Abstract
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Renata Bunoiu
University of Lorraine, Metz, France
Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babes-Bolyai, University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
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R. Bunoiu, R. Precup, Localization and multiplicity for stationary Stokes systems with variable viscosity, Communications in Mathematical Analysis and Applications, (2025), 1-28, https://dx.doi.org/10.2139/ssrn.4887415
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Paper (preprint) in HTML form
Localization and multiplicity for stationary Stokes systems with variable viscosity
Abstract
In this paper we discuss the localization and the multiplicity of solutions for the stationary Stokes system with variable viscosity and a reaction force term. The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems.
1 Introduction
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 , of an incompressible fluid with variable viscosity , where represents the velocity of the fluid and 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 can be seen as a correction of the external force , in order to keep the incompressibility of the fluid. Through the system (2.6) we associate to the function , which can be seen as a correction of the velocity and then we define , which solves the diffusion problem (2.7) and so can be interpreted as the recovered velocity for problem (2.1), under the external force 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 (see (2.11)). Thus the external force is sensitive to the recovered velocity incorporating both velocity and pressure (the last one by means of , 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 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 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 -periodic function on each spatial direction, with 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 -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.
2 Statement of the problem and preliminary results
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
(2.1) |
Here () is a bounded open set, with and . Physically, there are relevant the cases and where stands for the variable viscosity of the fluid, for the external force, the unknown functions and are the velocity and pressure, respectively, while the condition means that the fluid is incompressible. On the boundary of the domain 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 with , being unique up to an additive constant, and
(2.2) |
where . By we mean and Also is the unique function in satisfying
(2.3) |
Indeed, for any one has and for all Thus we may define , the velocity operator given by
(2.4) |
where is the unique solution in solving problem (2.3).
In addition, according for instance to [14, Theorem 5.3], we have that if is defined by
then
and minimizes in the functional
Denoting we define the operator where with solving the Dirichlet problem
(2.5) |
Letting for as in (2.1), one has
(2.6) |
We define and let Pr be the projection operator on . 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
(2.7) |
or alternatively
(2.8) |
Indeed, if solves (2.8), then the pair where Pr and 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), can be seen as an intrinsic correction of the external force in order to keep the incompressibility of the fluid, in case of system (2.8), we may see the term as a correction of the velocity due to the pressure and which is also necessary to express the incompressibility. Thus can be seen as the recovered velocity under the external force and a constant pressure, or equivalently as the density of a diffusion process governed by the source density As regards the relationship between and namely the equality by which the pressure is covered into a velocity, we note that 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
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 can be seen as a reaction force, and for the case of the inertial 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 consider nonlinear reaction forces of the type , with . Thus the external force is sensitive to the recovered velocity and incorporates both the velocity and the pressure of the Stokes system, by means of the correction function (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
(2.9) |
we are able to localize two solutions and in the following form
(2.10) | |||||
where is a subdomain of the vectors belong to with and the inequalities are understood component-wise, then the corresponding solutions of the reaction Stokes system
(2.11) |
are distinct. Indeed, if the solutions are not distinct we have which is clearly excluded by the localization relations (2.10), in view of the relation
In what follows, the localization of solutions will be given in the set defined by
(2.12) |
For 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 defined by
which will allow to characterize the solution 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 for a.e. , then the weak maximum principle (see for instance [13, Theorem 8.1]) guarantees that the solution of problem (2.8) satisfies a.e. in
(R2) If then, according to [13, Theorem 8.15], the solution of problem (2.8) belongs to and there is a constant independent of which satisfies such that
(R3) For each with in and on a subset of nonzero measure of there exists such that the solution of problem (2.8) verifies for a.e. in the fixed subdomain For which satisfies the constant can be chosen to be independent of (see [15, Theorem 12.1.2]).
For a square matrix with nonnegative entries, the spectral radius is the maximum of the modulus of its eigenvalues i.e., of all with There are known the following characterizations of such kind of matrices:
(a) tends to the zero matrix as
(b) The matrix is invertible and the entries of are also nonnegative.
3 Main results
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 and 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.
3.1 Localization of the recovered velocity
We start by giving a localization of the solution for the linear diffusion problem (2.8), which for the classical Stokes system (2.1) represents a localization of the recovered velocity in terms of some bounds of the external force
Proposition 1
If
then the solution of (2.8) satisfies
Here is the characteristic function of the domain and are the constants given by the remarks (R3), (R2), associated to the functions and constant respectively.
Proof. From the relation on according to (R1), one has on Next, using (R2), from the relation on we have
Similarly, using (R3), since on we obtain
We now state and prove an existence and localization theorem for the semi-linear reaction-diffusion problem (2.9) which consequently leads to an existence and localization result for the reaction Stokes system (2.11). The assumptions on the data are as follows:
- (h1)
-
There are two functions such that
(3.1) - (h2)
-
There exist constants such that for each
(3.2) and the spectral radius of the matrix
is strictly less than one. Here is the first eigenvalue of the Dirichlet problem for the operator
Remark 2
(a) Condition (3.1), more exactly the requirement on shows that the external force does not react in the exterior of namely in the vicinity of the boundary For example, with on is such a function.
(b) We underline the local form of the conditions imposed on which are given only on a vector interval This makes possible for a given function these conditions to be satisfied for several intervals leading then to multiple solutions.
Theorem 3
Proof. (a) Existence, uniqueness and localization. Let us notice that system (2.9) is equivalent to the fixed point equation where We apply Perov’s vector version of Banach’s fixed point theorem. Perov’s vectorial approach, initiated by Perov ([21]) in connection with the contraction principle, was extended for instance in [22] and [3] for other results from nonlinear functional analysis. We first prove the invariance condition To this aim, let be any element of Using (h1) we have
whence
Hence
Next we prove that is a Perov contraction on To show this, let and denote Then one has
in the weak sense. Thus, multiplying the -th equation by and integrating over yield
(3.3) |
The left-hand side is greater than while for the right-hand side, in view of the fact that does not depend on for and using (h2), we have
where for the last inequality Poincaré’s inequality with the best constant has been applied twice. Hence
These inequalities can be put together under the form of the matrix inequality
where the spectral radius of the matrix is less than one. Thus is a Perov contraction on This implies that has a unique fixed point in
(b) Nash equilibrium. In order to prove that the solution is a Nash equilibrium, we use an iterative approximation scheme. We start with some fixed elements in At each step (), the elements having been determined at step , first we apply Ekeland’s principle to and find a such that
Next we apply Ekeland’s principle to and obtain an with
After steps we obtain where for each one has
(3.4) |
Our next goal is to prove the convergence of the sequences To this aim we use a technique first suggested in [24] and extended in [26]. For and we let
From (3.4), one has in as We have
(3.5) |
which yields
(3.6) | |||
Denote
One clearly has,
(3.7) |
With the notations before, inequality (3.6) writes
By denoting where for and for and the above inequality can be written in the form
Since as we clearly have that as so the matrix is invertible and its inverse is nonnegative. Consequently
It follows that the sequences converge to zero as uniformly with respect to Thus, for each the sequence is Cauchy in and therefore convergent to some Passing to the limit in (3.4) we obtain
(3.8) |
(3.9) |
Now (3.8) shows that solves (2.9) in and the uniqueness of solution implies that where is the solution given by Perov’s theorem. Finally, (3.9) shows that is a Nash equilibrium with respect to the partial energy functionals
In terms of the Stokes system, Theorem 3 immediately yields the following localization and multiplicity result.
Theorem 4
(a) Under conditions (h1) and (h2), problem (2.11) has a unique (modulo an additive constant in pressure) solution such that the recovered velocity belongs to and is a Nash equilibrium with respect to the associated partial energy functionals.
(b) In case that conditions (h1) and (h2) are satisfied for two pairs of vectors and with problem (2.11) has two distinct solutions with and
3.2 Localization of the velocity
The results established in Section 3.1 clearly yield localization of the velocity also in the particular case that is for a uniform pressure. In this section we give an answer to the question concerning the type of forces which guarantee us to have a uniform pressure. More precisely, we give a sufficient condition for the external force in order to guarantee that namely the equality
(3.10) |
We underline the dependence of the sufficient condition (3.10) on the viscosity by means of the operator and the fact that in case of a constant viscosity this condition reduces to the self property of the force namely the force is irrotational. For the same condition yielding constant pressure in the case of periodic boundary conditions, we refer the reader to [18, pag. 84].
Our aim is to investigate the localization of the velocity for planar Stokes systems with an additional reaction force depending on the velocity itself, namely
(3.11) |
Here and We shall consider the special type of the reaction additional force, defined by
(3.12) |
where and the hypothesis on will be made precise later.
We start with some preliminary results. We analyze first system (2.1) with a given non reactive external force satisfying the sufficient condition (3.10). One has the following result.
Lemma 5
If then and , where is the velocity operator defined in (2.4).
Proof. The function satisfies
In particular, since satisfies the identity (2.3). It remains to prove that that is It suffices to prove that the function is zero on any subset Since is dense in it is enough to show that
(3.13) |
From the equality one has, for any
The symbol is used to denote the value of a distribution on a test function. Thus
Now let be arbitrary. Denote Then Consider a sequence with in Since is uniformly Lipschitz continuous and one has (see [13, Theorem 8.8]). Consequently
We have
From in one has in Passing to the limit gives
Thus (3.13) is proved.
Remark 6
If is such that then denoting one has the following expression of
where This formula shows that for an external force to make the pressure constant it has to depend on the viscosity.
Note that since is if then (see [13, Theorem 8.12]), which together with gives
We show now that a localization of immediately implies the localization of
Proposition 7
Assume that Let and be such that where and is such that on If
(3.14) |
(3.15) |
then the velocity satisfies
Proof. From Lemma 5, one has Next, from (3.14) and the remark (R2), we have
From (3.15) and in we have for a.e. whence
Let us mention that in the particular case of a constant viscosity one has and assumptions (3.14), (3.15) read as
Next we consider planar Stokes systems with a velocity-dependent reaction force. It is convenient that besides the property to have where for a given vector is the vector defined by We have the two propositions below.
Proposition 8
If is such that
then
Proof. According to Lemma 5, Clearly, one has Hence
Let us now consider a closed subspace of namely
Also denote
Hence one has
We are now in position to consider the planar Stokes system (3.11) with the special type of reaction additional force defined by (3.12). We assume that is of class and
(3.16) |
One has the following result.
Proposition 9
Under assumption (3.16), belongs to for every
Assume that i.e., and We give the fixed point formulation of problem (3.11) (under hypothesis (3.12)). Under the above conditions, we have
Hence
and the following result holds.
Proposition 10
(a) Problem (3.11) has a solution if and only if the operator has a fixed point.
(b) If solves (3.11), then is a fixed point of and (constant).
(c) If is a fixed point of then solves (3.11) for any constant
Let be defined by
Hence, if then
It is easy to see that the set is closed in
Let us consider the matrix with entries
where
(3.17) |
and is the first eigenvalue of in with the Dirichlet boundary condition.
For a function denote
Also, let us consider the following bounds of given only locally:
(3.18) |
The main result in this section is the following localization theorem, also able to produce multiple solutions, depending on the oscillatory properties of function .
Theorem 11
Proof. We follow the same ideas as in the proof of Theorem 3. Let Since on and on we have a.e. in Then using the positivity and monotonicity properties of we obtain
For the lower estimation in since on we have a.e. in and thus a.e. in Then
(3.20) |
Hence due to (3.19).
Now we prove that the operator satisfies on the contraction condition in the sense of Perov. To this aim, let and denote Then
in the weak sense, for These give
in the weak sense. Thus, for each multiplying by and integrating over yield
(3.21) | |||||
(3.22) | |||||
(3.23) | |||||
(3.24) |
One has
Using twice Poincaré’s inequality we deduce
Therefore
Since the spectral radius of is less than one, we may apply Perov’s fixed point theorem to deduce the existence and uniqueness of with
Remark 12
If the functions are nondecreasing in both variables in then for and condition (3.19) becomes
Let us notice that such vectors always exist if the asymptotic conditions below are satisfied:
One also has the following variational characterizations of the localized solution.
Theorem 13
Under the assumptions of Theorem 11,
(a) is a Nash equilibrium in for the pair of partial energy functionals
(b) if in addition is of potential type, i.e., then minimizes in
the functional
Proof. The proof is similar to that of Theorem 4.1 in [4] and we omit it.
4 Stokes systems with variable periodic viscosity
This section is devoted to the Stokes system with variable periodic viscosity (see [1]). We follow the same goals as in the previous section, namely the localization and multiplicity of solutions. The physical motivation consists in the incompressible flow in a domain filled with a mixture of fluids having a highly heterogeneous viscosity denoted which is a periodically varying function of the space variable with small period . Mathematically, we have a family of equations depending on the small parameter and the problem consists in an asymptotic analysis, as aimed to give an averaged description of the process. As before, let be a function representing the given external force. Let be the unitary -cube and a -periodic function belonging to . We define by the -periodic scaled function, which represents the viscosity of a complex mixture of fluids having different viscosities. We assume that the viscosity is such that and
(4.1) |
Note that according to remarks (R2), (R3) from Section 2 and in view of (4.1), the constants and do not depend on Consequently, all the estimations in Section 3 remain valid and we immediately obtain the following versions of the results in Section 3, for the -parametrized problems and the homogenized ones. The convergence results and the localization of the solution for the homogenized problems are obtained as in [4, Theorem 3.2].
Thus we discuss below a series of parametrized problems.
1a. We start with the analogue of system (2.1), with the given external force , namely
(4.2) |
where and represent the velocity and the pressure of the mixture. For a fixed , system (4.2) is equivalent, as in the nonparametric case, to the diffusion problem
(4.3) |
The following analogue of Proposition 1 holds.
Theorem 14
If
then the solution of (4.3) belongs to . There is a function such that
belongs to and it is the unique solution of the following homogenized problem
(4.4) |
where the entries of the positive definite homogenized matrix are given for by
(4.5) |
and are the unique solutions of the local problems
(4.6) |
1b. We consider now the analogue of system (2.11), with an -dependent reaction force, namely
(4.7) |
For a fixed , system (4.7) is equivalent with the diffusion problem
(4.8) |
Let the operator be defined by . The analogue of Theorem 3 is the following.
Theorem 15
Assume that satisfies (h1) and (h2) with
for every and some Then problem (4.8) has a unique solution Moreover, is in a Nash equilibrium with respect to the partial energy functionals of the system. There is a function such that
belongs to and it is the unique solution of the homogenized problem
(4.9) |
where the matrix is defined in (4.5).
So for both problems (4.2) and (4.7) we localized the recovered velocity , as far as the homogenized recovered velocities and , respectively.
For fixed , system (4.10) is equivalent with the diffusion problem
(4.11) |
where Moreover, if in , there is a function such that for one has
(4.12) |
and is the unique solution of the homogenized problem
(4.13) |
where the matrix is defined in (4.5).
For the case of an -dependent non reactive force strongly converging in to , assuming that and denoting we may state the following localization result for problem (4.10), the analogue of Proposition 7.
Theorem 16
2b. For the analogue of the planar system (3.11) we consider, without loss of generality, the case
(4.16) |
where the reaction force is of the form
Using the notations from Section 3.2 and the assumptions (3.16) on the given function we have the following result for the system (4.16), which is the analogue of Theorem 11.
Theorem 17
Assume that is as in Theorem 11,
(4.17) |
and the spectral radius of the matrix is strictly less than one. Then, there exists such that for any , problem (4.16) has a unique solution in , where is the space of the functions with mean value zero on . Moreover, there is a function such that
belongs to and it is the unique solution of the homogenized problem
(4.18) |
where the matrix is defined in (4.5) and .
5 Conclusions
We addressed in this paper the stationary Stokes system with variable viscosity and a reaction force term. We gave appropriate conditions on the reaction force term in order to obtain the localization and multiplicity of solutions. The method used combines concepts and results from the linear theory of PDEs and nonlinear functional analysis. In particular, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland’s variational principle The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems. In this context, a significant gain is that our method makes possible the emergence of finitely or infinitely many solutions.
As a perspective, let us mention that the method can be applied to other classes of partial differential equations.
References
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