1. Introduction

A fluid-mechanical theory for two-phase mixtures of fluids faces a well-known mathematical difficulty: the movement of the interfaces is naturally amenable to a Lagrangian description, while the bulk fluid flow is usually considered in the Eulerian framework [Reference Feireisl, Petzeltova, Rocca and Schimperna10]. Recently, the phase-field methods, or sometimes called the diffuse interface approaches, has been introduced to overcome this difficulty by postulating the existence of a ‘diffuse’ interface spread over a possibly narrow region covering the ‘real’ sharp interface boundary. Now, these models become one of the major tools to study a variety of interfacial phenomena. As the underlying physical problem still conceptually consists of sharp interfaces, the dynamics of the phase variable remains to a considerable extent purely fictitious [Reference Feireisl, Petzeltova, Rocca and Schimperna10]. Typically, different variants of Allen–Cahn, Cahn–Hilliard or other types of dynamics were used to describe the models, see previous studies [Reference Anderson, McFadden and Wheeler1, Reference Gurti, Polignone and Vinals14, Reference Liu, Shen and Yang20, Reference Lowengrub and Truskinovsky21] for example.

One of the well-known diffuse interface model for two-phase flow is the coupled Navier–Stokes/Allen–Cahn system. In this model, the interfaces between the phases are assumed to be of ‘diffuse’ nature, that is, sharp interfaces are replaced by narrow transition layers. These regions are located by a phase field variable $\chi$ governed by the Allen–Cahn equation, while the dynamics are described by the Navier–Stokes equations. In [Reference Blesgen2], Blesgen proposed this model and developed a thermodynamically and mechanically consistent set of partial differential equations extending the Navier–Stokes equations to a compressible binary Allen–Cahn mixture. As Blesgen [Reference Blesgen2] has pointed out before, the Navier–Stokes–Allen–Cahn model can be seen as a first step towards incorporating transport mechanism into the description of phase-formation processes.

During the past few years, many authors studied the properties of solutions for the incompressible Navier–Stokes system with matched density. For example, Favre and Schimperna [Reference Favre and Schimperna9] considered the existence of weak solution in three-dimensional (3D) and well-posedness of strong solution in two-dimensional (2D), Xu et al. [Reference Xu, Zhao and Liu29] studied the existence of axisymmetric solutions, Zhao et al. [Reference Zhao, Guo and Huang31] studied the vanishing viscosity limit, Gal and Grasselli [Reference Gal and Grasselli11, Reference Gal and Grasselli12] considered the asymptotic behaviour and attractors. For the system based on the incompressible Navier–Stokes system with different densities, Li et al. [Reference Li and Huang18, Reference Li, Ding and Huang19] studied the local well-posedness and blow-up criterion of strong solutions. Moreover, by using an energetic variational approach, Jiang et al. [Reference Jiang, Li and Liu15] derived a different model of Navier–Stokes–Allen–Cahn, then proved the existence of weak solutions in 3D, the well-posedness of strong solutions in 2D, and studied the long-time behaviour of the strong solutions.

As far as we know, there are also some classical results are available for the initial-boundary value problem of compressible Navier–Stokes–Allen–Cahn system. In [Reference Feireisl, Petzeltova, Rocca and Schimperna10], for the initial-boundary value problem of 3D compressible model, by using Faedo–Galerkin approximation, Feireisl et al. proved the existence of global-in-time weak solutions without any restriction on the size of initial data for the exponent of pressure $\gamma >6$. This result was extended to $\gamma >2$ in Chen et al. [Reference Chen, Wen and Zhu4]. Moreover, supposed that the boundary conditions of the model are of mixed type (Neumann–Dirichlet) and may be nonhomogeneous, the density is Hölder continuous for instance, Kotschote [Reference Kotschote17] established the local well-posedness of solutions for sufficiently smooth data, in a general $C^2$ bounded domain of $\mathbb {R}^n$ ($n\geq 1$). Ding et al. [Reference Ding, Li and Luo5, Reference Ding, Li and Tang6] proved the existence and uniqueness of global classical solution, the existence of weak solutions and the existence of unique global strong solution of the initial-boundary value problem of one-dimensional (1D) compressible Navier–Stokes–Allen–Cahn systems for the initial data without vacuum states. Recently, Zhu and his authors [Reference Luo, Yin and Zhu22, Reference Yin and Zhu30] paid their attention to the large time behaviour of solutions for the Cauchy problem and inflow problem of 1D compressible Navier–Stokes–Allen–Cahn equations, respectively. In [Reference Luo, Yin and Zhu22], Luo, Yin and Zhu proved that the solutions to the Cauchy problem of 1D compressible Navier–Stokes/Allen–Cahn system tend time-asymptotically to the rarefaction wave, where the strength of the rarefaction wave is not required to be small. In addition, for the inflow problem of 1D compressible Navier–Stokes/Allen–Cahn system, Yin and Zhu [Reference Yin and Zhu30] analysed the large-time behaviour of the solution, proved the existence of the stationary solution and the asymptotic stability of the nonlinear wave. However, to our knowledge, there's no result on the Cauchy problem for the 3D compressible Navier–Stokes–Allen–Cahn system.

In this paper, we consider the Cauchy problem of 3D compressible Navier–Stokes–Allen–Cahn equations [Reference Blesgen2, Reference Kotschote17, Reference Witterstein28]

(1.1)\begin{equation} \left\{\begin{array}{@{}l} \rho_t+\nabla\cdot(\rho u)=0,\\ \partial_t(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla p-\mu\Delta u-(\mu+\lambda)\nabla\text{div}\,u\\ \qquad\qquad\qquad={-}\ell\nabla\cdot\left(\nabla\chi\otimes\nabla\chi={-}\dfrac{|\nabla\chi|^2}2 \mathbb{I}_3\right),\\ \partial_t(\rho\chi)+\text{div}(\rho\chi u)={-}\omega, \\ \rho\omega={-}\ell\Delta\chi+\rho\dfrac{\partial\Phi(\rho,\chi)}{\partial\chi},\\ (\rho,u,\chi)|_{t=0}=(\rho_0,u_0,\chi_0), \end{array} \right. \end{equation}

where $\rho$ denotes the total fluid density, $u$ implies the mean velocity of the fluid mixture, $\chi$ is the concentration of one selected constituent and the pressure $p=p(\rho )$ is a smooth function in a neighbourhood of $1$ with $p'(1)=1$, respectively. Moreover, $\omega$ is the chemical potential and $\mathbb {I}_3$ denotes a $3\times 3$ identity matrix. $\mu$ and $\lambda$ are two viscosity coefficients, which satisfy

\[ \mu>0,\quad 2\mu+3\lambda\geq0. \]

The specific free energy $f(\rho ,\chi )$ can be defined as

(1.2)\begin{equation} \Phi(\rho,\chi)=\frac{\rho^{\gamma-1}}{\gamma-1}+\frac1{\ell}\left(\frac{\chi^4}4-\frac{\chi^2}2\right), \end{equation}

where $\gamma >1$ is the adiabatic constant and the constant $\sqrt {\ell }$ denotes the thickness of the interfacial region. In this paper, we take

\[ \rho\frac{\partial \Phi(\rho,\chi)}{\partial\chi}=\frac{\rho}{\ell}(\chi^3-\chi). \]

For simplicity, we let $\ell \equiv 1$ throughout the rest of this paper.

Notation 1.1 In the following, we use $H^k(\mathbb {R}^3$ $(k\in \mathbb {R})$ to denote the usual Sobolev spaces with norm $\|\cdot \|_{H^s}$ and $L^p(\mathbb {R}^3)$, $1\leq p\leq \infty$ to denote the usual $L^p$ spaces with norm $\|\cdot \|_{L^p}$. We also introduce the homogeneous negative index Sobolev space $\dot {H}^{-s}(\mathbb {R}^3)$:

\[ \dot{H}^{{-}s}(\mathbb{R}^3):=\{f\in L^2(\mathbb{R}^3):\||\xi|^{{-}s}\hat{f}(\xi)\|_{L^2}<\infty\} \]

endowed with the norm $\|f\|_{\dot {H}^{-s}}:=\||\xi |^{-s}\hat {f}(\xi )\|_{L^2}$. The symbol $\nabla ^l$ with an integer $l\geq 0$ stands for the usual spatial derivatives of order $l$. For instance, define

\[ \nabla^lz=\{\partial_x^{\alpha}z_i||\alpha|=l,i=1,2,3\},\quad z=(z_1,z_2,z_3). \]

If $l<0$ or $l$ is not a positive integer, $\nabla ^l$ stands for $\Lambda ^l$ defined by

(1.3)\begin{equation} \Lambda^sf(x)=\int_{\mathbb{R}^3}|\xi|^s\hat{f}(\xi)\,{\rm e}^{2\pi ix\cdot\xi}\,{\rm d}\xi, \end{equation}

where $\hat {f}$ is the Fourier transform of $f$ (see [Reference Grafakos13]). Moreover, we use the notation $A\lesssim B$ to mean that $A\leq cB$ for a universal constant $c>0$ that only depends on the parameters coming from the problem and the indexes $N$ and $s$ coming from the regularity on the data.

For this system, we first show the local well-posedness in the following lemma.

Lemma 1.1 Let the initial data$(\varrho _0,u_0, \chi _0)\in H^3(\mathbb {R}^3)\times H^3(\mathbb {R}^3)\times H^{4}(\mathbb {R}^3)$. Then there exit positive constants $\nu _0>0$ and $T>0$ depending only on $\|(\varrho _0,u_0, \chi _0,\nabla \chi _0)\|_{H^3}$ such that the system (1.4) has a unique solution

\[ (\varrho,u,\chi,\nabla\chi)\in L^{\infty}(0,T;H^3),\quad(u,\chi,\nabla\chi)\in L^2(0,T;H^4), \]

satisfying

\begin{align*} &\|(\varrho,u,\chi,\nabla\chi)(t)\|_{H^3}, \left(\nu_0\int_0^t\|\nabla(u,\chi,\nabla\chi)(\tau)\|_{H^3}^2\,{\rm d}\tau\right)^{1/2}\\ &\quad \leq C(\|(\varrho_0,u_0,\chi_0,\nabla\chi_0)\|_{H^3}),\quad\forall t\in[0,T]. \end{align*}

Lemma 1.1 can be proved by using the Banach fixed point theorem. One will sketch the proof in the second part of § 2.

Next, denote $\varrho =\rho -1$, rewrite (1.1) in the perturbation form as

(1.4)\begin{equation} \left\{\begin{array}{@{}l} \varrho_t+\text{div}\,u={-}\varrho\text{div}\,u-u\cdot\nabla\varrho,\\ u_t-\mu\Delta u-(\mu+\lambda)\nabla\text{div}\,u+\nabla\varrho\\ \quad ={-}u\cdot\nabla u-h(\varrho)(\mu\Delta u+(\mu+\lambda)\nabla\text{div}\,u)\\ \qquad\,-g(\varrho)\nabla\varrho-\phi(\varrho)\text{div}\left(\nabla\chi\otimes\nabla\chi-\dfrac{|\nabla\chi|^2}2 \mathbb{I}_3\right),\\ \chi_t-\Delta\chi ={-}u\cdot\nabla\chi-\varphi(\varrho)\Delta\chi-\phi(\varrho)(\chi^3-\chi),\\ (\varrho,u,\chi)|_{t=0}=(\varrho_0,u_0,\chi_0)=(\rho_0-1,u_0,\chi_0), \end{array} \right. \end{equation}

where

\[ h(\varrho)=\frac{\varrho}{\varrho+1},\quad g(\varrho)=\frac{p'(\varrho+1)}{\varrho+1}-1,\quad\phi(\varrho)=\frac1{\varrho+1},\quad \varphi(\varrho)=\frac{\varrho(\varrho+2)}{(\varrho+1)^2}, \]

The main purpose of this paper is to study the small initial data global well-posedness and optimal decay estimates of strong solutions for system (1.1) in the whole space $\mathbb {R}^3$. We use a general energy method, Kato–Ponce inequality and the Gagliardo–Nirenberg interpolation techniques to obtain the global well-posedness and the optimal time-decay rates of the solution to system (1.1) when the initial data are sufficiently small.

Next, one state the global well-posedness and decay estimates of solutions for system (1.4):

Theorem 1.1 Let $N\geq 3$, assume that $(\varrho _0,u_0, \chi _0)\in H^N(\mathbb {R}^3)\times H^N(\mathbb {R}^3)\times H^{N+1}(\mathbb {R}^3)$, and there exists a constant $\delta _0>0$ such that if

(1.5)\begin{equation} \|\varrho_0\|_{H^3}+\|u_0 \|_{H^3}+\| \chi_0\|_{H^3}+\| \nabla \chi_0\|_{H^3}\leq\delta_0, \end{equation}

then there exists a unique global solution $(\varrho ,u,\chi )$ satisfying that for all $t\geq 0$,

(1.6)\begin{align} &\|\varrho(t)\|_{H^N}^2+\|u(t)\|_{H^N}^2+\| \chi\|_{H^N}^2+\|\nabla \chi\|_{H^N}^2\nonumber\\ &\quad+\int_0^t(\|\nabla u(s)\|_{H^N}^2+\|\chi\|_{H^N}^2+\|\nabla \chi\|_{H^N}^2)\,{\rm d}s\nonumber\\ &\qquad\leq C(\|\varrho_0\|_{H^N}^2+\|u_0\|_{H^N}^2+\| \chi_0\|_{H^N}^2+\| \nabla\chi_0\|_{H^N}^2). \end{align}

If further, $(\varrho , u_0, \chi _0,\nabla \chi _0)\in \dot {H}^{-s}(\mathbb {R}^3)$ for some $s\in [0,\frac {3}{2})$, then for all $t\geq 0$,

(1.7)\begin{equation} \|\Lambda^{{-}s}\varrho(t)\|_{L^2}^2+\|\Lambda^{{-}s}u(t)\|_{L^2}^2+\|\Lambda^{{-}s} \chi(t)\|_{L^2}^2+\|\Lambda^{{-}s} \nabla\chi(t)\|_{L^2}^2\leq C, \end{equation}

and

(1.8)\begin{align} &\|\nabla^l\varrho(t)\|_{H^{N-l}}+\|\nabla^lu(t)\|_{H^{N-l}}+\|\nabla^l \chi(t)\|_{H^{N-l}}\nonumber\\ &\quad+ \|\nabla^{l+1} \chi(t)\|_{H^{N-l}}\leq C(1+t)^{-({l+1})/2},\quad\text{for}\ l=0,1,\ldots, N-1. \end{align}

Note that the Hardy–Littlewood-Sobolev theorem implies that for $p\in (1,2]$, $L^{p}(\mathbb {R}^3)\subset \dot {H}^{-s}(\mathbb {R}^3)$ with $s=3(\frac {1}{p}-\frac {1}{2})\in [0,\frac {3}{2})$. Then, on the basis of theorem 1.1, we easily obtain the following corollary of the optimal decay estimates.

Corollary 1.1 Under the assumptions of theorem 1.1, if we replace the $\dot {H}^{-s}(\mathbb {R}^3)$ assumption by

\[ (\varrho,u_0,\chi_0,\nabla\chi_0)\in L^{p}(\mathbb{R}^3),\quad 1< p\leq 2, \]

then the following decay estimate holds:

(1.9)\begin{align} &\|\nabla^l\varrho(t)\|_{H^{N-l}}+\|\nabla^lu(t)\|_{H^{N-l}}+\|\nabla^l \chi(t)\|_{H^{N-l}}+\|\nabla^{l+1} \chi(t)\|_{H^{N-l}}\nonumber\\ &\quad \leq C(1+t)^{-\left[{3}/{2}\left({1}/{p} -{1}/{2}\right)+{l}/{2}\right]},\quad\text{for}\ l=0,1,\ldots, N-1. \end{align}

Remark 1.1 The compressible Hall-magnetohydrodynamics system [Reference Fan, Alsaedi, Hayat, Nakamura and Zhou7] has the following form:

(1.10)\begin{equation} \left\{\begin{array}{@{}l} \rho_t+\text{div}(\rho u)=0,\\ (\rho u)_t+\text{div}(\rho u\otimes u)+\nabla p-\mu\Delta u-(\lambda+\mu)\nabla\text{div}\,u=(\nabla\times b)\times b,\\ b_t+\nabla\times(b\times u)+\nabla\times\left(\dfrac{(\nabla\times b)\times b}{\rho}\right)=\Delta b,\\ \text{div}\,b=0,\\ (\rho,u,b)|_{t=0}=(\rho_0,u_0,b_0), \end{array} \right. \end{equation}

where $\rho$ is the density of the fluid, $u$ is the fluid velocity field and $b$ is the magnetic field. In [Reference Fan, Alsaedi, Hayat, Nakamura and Zhou7], the authors considered the local well-posedness, small initial data global well-posedness and large time behaviour of strong solutions for system (1.10). It is worth pointing out that the structures of systems (1.1) and (1.10) are different. Especially, the principle part of (1.10)$_3$ is a linear term $-\Delta b$, however, it is a nonlinear function $-({\ell }/{\rho })\Delta \chi$ in (1.1)$_3$. One can't use the tools of the dissipative equation with linear principle part to study the properties of this equation. Here, we borrow a linear term from the right-hand side of (1.1)$_3$, rewrite this equation as a second-order PDE

(1.11)\begin{equation} \chi_t-\Delta\chi ={-}u\cdot\nabla\chi-\varphi(\varrho)\Delta\chi-\phi(\varrho)(\chi^3-\chi), \end{equation}

where $\varrho =\rho -1$, $\phi (\varrho )=1/{(\varrho +1)}$ and $\varphi (\varrho )=({\varrho (\varrho +2)})/({(\varrho +1)^2})$. It is worth pointing out that there exists a linear principle part $-\Delta \chi$ in (1.11). Then, by using the pure energy method, one can obtain suitable energy estimates. Moreover, the right-hand term of system (1.1)$_2$ is different from system (1.10)$_2$. One can use the usual Sobolev embedding inequality, Kato–Ponce inequality together with the Gagliardo–Nirenberg interpolation techniques to deal with this term.

The structure of this paper is organized as follows. In § 2, we introduce some preliminary results and give a brief proof on the local well-posedness. Section 3 is devoted to establish some refined energy estimates for the solution. In § 4, we derive the evolution of the negative Sobolev norms of the solution. Finally, the proof of theorem 1.1 is postponed in § 5.