j\ge j_0-1,$ we have

$$\begin{aligned} \langle \nabla f_j, \nabla \sigma _j \rangle&= -\langle \nabla \dot{\Delta }_{j}\textrm{div}\,((v^{*}+w)\sigma +\sigma ^*w) ,\nabla \sigma _{j} \rangle \\&\lesssim 2^{j}\Vert \dot{\Delta }_{j}(\textrm{div}(v^{*}+w)\,\sigma + \nabla \sigma ^{*}\cdot w)\Vert _{L^2}\Vert \nabla \sigma _j\Vert _{L^2}\\&\quad + 2^{j}\Vert [\dot{\Delta }_{j},(v^{*}+w)\cdot \nabla ]\sigma + [\dot{\Delta }_{j},\sigma ^{*}\textrm{div}\,]w\Vert _{L^2}\Vert \nabla \sigma _j\Vert _{L^{2}}\\&\quad - \langle \nabla ((v^{*}+w)\cdot \nabla \sigma _j), \nabla \sigma _j \rangle - \langle \nabla (\sigma ^* \textrm{div}\, w_j),\nabla \sigma _j \rangle \\&\lesssim _{\epsilon , j_0} 2^{-j(s-1)} \Vert (\sigma ^*,v^*, w)\Vert _{\dot{B}^{\frac{5}{2}}_{2,1}} \Vert (\sigma ,w)\Vert _{\dot{B}^{s}_{2,\infty }}\Vert \nabla \sigma _j\Vert _{L^{2}}\\&\quad +\Vert \nabla v^*\Vert _{L^\infty } \Vert \nabla \sigma _j\Vert _{L^2}^2 + \Vert (\sigma ^*,\nabla \sigma ^*,\sigma ,\nabla \sigma )\Vert _{L^\infty }\Vert \nabla ^2 w_j\Vert _{L^2} \Vert \nabla \sigma _j\Vert _{L^2}\\&\lesssim \delta 2^{-j(s-1)} \Vert (\sigma ,w)\Vert _{\dot{B}^{s}_{2,\infty }}\Vert \nabla \sigma _j\Vert _{L^2} + \delta \Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2}^2, \end{aligned}$$

where we use the identities:

$$\begin{aligned} \langle v^* \cdot \nabla \partial _k \sigma _j, \partial _k \sigma _j \rangle =-\frac{1}{2}\langle \textrm{div}\,v^* \partial _k \sigma _j, \partial _k \sigma _j \rangle \quad {\textrm{for}}\ 1\le k \le 3. \end{aligned}$$

Using Proposition 2.1(i), (v) and (vi), we have

$$\begin{aligned} \langle \nabla f_j, w_j \rangle&\lesssim 2^{-j(s-1)} \Vert \textrm{div}\,((v^{*}+w)\sigma +\sigma ^{*}w)\Vert _{\dot{B}^{s-1}_{2,\infty }}\Vert \nabla w_j\Vert _{L^2}\\&\lesssim 2^{-j(s-1)}\Vert (\sigma ^*,\sigma ,v^*)\Vert _{\dot{B}^{\frac{3}{2}}_{2,1}}\Vert (\sigma ,w)\Vert _{\dot{B}^{s}_{2,\infty }}\Vert \nabla w_j\Vert _{L^2}\\&\lesssim \delta 2^{-j(s-1)} \Vert (\sigma ,w)\Vert _{\dot{B}^{s}_{2,\infty }}\Vert \nabla w_j\Vert _{L^2}. \end{aligned}$$

Similarly to the estimate of $\langle \nabla f_j, w_j \rangle ,$ by applying Proposition 2.1(v), we have, for any $j\ge j_0-1,$

$$\begin{aligned}&\sum _{i=1}^{2}(\kappa \langle g_{j}^i, \nabla \sigma _j \rangle + \langle \nabla g^i_j , \nabla w_j \rangle ) \\&\quad \lesssim _{\kappa } \delta 2^{-j(s-1)} \Vert (\sigma ,w)\Vert _{\dot{B}^{s}_{2,\infty }}\Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2} + \delta \Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2}^2. \end{aligned}$$

We next consider bounds for $\kappa \langle g_{j}^3, \nabla \sigma _j \rangle + \langle \nabla g^3_j , \nabla w_j \rangle .$ Let $\Psi (\zeta )=1/(\zeta +\rho _\infty ).$ If $\epsilon >0$ is sufficiently small, then, by Proposition 2.1(v), (vi), we have

$$\begin{aligned}&\kappa \langle g_{j}^3, \nabla \sigma _j \rangle + \langle \nabla g^3_j , \nabla w_j \rangle \\&\quad \lesssim _{\kappa } 2^{-j(s-1)} \left\| (\Psi (\sigma ^*+\sigma )-\Psi (\sigma ^*)) {\mathcal {A}}(v^{*}+w)\right\| _{\dot{B}^{s-1}_{2,\infty }}\Vert (\nabla \sigma _j ,\nabla ^2 w_j) \Vert _{L^2}\\&\quad \lesssim \delta 2^{-j(s-1)}\Vert \sigma \Vert _{\dot{B}^{s}_{2,\infty }} \Vert (\nabla \sigma _j, \nabla ^2 w_j)\Vert _{L^2}. \end{aligned}$$

To estimate $\kappa \langle g_{j}^4, \nabla \sigma _j \rangle + \langle \nabla g^4_j , \nabla w_j \rangle ,$ we introduce the function $h = \Psi (\sigma ^*)-\Psi (0).$ By Proposition 2.1(iii), (v), (vi) and Lemma 2.3, we have

$$\begin{aligned}&\kappa \langle g_{j}^4, \nabla \sigma _j \rangle + \langle \nabla g^4_j , \nabla w_j \rangle \\&\quad = \langle \dot{\Delta }_{j}(h {\mathcal {A}}w), \kappa \nabla \sigma _j - \Delta w_j \rangle \lesssim \Big (2^j\Vert [\dot{\Delta }_{j}, h \nabla ]w\Vert _{L^2} + \Vert \nabla \left( h \nabla w_j \right) \Vert _{L^2} \\&\qquad + \sum _{1\le k,l \le 3}\Vert \dot{\Delta }_{j}\left( \partial _{k} h \,\partial _{l} w \right) \Vert _{L^2}\Big )\Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2}\\&\quad \lesssim _{\epsilon . j_0} 2^{-j(s-1)}\left( \Vert \sigma ^{*}\Vert _{\dot{B}^{\frac{5}{2}}_{2,1}}\Vert w\Vert _{\dot{B}^{s}_{2,\infty }} + \Vert (\sigma ^*,\nabla \sigma ^*)\Vert _{L^{\infty }}\Vert \nabla ^2 w_j\Vert _{L^2}\right) \\&\qquad \times \Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2} + 2^{-j(s-1)} \sum _{1\le k,l \le 3}\Vert \partial _{k} h\, \partial _{l} w \Vert _{\dot{B}^{s-1}_{2,\infty }}\Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2} \\&\quad \lesssim \delta 2^{-j(s-1)}\Vert w\Vert _{\dot{B}^{s}_{2,\infty }}\Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2} + \delta \Vert (\nabla \sigma _j,\nabla ^2 w_j)\Vert _{L^2}^2. \end{aligned}$$

Let $\kappa =\kappa (j_0)>0$ be small enough. Then, for any $j\ge j_0-1,$ we have

$$\begin{aligned} {\mathcal {E}}_{j}(t) \equiv \Vert \nabla (\sigma _j,w_j)\Vert _{L^2}^2 + \kappa \langle \nabla \sigma _j, w_j \rangle \sim _{j_0} \Vert \nabla (\sigma _j,w_j)\Vert _{L^2}^2. \end{aligned}$$

(51)

It then follows that there exists a constant $c_1=c_1(j_0)>0$ such that

$$\begin{aligned} \frac{d}{dt}{\mathcal {E}}_{j} +2 c_1 \widetilde{{\mathcal {E}}_{j}} \lesssim d_j \widetilde{{\mathcal {E}}_j}^{\frac{1}{2}}, \end{aligned}$$

where $\widetilde{{\mathcal {E}}_j} \equiv \Vert (\nabla \sigma _j, \nabla ^2 w_j)\Vert ^2_{L^2},$ $d_j \equiv \delta 2^{-j(s-1)} \Vert (\sigma ,w)\Vert _{\dot{B}^{s}_{2,\infty }}.$ From Young’s inequality and the fact that ${\mathcal {E}}_{j} \lesssim _{j_0} \widetilde{{\mathcal {E}}_{j}}$ for any $j\ge j_0-1,$ we deduce that

$$\begin{aligned} \frac{d}{dt}{\mathcal {E}}_{j} + c_1 {\mathcal {E}}_{j} \lesssim _{j_0} d_j^2. \end{aligned}$$

Therefore, Grönwall’s inequality and the relation (51) ensure that

$$\begin{aligned}&2^{2js}\Vert (\sigma _j,w_j)(t)\Vert _{L^2}^2 \\&\quad \lesssim _{j_0} e^{-c_1 t} \Vert (\sigma _0,w_0)\Vert ^{2}_{\dot{B}^{s}_{2,\infty }} + \delta ^{2} \int ^{t}_{0} e^{-c_1(t-\tau )} \Vert (\sigma ,w)(\tau )\Vert ^{2}_{\dot{B}^{s}_{2,\infty }} d\tau \\&\quad \lesssim e^{-c_1 t} \Vert (\sigma _0,w_0)\Vert ^{2}_{\dot{B}^{s}_{2,\infty }}+ \delta ^{2} {\mathcal {D}}_{\epsilon ,s_0}(T)^{2} \int ^{t}_{0} e^{-c_1(t-\tau )} (1+\tau )^{-(s-s_0)} d\tau \\&\quad \lesssim (1+t)^{-(s-s_0)} \big ( \Vert (\sigma _0,w_0)\Vert _{\dot{B}^{s}_{2,\infty }} + \delta {\mathcal {D}}_{\epsilon ,s_0}(T) \big )^{2}, \end{aligned}$$

where $j\ge j_0.$ Hence, we obtain

$$\begin{aligned} (1+t)^{\frac{s-s_0}{2}} \Vert (\sigma _H, w_H)(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{j_0} \Vert (\sigma _0, w_0)\Vert _{\dot{B}^{s}_{2,\infty }} + \delta {\mathcal {D}}_{\epsilon ,s_0}(T). \end{aligned}$$

$\square$

Propositions 5.2 and 5.5 derive the weak-type estimate (36). This completes the proof of Theorem 5.1.

Next, we show Theorem 1.5 and the decay estimate (12) in Theorem 1.6(ii). Since we have the interpolation inequality (see Proposition 2.1(iii))

$$\begin{aligned} \Vert u\Vert _{\dot{H}^{s}} \lesssim _{s_1,s_2,\theta } \Vert u\Vert _{\dot{B}^{s_1}_{2,\infty }}^{1-\theta } \Vert u\Vert _{\dot{B}^{s_2}_{2,\infty }}^{\theta }\quad {\textrm{with}}\ s=(1-\theta ) s_1 + \theta s_2,\ \theta \in (0,1), \end{aligned}$$

(52)

Propositions 5.4 and 5.5 allow us to complete the proof of the estimate (11) when $(p,s)\ne (2,0)$ and the estimate (12) in Theorem 1.6(ii). The rest of the proof of Theorem 1.5 is established by the following proposition.

Proposition 5.6

If $\delta =\delta _1+\delta _2$ is small enough and $(\sigma _0,w_0)\in L^2,$ then the energy estimate

$$\begin{aligned} \Vert (\sigma , w)(t)\Vert _{H^3}^2+\int ^{t}_{0}\big (\Vert \nabla w(\tau )\Vert _{H^3}^2+\Vert \nabla \sigma (\tau )\Vert _{H^2}^2\big ) d\tau \lesssim \Vert (\sigma _0,v_0)\Vert _{H^3}^2 \end{aligned}$$

holds for any $t\ge 0.$

Proof

Let $\kappa >0$ be sufficiently small. We set

$$\begin{aligned} {\mathcal {E}}_1(t) = \sum _{|\alpha _1| \le 3} \Vert \partial _{x}^{\alpha _1}(\sigma ,w)(t)\Vert _{L^{2}}^2 + \sum _{|\alpha _2|\le 2}\kappa \langle \partial _{x}^{\alpha _2}\nabla \sigma (t), \partial _{x}^{\alpha _2} w(t) \rangle . \end{aligned}$$

Since $(\sigma ,w)$ satisfies Eq. (17), we have the following inequality

$$\begin{aligned} {\mathcal {E}}_1(t)-{\mathcal {E}}_1(0) + \int _{0}^{t}\widetilde{{\mathcal {E}}}_1(\tau )\,d\tau&\lesssim \int _{0}^{t}\sum _{|\alpha _1|\le 3}\left( \langle \partial _{x}^{\alpha _1} f, \partial _{x}^{\alpha _1} \sigma \rangle + \langle \partial _{x}^{\alpha _1} g, \partial _{x}^{\alpha _1} w \rangle \right) \nonumber \\&\quad +\sum _{|\alpha _2|\le 2}\kappa \left( \langle \partial _{x}^{\alpha _2}\nabla f,\partial _{x}^{\alpha _2} w \rangle + \langle \partial _{x}^{\alpha _2}g, \partial _{x}^{\alpha _2}\nabla \sigma \rangle \right) \,d\tau , \end{aligned}$$

(53)

where $t\ge 0$ and

$$\begin{aligned} \widetilde{{\mathcal {E}}}_1(t)&= \sum _{|\alpha _1|\le 3} \left( \nu \Vert \partial _{x}^{\alpha _1}\nabla w(t)\Vert _{L^2}^2 + (\nu + \nu ')\Vert \partial _{x}^{\alpha _1} \textrm{div}\,w(t)\Vert _{L^2}^2\right) \\&\quad + \sum _{|\alpha _2|\le 2} \kappa \left( \Vert \partial _{x}^{\alpha _2}\nabla \sigma (t)\Vert _{L^2}^2 +\Vert \partial _{x}^{\alpha _2}\textrm{div}\,w(t)\Vert _{L^2}^2\right) . \end{aligned}$$

We also have

$$\begin{aligned} {\mathcal {E}}_1(t) \sim \Vert (\sigma ,w)(t)\Vert _{H^{3}}^{2},\quad \widetilde{{\mathcal {E}}}_1(t) \sim \Vert \nabla \sigma (t)\Vert _{H^{2}}^2 + \Vert \nabla w(t)\Vert _{H^3}^2 \end{aligned}$$

for $t\ge 0.$ Next, we establish the estimate of the right-hand of (53). By Proposition 2.1(ii), (vi), we have

$$\begin{aligned} \langle f, \sigma \rangle + \kappa \langle \nabla f, w\rangle \lesssim \Vert (\sigma ^*,v^*,w)\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }} \Vert (\nabla \sigma ,\nabla w, \nabla ^2 w)\Vert _{L^2}^{2}\lesssim \delta \widetilde{{\mathcal {E}}}_1. \end{aligned}$$

Using the identities

$$\begin{aligned} \langle \textrm{div}((v^*+w)\partial _{x}^{\alpha _1}\sigma ),\partial _{x}^{\alpha _1} \sigma \rangle =\frac{1}{2}\langle \textrm{div}(v^*+w) \partial _{x}^{\alpha _1} \sigma , \partial _{x}^{\alpha _1} \sigma \rangle , \end{aligned}$$

for any $1\le |\alpha _1|\le 3,$ we have

$$\begin{aligned}&\langle \partial _{x}^{\alpha _1} f, \partial _{x}^{\alpha _1} \sigma \rangle \nonumber \\&\quad = \frac{1}{2} \langle \textrm{div}(v^*+w)\, \partial _{x}^{\alpha _1} \sigma , \partial _{x}^{\alpha _1} \sigma \rangle + \langle \partial _{x}^{\alpha _1} \textrm{div}(\sigma ^* w),\partial _{x}^{\alpha _1} \sigma \rangle \nonumber \\&\qquad +\sum _{0< \beta \le \alpha _1} \langle \textrm{div}(\partial ^{\beta }_{x}(v^*+w)\, \partial _{x}^{\alpha _1-\beta } \sigma ), \partial _{x}^{\alpha _1} \sigma \rangle \nonumber \\&\quad \lesssim \Vert \textrm{div}(v^{*}+w)\Vert _{L^\infty } \Vert \nabla \sigma \Vert _{H^{2}}^2 + \Vert \sigma ^*\Vert _{H^{4}}\Vert \nabla w\Vert _{H^3}\Vert \nabla \sigma \Vert _{H^{2}}\nonumber \\&\qquad + \Vert \nabla v^*\Vert _{H^{4}} \Vert \nabla \sigma \Vert _{H^{2}}^2 + \Vert w\Vert _{H^3}\Vert \nabla \sigma \Vert _{H^{2}}^2+\Vert \nabla w\Vert _{H^3}\Vert \sigma \Vert _{H^3}\Vert \nabla \sigma \Vert _{H^{2}} \nonumber \\&\quad \lesssim \delta \widetilde{{\mathcal {E}}}_1. \end{aligned}$$

(54)

We also have

$$\begin{aligned} \langle \partial _{x}^{\alpha _2}\nabla f, \partial _x^{\alpha _2}w \rangle \lesssim \delta \widetilde{{\mathcal {E}}}_1\quad {\textrm{for}}\ 0\le |\alpha _2|\le 2. \end{aligned}$$

Similarly to the estimate of (54) by applying Proposition 2.1(v), we have

$$\begin{aligned} \sum _{|\alpha _1|\le 3} \langle \partial _{x}^{\alpha _1} g, \partial _{x}^{\alpha _1} w \rangle +\sum _{|\alpha _2|\le 2}\kappa \langle \partial _{x}^{\alpha _2}g, \partial _{x}^{\alpha _2}\nabla \sigma \rangle \lesssim \delta \widetilde{{\mathcal {E}}}_1. \end{aligned}$$

If $\delta$ is sufficiently small, then

$$\begin{aligned} {\mathcal {E}}_1(t)-{\mathcal {E}}_1(0) + c_0 \int _{0}^{t} \widetilde{{\mathcal {E}}}_1(\tau )\,d\tau \le 0, \end{aligned}$$

where $t\ge 0$ and $c_0>0$ is a constant. Hence, we obtain

$$\begin{aligned} \Vert (\sigma , w)(t)\Vert _{H^3}^2+\int ^{t}_{0}\Vert \nabla w(\tau )\Vert _{H^3}^2+\Vert \nabla \sigma (\tau )\Vert _{H^2}^2 d\tau \lesssim \Vert (\sigma _0,v_0)\Vert _{H^3}^2 \end{aligned}$$

for any $t\ge 0.$ $\square$

5.3 The proof of optimality

This section is devoted to the proof of Theorem 1.6. We first show the following lemma which derives the estimate of $N_{L}$ defined in (43):

$$\begin{aligned} N_L(t)&= \int ^{t}_{0} e^{(t-\tau )A}_{L} \begin{bmatrix} 0 \\ h \end{bmatrix}(\tau )\,d\tau +\int ^{t}_{0} e^{(t-\tau )A}_{L} \begin{bmatrix} 0 \\ \gamma _1^{-1}\sigma F(x)\end{bmatrix}(\tau )\,d\tau \\&=N_{L}^1(t)+N_{L}^{2}(t). \end{aligned}$$

Recall that $\delta _1$ is given by

$$\begin{aligned} \delta _1=\Vert \sigma ^*\Vert _{\dot{B}^{-\frac{1}{2}}_{2,\infty }\cap \dot{H}^{4}} + \Vert v^*\Vert _{\dot{B}^{\frac{1}{2}}_{2,\infty }\cap \dot{H}^{5}}. \end{aligned}$$

Lemma 5.7

Let an initial perturbation $U_0=(\sigma _0,w_0)$ be sufficiently small in $\dot{B}^{1/2}_{2,\infty }\cap H^3$ norm. If $U_0\in L^1$ and $-3/2<s<3/2,$ then there exists a small $d>0$ such that

$$\begin{aligned} \left\| N_L(t) \right\| _{\dot{H}^{s}} \lesssim _{s,d,U_0} \delta _1 (1+t)^{-\frac{s}{2}-\frac{3}{4}} + (1+t)^{-\frac{s}{2}-\frac{3}{4}-d}. \end{aligned}$$

In the case $1<p<2,$ $-3/2<s<3/2$ with $s/2 + 3/2\left( 1/p - 1/2 \right) > 0,$ if $U_0 \in L^{p,\infty },$ then there exists a small $d>0$ such that

$$\begin{aligned} \left\| N_L(t) \right\| _{\dot{H}^{s}} \lesssim _{s,d,U_0} \delta _1 (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p} - \frac{1}{2} \right) } + (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p} - \frac{1}{2} \right) -d}. \end{aligned}$$

Proof

Theorem 1.5 and the decay estimate (12) in Theorem 1.6(ii) imply the decay estimate of the perturbation $U=(\sigma ,w)$:

$$\begin{aligned} \Vert U(t)\Vert _{\dot{H}^{s}} \lesssim _{s,p} (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{2}\right) }\Vert U_0\Vert _{\Lambda _p\cap H^{3}}, \end{aligned}$$

where $-3/2<s<3/2,$ $1\le p \le 2$ with $s/2+3/2(1/p-1/2)>0,$ and $\Lambda _p$ is the function space defined in Theorem 5.4. Let us denote $s_0(p)=-3(1/p-1/2).$ Let $\psi =(\psi _1,\ldots ,\psi _4)^{\textsf{T}} \in {\mathcal {S}}^4.$ Fix a real number $\alpha _0$ that satisfies $s-s_0(p)<\alpha _0<s-s_0(p)+2,$ $s+1/2< \alpha _0 < s+5/2.$ Proposition 2.1(ii) then yields

$$\begin{aligned} \left\langle N_L^{1}(t), \psi \right\rangle&\lesssim \int ^{t}_{\frac{t}{2}} \Vert h(\tau )\Vert _{\dot{B}^{s-2 }_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\quad +\int ^{\frac{t}{2}}_{0} \Vert h(\tau )\Vert _{\dot{B}^{s-\alpha _0}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{\alpha _0-s}_{2,1}}\,d\tau . \end{aligned}$$

Lemmas 4.1(ii) and 5.3 imply that if $-1/2<s<3/2$ and $s/2+3/2(1/p-1/2)>0,$ then for any small $d,\eta >0$ with $d\le \eta ,$ we have

$$\begin{aligned}&\int ^{t}_{\frac{t}{2}} \Vert h(\tau )\Vert _{\dot{B}^{s-2 }_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\quad \lesssim _{s} \delta _1 \int ^{t}_{\frac{t}{2}} \Vert U(\tau )\Vert _{\dot{B}^{s}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\qquad + \int ^{t}_{\frac{t}{2}} \Vert U(\tau )\Vert _{\dot{B}^{\frac{1}{2}+\eta }_{2,\infty }\cap \dot{B}^{\frac{3}{2}-\eta }_{2,\infty }} \Vert U(\tau )\Vert _{\dot{B}_{2,\infty }^{s-\eta }\cap \dot{B}^{s+\eta }_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\quad \lesssim _{s,\eta ,U_0} \big (\delta _1 (1+t)^{-\frac{s-s_0}{2}} + (1+t)^{-\frac{s-s_0}{2}-d} \big ) \int ^{\infty }_{0}\Vert e^{\tau A^{*}}_{L}\psi \Vert _{\dot{B}^{2-s}_{2,1}}\,d\tau \\&\quad \lesssim _{s,d} \big (\delta _1 (1+t)^{-\frac{s-s_0}{2}} + (1+t)^{-\frac{s-s_0}{2}-d}\big ) \Vert \psi \Vert _{\dot{B}^{-s}_{2,1}}. \end{aligned}$$

By Proposition 2.1(vi), Lemmas 4.1(i) and 5.3, for any small $d,\eta >0$ with $d\le \eta ,$ we have

$$\begin{aligned}&\int ^{\frac{t}{2}}_{0} \Vert h(\tau )\Vert _{\dot{B}^{s-\alpha _0}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{\alpha _0-s}_{2,1}}\,d\tau \\&\quad \lesssim _{s} \delta _1 \int ^{\frac{t}{2}}_0 \Vert U(\tau )\Vert _{\dot{B}^{s-\alpha _0+2}_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{\alpha _0-s}_{2,1}}\,d\tau \\&\qquad + \int ^{t}_{\frac{t}{2}} \Vert U(\tau )\Vert _{\dot{B}^{\frac{1}{2}+\eta }_{2,\infty }\cap \dot{B}^{\frac{3}{2}-\eta }_{2,\infty }} \Vert U(\tau )\Vert _{\dot{B}_{2,\infty }^{s-\alpha _0+2-\eta }\cap \dot{B}^{s-\alpha _0+2+\eta }_{2,\infty }} \Vert e^{(t-\tau )A^{*}}_{L}\psi \Vert _{\dot{B}^{\alpha _0-s}_{2,1}}\,d\tau \\&\quad \lesssim _{s,d} \big (\delta _1 (1+t)^{-\frac{s-s_0}{2}} + (1+t)^{-\frac{s-s_0}{2}-d}\big ) \Vert \psi \Vert _{\dot{B}^{-s}_{2,1}}. \end{aligned}$$

By Proposition 2.1(vi), Lemma 4.1(i) with $\alpha =s+3/2,$ we have

$$\begin{aligned} \Vert N_L^2(t)\Vert _{\dot{B}^{s}_{2,\infty }}&\lesssim _{j_0} \int ^{t}_{0} (1+t-\tau )^{-\frac{s}{2}-\frac{3}{4}}\Vert \sigma (\tau )F\Vert _{\dot{B}^{-\frac{3}{2}}_{2,\infty }}d\tau \\&\lesssim _{\epsilon ,p} \Vert F\Vert _{\dot{B}^{-s}_{2,1}} \int ^{t}_{0} (1+t-\tau )^{-\frac{s}{2}-\frac{3}{4}}\Vert \sigma (\tau )\Vert _{\dot{B}^{s}_{2,\infty }}d\tau \\&\lesssim _{\epsilon ,p}\delta _1 (1+t)^{-\frac{s-s_{0}}{2}}. \end{aligned}$$

Since $\psi \in {\mathcal {S}}$ is arbitrary, we have

$$\begin{aligned} \Vert N_L(t)\Vert _{\dot{B}^{s}_{2,\infty }} \lesssim _{s,d} \delta _1 (1+t)^{-\frac{s-s_0}{2}} + (1+t)^{-\frac{s-s_0}{2}-d}. \end{aligned}$$

By the interpolation inequality in Proposition 2.1(iii), we have

$$\begin{aligned} \Vert N_L(t)\Vert _{\dot{H}^{s}} \lesssim _{s,d} \delta _1 (1+t)^{-\frac{s-s_0}{2}} + (1+t)^{-\frac{s-s_0}{2}-d} \end{aligned}$$

(55)

for $-1/2<s<3/2,$ $1\le p\le 2$ with $s/2+3/2(1/p-1/2)>0.$ As in the proof of the inequality (46), we obtain the inequality (55) for $-3/2<s\le -1/2,$ $1\le p\le 2$ with $s/2+3/2(1/p-1/2)>0.$ $\square$

Next, we prove Theorem 1.6. The proof of optimality in Theorem 1.6 below is inspired by the argument in Kawashima et al. [12].

Proof of Theorem 1.6

Let an initial value $U_0=(\sigma _{0},w_0)^{\textsf{T}}$ satisfy $U_0\in L^1$ and $\Vert U_0\Vert _{H^{3}}\ll \delta _2.$ Then, the decay estimate (11) with $p=1$ holds. Lemma 5.7 ensures that there exists a small $d>0$ such that

$$\begin{aligned} \Vert U\Vert _{\dot{H}^{s}}&\ge \Vert e^{tA}_{L}U_0\Vert _{\dot{H}^{s}} - \left\| N_L(t) \right\| _{\dot{H}^{s}} \\&\gtrsim _{s,U_0} \Vert e^{tA}_{L}U_0\Vert _{\dot{H}^{s}} - \delta _1(1+t)^{-\frac{s}{2}-\frac{3}{4}} -(1+t)^{-\frac{s}{2}-\frac{3}{4}-d}, \end{aligned}$$

where $-3/2<s<3/2.$ We recall the spectral resolution in (22), (23):

$$\begin{aligned} e^{t\hat{A}(\xi )} = e^{\lambda _{+}t}P_{+}(\xi ) + e^{\lambda _{-}t}P_{-}(\xi ) + e^{\lambda _0t}P_0(\xi ). \end{aligned}$$

Since $\lambda _{\pm }(\xi )\sim i|\xi |$ as $\xi \rightarrow 0,$ we have the asymptotic behavior of $P_{\pm }(\xi )$:

$$\begin{aligned} P_{\pm }(\xi ) \sim Q_{\pm }(\xi ) + O(|\xi |)\quad {\textrm{as}}\ |\xi |\rightarrow 0,\quad Q_{\pm }(\xi )=\frac{1}{2}\begin{bmatrix} 1 &{}\mp \frac{\xi }{|\xi |}\\ \mp \frac{\xi ^{\textsf{T}}}{|\xi |} &{}\frac{\xi \otimes \xi }{|\xi |^{2}} \end{bmatrix}. \end{aligned}$$

(56)

Let

$$\begin{aligned} e^{tA_{0}}_L U_0 \equiv \dot{S}_{j_0}{\mathcal {F}}^{-1}\left[ \left( e^{\lambda _{+}t}Q_{+}(\xi ) + e^{\lambda _{-}t}Q_{-}(\xi ) + e^{\lambda _0t}P_0(\xi )\right) \widehat{U_0}\right] , \end{aligned}$$

where $\dot{S}_{j}$ is the low frequency cut-off operator defined in (13). Then, there exist constants $c,c_0>0$ such that

$$\begin{aligned} \Vert (e^{tA}_{L}- e^{tA_0}_{L})U_0\Vert _{\dot{H}^{s}}^{2} \lesssim \int _{|\xi |\le c_0} |\xi |^{2(s+1)} e^{-c t|\xi |^{2}} d\xi \ \Vert U_0\Vert _{L^1} \lesssim _{j_0} (1+t)^{-s-\frac{5}{2}} \Vert U_0\Vert _{L^1}. \end{aligned}$$

As real orthogonal projections $Q_{\pm },$ $P_0$ satisfy

$$\begin{aligned} Q_{+}+Q_{-}+P_0={\textrm{I}}_{4},\quad Q_{\pm } Q_{\mp }=Q_{\pm } P_{0}=0 \end{aligned}$$

and $\widehat{U_0}(\xi )$ is continuous at $\xi =0,$ we obtain the following estimate for sufficiently small $j_{0}\in {\mathbb {Z}},$

$$\begin{aligned}&\Vert e^{tA_0}_{L}U_0\Vert _{\dot{H}^{s}}^2 \\&\quad = \Vert \dot{S}_{j_0}{\mathcal {F}}^{-1}[e^{\lambda _{+}t}Q_{+}{\widehat{U_0}}]\Vert _{\dot{H}^{s}}^2 {+} \Vert \dot{S}_{j_0}{\mathcal {F}}^{-1}[e^{\lambda _{-}t}Q_{-}{\widehat{U_0}}]\Vert _{\dot{H}^{s}}^2 {+} \Vert \dot{S}_{j_0}{\mathcal {F}}^{-1}[e^{\lambda _{0}t}P_{0}{\widehat{U_0}}]\Vert _{\dot{H}^{s}}^2\\&\quad \gtrsim \left\| \dot{S}_{j_0}e^{ct\Delta }U_0\right\| _{\dot{H}^{s}}^2\gtrsim _{j_0,U_0} \left| \widehat{U_0}(0) \right| ^2 \left\| \dot{S}_{j_0}e^{ct\Delta }\right\| _{\dot{H}^{s}}^2 \gtrsim M(U_0)^2 (1+t)^{-s-\frac{3}{2}}, \end{aligned}$$

where $M(U_0) \equiv |\int U_0 dx|,$ $-3/2<s<3/2.$ Here, we use the identity

$$\begin{aligned} |V|^{2} = |Q_+V|^{2} + |Q_-V|^{2} + |P_0V|^2\quad \text {for any}\ V\in {\mathbb {C}}^4. \end{aligned}$$

Therefore, if $\delta _{1}=\delta _{1}(U_0)>0$ is sufficiently small, then we have

$$\begin{aligned} \Vert U(t)\Vert _{\dot{H}^{s}} \gtrsim _{s,U_0} (1+t)^{-\frac{s}{2}-\frac{3}{4}}\quad \text {for}\ t \gg 1, \end{aligned}$$

where $-3/2<s<3/2.$ We next treat the case $1<p< 2,$ $-3/2<s<3/2$ with $s/2 + 3/2\left( 1/p - 1/2 \right) > 0.$ Let $U_0 = \delta _0 \dot{S}_{j_0} (|\cdot |^{-3/p})\varvec{e}_1$ with $\varvec{e}_1=(1,0,0,0)^{\textsf{T}}$ and small $\delta _0>0.$ Then, we have $U_0\in L^{p,\infty }$ and $\Vert U_0\Vert _{H^3}\lesssim _{j_0} \delta _0,$ since ${\mathcal {F}}(|\cdot |^{-3/p})\sim |\cdot |^{3/p-3}.$ By (56), there exist constants $c,c_0>0$ such that

$$\begin{aligned} \Vert (e^{tA}_{L}- e^{tA_0}_{L})U_0\Vert _{\dot{H}^{s}}^{2}&\lesssim \int _{|\xi |\le c_0} |\xi |^{2\left( s+\frac{3}{p}-2\right) } e^{-c t|\xi |^{2}} d\xi \lesssim _{j_0} (1+t)^{-s-3\left( \frac{1}{p}-\frac{1}{2}\right) - 1}. \end{aligned}$$

(57)

Let $j_0$ be sufficiently small. Then, there exists a constant $c_1>0$ such that

$$\begin{aligned}&\Vert e^{tA_0}_{L}U_0\Vert _{\dot{H}^{s}}^2 \nonumber \\&\quad = \Vert \dot{S}_{j_0}{\mathcal {F}}^{-1}[e^{\lambda _{+}t}Q_{+}{\widehat{U_0}}]\Vert _{\dot{H}^{s}}^2 {+} \Vert \dot{S}_{j_0}{\mathcal {F}}^{-1}[e^{\lambda _{-}t}Q_{-}{\widehat{U_0}}]\Vert _{\dot{H}^{s}}^2 {+} \Vert \dot{S}_{j_0}{\mathcal {F}}^{-1}[e^{\lambda _{0}t}P_{0}{\widehat{U_0}}]\Vert _{\dot{H}^{s}}^2\nonumber \\&\quad \gtrsim \int _{|\xi |\le c_1} |\xi |^{2\left( s+\frac{3}{p}-3\right) } e^{-c t|\xi |^{2}} d\xi \gtrsim (1+t)^{-s-3\left( \frac{1}{p}-\frac{1}{2}\right) }. \end{aligned}$$

(58)

Hence, the corresponding global solution $U=(\sigma ,w)^{\textsf{T}}$ satisfies

$$\begin{aligned} \Vert U(t)\Vert _{\dot{H}^s} \gtrsim (1+t)^{-\frac{s}{2}-\frac{3}{2}\left( \frac{1}{p} - \frac{1}{2} \right) }\quad {\textrm{for}}\ t \gg 1. \end{aligned}$$

$\square$