Global well-posedness and L 2 decay estimate of smooth solutions for the 3-D incompressible Navier–Stokes–Allen

Abstract

We consider the Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.

Keywords

Navier–Stokes–Allen–Cahn global well-posedness time-decay Fourier-splitting

1. Introduction

In this paper, we study the three-dimensional global existence and time-decay estimate to the following incompressible Navier–Stokes–Allen–Cahn system: $\begin{matrix} (1.1) & \{\begin{array}{l} u_{t} + (u \cdot \nabla) u + \nabla p = μ Δ u - \nabla χ Δ χ, x \in R^{3}, t > 0, \\ div u = 0, \\ χ_{t} + u \cdot \nabla χ = - Σ, Σ = (χ^{3} - χ) - Δ χ, \end{array} \end{matrix}$ where $u = (u_{1}, u_{2}, u_{3})$ , $χ : R^{3} \to R$ , p are the velocity, concentration difference and pressure, respectively, and Σ denotes the chemical potential. We will consider the problem with the initial condition $\begin{matrix} (1.2) & (u, χ) (x, 0) = (u_{0}, χ_{0}) (x), x \in R^{3}, \end{matrix}$ and the far field behavior $\begin{matrix} (1.3) & u (x, 0) \to 0, | χ (x, 0) | \to 1, as | x | \to \infty . \end{matrix}$

For the Navier–Stokes equation, the well-posedness and large time asymptotic behavior of the solutions for compressible and incompressible models have been widely studied by many mathematicians ([25,26,30–33]). On this basis, it will form a new system by introducing Allen–Cahn equation to the Navier–Stokes type equations. Among them, the Navier–Stokes equation mainly describes the dynamic characteristics of the fluid, such as velocity; while the Allen–Cahn equation describes the interaction between two fluids near their interface, such as the change of concentration caused by diffusion.

In recent years, two-phase flows plays an important role in many industrial fields such as chemical engineering, bubble dynamics, microchannels, nuclear reactor, cavitation prediction, attracting the attention of many engineers and researchers both domestically and internationally in this field. We refer the readers to [1,3,11,15,21] and references therein.

It is well-known that the Navier–Stokes–Allen–Cahn system was first proposed by Blesgen [2]. For the incompressible Navier–Stokes–Allen–Cahn system with matched density, Xu–Zhao–Liu [34] proved the existence of axisymmetric solutions in $R^{3}$ by using variable separation and adding perturbation terms. Later on, in a bounded domain Ω $(Ω \subset R^{n}, n = 2, 3)$ , Zhao–Guo–Huang [36] obtained the existence of global weak solution in 2-D and 3-D and the well-posedness of global strong solution in 2-D. If the initial density $ρ_{0}$ has a positive lower bound, Li–Ding–Huang [22] established the blow-up criterion for strong solutions and Li–Huang [23] proved the existence and uniqueness of local strong solutions. Besides, Jiang–Li–Liu [17] showed the existence of global weak solutions in 3-D, the well-posedness of global strong solutions in 2-D, and investigated the long time behavior of the 2-D strong solutions. Huo–Teng [16] studied the existence and uniqueness of global smooth solution of (1.1) under the assumptions of small initial data in 2-D and investigated the large time behavior of smooth solution through using the Fourier-splitting method. Furthermore, in the theory of infinite dimensional dissipative dynamical systems, the authors [13,14,27] considered the existence of the trajectory attractor and analyzed the asymptotic behavior of the solutions.

Let us turn to the compressible Navier–Stokes–Allen–Cahn system. If the initial data $ρ_{0}$ without vacuum states, Ding–Li–Luo [10] obtained the existence of global weak solutions, the existence and uniqueness of global strong solution and global classical solution in 1-D bounded region by means of energetic variational method. Thereafter, when the initial vacuum is allowed, Chen–Guo [4] proved the existence and uniqueness of global strong solution and global classical solution. Ding–Li–Tang [9] studied the existence and uniqueness of global strong solutions with free boundary in 1-D. Moreover, the authors [12] constructed the existence of global weak solutions without any restriction on the size of initial data for the exponent of pressure $γ > 6$ in 3-D, this result was recently extended to $γ > 2$ by Chen–Wen–Zhu [5]. For the Cauchy problem and inflow problem, Luo–Yin–Zhu [24] proved that the solutions to the Cauchy problem tend time-asymptotically to the rarefaction wave in 1-D. Yin–Zhu [35] established the existence of the stationary solution and the asymptotic stability of the nonlinear wave. On the other hand, the authors [7,37] proved the existence and uniqueness of global smooth solution in 3-D and considered the long time behavior for higher-order spatial derivatives of the solution by means of energy estimates and negative Sobolev norm estimates.

Finally, we turn to the non-isentropic compressible Navier–Stokes–Allen–Cahn system. Based on the energy estimates method, Chen–He–Huang–Shi [6] studied the global existence and uniqueness of strong solutions in 1-D. Kotschote [19] certificated the existence and uniqueness of local strong solutions for arbitrary initial data in $Ω \in R^{n}$ , $n ⩾ 1$ . Also, Kotschote [20] investigated the stability of travelling wave solutions to the Navier–Stokes–Allen–Cahn system by linearizing the system around a traveling waves. Recently, Chen–Li–Tang [8] established the global well-posedness of the Cauchy problem and obtained the optimal time-decay rate of the density, velocity and the temperature.

Motivated by the results mentioned above, the main purpose of this paper is to study the existence and uniqueness of global smooth solution for the system (1.1)–(1.3) under the assumptions of small initial data. Furthermore, we establish the time decay-rates of smooth solution to the Cauchy problem through using Fourier-splitting method which has been developed by Schonbek [30,31].

Then we can state our main theorems as follows.

Theorem 1.1.
Assume that $\begin{matrix} u_{0} \in H^{s} (R^{3}), \nabla χ_{0} \in H^{s} (R^{3}), χ_{0}^{2} - 1 \in H^{s} (R^{3}), \end{matrix}$ for an integer $s ⩾ 3$ , $div u_{0} = 0$ and there exists a positive constant $η_{0}$ such that $\begin{matrix} (1.4) & ‖ u_{0} ‖_{H^{2}} + ‖ \nabla χ_{0} ‖_{H^{2}} + {‖ χ_{0}^{2} - 1 ‖}_{L^{2}}^{2} ⩽ η_{0} . \end{matrix}$ Then the Cauchy problem ( 1.1 )–( 1.3 ) admits a unique solution $(u, χ)$ on $[0, \infty)$ satisfying $\begin{aligned} u \in L^{\infty} ([0, \infty); H^{s} (R^{3})), \nabla χ \in L^{\infty} ([0, \infty); H^{s} (R^{3})), \\ χ^{2} - 1 \in L^{\infty} ([0, \infty); H^{s} (R^{3})), \nabla u \in L^{2} ([0, \infty); H^{s} (R^{3})), \\ \nabla χ \in L^{2} ([0, \infty); H^{s + 1} (R^{3})), χ^{2} - 1 \in L^{2} ([0, \infty); H^{s + 1} (R^{3})), \end{aligned}$ and $\begin{aligned} ‖ u ‖_{H^{s}}^{2} + ‖ \nabla χ ‖_{H^{s}}^{2} + {‖ χ^{2} - 1 ‖}_{H^{s}}^{2} + \int_{0}^{t} ‖ \nabla u ‖_{H^{s}}^{2} d s + \int_{0}^{t} ‖ \nabla χ ‖_{H^{s + 1}}^{2} d s \\ + \int_{0}^{t} {‖ χ^{2} - 1 ‖}_{H^{s + 1}}^{2} d s ⩽ C (‖ u_{0} ‖_{H^{s}}^{2} + ‖ \nabla χ_{0} ‖_{H^{s}}^{2} + {‖ χ_{0}^{2} - 1 ‖}_{H^{s}}^{2}), \end{aligned}$ for any $t > 0$ , where C is a positive constant independent of u and χ.
Theorem 1.2.
Let $u_{0}, \nabla χ_{0}, χ_{0}^{2} - 1 \in L^{1} (R^{3}) \cap L^{2} (R^{3})$ , $\nabla^{k} (u_{0}, \nabla χ_{0}, χ_{0}^{2} - 1) \in L^{2} (R^{3}) (k ⩾ 0)$ . Then the solution $(u, χ)$ of problem ( 1.1 )–( 1.3 ) satisfies $\begin{matrix} {‖ \nabla^{k} (u, \nabla χ, χ^{2} - 1) ‖}_{L^{2}}^{2} ⩽ C {(t + 1)}^{- k - \frac{3}{2}}, \end{matrix}$ where the constant C depends only on $‖ u_{0} ‖_{L^{1}}$ , $‖ \nabla χ_{0} ‖_{L^{1}}$ and $‖ χ_{0}^{2} - 1 ‖_{L^{1}}$ .
Remark 1.3.
The following comments are about Theorems 1.1 and 1.2.

For the global existence of the smooth solution, we only assume that the $H^{2}$ -norm of initial date is small, while the higher-order Sobolev norms can be arbitrarily large. Particularly, our assumptions are weaker than [7,37].

Compared with [7,37], we use another simpler method to estimate the $L^{2}$ decay rates of smooth solution for 3-D incompressible Navier–Stokes–Allen–Cahn model. Furthermore, the decay rate of global solution is optimal, and our results supplement the situation of $p = 1$ in the corollary of [7,37].

Note that here we obtain the $L^{2}$ optimal rate of the smooth solution. Then the general optimal $L^{q} (q > 2)$ decay rates of the solution can be obtained by Gagliardo–Nirenberg inequality. That is, $\begin{matrix} {‖ \nabla^{k} (u, \nabla χ, χ^{2} - 1) ‖}_{L^{q}}^{2} ⩽ C {(t + 1)}^{- k - 3 (1 - \frac{1}{q})} . \end{matrix}$

When $χ = 1$ , the large time behavior of the solution to the incompressible Navier–Stokes–Allen–Cahn system is consistent with the classical incompressible Navier–Stokes system. In particular, Schonbek [31] pointed out that if $u_{0} \in L^{2} (R^{3}) \cap L^{p} (R^{3})$ , $1 ⩽ p < 2$ , then $\begin{matrix} ‖ u ‖_{L^{2} (R^{3})}^{2} ⩽ C {(t + 1)}^{- (\frac{3}{p} - \frac{3}{2})} . \end{matrix}$ Moreover, Schonbek [32] established the decay rate of the homogeneous $H^{m} (m ⩾ 3)$ norms for solutions to the Navier–Stokes equations in two dimensions, that is $\begin{matrix} {‖ \nabla^{α} u ‖}_{L^{2} (R^{2})}^{2} ⩽ C {(t + 1)}^{- (| α | + 1)}, \end{matrix}$ where $u_{0} \in H^{m} (R^{2}) \cap L^{1} (R^{2})$ , $| α | ⩽ m$ .

For the 3-D incompressible Navier–Stokes system, taking $\nabla^{k}$ to the equation, multiplying the resulting equation by $\nabla^{k} u$ and integrating over $R^{3}$ (by part), we have $\begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ \nabla^{k} u ‖}_{L^{2}}^{2} + μ {‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2} = - \int_{R^{3}} \nabla^{k} [(u \cdot \nabla) u] \cdot \nabla^{k} u d x . \end{matrix}$ Based on the estimation of $I_{6}$ and the decay results in [31], utilizing the same method as in Section 5, we can obtain the same decay rate as in Theorem 1.2. At this point, it is only necessary to assume that the initial data $‖ u_{0} ‖_{H^{1}}$ is small enough.

The notations applied in this article are as follows.

$L^{p} (R^{3})$ denotes the usual Lebesgue spaces on $R^{3}$ , with norms $\begin{matrix} ‖ u ‖_{p} : = {(\int_{R^{3}} | u |^{p} d x)}^{\frac{1}{p}} . \end{matrix}$

$W_{p}^{k} (R^{3}) = {u \in L^{p} (R^{3}) : \nabla^{l} u \in L^{p} (R^{3}) (1 ⩽ l ⩽ k)}$ denotes the usual Sobolev spaces on $R^{3}$ , with norms $\begin{matrix} ‖ u ‖_{W_{p}^{k}}^{p} : = \int_{R^{3}} (| u |^{p} + | \nabla u |^{p} + \dots + {| \nabla^{k} u |}^{p}) d x . \end{matrix}$

$H^{k} (R^{3}) = W_{2}^{k} (R^{3})$ with the norms $\begin{matrix} ‖ u ‖_{H^{k}}^{2} : = \int_{R^{3}} (| u |^{2} + | \nabla u |^{2} + \dots + {| \nabla^{k} u |}^{2}) d x . \end{matrix}$

$\nabla^{m}$ denotes the summation of all terms $\nabla^{α}$ with the multi-index α satisfying $| α | = m$ .

$S (t) = {x \in R^{3}; | x | < r (t)}$ , where $r (t) = {[\frac{3}{2 a (t + 1)}]}^{\frac{1}{2}}$ and $a = min (μ, 1)$ .

The rest of this paper is organized as follows. Section 2 is devoted to obtaining some energy estimates that play an important role for establishing the global existence of solution. Section 3 combines all the estimates that we derived in Section 2 to prove Theorem 1.1. Section 4 is dedicated to utilizing Fourier transform and studying its fundamental properties regarding the upper bound of decay. Section 5 mainly focuses on investigating time-decay rate of velocity and concentration difference by means of Fourier-splitting method.
2. A priori estimate

In this section, we assume a priori that for sufficiently small positive constant $η_{1}$ satisfying $\begin{matrix} (2.1) & \sqrt{φ_{0}^{2}} : = ‖ u ‖_{H^{2}} + ‖ \nabla χ ‖_{H^{2}} + {‖ χ^{2} - 1 ‖}_{L^{2}}^{2} ⩽ \frac{η_{1}}{C}, \end{matrix}$ where $C ⩾ 1$ is a fixed constant.

Lemma 2.1 (Kato–Ponce inequality [18]).

Let $s > 0$ and $r \in (1, \infty)$ . Then $\begin{matrix} {‖ \nabla^{s} (f g) ‖}_{L^{r}} ⩽ C (‖ g ‖_{p_{1}} {‖ \nabla^{s} f ‖}_{q_{1}} + {‖ \nabla^{s} g ‖}_{p_{2}} ‖ f ‖_{q_{2}}), \end{matrix}$ with $q_{1}, p_{2} \in (1, \infty)$ and $p_{1}, q_{2} \in [1, \infty]$ such that $\begin{matrix} \frac{1}{r} = \frac{1}{p_{1}} + \frac{1}{q_{1}} = \frac{1}{p_{2}} + \frac{1}{q_{2}} . \end{matrix}$

Lemma 2.2 (Gagliardo–Nirenberg inequality [28]).

Let l, s and k be any real numbers satisfying $0 ⩽ l$ , $s < k$ , and let $p, r, q \in [1, \infty]$ and $0 ⩽ θ ⩽ 1$ such that $\begin{matrix} \frac{l}{3} - \frac{1}{p} = (\frac{s}{3} - \frac{1}{r}) (1 - θ) + (\frac{k}{3} - \frac{1}{q}) θ . \end{matrix}$ Then, for any $u \in W_{q}^{k} (R^{3})$ , we have $\begin{matrix} {‖ \nabla^{l} u ‖}_{L^{p}} ⩽ C {‖ \nabla^{s} u ‖}_{L^{r}}^{1 - θ} {‖ \nabla^{k} u ‖}_{L^{q}}^{θ} . \end{matrix}$ In particular, we require that $0 < θ < 1$ when $p = \infty$ .

Lemma 2.3 ([29]).

Let $f (φ)$ be smooth function of φ, with bounded derivatives of any order, and $‖ φ ‖_{L^{\infty}} ⩽ C$ . Then for any integer $m ⩾ 1$ , we have $\begin{matrix} {‖ \nabla^{m} f (φ) ‖}_{L^{p}} ⩽ C {‖ \nabla^{m} φ ‖}_{L^{p}}, \end{matrix}$ for any $1 ⩽ p ⩽ \infty$ , where C may depend on f and m.

Lemma 2.4.
Assume the condition ( 2.1 ) holds, then $\begin{matrix} (2.2) & \begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{R^{3}} (| u |^{2} + | \nabla χ |^{2} + {| χ^{2} - 1 |}^{2}) d x + μ ‖ \nabla u ‖_{L^{2}}^{2} \\ + ‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2} + \frac{3}{2} {‖ χ^{2} - 1 ‖}_{L^{2}}^{2} + {‖ \nabla (χ^{2} - 1) ‖}_{L^{2}}^{2} ⩽ 0 . \end{aligned} \end{matrix}$
Proof.
Under the incompressible condition $div u = 0$ , it holds that $\begin{matrix} \int_{R^{3}} (u \cdot \nabla) u \cdot u d x = 0, \forall t > 0 . \end{matrix}$ By multiplying ${(1.1)}_{1}$ by u and integrating over $R^{3}$ (by part), we obtain $\begin{matrix} (2.3) & \frac{1}{2} \frac{d}{d t} \int_{R^{3}} | u |^{2} d x + μ ‖ \nabla u ‖_{L^{2}}^{2} + \int_{R^{3}} \nabla χ Δ χ \cdot u d x = 0 . \end{matrix}$

We can rewrite ${(1.1)}_{3}$ and ${(1.1)}_{4}$ as $\begin{matrix} (2.4) & χ_{t} + u \cdot \nabla χ + (χ^{3} - 3 χ) + 2 χ - Δ χ = 0 . \end{matrix}$ By multiplying (2.4) by $- Δ χ$ , we get $\begin{matrix} (2.5) & \begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{R^{3}} | \nabla χ |^{2} d x + 2 ‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2} - \int_{R^{3}} \nabla χ Δ χ \cdot u d x \\ = - 3 \int_{R^{3}} (χ^{2} - 1) | \nabla χ |^{2} d x = I_{1} . \end{aligned} \end{matrix}$ By Hölder’s inequality, Gagliardo–Nirenberg’s inequality, Lemma 2.3 and (2.1), $I_{1}$ is estimated as follows $\begin{aligned} | I_{1} | & ⩽ C {‖ χ^{2} - 1 ‖}_{L^{\infty}} ‖ \nabla χ ‖_{L^{2}}^{2} \\ ⩽ C {‖ \nabla (χ^{2} - 1) ‖}_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} (χ^{2} - 1) ‖}_{L^{2}}^{\frac{1}{2}} ‖ \nabla χ ‖_{L^{2}}^{2} \\ ⩽ C ‖ \nabla χ ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{2} χ ‖}_{L^{2}}^{\frac{1}{2}} ‖ \nabla χ ‖_{L^{2}}^{2} \\ ⩽ C ‖ \nabla χ ‖_{H^{1}} ‖ \nabla χ ‖_{L^{2}}^{2} ⩽ η_{1} ‖ \nabla χ ‖_{L^{2}}^{2} . \end{aligned}$

We can rewrite (2.4) as $\begin{matrix} (2.6) & \begin{aligned} {(χ^{2} - 1)}_{t} + u \cdot \nabla (χ^{2} - 1) - Δ (χ^{2} - 1) + 2 | \nabla χ |^{2} + 2 (χ^{2} - 1) χ^{2} \\ = {(χ^{2} - 1)}_{t} + u \cdot \nabla (χ^{2} - 1) - Δ (χ^{2} - 1) + 2 | \nabla χ |^{2} \\ + 2 (χ^{2} - 1) (χ^{2} - 1) + 2 (χ^{2} - 1) = 0 . \end{aligned} \end{matrix}$ By multiplying (2.6) by $χ^{2} - 1$ and integrating over $R^{3}$ (by part), we have $\begin{matrix} (2.7) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ (χ^{2} - 1) ‖}_{L^{2}}^{2} + 2 {‖ χ^{2} - 1 ‖}_{L^{2}}^{2} + {‖ \nabla (χ^{2} - 1) ‖}_{L^{2}}^{2} \\ = - 2 \int_{R^{3}} (χ^{2} - 1) (| \nabla χ |^{2} + {(χ^{2} - 1)}^{2}) d x = I_{2} . \end{aligned} \end{matrix}$ $I_{2}$ is estimated as follows $\begin{aligned} | I_{2} | & ⩽ C {‖ χ^{2} - 1 ‖}_{L^{\infty}} (‖ \nabla χ ‖_{L^{2}}^{2} + {‖ χ^{2} - 1 ‖}_{L^{2}}^{2}) \\ ⩽ C ‖ \nabla χ ‖_{H^{1}} (‖ \nabla χ ‖_{L^{2}}^{2} + {‖ χ^{2} - 1 ‖}_{L^{2}}^{2}) \\ ⩽ η_{1} (‖ \nabla χ ‖_{L^{2}}^{2} + {‖ χ^{2} - 1 ‖}_{L^{2}}^{2}) . \end{aligned}$ Let $η_{1} ⩽ \frac{1}{2}$ , combining (2.3), (2.5) and (2.7), thus (2.2) follows. □
Lemma 2.5.
Assume the condition ( 2.1 ) holds, then $\begin{matrix} (2.8) & \begin{aligned} \frac{1}{2} \frac{d}{d t} ({‖ \nabla^{k} u ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}}^{2}) + \frac{μ}{2} {‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2} \\ + \frac{1}{2} ({‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 2} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}}^{2}) ⩽ 0, \end{aligned} \end{matrix}$ for $k = 1, \dots, s$ ( $s ⩾ 2$ ).
Proof.
Applying ∇ to (2.4), we obtain $\begin{matrix} (2.9) & \nabla χ_{t} + \nabla (u \cdot \nabla χ) + 3 (χ^{2} - 1) \nabla χ + 2 \nabla χ - \nabla Δ χ = 0 . \end{matrix}$ Applying $\nabla^{k}$ to (2.9), multiplying the resulting equation by $\nabla^{k + 1} χ$ and integrating over $R^{3}$ (by part), we have $\begin{matrix} (2.10) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + 2 {‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 2} χ ‖}_{L^{2}}^{2} \\ = \int_{R^{3}} \nabla^{k} (u \cdot \nabla χ) Δ \nabla^{k} χ d x - 3 \int_{R^{3}} \nabla^{k} [(χ^{2} - 1) \nabla χ] \nabla^{k + 1} χ d x = I_{3} + I_{4} . \end{aligned} \end{matrix}$

Applying $\nabla^{k}$ to ${(1.1)}_{1}$ , multiplying the resulting equation by $\nabla^{k} u$ and integrating over $R^{3}$ (by part), we get $\begin{matrix} (2.11) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \nabla^{k} u ‖}_{L^{2}}^{2} + μ {‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2} \\ = - \int_{R^{3}} \nabla^{k} (\nabla χ Δ χ) \cdot \nabla^{k} u d x - \int_{R^{3}} \nabla^{k} [(u \cdot \nabla) u] \cdot \nabla^{k} u d x = I_{5} + I_{6} . \end{aligned} \end{matrix}$

Applying $\nabla^{k}$ to (2.6), multiplying the resulting equation by $\nabla^{k} (χ^{2} - 1)$ and integrating over $R^{3}$ (by part), we obtain $\begin{matrix} (2.12) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}}^{2} + 2 {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}}^{2} \\ = - \int_{R^{3}} \nabla^{k} [u \cdot \nabla (χ^{2} - 1)] \nabla^{k} (χ^{2} - 1) d x - 2 \int_{R^{3}} \nabla^{k} (| \nabla χ |^{2}) \nabla^{k} (χ^{2} - 1) d x \\ - 2 \int_{R^{3}} \nabla^{k} [(χ^{2} - 1) (χ^{2} - 1)] \nabla^{k} (χ^{2} - 1) d x = \sum_{i = 7}^{9} I_{i} . \end{aligned} \end{matrix}$

From using Hölder’s inequality, Gagliardo–Nirenberg’s inequality, Young’s inequality, Kato–Ponce’s inequality and (2.1), the term $I_{i} (3 ⩽ i ⩽ 9)$ are estimated as follows $\begin{array}{c} \begin{aligned} | I_{3} | & ⩽ C {‖ \nabla^{k + 2} χ ‖}_{L^{2}} {‖ \nabla^{k} (u \cdot \nabla χ) ‖}_{L^{2}} \\ ⩽ C {‖ \nabla^{k + 2} χ ‖}_{L^{2}} ({‖ \nabla^{k} u ‖}_{L^{6}} ‖ \nabla χ ‖_{L^{3}} + ‖ u ‖_{L^{\infty}} {‖ \nabla^{k + 1} χ ‖}_{L^{2}}) \\ ⩽ C {‖ \nabla^{k + 2} χ ‖}_{L^{2}} ({‖ \nabla^{k + 1} u ‖}_{L^{2}} ‖ \nabla χ ‖_{L^{3}} + ‖ u ‖_{H^{2}} {‖ \nabla^{k + 1} χ ‖}_{L^{2}}) \\ ⩽ η_{1} ({‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 2} χ ‖}_{L^{2}}^{2}), \end{aligned} \\ \begin{aligned} | I_{4} | & ⩽ C {‖ \nabla^{k + 1} χ ‖}_{L^{2}} {‖ \nabla^{k} [(χ^{2} - 1) \nabla χ] ‖}_{L^{2}} \\ ⩽ C {‖ \nabla^{k + 1} χ ‖}_{L^{2}} ({‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{6}} ‖ \nabla χ ‖_{L^{3}} + {‖ χ^{2} - 1 ‖}_{L^{\infty}} {‖ \nabla^{k + 1} χ ‖}_{L^{2}}) \\ ⩽ C {‖ \nabla^{k + 1} χ ‖}_{L^{2}} ({‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}} ‖ \nabla χ ‖_{L^{3}} + ‖ \nabla χ ‖_{H^{1}} {‖ \nabla^{k + 1} χ ‖}_{L^{2}}) \\ ⩽ 2 η_{1} ({‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}}^{2}), \end{aligned} \\ \begin{aligned} | I_{5} | & ⩽ C {‖ \nabla^{k} u ‖}_{L^{6}} {‖ \nabla^{k} (\nabla χ Δ χ) ‖}_{L^{6 / 5}} \\ ⩽ C {‖ \nabla^{k + 1} u ‖}_{L^{2}} ({‖ \nabla^{k + 1} χ ‖}_{L^{2}} ‖ Δ χ ‖_{L^{3}} + ‖ \nabla χ ‖_{L^{3}} {‖ \nabla^{k + 2} χ ‖}_{L^{2}}) \\ ⩽ η_{1} ({‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 2} χ ‖}_{L^{2}}^{2}), \end{aligned} \\ \begin{aligned} | I_{6} | & ⩽ C {‖ \nabla^{k} u ‖}_{L^{6}} {‖ \nabla^{k} [(u \cdot \nabla) u] ‖}_{L^{6 / 5}} \\ ⩽ C {‖ \nabla^{k + 1} u ‖}_{L^{2}} ({‖ \nabla^{k} u ‖}_{L^{2}} ‖ \nabla u ‖_{L^{3}} + ‖ u ‖_{L^{3}} {‖ \nabla^{k + 1} u ‖}_{L^{2}}) \\ ⩽ C {‖ \nabla^{k + 1} u ‖}_{L^{2}} (‖ u ‖_{L^{2}}^{\frac{1}{k + 1}} {‖ \nabla^{k + 1} u ‖}_{L^{2}}^{\frac{k}{k + 1}} {‖ \nabla^{α} u ‖}_{L^{2}}^{\frac{k}{k + 1}} {‖ \nabla^{k + 1} u ‖}_{L^{2}}^{\frac{1}{k + 1}} + ‖ u ‖_{L^{3}} {‖ \nabla^{k + 1} u ‖}_{L^{2}}) \\ ⩽ 2 η_{1} {‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2}, \end{aligned} \end{array}$ where α is defined by $\begin{aligned} 0 = (\frac{α}{3} - \frac{1}{2}) \frac{k}{k + 1} + (\frac{k + 1}{3} - \frac{1}{2}) \frac{1}{k + 1} \Rightarrow α = \frac{k + 1}{2 k} \in (\frac{1}{2}, 1] . \\ \begin{aligned} | I_{7} | & ⩽ C {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} {‖ \nabla^{k} [u \cdot \nabla (χ^{2} - 1)] ‖}_{L^{2}} \\ ⩽ C {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} ({‖ \nabla^{k} u ‖}_{L^{6}} {‖ \nabla (χ^{2} - 1) ‖}_{L^{3}} + ‖ u ‖_{L^{\infty}} {‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}}) \\ ⩽ C {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} ({‖ \nabla^{k + 1} u ‖}_{L^{2}} ‖ \nabla χ ‖_{L^{3}} + ‖ u ‖_{H^{2}} {‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}}) \\ ⩽ η_{1} ({‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2} + {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}}^{2}), \end{aligned} \\ \begin{aligned} | I_{8} + I_{9} | & ⩽ C {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} ({‖ \nabla^{k} | \nabla χ |^{2} ‖}_{L^{2}} + {‖ \nabla^{k} [(χ^{2} - 1) (χ^{2} - 1)] ‖}_{L^{2}}) \\ ⩽ C {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} ({‖ \nabla^{k + 1} χ ‖}_{L^{2}} ‖ \nabla χ ‖_{L^{\infty}} + {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} {‖ χ^{2} - 1 ‖}_{L^{\infty}}) \\ ⩽ C {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} ({‖ \nabla^{k + 1} χ ‖}_{L^{2}} ‖ \nabla χ ‖_{H^{2}} + {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}} ‖ \nabla χ ‖_{H^{1}}) \\ ⩽ 2 η_{1} ({‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}}^{2}) . \end{aligned} \end{aligned}$ Let $η_{1} ⩽ min (\frac{1}{10}, \frac{μ}{10})$ , combining (2.10), (2.11) and (2.12), the estimate (2.8) follows. Thus, the proof is completed. □

3. Proof of Theorem 1.1

In this section, we can state the proof of Theorem 1.1 through the preparation in the previous section.

Step 1. We first prove the existence of local solution.

Summing up the estimate (2.8) of Lemma 2.5 from $k = 1$ to s, and combining the resulting equation and (2.2), we obtain $\begin{matrix} (3.1) & \begin{array}{r} \frac{d}{d t} Ψ^{s} (t) + Γ^{s} (t) ⩽ 0, \end{array} \end{matrix}$ where $\begin{array}{c} Ψ^{s} (t) ≃ ‖ u ‖_{H^{s}}^{2} + ‖ \nabla χ ‖_{H^{s}}^{2} + {‖ χ^{2} - 1 ‖}_{H^{s}}^{2}, \\ Γ^{s} (t) ≃ ‖ \nabla u ‖_{H^{s}}^{2} + ‖ \nabla χ ‖_{H^{s + 1}}^{2} + {‖ χ^{2} - 1 ‖}_{H^{s + 1}}^{2} . \end{array}$

Now taking $s = 2$ in (3.1), we get $\begin{aligned} ‖ u ‖_{H^{2}}^{2} + ‖ \nabla χ ‖_{H^{2}}^{2} + {‖ χ^{2} - 1 ‖}_{L^{2}}^{2} \\ ⩽ ‖ u ‖_{H^{2}}^{2} + ‖ \nabla χ ‖_{H^{2}}^{2} + {‖ χ^{2} - 1 ‖}_{H^{2}}^{2} \\ ⩽ C (‖ u_{0} ‖_{H^{2}}^{2} + ‖ \nabla χ_{0} ‖_{H^{2}}^{2} + {‖ χ_{0}^{2} - 1 ‖}_{H^{2}}^{2}) \\ ⩽ C (‖ u_{0} ‖_{H^{2}}^{2} + ‖ \nabla χ_{0} ‖_{H^{2}}^{2} + {‖ χ_{0}^{2} - 1 ‖}_{L^{2}}^{2} + {‖ \nabla (χ_{0}^{2} - 1) ‖}_{H^{1}}^{2}) \\ ⩽ C (‖ u_{0} ‖_{H^{2}}^{2} + ‖ \nabla χ_{0} ‖_{H^{2}}^{2} + {‖ χ_{0}^{2} - 1 ‖}_{L^{2}}^{2} + ‖ \nabla χ_{0} ‖_{H^{1}}^{2}) \\ ⩽ C (‖ u_{0} ‖_{H^{2}}^{2} + ‖ \nabla χ_{0} ‖_{H^{2}}^{2} + {‖ χ_{0}^{2} - 1 ‖}_{L^{2}}^{2}), \end{aligned}$ which yields the priori estimate (2.1). Moreover, we get that for any $t ⩾ 0$ , there holds $\begin{aligned} ‖ u ‖_{H^{s}}^{2} + ‖ \nabla χ ‖_{H^{s}}^{2} + {‖ χ^{2} - 1 ‖}_{H^{s}}^{2} \\ + \int_{0}^{t} ‖ \nabla u ‖_{H^{s}}^{2} d s + \int_{0}^{t} ‖ \nabla χ ‖_{H^{s + 1}}^{2} d s + \int_{0}^{t} {‖ χ^{2} - 1 ‖}_{H^{s + 1}}^{2} d s \\ ⩽ C (‖ u_{0} ‖_{H^{s}}^{2} + ‖ \nabla χ_{0} ‖_{H^{s}}^{2} + {‖ χ_{0}^{2} - 1 ‖}_{H^{s}}^{2}) . \end{aligned}$

Hence, the local existence result of smooth solutions can be established and the solutions satisfies the following: $\begin{matrix} (3.2) & \begin{aligned} u \in L^{\infty} ([0, T]; H^{s} (R^{3})), \nabla χ \in L^{\infty} ([0, T]; H^{s} (R^{3})), \\ χ^{2} - 1 \in L^{\infty} ([0, T]; H^{s} (R^{3})), \nabla u \in L^{2} ([0, T]; H^{s} (R^{3})), \\ \nabla χ \in L^{2} ([0, T]; H^{s + 1} (R^{3})), χ^{2} - 1 \in L^{2} ([0, T]; H^{s + 1} (R^{3})) . \end{aligned} \end{matrix}$

Step 2. Next, we prove the uniqueness of the local smooth solution.

Let $(u_{1}, p_{1}, χ_{1})$ and $(u_{2}, p_{2}, χ_{2})$ be two solutions of (1.1), and introduce $(\bar{u}, \bar{p}, \bar{χ})$ , where $\bar{u} = u_{1} - u_{2}$ , $\bar{p} = p_{1} - p_{2}$ , $\bar{χ} = χ_{1} - χ_{2}$ . Then $\begin{matrix} (3.3) & \{\begin{array}{l} {\bar{u}}_{t} + (u_{1} \cdot \nabla) \bar{u} + (\bar{u} \cdot \nabla) u_{2} + \nabla \bar{p} - μ Δ \bar{u} = - \nabla χ_{1} Δ \bar{χ} - \nabla \bar{χ} Δ χ_{2}, \\ div \bar{u} = 0, \\ {\bar{χ}}_{t} + u_{1} \cdot \nabla \bar{χ} + \bar{u} \cdot \nabla χ_{2} + (χ_{1}^{2} - 1) \bar{χ} + (χ_{1}^{2} - χ_{2}^{2}) χ_{2} - Δ \bar{χ} = 0, \\ \bar{u} |_{t = 0} = 0, \bar{χ} |_{t = 0} = 0 . \end{array} \end{matrix}$ By multiplying ${(3.3)}_{1}$ by $\bar{u}$ , multiplying ${(3.3)}_{3}$ by $\bar{χ}$ , and integrating over $R^{3}$ , we get $\begin{matrix} (3.4) & \begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ \bar{u} ‖_{L^{2}}^{2} + ‖ \bar{χ} ‖_{L^{2}}^{2}) + μ ‖ \nabla \bar{u} ‖_{L^{2}}^{2} + ‖ \nabla \bar{χ} ‖_{L^{2}}^{2} \\ = - ((\bar{u} \cdot \nabla) u_{2}, \bar{u}) - (\nabla χ_{1} Δ \bar{χ}, \bar{u}) - (\nabla \bar{χ} Δ χ_{2}, \bar{u}) - (\bar{u} \cdot \nabla χ_{2}, \bar{χ}) \\ - ((χ_{1}^{2} - 1) \bar{χ}, \bar{χ}) - ((χ_{1}^{2} - χ_{2}^{2}) χ_{2}, \bar{χ}) = \sum_{i = 10}^{15} I_{i} . \end{aligned} \end{matrix}$ By Hölder’s inequality, Young’s inequality, Gagliardo–Nirenberg’s inequality, Minkowski’s inequality and (3.2), we have $\begin{aligned} \begin{aligned} | I_{10} | & ⩽ C ‖ \nabla u_{2} ‖_{L^{2}} ‖ \bar{u} ‖_{L^{4}}^{2} ⩽ C ‖ \nabla u_{2} ‖_{L^{2}} ‖ \bar{u} ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla \bar{u} ‖_{L^{2}}^{\frac{3}{2}} \\ ⩽ C ‖ \nabla u_{2} ‖_{L^{2}}^{4} ‖ \bar{u} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{u} ‖_{L^{2}}^{2} ⩽ C ‖ \nabla u_{2} ‖_{L^{2}}^{2} ‖ \bar{u} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{u} ‖_{L^{2}}^{2}, \end{aligned} \\ \begin{aligned} | I_{11} | & = | \int_{R^{3}} \nabla χ_{1} Δ \bar{χ} \cdot \bar{u} d x | = | \int_{R^{3}} \nabla (\nabla χ_{1} \cdot \bar{u}) \nabla \bar{χ} d x | \\ ⩽ C {‖ \nabla^{2} χ_{1} ‖}_{L^{4}} ‖ \bar{u} ‖_{L^{4}} ‖ \nabla \bar{χ} ‖_{L^{2}} + C ‖ \nabla χ_{1} ‖_{L^{\infty}} ‖ \nabla \bar{u} ‖_{L^{2}} ‖ \nabla \bar{χ} ‖_{L^{2}} \\ ⩽ C {‖ \nabla^{2} χ_{1} ‖}_{L^{2}}^{\frac{1}{4}} {‖ \nabla^{3} χ_{1} ‖}_{L^{2}}^{\frac{3}{4}} ‖ \bar{u} ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla \bar{u} ‖_{L^{2}}^{\frac{3}{4}} ‖ \nabla \bar{χ} ‖_{L^{2}} + η_{1} ‖ \nabla \bar{u} ‖_{L^{2}} ‖ \nabla \bar{χ} ‖_{L^{2}} \\ ⩽ C {‖ \nabla^{2} χ_{1} ‖}_{L^{2}}^{\frac{1}{4}} ‖ \bar{u} ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla \bar{u} ‖_{L^{2}}^{\frac{3}{4}} ‖ \nabla \bar{χ} ‖_{L^{2}} + η_{1} ‖ \nabla \bar{u} ‖_{L^{2}} ‖ \nabla \bar{χ} ‖_{L^{2}} \\ ⩽ C {‖ \nabla^{2} χ_{1} ‖}_{L^{2}}^{2} ‖ \bar{u} ‖_{L^{2}}^{2} + (ε + \frac{η_{1}}{2}) ‖ \nabla \bar{u} ‖_{L^{2}}^{2} + (ε + \frac{η_{1}}{2}) ‖ \nabla \bar{χ} ‖_{L^{2}}^{2}, \end{aligned} \\ \begin{aligned} | I_{12} | & ⩽ ‖ \nabla \bar{χ} ‖_{L^{2}} ‖ Δ χ_{2} ‖_{L^{4}} ‖ \bar{u} ‖_{L^{4}} \\ ⩽ C ‖ Δ χ_{2} ‖_{L^{2}}^{\frac{1}{2}} {‖ \nabla^{3} χ_{2} ‖}_{L^{2}}^{\frac{3}{2}} ‖ \bar{u} ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla \bar{u} ‖_{L^{2}}^{\frac{3}{2}} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2} \\ ⩽ C ‖ Δ χ_{2} ‖_{L^{2}}^{\frac{1}{2}} ‖ \bar{u} ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla \bar{u} ‖_{L^{2}}^{\frac{3}{2}} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2} \\ ⩽ C {‖ \nabla^{2} χ_{2} ‖}_{L^{2}}^{2} ‖ \bar{u} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{u} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2}, \end{aligned} \\ \begin{aligned} | I_{13} | & ⩽ C ‖ \nabla χ_{2} ‖_{L^{2}} ‖ \bar{u} ‖_{L^{4}} ‖ \bar{χ} ‖_{L^{4}} ⩽ C ‖ \nabla χ_{2} ‖_{L^{2}} ‖ \bar{u} ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla \bar{u} ‖_{L^{2}}^{\frac{3}{4}} ‖ \bar{χ} ‖_{L^{2}}^{\frac{1}{4}} ‖ \nabla \bar{χ} ‖_{L^{2}}^{\frac{3}{4}} \\ ⩽ C ‖ \nabla χ_{2} ‖_{L^{2}}^{4} ‖ \bar{u} ‖_{L^{2}}^{2} + C ‖ \nabla χ_{2} ‖_{L^{2}}^{4} ‖ \bar{χ} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{u} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2} \\ ⩽ C ‖ \nabla χ_{2} ‖_{L^{2}}^{2} ‖ \bar{u} ‖_{L^{2}}^{2} + C ‖ \nabla χ_{2} ‖_{L^{2}}^{2} ‖ \bar{χ} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{u} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2}, \end{aligned} \\ \begin{aligned} | I_{14} | & ⩽ C {‖ χ_{1}^{2} - 1 ‖}_{L^{2}} ‖ \bar{χ} ‖_{L^{4}}^{2} ⩽ C {‖ χ_{1}^{2} - 1 ‖}_{L^{2}} ‖ \bar{χ} ‖_{L^{2}}^{\frac{1}{2}} ‖ \nabla \bar{χ} ‖_{L^{2}}^{\frac{3}{2}} \\ ⩽ C {‖ χ_{1}^{2} - 1 ‖}_{L^{2}}^{4} ‖ \bar{χ} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2} ⩽ C {‖ χ_{1}^{2} - 1 ‖}_{L^{2}}^{2} ‖ \bar{χ} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2}, \end{aligned} \\ \begin{aligned} | I_{15} | & = | \int_{R^{3}} (χ_{1} + χ_{2}) (χ_{1} - χ_{2}) χ_{2} \bar{χ} d x | = | \int_{R^{3}} (χ_{1} + χ_{2}) χ_{2} {\bar{χ}}^{2} d x | \\ ⩽ C ‖ χ_{1} + χ_{2} ‖_{L^{6}} ‖ χ_{2} ‖_{L^{6}} ‖ \bar{χ} ‖_{L^{6}} ‖ \bar{χ} ‖_{L^{2}} \\ ⩽ C (‖ \nabla χ_{1} ‖_{L^{2}} + ‖ \nabla χ_{2} ‖_{L^{2}}) ‖ \nabla χ_{2} ‖_{L^{2}} ‖ \nabla \bar{χ} ‖_{L^{2}} ‖ \bar{χ} ‖_{L^{2}} \\ ⩽ C ‖ \nabla χ_{2} ‖_{L^{2}}^{2} ‖ \bar{χ} ‖_{L^{2}}^{2} + ε ‖ \nabla \bar{χ} ‖_{L^{2}}^{2} . \end{aligned} \end{aligned}$ Let $ε = min (\frac{μ}{12}, \frac{1}{12})$ and $η_{1} = min (\frac{μ}{6}, \frac{1}{6})$ , then $\begin{aligned} \frac{d}{d t} (‖ \bar{u} ‖_{L^{2}}^{2} + ‖ \bar{χ} ‖_{L^{2}}^{2}) + μ ‖ \nabla \bar{u} ‖_{L^{2}}^{2} + ‖ \nabla \bar{χ} ‖_{L^{2}}^{2} \\ ⩽ C (‖ \nabla u_{2} ‖_{L^{2}}^{2} + {‖ \nabla^{2} χ_{1} ‖}_{L^{2}}^{2} + {‖ \nabla^{2} χ_{2} ‖}_{L^{2}}^{2} + ‖ \nabla χ_{2} ‖_{L^{2}}^{2} + {‖ χ_{1}^{2} - 1 ‖}_{L^{2}}^{2}) \cdot (‖ \bar{u} ‖_{L^{2}}^{2} + ‖ \bar{χ} ‖_{L^{2}}^{2}) . \end{aligned}$ Next, through using Gronwall’s inequality, we deduct that $\begin{matrix} (3.5) & \begin{aligned} ‖ \bar{u} ‖_{L^{2}}^{2} + ‖ \bar{χ} ‖_{L^{2}}^{2} & ⩽ (‖ {\bar{u}}_{0} ‖_{L^{2}}^{2} + ‖ {\bar{χ}}_{0} ‖_{L^{2}}^{2}) exp {C \int_{0}^{t} (‖ \nabla u_{2} ‖_{L^{2}}^{2} + {‖ \nabla^{2} χ_{1} ‖}_{L^{2}}^{2} \\ + {‖ \nabla^{2} χ_{2} ‖}_{L^{2}}^{2} + ‖ \nabla χ_{2} ‖_{L^{2}}^{2} + {‖ χ_{1}^{2} - 1 ‖}_{L^{2}}^{2}) d τ} . \end{aligned} \end{matrix}$ The intergal in the right-hand side of inequality above is finite. Also, as $(\bar{u}, \bar{χ}) |_{t = 0} = (0, 0)$ , we obtain that $\begin{matrix} ‖ \bar{u} ‖_{L^{2}}^{2} + ‖ \bar{χ} ‖_{L^{2}}^{2} = 0 . \end{matrix}$ So, we conclude that $u_{1} = u_{2}$ , $χ_{1} = χ_{2}$ .

Therefore, we obtain the following global existence result:

Lemma 3.1.
Under assumptions of Theorem 1.1 , there exist T and a unique smooth solution $(u, χ)$ to the Cauchy problem ( 1.1 )–( 1.3 ) satisfying ( 3.2 ).

Step 3. By utilizing Lemma 3.1, we shall prove that the local smooth solution $(u, χ)$ is indeed globally defined for all time. Proof.
Record the maximal existing time as $T^{}$ in Lemma 3.1. Now we claim that $T^{} ⩾ T$ for any $T > 0$ given in (3.2). Otherwise, if $T^{} < T$ , then all the priori estimates in Section 2 are valid with T is replaced by $T^{}$ . Hence, based on the prior estimates in Section 2, $(u, χ) (x, T^{})$ satisfies (3.2) at time $t = T^{}$ . Reusing Lemma 3.1, there exists a $T_{1}^{} > 0$ such that the smooth solution $(u, χ)$ exists on $[0, T^{} + T_{1}^{}]$ , which contradicts the fact that $T^{}$ is the maximal existing time of the smooth solution $(u, χ)$ . Therefore, it holds that $T^{*} ⩾ T$ . The proof of Theorem 1.1 is completed. □

4. Preliminaries on upper bound of decay

Lemma 4.1.
Assume that $u_{0}, \nabla χ_{0}, χ_{0}^{2} - 1 \in L^{1} (R^{3})$ . Then $\begin{matrix} (4.1) & | \hat{\nabla χ} | ⩽ C, | \hat{χ^{2} - 1} | ⩽ C, | \hat{u} | ⩽ C | ξ |^{- 1}, \forall ξ \in S (t), t > 0, \end{matrix}$ where the constant C depends only on $‖ u_{0} ‖_{L^{1}}$ , $‖ \nabla χ_{0} ‖_{L^{1}}$ , $‖ χ_{0}^{2} - 1 ‖_{L^{1}}$ .
Proof.
Applying the Fourier transform to (2.9), we obtain $\begin{matrix} \hat{\nabla χ_{t}} + 2 \hat{\nabla χ} + | ξ |^{2} \hat{\nabla χ} = G (ξ, t) = - \hat{\nabla (u \cdot \nabla χ)} - 3 \hat{(χ^{2} - 1) \nabla χ}, \end{matrix}$ where $\hat{.}$ is the Fourier transform. Then we can get $\begin{matrix} \frac{d}{d t} [e^{(| ξ |^{2} + 2) t} \hat{\nabla χ}] = e^{(| ξ |^{2} + 2) t} G (ξ, t), \forall ξ \in R^{3}, t > 0 . \end{matrix}$ Integrating it in time, we have $\begin{matrix} (4.2) & \hat{\nabla χ} = e^{- (| ξ |^{2} + 2) t} \hat{\nabla χ_{0}} + e^{- (| ξ |^{2} + 2) t} \int_{0}^{t} e^{(| ξ |^{2} + 2) s} G (ξ, s) d s . \end{matrix}$ By Hölder’s inequality, we deduce that $\begin{aligned} | \hat{\nabla (u \cdot \nabla χ)} | & ⩽ {‖ \hat{\nabla (u \cdot \nabla χ)} ‖}_{L^{\infty}} ⩽ {‖ \nabla (u \cdot \nabla χ) ‖}_{L^{1}} \\ ⩽ C ‖ u ‖_{L^{2}} {‖ \nabla^{2} χ ‖}_{L^{2}} + C ‖ \nabla u ‖_{L^{2}} ‖ \nabla χ ‖_{L^{2}} ⩽ C \end{aligned}$ and $\begin{aligned} | \hat{(χ^{2} - 1) \nabla χ} | & ⩽ {‖ \hat{(χ^{2} - 1) \nabla χ} ‖}_{L^{\infty}} ⩽ {‖ (χ^{2} - 1) \nabla χ ‖}_{L^{1}} \\ ⩽ C {‖ χ^{2} - 1 ‖}_{L^{2}} ‖ \nabla χ ‖_{L^{2}} ⩽ C . \end{aligned}$ Therefore, we conclude that $\begin{matrix} (4.3) & | G (ξ, t) | ⩽ C . \end{matrix}$ By substituting (4.3) into (4.2), in view of $\nabla χ_{0} \in L^{1} (R^{3})$ , we have $\begin{aligned} | \hat{\nabla χ} | & ⩽ e^{- (| ξ |^{2} + 2) t} | \hat{\nabla χ_{0}} | + C e^{- (| ξ |^{2} + 2) t} \int_{0}^{t} e^{(| ξ |^{2} + 2) s} d s \\ ⩽ ‖ \nabla χ_{0} ‖_{L^{1}} + C \frac{1}{| ξ |^{2} + 2} (1 - e^{- (| ξ |^{2} + 2) t}) \\ ⩽ ‖ \nabla χ_{0} ‖_{L^{1}} + \frac{1}{2} C ⩽ C . \end{aligned}$

Applying the Fourier transform to (2.6), we get $\begin{aligned} \hat{{(χ^{2} - 1)}_{t}} + 2 \hat{(χ^{2} - 1)} + | ξ |^{2} \hat{(χ^{2} - 1)} & = H (ξ, t) \\ = - \hat{u \cdot \nabla (χ^{2} - 1)} - 2 \hat{{(χ^{2} - 1)}^{2}} - 2 \hat{| \nabla χ |^{2}} . \end{aligned}$ Then $\begin{matrix} \frac{d}{d t} [e^{(| ξ |^{2} + 2) t} \hat{(χ^{2} - 1)}] = e^{(| ξ |^{2} + 2) t} H (ξ, t), \forall ξ \in R^{3}, t > 0 . \end{matrix}$ Integrating it in time yields $\begin{matrix} (4.4) & \hat{χ^{2} - 1} = e^{- (| ξ |^{2} + 2) t} \hat{(χ_{0}^{2} - 1)} + e^{- (| ξ |^{2} + 2) t} \int_{0}^{t} e^{(| ξ |^{2} + 2) s} H (ξ, s) d s . \end{matrix}$ By Hölder’s inequality, we deduce that $\begin{array}{c} \begin{aligned} | \hat{u \cdot \nabla (χ^{2} - 1)} | & ⩽ {‖ \hat{u \cdot \nabla (χ^{2} - 1)} ‖}_{L^{\infty}} \\ ⩽ {‖ u \cdot \nabla (χ^{2} - 1) ‖}_{L^{1}} \\ ⩽ C ‖ u ‖_{L^{2}} {‖ \nabla (χ^{2} - 1) ‖}_{L^{2}} ⩽ C, \end{aligned} \\ | \hat{{(χ^{2} - 1)}^{2}} | ⩽ {‖ \hat{{(χ^{2} - 1)}^{2}} ‖}_{L^{\infty}} ⩽ {‖ {(χ^{2} - 1)}^{2} ‖}_{L^{1}} ⩽ C {‖ χ^{2} - 1 ‖}_{L^{2}}^{2} ⩽ C \end{array}$ and $\begin{matrix} \hat{| \nabla χ |^{2}} ⩽ {‖ \hat{| \nabla χ |^{2}} ‖}_{L^{\infty}} ⩽ ‖ | \nabla χ |^{2} ‖_{L^{1}} ⩽ C ‖ \nabla χ ‖_{L^{2}}^{2} ⩽ C . \end{matrix}$ Hence, we conclude that $\begin{matrix} (4.5) & | H (ξ, t) | ⩽ C . \end{matrix}$ By substituting (4.5) into (4.4) and in view of $χ_{0}^{2} - 1 \in L^{1} (R^{3})$ , we deduce that $\begin{aligned} | \hat{χ^{2} - 1} | & ⩽ e^{- (| ξ |^{2} + 2) t} | \hat{χ_{0}^{2} - 1} | + C e^{- (| ξ |^{2} + 2) t} \int_{0}^{t} e^{(| ξ |^{2} + 2) s} d s \\ ⩽ {‖ χ_{0}^{2} - 1 ‖}_{L^{1}} + C \frac{1}{| ξ |^{2} + 2} (1 - e^{- (| ξ |^{2} + 2) t}) \\ ⩽ {‖ χ_{0}^{2} - 1 ‖}_{L^{1}} + \frac{1}{2} C ⩽ C . \end{aligned}$

Applying the Fourier transform to ${(1.1)}_{1}$ , we have $\begin{matrix} \frac{d}{d t} \hat{u} + μ | ξ |^{2} \hat{u} = W (ξ, t) = - (\hat{(u \cdot \nabla) u} + \hat{\nabla p} + \hat{\nabla χ Δ χ}), \end{matrix}$ then $\begin{matrix} \frac{d}{d t} (e^{μ | ξ |^{2} t} \hat{u}) = e^{μ | ξ |^{2} t} W (ξ, t), \forall ξ \in R^{3}, t > 0 . \end{matrix}$ Integrating it in time, we obtain $\begin{matrix} (4.6) & \hat{u} = e^{- μ | ξ |^{2} t} \hat{u_{0}} + e^{- μ | ξ |^{2} t} \int_{0}^{t} e^{μ | ξ |^{2} s} W (ξ, s) d s . \end{matrix}$ For the term $| W (ξ, t) |$ , note that $\begin{matrix} (4.7) & \begin{aligned} | \hat{u \cdot \nabla u} | & = | \sum_{j = 1}^{3} i ξ_{j} (\hat{u_{j} u}) | ⩽ | \sum_{j = 1}^{3} i ξ_{j} \int_{R^{3}} e^{- i ξ y} (u_{j} u) d y | \\ ⩽ \sum_{j = 1}^{3} | ξ_{j} | \int_{R^{3}} | u_{j} | | u | d y ⩽ | ξ | \int_{R^{3}} | u |^{2} d y ⩽ C | ξ |, \forall ξ \in R^{3}, t > 0 . \end{aligned} \end{matrix}$ By using Hölder’s inequality and Young’s inequality, we have $\begin{matrix} | \hat{\nabla χ Δ χ} | ⩽ ‖ | \hat{\nabla χ Δ χ} | ‖_{L^{\infty}} ⩽ ‖ | \nabla χ Δ χ | ‖_{L^{1}} ⩽ ‖ \nabla χ ‖_{L^{2}} ‖ Δ χ ‖_{L^{2}} ⩽ ‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2} . \end{matrix}$ The estimate for the pressure term is obtained from the equation $\begin{matrix} (4.8) & Δ p = - div ((u \cdot \nabla) u) - div (\nabla χ Δ χ) . \end{matrix}$ Applying the Fourier transform to (4.8), we deduce that $\begin{aligned} | ξ |^{2} | \hat{p} | & ⩽ \sum_{i, j = 1}^{3} | ξ_{i} ‖ ξ_{j} ‖ \hat{u_{i} u_{j}} | + | \hat{div (\nabla χ Δ χ)} | \\ ⩽ \sum_{i, j = 1}^{3} | ξ_{i} | | ξ_{j} | | \int_{R^{3}} e^{- i ξ y} u_{i} u_{j} d y | + | ξ | ‖ \nabla χ Δ χ ‖_{L^{1}} \\ ⩽ | ξ |^{2} \int_{R^{3}} | u |^{2} d y + C | ξ | (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) \\ ⩽ C | ξ |^{2} + C | ξ | (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}), \end{aligned}$ that is, $\begin{matrix} | ξ | | \hat{p} | ⩽ C | ξ | + C (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) . \end{matrix}$ Therefore, we get $\begin{matrix} (4.9) & | W (ξ, t) | ⩽ | \hat{(u \cdot \nabla) u} | + | ξ | | \hat{p} | + | \hat{\nabla χ Δ χ} | ⩽ C | ξ | + C (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) . \end{matrix}$ By substituting (4.9) into (4.6) and recalling that $u_{0} \in L^{1} (R^{3})$ , we have $\begin{aligned} | \hat{u} | & ⩽ e^{- μ | ξ |^{2} t} | {\hat{u}}_{0} | + e^{- μ | ξ |^{2} t} \int_{0}^{t} e^{μ | ξ |^{2} s} | W | d s \\ ⩽ e^{- μ | ξ |^{2} t} ‖ u_{0} ‖_{L^{1}} + C e^{- μ | ξ |^{2} t} \int_{0}^{t} e^{μ | ξ |^{2} s} | ξ | d s + C \int_{0}^{t} e^{- μ | ξ |^{2} (t - s)} (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) d s \\ ⩽ ‖ u_{0} ‖_{L^{1}} + C | ξ |^{- 1} (1 - e^{- μ | ξ |^{2} t}) + C \int_{0}^{t} (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) d s \\ ⩽ ‖ u_{0} ‖_{L^{1}} + C | ξ |^{- 1} + C . \end{aligned}$ Thus, for every $ξ \in S (t)$ , we have $\begin{aligned} | ξ | | \hat{u} | & ⩽ | ξ | ‖ u_{0} ‖_{L^{1}} + C + C | ξ | \\ ⩽ {[\frac{3}{2 a (t + 1)}]}^{\frac{1}{2}} ‖ u_{0} ‖_{L^{1}} + C + {[\frac{3}{2 a (t + 1)}]}^{\frac{1}{2}} C ⩽ C, \end{aligned}$ that is, $\begin{matrix} | \hat{u} | ⩽ C | ξ |^{- 1}, \forall ξ \in S (t) . \end{matrix}$ □
5. Proof of Theorem 1.2

Applying Plancherel’s theorem to (2.2) and letting $a = min (μ, 1)$ , we have $\begin{aligned} \frac{d}{d t} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ ⩽ - 2 a \int_{R^{3}} | ξ |^{2} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ = - 2 a \int_{{S (t)}^{c}} | ξ |^{2} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ - 2 a \int_{S (t)} | ξ |^{2} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ ⩽ - \frac{3}{t + 1} \int_{{S (t)}^{c}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ - 2 a \int_{S (t)} | ξ |^{2} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ = - \frac{3}{t + 1} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ + \int_{S (t)} (\frac{3}{t + 1} - 2 a | ξ |^{2}) (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ . \end{aligned}$ Then $\begin{matrix} (5.1) & \begin{aligned} \frac{d}{d t} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ + \frac{3}{t + 1} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ ⩽ \frac{3}{t + 1} \int_{S (t)} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ . \end{aligned} \end{matrix}$ By multiplying the factor ${(t + 1)}^{3}$ to (5.1), we get $\begin{matrix} (5.2) & \begin{aligned} \frac{d}{d t} [{(t + 1)}^{3} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ] \\ ⩽ 3 {(t + 1)}^{2} \int_{S (t)} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ . \end{aligned} \end{matrix}$ By using the transformation of spherical coordinates and (4.1), we obtain $\begin{matrix} (5.3) & \begin{aligned} \frac{d}{d t} [{(t + 1)}^{3} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ] \\ ⩽ 3 {(t + 1)}^{2} \int_{S (t)} | \hat{u} |^{2} d ξ + 3 {(t + 1)}^{2} \int_{S (t)} (| \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ ⩽ C {(t + 1)}^{2} \int_{S (t)} | ξ |^{- 2} d ξ + C {(t + 1)}^{2} \int_{S (t)} d ξ \\ ⩽ C {(t + 1)}^{2} \int_{0}^{r (t)} r^{- 2} r^{2} d r + C {(t + 1)}^{2} \int_{0}^{r (t)} r^{2} d r \\ ⩽ C {(t + 1)}^{\frac{3}{2}} . \end{aligned} \end{matrix}$ Integrate it in time to yield $\begin{aligned} {(t + 1)}^{3} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ \\ ⩽ \int_{R^{3}} (| {\hat{u}}_{0} |^{2} + | {\hat{\nabla χ}}_{0} |^{2} + {| {\hat{χ^{2} - 1}}_{0} |}^{2}) d ξ + C {(t + 1)}^{\frac{5}{2}} . \end{aligned}$ By virtue of $u_{0}, \nabla χ_{0}, χ_{0}^{2} - 1 \in L^{2} (R^{3})$ , applying Plancherel’s theorem, it follows that $\begin{matrix} (5.4) & \int_{R^{3}} (| u |^{2} + | \nabla χ |^{2} + {| χ^{2} - 1 |}^{2}) d ξ ⩽ C {(t + 1)}^{- 3} + C {(t + 1)}^{- \frac{1}{2}} ⩽ C {(t + 1)}^{- \frac{1}{2}}, t > 0 . \end{matrix}$

Next, let us use the conclusion of the previous step to obtain the sharp decay rates of u, $\nabla χ$ and $χ^{2} - 1$ .

Substituting (5.4) into (4.7), it follows that $\begin{matrix} (5.5) & | \hat{u \cdot \nabla u} | ⩽ | ξ | \int_{R^{3}} | u |^{2} d y ⩽ C | ξ | {(t + 1)}^{- \frac{1}{2}}, \forall ξ \in R^{3}, t > 0 . \end{matrix}$ Then $\begin{matrix} (5.6) & | W (ξ, t) | ⩽ C | ξ | {(t + 1)}^{- \frac{1}{2}} + C (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) . \end{matrix}$

Substituting (5.6) into (4.6), we get $\begin{matrix} (5.7) & \begin{aligned} | \hat{u} | & ⩽ e^{- μ | ξ |^{2} t} | {\hat{u}}_{0} | + \int_{0}^{t} e^{- μ | ξ |^{2} (t - s)} | W | d s \\ ⩽ ‖ u_{0} ‖_{L^{1}} + C \int_{0}^{t} | ξ | {(s + 1)}^{- \frac{1}{2}} d s + C \int_{0}^{t} (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) d s \\ ⩽ ‖ u_{0} ‖_{L^{1}} + C | ξ | {(t + 1)}^{\frac{1}{2}} + C \int_{0}^{t} (‖ \nabla χ ‖_{L^{2}}^{2} + ‖ Δ χ ‖_{L^{2}}^{2}) d s ⩽ C, \forall ξ \in S (t) . \end{aligned} \end{matrix}$ By substituting (5.7) into (5.2), we have $\begin{matrix} (5.8) & \begin{aligned} \frac{d}{d t} [{(t + 1)}^{3} \int_{R^{3}} (| \hat{u} |^{2} + | \hat{\nabla χ} |^{2} + {| \hat{χ^{2} - 1} |}^{2}) d ξ] \\ ⩽ C {(t + 1)}^{2} \int_{S (t)} d ξ ⩽ C {(t + 1)}^{2} \int_{0}^{r (t)} r^{2} d r ⩽ C {(t + 1)}^{\frac{1}{2}} . \end{aligned} \end{matrix}$ Integrating it in time yields $\begin{matrix} \int_{R^{3}} (| u |^{2} + | \nabla χ |^{2} + {| χ^{2} - 1 |}^{2}) d x ⩽ C {(t + 1)}^{- \frac{3}{2}} . \end{matrix}$

Finally, we will use our previous results to obtain the time-decay rates of higher-order norm.

From (2.8), we obtain $\begin{matrix} (5.9) & \begin{aligned} \frac{d}{d t} ({‖ \nabla^{k} u ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k} (χ^{2} - 1) ‖}_{L^{2}}^{2}) \\ ⩽ - a ({‖ \nabla^{k + 1} u ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 2} χ ‖}_{L^{2}}^{2} + {‖ \nabla^{k + 1} (χ^{2} - 1) ‖}_{L^{2}}^{2}) . \end{aligned} \end{matrix}$

Set $\begin{matrix} \tilde{S} (t) = {x \in R^{3}; | x | < \tilde{r} (t)}, \tilde{r} (t) = {[\frac{p}{a (t + 1)}]}^{\frac{1}{2}}, \end{matrix}$ where p is a sufficiently large positive integer. Let $\begin{matrix} {| Λ^{k} z |}^{2} = {| \nabla^{k} u |}^{2} + {| \nabla^{k + 1} χ |}^{2} + {| \nabla^{k} (χ^{2} - 1) |}^{2} . \end{matrix}$ Using the same method as in Section 4, it is easy to verify that $\begin{matrix} | \hat{z} |^{2} ⩽ C, \forall ξ \in \tilde{S} (t), t > 0 . \end{matrix}$ Applying Plancherel’s theorem to (5.9), then $\begin{aligned} \frac{d}{d t} {‖ \hat{Λ^{k} z} ‖}_{L^{2}}^{2} & ⩽ - \frac{p}{t + 1} \int_{{\tilde{S} (t)}^{c}} {| \hat{Λ^{k} z} |}^{2} d ξ - \int_{\tilde{S} (t)} | ξ |^{2} {| \hat{Λ^{k} z} |}^{2} d ξ \\ = - \frac{p}{t + 1} \int_{R^{3}} {| \hat{Λ^{k} z} |}^{2} d ξ + \int_{\tilde{S} (t)} (\frac{p}{t + 1} - | ξ |^{2}) {| \hat{Λ^{k} z} |}^{2} d ξ . \end{aligned}$ It implies that $\begin{matrix} (5.10) & \frac{d}{d t} {‖ \hat{Λ^{k} z} ‖}_{L^{2}}^{2} + \frac{p}{t + 1} \int_{R^{3}} {| \hat{Λ^{k} z} |}^{2} d ξ ⩽ \frac{p}{t + 1} \int_{\tilde{S} (t)} {| \hat{Λ^{k} z} |}^{2} d ξ . \end{matrix}$ By multiplying the factor ${(t + 1)}^{p}$ to (5.10), we deduce that $\begin{matrix} (5.11) & \begin{aligned} \frac{d}{d t} [{(t + 1)}^{p} {‖ \hat{Λ^{k} z} ‖}_{L^{2}}^{2}] & ⩽ p {(t + 1)}^{p - 1} \int_{\tilde{S} (t)} {| \hat{Λ^{k} z} |}^{2} d ξ \\ ⩽ p {(t + 1)}^{p - 1} \int_{\tilde{S} (t)} | ξ |^{2 k} | \hat{z} |^{2} d ξ ⩽ C {(t + 1)}^{p - 1} \int_{0}^{\tilde{r} (t)} {\tilde{r}}^{2 k + 2} d \tilde{r} \\ ⩽ C {(t + 1)}^{p - k - \frac{5}{2}} . \end{aligned} \end{matrix}$ Integrating (5.11) in time, we have $\begin{matrix} {(t + 1)}^{p} \int_{R^{3}} {| \hat{Λ^{k} z} |}^{2} d ξ ⩽ \int_{R^{3}} {| {\hat{Λ^{k} z}}_{0} |}^{2} d ξ + C {(t + 1)}^{p - k - \frac{3}{2}} . \end{matrix}$ In particular, choosing $p ⩾ k + 2$ , it follows that $\begin{matrix} \int_{R^{3}} {| Λ^{k} z |}^{2} d x ⩽ C {(t + 1)}^{- k - \frac{3}{2}} . \end{matrix}$ This finishes the proof. □

Data availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of interest

The authors declare that there is no conflict of interest. We also declare that this manuscript has no associated data.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 12071098) and the Fundamental Research Funds for the Central Universities (No. 2022FRFK060022).

References

Azbel, Two Phase Flows in Chemical Engineering, Cambridge Univ. Press, 1981.

Blesgen, A generalization of the Navier–Stokes equations to two-phase flows, J. Phys. D Appl. Phys. 32 (1999), 1119–1123. doi:10.1088/0022-3727/32/10/307.

Bonometti and

Magnaudet, An interface-capturing method for incompressible two-phase flows. Validation and application to bubble dynamics, J. Multiphase Flow 33 (2007), 109–133. doi:10.1016/j.ijmultiphaseflow.2006.07.003.

M.T.

Chen and

X.W.

Guo, Global large solutions for a coupled compressible Navier–Stokes/Allen–Cahn system with initial vacuum, Nonlinear Anal. 37 (2017), 350–373. doi:10.1016/j.nonrwa.2017.03.001.

S.M.

Chen,

H.Y.

Wen and

C.J.

Zhu, Global existence of weak solution to compressible Navier–Stokes/Allen–Cahn system in three dimensions, J. Math. Anal. Appl. 477 (2019), 1265–1295. doi:10.1016/j.jmaa.2019.05.012.

Y.Z.

Chen,

Q.L.

He,

Huang and

X.D.

Shi, Global strong solution to a thermodynamic compressible diffuse interface model with temperature dependent heat conductivity in 1D, Math. Methods Appl. Sci. 44 (2021), 12945–12962. doi:10.1002/mma.7597.

Y.Z.

Chen,

Hong and

X.D.

Shi, Stability of the phase separation state for compressible Navier–Stokes/Allen–Cahn system, 2021, arXiv:2105.07098.

Y.Z.

Chen,

H.L.

Li and

H.Z.

Tang, Global existence and optimal time decay rates of 3D non-isentropic compressible Navier–Stokes–Allen–Cahn system, J. Differ. Equ. 334 (2022), 157–193. doi:10.1016/j.jde.2022.06.018.

Ding,

Li and

Tang, Strong solutions to 1D compressible Navier–Stokes/Allen–Cahn system with free boundary, Math. Methods Appl. Sci. 42 (2019), 4780–4794. doi:10.1002/mma.5692.

10.

S.J.

Ding,

Y.H.

Li and

W.L.

Luo, Global solutions for a coupled compressible Navier–Stokes/Allen–Cahn system in 1D, J. Math. Fluid Mech. 15 (2013), 335–360. doi:10.1007/s00021-012-0104-3.

11.

Dreyfus,

Tabeling and

Willaime, Ordered and disordered patterns in two-phase flows in microchannels, Phys. Rev. Lett. 90 (2003), 144505. doi:10.1103/PhysRevLett.90.144505.

12.

Feireisl,

Petzeltová,

Rocca and

Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Meth. Appl. Sci. 20 (2010), 1129–1160. doi:10.1142/S0218202510004544.

13.

Gal and

Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discret. Contin. Dyn. Syst. 28 (2010), 1–39. doi:10.3934/dcds.2010.28.1.

14.

Gal and

Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. 31 (2010), 655–678. doi:10.1007/s11401-010-0603-6.

15.

J.J.

Ginoux, Two-Phase Flows and Heat Transfer with Application to Nuclear Reactor Design Problems, Hemisphere Publishing Corporation, 1978.

16.

W.W.

Huo and

K.M.

Teng, Global well-posedness and

L^{2}

decay estimate of smooth solutions for 2D incompressible Navier–Stokes–Allen–Cahn system, Z. Angew. Math. Phys. 73 (2022), 230. doi:10.1007/s00033-022-01864-z.

17.

Jiang,

Y.H.

Li and

Liu, Two-phase incompressible flows with variable density: An energetic variational approach, Discret. Contin. Dyn. Syst. 37 (2017), 3243–3284. doi:10.3934/dcds.2017138.

18.

Kato and

Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907. doi:10.1002/cpa.3160410704.

19.

Kotschote, Strong solutions of the Navier–Stokes equations for a compressible fluid of Allen–Cahn type, Arch. Rational Mech. Anal. 206 (2012), 489–514. doi:10.1007/s00205-012-0538-z.

20.

Kotschote, Spectral analysis for travelling waves in compressible two-phase fluids of Navier–Stokes–Allen–Cahn type, J. Evol. Equ. 17 (2017), 359–385. doi:10.1007/s00028-016-0380-0.

21.

R.F.

Kunz,

D.A.

Boger,

D.R.

Stinebring,

T.S.

Chyczewski,

J.W.

Lindau,

H.J.

Gibeling,

Venkateswaran and

T.R.

Govindan, A preconditioned Navier–Stokes method for two-phase flows with application to cavitation prediction, Comput. Fluids 29 (2000), 849–875. doi:10.1016/S0045-7930(99)00039-0.

22.

Y.H.

Li,

S.J.

Ding and

M.X.

Huang, Blow-up criterion for an incompressible Navier–Stokes/Allen–Cahn system with different densities, Discret. Contin. Dyn. Syst. Ser. B 21 (2016), 1507–1523. doi:10.3934/dcdsb.2016009.

23.

Y.H.

Li and

M.X.

Huang, Strong solutions for an incompressible Navier–Stokes/Allen–Cahn system with different densities, Z. Angew. Math. Phys. 69 (2018), 68. doi:10.1007/s00033-018-0967-0.

24.

Luo,

H.Y.

Yin and

C.J.

Zhu, Stability of the rarefaction wave for a coupled compressible Navier–Stokes/Allen–Cahn system, Math. Methods Appl. Sci. 41 (2018), 4724–4736. doi:10.1002/mma.4925.

25.

Matsumura and

Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 337–342.

26.

Matsumura and

Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), 67–104.

27.

T.T.

Medjo, A non-autonomous two-phase flow model with oscillating external force and its global attractor, Nonlinear Anal. 75 (2012), 226–243. doi:10.1016/j.na.2011.08.024.

28.

Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa 13 (1959), 115–162.

29.

X.K.

Pu and

B.L.

Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinet. Relat. Models 9 (2016), 165–191. doi:10.3934/krm.2016.9.165.

30.

M.E.

Schonbek,

L^{2}

decay for weak solutions of the Navier–Stokes equations, Arch. Rational Mech. Anal. 88 (1985), 209–222. doi:10.1007/BF00752111.

31.

M.E.

Schonbek, Large time behavior of solutions to the Navier–Stokes equations, Comm. Partial Differ. Equ. 11 (1986), 733–763. doi:10.1080/03605308608820443.

32.

M.E.

Schonbek, Large time behaviour of solutions to the Navier–Stokes equations in H spaces, Comm. Partial Differ. Equ. 20 (1995), 103–117. doi:10.1080/03605309508821088.

33.

Watanabe, Stability of stationary solutions to the three-dimensional Navier–Stokes equations with surface tension, Adv. Nonlinear Anal. 12(1) (2023), 20220279, 35.

34.

Xu,

Zhao and

Liu, Axisymmetric solutions to coupled Navier–Stokes/Allen–Cahn equations, SIAM J. Math. Anal. 41 (2010), 2246–2282. doi:10.1137/090754698.

35.

H.Y.

Yin and

C.J.

Zhu, Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier–Stokes/Allen–Cahn system, J. Differ. Equ. 266 (2019), 7291–7326. doi:10.1016/j.jde.2018.11.034.

36.

L.Y.

Zhao,

B.L.

Guo and

H.Y.

Huang, Vanishing viscosity limit for a coupled Navier–Stokes/Allen–Cahn system, J. Math. Anal. Appl. 384 (2011), 232–245. doi:10.1016/j.jmaa.2011.05.042.

37.

X.P.

Zhao, Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems, Proc. R. Soc. Edinb. Sect. A, Math. 152 (2022), 1291–1322. doi:10.1017/prm.2021.58.

Global well-posedness and L 2 decay estimate of smooth solutions for the 3-D incompressible Navier–Stokes–Allen–Cahn system

Abstract

Keywords

1. Introduction

Lemma 2.1 (Kato–Ponce inequality [18]).

Lemma 2.2 (Gagliardo–Nirenberg inequality [28]).

Lemma 2.3 ([29]).

Data availability

Conflict of interest

Footnotes

Acknowledgements

References