Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains

1.   Introduction

Fluid flowing in porous media is widely found in nature. It is a branch of various engineering and disciplines, involving exploration and exploitation of various underground fluid resources such as oil, natural gas and coalbed methane. The Brinkman-Forchheimer equation is a mathematical model that describes the motion of fluids in saturated porous media, so it has been an active research frontier in recent decades. The asymptotic behavior of the deterministic Brinkman-Forchheimer equations has been widely studied. For example, in the autonomous case, B. Wang and S. Lin ^[1] and D. Ugurlu ^[2] proved the existence of global attractors for the 3D Brinkman-Forchheimer equation. Moreover, X. G. Yang ^[3] studied the structure and stability of pullback attractors for three dimensional Brinkman-Forchheimer equation with delay. The uniform attractors for the non-autonomous Brinkman-Forchheimer equation with delay were obtained by Kang in ^[4]. In addition, the trajectory attractor and the approximation for the convective Brinkman-Forchheimer equations were obtained by C. Zhao et al. in ^[5,6].

During the past two decades, the mathematical theories of random dynamical systems ^[7] have made substantial progress in describing the asymptotic behavior of solutions for some dissipative dynamical systems. For example, in ^{[8,9,10,11,12]}, the authors have considered the asymptotic behavior of solutions for some dissipative random dynamical systems. In particular, the existence of attractors on unbounded domains has been studied extensively by many authors, see, e.g., ^{[9,10,13,14,15,16,17,18]}. Since Sobolev embeddings are no longer compact on unbounded domains, this is the main difficulty in proving the existence of attractors of equations defined on unbounded domains.

In ^[14], we studied the asymptotic behavior of the stochastic non-autonomous Brinkman-Forchheimer equations driven by linear multiplicative noise in unbounded domains. The existence of random attractors was obtaibed by transforming the stochastic equation into a pathwise random one. Comparing with ^[14], if we study the asymptotic behavior of stochastic Brinkman-Forchheimer equations driven by additive noise, different transformations will usually be used. This transformation will lead to more difficult calculations when we prove that the uniform estimates and the pullback asymptotic compactness of the solutions.

In this paper, we consider the following stochastic Brinkman-Forchheimer equations with additive noise:

where $u = (u_1, u_2, u_3)$ is the unknown velocity vector. $p = p(x, t)$ is the unknown pressure. $\nu > 0$ and $\alpha > 0$ denote the Brinkman kinematic viscosity and the Darcy coefficient respectively. $\beta > 0$ and $\gamma > 0$ are the Forchheimer coefficients. $g(x)$ is a force field. $h\in (H^{2}(\mathcal{O}))^{3}\bigcap(H_{0}^{1}(\mathcal{O}))^{3}$ and $w(t), t\in{\mathbb{R}}$ is a two-sided real-valued Wiener process on a probability space. The domian $\mathcal{O}\subset\mathbb{R}^{3}$ can be an arbitrary open set (bounded or unbounded) with smooth boundary $\partial{\mathcal{O}}$, and it satisfies the Poincar$\acute{{\mathrm{e}}}$ inequality: there exists a constant $\lambda_1 > 0$ such that

The purpose of this article is to study the asymptotic behavior of the 3D stochastic Brinkman-Forchheimer equation (1.1) on unbounded domains. We first establish a continuous random dynamical system for (1.1), see (3.46). To this end, we need to convert (1.1) into a deterministic equation (with a random parameter) (3.6) and (3.7) and obtain the existence, regularity and stability of weak solution to (3.6) and (3.7), see Theorems 3.1 and 3.2. The difficulty is the convergence of the nonlinear term, and we will use a truncation argument analogously to ^[13,19]. Next, we establish the existence of a unique $\mathcal{D}$-random attractor for (1.1), see Theorem 4.1. Since the Sobolev compact embeddings on unbounded domains are not compact, we will use the idea of energy equations, which was introduced by J. Ball ^[20]. Comparing with ^[10,18], we replace the advection term $(u\cdot \nabla)u$ by the damping term $\alpha{u}+\beta{|u|u}+\gamma{|u|^2 u}$, and deal with three dimensional case, which will be much harder to deal with.

This paper is organized as follows. Some basic concepts, a number of spaces and some inequalities are given in Section 2. Then a continuous random dynamical system for (1.1) is established in Section 3. The existence of a pullback random attractor is proved for (1.1) in Section 4. Finally, we summarize the main results and give some perspective on the next research.

2.   Preliminary

We recall some basic concepts (see ^{[7,8,9,11,12]}), and introduce some spaces and inequalities.

2.1.   Some basic concepts

Let $(X, \|\cdot\|_{X})$ be a separable Hilbert space with Borel $\sigma$-algebra $\mathcal{B}(X)$, and let $(\Omega, \mathcal{F}, P)$ be a probability space.

Definition 2.1. $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\in\mathbb{R}})$ is called a metric dynamical system if $\theta : \mathbb{R}\times\Omega\to\Omega$ is $(\mathcal{B}(\mathbb{R})\times\mathcal{F}, \mathcal{F})$-measurable, $\theta_{0}$ is the identity on $\Omega$, $\theta_{s+t} = \theta_{s}\circ\theta_{t}$ for all $s, t\in\mathbb{R}$ and $\theta_{t}(P) = P$ for all $t\in\mathbb{R}$.

Definition 2.2. A mapping $\phi: \mathbb{R}^{+}\times\Omega\times X\to X$ is called a continuous random dynamical system on $X$ over $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\in\mathbb{R}})$, if $\phi$ is $(\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal{B}(X), \mathcal{B}(X))$-measurable and satisfies, for $P$-$a.e.$ $\omega\in \Omega$,

(ⅰ)$\; \; \; \phi(0, \omega, \cdot)$ is the identity on $X$;

(ⅱ)$\; \; \phi(t+s, \omega, \cdot) = \phi(t, \theta_{s}\omega, \phi(s, \omega, \cdot))$ for all $t, s\in\mathbb{R}^{+}$;

(ⅲ)$\; \phi(t, \omega, \cdot) : X\to X$ is continuous for all $t\in\mathbb{R}^{+}$.

Definition 2.3. A random bounded set $\{D(\omega)\}_{\omega\in\Omega}$ of $X$ is called tempered with respect to $(\theta_{t})_{t\in\mathbb{R}}$ if for $P$-$a.e.$ $\omega\in\Omega$,

where $d(D) = sup_{x\in D}\|x\|_{X}$.

Definition 2.4. Let $\mathcal{D}$ be a collection of some families of nonempty subsets of $X$. Then $\{K(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ is said to be a random absorbing set for $\phi$ in $\mathcal{D}$ if for every $D = \{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and $P$-$a.e.$ $\omega\in\Omega$, there exsits $t_{D}(\omega) > 0$ such that

Definition 2.5. Let $\mathcal{D}$ be a collection of some families of nonempty subsets of $X$. Then $\phi$ is called $\mathcal{D}$-pullback asymptotically compact in $X$ if for $P$-$a.e.$ $\omega\in\Omega$, $\{\phi(t_{n}, \theta_{-t_{n}}\omega, x_{n})\}_{n = 1}^{\infty}$ has a convergent subsequence in $X$ for any sequence $t_{n}\to+\infty$, and $x_{n}\in D(\theta_{-t_{n}}\omega)$ with any $\{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$.

Definition 2.6. A random set $\{\mathcal{A}(\omega)\}_{\omega\in\Omega}$ of $X$ is called a $\mathcal{D}$-random attractor (or $\mathcal{D}$-pullback attractor) for $\phi$ if the following conditions are satisfied, for $P$-a.e. $\omega\in\Omega$,

(ⅰ)$\; \; \mathcal{A}(\omega)$ is compact and the mapping $\omega\to d(x, \mathcal{A}(\omega))$ is measurable for every $x\in X$;

(ⅱ)$\; \{\mathcal{A}(\omega)\}_{\omega\in\Omega}$ is invariant, that is,

(ⅲ)$\; \{\mathcal{A}(\omega)\}_{\omega\in\Omega}$ attracts every set in $\mathcal{D}$, i.e., for every $D = \{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$,

Theorem 2.1. (see ^[9], Proposition 2.7) Assume that $\phi$ is a continuous RDS which has a random absorbing set $\{K(\omega)\}_{\omega\in\Omega}$. If $\phi$ is $\mathcal{D}$-pullback asymptotically compact, then $\phi$ has a unique $\mathcal{D}$-random attractor $\{\mathcal{A}(\omega)\}_{\omega\in\Omega}$ which is given by

2.2.   Some generic functional spaces and inequalities

Denote ${\textbf{L}}^{p}(\mathcal{O}) = (L^{p}(\mathcal{O}))^3$ and use $\Vert\cdot \Vert _{p}$ to denote the norm in ${\textbf{L}}^{p}(\mathcal{O})$. Denote $\mathcal{V}: = \{ u|u\in{(C_0^\infty(\mathcal{O}))^3}, \;$div$\ u = 0\}$. $H$ is the closure of $\mathcal{V}$ in ${\textbf{L}}^2(\mathcal{O})$ topology, $\Vert\cdot \Vert _{H}$ and $(\cdot, \cdot)$ denote the norm and inner product in $H$ respectively, where

$\mathit{V}$ is the closure of $\mathcal{V}$ in $(\mathit{H}_{0}^{1}(\mathcal{O}))^{3}$ topology, $\Vert\cdot \Vert _{V}$ and $((\cdot, \cdot))$ denote the norm and inner product in $V$ respectively, where

By (1.2), $V\hookrightarrow H\equiv H'\hookrightarrow V'$, $H'$ and $V'$ are dual spaces of $H$ and $V$ respectively, where the injection is dense and continuous. $\Vert\cdot \Vert _{*}$ and $\left \langle\cdot, \cdot \right \rangle$ denote the norm in $V'$ and the dual product between $V$ and $V'$ respectively.

Denote by $P$ the Helmholtz-Leray orthogonal projection in ${\textbf{L}}^{2}(\mathcal{O})$ onto the space $H$. Set $A:D(A)\subset {\textbf{L}}^{2}(\mathcal{O})\to {\textbf{L}}^{2}(\mathcal{O})$, where $D(A) = (H^{2}(\mathcal{O}))^{3}\cap V$ and $Au = -P\Delta u$.

In addition, the Ladyzhenskaya's inequality is as follows:

3.   Random dynamical systems for (1.1)

In this section, we establish a continuous random dynamical system for the 3D stochastic BF equations.

Applying the Helmholtz-Leray projection $P$ onto the first equation in (1.1), we obtain the following abstract formulation of the 3D stochastic BF equations:

with initial datum $u(0) = u_{0}$.

In the following, we consider the probability space $(\Omega, \mathcal{F}, P)$ where

$\mathcal{F}$ is the Borel $\sigma$-algebra induced by the compact-open topology of $\Omega$, and $P$ is the corresponding Wiener measure on $(\Omega, \mathcal{F})$. Then we will identify $\omega$ with

Define the time shift by

Then $(\Omega, \mathcal{F}, P, (\theta_{t})_{t\in\mathbb{R}})$ is a metric dynamical system.

Consider the one dimensional Ornstein-Uhlenbeck equation

where $\mu > 0$. One may easily check that a solution to (3.2) is given by

Note that $y(\theta_{t}\omega)$ is $P$-$a.e.$ continuous and the random variable $|y(\omega)|$ is tempered (see ^[7,8,12]). Therefore, it follows from Proposition 4.3.3 in ^[7] that there exists a tempered function $R(\omega) > 0$ such that

where $p\geq2$ and $R(\omega)$ satisfies, for $P$-$a.e.$ $\omega\in\Omega$,

Then it follows from (3.3) and (3.4) that, for $P$-$a.e.$ $\omega\in\Omega$,

Putting $z(\theta_{t}\omega) = hy(\theta_{t}\omega)$, by (3.2) we have

In addition, by $h\in (H^{2}(\mathcal{O}))^{3}\bigcap(H_{0}^{1}(\mathcal{O}))^{3}$ and Ladyzhenskaya's inequalities, we get $z(\theta_{t}\omega)\in L^{2}(0, T;V)\cap L^{3}(0, T;{\rm{\bf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{\rm{\bf{L}}}^{4}(\mathcal{O}))$ and $Az(\theta_{t}\omega)\in L^{\infty}(0, T;H)$.

Now, let us study (3.1) by means of the classical change of variable $v(t, \omega) = u(t, \omega)-z(\theta_{t}\omega)$, then $v(t, \omega)$ satifies

In what follows, we give the definition of weak solutions of problems (3.6) and (3.7).

Definition 3.1. Let $T > 0$, assume that $v_{0}\in H$ and $g\in V'$. We shall say that $v(x, t)\in L^{\infty}(0, T;H)\cap L^{2}(0, T;V)\cap L^{3}(0, T;{\rm{\bf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{\rm{\bf{L}}}^{4}(\mathcal{O}))$ is a weak solution to (3.6) and (3.7), if it satisfies, for $P$-$a.e.$ $\omega\in\Omega$,

where (3.8) holds for all $\xi\in V$ in the sense of $D'(0, T)$.

Since (3.6) and (3.7) is a deterministic equation with a random parameter, we will use the standard Faedo-Galerkin methods in ^[21] to show the existence of weak solutions to (3.6) and (3.7) in following.

Theorem 3.1. For any $T > 0$ and $v_{0}\in H$, $g\in V'$, for $P$-$a.e.$ $\omega\in\Omega$, then problems (3.6) and (3.7) possesses a weak solution $v(x, t)\in L^{\infty}(0, T;H)\cap L^{2}(0, T;V)\cap L^{3}(0, T;{\rm{\bf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{\rm{\bf{L}}}^{4}(\mathcal{O}))$. Moreover, $v\in C([0, T];H)$.

Proof. Step 1: Constructing the approximated solution of (3.6) and (3.7).

Since $V$ is a subspace of $(H_{0}^{1}(\mathcal{O}))^{3}$, then it is separable. Recalling that $\mathcal{V}$ is dense in $V$ and $H$, so there exists a sequence of linearly independent elements $\{\nu_{i}\}_{i\geq1}\subset \mathcal{V}$ are dense in $V$ and $H$. Applying the Gram-Schmidt orthonormalization process, one can obtain an orthonormal basis $\{w_{j}\}_{j = 1}^{\infty}\subset \mathcal{V}$ of $H$ such that the linear combinations of these elements are dense in $V$.

Let $V_{m}$ = span$\{w_{1}, \cdots, w_{m}\}$ and the projector $P_{m}:H\to V_{m}$ be given by

We construct the approximated solution $v_{m}(t) = \sum\limits_{j = 1}^{m} h_{j, m}(t)w_{j}$ satisfying the following Cauchy problem

The problem (3.10) is a well-known ordinary functional differential equations with respect to the unknown variables $\{h_{j, m}(t)\}_{j = 1}^{m}$, which has a unique local solution (in an interval [0, $t^{*}$] with 0$< t^{*}\le T$). In fact, the global solution $(t^{*} = T)$ can be deduced by the a $priori$ estimates below.

Step 2: Establishing $a$ $priori$ estimates for $\{v_{m}\}$.

Multiplying the first equation in (3.10) by $h_{j, m}(t)$ and summing in $j$, we obtain that for a.e. $t\in [0, T]$,

the above inequality is obtained by using the Young's inequality and Poincar$\acute{{\mathrm{e}}}$ inequality.

Integrating (3.11) over $[0, t]$ with the time variable, we find

Since $z(\theta_{t}\omega)\in L^{2}(0, T;V) \cap L^{3}(0, T;{{\textbf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{{\textbf{L}}}^{4}(\mathcal{O}))$, we have

Moreover, $\int_{\mathcal{O}}|v_{m}|^{3}dx = \int_{\mathcal{O}}|v_{m}+z(\theta_{t}\omega)-z(\theta_{t}\omega)|^{3}dx\le\int_{\mathcal{O}}(|v_{m}+z(\theta_{t}\omega)|$ $+|z(\theta_{t}\omega)|)^{3}dx\le 4\|v_{m}+z(\theta_{t}\omega)\|_{3}^{3}+4\|z(\theta_{t}\omega)\|_{3}^{3}$, then we get

Thus, $v_{m}\in L^{3}(0, T;{{\textbf{L}}}^{3}(\mathcal{O}))$. Similarly, $v_{m}\in L^{4}(0, T;{{\textbf{L}}}^{4}(\mathcal{O}))$.

In conclusion,

Since the domain $\mathcal{O}$ maybe unbounded and the boundary $\partial\mathcal{O}$ has no any regularity assumption, the compact injection $V\hookrightarrow H$ may not hold. So the way of proving a compactness property on bounded domain is no longer valid here. Next, we will use Corollary 2.34 in ^[19] to obtain the local compactness result. Based on the estimates (3.15), we just need to prove that the following condition holds, i.e.,

From (3.10), for any $0\le t\le t+a\le T$, one has

Multiplying $h_{j, m}(t+a)-h_{j, m}(t)$ and summing in $j$, one has

where $G_{m}(s) = \nu\|v_{m}(s)\|_{V}+\nu\|z(\theta_{s}\omega)\|_{V}+\frac{\alpha}{\sqrt{\lambda_{1}}}\|v_{m}(s)+z(\theta_{s}\omega)\|_{H}+\frac{\mu}{\sqrt{\lambda_{1}}}\|z(\theta_{s}\omega)\|_{H}+\|g\|_{*}$.

Hence,

Thanks to the Fubini theorem, one has

where

Then using the Young's inequality and the fact that $0\le \overline{s}-\overline{s-a}\le a$, we derive that

By simple computation shows that

Combining (3.13), (3.15), (3.22) and (3.23), one achieves (3.16).

Step 3: Passing to limit for deriving the global solution of (3.6) and (3.7) by a truncation argument.

Combining the preceding uniform estimates (3.13) and (3.15), we can deduce that there exists a subsequence $v_{m}$ (without relabeling) such that, when $m\to\infty$,

with $v\in L^{\infty}(0, T;H)\cap L^{2}(0, T;V)$ and $v+z\in L^{2}(0, T;H)$.

Next, we split into four steps to obtain the weak solution.

(Ⅰ): Using a truncation argument analogously to ^[13,19,22], we prove that

For any bounded subset $\mathcal{K}\subset\mathcal{O}$, there exists a bounded open ball $B_{R}$ such that $\mathcal{K}\subset B_{R}$. Denote $\widetilde{\mathcal{K}} = \mathcal{O}\cap B_{2R}$ and then the compact injection $(H_{0}^{1}(\widetilde{\mathcal{K}}))^{3}\subset {\textbf{L}}^{2}(\widetilde{\mathcal{K}})$ holds. Define a blob function $\rho\in C^{\infty}(\mathbb{R}^{+})$ with

Define $v_{m}^{R}(x) = v_{m}(x)\rho(\frac{|x|^{2}}{R^{2}})$, then by (3.15) and (3.16), one has

From Corollary 2.34 in ^[19], one obtains that

Notice that $v_{m}^{R}(x) = v_{m}(x)$ for $x\in\mathcal{K}$, one achieves (3.29) immediately.

(Ⅱ): Passing to the limit of (3.10).

By Lemma $1.3$ in ^[23] and (3.29), we can extract a subsequence of $\{v_{m}\}$ such that the limit of (3.27) and (3.28) satisfy

Let $\psi\in C^{1}([0, T])$ with $\psi(T) = 0$. From (3.10), one has

Collecting (3.24)–(3.28) and (3.34) together, and then taking the limit $m\to\infty$, we have

for any $w\in\{w_{j}\}_{j = 1}^{\infty}$. Since $\{w_{j}\}_{j = 1}^{\infty}$ is dense in $V$, then $v$ satisfies the first equation of (3.8) by taking $\psi\in C_{0}^{\infty}(0, T)$ in (3.36).

(Ⅲ): Proving that $v\in C([0, T];H)$.

For all $\xi\in V$, we have

Then we have $\frac{\partial v}{\partial t}\in L^{2}(0, T;V')+L^{\frac{3}{2}}(0, T;{{\textbf{L}}}^{\frac{3}{2}}(\mathcal{O}))+L^{\frac{4}{3}}(0, T;{{\textbf{L}}}^{\frac{4}{3}}(\mathcal{O}))$. Combining the fact that $v\in L^{2}(0, T;V)\cap L^{3}(0, T;{{\textbf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{{\textbf{L}}}^{4}(\mathcal{O}))$, it follows from the similar calculation process of Theorem 3.6 in ^[24] that $v$ satisfies energy equality and hence $v\in C([0, T];H)$.

(Ⅳ): Checking the initial data $v(0) = v_{0}$.

For any $\psi\in C^{\infty}([0, T])$ with $\psi(T) = 0$ and $w\in V$, since $v$ satisfies the first equation of (3.8), then

After integrating by parts, one has

Comparing (3.36) and (3.39), we obtain that

which means that $v(0) = v_{0}$.

Furthermore, we also obtain the following theorem about the stability of (3.6) and (3.7).

Theorem 3.2. For any $T\geq0$ and given functions $(v_{0i}, g_{i})\in H\times V'$ for $i = 1, 2$, then problems (3.6) and (3.7) possesses two weak solutions $\{v_{i}\}_{i = 1, 2}\in L^{\infty}(0, T;H)\cap L^{2}(0, T;V)\cap L^{3}(0, T;{\bf{L}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{\bf{L}}^{4}(\mathcal{O}))$ with respect to $\{(v_{0i}, g_{i})\}_{i = 1, 2}$, and the following stability estimate holds:

Proof. Setting $w = v_{1}-v_{2}$, we have

By Lemma 4.4 in ^[25], we derive that

Integrating the above inequality from 0 to $t$, we get

Thus,

Since $u(t, \omega, u_{0}) = v(t, \omega, v_{0})+z(\theta_{t}\omega)$, one can easily obtain that $u(t, \omega)$ is a unique solution to problem (3.1). We now define a mapping $\phi :\mathbb{R}^{+}\times \Omega\times H \to H$ by

where $v_{0} = u_{0}-z(\omega)$. Then $\phi$ satisfies conditions (ⅰ)–(ⅲ) in Definition $2.2$. Therefore, $\phi$ is a continuous random dynamical system associated with problem (3.1).

4.   Existence of a unique $\mathcal{D}$-random attractor

Let $\mathcal{D}$ be the collection of all tempered families of subsets $\{D(\omega)\}_{\omega\in\Omega}$ of $H$, i.e., for every $\omega\in\Omega$

where $\lambda_1$ is Poincar$\acute{{\mathrm{e}}}$ constant in (1.2) and $\|D(\theta_{-t}\omega)\|_{H} = sup_{x\in D(\theta_{-t}\omega)}\|x\|_{H}$.

Lemma 4.1. Assume that $D = \{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$. Then for every $\omega\in\Omega$, there exist $T = T(D, \omega) > 0$ and a tempered function $r:\Omega\to\mathbb{R}^{+}$ such that

for all $t\ge T$ and $v_{0}(\theta_{-t}\omega)\in D(\theta_{-t}\omega)$.

Proof. From (3.6), for all $\varphi\in V$, we have

Choosing $\varphi = v$, we have

Using the Young's inequality, we get

where the last inequality is obtained by $\|z(\theta_{t}\omega)\|_{p}^{p} = \|h(x)\|_{p}^{p}|y(\theta_{t}\omega)|^{p}\le c|y(\theta_{t}\omega)|^{p}$.

By Poincar$\acute{{\mathrm{e}}}$ inequality, we get

Multiplying both sides of (4.3) by $e^{\frac{\lambda_{1}\nu}{4}t}$ and integrating over $(0, s)$, we obtain

Replacing $s$ and $\omega$ by $t$ and $\theta_{-t}\omega$, then we obtain

Since $v_{0}(\theta_{-t}\omega)\in D(\theta_{-t}\omega)$ and $\{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$, we get

Since $|y(\theta_{\tau}\omega)|$ is tempered, then by (3.5), we have

It implies that

Taking into account (4.5)–(4.7), then there exists $T(D, \omega) > 0$ such that, for $t\geq T$

and

where

Then, we have

Thus, we can choose $r(\omega) = \sqrt{2r_{0}(w)}$ and $r(\omega)$ is tempered from (4.11). This completes the proof.

Proposition 4.1. Assume that $D = \{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$. Then the random dynamical system $\phi$ associated with problem (3.1) has a random absorbing set $K\in\mathcal{D}$.

Proof. By (3.46), we get

and $v_{0}(\omega) = u_{0}(\omega)-z(\omega)$, then

Since $D\in\mathcal{D}$ and $|z(\omega)|$ is tempered, we can easily get $v_{0}(\omega)\in D^{*}(\omega)$ for some $D^{*}\in\mathcal{D}$. Then by Lemma 4.1, there exists $T = T(D^{*}, \omega) > 0$ such that

for all $t\geq$T and $v_{0}(\theta_{-t}\omega)\in D^{*}(\theta_{-t}\omega)$. Combining (4.12) and (4.13), we obtain

for all $t\geq$T and $u_{0}(\theta_{-t}\omega)\in D(\theta_{-t}\omega)$. It implies that there exists a random absorbing set of $\phi$ in $\mathcal{D}$.

In order to show that $\phi$ is $\mathcal{D}$-pullback asymptotically compact in $H$, we need the following lemma.

Lemma 4.2. For any sequence $\{x_{n}\}\subset H$ such that $x_{n}\rightharpoonup x_{0}$ in $H$, then for $P$-$a.e.$ $\omega\in\Omega$,

and

Proof. Denote by $v_{n}(t) = v(t, \omega, x_{n})$ and $v(t) = v(t, \omega, x_{0})$ the corresponding solutions to problem (3.6) and (3.7). Observe that by Theorem 3.1, one has uniform bounds of $v_{n}$ ans $v$ in $L^{\infty}(0, T;H)\cap L^{2}(0, T;V)\cap L^{3}(0, T;{{\textbf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{{\textbf{L}}}^{4}(\mathcal{O}))$, then $v_{n}+z$ and $v+z$ are uniformly bounded in $L^{2}(0, T;H)\cap L^{3}(0, T;{{\textbf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, T;{{\textbf{L}}}^{4}(\mathcal{O}))$, and $v_{n}$ belongs to $C([0, T];H)$. Then there exists a subsequence $\{n\}$ (without relabeling) such that, when $n\to \infty$,

and

Using a truncation argument analogously to step $3$ (Ⅰ) in Theorem 3.1, we obtain that

Proceeding similarly to Theorem 3.1, we can prove that $\chi = |v(t)+z(\theta_{t}\omega)|(v(t)+z(\theta_{t}\omega))$ and $\zeta = |v(t)+z(\theta_{t}\omega)|^{2}(v(t)+z(\theta_{t}\omega))$. Hence, (4.18) and (4.19) hold.

Lemma 4.3. Assume that $D = \{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$. Then for $P$-$a.e.$ $\omega\in\Omega$, any sequence $t_{n}\to +\infty$ and $x_{n}\in D(\theta_{-t_{n}}\omega)$, the sequence $\{v(t_{n}, \theta_{-t_{n}}\omega, x_{n})\}_{n = 1}^{\infty}$ is precompact in $H$.

Proof. By Lemma 4.1, one knows that $\{v(t_{n}, \theta_{-t_{n}}\omega, x_{n})\}_{n = 1}^{\infty}$ is bounded in $H$, thus we can get a subsequence of $\{n\}$ (without relabeling) and a $w_{0}\in H$ such that

By the lower semi-continuity of the norm, we have

In order to prove that (4.20) is actually strong convergence, we need to show that

Replacing $s$ and $\omega$ by $t_{n}-m$ and $\theta_{-t_{n}}\omega$ in (4.4) for any fixed $m > 0$, we get

Since $t_{n}\to +\infty$ and $x_{n}\in D(\theta_{-t_{n}}\omega)$, there exists $N_{m} > 0$ such that for all $n\geq N_{m}$,

Then $\{ v(t_{n}-m, \theta_{-t_{n}}\omega, x_{n})\}_{n = 1}^{\infty}$ is bounded in $H$. That means there exists a subsequence of $\{n\}$ (without relabeling) such that

By (4.25) and Lemma 4.2, we obtain

From (4.20) and (4.26), we obtain

Choosing $\xi = e^{\frac{\lambda_{1}\nu}{2}(s-t)}v$ in (3.8), we get

Moreover, we integrate (4.28) on $[0, t]$ and deduce that

After replacing $\omega$ by $\theta_{-t}\omega$, we can easily obtain

Applying (4.30) for $t = m$, $v_{0}(\theta_{-t}\omega) = v_{n, m} : = v(t_{n}-m, \theta_{-t_{n}}\omega, x_{n})$ and (4.24), we have

From (4.25), we have $v_{n, m}\rightharpoonup w_{m} \; {\mathrm{in}} \; H$. Then, by Lemma 4.2 we get

Since $\lambda_{1}\nu-\frac{\lambda_{1}\nu}{4} > 0$, we get that ${\int_{0}^{m}} e^{\frac{\lambda_{1}\nu}{2}(s-m)}(\nu\|v(s, \theta_{-m}\omega, v_{n, m})\|_{V}^{2}-\frac{\lambda_{1}\nu}{4} \|v(s, \theta_{-m}\omega, v_{n, m})\|_{H}^{2})ds$ defines a norm in $L^{2}(0, m;V)$, thus

Similarly, since $e^{\frac{\lambda_{1}\nu}{2}(s-m)}\nu z(\theta_{s-m}\omega)\in L^{2}(0, m;V)$, we have

Moreover, from (4.17), since $({\int_{0}^{m}} e^{\frac{\lambda_{1}\nu}{2}(s-m)}\|v(s, \theta_{-m}\omega, v_{n, m})+z(\theta_{s-m}\omega)\|_{L^{p}}^{p}ds)^\frac{1}{p}$ defines an equivalent norm in $L^{p}(0, m;{\textbf{L}}^{p}(\mathcal{O}))$, we have

Since $z(\theta_{s-m}\omega)\in L^{2}(0, m;H)\cap L^{3}(0, m;{{\textbf{L}}}^{3}(\mathcal{O}))\cap L^{4}(0, m;{{\textbf{L}}}^{4}(\mathcal{O}))$, (4.18) and (4.19), we have

Since $e^{\frac{\lambda_{1}\nu}{2}(s-m)}(g+\mu z(\theta_{s-m}\omega)\in L^{2}(0, m;V')$, we have

Letting $n\to\infty$ in (4.31) and applying (4.33)–(4.37), we have

Applying (4.30) for $t = m$ and $v_{0}(\theta_{-t}\omega) = w_{m}$, we have

Combining (4.38) and (4.39), we have

Letting $m\to\infty$ in (4.40), and noticing that $v(t_{n}, \theta_{-t_{n}}\omega, x_{n}) = v(m, \theta_{-m}\omega, v_{n, m})$ and $w_{0} = v(m, \theta_{-m}\omega, w_{m})$, we have

which implies (4.22).

Lemma 4.4. Assume that $D = \{D(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$. Then the random dynamical system $\phi$ is $\mathcal{D}$-pullback asymptotically compact in $H$.

Proof. For $P$-$a.e.$ $\omega\in\Omega$, $t_{n}\to\infty$ and $x_{n}\in D(\theta_{-t_{n}}\omega)$, we will show that $\{\phi(t_{n}, \theta_{-t_{n}}\omega, x_{n})\}_{n = 1}^{\infty}$ is precompact. It follows from (3.46) that

and

Since $D\in\mathcal{D}$ and $|z(\omega)|$ is tempered, there exists a $D^{*}\in\mathcal{D}$ such that $x_{n}-z(\theta_{-t_{n}}\omega)\in D^{*}(\theta_{-t_{n}}\omega)$. Then by Lemma 4.3, it is easily to get that $\{v(t_{n}, \theta_{-t_{n}}\omega, y_{n})\}_{n = 1}^{\infty}$ has a subsequence $\{n'\}\subset\{n\}$, which satisfying

Combining (4.42) and (4.44), we have

It means that $\phi$ is $\mathcal{D}$-pullback asymptotically compact in $H$.

Theorem 4.1. The random dynamical system $\phi$ corresponding to problem $(3.1)$ has a unique $\mathcal{D}$-random attractor $\mathcal{A} = \{\mathcal{A}(\omega)\}_{\omega\in\Omega}$ in $H$.

Proof. By Proposition 4.1, we can get $\phi$ has a family of random absorbing set $\{K(\omega)\}_{\omega\in\Omega}$ in $\mathcal{D}$. Moreover, $\phi$ is $\mathcal{D}$-pullback asymptotically compact in $H$ by Lemma 4.4. Hence, by Theorem 2.1, it is easily to get that the existence of a unique $\mathcal{D}$-random attractor for $\phi$.

5.   Conclusions

In the previous sections, we mainly studied the long time behavior of the stochastic Brinkman-Forchheimer equations driven by additive noise on unbounded domains, and have obtained the existence of a unique pullback random attractor. This provides a new result for the study of Brinkman-Forchheimer equations, which has important significance for the study of porous media fluids in the future.

From a practical point of view, it is also common for a fluid to be affected by a nonlinear random disturbance. Therefore, it is of great significance to study the long time behavior of the stochastic differential equations driven by nonlinear color noise. To obtain more research results for the study of Brinkman-Forchheimer equations, in the next research, we may consider that the dynamics for the stochastic Brinkman-Forchheimer equations driven by nonlinear color noise on the unbounded domains.

Acknowledgments

The work is supported by the NSFC (11601021, 11831003, 11771031, 12171111) and Beijing Municipal Education Commission Foundation (KZ202110005011).

Conflict of interest

The authors declare there is no conflicts of interest.