Long Term Behavior of 2D and 3D Non-autonomous Random Convective Brinkman–Forchheimer Equations Driven by Colored Noise

Abstract

The long time behavior of Wong–Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman–Forchheimer (CBF) equations with non-linear diffusion terms on some bounded and unbounded domains is discussed in this work. To establish the existence of pullback random attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail-estimates are exploited to prove AC. In the literature, CBF equations are also known as Navier–Stokes equations (NSE) with damping, and it is interesting to see that the modification in NSE by linear and nonlinear damping provides better results than that available for NSE in 2D and 3D. The presence of linear damping term helps to establish the results in the whole space $\mathbb {R}^d$. The nonlinear damping term supports to obtain the results in 3D and to cover a large class of nonlinear diffusion terms also. In addition, we prove the existence of a unique pullback random attractor for stochastic CBF equations driven by additive white noise. Finally, for additive as well as multiplicative white noise cases, we establish the convergence of solutions and upper semicontinuity of pullback random attractors for Wong–Zakai approximations of stochastic CBF equations towards the pullback random attractors for stochastic CBF equations when the correlation time of colored noise converges to zero.

Pullback Random Attractors for Wong–Zakai Approximations of 2D Stochastic NSE on Unbounded Poincaré Domains

In this appendix, we discuss the existence and uniqueness of $\mathfrak {D}$-pullback random attractors for Wong–Zakai approximations of 2D stochastic NSE on Poincaré domains $\mathcal {O}$ (that is, satisfying Assumption 1.7) with nonlinear diffusion term. Consider the Wong–Zakai approximations of 2D NSE on $\mathcal {O}$ as

$$\begin{aligned} \left\{ \begin{aligned}\frac{\partial \varvec{u}}{\partial t}-\nu \Delta \varvec{u}+(\varvec{u}\cdot \nabla )\varvec{u}+\nabla p&=\varvec{f}(t) + S(t,x,\varvec{u})\mathcal {Z}_{\delta }(\vartheta _t\omega ),{} & {} \text{ in } \ \mathcal {O}\times (\mathfrak {s},\infty ), \\ \nabla \cdot \varvec{u}&=0,{} & {} \text{ in } \ \mathcal {O}\times (\mathfrak {s},\infty ), \\ \varvec{u}&={{\textbf {0}}}{} & {} \text{ on } \ \partial \mathcal {O}\times (\mathfrak {s},\infty ), \\ \varvec{u}|_{t=\mathfrak {s}}&=\varvec{u}_{\mathfrak {s}},{} & {} x\in \mathcal {O} \text{ and } \mathfrak {s}\in \mathbb {R}, \end{aligned} \right. \end{aligned}$$

(A.1)

where $\nu$ is the coefficient of kinematic viscosity of the fluid. In the work [23], authors proved the existence of a unique $\mathfrak {D}$-pullback random random attractor for the system (A.1) under Assumption 1.3. Here, we assume that the following conditions are satisfied:

Assumption A.1

Let $S:\mathbb {R}\times \mathcal {O}\times \mathbb {R}^2\rightarrow \mathbb {R}^2$ be a continuous function such that for all $t\in \mathbb {R}$, $x\in \mathcal {O}$ and ${\textbf {u}}\in \mathcal {O}$

$$\begin{aligned} |S(t,x,{\varvec{u}})|&\le |\mathcal {S}_1(t,x)||{\varvec{u}}|^{q-1}+|\mathcal {S}_2(t,x)|, \end{aligned}$$

where $1\le q<2$, $\mathcal {S}_1\in \text {L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{\frac{2}{2-q}}(\mathcal {O}))$ and $\mathcal {S}_2\in \text {L}^{\infty }_{loc }(\mathbb {R};\mathbb {L}^{2}(\mathcal {O}))$. Furthermore, suppose that $S(t,x,\varvec{u})$ is locally Lipschitz continuous with respect to $\varvec{u}$.

Remark A.2

Note that Assumption A.1 is clearly different from Assumption 1.3. Therefore, it is worth to prove the results under Assumption A.1.

Assumption A.3

We assume that the external forcing term $\varvec{f}\in \text {L}^2_{\text {loc}}(\mathbb {R};\mathbb {L}^2(\mathcal {O}))$ satisfies

$$\begin{aligned} \int _{-\infty }^{\mathfrak {s}} e^{\nu \lambda \xi }\Vert \varvec{f}(\cdot ,\xi )\Vert ^2_{\mathbb {L}^2(\mathcal {O})}\textrm{d}\xi <\infty , \ \ \text { for all } \mathfrak {s}\in \mathbb {R}. \end{aligned}$$

and for every $c>0$

$$\begin{aligned} \lim _{\tau \rightarrow -\infty }e^{c\tau }\int _{-\infty }^{0} e^{\nu \lambda \xi }\Vert \varvec{f}(\cdot ,\xi +\tau )\Vert ^2_{\mathbb {L}^2(\mathcal {O})}\textrm{d}\xi =0, \end{aligned}$$

where $\lambda$ and $\nu$ are given in (1.10) and (A.1), respectively.

Since, 2D CBF equations with $r=1$ is a linear perturbation of 2D NSE, one can prove the next theorem using the same arguments as it is carried for 2D random CBF equations (2.12) with $d=2$ and $r=1$ in Sect. 4.2. Moreover, there are only minor changes, hence we omit the proof here.

Theorem A.4

Assume that Assumptions A.1 and A.3 hold true. Then there exists a unique $\mathfrak {D}$-pullback random attractor for the system (A.1), in $\mathbb {L}^2(\mathcal {O})$.