The stochastic convective Brinkman-Forchheimer (SCBF) equations in an open connected set $\mathcal{O}\subseteq\mathbb{R}^d$ ($d\in \{2,3,4\}$) or torus are considered in this work. We show the existence of a pathwise unique strong solution (in the probabilistic sense) satisfying the energy equality (Itô formula) to SCBF equations perturbed by multiplicative Gaussian noise. We exploited a monotonicity property of the linear and nonlinear operators as well as a stochastic generalization of the Minty-Browder technique in the proofs. The energy equality is obtained by approximating the solution using approximate functions constituting the elements of eigenspaces of a compact operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. We further discuss the global in time regularity results of such strong solutions on the torus. The exponential stability results (in mean square and pathwise sense) for the stationary solutions is also established in this work for large effective viscosity. Moreover, a stabilization result of the stochastic convective Brinkman-Forchheimer equations by using a multiplicative noise is obtained. Finally, when $\mathcal{O}$ is a bounded domain, we establish the existence of a unique invariant measure for the SCBF equations with multiplicative Gaussian noise, which is both ergodic and strongly mixing, using the exponential stability of strong solutions.
We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.
In this work, we study the Brinkman-Forchheimer equations driven under slip boundary conditions of friction type. We prove the existence and uniqueness of weak solutions by means of regularization combined with the Faedo-Galerkin approach. Next we discuss the continuity of the solution with respect to Brinkman's and Forchheimer's coefficients. Finally, we show that the weak solution of the corresponding stationary problem is stable.
We give a comprehensive study of the 3D Navier-Stokes-Brinkman-Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the corresponding uniform attractors
The nonlinear Brinkman-Forchheimer-extended Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with nonconstant porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy's range.
The purpose of this paper is to study boundary value problems of transmission type for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in two complementary Lipschitz domains on a compact Riemannian manifold of dimension 2 or 3. We exploit a layer potential method combined with a fixed point theorem in order to show existence and uniqueness results when the given data are suitably small in $L^2$-based Sobolev spaces.
We consider three-dimensional (3D) Boussinesq convection system of an incompressible fluid in a closed sample of a porous medium. Specifically, we introduce and analyze a 3D Brinkman-Forchheimer-Bénard convection problem describing the behavior of an incompressible fluid in a porous medium between two plates heated from the bottom and cooled from the top. We show the existence and uniqueness of global in-time solutions, and the existence of absorbing balls in $L^2$ and $H^1$. Eventually, we comment on the applicability of a data assimilation algorithm to our system.
We prove a robustness of regularity result for the $3$D convective Brinkman-Forchheimer equations $$ \partial_tu -μΔu + (u \cdot \nabla)u + \nabla p + αu + β\abs{u}^{r - 1}u = f, $$ for the range of the absorption exponent $r \in [1, 3]$ (for $r > 3$ there exist global-in-time regular solutions), i.e. we show that strong solutions of these equations remain strong under small enough changes of the initial condition and forcing function. We provide a smallness condition which is similar to the robustness conditions given for the $3$D incompressible Navier-Stokes equations by Chernyshenko et al. …
The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in ${\mathbb R}^3$, one of them is a bounded Lipschitz domain $Ω$ with connected boundary, and another one is the exterior Lipschitz domain ${\mathbb R}^3\setminus \overline{Ω}$. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, when…
In this paper we give a simple proof of the existence of global-in-time smooth solutions for the convective Brinkman-Forchheimer equations (also called in the literature the tamed Navier-Stokes equations) $$ \partial_tu -μΔu + (u \cdot \nabla)u + \nabla p + αu + β|u|^{r - 1}u = 0 $$ on a $3$D periodic domain, for values of the absorption exponent $r$ larger than $3$. Furthermore, we prove that global, regular solutions exist also for the critical value of exponent $r = 3$, provided that the coefficients satisfy the relation $4μβ\geq 1$. Additionally, we show that in the critical case every wea…
A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy-Forchheimer-Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observatio…
The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in $L_p$-based Besov spaces on a bounded Lipschitz domain in ${\mathbb R}^3$, with $p$ in a neighborhood of $2$. This system is obtained by adding the semilinear term $|{\bf u}|{\bf u}$ to the linear Brinkman equation. {First, we provide some results about} equivalence between the Gagliardo and non-tangential traces, as well as between the weak canonical conormal derivatives and the non-tangential conormal derivatives. Various mapping and invertibility p…
In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: \begin{align*} \boldsymbol{u}_t-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{F}:=f \boldsymbol{g}, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} in a bounded domain $Ω\subset\mathbb{R}^2$ with smooth boundary $\partialΩ$, where $α,β,μ>0$ and $r\in[1,3]$. The investigated inverse problem consists of reconstructing the vector-valued velocity function $\boldsymbol{u}$, the pressure field $p$ and t…
In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: \begin{align*} \boldsymbol{u}_t-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{F}:=\boldsymbol{f} g, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} in bounded domains $Ω\subset\mathbb{R}^d$ ($d=2,3$) with smooth boundary, where $α,β,μ>0$ and $r\in[1,\infty)$. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideratio…
In this work, we carry out the asymptotic analysis of the two dimensional convective Brinkman-Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and Poincaré domains (using asymptotic compactness property). In Poincaré domains, for $r=1,2$ and $3$, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors for the 2D CBF equations. We con…
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, Poincaré 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 Poincaré domains. Furthermore, we deduce the existence of a unique invariant measure in $\mathbb{H}^1$ for the 2D SCBF equations defined on Poincaré domains. In addition, a remark on the extension of these…
In this paper, we study analytically the long-time behavior of three-dimensional Brinkman-Forchheimer-extended Darcy model, in the context that the parameters related to the damping nonlinear term are unknown. This work is inspired by the approach firstly introduced for two-dimensional Navier-Stokes equations by Carlson, Hudson and Larios. We show estimates in L2 and H1 for large-time error between the true solution and the assimilated solution, which is constructed with the unknown damping parameters and observational measurements obtained continuously in time from a continuous data assimilat…
The purpose of this paper is the mathematical analysis of the weak solution of the mixed Dirichlet-Robin boundary value problem for the nonlinear Darcy-Forchheimer- Brinkman system in a bounded, two-dimensional Lipschitz domain, and the application of the corresponding results to the study of the lid-driven flow problem of an incompressible viscous fluid located in a square cavity filled with a porous medium. First we obtain a well-posedness result for the linear Brinkman system with Dirichlet-Neumann boundary conditions, employing a variational approach for the corresponding boundary integral…
Slightly compressible Brinkman-Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified.
In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtaining improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Speci…
This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in periodic domains: $$\frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ 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 $2βμ\geq 1$ for $r=n=3$). After perturbing the above system with a…
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)$: $$\frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0,$$ 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…
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 litera…
A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn-Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman-Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally n…
We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associat…
We consider a diffuse interface model for phase separation of an isothermal incompressible binary fluid in a Brinkman porous medium. The coupled system consists of a convective Cahn-Hilliard equation for the phase field $ϕ$, i.e., the difference of the (relative) concentrations of the two phases, coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for the fluid velocity $\mathbf{u}$. This equation incorporates a diffuse interface surface force proportional to $ϕ\nabla μ$, where $μ$ is the so-called chemical potential. We analyze the well-posedness of the resulting Cahn-Hil…
The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $φ$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $μ\nablaφ$, where $μ$ is the chemical poten…
