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In this article, we consider a novel version of three-dimensional (3D) globally modified Navier-Stokes (GMNS) system introduced by [Caraballo et. al., Adv. Nonlinear Stud. (2006), 6:411-436], which is very significant from the perspective of deterministic as well as stochastic partial differential equations. Our focus is on examining a stochastic version of the suggested 3D GMNS equations that are perturbed by an infinite-dimensional additive noise. We can consider a rough additive noise (Lebesgue space valued) with this model, which is not appropriate to consider with the system presented in [Caraballo et. al., Adv. Nonlinear Stud. (2006), 6:411-436]. One of the technical problems associated with the rough noise is overcome by the use of the corresponding Cameron-Martin (or reproducing kernel Hilbert) space. This article aims to accomplish three objectives. Firstly, we establish the existence and uniqueness of weak solutions (in the analytic sense) of the underlying stochastic system using a Doss-Sussman transformation and a primary ingredient Minty-Browder technique. Secondly, we demonstrate the existence of random attractors for the underlying stochastic system in the natural space of square integrable divergence-free functions. Finally, we show the existence of an invariant measure for the underlying stochastic system for any viscosity coefficient $ \nu > 0 $ and uniqueness of invariant measure for sufficiently large $\nu$ by using the exponential stability of solutions. A validation of the proposed version of 3D GMNS equations has also been discussed in the appendix by establishing that the sequence of weak solutions of 3D GMNS equations converges to a weak solution of 3D Navier-Stokes equations as the modification parameter goes to infinity.

Here we construct infinitely many H\"older continuous global-in-time and stationary solutions to the stochastic Euler and hypodissipative Navier-Stokes equations in the space $C(\mathbb{R};C^{\vartheta})$ for $0<\vartheta<\frac{5}{7}\beta$, with $0<\beta< \frac{1}{24}$ and $0<\beta<\min\left\{\frac{1-2\alpha}{3},\frac{1}{24}\right\}$ respectively. A modified stochastic convex integration scheme, using Beltrami flows as building blocks and propagating inductive estimates both pathwise and in expectation, plays a pivotal role to improve the regularity of H\"older continuous solutions for the underlying equations. As a main novelty with respect to the related literature, our result produces solutions with noteworthy H\"older exponents., Comment: 38 pages. arXiv admin note: substantial text overlap with arXiv:2401.09894 by other authors

Background: In the absence of prospective data on diabetic foot ulcers (DFU), cross-sectional associations with causal risk factors (peripheral neuropathy, and peripheral arterial disease (PAD)) could be used to establish the validity of plantar thermography for DFU risk stratification. Methods: First, we investigated the associations between the intrinsic clusters of plantar thermographic images with several DFU risk factors using an unsupervised deep-learning framework. We then studied associations between obtained thermography clusters and DFU risk factors. Second, to identify those associations with predictive power, we used supervised learning to train Convolutional Neural Network (CNN) regression/classification models that predicted the risk factor based on the thermograph (and visual) input. Findings: Our dataset comprised 282 thermographs from type 2 diabetes mellitus patients (aged 56.31 +- 9.18 years, 51.42 % males). On clustering, we found two overlapping clusters (silhouette score = 0.10, indicating weak separation). There was strong evidence for associations between assigned clusters and several factors related to diabetic foot ulceration such as peripheral neuropathy, PAD, number of diabetes complications, and composite DFU risk prediction scores such as Martins-Mendes, PODUS-2020, and SIGN. However, models predicting said risk factors had poor performances. Interpretation: The strong associations between intrinsic thermography clusters and several DFU risk factors support the validity of using thermography for characterising DFU risk. However, obtained associations did not prove to be predictive, likely due to, spectrum bias, or because thermography and classical risk factors characterise incompletely overlapping portions of the DFU risk construct. Our findings highlight the challenges in standardising ground truths when defining novel digital biomarkers., Comment: 13 pages, 2 Figures, 1 Table. Supplementary files and link to code to be uploaded

In this article, we discuss the existence and asymptotically autonomous robustness (AAR) (almost surely) of random attractors for 3D stochastic globally modified Navier-Stokes equations (SGMNSE) on Poincar\'e domains (which may be bounded or unbounded). Our aim is to investigate the existence and AAR of random attractors for 3D SGMNSE when the time-dependent forcing converges to a time-independent function under the perturbation of linear multiplicative noise as well as additive noise. The main approach is to provide a way to justify that, on some uniformly tempered universe, the usual pullback asymptotic compactness of the solution operators is uniform across an infinite time-interval $(-\infty,\tau]$. The backward uniform ``tail-smallness'' and ``flattening-property'' of the solutions over $(-\infty,\tau]$ have been demonstrated to achieve this goal. To the best of our knowledge, this is the first attempt to establish the existence as well as AAR of random attractors for 3D SGMNSE on unbounded domains., Comment: arXiv admin note: text overlap with arXiv:2208.06808

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-dimensional torus: \begin{align*} \frac{\partial\boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta\vert \boldsymbol{y}\vert ^{r-1}\boldsymbol{y}+\gamma\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0, \end{align*} where $d\in\{2,3\}$, $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$. We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of $m$-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.

In this article, we establish the \textsl{Wong-Zakai approximation} result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection-diffusion equation, the Cahn-Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo-Galerkin approximation method, compactness arguments, and Prokhorov's and Skorokhod's representation theorems to ensure the existence of a \textsl{probabilistically weak solution} for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada-Watanabe theorem allows us to conclude the existence of a \textsl{probabilistically strong solution} (analytically weak solution) for the approximating system. Subsequently, we establish the Wong-Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong-Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work's functional framework.

The two dimensional stochastic Euler equations (EE) perturbed by a linear multiplicative noise of It\^o type on the bounded domain $\mathcal{O}$ have been considered in this work. Our first aim is to prove the existence of \textsl{global weak (analytic) solutions} for stochastic EE when the divergence free initial data $\boldsymbol{u}^*\in\mathbb{H}^1(\mathcal{O})$, and the external forcing $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{H}^1(\mathcal{O}))$. In order to prove the existence of weak solutions, a vanishing viscosity technique has been adopted. In addition, if $\mathrm{curl}\ \boldsymbol{u}^*\in\mathrm{L}^{\infty}(\mathcal{O})$ and $\mathrm{curl}\ \boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$, we establish that the global weak (analytic) solution is unique. This work appears to be the first one to discuss the existence and uniqueness of global weak (analytic) solutions for stochastic EE driven by linear multiplicative noise. Secondly, we prove the existence of a \textsl{pullback stochastic weak attractor} for stochastic \textsl{non-autonomous} EE using the abstract theory available in the literature. Finally, we propose an abstract theory for \textsl{weak asymptotic autonomy} of pullback stochastic weak attractors. Then we consider the 2D stochastic EE perturbed by a linear multiplicative noise as an example to discuss that how to prove the weak asymptotic autonomy for concrete stochastic partial differential equations. As EE do not contain any dissipative term, the results on attractors (deterministic and stochastic) are available in the literature for dissipative (or damped) EE only. Since we are considering stochastic EE without dissipation, all the results of this work for 2D stochastic EE perturbed by a linear multiplicative noise are totally new.

The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential \begin{equation*} \frac{\partial \boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta|\boldsymbol{y}|^{r-1}\boldsymbol{y}+\nabla p+\Psi(\boldsymbol{y})\ni\boldsymbol{g},\ \nabla\cdot\boldsymbol{y}=0, \end{equation*} in a $d$-dimensional torus is considered in this work, where $d\in\{2,3\}$, $\mu,\alpha,\beta>0$ and $r\in[1,\infty)$. For $d=2$ with $r\in[1,\infty)$ and $d=3$ with $r\in[3,\infty)$ ($2\beta\mu\geq 1$ for $d=r=3$), we establish the existence of \textsf{\emph{a unique global strong solution}} for the above multi-valued problem with the help of the \textsf{\emph{abstract theory of $m$-accretive operators}}. %for nonlinear differential equations of accretive type in Banach spaces. Moreover, we demonstrate that the same results hold \textsf{\emph{local in time}} for the case $d=3$ with $r\in[1,3)$ and $d=r=3$ with $2\beta\mu<1$. We explored the $m$-accretivity of the nonlinear as well as multi-valued operators, Yosida approximations and their properties, and several higher order energy estimates in the proofs. For $r\in[1,3]$, we {quantize (modify)} the Navier-Stokes nonlinearity $(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}$ to establish the existence and uniqueness results, while for $r\in[3,\infty)$ ($2\beta\mu\geq1$ for $r=3$), we handle the Navier-Stokes nonlinearity by the nonlinear damping term $\beta|\boldsymbol{y}|^{r-1}\boldsymbol{y}$. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.

This article is concerned with the \emph{asymptotically autonomous robustness} (almost surely and in probability) of non-autonomous random attractors for two stochastic versions of 3D convective Brinkman-Forchheimer (CBF) equations defined on the whole space $\mathbb{R}^3$: $$\frac{\partial\boldsymbol{v}}{\partial t}-\mu \Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v} +\alpha\boldsymbol{v}+ \beta|\boldsymbol{v}|^{r-1}\boldsymbol{v}+\nabla p=\boldsymbol{f}(t)+``\mbox{stochastic terms}",\quad \nabla\cdot\boldsymbol{v}=0,$$ with initial and boundary vanishing conditions, where $\mu,\alpha,\beta >0$, $r\geq1$ and $\boldsymbol{f}(\cdot)$ is a given time-dependent external force field. By the asymptotically autonomous robustness of a non-autonomous random attractor $ \mathscr{A}=\{ \mathscr{A}(\tau,\omega): \tau\in\mathbb{R}, \omega\in\Omega\}$ we mean its time-section $\mathscr{A}(\tau,\omega)$ is robust to a time-independent random set as time $\tau$ tends to negative infinity according to the Hausdorff semi-distance of the underlying space. Our goal is to study this topic, almost surely and in probability, for the non-autonomous 3D CBF equations when the stochastic term is a linear multiplicative or additive noise, and the time-dependent forcing converges towards a time-independent function. Our main results contain two cases: i) $r\in(3,\infty)$ with any $\beta,\mu>0$; ii) $r=3$ with $2\beta\mu\geq1$. The main procedure to achieve our goal is how to justify that the usual pullback asymptotic compactness of the solution operators is uniform on some \emph{uniformly} tempered universes over an \emph{infinite} time-interval $(-\infty,\tau]$. This can be done by a method based on Kuratowski's measure of noncompactness., Comment: arXiv admin note: text overlap with arXiv:2208.06808

This article concerns the random dynamics and asymptotic analysis of the well known mathematical model, the Navier-Stokes equations. We consider the two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a \textsl{linear multiplicative white noise of It\^o type} on the whole space $\mathbb{R}^2$. Firstly, we prove that the non-autonomous 2D SNSE generates a bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $\mathbb{L}^2(\mathbb{R}^2)$, then there exists a unique bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-pullback random attractor for non-autonomous SNSE which is compact and attracting not only in $\mathbb{L}^2$-norm but also in $\mathbb{H}^1$-norm. Next, as a consequence of the existence of pullback random attractors, we prove the existence of a family of invariant sample measures for non-autonomous random dynamical system generated by 2D non-autonomous SNSE. Moreover, we show that the family of invariant sample measures satisfies a stochastic Liouville type theorem. Finally, we discuss the existence of an invariant measure for the random cocycle associated with 2D autonomous SNSE. We prove the uniqueness of invariant measures for $\boldsymbol{f}=\mathbf{0}$ and for any $\nu>0$ by using the linear multiplicative structure of the noise coefficient and exponential stability of solutions. The above results for SNSE defined on $\mathbb{R}^2$ are totally new, especially the results on bi-spatial random attractors and stochastic Liouville type theorem for 2D SNSE with linear multiplicative noise are obtained in any kind of domains for the first time.

The asymptotically autonomous robustness of random attractors of stochastic fluid equations defined on \emph{bounded} domains has been considered in the literature. In this article, we initially consider this topic (almost surely and in probability) for a non-autonomous stochastic 2D Navier-Stokes equation driven by additive and multiplicative noise defined on some \emph{unbounded Poincar\'e domains}. There are two significant keys to study this topic: what is the asymptotically autonomous limiting set of the time-section of random attractors as time goes to negative infinity, and how to show the precompactness of a time-union of random attractors over an \emph{infinite} time-interval $(-\infty,\tau]$. We guess and prove that such a limiting set is just determined by the random attractor of a stochastic Navier-Stokes equation driven by an autonomous forcing satisfying a convergent condition. The uniform "tail-smallness" and "flattening effecting" of the solutions are derived in order to justify that the usual asymptotically compactness of the solution operators is \emph{uniform} over $(-\infty,\tau]$. This in fact leads to the precompactness of the time-union of random attractors over $(-\infty,\tau]$. The idea of uniform tail-estimates due to Wang \cite{UTE-Wang} is employed to overcome the noncompactness of Sobolev embeddings on unbounded domains. Several rigorous calculations are given to deal with the pressure terms when we derive these uniform tail-estimates.

This article concerns the long-term random dynamics in regular spaces for a non-autonomous Navier-Stokes equation defined on a bounded smooth domain $\mathcal{O}$ driven by multiplicative and additive noise. For the two kinds of noise driven equations, we demonstrate the existence of a unique pullback attractor which is backward compact and asymptotically autonomous in $\mathbb{L}^2(\mathcal{O})$ and $\mathbb{H}_0^1(\mathcal{O})$, respectively. The backward-uniform flattening property of the solution is used to prove the backward-uniform pullback asymptotic compactness of the non-autonomous random dynamical systems in the regular space $\mathbb{H}_0^1(\mathcal{O})$.

In this work, we demonstrate the Wong-Zakai approximation results for two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains. Even though the existence and uniqueness of a pathwise strong solution to SCBF equations is known, the existence of a unique solution to the approximating system is not immediate from the solvability results of SCBF equations, and we prove it by using Faedo-Galerkin approximation technique and monotonicity arguments. Moreover, as an application of the Wong-Zakai approximation, we obtain the support of the distribution of solutions to SCBF equations.

The asymptotic behavior of solutions of two dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations in unbounded domains is discussed in this work (for example, Poincar\'e domains). We first prove the existence of $\mathbb{H}^1$-random attractors for the stochastic flow generated by 2D SCBF equations (for the absorption exponent $r\in[1,3]$) perturbed by an additive noise on Poincar\'e domains. Furthermore, we deduce the existence of a unique invariant measure in $\mathbb{H}^1$ for the 2D SCBF equations defined on Poincar\'e domains. In addition, a remark on the extension of these results to general unbounded domains is also discussed. Finally, for 2D SCBF equations forced by additive one-dimensional Wiener noise, we prove the upper semicontinuity of the random attractors, when the domain changes from bounded to unbounded (Poincar\'e).

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 bounded and unbounded ($\mathbb{R}^d$ for $d=2,3$) domains is discussed in this work. To establish the existence of random pullback 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 \emph{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. The presence of linear damping term helps to establish the results in the whole domain $\mathbb{R}^d$. The nonlinear damping term supports to obtain better results in 3D and also for a large class of nonlinear diffusion terms. Moreover, we prove the existence of a unique random pullback attractor for stochastic CBF equations with additive white noise. Finally, for additive as well as multiplicative noise case, we establish the convergence of solutions and upper semicontinuity of random pullback attractors for Wong-Zakai approximations of stochastic CBF equations towards the random pullback attractors for stochastic CBF equations when correlation time of colored noise converges to zero.

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counter part with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$\frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2\beta\mu\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with multiplicative white noise. Finally, we establish the upper semicontinuity of the random pullback attractor, that is, the random pullback attractor converges towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the solution is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $\alpha\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations.

In this paper, we consider an inverse problem for three dimensional viscoelastic fluid flow equations, which arises from the motion of Kelvin-Voigt fluids in bounded domains (a hyperbolic type problem). This inverse problem aims to reconstruct the velocity and kernel of the memory term simultaneously, from the measurement described as the integral over determination condition. By using the contraction mapping principle in an appropriate space, a local in time existence and uniqueness result for the inverse problem of Kelvin-Voigt fluids are obtained. Furthermore, using similar arguments, a global in time existence and uniqueness results for an inverse problem of Oseen type equations are also achieved.

This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in periodic domains: $$\frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove that the global attractor of the above system is a singleton under small forcing intensity ($r\geq 1$ for $n=2$ and $r\geq 3$ for $n=3$ with $2\beta\mu\geq 1$ for $r=n=3$). After perturbing the above system with additive or multiplicative white noise, the random attractor does not have a singleton structure. But we obtain that the random attractor for 2D stochastic CBF equations with additive and multiplicative white noise converges towards the deterministic singleton attractor for $1\leq r\leq 2$ and $1\leq r<\infty$, respectively, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). Interestingly in the case of 3D stochastic CBF equations perturbed by multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for $3\leq r\leq 5$ ($2\beta\mu\geq 1$ for $r=3$), when the coefficient of random perturbation converges to zero.

The large time behavior of the deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations for $r\geq3$ ($r>3$, for any $\mu$ and $\beta$, and $r=3$ for $2\beta\mu\geq1$), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Finally, we establish the upper semicontinuity of random attractor for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero.

This work is concerned about the asymptotic behavior of the solutions of the two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations driven by white noise with nonlinear diffusion terms. We prove the existence and uniqueness of weak pullback mean random attractors for the 2D SCBF equations (for $r\geq1$) as well as 3D SCBF equations (for $r>3$, any $\mu,\beta>0$ and for $r=3$, $2\mu\beta\geq1$) in Bochner spaces, when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we establish the existence of weak pullback mean random attractors for a class of locally monotone stochastic partial differential equations.

In this work, we discuss the large time behavior of the solutions of the two dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations in bounded domains. Under the functional setting $\V\hookrightarrow\H\hookrightarrow\V'$, where $\H$ and $\V$ are appropriate separable Hilbert spaces and the embedding $\V\hookrightarrow\H$ is compact, we establish the existence of random attractors in $\H$ for the stochastic flow generated by the 2D SCBF equations perturbed by small additive noise. We prove the upper semicontinuity of the random attractors for the 2D SCBF equations in $\H$, when the coefficient of random term approaches zero. Moreover, we obtain the existence of random attractors in a more regular space $\V$, using the pullback flattening property. The existence of random attractors ensures the existence of invariant compact random set and hence we show the existence of an invariant measure for the 2D SCBF equations.

In this work, we consider the two and three-dimensional stochastic convective Brinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additive white noise $$\mathrm{d}\boldsymbol{u}-[\mu \Delta\boldsymbol{u}-(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}-\alpha\boldsymbol{u}-\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}-\nabla p]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\mathrm{d}\mathrm{W},\ \nabla\cdot\boldsymbol{u}=0,$$ for $r\in[1,\infty),$ $\mu,\alpha,\beta>0$ in unbounded domains (like Poincar\'e domains) $\mathcal{O}\subset\mathbb{R}^d$ ($d=2,3$) where $\mathrm{W}(\cdot)$ is a Hilbert space valued Wiener process on some given filtered probability space, and discuss the asymptotic behavior of its solution. For $d=2$ with $r\in[1,\infty)$ and $d=3$ with $r\in[3,\infty)$ (for $d=r=3$ with $2\beta\mu\geq 1$), we first prove the existence and uniqueness of a weak solution (in the analytic sense) satisfying the energy equality for SCBF equations driven by an irregular additive white noise in Poincar\'e domains by using a Faedo-Galerkin approximation technique. Since the energy equality for SCBF equations is not immediate, we construct a sequence which converges in Lebesgue and Sobolev spaces simultaneously and it helps us to demonstrate the energy equality. Then, we establish the existence of random attractors for the stochastic flow generated by the SCBF equations. One of the technical difficulties connected with the irregular white noise is overcome with the help of the corresponding Cameron-Martin space (or Reproducing Kernel Hilbert space). Finally, we address the existence of a unique invariant measure for 2D and 3D SCBF equations defined on Poincar\'e domains (bounded or unbounded). Moreover, we provide a remark on the extension of the above mentioned results to general unbounded domains also.