In fluid mechanics, Stokes system describes the linear flow of a viscous fluid. The case of a constant viscosity was extensively studied in the literature (see for instance [14], [10], and [27]). The aim of this paper is to study the localization, multiplicity and homogenization for stationary Stokes systems with variable viscosity in bounded domains. The viscosity is a characteristic of the nature of the fluid and in our case it is supposed to depend on the position. This describes the variable properties of the fluid, as for instance in the case of complex multiphase flow. As applications of such flow one can mention a wide range of engineering problems such as power generation, water treatment, water desalination, refrigeration and air conditioning, and carbon capture and sequestration. We may refer the reader to [20], [8], [11] for the Stokes flow with variable viscosity in thin domains, and to [1], [2] for homogenization results.

We start our study with the Stokes system (2.1) in pseudo-stress-velocity formulation (see for instance [6], [7], and [16]), modeling the steady-state motion under an external force $f$, of an incompressible fluid with variable viscosity $\mu$, where $u$ represents the velocity of the fluid and $p$ is the pressure. Our first objective is to study the localization and the multiplicity of the solution for a system similar to (2.1), if the right-hand side in the first equation depends in a nonlinear manner on the velocity (see (2.11)). From the physical point of view, this corresponds to a reaction external force. A first practical application of our study is to find a velocity dependent reaction force which guarantees that each component of the velocity stays bounded between two a priori given bounds. A second application is to find bounds for the velocity when the reaction external force is given.

In general, two essential tools for proving the localization and multiplicity are the maximum principle and Moser-Harnack type inequalities (see [23]). In order to compensate the loss of these tools in the case of Stokes system, the idea is to associate to system (2.1) (and thus to system (2.11)) a diffusion problem. To do so, we notice that in (2.1) the term $\nabla p$ can be seen as a correction of the external force $f$, in order to keep the incompressibility of the fluid. Through the system (2.6) we associate to $\nabla p$ the function $q$, which can be seen as a correction of the velocity $u$ and then we define $w=u+q$, which solves the diffusion problem (2.7) and so $w$ can be interpreted as the recovered velocity for problem (2.1), under the external force $f$ and a constant pressure. Consequently, following this idea, we can match a Stokes system with variable viscosity and a corresponding diffusion problem with diffusion coefficients varying with space (see for instance [5]). Once we do this, we prove localization and multiplicity results for the diffusion problem, which will give localization and multiplicity results for the recovered velocity of the Stokes system.

Except for the case of the Navier-Stokes system, where the velocity-dependent convective term can be seen as a reaction force, and for the case of the Coriolis force (see, e.g., [9], [17]), up to our knowledge, there are few mathematical studies in the literature on Stokes systems with a more general reaction force. In this paper we first consider nonlinear reaction forces of the type $\ h\left(x,u\left(x\right)+q\left(x\right)\right)$ (see (2.11)). Thus the external force $h$ is sensitive to the recovered velocity $\ w=u+q\$ incorporating both velocity $u$ and pressure $p$ (the last one by means of $q$, which is a correction of the velocity). The main result concerning this first system in given in Theorem 4. Then, we consider the special case of planar Stokes systems with a constant pressure, namely such that $\nabla p=0.$ We start by giving a sufficient condition on the right-hand side, which guarantees that the pressure of the planar Stokes system is constant. Then we are able to discuss the case of velocity-dependent reaction forces, to localize the velocity $u$ itself, and to obtain multiple solutions with distinct velocity components. The main result concerning this second case in given in Theorem 11. Following the ideas in our previous work [4], we end this paper by giving localization and multiplicity results for the two previous cases, under the hypothesis that the viscosity is highly heterogeneous, being an $\varepsilon$-periodic function on each spatial direction, with $\varepsilon$ a small positive parameter that we will let tend to zero. From the physical point of view, this situation modelizes for instance a particular mixture of fluids. We refer to [1] for the homogenization of the Stokes system in pseudo-stress-velocity formulation with $\varepsilon$-periodic viscosity. For homogenization problems with nonlinear reaction terms we refer for instance to [19] and [12]. Localization and multiplicity results for a nonlinear right-hand side in the homogenization context are given in Theorem 15 and Theorem 17.

The paper is organized as follows. In Section 2 we state the problem and we give some preliminary results. Section 3 deals with the localization and multiplicity results for the two cases under study. In Section 4 we give analogous results in the periodic homogenization context.

We consider the mathematical model for the steady-state motion of an incompressible fluid with variable viscosity given by the pseudo-stress-velocity formulation Stokes system (2.1) below

Here $\Omega\subset\mathbb{R}^{n}\$ ($n\geq 2$)  is a $C^{2}$ bounded open set, $\mu\in C^{1}\left(\overline{\Omega}\right)$ with $0<\underline{\mu}\leq\mu\left(x\right)\leq\overline{\mu}$ and $f=\left(f_{1},..,f_{n}\right)\in L^{2}\left(\Omega;\mathbb{R}^{n}\right)$. Physically, there are relevant the cases $n=2$ and $n=3,$ where $\mu\left(x\right)$ stands for the variable viscosity of the fluid, $f\left(x\right)$ for the external force, the unknown functions $u$ and $p$ are the velocity and pressure, respectively, while the condition $\operatorname{div}\,u=\sum_{i=1}^{n}\partial_{x_{i}}u_{i}=0$ means that the fluid is incompressible. On the boundary of the domain $\Omega$ we consider no-slip condition.

It is known (see, e.g., [10], [14], [18], [25], [27]) that a weak solution exists and is unique, that is a pair $\left(u,p\right)$ with $u\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right),$ $\operatorname{div}\,u=0,$ $p\in L^{2}\left(\Omega\right)$, $p$ being unique up to an additive constant, and

where $~{}\left(\left(u,v\right)\right)_{H_{0}^{1}}=\int_{\Omega}\mu\left(x\right)\nabla u\cdot\nabla v\,dx$. By $\ \nabla u\cdot\nabla v$ we mean $\ \sum\limits_{i=1}^{n}\nabla u_{i}\cdot\nabla v_{i}$ and $\ \left(f,v\right)_{L^{2}}=\sum\limits_{i=1}^{n}\left(f_{i},v_{i}\right)_{L^{2}}.~{}$Also $u$ is the unique function in $V=\left\{v\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right):\ \operatorname{div}\,v=0\right\}$ satisfying

Indeed, for any $p\in L^{2}\left(\Omega\right)$ one has $\nabla p$ $\in H^{-1}\left(\Omega;\mathbb{R}^{n}\right)$ and $\left(\nabla p,v\right)=\left(p,\operatorname{div}\,v\right)_{L^{2}}=0$ for all $v\in V.$ Thus we may define $S$, the velocity operator given by

where $u_{f}$ is the unique solution in $V$ solving problem (2.3).

In addition, according for instance to [14, Theorem 5.3], we have that if $J\left(v,q\right)$ is defined by

then

and $u$ minimizes in $V$ the functional

Denoting $\ Lv:=-\operatorname{div}\left(\mu\left(x\right)\nabla v\right),$ we define the operator $\ L^{-1}:H^{-1}\left(\Omega;\mathbb{R}^{n}\right)\rightarrow H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)$ where $\ L^{-1}f=v_{f},\$ with $\ v_{f}\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)$ solving the Dirichlet problem

Letting $\ q:=L^{-1}\nabla p$ for $\nabla p$ as in (2.1), one has

We define $w=u+q$ and let  Pr${}_{V}\$ be the projection operator on $V$. Then, by using (2.6), the system (2.1) can be rewritten under the equivalent form of a diffusion problem with a variable diffusion coefficient

Indeed, if $w\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)$ solves (2.8), then the pair $\left(u,p\right)$ where $u=$Pr${}_{V}w$ and $\nabla p=L\left(w-u\right)$ solves the Stokes system (2.1). Let us notice that the existence and the uniqueness of the solution of problem (2.8) is a direct consequence of the Lax-Milgram theorem.

If in the case of system (2.1), $\nabla p$ can be seen as an intrinsic correction of the external force $f$ in order to keep the incompressibility of the fluid, in case of system (2.8), we may see the term $q$ as a correction of the velocity due to the pressure and which is also necessary to express the incompressibility. Thus $w=u+q$ can be seen as the recovered velocity under the external force $f$ and a constant pressure, or equivalently as the density of a diffusion process governed by the source density $f.$ As regards the relationship between $p$ and $q,$ namely the equality $q=L^{-1}\nabla p,$ by which the pressure $p$ is covered into a velocity, we note that $q$ appears as the solution of the Dirichlet problem (2.6), that is, in terms of diffusion, as a diffusion density due to the source density $\nabla p.$

By associating a diffusion problem to the Stokes model, we are able to compensate the loss in case of the Stokes problem of the maximum principle and of Moser-Harnack type inequalities, some essential tools for the localization and multiplicity of solutions (see [23]), which are the main objectives of our paper. By means of this association, we can however discuss the multiplicity of solutions for the Stokes problem with a reaction force. Let us notice that except for the case of the Navier-Stokes system, where the velocity-dependent convective term $(u\cdot\nabla)u$ can be seen as a reaction force, and for the case of the inertial Coriolis force $\omega\times u$ (see, e.g., [9], [17]), up to our knowledge, there are few mathematical studies in the literature on Stokes systems with a more general reaction force. In this paper we consider nonlinear reaction forces of the type $\ h\left(x,u\left(x\right)+q\left(x\right)\right)$, with $q=L^{-1}\nabla p$. Thus the external force $h$ is sensitive to the recovered velocity $\ w=u+q\$ and incorporates both the velocity and the pressure of the Stokes system, by means of the correction function $q$ (see (2.11)).

Coming back to the association between a diffusion problem and the Stokes system, we remark that if for the reaction-diffusion problem

we are able to localize two solutions $w$ and $\overline{w}$ in the following form

where $\Omega^{\prime}\subset\subset\Omega$ is a subdomain of $\Omega,$ the vectors $r,\overline{r},R,\overline{R}$ belong to $\mathbb{R}_{+}^{n}$ with $0<r<R\lvertneqq\overline{r}<\overline{R},$ and the inequalities are understood component-wise, then the corresponding solutions $\left(u,p\right),$ $\left(\overline{u},\overline{p}\right)$ of the reaction Stokes system

are distinct. Indeed, if the solutions are not distinct we have $\ w=u+q=\overline{u}+\overline{q}=\overline{w}$ $\left(u=\text{Pr}_{V}w,\,\overline{u}=\text{Pr}_{V}\overline{w},\,q=L^{-1}\nabla p,\,\overline{q}=L^{-1}\nabla\overline{p}\right),$ which is clearly excluded by the localization relations (2.10), in view of the relation $\ R\lvertneqq\overline{r}.$

In what follows, the localization of solutions will be given in the set $D_{rR}$ defined by

For $i\in\left\{1,..,n\right\},$ we shall also use the notation

Notice that in general system (2.9) does not have a variational form. However, each of its equations has one. Thus we may consider the partial energy functionals of (2.9), namely the functionals $\ J_{i}:H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)\rightarrow\mathbb{R}\ \ \left(i=1,..,n\right),$ defined by

which will allow to characterize the solution $w$ of the stationary model (2.9) in terms of Nash equilibrium.

To end this introductory section we recall some basic results from the theory of linear elliptic equations and the theory of matrices.

(R1) If $f(x)\geq 0$ for a.e. $x\in\Omega$, then the weak maximum principle (see for instance [13, Theorem 8.1]) guarantees that the solution $w=L^{-1}f$ of problem (2.8) satisfies $w\left(x\right)\geq 0$ a.e. in $\Omega.$

(R2) If $f\in L^{\infty}(\Omega)$ then, according to [13, Theorem 8.15], the solution $w$ of problem (2.8) belongs to $L^{\infty}(\Omega)$ and there is a constant $\Gamma_{f}$ independent of $\mu$ which satisfies $0<\underline{\mu}\leq\mu\left(x\right)\leq\overline{\mu},$ such that

(R3) For each $f\in L^{\infty}\left(\Omega\right)$ with $f\geq 0$ in $\Omega$ and $f>0$ on a subset of nonzero measure of $\Omega,$ there exists $\gamma_{f}>0$ such that the solution $w$ of problem (2.8) verifies $\ w\left(x\right)\geq\gamma_{f}\$ for a.e. $x$ in the fixed subdomain $\Omega^{\prime}\subset\subset\Omega.$ For $\mu$ which satisfies $0<\underline{\mu}\leq\mu\left(x\right)\leq\overline{\mu},$ the constant $\ \gamma_{f}$   can be chosen to be independent of $\mu$ (see [15, Theorem 12.1.2]).

For a square matrix $\mathcal{M}$ with nonnegative entries, the spectral radius $\rho\left(\mathcal{M}\right)$ is the maximum of the modulus of its eigenvalues $\lambda,$ i.e., of all $\lambda\in\mathbb{C}$ with $\ \det\left(\mathcal{M}-\lambda I\right)=0.$ There are known the following characterizations of such kind of matrices:

(a) $\mathcal{M}^{k}$ tends to the zero matrix as $k\rightarrow\infty;$

(b) The matrix $I-\mathcal{M}$ is invertible and the entries of $\left(I-\mathcal{M}\right)^{-1}$ are also nonnegative.

We give now the main results of the paper. In Section 3.1, the localization and multiplicity results are proven for the recovered velocity of system (2.11), under the hypotheses $\mathbf{(h1)}$ and $\mathbf{(h2)}$ below on the nonlinear right-hand side data function. In Section 3.2 we deal with the case of a uniform pressure. In this case, the recovered velocity and the velocity of the Stokes system coincide and thus the localization and the multiplicity results are obtained for the velocity itself. We first give the sufficient condition (3.10) in order the pressure of a Stokes system to be uniform and then we prove localization results for the system (3.11), under the hypotheses (3.16) on the right-hand side data function. These latest results are proven in the planar case.