Attractors for a Navier–Stokes–Allen–Cahn system with unmatched densities

1.   Introduction

This paper considers the following Navier–Stokes–Allen–Cahn (NSAC) system modeling for two-phase flow with unmatched densities and viscosities, reading as follows:

where $\Omega\subset\mathbb{R}^2$ is a bounded domain with a smooth boundary. Considering the no-slip boundary condition for $\boldsymbol{u}$, the homogeneous Neumann boundary condition for $\phi$

and the initial conditions

Here, $\boldsymbol{u} = \boldsymbol{u}(x, t)$ represents the volume-averaged fluid velocity field, while $P = P(x, t)$ denotes the pressure. The viscosity of the mixture, $\nu$, is not constant, and $\rho$ denotes the density of the mixture, which depends on the phase function $\phi$. $D$ is the symmetric gradient, which has the following form: $D = \frac{1}{2}(\nabla+\nabla^{T})$. The chemical potential is defined by

and $\overline{X} = \frac{1}{|\Omega|}\int_\Omega X\mathrm{d}x$ denotes the spatial average of the term $X$. As an example, though not the only possibility, we can consider the averaged density

and the averaged viscosity of the binary fluids

where $\rho_1$ and $\rho_2$ denote the densities of the two fluids, and $\nu_1$ and $\nu_2$ represent their respective viscosities. The function $\Psi$ is the double-well free energy density, also known as the Flory–Huggins potential, which is given by

for every $s\in[-1, 1],$ where $\theta$ and $\theta_0$ are two positive constants representing the absolute temperature of the mixture and the critical temperature, respectively, and they satisfy $0 < \theta < \theta_{0}$.

Investigating the dynamics of two-phase flows is one of the most attractive and important problems within the hydrodynamic theory of fluids, with the Allen–Cahn equation playing a fundamental role (see ^[1,2]). The interface between two fluids is a $(d-1)$-dimensional manifold, posing great challenges both to the theoretical analysis and to the computational applications. Recently, a method called the diffuse-interface approach has emerged as a powerful technique for the study of interface theory (see ^[3,4,5,6]). The diffuse-interface method introduces a labeling function to replace the sharp interfaces with transition layers of width $\varepsilon > 0$, where $\varepsilon$ is a small parameter. Under this framework, the dynamics of interfaces between two fluids recognized as level sets of the order parameter can be naturally described (see ^[7]). Within the diffuse-interface framework, the phase function $\phi$ represents the contrast between local concentrations of the two fluids.

Two commonly used model equations in the study of the evolution of binary fluid systems with mass conservation are the following.

(1) Mass-conserving Allen–Cahn equation (see ^[8])

(2) Cahn–Hilliard equation (see ^[9])

where $\mu$ represents the chemical potential, and $m$ is a physically relevant constant.

The transport equation, in contrast to the Allen–Cahn and Cahn–Hilliard equations, does not include a diffusive term, and thus fails to maintain the proper shape of the diffuse interface along the normal direction, which motivates us to study the incompressible Navier–Stokes equations coupled with either the Allen–Cahn equation (1.6) or the Cahn–Hilliard equation (1.7), without dropping the crucial conservation laws (see (4.1)).

Recent research on incompressible binary fluid mixtures has led to significant findings. For the phase-field model of binary fluids, including the case of equal densities or small density contrasts, which can be approximated by the Boussinesq equations, we refer readers to ^{[3,10,11,12,13]} and references therein. We also refer interested readers to ^[14] for a nice reference about the compressible Navier–Stokes equations with Onsager's regularity. Nevertheless, in most physical models, the density differences between two fluids are non-negligible. Significant contributions to the Navier–Stokes–Allen–Cahn models with constant density have been made by the authors in ^[15,16,17]. However, the authors chose the potential as the classical Landau double-well form as well as the lack of mass conservation (see e.g. ^[18] and ^[19] and references therein).

The model (1.1) was derived by Onsager (see ^[20,21]), and we also refer interested readers to ^[22] for the Navier–Stokes–Cahn–Hilliard system. We also mention that there are several works for coupled nonlinear parabolic systems (see ^[23,24,25]) by introducing the so-called potential well method for the global existence of weak solutions (see ^[26,27,28]). Such systems are particularly relevant in the modeling of two-phase fluid systems, where the complex interplay between the phases often results in nonlinear coupling. The authors in ^[21] demonstrated both the well-posedness and the existence of the global attractor associated with system (1.1) in the 2-dimensional case. Their analysis focused on a specific case when the potential is smooth. The system they considered lacks a mass-conserving law and has constant viscosity. As a result, they ensured that the phase function $\phi$ remained confined within the physical range $[-1, 1]$, which is essential to their analysis. The well-posedness, regularity, and existence of the global attractor for the Navier–Stokes–Cahn–Hilliard system were established by the authors in ^[29] and ^[30]. The system they considered has matched density, and consequently, the uniqueness of the weak solution was easily obtained. This, together with the higher-order regularity of the phase function $\phi$ in (1.7), ensured that the dynamical system they constructed was on a lower-order regularity space ${\bf H}_\sigma\times H^1(\Omega)$, and they obtained the compact absorbing set by dissipativity estimates in ${\bf V}_\sigma\times H^2(\Omega)$ (see definition in section 2). Nevertheless, compared to the NSCH system, the NSAC system (1.1) contains only second-order diffusion terms. As a result, the regularity of $\phi$ is lower and we need more delicate estimates for $\phi$ (for more details, we refer to Proposition 4.1 in this paper and the argument of absorbing set in Theorem 4.1 in ^[30]). Moreover, since there is currently no theoretical proof of the uniqueness of weak solutions for the NSAC systems with unmatched densities, we can only consider strong solutions and construct the global attractor in a higher-regularity space ${\bf V}_\sigma\times H^2(\Omega)$, and therefore we need a much higher estimate of solutions in ${\bf H}_\sigma^2\times H^3(\Omega)$ to get the existence of a compact absorbing set (see Proposition 4.4 and Proposition 4.5 for more details).

The authors in ^[31] established the existence of a global weak solution of (1.1) in both 2-dimensional and 3-dimensional cases, together with the uniqueness of weak solutions with matched densities in the 2-dimensional case. Additionally, they proved the existence and uniqueness of strong solutions in the 2-dimensional case and derived several entropy estimates. However, there is no successful method that gives the uniqueness of weak solutions to system (1.1) with unmatched densities in the 2-dimensional case.

Before concluding this introduction, we give some additional remarks about our work. This study investigates the long-time behavior of solutions to the NSAC system (1.1). The system we considered here is more closely related to the actual physical model since the differences between densities and viscosities are not dropped. We also consider the system added a nonlinear term $(1/2)\rho^{\prime}(\phi)|\boldsymbol{u}|^2$ representing the force which effectively models the impact of macroscopic fluid effects on the microscopic description arising from density differences (see ^[31]). Building on the framework established in ^[31], we demonstrate the dissipativity in the complete metric space $\mathcal{H}_m$. Due to Theorem 3.2, the existence of strong solutions provided in ^[31], we focus our analysis on strong solutions and construct an absorbing set on a suitable phase space $\mathbb{Y}_m$ (refer to Section 3 for details). Moreover, because the chosen Flory–Huggins potential has singular derivatives, the uniform bound of $F^{\prime\prime}(\phi)$ as time $t$ away from zero is obtained by a corollary of Theorem 3.3, and this result enables us to derive compactness of the trajectories by proving dissipativity in a higher-regularity function space. Finally, applying the interpolation techniques, we demonstrate the continuity of the semigroup on the phase space $\mathbb{Y}_m$ and obtain the existence of the global attractor. For further research, one may get a higher regularity of the global attractor by the framework in ^[32]. This analysis lies beyond the framework of the present study and will be investigated in future work.

The plan of this paper reads as follows: In section 2, we present the function spaces, several inequalities in analysis, the theory of elliptic and the Stokes problems, as well as some Gronwall-type lemmas. In section 3, we recall the well-posedness results shown in ^[31], and we introduce the dynamical system in a suitable phase space generated by (1.1)-(1.3). Section 4 gives the existence of the global attractor, demonstrating the existence of a bounded and compact absorbing set in the phase space together with the continuity of the semigroup.

2.   Preliminaries

2.1.   Function spaces

Throughout this paper, the notation $C = C(a_1, a_2, ..., a_N)$ indicates that the constant $C$ is a positive constant depending on the quantities $a_1, a_2, ..., a_N$. The boldface letter (e.g., ${\bf L}$) denotes the space of vector fields. If $X$ is a metric space, $B_X(R)$ denotes the closed ball in $X$ with radius $R$, centered at the origin. In this paper, $A:B$ is defined as the inner product of two matrices $A$ and $B$, given by $A:B = tr(A^TB)$, and we denote norms $\|\cdot\|_{L^p(\Omega)}, \|\cdot\|_{H^1(\Omega)}...$ by $\|\cdot\|_{L^p}, \|\cdot\|_{H^1}...$ ($1\leqslant p\leqslant +\infty$) unless otherwise specified.

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with a smooth boundary $\partial\Omega$. We denote by ${\bf H}_{\sigma}$ the closure of $C_{0, \sigma}^\infty(\Omega)$ in ${\bf L}^2(\Omega)$, ${\bf V}_{\sigma}$ the closure of $C_{0, \sigma}^\infty(\Omega)$ in ${\bf H}^1(\Omega)$, and ${\bf H}^2_{\sigma}$ the closure of $C_{0, \sigma}^\infty(\Omega)$ in ${\bf H}^2(\Omega)$, where

Then they are Hilbert spaces, and for convenience, we may still use $\|\cdot\|_{L^2}$, $\|\cdot\|_{H^1}$ and $\|\cdot\|_{H^2}$ for the norms in those spaces.

The Korn inequality related to the symmetric gradient reads as follows:

We also recall the following inequalities in the 2D case (see^[31]):

We recall the following lemma and refer interested readers to ^[31] for a detailed proof.

Lemma 2.1 (^[31]). Let $\begin{aligned}f\in H^{1}(\Omega), g\in L^{p}(\Omega)\end{aligned}$ where $\Omega\subset\mathbb{R}^2$ is a bounded domain with a smooth boundary and $p > 2$. Then

for some $C = C(\Omega)$.

In the following, we recall an important differential inequality in order to obtain the dissipativities later (see ^[29,32] for more details).

Lemma 2.2 (Uniform Gronwall lemma in logarithm). Assume $f > 0$ is absolutely continuous on $[0, \infty)$ and $g, h > 0$ are both locally integrable on $[0, \infty)$, satisfying

and in addition the uniform bounds: for every $t\geq0$,

for some $r$, $a_1, a_2, a_3 > 0$. Then for every $t\geq r$,

2.1.1.   Stokes problems and Neumann problems

Now we recall two lemmas for the Stokes problem and the elliptic estimate of Neumann problems (see ^[3] and ^[31]).

Lemma 2.3. Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with a smooth boundary. $\nu\in W^{2, \infty}(\mathbb{R})$, and satisfies $0 < \nu_*\leq\nu(s)\leq\nu^*$ for all $s\in \mathbb{R}$. $\varphi\in W^{1, r}(\Omega)$, with $r > 2$. The force $\boldsymbol{g}\in {\bf L}^p(\Omega)$, with $p\in(1, \infty)$. Assume that $\boldsymbol{u}\in{\bf V}_\sigma$ is a weak solution of

in the following sense

Then,

for some positive constant $C = C(p, \Omega)$ and $\frac{1}{p} = \frac{1}{2}+\frac{1}{r}$.

Lemma 2.4. Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with a smooth boundary. Assume that $\varphi$ is the solution to the Neumann problem:

Then we have:

(a) If $g\in L^p(\Omega)$, $p\in[2, \infty]$, then

(b) If $g\in H^1(\Omega)$, then

and for every $p\geq 2$, there exists a positive constant $C = C(p, \Omega)$, such that

3.   Well-posedness and dynamical system

First we assume that the density and the viscosity $\rho, \nu\in C^2([-1, 1])$ satisfy

for every $s\in[-1, 1]$.

Next, we recall the well-posedness and regularity theorems given in ^[31].

Theorem 3.1 (^[31]). Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with a smooth boundary, $(\boldsymbol{u}_0, \phi_0)\in{\bf H}_{\sigma}\times (H^1(\Omega)\cap L^\infty(\Omega))$ with $\|\phi_0\|_{L^\infty} \leq 1$ and $|\overline{\phi}_0| < 1$. Then there exists a weak solution $(\boldsymbol{u}, \phi)$ to problem (1.1)-(1.3) on the interval $[0, \infty)$, satisfying:

(a) For every $T > 0$,

(b) The pair $(\boldsymbol{u}, \phi)$ solves the problem in the following sense:

for every $T > 0$, $\boldsymbol{w}\in {\bf V}_\sigma$, $\eta\in C^1([0, T])$ with $\eta(T) = 0$, and

$\boldsymbol{u}(\cdot, 0) = \boldsymbol{u}_0$, $\phi(\cdot, 0) = \phi_0$ in $\Omega$, and $\partial_{\boldsymbol{n}}\phi = 0$ almost everywhere on $\partial\Omega\times(0, T)$.

(c) Set the total energy of the system by

then the weak solutions satisfy the energy inequality as follows:

for all $t > 0$.

Theorem 3.2 (^[31]). Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with a smooth boundary. Assume that $(\boldsymbol{u}_0, \phi_{0})\in{\bf V}_\sigma(\Omega)\times H^{2}(\Omega)$ such that $\|\phi_0\|_{L^\infty}\leq1$, $|\overline{\phi}_0| < 1$, $\mu_{0} = \Psi^{\prime}(\phi_{0})-\Delta\phi_{0}\in H^{1}(\Omega)$ and $\partial_{\boldsymbol{n}}\phi_{0} = 0$ on $\partial\Omega$. Then there is a strong solution $(\boldsymbol{u}, \phi)$ to problem (1.1)-(1.3) on the interval $[0, \infty)$, satisfying:

(a) For every $T > 0$ and for every $p\in (2, \infty)$,

The solution $(\boldsymbol{u}, \phi)$ solves the system (1.1) almost everywhere in $\Omega\times(0, \infty)$. Moreover, $\boldsymbol{u}(\cdot, 0) = \boldsymbol{u}_0$, $\phi(\cdot, 0) = \phi_0$ in $\Omega$, $\partial_{\boldsymbol{n}}\phi = 0$ a.e. on $\partial\Omega\times(0, T)$.

(b) If in addition there exists $\eta_1 = \eta_1(E(\boldsymbol{u}_0, \phi_0), \|\boldsymbol{u}_0\|_{{\bf V}_\sigma}, \|\phi_0\|_{H^2}, \|F^{\prime}(\phi_0)\|_{L^2}, \theta, \theta_0)$, such that $\|\rho^{\prime}\|_{L^\infty(-1, 1)}\leq\eta_1$ and $F^{\prime\prime}(\phi_0)\in L^1(\Omega)$, then for every $T > 0$,

and furthermore, the strong solution that satisfies (3.4) is unique.

Theorem 3.3 (^[31]). Let the assumptions of Theorem 3.2 be satisfied. Assume in addition that $\|\rho^{\prime}\|_{L^\infty(-1, 1)}\leq\eta_1$. If $(\boldsymbol{u}, \phi)$ is the strong solution of system (1.1), then for every $\xi > 0$, there exists a positive constant $\delta(\xi)$, such that the absolute value of $\phi$ is away from one:

for every $x\in\overline{\Omega}$ and $t\geq\xi$.

Remark 3.4. In contrast to the Navier–Stokes–Cahn–Hilliard system, the phase function $\phi$ can approach $\pm1$ arbitrarily closely as $t$ goes to zero (see ^[31] and ^[30] for a detailed discussion).

For any $m\in(-1, 1)$, we define the following spaces:

where

Then $\mathcal{H}_m$ and $\mathbb{Y}_m$ are two complete metric spaces.

According to Theorems 3.1-3.2, the problem (1.1)-(1.3) generates a dynamical system: for each $t\geq0$,

in the following sense

where $(\boldsymbol{u}(t), \phi(t))$ is the unique solution of problem (1.1)-(1.3). The dynamical system is a semigroup $S(t)$ on $\mathbb{Y}_m$ satisfying:

4.   The Global Attractor

In this section, we will prove the existence of the global attractor $\mathcal{A}_m$ of the semigroup $S(t)$ on the phase space $\mathbb{Y}_m$.

4.1.   Absorbing Set

Proposition 4.1. There is a bounded set $B_0\subset \mathcal{H}_m$, such that for any bounded subset $B$ of $\mathcal{H}_m$, there exists $t_0(B) > 0$, which depends only on the $\mathcal{H}_m$-bounds of $B$, satisfying

where $(\boldsymbol{u}, \phi)$ is the weak solution of (1.1) subject to the initial value $(\boldsymbol{u}_0, \phi_0)\in B$.

Proof. Let us fix $R > 0$. We consider $(\boldsymbol{u}_0, \phi_0)\in B_{\mathcal{H}_m}(R)\subset\mathcal{H}_m$. First we integrate the equation $(1.1)_3$ over $\Omega$ to obtain the mass conservation: for every $t\geq0$,

and we define

By Theorem 3.1, we recall the energy identity:

for every $t > 0$. For $(1.1)_3$, we take the $L^2$-inner-product with $\phi-\overline{\phi} = \phi-m$ to obtain

By multiplying the above equation by $\varepsilon$ and summing with (4.2), we arrive at

where $C = C(\theta_0, m, \Omega, \varepsilon)$ and $C_1 = C_1(\rho, m, \Omega)$. Then by the Korn inequality and (3.1), we obatin

Next we need an inequality, which can be found in ^[33]:

for some $\beta$, $C_0 > 0$, depending only on $F$ and $m$. Then we obatin

Taking $\varepsilon = \frac{1}{4C_1}\nu_*$, and $\Psi^*: = \max_{s\in[-1, 1]}|\Psi(s)|, \Psi_*: = \max_{s\in[-1, 1]}|\Psi(s)|$ we obtain

where $\tilde{C_1} = \tilde{C_1}(F, \theta_0, m, \nu, \Omega)$. By the definition of $E(t)$, we obtain

where $\alpha = \alpha(\lambda_1, \Psi_*)$, depends on parameters of system (1.1), and $\lambda_1$ is the first eigenvalue of the Stokes operator ${\bf A}$, while $K^2_0: = \tilde{C_1}+|\Omega|\Psi^*$. By the Gronwall lemma, for each $t\geq 0$,

where $(K_1^{\prime})^2 = \frac{K^2_0}{\alpha}$. Thus by the definition of the energy $E(t)$ again, we obtain the crucial inequality:

As $(u_0, \phi_0)\in B_{\mathcal{H}_m}(R)$, when $t\geq t_e(R)$, for some $t_e(R) = t_e(\rho, R)$,

and

where $K^2_1: = K^2+1$, depends only on parameters of system (1.1). Then we can finish the proof by taking $B_0 = B_{\mathcal{H}_m}(K_1)$. □

Remark 4.2. As a direct consequence of the above proposition, we indeed obtain the dissipativity in the weak solution space. This may allow us to construct the so-called trajectory attractor for the weak solutions of the NSAC system (1.1) without uniqueness. On the other hand, (4.5) together with (4.4) yields, for all $t\geq 0$,

for some $M_0$ that depends only on the parameters of the system and $K_1$.

Lemma 4.3. The following estimates hold for all $t\geq t_e(R)$:

and

for some $C_1, C_2$ dependent on parameters of (1.1) and $K_1$.

Proof. Multiply (1.4) by $-\Delta\phi$ and integrate over $\Omega$ to obtain

As $t\geq t_e(R)$, and by (4.5),

Thus,

Then by (4.5), we obatin

for any $t\geq t_e(R)$. Also by (2.4) and (4.5),

We can infer from $(1.1)_3$ that

Denoting $(\rho^{\prime})^{*} = \operatorname*{max}_{s\in[-1, 1]}|\rho^{\prime}(s)|$, we obtain that

Thus, we obtain

where $C = C(K_1)$. Consequently, we obtain

which implies

for all $t\geq t_e(R)$.

Since

we obtain that for each $t\geq t_e(R)$,

where the parameters $C_1, C_2$ depend only on parameters of the system (1.1) and $K_1$. □

Proposition 4.4. The dynamical system $(\mathbb{Y}_m, S(t))$ possesses a bounded absorbing set $B_1$, i.e., for any bounded set $B\subset\mathbb{Y}_m$, there exists $\ t_1(B) > 0$, depending only on the $\mathbb{Y}_m$-bound of $B$, such that for any $t\geq t_1(B)$, $S(t)B\subset B_1$.

Proof. This part of the proof is similar to the higher regular estimates of the NSAC system in ^[31], whereas keep in mind that (4.6) is valid for any $t > t_e(R)+1$.

Multiplying (1.1) by $\partial_t\boldsymbol{u}$, integrating over $\Omega$, we get

where

we get

Differentiating $(1.1)_3$ with respect to $t$, multiplying by $\partial_t\phi$ and integrating over $\Omega$, we arrive at

By summing up (4.10) and (4.11), we obtain that

Then by (2.2)-(2.6) and Lemma 2.1, as well as a fundamental inequality: for each $x, y > 0$,

we obtain that

where

and

From Lemma 2.3, for the Stokes problem with the force term $\boldsymbol{g} = -\rho(\phi)\partial_t\boldsymbol{u}-\rho(\phi)\boldsymbol{u}\cdot\nabla\boldsymbol{u}-\Delta\phi\nabla\phi$, we have, for any $\varepsilon\in (0, 1)$,

By the Sobolev embedding $W^{2, 1+\varepsilon}(\Omega)\hookrightarrow W^{1, p}(\Omega)$ for every $p\in(2, \infty)$ and $\frac{1}{p} = \frac{1}{1+\varepsilon}-\frac{1}{2}$,

Consider the elliptic problem

and Lemma 2.4,

for any $p\in[2, \infty)$. By $(1.1)_3$, we have

Note that $\overline{\mu} = \overline{F^\prime(\phi)-\theta_0\phi}$, and thus $|\overline{\mu}|\leq C(1+\|F^\prime(\phi)\|_{L^1})$. Taking the $L^2$-inner product of (1.1) with $\phi-m$, we obtain

By (4.3),

Thus,

Then by (4.20)

Consequentially, applying the following inequality: for each $x, y > 0$,

we obtain that

Similarly, we can obtain that

Then by (4.14),

Due to (4.16), we have

where using (4.5) and (4.6), we obtain

and

Then by the uniform Gronwall lemma in logarithm, we get that, for all $t\geq t_e(R)+1: = t_1(R)$,

On the other hand, from (4.7) and (4.16), for every $t\geq t_e(R)+1: = t_1(R)$,

where $K_2$ depends only on the parameters of the system. Therefore, we can finish the proof by taking $B_1 = B_{\mathbb{Y}_m}(K_2)$. □

4.2.   Further Dissipativity

Proposition 4.5. For every $R > 0$ and for any $(\boldsymbol{u}_0, \phi_0)\in B_{\mathbb{Y}_m}(R)$ there exists $t_2 > 0$ and a bounded subset $B_2$ of ${\bf H}_\sigma^2\times H^3(\Omega)$, such that

for all $t\geq t_2,$ and therefore $B_1\cap B_2$ is the compact absorbing set for $S(t)$ on phase space $\mathbb{Y}_m$.

Recall that the set $B_1$ was obtained in Proposition 4.4.

Proof. Recall that ${\bf A}$ is the Stokes operator, and there exists $\Pi\in L^2(0, T; L^2(\Omega))$. Then $-\Delta\boldsymbol{u}+\nabla\Pi = {\bf A}\boldsymbol{u}$ a.e. in $\Omega\times(0, \infty)$ with the following estimates:

Multiplying $(1.1)_1$ by ${\bf A}\boldsymbol{u}$ and taking integration over $\Omega$, we obtain

where

and

Then we obtain

By (2.2), (4.5), (4.24), and the Young inequality, we obtain

and

By (2.2), (2.4), (4.5), (4.24), (4.25), and the global bound of $\phi$ in $L^\infty(\Omega)$, we obtain

and

Thus, by the Sobolev embeddings, (2.6) and (4.20), we arrive at

Adding (4.27) and (4.23), we obtain that as $t\geq t_2(R)$,

where $C = C(K_1, K_2)$ and depends on the parameters of system (1.1). Integrating (4.28), and by (4.24), we get there exists $M_1$ depending on $K_1, K_2$, such that

for every $t\geq t_2.$

Multiplying $(1.1)_1$ with $-\Delta\boldsymbol{u}_t$ and taking integration over $\Omega$, we obtain

which yields

By (2.5) and (4.24), and the Sobolev embeddings, we obtain

and

By the same argument in (4.27), the Agmon inequality together with the Sobolev embeddings, we obtain

and

From the above estimates, we obtain

On the other hand, differentiating $(1.1)_3$ with respect to $t$, multiplying $-\Delta\phi_t$, integrating over $\Omega$, we obtain

By Theorem 3.3, there exists $\delta = \delta(1) > 0$, such that for every $x\in \overline{\Omega}$ and $t\geq 1$,

Then $\|F^{\prime\prime}(\phi)\|_{L^\infty}\leq C(\delta)$. By (4.5), (4.24), and (4.27), we obtain

and by the Agmon inequality (2.3), we have that

From the above estimates, we obtain

Adding (4.31) into the above inequality, we obtain

Taking the gradient of $(1.1)_3$, we obtain

and consequently,

Then by (4.33), we arrive at

where $\Lambda(t) = \frac{1}{2}\int_{\Omega}\frac{\nu(\phi)}{2}|\Delta \boldsymbol{u}|^{2}\mathrm{d}x+\|\nabla\phi_{t}\|_{L^2}^{2}$, and

By (4.29), we obtain

Then by the uniform Gronwall lemma (e.g., see ^[32]), we can obtain that

and by (4.34),

for every $t\geq t_{2}+1 = t_e(R)+2$, and

for each $t\geq t_{3}: = t_2+1$. Thus, we obtain that $B_1\cap B_2$ is a compact absorbing set by the compact Sobolev embedding theorem, where $B_1 = B_{\mathbb{Y}_m}(K_2)$ and $B_2 = B_{{\bf H}_\sigma^2\times H^3(\Omega)}(K_3)$. □

The proposition above directly leads to the following corollary.

Corollary 4.6. For each bounded set $B\subset\mathbb{Y}_m$, there exists $t_3 = t_3(B) > t_e(R)+2$, such that $\bigcup_{t\geq t_2}S(t)B$ is relatively compact in $\mathbb{Y}_m$.

4.3.   Continuity of the semigroup

In this section, we will prove that for every $t > 0$, the semigroup $S(t)$ is continuous on the phase space $\mathbb{Y}_m$. Let $\{(\boldsymbol{u}_0^n, \phi_0^n)\}_{n\geq1}$ be a sequence in $\mathbb{Y}_m$ and $(\boldsymbol{u}_0, \phi_0)\in\mathbb{Y}_m$, such that $(\boldsymbol{u}_0^n, \phi_0^n)\rightarrow (\boldsymbol{u}_0, \phi_0)$ in the strong topology of $\mathbb{Y}_m$. Let $(\boldsymbol{u}^n(t), \phi^n(t)) = S(t)(\boldsymbol{u}_0^n, \phi_0^n), (\boldsymbol{u}(t), \phi(t)) = S(t)(\boldsymbol{u}_0, \phi_0)$. Then by assumption, there exists $\tilde{M}_1 > 0$, such that, for all $n = 1, 2, \cdots$,

Proposition 4.7. For any $t\geqslant 0$, there is a constant $C = C(\tilde{M}_1, t), \mathit{\text{}\; }\tilde{M} = \tilde{M}(\tilde{M}_1, t)$, such that

Proof. Let $(\boldsymbol{u}_1, \phi_1), (\boldsymbol{u}_2, \phi_2)$ be the corresponding strong solutions initiated from $(\boldsymbol{u}_{01}, \phi_{01}), (\boldsymbol{u}_{02}, \phi_{02})$ and let $(\boldsymbol{u}, \phi) = (\boldsymbol{u}_1, \phi_1)-(\boldsymbol{u}_2, \phi_2)$.

Following the proof of Theorem 4.1 in ^[31], and the uniqueness of the strong solution, we obtain

where

and

moreover, $A$ and $B$ are both in $L^{1}(0, T)$ for any $T > 0.$ Let $\tilde{M} = \max\big\{\int_{0}^{t}A(s)\mathrm{d}s, \int_{0}^{t}B(s)\mathrm{d}s\big\}$ which depends on $\tilde{M}_1$ following the proof in ^[31], and use the Gronwall lemma to obtain

Then

Following the proof of Proposition 4.5, we obtain that for each $\tau, t:0 < \tau < t\leq T$,

Note that in the proof of Proposition 4.5, $M_1$ in (4.29) is independent of the initial value $(\boldsymbol{u}_0, \phi_0)$ and (4.29) is valid for $t\geq t_2+1$. However, here in (4.37), $C(\tilde{M}_1)$ depends on the initial data, but we need the inequality for $t > 0$. Then there is a $\tau_{0}\in(0, \tau)$ such that $(\boldsymbol{u}(\tau_0), \phi(\tau_0))$ satisfies $\boldsymbol{u}(\tau_0)\in \mathbb{V}_{\sigma}, \phi(\tau_{0})\in H^{2}(\Omega), \partial_{n}\phi(\tau_{0}) = 0\mathrm{\; on\; }\partial\Omega$, $F^{\prime\prime}(\phi(\tau_0))\in L^2(\Omega)$ and

Taking $(\boldsymbol{u}(\tau_0), \phi(\tau_0))$ as the initial datum, there exists a global strong solution $(\tilde{\boldsymbol{u}}(t), \tilde{\phi}(t))\; \text{on }\; [\tau_{0}, \infty)$, for each $t\geq \tau_{0}$, there exists a constant $C_{0} = C_{0}(\tilde{M}_{1})$,

where

Also by (4.35), there is a constant $C_1 = C_1(\tilde{M}_1)$, such that

Using the uniform Gronwall lemma again, we obtain

and

for every $t\geq \tau$, where $\tilde{M}^{\prime}_{2} = \tilde{M}^{\prime}_2(\tau, \tilde{M}_1)$ and $\tilde{M}_{2} = \tilde{M}_{2}(\tau, \tilde{M}_1)$. Then we obtain that for every $\tau > 0,$ $i = 1, 2,$

by viewing $(\boldsymbol{u}_{01}, \phi_{01}), (\boldsymbol{u}_{02}, \phi_{02})$ as initial values of the trajectories. Now let $t^* > 0$, $\tau = \frac{1}{2}t^*$, $t^* > \tau > 0$. By interpolation,

Replacing $(\boldsymbol{u}_1, \phi_1)\; \text{by }\; (\boldsymbol{u}_n, \phi_n)\; \text{, }\; (\boldsymbol{u}_2, \phi_2)\; \text{by }\; (\boldsymbol{u}, \phi)$, we obtain

where $C = C(t^*, \tilde{M}_1)$ and $\tilde{M} = \tilde{M}(t^*, \tilde{M}_1)$. □

Thanks to these propositions above, we obtain the following theorem by Theorem 1.1 in ^[32].

Theorem 4.8. Let the assumptions of Theorem 3.2 and Theorem 3.3 hold. The dynamical system $(\mathbb{Y}_m, S(t))$ possesses a unique global attractor $\mathcal{A}_m\subset\mathbb{Y}_m$, which is a connected compact set and has the following properties:

(a) $\mathcal{A}_m$ is strictly invariant in $\mathbb{Y}_m$, i.e., $S(t)\mathcal{A}_m = \mathcal{A}_m,$ for every $t\geq 0$;

(b) $\mathcal{A}_m$ is an attracting set for $S(t)$ on $\mathbb{Y}_m$, i.e., for every bounded ball $B_{\mathbb{Y}_m}$ in the phase space $\mathbb{Y}_m$

where the Hausdorff semi-distance between sets is defined by $\operatorname{\mathrm{dist}}(A, B): = \operatorname*{sup}_{a\in A}\operatorname*{inf}_{b\in B}\operatorname{\mathrm{dist}}_{\mathbb{Y}_m}(a, b)$.

Note that from Proposition 4.5, we know that the obtained global attractor $\mathcal{A}_m$ is bounded in ${\bf H}_\sigma^2\times H^3(\Omega)$.

Author contributions

Chunyou Sun: Methodology, Writing-review & editing; Junyan Tan: Writing-original draft, Writing-review & editing.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partly supported by the NSFC (Grants No. 12271227)

Conflict of interest

The authors declare there is no conflict of interest.