Posts by Radu Precup

Abstract

We analyze a general class of coupled systems of stationary Navier-Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in \({R^N}\), \({N≤3}\), the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory.

Authors

Mirela Kohr
Faculty of Mathematics and Computer Science, Babeş–Bolyai University, Cluj-Napoca, Romania

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Navier–Stokes equations; multidisperse porous media; fixed point index

Paper coordinates

M. Kohr, R. Precup, Localization of energies in Navier–Stokes models with reaction terms, Analysis and Applications, 22 (2024) no. 6, pp. 1053-1073, https://doi.org/10.1142/S0219530524500118

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Analysis and Applications

Publisher Name

World Scientific

Print ISSN

0219-5305

Online ISSN

1793-6861

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Localization of energies in Navier-Stokes models with reaction terms

Localization of energies in Navier-Stokes models with reaction terms

Mirela Kohr, Radu Precup M. Kohr, Faculty of Mathematics and Computer Science, Babeş–Bolyai University, 1 M. Kogălniceanu Str., 400084 Cluj-Napoca, Romania mkohr@math.ubbcluj.ro R. Precup, Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş–Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania r.precup@math.ubbcluj.ro
Abstract.

We analyze a general class of coupled systems of stationary Navier-Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in N, N3, and localization results for the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory.

Key words and phrases:
Navier-Stokes equations; multidisperse porous media; fixed point index
1991 Mathematics Subject Classification:
35Q30, 76D05

1. Introduction

The analysis of systems of Navier-Stokes coupled equations developed in this paper has been inspired by the models of Nield and Kuznetsov [33], [34], [36] related to the momentum transfer and convection in bidisperse and tridisperse porous media.

A monodisperse porous medium (Mono-DPM) is understood as a standard porous medium whose structure consists of a fluid phase and a solid phase (see, e.g., [32] for further physical details).

1.1. Bidisperse porous media

Starting from the previous description of a monodisperse porous medium (Mono-DPM), one may consider a bidisperse porous medium (BDPM) as a (Mono-DPM) where a different porous medium replaces the corresponding solid phase. A BDPM is thus a standard porous medium consisting of clusters of large particles that appear as agglomerations of small particles [7], [34], [36]. Having in view such a structure, one may characterize a BDPM by two phases, the f-phase due to the existing macro-pores (the voids separating the clusters), and the p-phase corresponding to the micro-pores (the voids within the clusters). The models of bidisperse porous media with very complex structures are encountered in many biological, medical and engineering applications. There are many references in this sense, among of them we refer to [31], [32], [33], [34], [36].

The first model of Nield and Kuznetsov [33] is linear and refers to the momentum transfer in a BDPM in the steady-case, and is given by the following couple of Brinkman type equations for the velocity field vf of the f-phase and the velocity field vp of the p-phase

(1.1) {G=μKfvf+ζ(vfvp)μ~fΔvfG=μKpvp+ζ(vpvf)μ~pΔvp,

where G=p is the pressure force due to the pressure field p (the same in both phases). In addition, μ is the fluid viscosity coefficient, Kp and Kf and μ~p and μ~f are the permeability coefficients and, respectively, the effective viscosity coefficients for the two phases, ζ is the coefficient for momentum transfer between the phases [33] (see also [34]). All variables are dimensional, as the superscript shows.

The next model of BDPM proposed by Nield and Kuznetsov [36] is semilinear and extends the linear model in [33], [34] by considering the semilinear terms of Forchheimer type |vp|vp and |vf|vf in the following system of coupled equations for the velocity fields vf and vp

(1.2) {G=μKfvf+ζ(vfvp)μ~fΔvf+cfρKf1/2|vf|vfG=μKpvp+ζ(vpvf)μ~pΔvp+cpρKp1/2|vp|vp,

where cf and cp are the corresponding Forchheimer coefficients for the two phases, and ρ is the fluid density. See also [6], [42], [43], [45], and [46] for further extensions and models based on forced, natural and mixed convection phenomena. For instance, Straughan [46] analyzed a model, which describes the thermal convection in an anisotropic BDPM, and with different pressures within the macro and micro phases. A similar consideration has been adopted in [6].

The model of Nield and Kuznetsov [36], where the steady-state momentum transfer is described by the semilinear system (1.2), and also the model of Straughan [46] have suggested us in [22] to consider a more general nonlinear system of two coupled Navier-Stokes type equations, as follows

(1.3) {μ1Δu1+η1u1+κ1(u1)u1+p1=h1α1|u1|q1u1γ1(u1u2)in Ωμ2Δu2+η2u2+κ2(u2)u2+p2=h2α2|u2|q1u2γ2(u2u1)in Ωdiv ui=0in Ω,i=1,2ui=0on Ω,i=1,2,

where ΩN is a bounded domain (N3), q1, ηi,κi,αi0 and μi,γi>0, i=1,2, are given constants whose meaning depends on the physical properties of fluid flow and porous media, while hi, i=1,2, are given data in some Sobolev spaces.

Note that all along this paper we assume that the fluid density is normalized at 1. Also, the case N=1 is not considered here.

1.2. Tridisperse porous media

A tridisperse porous medium (TDPM) can be defined as a monodisperse porous medium (Mono-DPM) in which the solid phase is replaced by a bidisperse porous medium (BDPM). Thus, the solid phase contains three types of pores. One type are the macro-pores, but there are also pores of smaller scale called meso-pores, and cracks or fissures of still smaller scale, which are called micro-pores (see [35]). The basic theory for thermal convection in a triple porosity (tridispersive) medium was developed by Nield & Kuznetsov [35] (see also [26], [15]). The authors in [8], [16] have considered different velocities Uf, Um and Uc and different pressures pf, pm and pc in the macro, meso and micro-pores.

There are many applications where more than two porosity scales are present in a medium, concerning the modeling of fluid flow in cellular biological media, as well as the flow through geological rock formations (cf. [35] and the references therein).

Nield and Kuznetsov [35] followed their model (1.1) (see [33]) and proposed a Darcy model for the steady-state momentum transfer in a TDPM, characterized by three phases determining three levels. At at first level there is a fluid phase, at the second level there is a Mono-DPM, and at the third level there is a BDPM. For each of the three phases there is a volume-averaged velocity (a Darcy velocity), vi at the ith-level, i=1,2,3. The authors assumed that the momentum equations for v1 and v2 are coupled, and those for v2 and v3 are coupled, but those for v1 and v3 are not directly coupled since the TDPM is assumed to be structured so that, due to the presence of the second phase, the first and the third phases are geometrically isolated from each other (cf. [26]). Therefore, adopting the Darcy model, Nield and Kuznetsov [35] (see also [26]) considered the following triplet of equations for steady-state momentum transfer

(1.4) {G=μK1v1+ζ12(v1v2)G=μK2v2+ζ12(v2v1)+ζ23(v2v3)G=μK3v3+ζ23(v3v2),

where, as in (1.1), it is assumed that the pressure forces pi, i=1,2,3, are the same and denoted G in all three phases. In addition, μ is the fluid viscosity, K1, K2 and K3 are the permeabilities of the three phases, and ζ12 and ζ23 are some coupling coefficients for momentum transfer. See also [15] for the free convection in a square cavity filled with a tridisperse porous medium.

Gentile and Straughan [16] considered different pressures in their analysis of the thermal convection in a tridisperse porous medium when only one temperature is employed and the horizontal layer, which contains the saturated porous medium, is heated from below.

1.3. Multidisperse porous media model

A multidisperse porous medium (MDPM) can be viewed as a standard porous medium consisting of a fluid phase (f-phase) and a solid phase containing several types of pores (provided by different porous media) located within each other.

Taking into account the (TDPM) model of Nield and Kuznetsov [35], where the steady-state momentum transfer is described by the linear system (1.4), the (BDPM) model of the same authors [36] described by the semilinear system (1.2), and the model of Gentile and Straughan [16], we consider a more general nonlinear coupled type Navier-Stokes system with reaction terms arising in the analysis of fluid flows in anisotropic multidisperse porous media (MDPM). This system has the form

(1.5) {div (Ai(x)ui)+η0i(x)ui+κ0i(x)(ui)ui+pi=hiδi(x)|ui|qi1uiΓi(x,u) in Ω,i=1,2,,m,div ui=0 in Ω,i=1,2,,m,ui=0 on Ω,i=1,2,,m,

where m, ΩN,N3, is an open bounded set, η0i,κ0i,δiL(Ω) are given functions, such that η0i(x)η, for some constant η0, and κ0i(x)0. In addition, hiH1(Ω)N and the entries aj(i) of the matrix Ai are nonnegative functions in L(Ω) with aj(i)=aj(i) and

(1.6) j,=1Naj(i)(x)ξjξμ|ξ|2,ξ=(ξ1,ξ2,,ξN)N

for some constant μ>0 related to the viscosity tensors Ai, i=1,2,,m. In addition, the coupling terms are

(1.7) Γi(x,u)=jiγij(x)|uiuj|rij1(uiuj).

We assume that 1qi,rij2/2=N/(N2) if N=3, and 1qi,rij<+ in case N=2. Also γijL(Ω) are given non-negative functions, i,j=1,2,,m.

The unknowns of system (1.5) are the velocity and pressure fields ui and pi, i=1,2,,m.

In order to analyze this system, we provide a deep localization analysis of two important related energies, the kinetic energy and the enstrophy (see Subsection 1.4).

1.3.1. Special particular cases

  • (i)

    If all matrices Ai are constant and equal to the identity matrix 𝕀N in N, γij=0 (and hence Γi=0), all coefficients η0i,δi0 and κ0i>0 are given constants, and m=2, then system (1.5) reduces to the one that have been analyzed in [22].

  • (ii)

    If m=3, η0i=κ0i=δi=rij1=0 and γ13=0, γ12=γ21>0 and γ23>0 are given constants, system (1.5) reduces to system (1.4).

  • (iii)

    If m=1 and all coefficients involved in system (1.5) are constants, η0i=0, δi=0 and γij=0, i,j=1, one obtains the well-known Navies-Stokes system in the isotropic case. If, in addition, κ0i=0, one obtains the Stokes system in the constant coefficient case.

    The Stokes and Navier-Stokes systems describe various models of fluid mechanics, engineering, biology, chemistry, and there is a huge list of references concerning the mathematical analysis of related boundary value problems and of their applications. Among of them, there are the books [2], [3], [9], [14], [27], [28], [32], [44], [47], [48].

    A layer potential approach has been employed by Fabes, Kenig and Verchota [12] in the analysis of the Dirichlet problem for the Stokes system on Lipschitz domains in N (see also [10], [18] for further applications of layer potentials in the analysis of boundary value problems for strongly elliptic differential operators). Well-posedness results in various function spaces for Dirichlet problems for the Stokes system with constant coefficients in Lipschitz domains in n have been obtained by Mitrea and Wright [30] (see also the references therein).

    Korobkov, Pileckas and Russo [25] analyzed the flux problem in the theory of steady Navier-Stokes equations with constant coefficients and non-homogeneous boundary conditions. Amrouche and Rodríguez-Bellido [1] have proved the existence of a very weak solution for the non-homogeneous Dirichlet problem for the compressible Navier-Stokes system in a bounded domain of the class C1,1 in 3.

  • (iv)

    If γij=0 and η0i=0, we obtain the anisotropic Navies-Stokes system. If, in addition, κ0i=0, (1.5) becomes the Stokes system in the anisotropic case. Several boundary value problem for anisotropic Stokes and Navier-Stokes systems in Lipschitz domains and in L2-based Sobolev spaces have been studied in [20], [21], [23], [24] by using variational techniques and fixed point theorems (see also [19]). Mazzucato and Nistor [29] obtained well-posedness and regularity results in Sobolev spaces for the linear elasticity equations in the anisotropic case with mixed boundary conditions on polyhedral domains.

1.4. Kinetic energy and enstrophy

Let ΩN be an open bounded set, N3.

The kinetic energy of a flow velocity field u in Ω has the following expression

(u):=12Ω|u|2𝑑x,

and the enstrophy is given by [13, p. 28]

E(u):=Ω|u|2𝑑x=i,j=1NΩ|uixj|2𝑑x.

The significance of the enstrophy is that it determines the rate of dissipation of the kinetic energy

(1.8) d(u)dt=νE(u).

Indeed, by assuming that the flow velocity and pressure fields u and p satisfy the unsteady Navier-Stokes system in Ω,

(1.11) {utνΔu+(u)u+p=0divu=0,

and the boundary condition u|Ω=0, multiplying in L2(Ω) the Navier-Stokes equation with u, integrating by parts, and using the boundary condition, we obtain that

12ddt|u(,t)|L2(Ω)2+ν|u(,t)|L2(Ω)2=0,

which is relation (1.8). This relation shows that the kinetic energy decays due to the viscosity with the rate νE(u). In the case Ω=3, this result has been obtained in [13, (1.9)]. Let us add that the enstrophy has an important role in the study of turbulent flows (see [13, Chapter 2] for further details).

In addition, if Ω=N, N3, and the velocity field u decays rapidly at infinity, then the enstrophy E(u) becomes

E(u)=Ω|ω|2𝑑x,

where ω=×u is the vorticity field. Note that this formula remains true even if ΩN, but the boundary terms that follow from the integration by parts vanish (see [13, Chapter 2, p.28]).

1.5. Physical significance of the localization of solutions

With reference to system (1.5), there are two matters of physical interest in relation to the velocities.

First requirement: Find suitable forces hi, in order to guarantee that the kinetic energies 12Ω|ui|2𝑑x or the enstrophies Ω|ui|2𝑑x stay bounded between a priori given bounds. Such a requirement is important in practice in case that these energies have to be limited.

Second requirement: The forces hi being given, find the bounds of the corresponding kinetic energies and enstrophies.

Obtaining localization results for solutions to nonlinear problems in general requires the manipulation of various inequalities such as Poincaré’s and Harnack’s inequalities and also abstract localization techniques such as Krasnoselskii-type methods and those that use the properties of the topological degree, in particular of the fixed point index. For such kind of results concerning various classes of nonlinear problems, we refer the reader to the papers [5, 37, 38, 39] and [41].

1.6. The outlook of the paper

The paper is structured as follows. First, in Section 2, we mention some well-known but useful results regarding the stationary Navies-Stokes equations in the incompressible case. We introduce the corresponding solution operator of the stationary Navier-Stokes equation and state some of its useful properties. The next Section 3 is devoted to localization results for the anisotropic Navier-Stokes equations with velocity-dependent reaction terms. By using the fixed point index theory we obtain localization results of the enstrophy and of kinetic energy under some suitable conditions. The Section 4 is devoted to the analysis of coupled systems of anisotropic Navier-Stokes equations. We obtain localization results for the corresponding enstrophy and kinetic energies.

1.7. Auxiliary results

We conclude this Introduction by some results about the fixed point index, which are used in the reasonings that follow.

Recall that, if X is a Banach space, U is a bounded open subset of X and T:U¯X is a compact operator (continuous with T(U¯) relatively compact), fixed point free on the boundary U of U, then by the fixed point index i(T,U) one means the Leray-Schauder degree D(IT,U)(see [11, 17]). Among the properties of the index, we mention the following ones which will be useful further:

  • (I1)

    if i(T,U)0, then T has at least one fixed point in U.

  • (I2)

    if VU is open and T has no fixed points on the boundary of V, then i(T,UV¯) =i(T,U)i(T,V).

  • (I3)

    if 0U and xλT(x) for all (x,λ)U×(0,1), then i(T,U)=1 (see [17, Theorem 12.7.3]).

  • (I4)

    if there exists a point x0X{0} such that xT(x)+λx0 for all xU and λ>0, then i(T,U)=0 (see [17, Theorem 12.7.11(ii)]).

2. The solution operator of the stationary Navier-Stokes equation

Consider the system

{div(A(x)u)+η0(x)u+κ0(x)(u)u+p=fin Ωdiv u=0in Ωu=0on Ω,

where ΩN,N3, is an open and bounded set, η0,κ0L(Ω), η0(x)η0,κ0(x)0, f=(f1,f2,,fN)H1(Ω)N, and the entries aij of the viscosity matrix A are nonnegative functions in L(Ω) with aij=aji and

(2.1) i,j=1Naij(x)ξiξjμ|ξ|2,ξ=(ξ1,ξ2,,ξN)N

for some constant μ>0. From the physical point of view, η0 is the permeability coefficient, κ0 is the viscosity coefficient of a viscous fluid whose flow inside a porous medium is described by the above system, and the constant μ is related to the viscosity tensor A (see [32] for further physical arguments).

The variational form of this problem is

a(u,v)+b(u,u,v)(p,divv)L2=(f,v),vH01(Ω)N,

where (u,p)V×L2(Ω). Here

a(u,v)=k=1NΩ(i,j=1Naij(x)ukxjvkxi+η0(x)ukvk)𝑑x,
b(u,v,w)=i,j=1NΩκ0(x)ujvixjwi𝑑x,(f,v)=k=1N(fk,vk),
V={v=(v1,v2,,vN)H01(Ω)N:divv=0},

and H01(Ω) is the closure of C0(Ω) in H1(Ω). On the space H01(Ω)N we consider the inner product a(u,v). The induced norm on V is denoted by |u|V. It is equivalent to the common norm

|u|H01=(Ω|u|2𝑑x)12.

One has VL2(Ω)NV and for uV,

c2|u|H012|u|V2μ|u|H012+η|u|L22,

for some constant c>0, which together with Poincaré’s inequality yields

|u|L21μλ1+η|u|V,

where λ1 is the first eigenvalue of the Laplacian with the homogeneous Dirichlet condition on the boundary. In addition, we recall that the constant μ comes from the ellipticity condition (1.6) being related to the viscosity tensor A, and η is a lower bound for the permeability coefficient η0(x).

For hL2(Ω)N, one has

|h|V=supvV|(h,v)||v|V|h|L2supvV|v|L2|v|V1μλ1+η|h|L2.

Hence 1μλ1+η is the embedding constant of both inclusions VL2(Ω)N and L2(Ω)NV. (Note that from the embedding VH01(Ω), one has H1(Ω)V.)

Also note that according to a well known result (see, e.g., [14], [47]), there are constants M0,M>0 depending on Ω, A,η0 such that the trilinear functional b:V3 satisfies the relations

b(u,v,w)+b(u,w,v) = 0,
|b(u,v,w)| κM0|u|H01|v|H01|w|H01,
|b(u,v,w)| κM|u|V|v|V|w|V,

where κ=|κ0|L. Also M=M0c3.

As in [22], one can define de solution operator

S:D0={fV:κM|f|V<1}V

which to each fD0 associates the unique ufD0 such that

(2.2) a(uf,v)+b(uf,uf,v)=(f,v),vV.

Also

|S(f)|V|f|V,fD0

and

|S(f)S(g)|V112κMρ|fg|V

for all f,gV satisfying |f|V,|g|Vρ, where ρ<12κM, that is the solution operator S is Lipschitz continuous on the ball of V centered at the origin and of radius ρ.

Note that after finding the solution ufV of the variational equation (2.2), the pressure pL2(Ω) of the Dirichlet problem for the Navier-Stokes type system mentioned at the beginning of Section 2 follows from the well known De Rham’s Theorem (cf., e.g., [47, Proposition 1.1, Chapter 1]).

3. Localization results for equations with velocity-dependent reaction terms

We discuss the localization of velocity for a problem of the type

(3.1) {div (A(x)u)+η0(x)u+κ0(x)(u)u+p=Φ(x,u)in Ωdiv u=0in Ωu=0on Ω,

where Φ is a reaction term dependent on velocity u. The problem reduces to the fixed point equation

(3.2) u=S(Φ(,u)),

for uV such that |Φ(,u)|V<1κM.

3.1. Localization of the enstrophy

Assume that

(H1):

Φ:Ω×V L2(Ω)N is L2-Carathéodory and

|Φ(x,u)|L2f(|u|H01), for uV,

and some increasing function f:[0,R0]+, where R0(0,μλ1+η2κMμλ1) and

(3.3) f(R0)μλ1R0.

Then for |u|H01R0, one has

|Φ(,u)|V 1μλ1+η|Φ(,u)|L21μλ1+ηf(|u|H01)
1μλ1+ηf(R0)μλ1R0μλ1+η<12κM.

Therefore, the operator T(u)=S(Φ(,u)) is well-defined and continuous on the set

DR0={uV:|u|H01R0}.

In addition T is compact due to the compact embedding of L2(Ω)N in V.

Lemma 1.

Assume that condition (𝐇𝟏) holds. Let α(0,R0] be such that

(3.4) f(α)<μλ1α.

Then

(3.5) |S(Φ(,u))|H01<αfor all uV with |u|H01α.
Proof.

Let uV with |u|H01α and denote v=S(Φ(,u)). Then

a(v,w)+b(v,v,w)=(Φ(,u),w)L2,wV.

For w=v, this gives

μ|v|H012 |v|V2=|v|V2+b(v,v,v)|Φ(,u)|L2|v|L2
|v|H01λ1f(α).

Hence using (3.4) one has

|v|H01f(α)μλ1<α.

   

Assume now that

(H2):

there exists eV with |e|L2=1 such that

(Φ(,u),e)L2g(|u|H01), for uV,

and some function g:++.

Lemma 2.

Assume that conditions (𝐇𝟏) and (𝐇𝟐) hold with β(0,R0], κMcβ1 and

(3.6) g(β)>(c2β+κM0β2)|e|H01.

Then

(3.7) uλe+S(Φ(,u))for all uV with |u|H01=β and λ0.
Proof.

Assume the contrary. There there exist uV with |u|H01=β and a number λ0 such that

u=λe+S(Φ(,u)).

Then

a(uλe,v)+b(uλe,uλe,v)=(Φ(,u),v)L2,vV.

Take v=e and find

a(u,e)λ|e|V2+b(uλe,uλe,e)=(Φ(,u),e)L2.

One has

b(uλe,uλe,e) = b(u,uλe,e)λb(e,uλe,e)
= b(u,u,e)λb(u,e,e)λb(e,u,e)+λ2b(e,e,e)
= b(u,u,e)λb(e,u,e).

Hence

(3.8) a(u,e)λ|e|V2+b(u,u,e)λb(e,u,e)=(Φ(,u),e)L2.

Since

b(e,u,e)+|e|V2 |e|V2κM|u|V|e|V2
(1κMcβ)|e|V20,

from (3.8) and (𝐇𝟐) we deduce that

g(β)a(u,e)+b(u,u,e)(c2β+κM0β2)|e|H01,

a contradiction to (3.6).    

Remark 3.

In case of the Brinkman equations, when κ=0, there are no restrictions on R0, so α,β can be any positive numbers satisfying (3.4) and (3.6), respectively.

Remark 4.

In case of equations with nonreactive force terms of solenoidal type, that is with Φ=h for a given function hV, we may take

e=h/|h|L2,f=|h|L2,g=(h,e)L2=|h|L2,

and then conditions (3.4) and (3.6) read as follows:

|h|L2 < μλ1α,
|h|L22 > (c2β+κM0β2)|h|H01.
Theorem 5.

(a) Under the assumptions of Lemma 1, problem (3.1) has a solution (u,p) with |u|H01<α.

(b) Under the assumptions of Lemmas 1 and 2, if in addition αβ, problem (3.1) has a solution (u,p) with

r<|u|H01<R,

where r=min{α,β} and R=max{α,β}.

(c) (two solutions) Under the conditions from (b), in case that α<β, a second solution exists with |u|H01<α.

(d) (multiple solutions) If the conditions in (b) hold for more pairs (αi,βi), then for each of them there is a solution (ui,pi) with

min{αi,βi}<|ui|H01<max{αi,βi}.
Proof.

As explained above, the operator T(u):=S(Φ(,u)) is well defined on DR0, maps DR0 into itself and is compact.

(a) Let Uα={uV:|u|H01<α}. From (3.5), operator T has no fixed points on the boundary Uα of Uα and so the fixed point index i(T,Uα) is defined. Using (3.5) and property (I3) of the index, we have i(T,Uα)=10, which guarantees that T has a fixed point in Uα.

(b) Let Uβ={uV:|u|H01<β}. From (3.7), which also holds for λ=0, the fixed point index i(T,Uβ) is defined, and using property (I4) of the index, we have i(T,Uβ)=0. Now we use property (I2) of the index: if β<α, then UβUα and

i(T,UαU¯β)=i(T,Uα)i(T,Uβ)=10=10,

hence based on property (I1), T has a fixed point with β<|u|H01<α. If by contrary, α<β, then UαUβ and

i(T,UβU¯α)=i(T,Uβ)i(T,Uα)=01=10,

whence again T has a fixed point, in this case with α<|u|H01<β.

Note that all the above indices are defined since T is fixed point free on Uα and Uβ.

(c) From (a) we have a solution u1 with |u1|H01<α, and from (b), a solution u2 with α<|u2|H01<β.

(d) The statement clearly holds in virtue of (b).    

3.2. Localization of the kinetic energy

Obviously, from Poincaré’s inequality, an upper bound for the enstrophy immediately gives an upper bound for the kinetic energy. We now try to obtain a lower bound of the kinetic energy. To this aim we introduce a new condition.

(H2’):

One has

(Φ(,u),e)L2g0(|u|L2), for uV,

and some function g0:++.

Theorem 6.

Assume the conditions (𝐇𝟏) and (𝐇𝟐) hold for a decreasing function g0 in (𝐇𝟐). Let αmin{R0,1κMc}, β<α|e|H01 be such that

(3.9) f(α)<μλ1α,
(3.10) g0(β)>(c2α+κM0α2)|e|H01.

Then problem (3.1) has a solution (u,p)V×L2(Ω), with |u|H01<α and

(3.11) β<|u|L2<αλ1.
Proof.

As above i(T,Uα)=1 whence it follows that a solution exists with |u|H01<α and hence the second inequality in (3.11) holds.

Let Vαβ={uV:|u|H01<α and |u|L2<β}. Clearly Vαβ is open bounded in V. We now prove that

uλe+T(u)for all uVαβand λ0,

and so i(T,Vαβ) is also defined and equal to 0. Assume the contrary, namely that u=λe+S(Φ(,u)) for some uVαβ and some λ0. As above, one has (3.8), while now

b(e,u,e)+|e|V2 |e|V2κM|u|V|e|V2
|e|V2κMc|u|H01|e|V2
(1κMcα)|e|V20.

Then

(3.12) g0(|u|L2)(Φ(,u),e)L2a(u,e)+b(u,u,e)(c2α+κM0α2)|e|H01.

The boundary of Vαβ has two parts: (p1) |u|L2=β and |u|H01<α; (p2) |u|L2β and |u|H01=α>0. In both cases cases |u|L2β and since g0 is decreasing, we have

g0(β)(c2α+κM0α2)|e|H01.

This inequality contradicts (3.10). Hence, according to property (I4), one has i(T,Vαβ)=0. Then all indices i(T,Uα),i(T,Vαβ),i(T,UαV¯αβ) are defined and one has

i(T,UαV¯αβ)=i(T,Uα)i(T,Vαβ)=10=1.

So our problem has a solution with uUαV¯αβ, whence β<|u|L2<αλ1.

Note that the set UαV¯αβ is nonempty. Indeed, in virtue of the assumption β<α|e|H01, it contains the element αε|e|H01e, for every ε>0 with β<αε|e|H01.    

Remark 7.

If (𝐇𝟐) holds for the special element e=ϕ, where ϕ with |ϕ|L2=1 is a function in V for which there is a number σ>0 such that a(u,ϕ)=σ(u,ϕ)L2 for all uV, i.e., ϕ is an eigenfunction, then a better condition than (3.10) is

g0(β)>σβ+κM0α2|ϕ|H01.

4. Energy-localization results for systems modelling fluid flow in multidisperce porous media

We now generalize the energy-localization results from the unidimensional case to the multidimensional one. The estimates found will also show the influence of the coupling terms.

4.1. Estimates of the enstrophy

Let us now consider the problem

(4.1) {div(Ai(x)ui)+η0i(x)ui+κ0i(x)(ui)ui+pi=hiδi(x)|ui|qi1uiΓi(x,u)in Ωdiv ui=0in Ωui=0on Ω(i=1,2,,m),

where   

Γi(x,u)=jiγij(x)|uiuj|rij1(uiuj)(i=1,2,,m),

1qi,rij2/2=N/(N2). Note that the special form of the coupling terms Γi(x,u) generalizes those of Nield and Kuznetsov [33], [35], [36] in the bidisperse and tridisperse cases (see also systems (1.1), (1.2), (1.3) and (1.4)).


Let αi>0 be given such that αiR0i (i=1,2,,m) (any positive numbers in the Brinkman case) and define the set

Uα:={u=(u1,u2,,um)Vm:|ui|H01<αi,i=1,2,,m}.

We shall guarantee that T(U¯α)Uα. To this aim, let uU¯α be arbitrary and let v=T(u). Then for each i, we have vi=Ti(u), whence

μi|vi|H012 ai(vi,vi)
= (hiδi|ui|qi1uiΓi(x,u),vi)
(hi,vi)+|δi||ui|L2qiqi|vi|L2+|Γi(.,u)|L2|vi|L2
|hi|H1|vi|H01+|δi|λ1cqiqiαiqi|vi|H01+1λ1|Γi(.,u)|L2|vi|H01,

with cqi the embedding constant of H01(Ω)L2qi(Ω). Also

|Γi(.,u)|L2 ji|γij|L|uiuj|L2rijrijji|γij|Lcijrij|uiuj|H01rij
ji|γij|Lcijrij(αi+αj)rij,

where cij are the embedding constants for the inclusions H01(Ω)L2rij(Ω). So

μi|vi|H01|hi|H1+|δi|Lλ1cqiqiαiqi+1λ1ji|γij|Lcijrij(αi+αj)rij.

Hence the desired inclusion holds if

(4.3) |hi|H1+|δi|Lλ1cqiqiαiqi+1λ1ji|γij|Lcijrij(αi+αj)rij<μiαi,

for all i. Then the fixed point index i(T,Uα) is defined and equals 1.

Furthermore, let βi<αi, with κiMciβi1 and consider the subset Vαβ of Uα defined as

Vαβ:={uUα:|ui|H01<βifor at least one i}.

Clearly it is open bounded in Vm. We now guarantee that

uλe+T(u)for all uVαβand λ0,

and so i(T,Vαβ) is also defined and equal to 0, where e=(e1,e2,,em),    eiV with |ei|L2=1(i=1,2,,m). Assume the contrary, namely that u=λe+T(u) for some uVαβ and some λ0. Note that if uVαβ, then |ui|L2=βi for some of the indices i (but for at least one) and |ui|H01=αi   for the other indices. Take one the the indices, let it be i, for which |ui|L2=βi. Then from ui=λei+Ti(u), we deduce (see the proof of Lemma 2)

hiδi(x)|ui|qi1uiΓi(x,u),ei ai(ui,ei)+bi(ui,ui,ei)
Ci1βi+Ci2βi2,

where Ci1=ci2|ei|H01 and Ci2=|κi|LM0|ei|H01. One has

(hiδi|ui|qi1uiΓi(x,u),ei) (hi,ei)|δi|L|ui|L2qiqi|Γi(.,u)|L2
(hi,ei)δicqiqiβiqiji|γij|Lcijrij(βi+αj)rij.

Thus we have a contradiction if we assume

(4.4) (hi,ei)>|δi|Lcqiqiβiqi+Ci2βi2+Ci2βi+ji|γij|Lcijrij(βi+αj)rij.
Theorem 8.

If conditions (4.3) and (4.4) hold for all i, then there is a solution (u,p) of problem (4.1), u=(u1,u2,,um), p=(p1,p2,,pm), with (ui,pi)H01(Ω)N×L2(Ω) and

(4.5) βi<|ui|H01<αifor i=1,2,,m.
Proof.

Since T has no fixed points on the boundaries of Uα,Vαβ,UαV¯αβ, all indices i(T,Uα), i(T,Vαβ), i(T,UαV¯αβ) are all defined and one has

i(T,UαV¯αβ)=i(T,Uα)i(T,Vαβ)=1,

while

UαV¯αβ={uVm:βi<|ui|H01<αi,i=1,2,,m}.

Thus T has a fixed point uUαV¯αβ, that is located as in (4.5).

After finding the fixed point u=(u1,u2,,um), each pressure piL2(Ω), i=1,2,,m, of problem (4.1) follows from De Rham’s Theorem (cf., e.g., [47, Proposition 1.1, Chapter 1]).    

Remark 9.

In virtue of the Poincaré inequality, |h|H11/λ1|h|L2(hL2(Ω)), a sufficient condition for (4.3) to hold is

(4.6) |hi|L2+|δi|Lcqiqiαiqi+ji|γij|Lcijrij(αi+αj)rij<μiαiλ1.
Remark 10 (case of solenoidal vector field).

If hiV, that is hi is a solenoidal vector field, and hi0, then one can take ei=hi/|hi|L2 and condition (4.4) reads as follows:

|hi|L2>|δi|Lcqiqiβiqi+Ci2βi2+Ci2βi+ji|γij|Lcijrij(βi+αj)rij.
Remark 11.

In particular, for a system as that considered in our previous paper [22], namely if

rij=1,Γi(x,u)=γi(x)(uiui+1)(i=1,2,,m),

where um+1=u1, the two conditions (4.3), (4.4) become

|hi|H1+|δi|Lλ1cqiqiαiqi+1λ1i=1m|γi|L(αi+αi+1)<μiαi,
(hi,ei)>|δi|Lcqiqiβiqi+Ci2βi2+Ci2βi+1λ1i=1m|γi|L(βi+αi+1).
Remark 12.

Let us give a physical interpretation of the inequalities (4.6) and (4.4).

  • (a)

    Condition (4.6) shows that to guarantee a small turbulence energy in the fluid (that is, small αi), the external force hi must be sufficiently small.

  • (b)

    Condition (4.4) says that a big turbulence occurs (that is βi is large) when the force hi is correspondingly large at least in some direction. In the case of a solenoidal force, this happens if the force is large enough.

4.2. Estimates of the kinetic energy

The upper estimate is immediately obtained from the upper estimate of enstrophy via Poincaré’s inequality. Lower estimates can be obtained in additional conditions as we have seen in the one-dimensional case. Let α and e as above and assume conditions (4.3). Let βi<αi|ei|H01 and define the open bounded set

Vαβ:={uUα:|ui|L2<βifor at least one i}.

To now guarantee that uλe+T(u)for all λ0 and uVαβ. Note that if uVαβ, then there is i such that either |ui|H01=α and |ui|L2βi or |ui|L2=βi and |ui|H01αi. Assume the contrary. Then for such u, λ and i, as above, we have

(hiδi(x)|ui|qi1uiΓi(x,u),ei)ai(ui,ei)+bi(ui,ui,ei).

As above one has

ai(ui,ei)+bi(ui,ui,ei)ci2αi|ei|H01+κiM0αi2|ei|H01

and since |ui|H01αi

(hiδi(x)|ui|qi1uiΓi(x,u),ei) (hi,ei)|δi|L|ui|L2qiqi|ei|L2|Γi(x,u)|L2
(hi,ei)|δi|Lcqiqiαiqi|Γi(x,u)|L2,

where cqi is the embedding constant of the inclusion H01(Ω)L2qi(Ω). Also from (4.1), one has

|Γi(.,u)|L2ji|γij|Lcijrij(αi+αj)rij.

Thus we have a contradiction if we assume

(4.7) (hi,ei)>ci2αi|ei|H01+κiM0αi2|ei|H01+|δi|Lcqiqiαiqi+ji|γij|Lcijrij(αi+αj)rij.
Theorem 13.

Assume that conditions (4.3) and (4.7) hold for all i. Then there exists a solution (u,p) of problem (4.1), u=(u1,u2,,um), p=(p1,p2,,pm), with (ui,pi)H01(Ω)N×L2(Ω) and

βi<|ui|L2<αiλ1,i=1,2,,m.

We conclude by emphasizing how the conditions required in our main results reflect the contribution of the various physical parameters of equations and coupled systems of Navier-Stokes type to left and right side estimates of the kinetic energy and enstrophy of the solutions. In this way, we have answered the two requirements stated in Section 1.5.

Acknowledgements

The authors are very thankful to reviewers for their valuable comments and remarks that led to an improved version of the paper.

M. Kohr acknowledges the support of the grant PN-III-P4-PCE-2021-0993 (cod PCE 69/2022), UEFSCDI, Romania.

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[1]. C. Amrouche and M. A. Rodríguez-Bellido , The Oseen and Navier–Stokes equations in a non-solenoidal framework, Math. Methods Appl. Sci. 39 (2016) 5066–5090. CrossrefGoogle Scholar

[2]. F. Boyer and P. Fabrie , Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models (Springer, New York, 2013). CrossrefGoogle Scholar

[3]. C. E. Brennen , Fundamentals of Multiphase Flows (Cambridge University Press, 2005). CrossrefGoogle Scholar

[4]. M. Bulíček, J. Málek and J. Žabenský , On generalized Stokes’ and Brinkman’s equations with a pressure- and shear-dependent viscosity and drag coefficient, Nonlinear Anal. Real World Appl. 26 (2015) 109–132. CrossrefGoogle Scholar

[5]. R. Bunoiu and R. Precup , Localization and multiplicity in the homogenization of nonlinear problems, Adv. Nonlinear Anal. 9 (2020) 292–304. CrossrefGoogle Scholar

[6]. F. Capone, M. Gentile and G. Massa , The onset of thermal convection in anisotropic and rotating bidisperse porous media, Z. Angew. Math. Phys. 72 (2021) 169. Crossref, Web of ScienceGoogle Scholar

[7]. Z. Q. Chen, P. Cheng and C. T. Hsu , A theoretical and experimental study on stagnant thermal conductivity of bi-dispersed porous media, Int. Commun. Heat Mass Transf. 27 (2000) 601–610. Crossref, Web of ScienceGoogle Scholar

[8]. C. Y. Cheng , Natural convection heat transfer about a vertical cone embedded in a tridisperse porous medium, Transp. Porous Media 107 (2015) 765–779. CrossrefGoogle Scholar

[9]. P. Constantin and C. Foias , Navier–Stokes Equations (The University of Chicago Press, Chicago, 1988). CrossrefGoogle Scholar

[10]. M. Costabel , Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (1988) 613–626. Crossref, Web of ScienceGoogle Scholar

[11]. K. Deimling , Nonlinear Functional Analysis (Springer, Berlin, 1985). CrossrefGoogle Scholar

[12]. E. Fabes, C. Kenig and G. Verchota , The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988) 769–793. CrossrefGoogle Scholar

[13]. C. Foias, O. Manley, R. Rosa and R. Temam , Navier–Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Vol. 83 (Cambridge University Press, Cambridge, 2001). CrossrefGoogle Scholar

[14]. G. P. Galdi , An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. (Springer, New York, 2011). CrossrefGoogle Scholar

[15]. M. Gentile and B. Straughan , Tridispersive thermal convection, Nonlinear Anal. Real World Appl. 42 (2018) 378–386. Crossref, Web of ScienceGoogle Scholar

[16]. M. Ghalambaz, H. Hendizadeh, H. Zargartalebi and I. Pop , Free convection in a square cavity filled with a tridisperse porous medium, Transp. Porous Media 116 (2017) 379–392. Crossref, Web of ScienceGoogle Scholar

[17]. A. Granas and J. Dugundji , Fixed Point Theory (Springer-Verlag, New York, 2003). CrossrefGoogle Scholar

[18]. G. C. Hsiao and W. L. Wendland , Boundary Integral Equations, 1st edn. (Springer-Verlag, Heidelberg, 2008); 2nd edn. (Springer, Cham, 2021). CrossrefGoogle Scholar

[19]. M. Kohr, M. Lanza de Cristoforis and W. L. Wendland , Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains, Potential Anal. 38 (2013) 1123–1171. Crossref, Web of ScienceGoogle Scholar

[20]. M. Kohr, S. E. Mikhailov and W. L. Wendland , Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier–Stokes systems in Lipschitz domains with transversal interfaces, Calc. Var. Partial Differential Equations 61 (2022) 198. Crossref, Web of ScienceGoogle Scholar

[21]. M. Kohr, S. E. Mikhailov and W. L. Wendland , On some mixed-transmission problems for the anisotropic Stokes and Navier–Stokes systems in Lipschitz domains with transversal interfaces, J. Math. Anal. Appl. 516 (2022) 126464. Crossref, Web of ScienceGoogle Scholar

[22]. M. Kohr and R. Precup , Analysis of Navier–Stokes models for flows in bidisperse porous media, J. Math. Fluid Mech. 25(2) (2023) 38. Crossref, Web of ScienceGoogle Scholar

[23]. M. Kohr and W. L. Wendland , Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds, Calc. Var. Partial Differential Equations 57 (2018) 165. CrossrefGoogle Scholar

[24]. M. Kohr and W. L. Wendland , Boundary value problems for the Brinkman system with coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach, J. Math. Pures Appl. 131 (2019) 17–63. CrossrefGoogle Scholar

[25]. M. V. Korobkov, K. Pileckas and R. Russo , On the flux problem in the theory of steady Navier–Stokes equations with non-homogeneous boundary conditions, Arch. Ration. Mech. Anal. 207 (2013) 185–213. Crossref, Web of ScienceGoogle Scholar

[26]. A. V. Kuznetsov and D. A. Nield , The onset of convection in a tridisperse porous medium, Int. J. Heat Mass Transf. 54 (2011) 3120–3127. Crossref, Web of ScienceGoogle Scholar

[27]. P. G. Lemarié-Rieusset , The Navier–Stokes Problem in the 21st Century (CRC Press, Boca Raton, 2016). CrossrefGoogle Scholar

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