ON THE EQUATIONS GOVERNING THE FLOW OF MECHANICALLY INCOMPRESSIBLE, BUT THERMALLY EXPANSIBLE, VISCOUS FLUIDS

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Mathematical Models and Methods in Applied SciencesVol. 18, No. 6 (2008) 813–857c© World Scientific Publishing Company

ON THE EQUATIONS GOVERNING THE FLOWOF MECHANICALLY INCOMPRESSIBLE, BUTTHERMALLY EXPANSIBLE, VISCOUS FLUIDS

ANGIOLO FARINA∗ and ANTONIO FASANO†

Dipartimento di Matematica “Ulisse Dini”,Universita degli Studi di Firenze,

Viale Morgagni 67/A, I-50134 Firenze, Italy∗[email protected]†[email protected]

ANDRO MIKELIC

Universite de Lyon, Lyon, F-69003, France

and

Universite Lyon 1, Institut Camille Jordan,UFR Mathematiques, Site de Gerland,

Bat. A, 50, avenue Tony Garnier,69367 Lyon Cedex 07, [email protected]

Received 1 March 2007Revised 28 June 2007

Communicated by N. Bellomo

In this paper, the stationary flow of a heat conducting viscous fluid, which is mechanicallyincompressible but thermally expansible is studied. The flow takes place in a boundeddomain and the discharge is prescribed. The thermodynamical modeling of this situationis discussed first. Then the stationary model with zero Eckert number and prove existenceof a solution is studied. Using these results, the Oberbeck–Boussinesq system obtained inthe zero expansivity limit is proved. Next, uniqueness for small data and the regularity ofthe weak solutions is proved. For the unique regular solution the higher order correctionsfor Boussinesq’ approximation and we constructed the error estimate with respect to thethermal expansivity coefficient is proved. The next order correction in this limit is anOseen type momentum equation coupled with a linear advection/diffusion equation forthe temperature. Such higher order correction seems to be new in the literature.

Keywords: Isochoric viscous fluids; heat conducting Navier–Stokes equations; justifica-tion of the Boussinesq approximation.

AMS Subject Classification: 35Q30, 76N10, 35Q35, 76R05

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814 A. Farina, A. Fasano & A. Mikelic

1. Introduction

In this paper, we want to study the non-isothermal flow of an isotropically thermallyexpansible Newtonian fluid (e.g. melted glass, melted polymer, etc.) in the presenceof gravity. This situation is typical of the gravity flow through nozzles in the earlystage of a fiber manufacturing process.

The fluid density, though varying with temperature, will be taken independent ofpressure. This physical condition has strong implications on the thermo-mechanicalbehavior of the system, as we shall see. Further, comments about the physical andmathematical settings of the problem will be made in Sec. 2, after completing thepresentation of the model.

The flow domain (see Fig. 1) is the set Ω = r < R(x3, φ) × [0, 2π]× (0, H),where R : [0, H ]×[0, 2π] → [Rmin, Rmax] is a C∞-map. For the reference problem wehave in mind R and H are lengths of comparable size. The boundary of Ω containsthree distinct parts (see again Fig. 1):

• Lateral boundary Γlat = r = R(x3, φ), x3 ∈ (0, H), φ ∈ [0, 2π].• Inlet boundary or upper boundary Γin = x3 = H and r ≤ R(H,φ), φ ∈ [0, 2π].• Outlet boundary Γout = x3 = 0 and r ≤ R(0, φ), φ ∈ [0, 2π].

The inlet and outlet temperatures of the fluid are prescribed. In particular, thefluid temperature on Γin is higher than that the one on Γout. The precise form ofthe boundary conditions will be specified later.

Our first goal is to give a thermodynamically consistent mathematical model forthe general case, emphasizing the influence of the various parameters (Sec. 2). This

Fig. 1. The domain Ω.

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Stationary Flow of Thermally Expansible Viscous Fluids 815

will motivate the mathematical analysis of the stationary system in non-dimensionalform:

div(ρ(ϑ)v) = 0, ρ(ϑ) = 1 − αϑ, (1.1)

ρ(ϑ) (v∇)v = Ar ϑe3 −1

Re

∇

(P +

23µ(ϑ) div v

)− div(2µ(ϑ)D (v ))

, (1.2)

cp1(ϑ) ρ(ϑ)v · ∇ϑ =1

Pediv(λ(ϑ)∇ϑ) , (1.3)

where α ≥ 0 is the thermal expansivity of the fluid and cp1 is a function of temper-ature which can be expressed by means of Helmholtz free energy, as we shall see.We refer the reader to see Sec. 2.1 for the notations which, on the other hand, arequite standard.a

Here is the list of the original results that will be obtained.We will provide an existence proof for the boundary value problem (1.1)–(1.3),

which does not require small data. First, we will show that velocity is in H1 andtemperature in W 1,3∩L∞ and then higher regularity is obtained by bootstrapping.In Sec. 4 uniqueness will be obtained.

For Eqs. (1.1)–(1.3) either Oberbeck–Boussinesq approximation or Stokes sys-tem can be derived taking particular limits of the non-dimensional parameters. Theuniqueness and regularity results concerning systems (1.1)–(1.3) will be exploitedto obtain a rigorous mathematical justification of the Oberbeck–Boussinesq approx-imation (in the limit of small expansivity parameter), a long debated question, andalso to compute the next order approximation (and in principle the higher orderstoo). To this end an analysis of the derivatives of all order with respect to theexpansivity parameter will be necessary. Most of the technical details are confinedin the Appendices. From the physical point of view we may say that the correctionobtained in the selected framework has the momentum equation of Oseen’s kind.The first-order corrections (6.29)–(6.33) are new to our knowledge. If we compareit to the equations for the first order correction obtained in the incompressibilitylimit (the zero Mach number limit) (see Ref. 14 or 22), we see that they are totallydifferent.

2. Mathematical Modeling

2.1. Definitions and basic equations

We start by recalling the mass balance, the momentum and energy equations, as wellas the Clausius–Duhem inequality in the Eulerian formalism. Following an approachsimilar to the one presented in Ref. 16, pp. 51–85, we denote by ρ,v, e, T, s the

aThe vector-valued operator (v ∇)( · ) stands forP

i vi ∂xi (·).

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density, velocity, specific internal energy, absolute temperature and specific entropy,satisfying the following system of equations:

Dρ

Dt= −ρ div v, (2.1)

ρDv

Dt= −ρge3 + div Π, (2.2)

ρDe

Dt= − div q +D (v ) : Π, (2.3)

ρDs

Dt+ div

(q

T

)≥ 0, (2.4)

where

• DDt = ∂

∂t + v · ∇ denotes the material derivative.• D(v ) = 1

2 (∇v + (∇v )rmT ) is the rate of strain tensor.• Π is the Cauchy stress tensor. Further, D(v) : Π =tr(D(v)ΠT) =

∑i,j DijΠij .

• q is the heat flux vector.• −ge3 is the gravity acceleration. Indeed e3 is the unit vector relative to the x3

axis directed upward.

Introducing now the specific Helmholtz free energy

ψ = e− Ts, (2.5)

inequality (2.4) becomes

ρ

(Dψ

Dt+ s

DT

Dt

)− Π : D(v) + q · ∇T

T≤ 0. (2.6)

We now consider the case in which the material is mechanically incompressiblebut thermally dilatable (i.e. the fluid can sustain only isochoric motion in isothermalconditions). Consistently, we assume:

A1. Denoting by J the determinant of the deformation gradient, we stipulate

J = f(T ), (2.7)

where f(T ) can be expressed in terms of the thermal expansion coefficient. Indeed,defining the thermal expansion coefficient as

1ρ

(∂ρ

∂T

)= −β, (2.8)

with β = β(T ), we have

β(T ) =1

f(T )df(T )dT

.

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In particular, because of (2.7), Eq. (2.1) reads as a constraint linking temperaturevariations with the divergence of the velocity field, namely

−βDTDt

+D(v) : I = 0, ⇔ −βDTDt

+ div v = 0.

Appending now such a constraint to inequality (2.6) we obtain

ρ

(Dψ

Dt+ s

DT

Dt

)− Π : D(v) + q · ∇T

T− p

(−βDT

Dt+D(v) : I

)≤ 0, (2.9)

where p is an arbitrary scalar quantity (Lagrange multiplier) identifiable with pres-sure.

Since in our case the specific volume is determined by temperature, ψ is ulti-mately a function of temperature only. Hence we assume:

A2. ψ = ψ(T ).Inserting the latter into (2.9), yields(

ρdψ

dT+ ρs+ pβ

)DT

Dt− (Π + pI) : D(v) + q · ∇T

T≤ 0. (2.10)

Now, inequality (2.10) has to hold true for all thermo-mechanical processes. Thus,we deduce

s = −dψdT

− p

ρβ,

(Π + pI) : D(v) ≥ 0,

q · ∇T ≤ 0,

(2.11)

which, therefore, are restrictions on the constitutive assumptions. In particular, weconsider a thermally conducting Newtonian viscous fluid assuming that:

A3. The Cauchy stress tensor Π is

Π = −pI + 2µD(v) − 2µ3

(D(v) : I) I, (2.12)

where µ = µ(T ) > 0 is the dynamic viscosity and Stokes assumption has been used.Inequality (2.11)2 is fulfilled since

(Π + pI) : D(v) = 2µ| D(v) |2 ≥ 0,

with

D(v) = D(v) − 13

div vI. (2.13)

A4. The heat flux vector is

q = −λ∇T, (2.14)

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where λ is the thermal conductivity, λ = λ(T ) > 0. Inequality (2.11)3 is fulfilledsince q · ∇T = −λ|∇T |2.From (2.11) and (2.5)

e = e(T, p) = ψ(T ) − Tdψ

dT− βpT

ρ,

and

De

Dt=

(− T

d2ψ

dT 2+ Tp

(β

ρ2

dρ

dT

)︸︷︷︸

− β2ρ

− pT

ρ

dβ

dT

)DTDt

−(βp

ρ

DT

Dt

)︸︷︷︸

pρ divv

− βT

ρ

Dp

Dt.

Therefore, once assumptions A1, A2, A3 and A4 are stated, the mathematicalmodels describing the flow (2.1)–(2.3) rewrites as

βDT

Dt= div v, (2.15)

ρDv

Dt= −ρge3 −∇p+ div

2µD(v) − 2µ

3div vI

, (2.16)

ρ

(−T d

2ψ

dT 2− β2

ρTp− pT

ρ

dβ

dT

)DT

Dt= div

(λ∇T

)+ βT

Dp

Dt+ 2µ|D(v)|2, (2.17)

where the term |D(v)|2, calculated on the basis of (2.13), can be also written as|D(v)|2− 1

3 ( div v)2. The last term in (2.17) represents mechanical energy convertedinto heat by the internal friction. We remark that in the framework of the mechan-ical incompressibility assumption, the term βp

ρDTDt (i.e. p/ρ div v ) is necessarily

compensated by the mechanical work associated with dilation. Thus, it does notappear in the energy balance.

The coefficient in front of DTDt represents, from the physical point of view, the

isobaric specific heat cp. The energy equation (2.17) is exactly the correspondinggeneral energy equation in the specific enthalpy formulation from Ref. 16, pp. 51–85,but with the particular choice

cp(T, p) = cp 1(T ) − β2

ρTp− pT

ρ

dβ

dT, with cp 1(T ) = −T d

2ψ(T )dT 2

.

The fluid we are modeling admits only the isobaric specific heat. Indeed, because ofthe assumption A1 any change of body’s temperature implies a change in volume.Hence it is not possible to work with the isochoric specific heat cv. Consequently,the form of the energy equation adopted here is necessarily different from the theorydeveloped in Refs. 24 and 25.

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Since variations with respect to pressure are generally quite small, we imposethat cp(T, p) is constant with respect to the pressure field p and require

dβ

dT= −β2,

implying cp(T, p) ≡ cp 1(T ) and that β is of the form

β(T ) =βR

1 + βR(T − TR),

with TR reference temperature and βR = β(TR). As a consequence, from (2.8), wehave the following law for the density:

ρ =ρR

1 + βR (T − TR). (2.18)

Note that the ratio β/ρ appearing in the definition of e is constant.Keeping cp(T, p) = cp1(T ), we will consider the linearized version of (2.18),

namely

ρ(T ) = Aρ −BρT. (2.19)

with

βR =Bρ

ρRand ρR = Aρ −BρTR. (2.20)

We remark that, from the mathematical point of view such a simplification is notcrucial and it is consistent with the data reported in the experimental literature.

We then introduce the rescaled dimensionless temperature, that is

ϑ =T − TR

Tw − TR, ⇔ T = TR[1 + ϑ(Tw − 1)], with Tw =

Tw

TR, (2.21)

where Tw, Tw > TR, is another characteristic temperature (the temperature at theinlet boundary). Note that (2.19) reads now

ρ(ϑ) = ρR − βRρRTR(Tw − 1)ϑ. (2.22)

Next, we introduce the hydraulic head

P = p+ ρRgx3. (2.23)

Hence −ρge3 −∇p in (2.16) becomes

−ρge3 −∇p = (ρR − ρ) ge3 −∇P = [ρRβRTR(Tw − 1)ϑ] ge3 −∇P.

We note that after eliminating the (large) hydrostatic pressure, the effect of thetemperature field on the flow is more easy to observe.

Concerning the shear viscosity µ we assume the well-known Vogel–Fulcher–Tamman’s (VFT) formula

logµ(ϑ) = −Cµ +Aµ

TR −Bµ + TR(Tw − 1)ϑ, Aµ, Bµ, Cµ > 0. (2.24)

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In particular, µ is monotonically decreasing with ϑ. For more details we refere.g. to Ref. 17, Chapter 6. We note that the temperature in the problems we areconsidering is such that the denominator in (2.24) is always positive. Consequently,µ is a given strictly positive and C∞ function of the temperature. In the polymerphysics, formula (2.24) is known as Williams–Landau–Ferry (WLF) relation.

Finally, we suppose cp1 and λ to be smooth functions of ϑ, bounded from aboveand below by positive constants.

2.2. Scaling and dimensionless formulation of the general model

The scaling of model (2.15)–(2.17) will be operated paying particular attention tothe specific problem we are interested in. We introduce the following dimensionlessquantities:

x =x

H, v =

v

VR, t =

t

tRwith tR =

H

VR,

µ =µ

µR, λ =

λ

λR, cp1 =

cp1

cpR,

ρ =ρ

ρR, P =

P

PR, ψ =

ψ

TRcpR,

H plays the role of a length scale. The choice of the reference velocity VR is adistinctive feature of the scaling here adopted, suggested by the particular flowconditions we are referring to.b We will return to this point later on. From thereference pressure PR point of view the flows of glass or polymer melts are essentiallydominated by viscous effects. Accordingly we setc

PR =µR

HVR . (2.25)

Concerning the reference quantities µR, λR and cpR we identify them with thevalues taken by the respective quantities µ, λ and cp1 for T = TR.

Suppressing tildes to keep notation simple, model (2.15)–(2.17) rewritesd

div v =|Kρ|ρ(ϑ)

(Tw − 1)Dϑ

Dt, (2.26)

ρ(ϑ)Dv

Dt=

(|Kρ| (Tw − 1)

Fr2

)ϑe3 −

(PR

ρRV 2R

)∇P

+1

ReDiv

2µ(ϑ)D(v) − 2µ(ϑ)

3div vI

, (2.27)

bThis makes our approach different from the ones presented in Refs. 7 and 15 where there is novelocity scale defined by exterior conditions.cNotice that PR → 0 as VR tends to 0 and, as a consequence p tends to the hydrostatic pres-sure. This is consistent with the fact that P “measures” the deviation of the pressure from thehydrostatic-one due to the fluid motion.dIn the phenomena we are considering Tw − 1 is small but not negligible. Typically Tw − 1 is oforder 10−1.

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ρ(ϑ)cp1(ϑ)Dϑ

Dt=

(|Kρ|PR

ρRcpRTR(Tw − 1)

)1 + (Tw − 1)ϑ

ρ(ϑ)

[DP

Dt− ρRgH

PRv3

]

+1

Pediv(λ∇ϑ) + 2

EcRe(Tw − 1)

µ(ϑ)(|D(v)|22 −

13(div v)2

),

(2.28)

with

cp1(ϑ) = −1 + (Tw − 1)ϑ(Tw − 1)2

d2ψ

dϑ2.

We list the non-dimensional characteristic numbers appearing in (2.26)–(2.28):

Kρ = −βRTR, thermal expansivity number (negative, according to the usual nota-tion) and we may write (2.22) as

ρ(ϑ) = 1 − |Kρ| (Tw − 1)ϑ. (2.29)

Fr =VR√gH

, Froude’s number.

νR =µR

ρR, kinematic viscosity.

Re =VRH

νR, Reynolds’ number, which, because of (2.25), can also be written as

Re =ρRV

2R

PR.

Pe = Re ·Pr =VRHρRcpR

λR, Peclet’s number, with cpR = ψR/TR.

Pr =µRcpR

λR, Prandtl’s number.

Ec =V 2

R

cpRTR, Eckert’s number.

As mentioned, we are interested in studying, vertical slow flows of very viscousheated fluids (molten glasses, polymer, etc.) which are thermally dilatable. So,introducing the so-called expansivity coefficient (or thermal expansion coefficient)

α = |Kρ| (Tw − 1). (2.30)

we will consider the system (2.26)–(2.28) in the realistic situation in which theparameter α is small. Typically (e.g. for molten glass) 10−3 α 10−2. Equation(2.29) can be rewritten as

ρ(ϑ) = 1 − αϑ.

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Next, we define the Archimedes’ number

Ar =|Kρ| (Tw − 1)

Fr2 =|Kρ| (Tw − 1)

V 2R

gH, ⇒ α = ArFr2.

So, system (2.26)–(2.28) rewrite ase

div v =ArFr2

ρ(ϑ)Dϑ

Dt, (2.31)

ρ(ϑ)Dv

Dt= Arϑe3 +

1Re

[−∇P + Div

(2µ(ϑ)D(v) − 2µ(ϑ)

3div vI

)], (2.32)

ρ(ϑ)cp1 (ϑ)Dϑ

Dt=

1 + (Tw − 1)ϑρ(ϑ)

EcAr(Tw − 1)2

[Fr2

ReDP

Dt− v3

]+

1Pe

div(λ∇ϑ)

+ 2Ec

Re(Tw − 1)µ(ϑ)

(|D(v)|2 − 1

3(div v)2

). (2.33)

Notice that the ratios EcRe and Fr2

Re which appear naturally in the equation abovecan be interpreted as ratios of characteristic times

EcRe

=(

νR

cpRTR

)1tR,

Fr2

Re=

(νR

gH

)1tR.

Note also that

EcAr = αgH

cp R TR= α

EcFr2 .

2.3. Definition of the mathematical problem and

Oberbeck–Boussinesq limit

We consider a flow regime such that Ar = O(1), Re = O(1), Pe = O(1) andEc 1 (e.g. Ec 10−9). The terms in energy equation (2.33) containing the Eckertare dropped. So (2.31) and (2.32) remain unchanged while (2.33) reads as follows:

ρ(ϑ)cp1(ϑ)Dϑ

Dt=

1Pe

div(λ(ϑ)∇ϑ

). (2.34)

However, we eventually consider system (2.31)–(2.34) in the limit in which theparameter α is small while Ar and Re are of order 1, meaning that the buoyancyand the viscous forces cannot be negligible.

eWe remark that

ArFr2

ρ(ϑ)=

α

ρ(ϑ)=

α

1 − αϑ.

Hence, if order α2 terms are neglected, we may approximate αρ(ϑ)

by α.

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We will show that in such a case the system (2.31)–(2.34) can be approxi-mated by a system similar to the Oberbeck–Boussinesq system. Indeed our choiceof the parameters takes us close to the conditions of the formal derivation of theOberbeck–Boussinesq system for bounded domain. Such conditions (see Ref. 16,pp. 86–91) are

VR ≈ 0, Ma ≈ 0, Re = O(1) and Ar = O(1), (2.35)

with Ma being Mach’s number. Under conditions (2.35), the non-isothermal com-pressible Navier–Stokes system can be approximated (formally) by the Oberbeck–Boussinesq system

div vOB = 0, (2.36)

DvOB

Dt= ArϑOBe3 −∇POB +

1Re

div2µ(ϑOB)D(vOB)

, (2.37)

cp1(ϑOB)DϑOB

Dt=

1Pe

div(λ(ϑOB)∇ϑOB

), (2.38)

where the pressure P is rescaled by ρRV2R .

For the mathematical theory of the systems (2.36)–(2.38), subject to variousboundary conditions, but with constant viscosity, we refer to Ref. 19, pp. 129–137.

We note that the same reasoning applies to the situations in which other bound-ary conditions are given, just modifying the definition of ϑ and of Archimedes’number Ar. For details we refer again to Ref. 16, pp. 89.

The existence and uniqueness for the complete non-stationary problem (2.36)–(2.38) in Ref. 5.

In the case of natural convection flows, differently from gravity related flows,there is no characteristic velocity and one takes VR = µR/(HρR). In this case therelevant characteristic number is the Grashof number, Gr = ArRe2. Instead ofimposing Ar = O(1), one requires Gr = O(1) and the condition VR ≈ 0 becomesµR/ρR ≈ 0. Then the Oberbeck–Boussinesq system is obtained.

The Oberbeck–Boussinesq approximation for the natural convection in the caseof a layer of fluid of thickness H , with the top and bottom surfaces held at constanttemperature, is derived in Ref. 15. The authors took the Chandrasekhar’s velocityVR =

√gLβ(Tw − TR) (linked to buoyancy) as characteristic. Then they introduced

the small parameter ε =(V 3

RρR

gµR

)1/3 and obtained formally the Oberbeck–Boussinesqequations at order O(ε3). This approach was developed further in Ref. 10 in orderto get a modified Oberbeck–Boussinesq system, with important viscous heating.

A similar derivation of the Oberbeck–Boussinesq approximation for the naturalconvection in the case of a layer of fluid, can be found in Ref. 7, pp. 43–53. Therefore,for the sake of definiteness, a perfect gas is considered but the scalings and the ordersof magnitude involved are similar to Ref. 15.

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Next, we mention a number of papers by Zeytounian on the thermocapillaryproblems with or without a free boundary. They are reviewed in Refs. 24 and25. He considered the natural convection in an infinite horizontal layer of viscous,thermally conducting and weakly expansible fluid, heated from below. The smallparameter is α, given by (2.30), and for

α ≈ 0, Fr ≈ 0, Ma ≈ 0, and Gr =αRe2

Fr2 = O(1), (2.39)

he got the Oberbeck–Boussinesq approximation. In his derivation, the viscous dis-sipation term is absent if

1 mm ≈(µ2

R

gρ2R

)1/3

H ≈ C0/ (g(Tw − TR)) . (2.40)

The formal analysis of Zeytounian carries over to the free boundary cases.

Remark 2.1. Let us note that the model (2.31)–(2.33) is criticized in Refs. 2 and 3,who assert that giving the density as a function of the temperature, i.e. assumptionA1, contradicts the Gibbs convexity inequalities generating a rest state which is notstable. They propose to replace the constraint ρ = ρ(ϑ) with the constraint ρ = ρ(s).However, we consider the system (2.31)–(2.34) whose rest state (in absence of anybody force) is linearly stable.

We have also to remark that, when Eckert’s number is very small, the rest stateinstability of the system (2.31)–(2.33) would show up after such a long time to be ofno interest in technical problems. Clearly, for some other problems (e.g. atmosphericflows) the presence of instabilities, pointed out in Refs. 2 and 3, is a real difficultyand the meaning of the stationary problems is not clear. For the study of the fullcompressible Navier–Stokes system we refer to Ref. 20.

We are now able to define the mathematical problem that will be analyzed.

Problem A:We consider the stationary version of system (2.31)–(2.34) in the domain Ω, (namelysystem (1.1)–(1.3)) with the boundary conditions

v = v1eg, ϑ = 1, on Γin, (2.41)

v = v2eg, ϑ = 0, on Γout , (2.42)

v = 0, − 1Pe

λ(ϑ)∇ϑ · n = q0ϑ+ S, on Γlat , (2.43)v1 ∈ C1

0 (Γin) ∩ C∞(Γin),

v2 ∈ C10 (Γout) ∩ C∞(Γout),

S ∈ C∞(Γlat), S ≥ 0,

(2.44)

∫Γin

ρ(1)v1 rdrdφ =∫

Γin

(1 − α)v1 rdrdφ =∫

Γout

v2 rdrdφ, (2.45)

since ρ(0) = 1.

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Remark 2.2. It is important to remark that condition ϑ = 0 on Γout can bereplaced with a more general one, namely ϑ = ϑ2(x1, x2), with ϑ2 ∈ L∞(Γout) and0 ≤ ϑ2 ≤ 1. The proof of existence and uniqueness remains practically unchanged.

3. Existence for the Stationary Problem

In this section, we prove the existence of at least one solution for Problem A. Westart by studying the energy equation for a given w = ρv ∈ H(Ω), wheref

H(Ω) =z ∈ L3(Ω)3 |divz = 0 in Ω, z · n = 0 on Γlat,

z · n|Γin = ρ(1)v1, z · n|Γout = ρ(0)v2 .

We expect ϑ not to exceed 1 and the heat flux at Γlat to be always outgoing. Inother words, ϑ should vary in

[− ‖S‖L∞(Γlat)

q0, 1

]. In view of this fact it is convenient

to modify the definitions of cp1 and λ. We extend it on the whole of R by setting

cp1(ϑ) =

1ϑ2

cp1(1), for ϑ > 1,(‖S‖L∞(Γlat)

q0ϑ

)2

cp1

(−‖S‖L∞(Γlat)

q0

), for ϑ < −

‖S‖L∞(Γlat)

q0,

(3.1)

λ(ϑ) =

λ(1), for ϑ > 1

λ

(−‖S‖L∞(Γlat)

q0

), for ϑ < −

‖S‖L∞(Γlat)

q0.

(3.2)

We will prove indeed ϑ ∈[− ‖S‖L∞(Γlat)

q0, 1

]so that extensions (3.1) and (3.2) never

come into the play. In particular, (3.1) entails that∫ ϑ

0cp1(s)ds is bounded.

Now, for givenw ∈ H(Ω), S ∈ L∞(Γlat), S ≥ 0

λ, cp1 ∈W 1,∞(R), λ ≥ λ0 and constants q0, Pe ≥ 0,(3.3)

we consider the auxiliary problem

Problem Θ. Find ϑ ∈ H1(Ω) ∩ L∞(Ω) such that

cp1(ϑ)w · ∇ϑ =1

Pediv (λ(ϑ)∇ϑ) , (3.4)

ϑ = 1 on Γin and ϑ = 0 on Γout, (3.5)

− 1Pe

λ(ϑ)∇ϑ · n = q0ϑ+ S, on Γlat. (3.6)

Existence is straightforward once we have an explicit bound for the H1-norm of ϑ.

fRecall that the stationary version of (2.31) is equivalent to div(ρv) = 0, since ρ = 1 − αϑ.

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Proposition 3.1. Under the stated assumptions, Problem Θ has at least one vari-ational solution in H1(Ω), satisfying the estimate

λ0

2Pe‖∇ϑ‖2

L2(Ω)3 +q04‖ϑ‖2

L2(Γlat)≤ A0 +B0‖w‖L1(Ω)3 , (3.7)

where

B0 = ‖cp1‖∞ max

1,‖S‖L∞(Γlat)

q0

,

and

A0 =√q0 + 44q0

‖S‖L2(Γlat) + 2√q0|Γlat| +

‖λ‖∞2Peλ0

|Ω|

+32‖ρ‖∞ ‖cp1‖∞‖v1‖L2(Γin).

Proof. It is enough to prove the estimate (3.7). Then the classical technique ofLeray and Lions gives existence (see e.g. Chapter 4, Sec. 9, from Ref. 12 for details).

In order to prove (3.7), we write the corresponding variational formulation

1Pe

∫Ω

λ∇ϑ∇ϕ dx+∫

Ω

cp1 w∇ϑϕ dx+∫

Γlat

q0ϑϕdΓ = −∫

Γlat

SϕdΓ, (3.8)

∀ ϕ ∈ V with

V =H1(Ω), ϕ = 0 on Γout ∪ Γin

.

Next, we test (3.8) by ϕ = ϑ− x3 and the desired estimate (3.7) follows.

Corollary 3.1. Under the assumptions of Proposition 3.1, we have the followingestimate:

‖ϑ‖2L2(Ω) ≤ 8 max

Peλ0,

1q0

‖∂x3R‖∞(

λ0

2Pe‖∇ϑ‖2

L2(Ω)3 +q04‖ϑ‖2

L2(Γlat)

)≤ 8 max

Peλ0,

1q0

‖∂x3R‖∞(

A0 +B0‖w‖L1(Ω)3). (3.9)

Proof. To estimate ‖ϑ‖L2(Ω), we exploit the fact that for any section S(x3) at agiven x3, we may write∫

S(x3)

ϑ2dx1dx2 =∫ x3

0

d

dη

(∫S(η)

ϑ2 dx1dx2

)dx3

=∫ x3

0

∫S(η)

2ϑ∂ϑ

∂ηdx1dx2dη +

∫ x3

0

∫∂S(η)

ϑ2 ∂R

∂ηdSdη.

Then, since ‖ϑ‖2L2(Ω) =

∫ 1

0(∫

S(x3)ϑ2dx1dx2)dx3, by means of fundamental inequal-

ities, we get (3.9).

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Next we are interested in L∞-estimates.

Lemma 3.1. Any variational solution ϑ ∈ H1(Ω) to Problem Θ, satisfying the apriori estimate (3.7), satisfies also

−‖S‖L∞(Γlat)

q0≤ ϑ ≤ 1 a.e. on Ω. (3.10)

Proof. We start by testing (3.8) by ϕ = (ϑ − 1)+. Then the right-hand side ofthe inequality (3.10) is immediate. Next, we test (3.8) by ϕ = (A − ϑ)+, whereA ≤ 0 is a suitable constant to be chosen. A simple direct calculation shows thatfor A ≤ − ‖S‖L∞ (Γlat)

q0, ϕ = 0. Hence the lower bound in (3.10) is obtained as well.

Under the condition that the interior angle between the lateral boundary r =R(x3, φ) ∈ C2 and the upper and lower surfaces is π/2, we can apply the ellipticregularity. This can be seen directly by extending the equation for x3 > 1 andx3 < 0. For more general situations see e.g. Ref. 1, Chapter 5, Sec. 4. Indeed, if uis the solution of the mixed boundary value problem

div(∇u− f) = 0 in Ω, (3.11)

∇u · n = g on Γlat, and u = 0 on Γout ∪ Γin . (3.12)

it is known that u possesses W 1,3(Ω) estimate

‖∇u‖L3(Ω)3 ≤ 3(Ω)‖ g ‖L3(Γlat) + ‖ f ‖L3(Ω)3, (3.13)

for some Ω-dependent constant 3(Ω). We thus have the following:

Lemma 3.2. In our setting

‖∇ϑ‖L3(Ω)3 ≤ E00 + E01‖w‖L3(Ω), (3.14)

where

E00 =|Γlat|1/3 3(Ω)Pe

λ0

(‖S‖L∞(Γlat) + q0 max

1,

‖S‖L∞(Γlat)

q0

)+

‖λ‖∞λ0

(|Ω|1/3 + 3(Ω)|Γlat|1/3) (3.15)

and

E01 =3(Ω)λ0

Pe ‖cp1‖∞ max

1,‖S‖L∞(Γlat)

q0

. (3.16)

Proof. Let

Λ =∫ ϑ

0

λ (s) ds and Cp =∫ ϑ

0

cp1(s)ds. (3.17)

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Then, recalling (1.3), we have that Λ1 = Λ − x3

∫ 1

0 λ(s) ds, fulfilsg

div(

1Pe

∇Λ1 − wCp

)= 0,

Λ1|Γin= Λ1|Γout

= 0,

1Pe

∇Λ1 · n∣∣∣∣Γlat

= −qoϑ− S +n · e3Pe

∫ 1

0

λ(s)ds.

(3.18)

Next we apply estimate (3.13)–(3.18) and estimates (3.15) and (3.16) as follow.

Let us now return to the continuity and momentum equations, determining themomentum field

w = ρv,

and the pressure P . The corresponding boundary value problem is as follows:For given ϑ determine w, P satisfying

div w = 0 in Ω, (3.19)

(w∇)(w

ρ

)= Arϑeg −∇

(P

Re+

2µ3Re

divw

ρ

)

+1

ReDiv

(2µ(ϑ)D

(w

ρ

))in Ω, (3.20)

w = 0 on Γlat, w = (1 − α)v1eg on Γin and w = v2eg on Γout. (3.21)

Our first step is to eliminate the non-homogeneous boundary conditions for thevelocity field. By Ref. 6, Vol. II, Lemma 4.1, pp. 24–26, there exists

ζ ∈ H2(Ω)3, ∇ζ ∈ L3(Ω)9, (3.22)

with curl(ζ) satisfying (3.21). Next, we adapt the well-known Hopf construction toour non-standard nonlinearities. We have

Proposition 3.2. Let us suppose that 1/ρ ∈ L∞(Ω), 1/ρ ≤ 1/ρmin. Then, forevery γ > 0 there exists a ξ, depending on γ and on ρmin, such that

ξ ∈ H1(Ω)3, div ξ = 0 in Ω, ξ = curl (ζ) on ∂Ω (3.23)

and ∣∣∣∣∫Ω

1ρ

(φ∇

)φ · ξ dx

∣∣∣∣ ≤ γ‖φ‖2H1(Ω)3 , ∀ φ ∈ H1

0 (Ω)3. (3.24)

gRecall that w = ρ(ϑ)v.

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Proof. The proof is a slight generalization of Hopf’s construction (see e.g. Ref. 18,Chapter II, Sec. 1 or 6, Vol. II, pp. 27–32). Here, we only mention the crucial steps.

First, Ref. 6, Vol. I, Lemma 6.2, pp. 172–173, for any ε > 0 there exists afunction θε ∈ C∞(Ω) such that:

• |θε(x)| ≤ 1 for all x ∈ Ω.

• θε = 1 if dist(x, ∂Ω) < 12κ1

exp− 2ε.

• θε = 0 if dist(x, ∂Ω) ≥ 2 exp− 1ε.

• |∇θε(x)| ≤ κ2εdist(x,∂Ω) for all x ∈ Ω,

where κ1 and κ2 are constants corresponding to Stein’s regularized distance, κ1 =123 · 20/3.

Following Ref. 18, pp. 175–178, and Ref. 6, Vol. II, pp. 27–32, we set

ψε = curl(θεζ ).

It is easy to check that ψε satisfies the boundary conditions and div ψε = 0. Fur-thermore,

‖φ⊗ ψε‖L2(Ω)9 ≤ χ(ε)‖φ‖H1(Ω)3 , ∀ φ ∈ H10 (Ω)3, (3.25)

where χ(ε) → 0 as ε→ 0. Exploiting (3.25), we haveh

∣∣∣∣∫Ω

1ρ(φ∇)φ · ψε dx

∣∣∣∣ =∣∣∣∣ ∫

Ω

φ⊗ ψε

ρ: ∇φ dx

∣∣∣∣≤ 1ρmin

‖∇φ‖L2(Ω)9‖φ⊗ ψε‖L2(Ω)9

≤ χ(ε)ρmin

‖ φ ‖2H1(Ω)3 . (3.26)

So, once γ is fixed, let εγ such that

χ (εγ)ρmin

≤ γ.

Hence, inequality (3.24) follows by setting

ξ = ψεγ = curl (θεγζ ).

hAs already specified in Sec. 2.1, A : B = tr(ABT).

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Now, let us introduce some useful constants:

γ =2µmin

ρmaxRe, A00 = 4

√6

ρ2min

, B00 =µmin

ρmax, C01 = 2 · 61/6|Ω|5/6,

B01 = 2 · 61/3 1ρ2min

‖ξ‖L6(Ω)3 , B02 =4µmax61/6

ρ3min

,

C00 =1

ρmin

(‖ξ‖2

L6(Ω)3 +2µmax

Re‖D(ξ)‖L2(Ω)9

).

(3.27)

Theorem 3.1. Let us assume that

1Re

(B00 − αB02‖∇ϑ‖L3(Ω)3

)− αB01‖∇ϑ‖L2(Ω)3 > 0, (3.28)

and

∆ =[

1Re

(B00 − αB02‖∇ϑ‖L3(Ω)3

)− αB01‖∇ϑ‖L2(Ω)3

]2

− 4A00α‖∇ϑ‖L2(Ω)3(C00 + ArC01‖ϑ‖L∞(Ω)

)> 0, (3.29)

where A00, B00, B01, B02, C00 and C01 are positive constants defined in formula(3.27). Then there is a solution w ∈ H1(Ω)3 for the problems (3.19)–(3.21), satis-fying the estimate

‖D(w )‖L2(Ω)9 ≤ ‖D(ξ )‖L2(Ω)9 +C00 + ArC01‖ϑ‖L∞(Ω)√

∆. (3.30)

Proof. Let

W = w − ξ,

where ξ is given by (3.23) and (3.24). Of course, W |∂Ω = 0. Then we have

(w∇)(w

ρ

)=

1ρ

[(( W + ξ )∇) W + (( W + ξ )∇)ξ

]+

[( W + ξ) · ∇1

ρ

]( W + ξ ),

D

(w

ρ

)=

1ρ(D( W ) +D(ξ )) + sym

(( W + ξ ) ⊗∇1

ρ

),

and Eq. (3.20) becomes

1ρ(( W + ξ )∇) W + (( W + ξ )∇)ξ

+(

( W + ξ ) · ∇1ρ

)( W + ξ ) − Div

(2µ(ϑ)ρRe

D( W ))

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= −Arϑeg + ∇[− P

Re+

2µ3Re

div

(W + ξ

ρ

)]+ Div

(2µ(ϑ)ρRe

D(ξ ))

+ Div(

2µRe

sym(

( W + ξ) ⊗∇1ρ

))in Ω. (3.31)

As in the case of the stationary isothermal incompressible Navier–Stokes system,the existence follows from the coercivity properties. This approach would also givethe required a priori estimate. In order to establish it, we test (3.31) with W and get∫

Ω

12 ρ

(ξ + W )∇| W |2dx︸︷︷︸(I)

+∫

Ω

1ρ(( W + ξ )∇)ξ · W dx︸︷︷︸

(II)

+∫

Ω

(( W + ξ ) · ∇1

ρ

)(( W + ξ ) · W ) dx︸︷︷︸

(III)

+∫

Ω

2µ(ϑ)ρRe

|D( W )|2 dx︸︷︷︸(IV)

= −∫

Ω

2µ(ϑ)ρRe

D(ξ ) : D( W ) dx︸︷︷︸(V)

−∫

Ω

2µRe

sym(

( W + ξ ) ⊗∇1ρ

): D( W ) dx︸︷︷︸

(VI)

−∫

Ω

Arϑeg · W dx︸︷︷︸(VII)

. (3.32)

Let us estimate term (I) in the nonlinear form defined by (3.32)∫Ω

12ρ

(ξ + W )∇| W |2 dx =∫

∂Ω

| W |22ρ

(ξ + W ) · n dΓ −∫

Ω

| W |22

(ξ + W )∇1ρdx

= −12

∫Ω

| W |2(ξ + W )∇1ρdx, (3.33)

since W |∂Ω = 0. Let us now consider (II) + (III)

(II) + (III) =∫

Ω

| W |2(ξ + W ) · ∇1ρdx+

∫Ω

( W + ξ ) ⊗ W : ∇ξ

ρdx

=∫

Ω

| W |2(ξ + W ) · ∇1ρdx−

∫Ω

1ρ(D( W ) ξ ) · ξ dx

−∫

Ω

1ρ(( W ∇) W ) · ξ dx. (3.34)

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Now, we are able to estimate the inertia term, namely (I) + (II) + (III). We have

|(I) + (II) + (III)|

=∣∣∣∣12

∫Ω

| W |2(ξ + W ) · ∇1ρdx−

∫Ω

1ρ(D( W ) ξ ) · ξ dx−

∫Ω

1ρ(( W ∇) W ) · ξ dx

∣∣∣∣≤ 1

2‖ W ‖2

L6(Ω)3

(‖ W‖L6(Ω)3 + ‖ξ‖L6(Ω)3

) ∥∥∥∥∇1ρ

∥∥∥∥L2(Ω)3

+1

ρmin‖D( W )‖L2(Ω)9 ‖ξ ‖2

L4(Ω)3 + γ‖∇ W‖2L2(Ω)9 , (3.35)

where for the first two integrals have been estimated using Holder inequality whilethe last exploiting (3.24).

Concerning the viscosity terms (V) and (VI), we have

|(V)| ≤ 2µmax

ρminRe‖D( W )‖L2(Ω)9 ‖D(ξ )‖L2(Ω)9 , (3.36)

and

|(VI)| ≤ 2µmax

ρminRe‖D( W )‖L2(Ω)9

(‖ W‖L6(Ω)3 + ‖ξ ‖L6(Ω)3

) ∥∥∥∥∇1ρ

∥∥∥∥L3(Ω)3

. (3.37)

Here it is worth to remark that:

• ‖D(φ )‖2L2(Ω)9 = 1

2‖∇φ ‖2L2(Ω)9 ∀φ ∈ H1

0 (Ω)3, div φ = 0. Consequently

‖∇φ ‖L2(Ω)9 =√

2‖D(φ) ‖L2(Ω)9 .

• ‖φ ‖L6(Ω)3 ≤ 6√

48‖∇φ ‖L2(Ω)9 , ∀φ ∈ H10 (Ω)3, see Ref. 11, Lemma 3, p. 10. We

thus estimate the cubic term ‖ W‖3L6 appearing in (3.35) as follows:

12‖ W‖3

L6(Ω)3 ≤ 12( 6√

48)3‖∇ W‖3L2(Ω)9 ≤ (

√2 )3

2( 6√

48)3︸︷︷︸4√

6

‖D( W )‖3L2(Ω)9 .

In particular, ‖ W‖L6(Ω)3 ≤ 2 6√

6‖D( W )‖L2(Ω)9 .

Finally, term (VII) is estimated as follows:

|(V II)| ≤ Ar ‖ϑ‖L6/5(Ω) ‖ W‖L6(Ω)

≤ Ar ‖ϑ‖L∞(Ω)|Ω|5/6 481/6 ‖∇ W‖L2(Ω)9

≤ Ar√

2 481/6︸︷︷︸2 6√6

|Ω|5/6 ‖ϑ‖L∞(Ω) ‖D( W )‖L2(Ω)9 ,

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and we recall that ∥∥∥∥∇1ρ

∥∥∥∥Lp(Ω)3

≤ α

ρ2min

‖∇ϑ‖Lp(Ω)3 .

Now, we note that (3.32) defines a nonlinear form I which can be bounded frombelow in this way

I ≥ −|(I)| − |(II)| − |(III)| + (IV) − |(V)| − |(VI)| − |(VII)|

≥ 2µmin

Re ρmax‖D( W )‖2

L2(Ω)9 − |(I)| − |(II)| − |(III)| − |(V)|

− |(VI)| − |(VII)|. (3.38)

After exploiting the above estimates, the r.h.s. of inequality (3.38) can be writtenas a function of X = ‖D(w )‖L2(Ω)9 . Indeed we have

I ≥ F (X) = −

α(

4√

6ρ2min

)︸︷︷︸

A00

‖∇ϑ‖L2(Ω)3

X3

+

1Re

(µmin

ρmax

)︸︷︷︸

B00

− α

Re

(4 6√

6µmax

ρ3min

)︸︷︷︸

B02

‖∇ϑ‖L3(Ω)3

−α

(2 3√

6 ‖ξ ‖L6(Ω)3

ρ2min

)︸︷︷︸

B01

‖∇ϑ‖L2(Ω)3

X2

−

1ρmin

‖ξ ‖2L4(Ω)3 +

2µmax

ρmin Re‖D(ξ )‖L2(Ω)9︸︷︷︸

C00

+Ar(2 |Ω|5/6

√6)

︸︷︷︸C01

‖ϑ‖L∞(Ω)

X , (3.39)

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namely

F (X) = −A00α‖∇ϑ‖L2(Ω)3X3

+[

1Re

(B00 − αB02‖∇ϑ‖L3(Ω)3

)− αB01‖∇ϑ‖L2(Ω)3

]X2

−(C00 + ArC01‖ϑ‖L∞(Ω)

)X. (3.40)

Now, it is easy to check that F has two positive roots X0 and X1, X0 ≤C00+Ar C01‖ϑ‖L∞(Ω)√

∆< X1, if and only if the conditions (3.28) and (3.29) are satisfied.

Next, it is worth to note that conditions (3.28) and (3.29) hold if α is not toolarge. In other words, we find out that there exits always an α0 > 0 (dependingon A00, B00, etc.) such that for α ≤ α0 (3.28) and (3.29) are fulfilled. So, forα ∈ (0, αo) the function F (X) is negative in (0, X0), then positive in (X0, X1) andthen negative X > X1.

Let φj be an orthonormal basis for the space

V (Ω) = z ∈ H10 (Ω)3 | div z = 0 in Ω,

and let N > 0 be an arbitrary integer. Then we search for a solution to the problem

Find wN =N∑

j=1

ζj φj(x) + ξ such that for j = 1, . . . , N,

∫Ω

(wN∇

) wN

ρφ j dx

= −∫

Ω

Ar ϑφj3 dx−

∫Ω

2Re

µ(ϑ)D(wN

ρ

): D(φj) dx. (3.41)

We note that (3.41) can be written as

Φj(ζ1, . . . , ζN ) = 0, j = 1, . . . , N, (3.42)

where Φ : RN → R

N is a continuous map. Furthermore,

Φ(δ1, . . . , δN) · (δ1, . . . , δN) ≥ F (‖D( W Nδ )‖L2(Ω)9),

where W Nδ =

∑Nj=1 δj

φj . By the above considerations, for all (δ1, . . . , δN), suchthat

|(δ1, . . . , δN )| = ‖D( W Nδ )‖L2(Ω)9 = X0

we have Φ(δ1, . . . , δN )·(δ1, . . . , δN ) ≥ 0. Now, Brouwer’s fixed point theorem impliesthat there is a solution (ζ1, . . . , ζN ) for (3.42), i.e. a solution wN =

∑Nj=1 ζj

φj(x)+ξfor (3.41), satisfying ‖D(wN − ξ)‖L2(Ω)9 ≤ X0.

After obtaining this lower bound, the existence of at least one solution followsthe lines of the classic proof for the steady Navier–Stokes system. We simply passto the limit N → +∞ as e.g. in Ref. 6 or in 18, Vol. II.

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We now define our iterative procedure:

(1) Let γ given by (3.28) and (3.29) and ξ be the corresponding vector valuedfunction from Hopf’s construction.

(2) For a given wm = W m + ξ, such that W m ∈ BR,

BR = z ∈ H10 (Ω)3 : div z = 0 in Ω and ‖D( W m)‖L2(Ω)9 ≤ R,

(with R > 0 that will be specified in the sequel) we calculate ϑm, a solution to(3.4)–(3.6).

(3) With such ϑm, we determine a solution wm+1 = W m+1 + ξ for the problem(3.19)–(3.21), satisfying the estimate (3.30).

The natural question arising in the iterative process is: does W m+1 remain in BR?Of course, the answer will depend on the particular R that has been selected.We have the following result:

Proposition 3.3. Let the constants A0 and B0 be given as in Proposition 3.1 andlet E00 and E01 be given by Lemma 3.2. Let the constants γ,B00, B01, B02, A00, C00

and C01 be given by formula (3.27). Let ξ be generalized Hopf’s lift, given by Propo-sition 3.2 and corresponding to γ. Let R be given by

R =√

2Reρmax

µmin

(2Ar|Ω|5/661/6 max

1,

‖S‖L∞(Γlat)

q0

+1

ρmin

(‖ξ‖2

L6(Ω)3 +2µmax

Re‖D(ξ)‖L2(Ω)9

)), (3.43)

Then for all α > 0 such that

∆1 =1

Re

B00 − αB02

[E00 + E01

(‖ξ ‖L3(Ω)3 + 481/12|Ω|1/6R

)]−αB01

√2Peλ0

[√A0 +

√B0(‖ξ ‖1/2

L1(Ω)3

+ |Ω|1/4√R

)]> 0 (3.44)

and

∆2 = ∆21 − 4A00α

√2Peλ0

[√A0 +

√B0

(‖ξ ‖1/2

L1(Ω)3 + |Ω|1/4√R

)]×

[C00 + ArC01 max

1,

‖S‖L∞(Γlat)

q0

]>

B200

2Re2 , (3.45)

Wm ∈ BR implies Wm+1 ∈ BR.

Remark 3.1. Expressions (3.44) and (3.45) define an α0 such that for all α ≤ α0

the sequence W m+1 remains in the ball BR.

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Proof. First of all let us fix a positive R. Then, after having taken ‖D( W m)‖L2(Ω)9

in BR, we rewrite estimates (3.7) and (3.14) in term of R, namely

‖∇ϑ‖L2(Ω)3 ≤√

2Peλ0

[√A0 +

√B0

(‖ξ ‖1/2

L1(Ω)3 + |Ω|1/4√R

)],

‖∇ϑ‖L3(Ω)3 ≤ E00 + E01

(‖ξ ‖L3(Ω)3 + 481/12|Ω|1/6R

).

Next, we determine α0 such that for any α ∈ (0, α0) conditions (3.44) and (3.45)are fulfilled. We note that (3.44) and (3.45) imply conditions (3.28) and (3.29).

Now, for α ∈ (0, α0), Theorem 3.1 entails that

‖D( Wm+1)‖L2(Ω)9 ≤C00 + ArC01‖ϑm‖L∞(Ω)√

∆2

(3.46)

Consequently, for R given by (3.43), ‖D( Wm+1 )‖L2(Ω)9 ∈ BR and the propositionis proven.

Theorem 3.2. There is a weak solution ϑ,v ∈W 1,3(Ω)×H1(Ω)3 for the Prob-lem A, such that

‖∇ϑ‖L2(Ω)3 ≤√

2Peλ0

[√A0 +

√B0(‖ξ ‖1/2

L1(Ω)3 + |Ω|1/4√R)

]−‖S‖L∞(Γlat)

q0≤ ϑ ≤ 1 and ‖D(v − ξ)‖L2(Ω)9 ≤ R,

(3.47)

where R is given by (3.43) and ξ is given by Proposition 3.2 with γ given by (3.27).

Proof. We start with some W 0 ∈ BR and calculate ϑ1 and W 1. By the previousProposition, there exists W 1 which is also in BR. By repeating the procedure, weconstruct a sequence wm, ϑmm∈N, such that W mm∈N is uniformly boundedin BR and ϑmm∈N is uniformly bounded in W 1,3(Ω) ∩ L∞(Ω). Therefore, thesesequences contain subsequences which converge. More precisely, without changingindices, there are w ∈ H1(Ω)3 and ϑ ∈W 1,3(Ω) ∩ L∞(Ω) such that

wm w weakly in H1(Ω)3 and wm → w strongly in Lq(Ω)3, ∀ q ∈ (1, 6),

ϑm ϑ weakly in W 1,3(Ω) and ϑm → ϑ strongly in Lq(Ω), ∀ q ∈ (1,+∞),

ϑm ϑ weak star in L∞(Ω).

Furthermore, the L∞-bounds on ϑm guarantee that the density ρ(ϑm) remainsuniformly bounded and bounded from below by a positive constant. With the aboveconvergence, it is easy to prove that the cluster point w, ϑ satisfies the equationsand boundary conditions of Problem A.

Now, we are in a position to pass to the limit when the expansivity parameter αtends to zero.

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First we remark that the a priori estimates stated above are independent of α,|α| ≤ α0, where α0 is the maximal positive α satisfying (3.44) and (3.45). Conse-quently, we have

Theorem 3.3. Let ϑ(α), v(α), α ∈ (0, α0), be a sequence of weak solutions toProblem A, satisfying the bounds (3.47). Then there exists ϑOB, vOB ∈ W 1,3(Ω)×H1(Ω)3 and a subsequence ϑ(αk), v(αk) such that

ϑ(αk) → ϑOB, uniformly on Ω,

ϑ(αk) ϑOB, weakly in W 1,3(Ω),

v(αk) vOB , weakly in H1(Ω)3.

Furthermore, ϑOB, vOB is a weak solution for the Eqs. (2.36)–(2.38), satisfyingthe boundary conditions (2.41) and (2.45) and the bounds (3.47).

With the a priori estimates from the existence proof this result is obvious. Never-theless, in order to say more we should establish uniqueness.

4. The Uniqueness for the Stationary Problem

The uniqueness proof for Problem A is quite involved. For this reason we only statethe main results and put the detailed calculations in the Appendix.

We prove uniqueness in two steps. At first we consider the Oberbeck–Boussinesqsystemi

div v = 0 in Ω, (4.1)

(v∇ )v = Ar ϑeg +1

Re[−∇P + Div(2µ(θ)D(v))], (4.2)

v |Γin = v1eg, v |Γout = v2eg, v |Γlat = 0, and∫

Γin

v1dΓ =∫

Γout

v2dΓ,

(4.3)

cp1(ϑ)v · ∇ϑ =1

Pediv(λ(ϑ)∇ϑ), (4.4)

ϑ|Γin = 1, ϑ|Γout = 0, − 1Pe

λ(ϑ)∇ϑ · n = q0ϑ+ S on Γlat , (4.5)

and derive some estimate that will be extended to Problem A with the purpose ofshowing uniqueness.

In the authors knowledge, uniqueness for problem (4.1)–(4.3) has not been con-sidered in mathematical literature. In analogy with the case of constant viscositywe expect uniqueness only for small data.

iIn system (4.1)–(4.5) the fluid density ρ is constant, namely ρ = ρR = 1.

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Theorem 4.1. (Uniqueness theorem for the Boussinesq system with temperaturedependent viscosity) Let us assume that our data satisfy the conditions

2Re

µmin −√

8 61/4(R + ‖D(ξ )‖L2(Ω)9) > 0. (4.6)

1Pe

−‖cp1‖L∞(R)

λ0(‖ξ‖L6(Ω)3 + 2 6

√6R)C3(Ω) > 0, (4.7)

where ξ corresponds to ρ = 1, γ = µmin/Re and C3(Ω) is the constant in the imbed-ding of the functions from H1(Ω), with zero trace at x3 = 0 in L3(Ω). Furthermore,let

2Ar 6√

6 |Ω|5 +2

Re‖µ′‖L∞(R)(R+ ‖D(ξ )‖L2(Ω)9)

2Re

µmin −√

8 61/4 (R+ ‖D(ξ )‖L2(Ω)9)

<1

Peλo

(4Bθ 23/461/8 + Aθ

) . (4.8)

Then the problems (4.1)–(4.5) admit a unique weak solution v, ϑ ∈ H1(Ω)3 ×(H1(Ω) ∩ L∞(Ω)), satisfying (3.47).

Next, we then generalize the results just obtained to the case of non-constantdensity ρ(ϑ) = 1 − αϑ, i.e. to Problem A. We have

Theorem 4.2. (Uniqueness theorem for Problem A) Let us assume that our datasatisfy conditions

2Re

ρminµmin − ρ2max2 · 31/4(R+ ‖D(ξ)‖L2(Ω)9) > α

4µmax

Re61/6‖∇θ(1)‖L3(Ω)3

+αρmax‖∇θ(1)‖L2(Ω)32 · 61/3(2 · 61/6R+ ‖ξ‖L6(Ω)3) (4.9)

and (4.7), where ξ corresponds to ρ = ρ(α, ϑ). Furthermore let2

Reµminρmin − 2ρ2

max31/4(R+ ‖D(ξ)‖L2(Ω)9)

−α4µmax

Re61/6(E00 + E01(‖ξ‖L3(Ω)3 + 481/12|Ω|1/6R))

− 2αρmax61/3(2 · 61/6R+ ‖ξ‖L6(Ω)3)

·√

2Peλ0

(√A0 +

√B0(‖ξ‖1/2

L1(Ω)3 + |Ω|1/4√R))

>2αRe

(µmax + 2‖µ′‖L∞(R)61/6)(R+ ‖D(ξ)‖L2(Ω)9)

Peλ0

(Aθ + 4Bθ23/461/8

)+µmax(2 · 61/6R+ ‖ξ‖L6(Ω)3)L3

+

2Re

ρmax‖µ′‖L∞(R)(R+ ‖D(ξ)‖L2(Ω)9)

× Peλ0

(Aθ + 4Bθ23/461/8

)+ αρmax

√2C6(Ω)L2(2 · 61/6R+ ‖ξ‖L6(Ω)3)

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×((R

√2 + ‖∇ξ‖L2(Ω)9)

√2 · 61/3 + 2 · 61/6R + ‖ξ‖L6(Ω)3)

+Arα(R + ‖ξ‖L2(Ω)3)L22241/4 + Arρmax

L2√2

+2αRe

‖µ′‖L∞(R)(R + ‖ξ‖L2(Ω)3)Peλ0

(Aθ + 4Bθ23/461/8

)·

(R+ ‖ξ‖L2(Ω)3)Peλ0

(Aθ + 4Bθ23/461/8

)+ L3(2 · 61/6R + ‖ξ‖L6(Ω)3)

+αρmax2 · 61/3 Pe

λ0

(Aθ + 4Bθ23/461/8

)· (R

√2 + ‖∇ξ‖L2(Ω)9)

× (R√

261/12 + ‖ξ‖L3(Ω)3)|Ω|1/6 +αPeλ0

(Aθ + 4Bθ23/461/8

)2

× (R+ ‖D(ξ)‖L2(Ω)9)(2 · 61/6R+ ‖ξ‖L6(Ω)3)|Ω|1/6. (4.10)

Then there is a unique weak solution v, ϑ ∈ H1(Ω)3 × (W 1,3(Ω) ∩ C(Ω)) forProblem A, satisfying the bounds (3.47).

For the proofs of Theorems 4.1 and 4.2 we refer to the Appendix.

5. The Regularity of Solutions

In this section we prove that, in fact, the solutions to Problem A are smooth forsmooth data. For C∞-regularity of the data we will establish interior C∞-regularityof solutions.

The regularity is proved by iterating the corresponding regularity properties forthe energy equation and for the coupled continuity and momentum equations. It isthe classical bootstrapping procedure in the elliptic regularity theory.

We start with the energy equations (3.4)–(3.6), for given w ∈ L6(Ω)3. Then wehave

Lemma 5.1. Let cp1 and λ ∈ C∞(R). Furthermore, let Γlat ∈ C∞ and S ∈C∞(Γlat). Then ϑ ∈W 2,6(Ω) ⊆ C1,1/2(Ω).

Proof. We write the energy equations (3.4)–(3.6) in the formj

div

1Pe

∇Λ − wCp(ϑ)

= 0 in Ω,

Λ|Γout= 0, Λ|Γin

=∫ 1

0

λ(s)ds,

− 1Pe

∇Λ · n = q0ϑ(Λ) on Γlat,

(5.1)

jRecall w = v ρ(ϑ).

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where Λ and Cp are given by (3.17). Then, using the elliptic potentials as in Ref. 1,Chapter 5, we have Λ ∈ W 1,6(Ω) and, consequently, ϑ ∈ W 1,6(Ω) ⊆ C0,1/2(Ω). Nowcp1 w∇ϑ ∈ L3(Ω) and the Elliptic Regularity Theory (see for instance Chapter 9 inRef. 8) gives ϑ ∈ W 2,3(Ω). The bootstrapping yields the desired regularity.

Next, we study the system (3.19)–(3.21). We have

Lemma 5.2. Let ϑ ∈ W 2,6(Ω), let Γlat ∈ C∞ and let vj ∈ C∞, j = 1, 2 satisfy(2.44). Then w, P ∈ W 2,q(Ω)3 ×W 1,q(Ω), ∀ q < +∞.

Proof. First, as in the case of the linear Stokes system, using Tartar’s argument(see e.g. Ref. 18, Remark 1.9, pp. 19–20) we conclude that there exists a pressurefield π ∈ L2(Ω) such that (3.20) holds in the sense of distributions.

Next, we write Eq. (3.20) in the form

−∆w + Re∇(ρ(ϑ)µ(ϑ)

π

)= F in Ω,

div w = 0 in Ω,

w = 0 on Γlat, w = v2e3 on Γout, w = (1 − α)v1e3 on Γin,

where

F = Re π(ρ(ϑ)µ(ϑ)

)′∇ϑ+ 2

(µ(ϑ)ρ(ϑ)

)′ρ(ϑ)µ(ϑ)

D(w)∇ϑ

+ρ(ϑ)µ(ϑ)

Div(

2µ(ϑ) sym(w ⊗∇1

ρ

))− Re

1µ(ϑ)

(w∇ )w

− ρ(ϑ)µ(ϑ)

Re w

(w · ∇1

ρ

)+ Arϑ

ρ(ϑ)µ(ϑ)

Ree3

and

π =P

Re+

2µ3Re

div(w

ρ

). (5.2)

The regularity is now established as for the steady-state incompressible Navier-Stokes equations. We apply Proposition 2.3, pp. 35–37 from Ref. 18. It is enoughto establish regularity of F .

As in the standard case, the lowest regularity comes from the term Re 1µ(ϑ)

(w∇ )w, which is from L3/2(Ω)3. Now, by the regularity for the Stokes system,we have w ∈ W 2,3/2(Ω)3 and π ∈ W 1,3/2(Ω)3. Next, we insert w, π with thisregularity, back to F . Again, the inertial term dictates the regularity and F ∈L3−δ(Ω)3, for all δ > 0. Consequently, w ∈ W 2,3−δ(Ω)3 and π ∈ W 1,3−δ(Ω)3, forall δ > 0. After the bootstrap which follows we get the required regularity.

Next, we go back to the energy equations (3.4)–(3.6). This same bootstrappingtechniques leads to the following regularity result.

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Theorem 5.1. Let the assumptions on the data from Lemmas 5.1 and 5.2 holdtrue. Then every weak solution v, P, ϑ ∈ H1(Ω)3 × L2(Ω) × (H1(Ω) ∩ L∞(Ω)) ofProblem A is an element of W 2,q(Ω)3×W 1,q(Ω)×W 2,q(Ω), ∀ q <∞. Furthermore,v, p, ϑ ∈ C∞(Ω)4.

6. The Boussinesq Limit

In this section, we reconsider the limit when the expansivity parameter α tendsto zero. We saw at the end of the section on the existence of a weak solutionthat the obtained a priori estimates allow passing to the Boussinesq limit. Havingjustified the Oberbeck–Boussinesq system as the limit equations when the expan-sivity parameter α tends to zero, the next question is: What is the accuracy of theapproximation?

The answer relies on the uniqueness and regularity results from the previoussections.

We start by studying the equations for the derivatives with respect to α. Forsimplicity, we assume that

V1, v2 are independent of α with V1 = v1(1 − α). (6.1)

Hence, by (2.45),∫Γout

dv2dα

rdrdφ = 0 and∫

Γin

d

dα((1 − α)v1) rdrdφ = 0.

Proposition 6.1. Let us suppose that the conditions (3.44)–(3.45), (4.7), (4.9) and(4.10), ensuring existence, uniqueness and regularity of a solution lying inside theball defined by the bounds (3.47). Furthermore, let the solution v, p, ϑ satisfy theinequalities

N = 1 − 5C6(Ω)max

Peλ0,

1q0

‖∂x3R‖∞

×(‖cp1‖L∞(R)‖v‖L3(Ω)3ρmax +

‖λ′‖L∞(R)

Pe‖∇ϑ‖L3(Ω)3

)> 0 , (6.2)

2Re

µmin

ρmax> 4

(32

)1/8

‖v ‖L4(Ω)3 + ArZΘ|Ω|1/3 +2

Re‖µ′‖∞‖D(v )‖L3(Ω)9ZΘ

+α

[‖v ‖2

L6(Ω)3ZΘ +2

Reµmax

ρmin‖v ‖L∞(Ω)3

(LΘ

Peλ0

+α‖∇ϑ‖L3(Ω)3

ρminZΘ

)

+2

Reµmax

ρmin‖D(v )‖L3(Ω)9ZΘ +

2Re

µmax

ρ2min

481/6‖∇ϑ‖L3(Ω)3

], (6.3)

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where

LΘ =‖cp1‖L∞(R)‖ϑ‖L∞(Ω)

N , (6.4)

ZΘ = 10C6(Ω)LΘ

√Pe2λ0

max

Peλ0,

1q0

‖∂x3R‖∞, (6.5)

and C6(Ω) is the embedding constant of the functions from H1(Ω), with zero traceat x3 = 0 in L6(Ω). Then derivatives of the solution, with respect to α, exist at allorders as continuous functions of α.

Proof. Recalling that ρ = 1 − αϑ, we rewrite the momentum equation (1.2) as

Div((ρ(ϑ)v) ⊗ v) = Ar ϑe3 −∇π +2

ReDiv(µ(ϑ)D(v )),

with π given by (5.2) and Eq. (1.3) as

div[(ρ(ϑ)v) Cp(ϑ) − 1

Peλ(ϑ)∇ϑ

]= 0,

with Cp defined by (3.17).

Next, introducing u, π, ζ =d

dαv, π, ϑ, by direct computation we find that

the system of PDEs for the first derivative u, π, ζ is

div(ρ(ϑ, α)u − αζv − ϑv) = 0, (6.6)

div[(ρ(ϑ, α)u − αζv − ϑv) ⊗ v+(ρ(ϑ)v) ⊗ u]

= Ar ζ e3 −∇π +2

ReDiv[µ(ϑ)D(u) + µ′(ϑ) ζ D(v)], (6.7)

div− 1

Pe(λ′(ϑ) ζ∇ϑ+ λ(ϑ)∇ζ)

+ (ρ(ϑ, α)u− αζv − ϑv)Cp(ϑ) + ρ(ϑ, α)v cp1(ϑ) ζ

= 0, (6.8)

while the boundary conditions are

ζ|Γout = 0, u|Γout = 0 , ζ|Γin = 0, u|Γin =V1

(1 − α)2e3 , (6.9)

u = 0, and − 1Pe

(λ(ϑ)∇ζ + λ′(ϑ)ζ∇ϑ) · n = q0ζ on Γlat. (6.10)

Now, since the discretized first derivative satisfies the same equation, it is enoughto prove that the system (6.6)–(6.10) is solvable and admits a unique solution.

In order to use the Spectral Theory, we introduce the functional spaces W =φ ∈ H1(Ω)3| div φ = 0 in Ω and T = φ ∈ H1(Ω)| φ = 0 on Γout ∪ Γin and theunknown function

q = ρ(ϑ, α)u − αζv.

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Hence Eq. (6.6) can be rewritten as

div q = div (ϑv), (6.11)

(6.8) as

div[− 1

Pe(λ′ ζ∇ϑ+ λ∇ζ) + qCp + ρv cp1 ζ

]= div (Cp(ϑ)ϑv), (6.12)

and (6.7) as

Div(q ⊗ v + v ⊗ (q + α ζ v)

)= Arζe3 −∇π +

2Re

Divµ(ϑ)D

(q + α ζ v

ρ

)+ µ′(ϑ) ζ D(v)

+ Div (ϑv ⊗ v). (6.13)

Due to the results obtained in the section on regularity and the Spectral The-ory, the unique solvability of the problem (6.11)–(6.13), (6.9), (6.10) is valid if thecorresponding homogeneous problem, for q, π, ζ, has only the trivial solution inW × L2

0(Ω) × T . The homogeneous system reads

div q = 0 , (6.14)

Div(q ⊗ v + v ⊗ (q + α ζ v)

)= Arζe3 −∇π +

2Re

Divµ(ϑ)D

(q + α ζ v

ρ

)+ µ′(ϑ) ζ D(v)

, (6.15)

div−λ(ϑ)

Pe∇ζ +

[ρ(ϑ, α)v cp1(ϑ) − λ′(ϑ)

Pe∇ϑ

]ζ + qCp(ϑ)

= 0 , (6.16)

ζ|Γout∪Γin = 0, q |Γin∪Γout = 0, (6.17)

q = 0 and − 1Pe

(λ(ϑ)∇ζ + λ′(ϑ)ζ∇ϑ) · n = q0ζ on Γlat. (6.18)

Let us suppose that a non-trivial solution for (6.14)–(6.18) exists. As a consequenceof (3.9) we have

‖ζ‖2H1(Ω) ≤ 5 max

Peλ0,

1q0

‖∂x3R‖∞(

λ0

Pe‖∇ζ‖2

L2(Ω)3 +q02‖ζ‖2

L2(Γlat)

). (6.19)

Then we test (6.16) by ζ and get

R− 5RC6(Ω)[‖cp1‖∞ρmax‖v ‖L3(Ω)3 +

‖λ′‖∞Pe

‖∇ϑ‖L3(Ω)3

]

· max

Peλ0,

1q0

‖∂x3R‖∞

≤ ‖ϑ‖L∞(Ω)‖cp1‖∞ · ‖q ‖L2(Ω)3

√Peλ0

R 12 , (6.20)

where

R =λ0

Pe‖∇ζ‖2

L2(Ω)3 + q0‖ζ‖2L2(Γlat)

,

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and (6.19) implies

‖∇ζ‖L2(Ω)3 ≤ LΘPeλ0

‖q‖L2(Ω)3 , (6.21)

with LΘ given in (6.4).Next, we test Eq. (6.15) with q and obtain

2Re

µmin

ρmax‖D(q )‖2

L2(Ω)9

≤ 2‖D(q )‖L2(Ω)9‖q ‖L4(Ω)3‖v ‖L4(Ω)3 + α‖D(q )‖L2(Ω)9‖v ‖2L6(Ω)3‖ζ ‖L6(Ω)

+2

Re‖µ′‖∞‖D(v )‖L3(Ω)9 ‖D(q )‖L2(Ω)9 · ‖ζ‖L6(Ω) + Ar‖ζ‖L2(Ω)‖q3‖L2(Ω)

+2αRe

µmax‖D(q )‖L2(Ω)9‖v ‖L∞(Ω)3 ·(‖∇ζ‖L2(Ω)3

ρmin+α‖∇ϑ‖L3(Ω)3‖ζ ‖L6(Ω)

ρ2min

)+

2αRe

µmax

ρmin‖D(q )‖L2(Ω)9 ‖ζ‖L6(Ω) · ‖D(v )‖L3(Ω)9

+2

Reµmax‖D(q )‖L2(Ω)9‖q ‖L6(Ω)3

α‖∇ϑ‖L3(Ω)3

ρ2min

. (6.22)

Using then (6.21), (6.22) and the interpolation inequality,

‖q ‖2L4(Ω)3 ≤ 4(3/2)1/4 ‖D( q )‖2

L2(Ω)9 ,

we get2

Reµmin

ρmax‖D(q )‖2

L2(Ω)9

≤ ‖D(q )‖2L2(Ω)9

4

(32

)1/8

‖v ‖L4(Ω)3 + ArZΘ|Ω|1/3

+2

Re‖µ′‖∞‖D(v )‖L3(Ω)9ZΘ + α

[‖v ‖2

L6(Ω)3ZΘ +2

Reµmax

ρmin‖v ‖L∞(Ω)3

·(LΘ

Peλ0

+α‖∇ϑ‖L3(Ω)3

ρminZΘ

)+

2Re

µmax

ρmin‖D(v )‖L3(Ω)9ZΘ

+2

Reµmax

ρ2min

481/6‖∇ϑ‖L3(Ω)3

]. (6.23)

Since (6.23) contradicts (6.3), we conclude that only the trivial solution is possible.Concerning the higher order derivatives, they are determined recursively

through the problems

div

ρ(ϑ, α)

dNv

dαN−N

dN−1ϑ

dαN−1v −

N−1∑k=1

(N

k

)dN−kv

dαN−k

(kdk−1ϑ

dαk−1+ α

dkϑ

dαk

)

−αdNϑ

dαNv = 0 in Ω, (6.24)

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Div

(ρ(ϑ, α)v ⊗ dNv

dαN+dN (ρv)dαN

⊗ v +N−1∑k=1

(N

k

)dk(ρv)dαk

⊗ dN−kv

dαN−k

)

= ArdNϑ

dαNe3 −∇dN π

dαN+

2Re

Divµ(ϑ)D

(dNv

dαN

)+ µ′(ϑ)

dNϑ

dαND(v)

+N−1∑k=1

(N

k

)dkµ(ϑ)dαk

D

(dN−kv

dαN−k

)+D(v)

N−1∑k=1

(N

k − 1

)dN−kµ′(ϑ)dαN−k

D

(dkϑ

dαk

)in Ω, (6.25)

div−λ(ϑ)

Pe∇dNϑ

dαN+

(ρ(ϑ, α)vcp1(ϑ) − λ′(ϑ)

Pe∇ϑ

)dNϑ

dαN+ Cp(ϑ)

dN (ρv)dαN

− 1Pe

N−1∑k=1

(N

k

)dkλ(ϑ)dαk

∇dN−kϑ

dαN−k− ∇ϑ

Pe

N−2∑k=0

(N − 1k

)dN−1−kλ′(ϑ)dαN−1−k

dk+1ϑ

dαk+1

+N−1∑k=1

(N

k

)dk(ρv)dαk

dN−k(Cp(ϑ))dαN−k

+ ρv

N−2∑k=0

(N − 1k

)dN−1−kcp1(ϑ)dαN−1−k

dk+1ϑ

dαk+1

= 0 in Ω, (6.26)

dNϑ

dαN= 0,

dNv

dαN= 0 on Γout,

dNϑ

dαN= 0,

dNv

dαN=

N !V2

(1 − α)N+1eg on Γin,

(6.27)

dNv

dαN= 0,

− 1Pe

(λ(ϑ)∇dNϑ

dαN+ λ′(ϑ)

dNϑ

dαN∇ϑ+ ∇ϑ

N−1∑k=1

(N − 1k − 1

)dN−kλ′(ϑ)dαN−k

dkϑ

dαk

+N−1∑k=1

(N

k

)dkλ(ϑ)dαk

∇dN−kϑ

dαN−k

)· n = q0

dNϑ

dαNon Γlat . (6.28)

In order to use the Spectral Theory, we introduce as before the unknown function

q(N) = ρ(ϑ, α)dNv

dαN− α

dNϑ

dαNv.

and use the functional spaces W and T .Once more, the unique solvability of the problem (6.24)–(6.28) is valid if the cor-

responding homogeneous problem, for q(N), π, dN ϑdαN , has only the trivial solution

in W × L20(Ω) × T .

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We see immediately that the corresponding homogeneous problem is again(6.14)–(6.18). Then under the same conditions we conclude existence and unique-ness for derivatives of all orders with respect to α.

With this result, we are ready to state the error estimate for Boussinesq’s limit.First, we write the first order correction (i.e. the system defining the first deriva-

tives u 0, π0, ζ0 = ddαv, π, ϑ|α=0), getting

divu 0 − ϑOBvOB

= 0 in Ω, (6.29)

−ϑOB(vOB∇ )vOB + (u 0∇ )vOB + (vOB∇)u 0

= Arθ0eg −∇ π0 +2

ReDiv

µ(ϑOB)D(u 0) + µ′(ϑOB)ζ0D(vOB)

in Ω,

(6.30)

div− λ(ϑOB)

Pe∇ζ0 +

[vOBcp1(ϑOB) − λ′(ϑOB)

Pe∇ϑOB

]ζ0

−ϑOBvOBCp(ϑOB) + u 0Cp(ϑOB)

= 0 in Ω, (6.31)

ζ0|Γout = 0, u 0|Γout = 0, θ0|Γin = 0, w0|Γin = V2eg, (6.32)

u 0 = 0 and − 1Pe

(λ(ϑOB)∇ζ0 + λ′(ϑOB)ζ0∇ϑOB) · n = q0ζ0 on Γlat.

(6.33)

Under the conditions of the preceding proposition, with α = 0, system (6.29)–(6.33) has a unique smooth solution. Hence we have established rigorously theO(α2) approximation for Problem A. Clearly, one could continue to any order.

The following theorem is, therefore, a straightforward corollary of Proposition 6.1.

Theorem 6.1. Let us suppose the assumptions of Proposition 6.1. Then we have

‖v − vOB − αu 0‖W 1,∞(Ω)3 ≤ Cα2, (6.34)

infK∈R

‖P − POB − αP 0 +K‖L∞(Ω)3 ≤ Cα2, (6.35)

‖ϑ− ϑOB − αζ0‖W 1,∞(Ω)3 ≤ Cα2, (6.36)

where P 0/Re = π0 − 2µ(ϑOB) div u 0/(3Re).

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Appendix A. Uniqueness Proof

In this appendix we give details of the uniqueness proof, i.e. we prove Theorems4.1 and 4.2. We start by establishing the stability with respect to the perturbationof the temperature field ϑ for the stationary incompressible Navier–Stokes system(4.1)–(4.3) with temperature dependent viscosity. We have the following estimate:

Proposition A.1. Let v (i) ∈ H1(Ω)3 be a solution for (4.1)–(4.3) correspondingto ϑ(i) ∈ H1(Ω) ∩ L∞(Ω), i = 1, 2, and satisfying the a priori estimate ‖D(v (i) −ξ )‖L2(Ω)9 ≤ R, where R is given by (3.43) and ξ is defined by Proposition 3.2with γ = µmin/Re and ρ = 1. Let

V = v (1) − v (2) and Θ = ϑ(1) − ϑ(2). (A.1)

Let us suppose that the data satisfy (4.6). Then we have

‖D(V )‖L2(Ω)9 ≤2Ar 6

√6 |Ω|5 +

2Re

‖µ′‖∞(R+ ‖D(ξ )‖L2(Ω)9)

2Re

µmin −√

861/4 (R+ ‖D(ξ )‖L2(Ω)9)‖Θ‖L∞(Ω). (A.2)

Corollary A.1. Let us assume (4.6) and ϑ ∈ H1(Ω) ∩ L∞(Ω). Then the Navier–Stokes system (4.1)–(4.3) has a unique solution in the ball

B =v ∈ H1(Ω)3 | ‖D(v )‖L2(Ω)9 ≤ R+ ‖D(ξ )‖L2(Ω)9

.

Proof. By estimate (A.2) we have ‖D(V )‖L2(Ω)9 = 0 if ϑ1 = ϑ2.

Proof of Proposition A.1. Here we essentially apply the same technique thatwe have used in Theorem 3.1. For every φ ∈ H1

0 (Ω)3, div φ = 0, V satisfies thevariational equation∫

Ω

[(V ∇)v (2) + (v (1)∇ ) V

]φ dx

= Ar∫

Ω

Θφ3 dx+2

Re

∫Ω

µ(ϑ(1))D(V ) : D(φ ) dx

+2

Re

∫Ω

(µ(ϑ(1)) − µ(ϑ(2))

)D(v(2)) : D(φ ) dx. (A.3)

Now we test (A.3) with φ = V and recall that, for all φ ∈ H10 (Ω)3, such that

div φ = 0, we have:

• ‖φ ‖L6(Ω)3 ≤ 2 6√

6 ‖D(φ )‖L2(Ω)9 (see proof of Theorem 3.1).

• ‖φ ‖L2(Ω)3 ≤ 1√2‖∇(φ )‖L2(Ω)9 ≤ ‖D(φ )‖L2(Ω)9 .

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We thus get ∫Ω

(v (1) ∇)V · V dx = 0, (A.4)∣∣∣∣∫Ω

(V ∇)v (2) · V dx

∣∣∣∣=

∣∣∣∣∫Ω

V ⊗ V : D(v (2)) dx∣∣∣∣

≤ ‖D(v(2))‖L2(Ω)9 · ‖V ‖2L4(Ω)3︸︷︷︸

≤‖V ‖1/2L2(Ω)3

‖V ‖3/2L6(Ω)3

≤√

8 61/4(R + ‖D(ξ )‖L2(Ω)9)‖D(V )‖2L2(Ω)9 , (A.5)

Ar∣∣∣∣∫

Ω

ΘV3 dx

∣∣∣∣ ≤ 2Ar 6√

6 |Ω|5‖D(V )‖L2(Ω)9‖Θ‖L∞(Ω) , (A.6)

2Re

∫Ω

µ(ϑ(1)) |D(V )|2 dx ≥ 2µmin

Re‖D(V )‖2

L2(Ω)9 , (A.7)

2Re

∣∣∣∣∫Ω

(µ(ϑ(1) ) − µ(ϑ(2) ))D(v (2)) : D(V ) dx∣∣∣∣

≤ 2Re

‖µ′‖L∞(R)‖D(V )‖L2(Ω)9(R+ ‖D(ξ )‖L2(Ω)9)‖Θ‖L∞(Ω). (A.8)

Let X = ‖D(V )‖L2(Ω)9 , then (A.3) defines a nonlinear form which is bounded frombelow by the function

G(X) = X

[2

Reµmin −

√861/4(R + ‖D(ξ )‖L2(Ω)9)

]X

−[2Ar 6

√6 |Ω|5 +

2Re

‖µ′‖L∞(R)(R+ ‖D(ξ )‖L2(Ω)9)]‖Θ‖L∞(Ω)

.

Supposing (4.6), G(X) is negative in an interval (0, X0) and then positive. Estimate(A.2) follows immediately.

Next, we study the stability of the temperature ϑ with respect to the perturba-tion of the velocity field v. In fact we need two stability results: for the H1-normand for the L∞-norm.

For given velocities v (j), j = 1, 2, the corresponding problems (4.4) and (4.5)can be rewritten as

div

1Pe

∇Λ( j) − v (j)Cp (ϑ(Λ(j)))

= 0 in Ω,

Λ(j)∣∣Γin

=∫ 1

0

λ(s) ds, Λ(j)∣∣∣Γout

= 0 ,

− 1Pe

∇Λ(j) · n = q0ϑ(Λ(j)) + S on Γlat ,

(A.9)

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where Cp and Λ(j) = Λ(ϑ(j)) have been introduced in Lemma 3.2 (see formula(3.17)). In particular, introducing

Λ = Λ(1) − Λ(2),

we have that Λ solvesdiv

1

Pe∇Λ − V Cp(ϑ(1)) − v (2)

[Cp(ϑ(1)) − Cp(ϑ(2))

]= 0 in Ω,

Λ|Γout∪Γin= 0, − 1

Pe∇Λ · n = q0(ϑ(1) − ϑ(2)) on Γlat,

(A.10)

where V has been defined in (A.1) and ϑ(j), j = 1, 2, stand for ϑ(Λ(j)). We startby stating a stability result for the H1-norm. We have

Proposition A.2. Let Λ(j) ∈ H1(Ω) ∩ L∞(Ω), j = 1, 2, be a solution for (A.9)corresponding to v (j) ∈ H1(Ω)3, j = 1, 2, and satisfying ‖D(v (j) − ξ)‖L2(Ω)9 ≤ R.If the data fulfill (4.7), then

‖∇Λ‖L2(Ω)3 ≤‖cp1‖L∞(R) max1, ‖S‖L∞(Γlat)/q0

1Pe

− ‖cp1‖L∞(R)

λ0(‖ξ‖L6(Ω)3 + 2 6

√6R)C3(Ω)

‖V ‖L2(Ω)3 . (A.11)

Note that the existence of Λ(j) is guaranteed by Proposition 3.1.

Corollary A.2. Let us assume (4.7) and V ∈ H1(Ω)3. Then the system (A.10)has a unique solution in H1(Ω) ∩ L∞(Ω).

Proof of Proposition A.2. The function Λ satisfies the variational equation

1Pe

∫Ω

∇Λ · ∇ϕ dx︸︷︷︸(I)

+∫

Γlat

q0[ϑ(Λ(1)) − ϑ(Λ(2))] ϕdΓ︸︷︷︸(II)

−∫

Ω

Cp(ϑ(Λ(1)))V ·∇ϕ dx︸︷︷︸(III)

−∫

Ω

[Cp(ϑ(Λ(1))) − Cp(ϑ(Λ(2)))]v (2) · ∇ϕ dx︸︷︷︸(IV)

= 0, (A.12)

for any ϕ ∈ H1(Ω), such that ϕ|Γout∪Γin= 0. We select ϕ = Λ and again exploit

coercivity. Then the corresponding terms can be estimated follows:

(I)1

Pe∫Ω∇Λ · ∇Λ dx =

1Pe

‖∇Λ‖2L2(Ω)3 .

(II) Since Λ(ϑ) is a monotone increasing function, [ϑ(Λ(1)) − ϑ(Λ(2))] Λ ≥ 0, andso ∫

Γlat

q0[ϑ(Λ(1)) − ϑ(Λ(2))] Λ dΓ ≥ 0.

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(III) Recalling (3.10), we have∣∣∣∣∫Ω

V Cp(ϑ(Λ(1)))∇Λ dx

∣∣∣∣ ≤ ‖∇Λ‖L2(Ω)3 ‖cp1‖L∞(R)

× max1, ‖S‖L∞(Γlat)/q0‖V ‖L2(Ω)3 . (A.13)

(IV) First, we note that

|Cp(ϑ(Λ(1))) − Cp(ϑ(Λ(2)))| ≤‖cp1‖L∞(R)

λ0|Λ|.

Then, since

‖v (2)‖L6(Ω)3 ≤ ‖ξ‖L6(Ω)3 + 2 6√

6 ‖D(v (2)−ξ)‖L2(Ω)3 ≤ ‖ξ‖L6(Ω)3 + 2 6√

6R,

we have

|IV| ≤‖cp1‖L∞(R)

λ0‖∇Λ‖L2(Ω)3‖v (2)‖L6(Ω)3‖Λ‖L3(Ω)

≤‖cp1‖L∞(R)

λ0

(‖ξ‖L6(Ω)3 + 2 6

√6R

)C3(Ω)‖∇Λ‖2

L2(Ω)3 . (A.14)

So, (A.12) is bounded from below by a quadratic function of ‖∇Λ‖L2(Ω)3 whichis negative in an interval which contains the origin. So, estimate (A.11 ) followsimmediately.

Now we study the stability of temperature (with respect to the perturbation of thevelocity field) in the L∞-norm. The results which follow will heavily rely on theL∞-bound for the elliptic equations of the typediv

∇U + F

= 0, in Ω,

U |Γout∪Γin= 0, −∇U · n = g on Γlat .

(A.15)

After adapting the Nash–Moser technique from Ref. 8, Chapter 8, Sec. 8.5, we have

Lemma A.1. Let F ∈ L4(Ω)3 and let g be a normal trace of the function G1 ∈H1(Ω)3. Then we obtain the following L∞-bound

‖U ‖L∞(Ω) ≤ C(Ω)(‖ F ‖L4(Ω) + ‖ G1‖L4(Ω)), (A.16)

where

C(Ω) =(

32

)9

(C6(Ω) + |Ω|−1/12)3 (C6(Ω) + |Ω|1/6),

and C6(Ω) is the constant in the imbedding of the functions from H1(Ω), with zerotrace at x3 = 0 in L6(Ω).

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Proposition A.3. Under the hypothesis of Proposition A.2 we have

‖Θ‖L∞(Ω) ≤Peλ0

(4Bθ23/461/8 + Aθ)‖D(V )‖L2(Ω)3 , (A.17)

where

Aθ =PeC(Ω)

λ0

PeC(Ω)

‖cp1‖2L∞(R)

λ0

(2R61/6 + ‖ξ‖L6(Ω)3

)2

+q0√2

×‖cp1‖L∞(R) max1, ‖S‖L∞(Γlat)/q0

1Pe

− 1λ0

‖cp1‖L∞(R)C3(Ω)(2R61/6 + ‖ξ‖L6(Ω)3)(A.18)

and

Bθ = C(Ω)‖cp1‖L∞(R) max1, ‖S‖L∞(Γlat)/q0. (A.19)

Proof. We note that Λ is a solution for problem (A.15) with

F = Pe −V Cp(ϑ(1)) − v (2)[Cp(ϑ(1)) − Cp(ϑ(2))],G1 · n = Pe q0 (ϑ(1) − ϑ(2)). (A.20)

Thus, from (A.16), we have

C(Ω)Pe

‖ F ‖L4(Ω)

≤[C(Ω)‖cp1‖L∞(R) max1, ‖S‖L∞(Γlat)/q0

]︸︷︷︸Bθ

‖V ‖L4(Ω)3

+C(Ω)‖cp1‖L∞(R)

λo‖v(2)‖L6(Ω)3 ‖Λ‖L12(Ω)

≤ Bθ

(‖V ‖1/4

L2(Ω)3 ‖V ‖3/4L6(Ω)3

)︸︷︷︸≤ 23/461/8‖D(V ) ‖L2(Ω)3

+ 2C(Ω)‖cp1‖L∞(R)

λo(2R 6

√6 + ‖ξ‖L6(Ω)3)‖Λ‖L12(Ω)

≤ Bθ 23/461/8‖D(V ) ‖L2(Ω)3 + C(Ω)‖cp1‖L∞(R)

λo(2R 6

√6 + ‖ξ‖L6(Ω)3)

×C6(Ω)1/2‖Λ‖1/2L∞(Ω)‖∇Λ‖1/2

L2(Ω)3 , (A.21)

C(Ω)Pe

‖ G1‖L2(Ω) ≤ C(Ω)q0λ0

‖Λ‖L2(Ω) ≤q0

λ0

√2C(Ω)‖∇Λ‖L2(Ω). (A.22)

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Inserting (A.21) and (A.22) into (A.16) yields

‖Λ ‖L∞(Ω) ≤ PeBθ 23/461/8‖D(V ) ‖L2(Ω)3

+PeC(Ω)‖cp1‖L∞(R)

λo(2R 6

√6 + ‖ξ‖L6(Ω)3)

·C6(Ω)1/2 ‖Λ‖1/2L∞(Ω)‖∇Λ‖1/2

L2(Ω)3︸︷︷︸interpolation inequality

+Peq0

λ0

√2C(Ω)‖∇Λ‖L2(Ω),

namely

‖Λ ‖L∞(Ω) ≤ 4PeBθ 23/461/8‖D(V ) ‖L2(Ω)3

+Pe

Pe

C(Ω)λ0

[PeC(Ω)

‖cp1‖2L∞(R)

λoC6(Ω)(2R61/6 + ‖ξ‖L6(Ω)3)2

]

+q0

λ0

√2

‖∇Λ‖L2(Ω)3 .

Now, recalling estimate (A.11) and that ‖Λ ‖L∞(Ω) ≥ λ0‖Θ ‖L∞(Ω), we get

λ0‖Θ ‖L∞(Ω) ≤ Pe (4Bθ 23/461/8‖D(V ) ‖L2(Ω)3 + Aθ‖V ‖L2(Ω)3)

≤ Pe(4Bθ23/461/8 + Aθ)‖D(V )‖L2(Ω)3

with Aθ defined by (A.18). Hence estimate (2.30) follows.

We are now in a position to prove Theorem 4.1 in Sec. 4.

Proof of Theorem 4.1. Applying in sequence (2.30) and (A.2), we obtain

‖Θ ‖L∞(Ω) ≤ M1‖D(V )‖L2(Ω)3 ≤ M1M2‖Θ ‖L∞(Ω),

namely

‖Θ ‖L∞(Ω) (1 −M1M2) ≤ 0.

So, if (1 −M1M2) > 0, i.e. if (4.8) is fulfilled, we conclude that ‖Θ‖L∞(Ω) = 0.

Remark A.1. It is obvious that the nonlinear application defined by the ProblemA is Frechet-differentiable with respect to v, P, ϑ, at the unique stationary solu-tion corresponding to α = 0. This opens the possibility to use the Implicit FunctionTheorem, which would give us existence of a unique solution for all α being from anopen interval containing 0. Unfortunately, our functional space for the temperatureare not practical to handle and proving that the Frechet derivative is a homeomor-phism leads to some complications. It turns to be easier to give a direct uniquenessproof using energy arguments and then establish dependence of the solution on α.

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Now, in order to prove Theorem 4.2 we have to consider ρ = 1 − αϑ. We startgeneralizing Proposition A.1 by considering the following system for the velocity:

div(ρ(ϑ)v ) = 0 in Ω, (A.23)

ρ(ϑ)(v∇)v = Ar ϑe3 +1

Re[−∇P + Div (2µ(ϑ)D (v ))] in Ω, (A.24)

v |Γin= −(1 − α)v1e3, v |Γout

= −v2e3, v |Γlat= 0. (A.25)

We have the following estimate:

Proposition A.4. Let v(i) ∈ H1(Ω)3 be a solution for (A.23)–(A.25) correspond-ing to θ(i) ∈ H1(Ω) ∩ L∞(Ω), i = 1, 2, and both satisfying the a priori estimate(3.47). Let V = v(1) −v(2) and Θ = θ(1) − θ(2). Let us suppose that the data satisfy(4.9). Then we have(

2Re

µminρmin − ρ2max2 · 31/4(R+ ‖D(ξ)‖L2(Ω)9) − α

4µmax

Re61/6‖∇θ(1)‖L3(Ω)3

−αρmax‖∇θ(1)‖L2(Ω)32 · 61/3(2 · 61/6R+ ‖ξ‖L6(Ω)3))‖D(V )‖2

L2(Ω)9

≤ 2Re

α‖D(V )‖L2(Ω)9µmax‖Θ‖L∞(Ω)(R+ ‖D(ξ)‖L2(Ω)9)

+µmax(2 · 61/6R+ ‖ξ‖L6(Ω)3)‖∇Θ‖L3(Ω)3 + ‖µ′‖L∞(R)

× 2 · 61/6‖∇θ(1)‖L3(Ω)3(R+ ‖D(ξ)‖L2(Ω)9)‖Θ‖L∞(Ω)

+

2Re

ρmax‖µ′‖L∞(R)

× (R+ ‖D(ξ)‖L2(Ω)9)‖Θ‖L∞(Ω)‖D(V )‖L2(Ω)9

+αρmax‖D(V )‖L2(Ω)9√

2‖Θ‖L6(Ω)(2 · 61/6R + ‖ξ‖L6(Ω)3)

×((√

2R+ ‖∇ξ‖L2(Ω)9)√

2 · 61/3 + 2 · 61/6R+ ‖ξ‖L6(Ω)3)

+Arα(R + ‖ξ‖L2(Ω)3)‖∇Θ‖2L2(Ω)‖ · 241/4

+Arρmax‖D(V )‖L2(Ω)9‖Θ‖L2(Ω) +2

Reα‖µ′‖L∞(R)‖Θ‖L∞(Ω)

× (R+ ‖D(ξ)‖L2(Ω)9) ·(‖Θ‖L∞(Ω)(R + ‖D(ξ)‖L2(Ω)9)

+ ‖∇Θ‖L3(Ω)3(2 · 61/6R+ ‖ξ‖L6(Ω)3))

+ αρmax2 · 61/3‖Θ‖L∞(Ω)(R√

2

+ ‖∇ξ‖L2(Ω)9)(R√

261/12 + ‖ξ‖L3(Ω)3)‖D(V )‖L2(Ω)9 |Ω|1/6

+α2‖Θ‖2L∞(Ω)(R+ ‖D(ξ)‖L2(Ω)9)(2R61/6 + ‖ξ‖L6(Ω)3)2|Ω|1/6. (A.26)

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Corollary A.3. Let us assume (4.9) and θ ∈ H1(Ω) ∩ L∞(Ω). Then the Navier–Stokes system (A.23)– (A.25) has a unique solution in the ball B = v ∈ H1(Ω)3 |‖D(v)‖L2(Ω)9 ≤ R + ‖D(ξ)‖L2(Ω)9.

Proof of Proposition A.4. The function V satisfies, for every divergence freeϕ ∈ H1

0 (Ω)3, div ϕ = 0, the variational equation∫Ω

(ρ(θ(1))

((V∇)v(2) + (v(1)∇)V

)+ (ρ(θ(1)) − ρ(θ(2)))

((v(2)∇)v(2)

))ϕ dx

= Ar∫

Ω

θϕ3 dx+2

Re

∫Ω

µ(θ(1))D(V ) : D(ϕ) dx

+2

Re

∫Ω

(µ(θ(1)) − µ(θ(2)))D(v(2)) : D(ϕ) dx. (A.27)

Now we test (A.27) with ϕ = ρ(θ(1))V + (ρ(θ(1)) − ρ(θ(2)))v(2). Using that

‖ϕ‖L2(Ω)3 ≤ ‖D(ϕ)‖L2(Ω)3 and ‖ϕ‖L6(Ω)3 ≤ 2 · 61/6‖D(ϕ)‖L2(Ω)3 , ∀ϕ ∈ H10 (Ω)3,

we get ∣∣∣∣∫Ω

ρ(θ(1))(V∇)v(2) · ρ(θ(1))v dx∣∣∣∣

≤ ρ2max2 · 31/4(R + ‖D(ξ)‖L2(Ω)9)‖D(V )‖2

L2(Ω)9 , (A.28)∣∣∣∣∫Ω

ρ(θ(1))(V∇)v(2)(ρ(θ(1)) − ρ(θ(2)))v(2) dx

∣∣∣∣≤ αρmax‖D(V )‖L2(Ω)9 ‖Θ‖L6(Ω)(2 · 61/6R + ‖ξ‖L6(Ω)3)

× (√

2R+ ‖∇ξ‖L2(Ω)9)2 · 61/6, (A.29)∣∣∣∣∫Ω

ρ(θ(1))(v(1)∇)V (ρ(θ(1)) − ρ(θ(2)))v(2) dx

∣∣∣∣≤ αρmax

√2‖D(V )‖L2(Ω)9 ‖Θ‖L6(Ω)(2 · 61/6R+ ‖ξ‖L6(Ω)3)2, (A.30)∣∣∣∣∫

Ω

ρ(θ(1))(v(1)∇)V ρ(θ(1))V dx

∣∣∣∣≤ αρmax‖D(V )‖2

L2(Ω)9‖∇θ(1)‖L2(Ω)32 · 61/3(2 · 61/6R+ ‖ξ‖L6(Ω)3), (A.31)∣∣∣∣∫Ω

(ρ(θ(1)) − ρ(θ(2)))(v(2)∇)v(2) · ρ(θ(1))V dx

∣∣∣∣≤ αρmax2 · 61/6‖Θ‖L∞(Ω) |Ω|1/6(R

√2 + ‖∇ξ‖L2(Ω)9)

× (2R61/6 + ‖ξ‖L6(Ω)3)‖D(V )‖L2(Ω)9 , (A.32)

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Stationary Flow of Thermally Expansible Viscous Fluids 855∣∣∣∣∫Ω

(ρ(θ(1)) − ρ(θ(2)))(v(2)∇)v(2) · (ρ(θ(1)) − ρ(θ(2)))v(2) dx

∣∣∣∣≤ α2|Ω|1/6‖Θ‖2

L∞(Ω)(R+ ‖D(ξ)‖L2(Ω)9)(2R61/6 + ‖ξ‖L6(Ω)3)2, (A.33)

Ar∣∣∣∣∫

Ω

Θ(ρ(θ(1))v3 + (ρ(θ(1)) − ρ(θ(2)))v(2)3 ) dx

∣∣∣∣≤ Ar

‖Θ‖L2(Ω)ρmax ‖D(V )‖L2(Ω)9

+α(R+ ‖ξ‖L2(Ω)3)‖∇Θ‖2L2(Ω)241/4

, (A.34)

2Re

∫Ω

ρ(θ(1))µ(θ(1))|D(V )|2 dx ≥ 2µminρmin

Re‖D(V )‖2

L2(Ω)9 , (A.35)

∣∣∣∣ 2Re

∫Ω

µ(θ(1))D(V ) : D((ρ(θ(1)) − ρ(θ(2)))v(2)) dx∣∣∣∣

≤ α2µmax

Re‖Θ‖L∞(Ω)(R+ ‖D(ξ)‖L2(Ω)9) + ‖∇Θ‖L3(Ω)3(2 · 61/6R

+ ‖ξ‖L6(Ω)3)‖D(V )‖L2(Ω)9 , (A.36)

∣∣∣∣ 2Re

∫Ω

µ(θ(1))D(V ) : V ⊗∇ρ(θ(1)) dx∣∣∣∣

≤ α4µmax

Re61/6‖D(V )‖2

L2(Ω)9‖∇θ(1)‖L3(Ω)3 , (A.37)

2Re

∣∣∣∣∫Ω

(µ(θ(1)) − µ(θ(2))

)D(v(2)) : D(V ρ(θ(1))) dx

∣∣∣∣ ≤ 2Re

‖µ′‖L∞(R)

· ‖D(V )‖L2(Ω)9‖Θ‖L∞(Ω)(R+ ‖D(ξ)‖L2(Ω)9)

· (ρmax + α2 · 61/6‖∇θ(1)‖L3(Ω)3), (A.38)

2Re

∣∣∣∣∫Ω

(µ(θ(1)) − µ(θ(2))

)D(v(2)) : D(v(2)(ρ(θ(1)) − ρ(θ(2))) dx

∣∣∣∣≤ α

2Re

‖µ′‖L∞(R)(R+ ‖D(ξ)‖L2(Ω)9)(‖Θ‖L∞(Ω)(R+ ‖D(ξ)‖L2(Ω)9)

+α‖∇Θ‖L3(Ω)3(2 · 61/6R+ ‖ξ‖L6(Ω)3))‖Θ‖L∞(Ω). (A.39)

After inserting (A.28) and (A.39) into (A.27) we get estimate (A.26).

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The stability of the temperature with respect to the perturbation of the velocityfield v is already established in Propositions A.2 and A.3. Using in addition theregularity estimate (3.13) we have

Proposition A.5. Under the conditions of Propositions A.2 and A.3 we have

‖∇Θ‖L2(Ω) ≤ L2‖D(V )‖L2(Ω)9 , (A.40)

‖Θ‖L∞(Ω) ≤Peλ0

(Aθ + 4Bθ23/461/8

)‖D(V )‖L2(Ω)9 , (A.41)

‖∇Θ‖L3(Ω) ≤ L3‖D(V )‖L2(Ω)9 , (A.42)

where

L2 =‖cp1‖L∞(R) max1, ‖S‖L∞(Γlat)/q0

1Pe − 1

λ0‖cp1‖L∞(R)(2R61/6 + ‖ξ‖L6(Ω)9)

, (A.43)

L3 =Peλ03(Ω)‖cp1‖L∞(R)max1, ‖S‖L2(Γlat)/q0

√2 · 61/12

+ (2 · 61/6R+ ‖ξ‖L6(Ω)3)L2. (A.44)

Theorem 4.2 now follows immediately.

Acknowledgments

The authors have the pleasure to thank Prof. K. R. Rajagopal for his comments onthe model. The research of A.M. was supported in part by the EZUS Lyon 1.

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Documents

ON THE EQUATIONS GOVERNING THE FLOW OF MECHANICALLY INCOMPRESSIBLE, BUT THERMALLY EXPANSIBLE, VISCOUS FLUIDS