Global Dynamics of the Compressible Fluid Model of the Korteweg Type in Hybrid Besov Spaces

Abstract

We are concerned with a system of equations governing the evolution of isothermal, viscous, and compressible fluids of the Korteweg type, which is used to describe a two-phase liquid–vapor mixture. It is found that there is a “regularity-gain" dissipative structure of linearized systems in case of zero sound speed $P^{\prime }\left({\rho }^{*}\right)=0$, in comparison with the classical compressible Navier–Stokes equations. First, we establish the global-in-time existence of strong solutions in hybrid Besov spaces by using Banach’s fixed point theorem. Furthermore, we prove that the global solutions with critical regularity are Gevrey analytic in fact. Secondly, based on Gevrey’s estimates, we obtain uniform bounds on the growth of the analyticity radius of solutions in negative Besov spaces, which lead to the optimal time-decay estimates of solutions and their derivatives of arbitrary order.

1. Introduction

The Korteweg system describes the dynamics of a liquid–vapor mixture, whose phases are separated by a hypersurface and the jump in the pressure across the hypersurface is proportional to the curvature. The theory formulation originated from the works of van der Waals [1] (and, later, Korteweg [2]) more than one century ago. The basic idea is to add to the classical compressible fluid equation a capillary term, which penalizes high variations of the density. The rigorous derivation of the corresponding equations, which we named the compressible Navier–Stokes–Korteweg system, is due to Dunn and Serrin in [3]. In the barotropic case, it reads as follows:

$$\left\{\begin{array}{c}{\partial }_t\rho +\mathrm{div}\left(\rho u\right)=0,\hfill \\ {\partial }_t\left(\rho u\right)+\mathrm{div}(\rho u\otimes u)+\nabla P\left(\rho \right)=\mathcal{A}u+\mathrm{div}K\left(\rho \right).\hfill \end{array}\right.$$

(1)

Here, $\rho =\rho (t,x)\in {\mathbb{R}}_{+}$ and $u=u(t,x)\in {\mathbb{R}}^d(d\ge 3)$ represent the unknown functions on $[0,+\infty )\times {\mathbb{R}}^d$, which stand for the density and velocity field of the fluid, respectively. The pressure $P=P\left(\rho \right)$ is a suitable smooth function of $\rho$ only since we neglect the thermal effect. The notation $\mathcal{A}u$ is represented by $\mathcal{A}u=\mu \Delta m+(\mu +\lambda )\nabla \mathrm{div}m$, where the Lamé coefficients $\lambda$ and $\mu$ (the bulk and shear viscosities) are density-dependent functions, respectively. Mathematically, in order to ensure the uniform ellipticity of $\mathcal{A}u$, they are assumed to satisfy

$$\lambda >0,\nu \triangleq \lambda +2\mu >0.$$

In the following, the Korteweg tensor is given by (see [4])

$$\mathrm{div}K\left(\rho \right)=\nabla (\rho \kappa \left(\rho \right)\Delta \rho +\frac{1}{2}(\kappa \left(\rho \right)+\rho {\kappa }^{\prime }\left(\rho \right)){|\nabla \rho |}^2)-\mathrm{div}(\kappa \left(\rho \right)\nabla \rho \otimes \nabla \rho ),$$

where $\nabla \rho \otimes \nabla \rho$ stands for the tensor product ${\left({\partial }_j\rho {\partial }_k\rho \right)}_{jk}$ and the capillarity coefficient $\kappa >0$ can depend on $\rho$ in general.

System (1) is supplemented with initial data

$${(\rho ,u)|}_{t=0}=({\rho }_0\left(x\right),u_0\left(x\right)).$$

(2)

In the present paper, we are interested in studying the Cauchy problem (1) and (2), where the initial date tends to a constant equilibrium $({\rho }^{*},0)$ with ${\rho }^{*}>0$.

Clearly, if the capillarity coefficient $\kappa \equiv 0$, then System (1) reduces to the classical Navier–Stokes system of viscous compressible flows. To the best of our knowledge, there are many excellent literature studies on the solutions to the compressible Navier–Stokes system in different settings. Here, we pay more attention to the Korteweg system (1) and the dissipation effect arising from the Korteweg tensor. From [3], the existence of smooth solutions was known since the works of Hattori and Li [5,6]. In contrast with the local existence, global solutions were obtained only for initial data close enough to the stable equilibrium $({\rho }^{*},0)$ with convex pressure profiles. Danchin and Desjardins [7] established the global existence of strong solutions in so-called critical Besov spaces, which are invariant by the scaling invariant property of the Korteweg system (1); if the system admits a solution $(\rho ,u)$, then so does $({\rho }_{\lambda },u_{\lambda })$, where

$${\rho }_{\lambda }=\rho ({\lambda }^{2}t,\lambda x),u_{\lambda }=\lambda u({\lambda }^{2}t,\lambda x),\lambda >0$$

(3)

provided that the pressure laws P have been changed to ${\lambda }^{2}P$. Bresch, Desjardins, and Lin [8] addressed the global-in-time existence of weak solutions in a periodic or strip domain. Wang and Tan [9] deduced various optimal $L^2$ and $L^p(p\ge 2)$ time-decay rates of smooth solutions and their spatial derivatives. Charve, Danchin, and the second author [10] established the global well-posedness result in more general $L^p$ critical spaces. They investigated the Gevrey analyticity of (1) and (2) (the first analyticity efforts on the compressible fluids). Chikami and Kobayashi [11] investigated the global well-posedness and optimal time-decay estimates in the Besov spaces of $L^2$-type. Kawashima, Shibata, and the second author [12] developed different $L^p$ energy methods (independent of the spectral analysis) and obtained the optimal decay rates of strong solutions in the $L^p$ critical framework. Murata and Shibata [13] addressed a statement on the global existence and decay estimates to (1) and (2) in the Besov spaces, where the $L^p$-$L^q$ maximal regularity was mainly employed.

On the other hand, owing to the fact that the Korteweg system was deduced by using van der Waals potential, there exists non-monotone pressure due to the phase transition (see details in Kobayashi and Tsuda [14]). It is important and interesting to investigate more physical cases where $P^{\prime }\left({\rho }^{*}\right)=0$ (zero sound speed) and $P^{\prime }\left({\rho }^{*}\right)<0$. In those cases, the pressure term could not provide low-order dissipation on perturbation solutions. Danchin and Desjardins [7] gave a Fourier study of the linearized system to (1). Kotschote [15] considered the initial boundary value problem of (1) in a bounded domain and proved the local existence and uniqueness of strong solutions. Tang and Gao [16] proved the stability of the weak solution in the periodic domain ${\mathbb{T}}^d(d=2,3)$ under non-monotone pressure laws. Huang, Hong, and Shi [17] studied a more general pressure law, including van der Waals equation of state, and proved the local-in-time existence of smooth solutions to the Cauchy problem of (1) and (2). Furthermore, global smooth solutions to the periodic problem were also established. Chikami and Kobayashi [11] observed the different behaviors of characteristic roots $P^{\prime }\left({\rho }^{*}\right)\ge 0$, whether the quantity ${\overline{\nu }}^2-4\overline{\kappa }{\rho }^{*}$ is positive, negative, or zero (see below), which indicates the parabolic smoothing is available in all frequency spaces for the case of zero sound speed. Consequently, they established the existence and large time behaviors of global solutions in critical Besov spaces.

The main objective of this paper is to improve those results in [11] in the case of zero sound speed $P^{\prime }\left({\rho }^{*}\right)=0$. We shall establish the global-in-time existence of strong solutions in hybrid Besov spaces. Furthermore, it is shown that the global solutions with critical regularity are Gevrey analytics. Finally, based on Gevrey’s estimates, we obtained uniform bounds on the growth of the analyticity radius of solutions in negative Besov spaces, which lead to the optimal time-decay estimates of solutions and their derivatives of arbitrary order, without additional smallness on the low frequencies of initial data.

2. Momentum Formulation and Main Results

As in [11], we rewrite (1) in the momentum formulation, such that those nonlinear terms satisfy the divergence form. For the sake of simplicity, we normalize the constant equilibrium ${\rho }^{*}$ to be one. Moreover, we introduce the viscosity coefficients $\overline{\mu }=\mu \left(1\right)$, $\overline{\lambda }=\lambda \left(1\right)$ and the capillarity coefficient $\overline{\kappa }=\kappa \left(1\right)$. Denoting $a=\rho -1$ as the density fluctuation and $m=\rho u$ of the momentum, the Cauchy problem reads as follows:

$$\left\{\begin{array}{c}{\partial }_{t}a+\mathrm{div}m=0,\hfill \\ {\partial }_{t}m-\overline{\mathcal{A}}m-\overline{\kappa }\nabla \Delta a=g(a,m),\hfill \\ {(a,m)|}_{t=0}=(a_0,m_0)\hfill \end{array}\right.$$

(4)

with

$$\overline{\mathcal{A}}m=\overline{\mu }\mathrm{div}D\left(m\right)+(\overline{\mu }+\overline{\lambda })\nabla \mathrm{div}m,a_0\triangleq {\rho }_0-1,m_0\triangleq {\rho }_0{u}_0.$$

Setting $\check{\kappa }\triangleq \kappa \left(1\right)+{\kappa }^{\prime }\left(1\right)$, one can write the nonlinear terms $g=\sum _{i=1}^6{g}_i$ as follows:

$$\left\{\begin{array}{c}{g}_1=\overline{\mu }\Delta \left(Q\left(a\right)m\right)+(\overline{\mu }+\overline{\lambda })\nabla \mathrm{div}\left(Q\left(a\right)m\right),\hfill \\ g_2=2\mathrm{div}\left(\tilde{\mu }\left(a\right)D\left((1-Q(a\left)\right)m\right)\right)+\nabla \left(\tilde{\lambda }\left(a\right)\mathrm{div}\left((1-Q(a\left)\right)m\right)\right),\hfill \\ g_3=\mathrm{div}\left(\left(Q\right(a)-1)m\otimes m\right),\hfill \\ g_4=\nabla \left(aG\left(a\right)\right),\hfill \\ g_5=\nabla ({\tilde{\kappa }}_1\left(a\right)\Delta a),\hfill \\ g_6=\frac{1}{2}\nabla \left(\left({\tilde{\kappa }}_2\left(a\right)+\check{\kappa }\right){|\nabla a|}^2\right)-\mathrm{div}\left(\left({\tilde{\kappa }}_3\left(a\right)+\overline{\kappa }\right)\nabla a\otimes \nabla a\right),\hfill \end{array}\right.$$

(5)

with $Q\left(a\right)=\frac{a}{a+1}$, $G\left(a\right)=P^{\prime }(1+\theta a)$ and

$$\left\{\begin{array}{c}\tilde{\mu }\left(a\right)=2\mu (a+1)-2\overline{\mu },\hfill \\ \tilde{\lambda }\left(a\right)=\lambda (a+1)-\overline{\lambda },\hfill \\ {\tilde{\kappa }}_1\left(a\right)=(a+1)\kappa (a+1)-\overline{\kappa },\hfill \\ {\tilde{\kappa }}_2\left(a\right)=\kappa (a+1)+(a+1){\kappa }^{\prime }(a+1)-\check{\kappa },\hfill \\ {\tilde{\kappa }}_3\left(a\right)=\kappa (a+1)-\overline{\kappa }.\hfill \end{array}\right.$$

(6)

In our analysis, those functions vanish at zero and are assumed to be real analytics. Before stating the result, let us introduce the following functional space

$$\begin{array}{c}\hfill E_T:=\left\{(a,m)|{(\nabla a,m)}^h\in {\mathcal{C}}_T\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)\cap {\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right);{(\nabla a,m)}^{\ell }\in {\mathcal{C}}_T\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)\cap {\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)\right\},\end{array}$$

(7)

which is equipped with the following norm

$$\begin{array}{c}\hfill {\parallel (a,m)\parallel }_{E_T}\triangleq {\parallel (\nabla a,m)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)\cap {\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}^h+{\parallel (\nabla a,m)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)\cap {\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}^{\ell }.\end{array}$$

(8)

Actually, $E_T$ represents a class of hybrid Besov spaces, see Appendix A for the detailed definition. Moreover, we write $E_{\infty }$ if $T=\infty$.

Our first result is the global well-posedness and Gevrey analyticity of the strong solutions to (4) in case of zero sound speed.

Theorem 1.

Suppose that $P^{\prime }\left(1\right)=0$ and $d\ge 3$. If $(a_0,m_0)\in {\dot{B}}_{2,1}^{\frac{d}{2}}\times {\dot{B}}_{2,1}^{\frac{d}{2}-1}$; moreover, $(a_0^{\ell },u_0^{\ell })\in {{\dot{B}}^{\frac{d}{2}-2}}_{2,\infty }\times {{\dot{B}}^{\frac{d}{2}-3}}_{2,\infty }$, there exists a constant $\eta >0$ depending on functions $\kappa ,\lambda ,\mu$, and d, such that

$${\mathcal{X}}_0:=\parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+{\parallel (\nabla a_0,m_0)\parallel }_{{\dot{B}}_{2,\infty }^{\frac{d}{2}-3}}^{\ell }\le \eta .$$

(9)

Then the Cauchy problem (4) and (5) admits a unique global-in-time solution $(a,m)$ in the space $E_{\infty }$ satisfying

$${\parallel (a,m)\parallel }_{E_T}\lesssim {\mathcal{X}}_0$$

(10)

for any $T>0$. Moreover, the solution $(a,m)$ satisfies $e^{\sqrt{c_{0}t}{\Lambda }_1}(a,m)\in E_{\infty }$, where $c_0=c_0(\overline{\lambda },\overline{\mu },\overline{\kappa },d)>0$ and ${\Lambda }_1$ stands for the Fourier multiplier with symbol ${\left|\xi \right|}_1=\sum _{i=1}^d\left|{\xi }_i\right|$.

Remark 1.

From Theorem 1, we see that the Korteweg tensor compensates for the lack of dissipation arising from the pressure term and stabilizes the system in case of zero sound speed. By introducing the ${\dot{B}}_{2,\infty }^s$-type space at low frequencies, one can improve the global existence results of (4) and (5) in [11] with rough initial data. Moreover, it is proved that the solution is Gevrey analytic in the hybrid Besov space, where the radius of analyticity grows as $\sqrt{t}$ in time.

Based on Gevrey’s estimates, we present a new derivation for the optimal decay of the solution and its arbitrary higher-order derivatives.

Theorem 2.

Let $(a,m)$ be the global solution to (4) and (5) addressed by Theorem 1 for $d\ge 3$. Suppose that the real number ${\sigma }_1$ fulfills $2-\frac{d}{2}<{\sigma }_1<\frac{d}{2}$. If in addition initial norm $\parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1-1}}^{\ell }$ is bounded, then the solution $(a,m)$ satisfies the decay estimates

$$\begin{array}{c}\hfill \parallel {\Lambda }^l{a\parallel }_{L^p}\lesssim t^{-\frac{{\tilde{\sigma }}_1}{2}-\frac{l}{2}},l>-{\tilde{\sigma }}_1;\end{array}$$

(11)

$$\begin{array}{c}\hfill \parallel {\Lambda }^l{m\parallel }_{L^p}\lesssim t^{-\frac{{\tilde{\sigma }}_1}{2}-\frac{1}{2}-\frac{l}{2}},l>-{\tilde{\sigma }}_1-1,\end{array}$$

(12)

for $t\ge 1$ and $p\ge 2$, where ${\tilde{\sigma }}_1\triangleq {\sigma }_1-\frac{d}{p}+\frac{d}{2}$.

Remark 2.

The key estimate of Theorem 2 lies in the uniform bounds on the growth of the analyticity radius in Besov spaces of the negative order and, thus, is totally different in contrast to the recent energy methods developed in [12,18,19,20].

We would like to give a rough sight of the linearized system of (4). By virtue of the Leray projector by $\mathcal{P}=\mathrm{Id}-\frac{\nabla }{\Delta }\mathrm{div}$, the corresponding linear system can be reformulated in terms of the $\mathcal{P}m$ (divergence-free part) and $\mathcal{Q}m=(I-\mathcal{P})m$ (compressible part):

$$\left\{\begin{array}{c}{\partial }_{t}a+\mathrm{div}\mathcal{Q}m=0,\hfill \\ {\partial }_t\mathcal{Q}m-\overline{\nu }\Delta \mathcal{Q}m-\overline{\kappa }\nabla \Delta a=0,\hfill \\ {\partial }_t\mathcal{P}m-\overline{\mu }\Delta \mathcal{P}m=0,\hfill \end{array}\right.$$

(13)

where we set $\overline{\nu }\triangleq 2\overline{\mu }+\overline{\lambda }$.

It is clear that $\mathcal{P}m$ just fulfills a linear heat equation. Denote $\mathcal{U}\triangleq {\Lambda }^{-1}\mathrm{div}m$. The coupling system for a and $\mathcal{Q}m$ reduces to

$$\left\{\begin{array}{c}{\partial }_{t}a+\Lambda \mathcal{U}=0,\hfill \\ {\partial }_t\mathcal{U}-\overline{\nu }\Delta \mathcal{U}-\overline{\kappa }{\Lambda }^{3}a=0.\hfill \end{array}\right.$$

(14)

Taking the Fourier transform with respect to $x\in {\mathbb{R}}^d$ leads to

$$\frac{d}{dt}\left(\begin{array}{c}\widehat{a}\\ \widehat{\mathcal{U}}\end{array}\right)=G\left(\xi \right)\left(\begin{array}{c}\widehat{a}\\ \widehat{\mathcal{U}}\end{array}\right)withG\left(\xi \right)=\left(\begin{array}{cc}0& \left|\xi \right|\\ -\overline{\kappa }{\left|\xi \right|}^3& \overline{\nu }{\left|\xi \right|}^2\end{array}\right),$$

(15)

where $\xi \in {\mathbb{R}}^d$ is the Fourier variable. It is not difficult to check that

$\left(i\right)$

If ${\overline{\nu }}^2\ge 4\overline{\kappa }$, then $G\left(\xi \right)$ has two real eigenvalues:

$${\lambda }_{\pm }=\frac{-\overline{\nu }\pm \sqrt{({\overline{\nu }}^2-4\overline{\kappa })}}{2}{\left|\xi \right|}^2;$$

$\left(ii\right)$

If ${\overline{\nu }}^2<4\overline{\kappa }$, then $G\left(\xi \right)$ has two complex conjugated eigenvalues:

$${\lambda }_{\pm }=\frac{-\overline{\nu }\pm \mathrm{i}\sqrt{4\overline{\kappa }-{\overline{\nu }}^2}}{2}{\left|\xi \right|}^2.$$

Let us underline that (14) is of the “regularity-gain” type, according to the new notion for the general hyperbolic-parabolic system with dispersion formulated in [21], which indicates that the solution enjoys parabolic regularization. That is, the pressure term cannot offer dissipation in case of zero sound speed, however, the three-order term originated from the Korteweg tensor stabilizes the system to be parabolic. In particular, the solution behaves as a combination of the heat kernel and Schrödinger wave in the case of the large capillary coefficient that ${\overline{\nu }}^2<4\overline{\kappa }$. The proof of Theorem 1 depends mainly on the energy approach in terms of Hoff’s viscous effective flux (see [22]), which has been developed by Haspot [23] in the critical regularity framework. To control the evolution of norm ${\parallel a\parallel }_{L_T^1\left({\dot{B}}_{2,1}^{d/2}\right)}$ at low frequencies, inspired by [11], a lower regularity than the scaling one is imposed, which enables us to use “hybrid” Besov norms with different regularity indices in the low and high frequencies. To overcome the technical difficulties with more rough initial data, Proposition A1 is well employed, which leads to the global existence in the ${\dot{B}}_{2,\infty }^s$-type space at low frequencies. Furthermore, the dissipation mechanism of the “regularity-gain” type allows us to establish the analyticity of solutions to (4) and (5).

For asymptotic behaviors of solutions, inspired by Oliver and Titi’s work [24] for incompressible fluids, we shall develop a decay framework based on Gevrey’s estimates:

$$\begin{array}{c}\hfill \parallel {\Lambda }^l{u\parallel }_{L^2}\lesssim \parallel {\Lambda }^l{u\parallel }_{{\dot{B}}_{2,1}^0}\le C_{l,{\sigma }_1}{t}^{-\frac{l}{2}-\frac{{\sigma }_1}{2}}{\parallel e^{\sqrt{t}{\Lambda }_1}u\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1}}\mathrm{for}l>-{\sigma }_1.\end{array}$$

(16)

Obviously, in order to obtain the optimal decay estimates (11) and (12), it suffices to establish uniform bounds on the growth of the analyticity radius in negative Besov norms

$$\begin{array}{c}\hfill \parallel e^{\sqrt{t}{\Lambda }_1}{u\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1}}\le Cfort>0.\end{array}$$

(17)

In other words, the decay problem turns into the analytical stability one. The argument of proof of Theorem 2 is of independent interest in the critical setting, which may apply to a wide range of dissipative systems of the “regularity-gain” type as in [21].

The rest of this paper unfolds as follows. In Section 3, we establish the global-in-time existence of strong solutions in hybrid Besov spaces. Section 4 is devoted to the Gevrey analyticity of solutions. In Section 5, we establish uniform bounds on the growth of the radius of analyticity, which leads to the optimal time-decay estimates of solutions and their derivatives of arbitrary order. In the last section (Appendix A and Appendix B ), we briefly present nonlinear Besov(–Gevrey) estimates for the product and composition of functions for the convenience of readers.

3. The Global Well-Posedness

Throughout the proof, $C>0$ stands for a generic “constant". For brevity, $f\lesssim g$ means that $f\le Cg$. It will also be understood that ${\parallel (f,g)\parallel }_X={\parallel f\parallel }_X+{\parallel g\parallel }_X$ for all $f,g\in X$. In this section, our task is to give the proof of Theorem 1. It is observed that the linear system of (4) admits the parabolic regularization in both high frequencies and low frequencies due to the Korteweg tensor. We perform the energy method in terms of the effective velocity and obtain the key priori estimates. Furthermore, the global existence of strong solutions to (4) and (5) are achieved by the standard fixed point argument.

For clarity, we denote the norm by

$$\mathcal{X}\left(T\right)\triangleq {\parallel (a,m)\parallel }_{E_T}$$

and prove the following priori estimates for solutions to (4) and (5).

Proposition 1.

Suppose $(a,m)$ is the solution of systems (4) and (5) for $d\ge 3$. There exists a constant $R>0$, such that if

$${\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\le R,$$

then

$$\mathcal{X}\left(T\right)\lesssim {\mathcal{X}}_0+C\left(1+\mathcal{X}\left(T\right)\right){\mathcal{X}}^2\left(T\right)$$

(18)

for any $T>0$, where the constant C is dependent on R.

Proof. 

The first step is devoted to the linear analysis. By using the Leray projector $\mathcal{P}\triangleq \mathrm{Id}-\frac{\nabla }{\Delta }\mathrm{div}$, the momentum can be written as $m=\mathcal{P}m+\mathcal{Q}m$, which is the sum of the incompressible part $\mathcal{P}m$ and the compressible part $\mathcal{Q}m$. From (4), one can obtain the coupling system between a and $\mathcal{Q}m$ satisfying

$$\left\{\begin{array}{c}{\partial }_t\nabla a+\Delta \mathcal{Q}m=0,\hfill \\ {\partial }_t\mathcal{Q}m-\overline{\nu }\Delta \mathcal{Q}m-\overline{\kappa }\Delta \nabla a=\mathcal{Q}g,\hfill \end{array}\right.$$

(19)

where we used the fact $\nabla \mathrm{div}m=\Delta \mathcal{Q}m$. To decouple the system (19), we introduce the effective velocity inspired by [23], i.e.,

$$w\triangleq \mathcal{Q}m+\alpha \nabla a,$$

where $\alpha$ satisfies $\alpha =\frac{\overline{\kappa }}{\overline{\nu }-\alpha }$. After simple calculations, we arrive at

$${\partial }_{t}w-(\overline{\nu }-\alpha )\Delta w=\mathcal{Q}g.$$

(20)

Duhamel’s principle implies that

$$w\left(t\right)=e^{(\overline{\nu }-\alpha )t\Delta }{w}_0+{\int }_0^t{e}^{(\overline{\nu }-\alpha )(t-\tau )\Delta }\mathcal{Q}g\left(\tau \right)d\tau .$$

(21)

Employing Fourier localizations on w and Plancherel’s theory leads to

$$\widehat{{\dot{\Delta }}_{j}w}\left(\xi \right)=e^{-(\overline{\nu }-\alpha )t{\left|\xi \right|}^2}\widehat{{\dot{\Delta }}_j{w}_0}\left(\xi \right)+{\int }_0^t{e}^{-(\overline{\nu }-\alpha )(t-\tau ){\left|\xi \right|}^2}\widehat{{\dot{\Delta }}_j\mathcal{Q}g}(\xi ,\tau )d\tau .$$

Then it follows that

$$\left|\widehat{{\dot{\Delta }}_{j}w}\left(\xi \right)\right|\lesssim e^{-k{2}^{2j}t}\left|\widehat{{\dot{\Delta }}_j{w}_0}\left(\xi \right)\right|+{\int }_0^t{e}^{-k{2}^{2j}(t-\tau )}\left|\widehat{{\dot{\Delta }}_j\mathcal{Q}g}(\xi ,\tau )\right|d\tau ,$$

(22)

where $k\triangleq \mathrm{Re}(\overline{\nu }-\alpha )$ is positive. Hence, multiplying (22) with $2^{(\frac{d}{2}-1)j}$ and summarizing on $j\ge j_0$ ($j_0\in \mathbb{Z}$ is the threshold of the frequency cut-off) yield

$${\parallel w\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel w\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}^h\le C_{\overline{\nu },\overline{\kappa }}\left(\parallel w_0{\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+{\parallel \mathcal{Q}g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\right).$$

(23)

On the other hand, bounding $\mathcal{Q}m$, we use the fact

$$\nabla a=\frac{w-\mathcal{Q}m}{\alpha },$$

(24)

so that the equation for $\mathcal{Q}m$ can be rewritten as

$${\partial }_t\mathcal{Q}m-\alpha \Delta \mathcal{Q}m=\frac{\overline{\kappa }}{\alpha }\Delta w+\mathcal{Q}g.$$

(25)

Hence, repeating calculations in (23), one can deduce

$$\begin{array}{c}{\parallel \mathcal{Q}m\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel \mathcal{Q}m\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}^h\le C_{\overline{\nu },\overline{\kappa }}\left(\parallel \mathcal{Q}{m}_0{\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+\parallel \frac{\overline{\kappa }}{\alpha }\Delta w+\mathcal{Q}g{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\right)\hfill \\ \hfill \le C_{\overline{\nu },\overline{\kappa }}\left(\parallel \mathcal{Q}{m}_0{\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+\parallel w_0{\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+{\parallel \mathcal{Q}g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\right).\end{array}$$

(26)

Therefore, it follows from (23) and (26) that

$$\begin{array}{c}{\parallel (\nabla a,\mathcal{Q}m)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel (\nabla a,\mathcal{Q}m)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}^h\le C_{\overline{\nu },\overline{\kappa }}({\parallel (\nabla a_0,\mathcal{Q}{m}_0)\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h\hfill \\ \hfill +{\parallel \mathcal{Q}g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h).\end{array}$$

(27)

To control the evolution of norm ${\parallel a\parallel }_{L_T^1\left({\dot{B}}_{2,1}^{d/2}\right)}$ at the low frequencies, we need to establish the ${\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{d/2-3}\right)$ estimates of $(\nabla a,m)$. The effective velocity still plays a key role in the low-frequency estimates. Similarly, one can obtain

$$\begin{array}{c}{\parallel (\nabla a,\mathcal{Q}m)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }+{\parallel (\nabla a,\mathcal{Q}m)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}^{\ell }\le C_{\overline{\nu },\overline{\kappa }}({\parallel (\nabla a_0,\mathcal{Q}{m}_0)\parallel }_{{\dot{B}}_{2,\infty }^{\frac{d}{2}-3}}^{\ell }\hfill \\ \hfill +{\parallel \mathcal{Q}g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }).\end{array}$$

(28)

The details are left to the interested readers. Finally, we see that $\mathcal{P}m$ fulfills a mere heat equation:

$${\partial }_t\mathcal{P}m-\overline{\mu }\Delta \mathcal{P}m=\mathcal{P}g.$$

(29)

Hence, it follows from Proposition A4 that

$${\parallel \mathcal{P}m\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel \mathcal{P}m\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}^h\le C_{\overline{\mu }}\left(\parallel \mathcal{P}{m}_0{\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+{\parallel \mathcal{P}g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\right)$$

(30)

and

$${\parallel \mathcal{P}m\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }+{\parallel \mathcal{P}m\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}^{\ell }\le C_{\overline{\mu }}\left(\parallel \mathcal{P}{m}_0{\parallel }_{{\dot{B}}_{2,\infty }^{\frac{d}{2}-3}}^{\ell }+{\parallel \mathcal{P}g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\right).$$

(31)

Combining (57)–(28), (30) and (31) leads to

$$\mathcal{X}\left(T\right)\le C_{\overline{\nu },\overline{\kappa }}\left({\mathcal{X}}_0+{\parallel g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\right).$$

(32)

In what follows, let us handle those nonlinear terms in g. Indeed, regarding those nonlinear estimates at high frequencies, the reader is referred to Proposition 3.1 in [11]. Here, we feel free to skip the details. Consequently, one can deduce that

$$\begin{array}{c}\hfill {\parallel g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\le C_R\left(1+\mathcal{X}\left(T\right)\right){\mathcal{X}}^2\left(T\right),\end{array}$$

(33)

where $C_R>0$ is a constant depending on R. What is left is to bound the nonlinear terms in the low-frequency part. For that end, we shall frequently use the following inequality, which is the direct consequence of Proposition A1 for $d\ge 3$:

$$\begin{array}{c}\hfill {\parallel fg\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}\lesssim {\parallel f\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel g\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}.\end{array}$$

(34)

Let us start to estimate ${\parallel g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }$. First of all, we focus on the pressure term $g_4$, which is a lower-order one with respect to the scaling of the Korteweg system. It follows from (34) that

$$\begin{array}{c}\hfill \parallel \nabla \left(aG\left(a\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\lesssim {\parallel aG\left(a\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}^{\ell }\lesssim {\parallel a\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel G\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}.\end{array}$$

By performing Proposition A2, we have

$$\begin{array}{c}\hfill {\parallel G\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\le C_R{\parallel a\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)},\end{array}$$

where $C_R$ is some constant depending on R. Noticing the real interpolation for $s<\tilde{s}$, $p,r\in [1,\infty ]$ and $\theta \in (0,1)$,

$$\begin{array}{c}\hfill {\parallel f\parallel }_{{\tilde{L}}_T^{\frac{1}{1-\theta }}\left({\dot{B}}_{p,r}^{\theta s+(1-\theta )\tilde{s}}\right)}\le C_{\theta ,s,\tilde{s}}{\parallel f\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{p,r}^s\right)}^{\theta }{\parallel f\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{p,r}^{\tilde{s}}\right)}^{1-\theta },\end{array}$$

(35)

which implies that

$$\begin{array}{c}{\parallel a\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\le \parallel a^h{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}+{\parallel a^{\ell }\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill \lesssim {\left({\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}^h\right)}^{\theta }{\left({\parallel a\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+2}\right)}^h\right)}^{1-\theta }+{\left({\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}^{\ell }\right)}^{\theta }{\left({\parallel a\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}}\right)}^{\ell }\right)}^{1-\theta }\lesssim \mathcal{X}\left(T\right).\end{array}$$

Therefore, we deduce that

$$\begin{array}{c}\hfill \parallel g_4{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\le C_R{\mathcal{X}}^2\left(T\right).\end{array}$$

(36)

As a matter of fact, other nonlinear terms can be estimated in a similar way. For the term with $g_1=\mu \Delta \left(Q\left(a\right)m\right)+(\mu +\lambda )\nabla \mathrm{div}\left(Q\left(a\right)m\right)$, it follows from (34) that

$$\begin{array}{ccc}\hfill {\parallel \mu \Delta \left(Q\left(a\right)m\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }& \lesssim & {\parallel \nabla \left(Q\left(a\right)m\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}\hfill \\ \hfill & \lesssim & {\parallel \nabla Q\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel m\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}+{\parallel \nabla m\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel Q\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_R{\mathcal{X}}^2\left(T\right).\hfill \end{array}$$

Another term in $g_1$ can be dealt with in the same way. Consequently, we have

$$\begin{array}{c}\hfill \parallel g_1{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\le C_R{\mathcal{X}}^2\left(T\right).\end{array}$$

(37)

Regarding $g_2$, we estimate $\mathrm{div}\left(\tilde{\mu }\left(a\right)D\left(Q\left(a\right)m\right)\right)$ as an example. Precisely,

$$\begin{array}{ccc}\hfill \parallel \mathrm{div}\left(\tilde{\mu }\left(a\right)D\left(Q\left(a\right)m\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }& \lesssim & \parallel \tilde{\mu }{\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel D\left(Q\left(a\right)m\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_R{\mathcal{X}\left(T\right)\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel m\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\hfill \\ \hfill & \le & C_R{\mathcal{X}}^3\left(T\right).\hfill \end{array}$$

(38)

Moreover, performing similar calculations for $\mathrm{div}\left(\tilde{\mu }\left(a\right)Dm\right)$ and $\nabla \left(\tilde{\lambda }\left(a\right)\mathrm{div}\left((1-Q(a)\right)m\right))$ enables us to obtain

$$\begin{array}{c}\hfill \parallel g_2{\parallel }_{{\tilde{L}}^1{\dot{B}}_{q,\infty }^{\frac{d}{q}-3}}^{\ell }\le C_R(1+\mathcal{X}\left(T\right)){\mathcal{X}}^2\left(T\right).\end{array}$$

For $g_3$, one can employ (34) again to obtain

$$\begin{array}{ccc}\hfill \parallel g_3{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }& \lesssim & {\parallel (Q\left(a\right)-1)m\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel m\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_R\left(1+{\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\right){\parallel m\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}^2\hfill \\ \hfill & \le & C_R(1+\mathcal{X}\left(T\right)){\mathcal{X}}^{}\left(T\right).\hfill \end{array}$$

(39)

Finally, we handle the Korteweg terms in low frequencies. It follows from (34) that

$$\begin{array}{c}\hfill \parallel g_5{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\lesssim \parallel {\tilde{\kappa }}_1{\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel \Delta a\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\le C_R{\mathcal{X}}^2\left(T\right).\end{array}$$

(40)

For $g_6$, it suffices to take a look at the term $\nabla \left({\tilde{\kappa }}_2\left(a\right){|\nabla a|}^2\right)$. Indeed,

$$\begin{array}{ccc}\hfill \parallel \nabla \left({\tilde{\kappa }}_2\left(a\right){|\nabla a|}^2\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }& \lesssim & \parallel {\tilde{\kappa }}_2{\left(a\right)\nabla a\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel \nabla a\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_R{\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel \nabla a\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}^2\hfill \\ \hfill & \le & C_R{\mathcal{X}}^3\left(T\right).\hfill \end{array}$$

Repeating the above calculations to $\check{\kappa }\nabla {|\nabla a|}^2$ and $\mathrm{div}(\left({\tilde{\kappa }}_3\left(a\right)+\overline{\kappa }\right)\nabla a\otimes \nabla a)$ leads to

$$\begin{array}{c}\hfill \parallel g_6{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\le C_R(1+\mathcal{X}\left(T\right)){\mathcal{X}}^2\left(T\right).\end{array}$$

(41)

Putting those estimates (36)–(41) together, we conclude that

$$\begin{array}{c}\hfill {\parallel g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\le C_R\left(1+\mathcal{X}\left(T\right)\right){\mathcal{X}}^2\left(T\right).\end{array}$$

(42)

Furthermore, combining (32), (33), and (42), we arrive at (18), which finishes the proof of Proposition 1. □

Based on Proposition 1, one can establish the global-in-time existence and uniqueness by the fixed point argument. Precisely, it follows from the standard Duhamel formula that

$$\left(\begin{array}{c}a\left(t\right)\\ m\left(t\right)\end{array}\right)=\mathcal{G}\left(t\right)\left(\begin{array}{c}{a}_0\\ m_0\end{array}\right)+{\int }_0^t\mathcal{G}(t-s)\left(\begin{array}{c}0\\ g\left(s\right)\end{array}\right)ds,$$

where $\mathcal{G}\left(t\right)$ is the semi-group associated with the following linear system:

$$\left\{\begin{array}{c}{\partial }_{t}a+\mathrm{div}m=0,\hfill \\ {\partial }_{t}m-\overline{\mu }\Delta m-\overline{\nu }\nabla \mathrm{div}m-\overline{\kappa }\nabla \Delta a=0.\hfill \end{array}\right.$$

(43)

Set

$$\left(\begin{array}{c}{a}_L\left(t\right)\\ m_L\left(t\right)\end{array}\right)\triangleq \mathcal{G}\left(t\right)\left(\begin{array}{c}{a}_0\\ m_0\end{array}\right).$$

Define the functional $\Psi (\tilde{a},\tilde{m})\left(t\right)$ in the neighborhood of zero in the space $E_T$ by

$$\Psi (\tilde{a},\tilde{m})\left(t\right)={\int }_0^t\mathcal{G}(t-s)\left(\begin{array}{c}0\\ g(a_L+\tilde{a},m_L+\tilde{m})\left(s\right)\end{array}\right)ds.$$

(44)

To obtain the existence part of Theorem 1, it is sufficient to show that $\Psi (\tilde{a},\tilde{m})$ has a fixed point for in $E_T$. For that end, the proof is divided into two steps: the stability of the closed ball $\mathbf{B}(0,r)$ for sufficient small r and contraction in that ball. Let $a=a_L+\tilde{a},m=m_L+\tilde{m}$. From Proposition 1, we obtain

$$\begin{array}{c}\hfill \parallel (a_L,m_L){\parallel }_{E_T}\le C{\mathcal{X}}_0\le C\eta \end{array}$$

(45)

and

$$\parallel \Psi (\tilde{a},\tilde{m}){\parallel }_{E_T}\le {C(\parallel g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel g\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }).$$

Let r be small, such that $r\le \frac{2}{3}R$. Assuming that $2C\eta \le r$ implies that

$${\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\le C\eta +r\le R.$$

Thus, by calculations in the proof of Proposition 1, we have

$$\begin{array}{ccc}& & \parallel \Psi (\tilde{a},\tilde{m}){\parallel }_{E_T}\hfill \\ \hfill & \le & C_R\left(1+\parallel (a_L+\tilde{a},m_L+\tilde{m}){\parallel }_{E_T}\right){\parallel (a_L+\tilde{a},m_L+\tilde{m})\parallel }_{E_T}^2\hfill \\ \hfill & \le & C_R(1+C\eta +r){(C\eta +r)}^2.\hfill \end{array}$$

(46)

Choosing a suitable $(r,\eta )$, such that

$$\begin{array}{c}\hfill r\le min\left\{1,\frac{1}{30C_R},\frac{2}{3}R\right\}\mathrm{and}\eta \le \frac{r}{2C},\end{array}$$

(47)

we, thus, have

$$\Psi \left(\mathbf{B}\right(0,r\left)\right)\subset \mathbf{B}(0,r).$$

Next, we prove the contraction property. Let $({\tilde{a}}_1,{\tilde{m}}_1)$ and $({\tilde{a}}_2,{\tilde{m}}_2)$ be in $\mathbf{B}(0,r)$. We are going to estimate $\parallel (\Psi (\widetilde{a_2},\widetilde{m_2})-\Psi (\widetilde{a_1},\widetilde{m_1})){\parallel }_{E_T}$. According to (44), we obtain

$$\begin{array}{ccc}\hfill & & \parallel (\Psi (\widetilde{a_2},\widetilde{m_2})-\Psi (\widetilde{a_1},\widetilde{m_1})){\parallel }_{E_T}\hfill \\ \hfill & \lesssim & \parallel g(a_2,m_2)-g(a_1,m_1){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel g(a_2,m_2)-g(a_1,m_1)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }.\hfill \end{array}$$

(48)

Bounding those nonlinear terms on the right-hand side of (48) follows the similar procedure of the a priori estimate in Proposition 1. Let us show the calculation for $g_3=a\nabla G\left(a\right)$:

$$\begin{array}{ccc}& & \parallel \nabla \left(a_{1}G\left(a_1\right)\right)-\nabla \left(a_{2}G\left(a_2\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\hfill \\ & \le & \parallel \nabla \left((a_1-a_2)G\left(a_1\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel \nabla \left(a_2(G\left(a_1\right)-G\left(a_2\right)\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\hfill \\ & \le & \parallel {\tilde{a}}_1-{\tilde{a}}_2{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\parallel G\left(a_1\right){\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}+\parallel a_2{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel G\left(a_1\right)-G\left(a_2\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\hfill \\ & \le & C_R(1+\parallel (a_1,a_2){\parallel }_{E_T})\parallel (a_1,a_2){\parallel }_{E_T}{\parallel {\tilde{a}}_1-{\tilde{a}}_2\parallel }_{E_T}\hfill \end{array}$$

where Proposition A3 is applied in the last inequality. On the other hand,

$$\begin{array}{ccc}& & \parallel \nabla \left(a_{1}G\left(a_1\right)\right)-\nabla \left(a_{2}G\left(a_2\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\hfill \\ & \le & \parallel \nabla \left((a_1-a_2)G\left(a_1\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }+{\parallel \nabla \left(a_2(G\left(a_1\right)-G\left(a_2\right)\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\hfill \\ & \le & C_R(1+\parallel (a_1,a_2){\parallel }_{E_T})\parallel (a_1,a_2){\parallel }_{E_T}{\parallel {\tilde{a}}_1-{\tilde{a}}_2\parallel }_{E_T}.\hfill \end{array}$$

In conclusion, one can obtain the following error estimate, provided that r and $\epsilon$ are small enough as in (47)

$$\begin{array}{ccc}\hfill & & \parallel (\Psi (\widetilde{a_2},{\tilde{m}}_2)-\Psi (\widetilde{a_1},{\tilde{m}}_1)){\parallel }_{E_T}\hfill \\ \hfill & \le & \left(1+\parallel (a_1,m_1){\parallel }_{E_T}+{\parallel (a_2,m_2)\parallel }_{E_T}\right)\left(\parallel (a_1,m_1){\parallel }_{E_T}+{\parallel (a_2,m_2)\parallel }_{E_T}\right)\hfill \\ \hfill & & \times \parallel ({\tilde{a}}_1-{\tilde{a}}_2,{\tilde{m}}_1-{\tilde{m}}_2){\parallel }_{E_T}\hfill \\ \hfill & \le & 4C_R(C\eta +r+1)(C\eta +r){\parallel ({\tilde{a}}_1-{\tilde{a}}_2,{\tilde{m}}_1-{\tilde{m}}_2)\parallel }_{E_T}\hfill \\ \hfill & \le & \frac{1}{2}{\parallel ({\tilde{a}}_1-{\tilde{a}}_2,{\tilde{m}}_1-{\tilde{m}}_2)\parallel }_{E_T}.\hfill \end{array}$$

(49)

Hence, according to the fixed point theorem, there exists a unique solution $(a,m)$ to (4)–(6), which obviously belongs to the space $E_T$ and

$$\begin{array}{c}\hfill {\parallel a\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\le {\parallel (a,m)\parallel }_{E_T}\le \frac{1}{20C_R}\end{array}$$

(50)

for any $T>0$, changing $C_R$ into a greater constant if necessary. Furthermore, we can obtain the existence and uniqueness of solutions in $[0,\infty )$ by the continuity argument.

4. Gevrey Analyticity

In this section, it is proved that the solution addressed by Theorem 1 is actually Gevrey analytic. The proof lies in the following a priori estimate, which indicates the evolution of Gevrey regularity in the critical space. Set $(A,U)=e^{\sqrt{c_{0}t}{\Lambda }_1}(a,m)$ for some $c_0>0$ (to be confirmed later) and

$$\mathcal{G}\left(T\right)\triangleq {\parallel (A,U)\parallel }_{E_T}.$$

Proposition 2.

Suppose that $(a,m)$ is the solution of systems (4) and (5) for $d\ge 3$. There exists a constant $R_0$, such that if

$${\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\le R_0,$$

then it holds that

$$\begin{array}{c}\hfill \mathcal{G}\left(T\right)\le {\mathcal{X}}_0+C\left(1+\mathcal{G}\left(T\right)\right){\mathcal{G}}^2\left(T\right),\end{array}$$

(51)

where $C>0$ is a constant depending on $R_0$.

Proof. 

Indeed, one can repeat the procedure leading to Proposition (18) after imposing the Gevrey multiplier $e^{\sqrt{c_{0}t}{\Lambda }_1}$ everywhere. It follows from (21) that ($W\triangleq e^{\sqrt{c_{0}t}{\Lambda }_1}w$)

$$W\left(t\right)=e^{\sqrt{c_{0}t}{\Lambda }_1+(\overline{\nu }-\alpha )t\Delta }{w}_0+{\int }_0^t{e}^{\sqrt{c_0(t-\tau )}{\Lambda }_1+(\overline{\nu }-\alpha )(t-\tau )\Delta }{e}^{\sqrt{c_0}(-\sqrt{t-\tau }+\sqrt{t}-\sqrt{\tau }){\Lambda }_1}\mathcal{Q}G\left(\tau \right)d\tau$$

with $G\triangleq e^{\sqrt{c_{0}t}{\Lambda }_1}g$. Employing the Fourier cut-off operator ${\dot{\Delta }}_j$ gives

$$\begin{array}{c}\widehat{{\dot{\Delta }}_{j}W}(\xi ,t)=e^{\sqrt{c_{0}t}{\left|\xi \right|}_1-(\overline{\nu }-\alpha )t{\left|\xi \right|}^2}\widehat{{\dot{\Delta }}_j{\omega }_0}\left(\xi \right)\hfill \\ \hfill +{\int }_0^t{e}^{\sqrt{c_0(t-\tau )}{\left|\xi \right|}_1-(\overline{\nu }-\alpha )(t-\tau ){\left|\xi \right|}^2}{e}^{\sqrt{c_0}(-\sqrt{t-\tau }+\sqrt{t}-\sqrt{\tau }){\left|\xi \right|}_1}\widehat{{\dot{\Delta }}_j\mathcal{Q}G(\xi ,\tau )}d\tau .\end{array}$$

Noting that $\xi \in 2^j\mathcal{C}$, it holds that

$$\left|e^{\sqrt{c_{0}t}{\left|\xi \right|}_1-(\overline{\nu }-\alpha )t{\left|\xi \right|}^2}\right|\le C{e}^{-\frac{1}{2}\mathrm{Re}(\overline{\nu }-\alpha )2^{2j}t},$$

furthermore, we have

$$\left|\widehat{{\dot{\Delta }}_{j}W}(\xi ,t)\right|=e^{-\frac{1}{2}k{2}^{2j}t}\left|\widehat{{\dot{\Delta }}_j{\omega }_0}\left(\xi \right)\right|+{\int }_0^t{e}^{-\frac{1}{2}k{2}^{2j}(t-\tau )}\left|\widehat{{\dot{\Delta }}_j\mathcal{Q}G(\xi ,\tau )}\right|d\tau ,$$

(52)

where $k=\mathrm{Re}(\overline{\nu }-\alpha )$ and the triangle inequality $\sqrt{t-\tau }-\sqrt{t}+\sqrt{\tau }\ge 0$ has been used. Then the routine calculation (restricted in high frequencies) yields

$${\parallel W\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel W\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}^h\lesssim \parallel w_0{\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+{\parallel \mathcal{Q}G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h.$$

(53)

Applying $e^{\sqrt{c_{0}t}{\Lambda }_1}$ to (25) and repeating the same calculation as (53) lead to

$${\parallel \mathcal{Q}U\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{p,1}^{\frac{d}{p}-1}\right)}^h+{\parallel \mathcal{Q}U\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{p,1}^{\frac{d}{p}+1}\right)}^h\lesssim \parallel \mathcal{Q}{m}_0{\parallel }_{{\dot{B}}_{p,1}^{\frac{d}{p}-1}}^h+\parallel w_0{\parallel }_{{\dot{B}}_{p,1}^{\frac{d}{p}-1}}^h+{\parallel \mathcal{Q}G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{p,1}^{\frac{d}{p}-1}\right)}^h.$$

Therefore, by (24), we deduce that

$${\parallel (\nabla A,U)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel (\nabla A,U)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}^h\lesssim \parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h+{\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h.$$

(54)

In the spirit of Gevrey’s estimates in high frequencies, we obtain

$${\parallel (\nabla A,U)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }+{\parallel (\nabla A,U)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}^{\ell }\lesssim \parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,\infty }^{\frac{d}{2}-3}}^{\ell }+{\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }.$$

(55)

Together (54) with (55), we conclude that

$$\mathcal{G}\left(T\right)\lesssim {\mathcal{X}}_0+{\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h+{\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }.$$

(56)

Next, we begin to estimate the nonlinear part G. Set $R_0\triangleq min\{R_Q,R_G,R_{\tilde{\mu }},$$R_{\tilde{\lambda }},R_{{\tilde{\kappa }}_i}\}$, which represents the minimum of the convergence radius of those analytic functions appearing in the nonlinear term g. Since the Gevrey estimates on high frequencies are very close to those in [10], we only give the sketch as follows. For $G_1$, without loss of generality, it suffices to deal with the term $\Delta \left(Q\right(a\left)m\right)$. From Corollary A1, we have

$$\begin{array}{c}\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\Delta \left(Q\left(a\right)m\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\lesssim \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left(\nabla Q\left(a\right)m\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}+{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left(Q\left(a\right)\nabla m\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\hfill \\ \lesssim \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\nabla Q\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}+\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{Q\left(a\right)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel \nabla U\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\\ \hfill \triangleq I_1+I_2.\end{array}$$

It follows from Proposition A6 that

$$I_2\lesssim {\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel U\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}.$$

On the other hand, the regularity for composite in $I_1$ is great than $d/2$, and Proposition A6 cannot be applied directly, so bounding $I_1$ is more elaborate. Indeed,

$$\nabla Q\left(a\right)=Q^{\prime }\left(a\right)\nabla a=\tilde{Q}\left(a\right)\nabla a+Q^{\prime }\left(0\right)\nabla a,$$

where $\tilde{Q}\left(a\right)=Q^{\prime }\left(a\right)-Q^{\prime }\left(0\right)$. Consequently, we arrive at

$$\begin{array}{ccc}\hfill \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\nabla Q\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}& \lesssim & \left(1+\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\tilde{Q}{\left(a\right)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\right){\parallel \nabla A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\hfill \\ \hfill & \le & C_{R_0}\left(1+{\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\right){\parallel \nabla A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)},\hfill \end{array}$$

where $C_{R_0}>0$ is some constant depending on $R_0$. Therefore, one can obtain

$$\begin{array}{c}\hfill \parallel G_1{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\le C_{R_0}\left(1+\mathcal{G}\left(T\right)\right){\mathcal{G}}^2\left(T\right).\end{array}$$

(57)

Bounding $G_2$ is similar to (57), so we have

$$\begin{array}{c}\hfill \parallel G_2{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\le C_{R_0}\left(1+\mathcal{G}\left(T\right)\right){\mathcal{G}}^2\left(T\right).\end{array}$$

(58)

Regarding $G_3$, it follows from Proposition A6 that

$$\begin{array}{ccc}\hfill \parallel G_3{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h& \lesssim & (1+\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{Q\left(a\right)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{)\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}^2\hfill \\ \hfill & \le & C_{R_0}\left((1+\mathcal{G}(T)\right){\mathcal{G}}^2\left(T\right).\hfill \end{array}$$

(59)

The term $G_4$ with respect to the pressure can be estimated as

$$\begin{array}{c}\parallel G_4{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\lesssim {\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left(aG\left(a\right)\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}^h\hfill \\ \hfill \lesssim {\parallel A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}G\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\le C_{R_0}{\mathcal{G}}^2\left(T\right).\end{array}$$

(60)

Regarding the high frequencies of capillary terms, it follows that

$$\begin{array}{c}\hfill \parallel G_5{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\lesssim \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\tilde{\kappa }}_1{\left(a\right)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel \Delta A\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\le C_{R_0}{\mathcal{G}}^2\left(T\right)\end{array}$$

(61)

and

$$\begin{array}{c}\hfill \parallel G_6{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\le C_{R_0}\left(1+\mathcal{G}\left(T\right)\right){\mathcal{G}}^2\left(T\right).\end{array}$$

(62)

Adding those estimates (57)–(62) together, we conclude that

$$\begin{array}{c}\hfill {\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}^h\le C_{R_0}\left(1+\mathcal{G}\left(T\right)\right){\mathcal{G}}^2\left(T\right).\end{array}$$

(63)

Bounding Gevrey regularity in ${\dot{B}}_{2,\infty }^s$ at low frequencies is new, we need the following Gevrey product inequalities:

$$\begin{array}{c}\hfill \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\left(fg\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}\lesssim {\parallel F\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel G\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\end{array}$$

(64)

with $(F,G)\triangleq (e^{\sqrt{c_{0}t}{\Lambda }_1}f,e^{\sqrt{c_{0}t}{\Lambda }_1}g)$, which can be regarded as the consequence of Corollary A1 by taking $p=2$ and $s_1=s_2=\frac{d}{2}-1$. We would like to mention that the case of $d=2$ is excluded from (64). This fact is different from the usual product estimate (34) (see [25]). Now, we start to estimate ${\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }$. Precisely,

$$\begin{array}{ccc}\hfill & & \parallel G_1{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\hfill \\ \hfill & \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left(\nabla \left(Q\right(a\left)m\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}+{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left(\mathrm{div}\left(Q\right(a\left)m\right)\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}\hfill \\ \hfill & \le & C_{R_0}\left({\parallel \nabla A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}+{\parallel \nabla U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\right)\hfill \\ \hfill & \le & C_{R_0}{\mathcal{G}}^2\left(T\right).\hfill \end{array}$$

(65)

For simplicity, we just pay attention to the term with $\mathrm{div}\left(\tilde{\mu }\left(a\right)D\left(Q\left(a\right)m\right)\right)$ in $G_2$:

$$\begin{array}{ccc}& & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\mathrm{div}\left(\tilde{\mu }\left(a\right)D\left(Q\left(a\right)m\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\hfill \\ \hfill & \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\tilde{\mu }{\left(a\right)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}D\left(Q\left(a\right)m\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & & +\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\tilde{\mu }{\left(a\right)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}D\left(Q\left(a\right)m\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \lesssim & {\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}-1}\right)}{\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel U\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}.\hfill \end{array}$$

Repeating the above calculations to other terms in $G_2$ again, one can obtain

$$\begin{array}{c}\hfill \parallel G_2{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\le C_{R_0}(1+\mathcal{G}\left(T\right)){\mathcal{G}}^2\left(T\right).\end{array}$$

(66)

Similarly, by taking advantage of (64) and Proposition A6, we obtain

$$\begin{array}{ccc}\hfill \parallel G_3{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }& \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left\{\left(Q\left(a\right)-1\right)(m\otimes m)\right\}{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}^{\ell }\hfill \\ \hfill & \lesssim & \left(1+{\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\right){\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}^2\hfill \\ \hfill & \lesssim & C_{R_0}(1+\mathcal{G}\left(T\right)){\mathcal{G}}^2\left(T\right).\hfill \end{array}$$

(67)

It follows from (64) that

$$\begin{array}{c}\hfill \parallel G_4{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\lesssim \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left(aG\left(a\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}^{\ell }\lesssim {\parallel A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}G\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}.\end{array}$$

(68)

Then using Proposition A6 yields

$$\begin{array}{c}\hfill \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{G\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\le C_{R_0}{\parallel A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)},\end{array}$$

which lead to

$$\begin{array}{c}\hfill \parallel G_4{\parallel }_{{\tilde{L}}^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\le C_{R_0}{\mathcal{G}}^2\left(T\right).\end{array}$$

(69)

Regarding the Korteweg terms in low frequencies, we have

$$\begin{array}{ccc}\hfill \parallel G_5{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }& \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left({\tilde{\kappa }}_1\left(a\right)\Delta a\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}^{\ell }\hfill \\ \hfill & \lesssim & {\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel \Delta A\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_{R_0}{\mathcal{G}}^2\left(T\right)\hfill \end{array}$$

(70)

and

$$\begin{array}{ccc}\hfill \parallel G_6{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }& \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left\{\left({\tilde{\kappa }}_2\left(a\right)+\check{\kappa }\right){|\nabla a|}^2\right\}{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}^{\ell }\hfill \\ \hfill & & +\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\left\{\left({\tilde{\kappa }}_3\left(a\right)+\overline{\kappa }\right)\nabla a\otimes \nabla a\right\}{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}^{\ell }\hfill \\ \hfill & \lesssim & {(1+\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{p,1}^{\frac{d}{p}}\right)}{)\parallel A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}}\right)}^2\hfill \\ \hfill & \le & C_{R_0}(1+\mathcal{G}\left(T\right)){\mathcal{G}}^2\left(T\right).\hfill \end{array}$$

(71)

Plugging (65)–(71) together, we eventually arrive at

$$\begin{array}{c}\hfill {\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-3}\right)}^{\ell }\le C_{R_0}(1+\mathcal{G}\left(T\right)){\mathcal{G}}^2\left(T\right).\end{array}$$

(72)

Hence, (51) is followed by adding those inequalities (56), (63) and (72), which completes the proof of Proposition 2. □

With the help of Proposition 2, it is not difficult to work out a similar fixed point argument as in Section 3 and to prove the Gevrey regularity part in Theorem 1.

5. The Asymptotic Behaviors

This section is devoted to establishing the large-time asymptotic behaviors for solutions to the Cauchy problem (4). The following Lemma enables us to initiate the decay framework based on Gevrey’s estimates (16) and (17), which developed Oliver–Titi’s argument to Besov spaces.

Lemma 1.

Let $\sigma \in \mathbb{R}$. For any $\zeta >-\sigma$ and any tempered distribution f, it holds for all $t>0$ and $c_0>0$, such that

$$\parallel {\Lambda }^{\zeta }{f\parallel }_{{\dot{B}}_{2,1}^0}^{\ell }\lesssim t^{-\frac{\zeta +\sigma }{2}}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}f\parallel }_{{\dot{B}}_{2,\infty }^{-\sigma }}^{\ell }$$

and

$$\parallel {\Lambda }^{\zeta }{f\parallel }_{{\dot{B}}_{2,1}^0}^h\lesssim t^{-\frac{\zeta +\sigma }{2}}{e}^{-a\sqrt{t}}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}f\parallel }_{{\dot{B}}_{2,\infty }^{-\sigma }}^h,$$

where $a>0$ is some constant.

The reader is referred to [26] for similar proof details. In light of Lemma 1, the proof of Theorem 2 lies in the uniform upper bounds on the growth of the analyticity radius of solutions in negative Besov norms (restricted in low frequencies).

5.1. Uniform Upper Bounds on the Growth of Analyticity Radius

Proposition 3.

Let $(a,m)$ be the solution constructed in Theorem 1 for $d\ge 3$. Suppose that the real number ${\sigma }_1$ fulfills $2-\frac{d}{2}<{\sigma }_1<\frac{d}{2}$. If $\parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1-1}}^{\ell }$ is additionally bounded, then it holds that

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}(\nabla a,m)\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1-1}}^{\ell }\le C\end{array}$$

(73)

for any $T\in [0,\infty ]$, where the constant C depends on the initial norm $\parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1-1}}^{\ell }$.

Proof. 

Denote $\mathcal{D}\left(T\right)\triangleq {\parallel (\nabla A,U)\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)\cap {\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}^{\ell }$. Perform a similar process leading to (54) and (55) to obtain

$$\begin{array}{c}\hfill \mathcal{D}\left(T\right)\le C\left(\parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1-1}}^{\ell }+{\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }\right).\end{array}$$

(74)

The next step is to estimate those nonlinear terms $g_1\sim g_6$. As a consequence of Corollary A1, the following Gevrey product inequalities will be frequently used in subsequent analysis:

$$\begin{array}{c}\hfill \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\left(fg\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}\lesssim {\parallel F\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}{\parallel G\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)};\end{array}$$

(75)

$$\begin{array}{c}\hfill \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\left(fg\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}\lesssim {\parallel F\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}\end{array}$$

(76)

for $2-\frac{d}{2}<{\sigma }_1<\frac{d}{2}$ with $d\ge 3$.

For $G_1$, we only need to consider the first term $e^{\sqrt{c_{0}t}{\Lambda }_1}\Delta \left(Q\left(a\right)m\right)$, since another term can proceed in a similar way. Precisely,

$$\begin{array}{c}\hfill \parallel G_1{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }\lesssim \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{(\nabla Q\left(a\right)m)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}+{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}(Q\left(a\right)\nabla m)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}\triangleq K_1+K_2.\end{array}$$

In light of (75), it follows that

$$\begin{array}{c}\hfill K_1\lesssim \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\nabla Q\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}{\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}.\end{array}$$

Keeping in mind that $-{\sigma }_1+2<\frac{d}{2}$, we employ Proposition A6 and (35) to give

$$\begin{array}{c}\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\nabla Q\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}\le C_{R_0}{\parallel A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+2}\right)}\hfill \\ \hfill \le C_{R_0}\left(\parallel A^{\ell }{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}+{\parallel A^h\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}\right)\le C_{R_0}(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)).\end{array}$$

On the other hand, it follows from the interpolation that

$$\begin{array}{c}\hfill {\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\lesssim \parallel U^{\ell }{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-2}\right)}+{\parallel U^h\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}}\right)}\lesssim \mathcal{G}\left(T\right).\end{array}$$

Hence, we deduce that

$$\begin{array}{c}\hfill K_1\le C_{R_0}\mathcal{G}\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\end{array}$$

(77)

Similarly,

$$\begin{array}{ccc}\hfill K_2& \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{Q\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}{\parallel \nabla U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}\hfill \\ \hfill & \le & C_{R_0}\left(\parallel A^h{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}+{\parallel A^{\ell }\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\right)\left(\parallel U^h{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}+{\parallel U^{\ell }\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}\right)\hfill \\ \hfill & \le & C_{R_0}\mathcal{G}\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\hfill \end{array}$$

(78)

Together (77) with (78), we arrive at

$$\begin{array}{c}\hfill \parallel G_1{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }\le C_{R_0}\mathcal{G}\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\end{array}$$

(79)

For $G_2$, it is enough to estimate

$$e^{\sqrt{c_{0}t}{\Lambda }_1}\mathrm{div}\left(\tilde{\mu }\left(a\right)D\left(Q\left(a\right)m\right)\right)$$

and other terms follow from similar steps. Keeping in mind that (76), one has

$$\begin{array}{c}\hfill \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\mathrm{div}\left(\tilde{\mu }\left(a\right)D\left(Q\left(a\right)m\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }\le C_{R_0}{\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}D\left(Q\left(a\right)m\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}.\end{array}$$

Thus, employing a similar procedure leading to (77) and (78) yields

$$\begin{array}{c}\hfill \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}\mathrm{div}\left(\tilde{\mu }\left(a\right)D\left(Q\left(a\right)m\right)\right){\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }\le C_{R_0}{\mathcal{G}}^2\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right)\end{array}$$

(80)

which implies that

$$\begin{array}{c}\hfill \parallel G_2{\left(t\right)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }\le C_{R_0}{\mathcal{G}}^2\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\end{array}$$

(81)

For $G_3$, with the help of (75) and (76), we have

$$\begin{array}{ccc}\hfill \parallel G_3{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }& \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{((Q\left(a\right)-1)m\otimes m)\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1}\right)}\hfill \\ \hfill & \le & C_{R_0}\left(1+{\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}\right){\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}{\parallel U\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_{R_0}\left(\mathcal{G}\left(T\right)+{\mathcal{G}}^2\left(T\right)\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\hfill \end{array}$$

(82)

It follows from (75) that

$$\begin{array}{ccc}\hfill \parallel G_4{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }& \lesssim & {\parallel A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}G\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_{R_0}(\parallel A^h{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,1}^{\frac{d}{2}+1}\right)}+\parallel A^{\ell }{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}\left)\right(\parallel A^h{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}+1}\right)}+\parallel A^{\ell }{\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)})\hfill \\ \hfill & \le & C_{R_0}\mathcal{G}\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\hfill \end{array}$$

(83)

Similarly, thanks to (75), we employ Proposition A6 once again to deduce that

$$\begin{array}{ccc}\hfill \parallel G_5{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }& \lesssim & \parallel e^{\sqrt{c_{0}t}{\Lambda }_1}{\tilde{\kappa }}_1{\left(a\right)\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}{\parallel \Delta A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_{R_0}\mathcal{G}\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\hfill \end{array}$$

(84)

Finally, the term $G_6$ can be similarly treated and, thus, we have

$$\begin{array}{ccc}\hfill \parallel G_6{\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }& \le & C_{R_0}{\parallel A\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{2,1}^{\frac{d}{2}}\right)}{\parallel \nabla A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{-{\sigma }_1+1}\right)}{\parallel \nabla A\parallel }_{{\tilde{L}}_T^2\left({\dot{B}}_{2,\infty }^{\frac{d}{2}-1}\right)}\hfill \\ \hfill & \le & C_{R_0}{\mathcal{G}}^2\left(T\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right).\hfill \end{array}$$

(85)

Combining with (79)–(85), we conclude that

$$\begin{array}{c}\hfill {\parallel G\parallel }_{{\tilde{L}}_T^1\left({\dot{B}}_{2,\infty }^{-{\sigma }_1-1}\right)}^{\ell }\le C_{R_0}\left(\mathcal{G}\left(T\right)+{\mathcal{G}}^2\left(T\right)\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right),\end{array}$$

(86)

furthermore, it follows from (74) that

$$\mathcal{D}\left(T\right)\le C\left(\parallel (\nabla a_0,m_0){\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1-1}}^{\ell }+C_{R_0}\left(\mathcal{G}\left(T\right)+{\mathcal{G}}^2\left(T\right)\right)\left(\mathcal{D}\left(T\right)+\mathcal{G}\left(T\right)\right)\right).$$

Theorem 1 enables us to find some constant $\tilde{\eta }(\tilde{\eta }\le \eta )$ sufficient small, such that if ${\mathcal{X}}_0\le \tilde{\eta }$, then

$$\mathcal{G}\left(T\right)\le C\tilde{\eta },C_{R_0}\left(\mathcal{G}\left(T\right)+{\mathcal{G}}^2\left(T\right)\right)\le \frac{1}{2C}.$$

Consequently, (73) is followed by the fact ${\parallel f\parallel }_{{\tilde{L}}_T^{\infty }\left({\dot{B}}_{p,\infty }^s\right)}\approx {\parallel f\parallel }_{L_T^{\infty }\left({\dot{B}}_{p,\infty }^s\right)}$. The proof of Proposition 3 is complete. □

5.2. Optimal Time-Decay Rates

The last subsection is to establish the optimal decay estimates of solutions. It follows from Sobolev embedding ${\dot{B}}_{2,1}^{\frac{d}{2}-\frac{d}{p}}\hookrightarrow L^p(p\ge 2)$ that

$$\begin{array}{c}\hfill \parallel {\Lambda }^l{(a,m)\parallel }_{L^p}\le \parallel {\Lambda }^{l+\frac{d}{2}-\frac{d}{p}}{(a,m)\parallel }_{{\dot{B}}_{2,1}^0}^{\ell }+{\parallel {\Lambda }^{l+\frac{d}{2}-\frac{d}{p}}(a,m)\parallel }_{{\dot{B}}_{2,1}^0}^h.\end{array}$$

(87)

We first show that solutions decay polynomially as fast as heat kernels in low frequencies. In fact, by taking $\sigma ={\sigma }_1,\zeta =l+\frac{d}{2}-\frac{d}{p}$ in Lemma 1 and Proposition 3, one can deduce that

$$\begin{array}{c}\hfill \parallel {\Lambda }^{l+\frac{d}{2}-\frac{d}{p}}{a\parallel }_{{\dot{B}}_{2,1}^0}^{\ell }\le C_{l,{\sigma }_1}{t}^{-\frac{{\tilde{\sigma }}_1+l}{2}}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}a\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1}}^{\ell }\le C{t}^{-\frac{{\tilde{\sigma }}_1+l}{2}}\end{array}$$

(88)

for $-{\tilde{\sigma }}_1<l$, where ${\tilde{\sigma }}_1={\sigma }_1-\frac{d}{p}+\frac{d}{2}$. Bounding the large time behavior of momentum m is totally similar:

$$\begin{array}{c}\hfill \parallel {\Lambda }^{l+\frac{d}{2}-\frac{d}{p}}{m\parallel }_{{\dot{B}}_{2,1}^0}^{\ell }\le C_{l,{\sigma }_1}{t}^{-\frac{{\tilde{\sigma }}_1+l}{2}-\frac{1}{2}}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}m\parallel }_{{\dot{B}}_{2,\infty }^{-{\sigma }_1-1}}^{\ell }\le C{t}^{-\frac{{\tilde{\sigma }}_1+l}{2}-\frac{1}{2}}\end{array}$$

(89)

for $-{\tilde{\sigma }}_1-1<l$.

On the other hand, it is shown that the decay of the high frequency of solutions is actually exponential at large times. Indeed, taking $\sigma =0$ in Lemma 1, we have

$$\begin{array}{c}\hfill \parallel {\Lambda }^{l+\frac{d}{2}-\frac{d}{p}}{a\parallel }_{{\dot{B}}_{2,1}^0}^h\le \parallel {\Lambda }^{\gamma }{a\parallel }_{{\dot{B}}_{2,1}^0}^h\le C_{\gamma }{t}^{-\frac{\gamma }{2}}{e}^{-a\sqrt{t}}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}a\parallel }_{{\dot{B}}_{2,\infty }^0}^h\end{array}$$

(90)

for $\gamma >max\{l+\frac{d}{2}-\frac{d}{p},0\}$. According to Theorem 1, we arrive at

$$\begin{array}{c}\hfill \parallel {\Lambda }^{l+\frac{d}{2}-\frac{d}{p}}{a\parallel }_{{\dot{B}}_{2,1}^0}^h\le C_{\gamma }{t}^{-\frac{\gamma }{2}}{e}^{-a\sqrt{t}}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}a\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^h\le C{t}^{-\frac{\gamma }{2}}{e}^{-a\sqrt{t}}.\end{array}$$

(91)

Similarly,

$$\begin{array}{c}\hfill \parallel {\Lambda }^{l+\frac{d}{2}-\frac{d}{p}}{m\parallel }_{{\dot{B}}_{2,1}^0}^h\le C_{\gamma }{t}^{-\frac{\gamma }{2}}{e}^{-a\sqrt{t}}{\parallel e^{\sqrt{c_{0}t}{\Lambda }_1}m\parallel }_{{\dot{B}}_{2,1}^{\frac{d}{2}-1}}^h\le C{t}^{-\frac{\gamma }{2}}{e}^{-a\sqrt{t}}.\end{array}$$

(92)

Finally, by combining (88)–(92), one can achieve (11) and (12) for $t\ge 1$ immediately by choosing a large enough $\gamma$. Hence, the proof of Theorem 2 is finished.

6. Conclusions

In this paper, we established the global-in-time existence and uniqueness of the compressible Navier–Stokes–Korteweg system for non-increasing pressures, compared to results presented in [10,18].

Moreover, we present the Gevrey analyticity in the critical space, which is new for the zero sound speed case. Based on the Gevrey analyticity, we also established a new Gevrey energy method, compared to the classical time-weight energy method, see [11,18], to obtain large time behaviors, which enables us to arrive at decay rates for any order derivatives without asking for ’smallness’ for the initial assumption.