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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ON NON-NEWTONIAN FLUIDS WITH CONVECTIVE EFFECTS

SIGIFREDO HERR ´ON, ´ELDER J. VILLAMIZAR-ROA

Abstract. We study a system of partial differential equations describing a steady thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor and the heat flux depend on temperature and satisfy the condi- tions ofp, q-coercivity withp > n+22n ,q > p(n+1)−nnp , respectively. Consider- ing Dirichlet boundary conditions for the velocity and a mixed and nonlinear boundary condition for the temperature, we prove the existence of weak so- lutions. We also analyze the existence and uniqueness of strong solutions for small and suitably regular data.

1. Introduction

This article analyzes a system of partial differential equations describing a steady thermoconvective flow of a non-Newtonian fluid in a bounded domain Ω of Rn, n= 2,3, with smooth enough boundary ∂Ω. The model is given by the system of PDEs

−div µ(·, θ)T(D(u))

+ div(u⊗u) +∇π=θf in Ω, div u= 0 in Ω,

−div(κ(·, θ)a(∇θ)) +u· ∇θ=g in Ω,

(1.1) where the unknowns are u : Ω → Rn, θ : Ω → R and π : Ω → R denoting the velocity, the temperature and the pressure of the fluid, respectively. The fieldf denotes the given external body forces andgrepresents the heat source. The symbol T:Mn×nsym →Mn×nsym denotes the extra stress tensor andaindicates the constitutive law for diffusivity. The symbolD(u) represents the symmetric part of the velocity gradient∇u, that is,D(u) = 12(∇u+∇Tu); the functions µ(·, θ)>0, κ(·, θ)>0 denote the kinematic viscosity and thermal conductivity, respectively. Equations (1.1)1 and (1.1)3 correspond to the momentum and heat equations respectively;

the second equation in (1.1) corresponds to the incompressibility condition. We assume that the functionsη →T(η) and χ→a(χ) are continuous inMn×nsym and Rnrespectively, and satisfy the following conditions for somep, q >1 (see notation in Section 2):

i) (Coercivity) There existτ1, α1>0 such that T(η) :η≥τ1|η|p,

a(χ)·χ≥α1|χ|q, (1.2)

2010Mathematics Subject Classification. 35Q35, 76D03, 76D05, 35D30, 35D35.

Key words and phrases. Non-Newtonian fluids; shear-dependent viscosity; weak solutions;

strong solutions; uniqueness.

c

2017 Texas State University.

Submitted January 12, 2017. Published June 28, 2017.

1

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for allη∈Mn×nsym, χ∈Rn.

(ii) (Polynomial growth) There existτ2, α2>0 such that

|T(η)| ≤τ2(1 +|η|)p−1,

|a(χ)| ≤α2|χ|q−1, (1.3) for allη∈Mn×nsym, χ∈Rn.

(iii) (Strict monotonicity)

(T(η)−T(ξ)) : (η−ξ)>0, ∀η,ξ∈Mn×nsym,η6=ξ,

(a(ς)−a(χ))·(ς−χ)>0, ∀ς,χ∈Rn, ς 6=x. (1.4) The general non-linear tensor function Tand constitutive law for the heat flux aallow to consider a large class of non-Newtonian fluids subjected to heat effects, which have physical motivations as described in [6, 19, 20] and references therein.

Typical prototypes of extra stress tensors used in applications areT1(η) = 2µ(1 +

|η|2)(p−2)/2η and T2(η) = 2µ(1 +|η|)p−2η with p > 1. In these cases, ifp = 2 and a is the identity, we obtain the classical Boussinesq equation (see [5, 8, 9, 10, 22, 23]). We also consider the following hypotheses on the viscosity and the thermal conductivity functions µ, κ. It is assumed that µ, κ : Ω×R → R are Carath´eodory functions (i.e., for each fixedθthe functionsx7→µ(x, θ), x7→κ(x, θ) are (Lebesgue) measurable in Ω and, the functions θ 7→ µ(x, θ), θ 7→ κ(x, θ) are continuous for almost everyx∈Ω) such that

0< µ1≤µ(x, θ)≤µ2 a.e. x∈Ω, ∀θ∈R,

0< κ1≤κ(x, θ)≤κ2 a.e. x∈Ω, ∀θ∈R. (1.5) System (1.1) is complemented with the mixed boundary conditions

u= 0 on∂Ω,

θ= 0 on Γ0, κ(·, θ)a(∇θ)·n+γθ=hon Γ :=∂Ω\Γ0, (1.6) where γ is a non-negative constant, n denotes the unit outward normal on the boundary ∂Ω, and Γ0 is a open subset of ∂Ω. Boundary conditions (1.6)2 in- clude several physical boundary conditions like those appearing in several natural convection problems [9, 22]. The existence of weak solutions in the case of Navier- Stokes equations for flows with shear-dependent viscosity is known inW1,p(Ω) for p≥2n/(n+ 2). For the case p≥3n/(n+ 2), the existence of weak solutions was obtained by Lions [18] and Ladyzhenskaya [17] by using monotone operators the- ory. In [21], using theL-truncation method, the authors obtained the existence of weak solutions forp≥2n/(n+ 1). This method is based on the construction of a special class of test functions, and a characterization of the pressure, which permit the almost everywhere convergence of D(um) toD(u), where um corresponds to a sequence of approximated solutions um of the original problem. However, this method only works for p≥2n/(n+ 1) because of the required L1-integrability of the nonlinear term (u· ∇)u. To consider the case p ≥ 2n/(n+ 2), in [11] the Lipschitz truncation method was applied, which permits controling the nonlinear term (u· ∇)u using a test function class smoother than the test functions used in the L-truncation method. On the other hand, focusing on the boundary- value problem (1.1)-(1.6), the existence of weak solutions forp > 2n/(n+ 1) and q > np/(p(n+ 1)−n) was obtained in [6]. Motivated by this facts, in the first

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part of this paper, we extend the results of [6] to the case p > 2n/(n+ 2) and q > np/(p(n+ 1)−n).

The second part of this article concentrates on the existence of regular solutions to the boundary value problem (1.1)-(1.6). In the case of Navier-Stokes equa- tions for flows with shear-dependent viscosity, there are few works concerning the regularity of weak solutions (cf. [2, 4, 7, 15] and some references therein). The most recent results for the steady Navier-Stokes equations for flows with shear- dependent viscosity are due to Arada [2]. In [2], the author assumed that T is a classical power law stress tensor of the form T(η) = T1(η) := 2µ(1 +|η|2)p−22 η or T(η) = T2(η) := 2µ(1 +|η|)p−2η, where µ > 0 is a viscosity coefficient and p >1. He proved the existence of strong solutionsu∈W2,q(Ω),q > n, by assum- ing that kfkq/µ is small enough. Some uniqueness results were also established.

However, to the best of our knowledge, there are no results of existence of strong solutions for the steady problem (1.1)-(1.6). In the second part of this paper, we will study the existence of a strong solution for small and suitably regular data by takingT=T1 orT=T2. To ease the exposition, we also simplify the boundary conditions on temperatureθ; however, a similar analysis can be adapted for other types of boundary data. Our approach is based on regularity results for the Stokes problem and the Laplace equation, as well as a fixed-point argument. Observe that T1 depends on the differentiable term |D(u)|2 while T2 depends merely on the Lipschitz continuous term|D(u)|; thus, in the caseT=T1we can use the classical regularity results for the Stokes system to solve the velocity equation for a fixed temperature. However, in the case T=T2, to overcome the difficulty caused by the lack of regularity ofT2, we first introduce a family of penalized problems, then, we establish the existence of penalized strong solutions and finally, we carry out the pass to the limit in the sequence of penalized problems, as the penalization term goes to zero.

This article is organized as follows. In Section 2, we introduce the notation.

Section 3 is devoted to the existence of weak solutions. In Section 4, we analyze the existence of strong solutions in both cases: with the differentiable stress tensor T1, and with the Lipschitz continuous stress tensorT2. In Section 4, we also give conditions on the data which ensure that the obtained strong solution agrees with weak solutions.

2. Notation

In this section, we establish some general notation to be used throughout this article. As usual,C0(Ω) denotes the set of allC-functions with compact support in Ω, while C0,σ(Ω) consists of functions Φ ∈ C0(Ω) such that divΦ = 0. For p, q >1 we set

Hq :=C0,σ(Ω)k·kq ={u∈Lq(Ω) : divu= 0, u·n= 0 on∂Ω}, Vp:=C0,σ(Ω)k∇·kp={u∈W1,p0 (Ω) : divu= 0},

Xq :={θ∈W1,q(Ω)∩L2(Γ) :θ= 0 on Γ0}.

HereVpandXq are Banach spaces with the normskD(u)kp andkθkXq =k∇θkq+ kθk2,Γ. As usual,k · kpdenotes theLp-norm. Notice that, due to the trace theorem, Xq = {θ ∈ W1,q(Ω) : θ = 0 on Γ0} if q ≥ 2n/(n+ 1). For x, y ∈ R we denote (x, y)+ = max{x, y}, x+ = max{x,0}, Sp = (|p−2|,2)+. Frequently, we will

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use the notationh·,·iX0 (or simplyh·,·iif there is no ambiguity) to represent the duality product between X0 and X, for the Banach space X. We also introduce the constants

2rp= 1 + (p−3)+−(p−4)+, γp= [(p,3)+−2](p,3)+−2 [(p,3)+−1](p,3)+−1.

Form∈Nand 1< p <∞, the standard Sobolev Spaces are denoted byWm,p(Ω) and their norms byk · km,p. In particular,W−1,p(Ω) denotes the dual ofW01,p(Ω).

We also consider the space

Vm,p={v∈W1,p0 (Ω)∩Wm,p(Ω) : divv= 0 in Ω},

equipped with the usual normk · km,p:=k · kWm,p(Ω). Notice thatV1,p=Vp. Also, forr, q > nandδ >0, let us denote byBδ the convex set defined by

Bδ ={[ξ, ω]∈V2,q×W2,r(Ω) :CEk∇ξk1,q≤δ, CE˜k∇ωk1,r ≤δ}, (2.1) where CE is the norm of the embedding of W1,q(Ω) into L(Ω) and CE˜ is the norm of the embedding ofW1,r(Ω) intoL(Ω). Also, we consider the spaceV2,q× (W2,r(Ω)∩W01,r(Ω)) endowed with the norm

k[ξ, ω]k1,q,r:= max{k∇ξk1,q,k∇ωk1,r}.

Throughout the paper,Mn×ndenotes the space of all realn×nmatrices andMn×nsym

its subspace of all symmetric n×n matrices. We use the following summation convention on repeated indices: η:ξ:=ηijξij forη:ξ∈Mn×n, (u⊗v)ij:=uivj foru,v∈Rn andu·v:=uivi. Also we set|u|:= (u·u)1/2 and|η|:= (η:η)1/2 foru∈Rn, η∈Mn×n. Finally, the letter C stands for several positive constants that may change line by line; alsoCP =CP(n, s,Ω) denotes the Poincar´e constant corresponding to the general Poincar´e inequalityk · ks≤CPk∇(·)ks.

3. Weak solutions

The aim of this section is to prove the existence of weak solutions to prob- lem (1.1)-(1.6) for the case n+22n < p ≤ n+12n , q > p(n+1)−nnp . The existence of weak solutions for p > n+12n was analyzed in [6]. We assume that f ∈L(Ω), g∈ (W1,q(Ω))0, h∈L2(Γ). First we establish the notion of weak solution to (1.1)-(1.6).

Definition 3.1. We say that a pair [u, θ]∈Vp×Xq is a weak solution to problem (1.1)-(1.6) if

Z

µ(·, θ)T(D(u)) :D(Φ)dx− Z

(u⊗u) :D(Φ)dx= Z

θf·Φdx,

∀Φ∈C0,σ(Ω), Z

κ(·, θ)a(∇θ)· ∇φ dx+ Z

φu· ∇θ dx+γ Z

Γ

θφ dΓ =hg, φi(W1,q(Ω))0+ Z

Γ

hφ dΓ,

∀φ∈C0(Ω).

The purpose of this section is to prove the following theorem on existence of weak solutions.

Theorem 3.2. Letp > n+22n ,q > p(n+1)−nnp ,f ∈L(Ω),g∈(W1,q(Ω))0, h∈L2(Γ).

There exists a weak solution[u, θ]∈Vp×Xq to problem (1.1)-(1.6).

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To prove Theorem 3.2, we first consider a suitable sequence of approximate prob- lems (see (3.1)-(3.2) below); we establish the existence of approximate solutions, as well as some a priori estimates. In a second step, we describe the passing to the limit of the sequence of approximate solutions. Finally, we analyze the almost everywhere convergence in Ω of [D(um),∇θm]→[D(u),∇θ] through the Lipschitz truncation method.

3.1. Approximate solutions. Form∈N andt >max{p−12p ,q(n+1)−2nnq }, we de- fine the approximated problem: Find a weak solution [um, θm] of the system

−div (µ(·, θm)T(D(um))) + div(um⊗um) + 1

m|um|t−2um+∇π=θmf in Ω, div um= 0 in Ω,

−div(κ(·, θm)a(∇θm)) +um· ∇θm=g in Ω,

(3.1) with the boundary conditions

um= 0 on∂Ω,

θm= 0 on Γ0, κ(·, θm)a(∇θm)·n+γθm=hon Γ :=∂Ω\Γ0. (3.2) Following the ideas presented in [11, 12], we obtain the existence of a weak solution [u, θ] of (1.1)-(1.6) as the limit of a sequence of weak solutions [um, θm] of (3.1)- (3.2). A weak solution of the system (3.1)-(3.2) is a pair [um, θm] ∈ Vp ×Xq satisfying

Z

µ(·, θm)T(D(um)) :D(Φ)dx− Z

(um⊗um) :D(Φ)dx + 1

m Z

|um|t−2um·Φdx

= Z

θmf·Φdx,

(3.3)

Z

κ(·, θm)a(∇θm)· ∇φ dx− Z

θmum· ∇φ dx+γ Z

Γ

θmφ dΓ

=hg, φi(W1,q(Ω))0+ Z

Γ

hφ dΓ,

(3.4)

for allΦ∈C0,σ(Ω), φ∈C0(Ω).

The following lemma provides the existence of a weak solution to (3.1)-(3.2).

Lemma 3.3. Let p >2n/(n+ 2),t≥2p0,q > p(n+1)−nnp . Assume thatf ∈L(Ω), g ∈(W1,q(Ω))0, h∈L2(Γ). Then, there exists a unique weak solution [um, θm]∈ (Vp∩Ht)×Xq of (3.3)-(3.4). Moreover, the following uniform estimates hold

τ1µ1

2 kumkp1,p+ 1

mkumktt≤C1kfkp0

kgkq(W0 1,q)0+khk22,Γp0/q

, (3.5) α1κ1

2 k∇θmkqq

2kθmk22,Γ≤C2

kgkq(W0 1,q)0+khk22,Γ

, (3.6)

for some constants C1, C2>0 independent onm.

Proof. The proof follows by standard arguments of the monotone operator theory (cf. [11, 12]). The uniform estimates (3.5)-(3.6) follow by taking Φ = um and

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φ=θmin (3.3) and (3.4), respectively, and using the assumptions onT,a,f, gand

h.

3.2. Existence of weak solutions. The existence of a weak solution to the prob- lem (1.1)-(1.6) will be obtained as the limit, asm goes to infinity, in the sequence of solutions [um, θm] of (3.3)-(3.4). We use the Lipschitz truncation method used previously in [11] in the context of incompressible fluids with shear-dependent vis- cosity (without heat effects). Following [11], we introduce the sequence of approx- imate pressuresπm, observing that in (3.3) we can consider test functions Φ from Vp∩Vr =Vr with r =np/[(n+ 2)p−2n]. Notice that for this value of r and 2n/(n+ 2)< p≤2n/(n+ 1), it holds thatVr ,→,→Ly for all y ∈[1,∞). Then, defining

hFm,Φi(W1,r 0 (Ω))0 :=

Z

µ(·, θm)T(D(um)) :D(Φ)dx− Z

(um⊗um) :D(Φ)dx + 1

m Z

|um|t−2um·Φdx− Z

θmf·Φdx, it holds that hFm,Φi(W1,r

0 (Ω))0 = 0, for all Φ ∈ C0,σ(Ω). Furthermore, Fm ∈ W−1,r0(Ω). Thus, because of the De Rham Theorem (cf. [1]), there existsπm ∈ Lr0(Ω) such that

hFm,Φi(W1,r

0 (Ω))0 =h−∇πm,Φi(W1,r 0 (Ω))0 =

Z

πmdivΦdx andkπmkr0 ≤C.

(3.7) Therefore, we obtain the following weak formulation (for the velocityum) equivalent to (3.3):

Z

µ(·, θm)T(D(um)) :D(Φ)dx− Z

(um⊗um) :D(Φ)dx + 1

m Z

|um|t−2um·Φdx

= Z

θmf·Φdx+ Z

πmdivΦdx,

(3.8)

for allΦ∈W1,r0 (Ω). Now we pass to the limit in (3.8) asm→ ∞. From the uniform estimates (3.5), (3.6) and (3.7) there exists a subsequence of ([um, πm, θm])m∈N⊆ Vp×Lr0(Ω)×Xq, still denoted by ([um, πm, θm])m∈N, and [u, π, θ,χ,χ1]∈Vp× Lr0(Ω)×Xq×Lp0(Ω)×Lq0(Ω) such that asm→ ∞the following holds

D(um)→D(u) weakly inLp, (3.9) [um, θm, πm]→[u, θ, π] weakly inVp×Xq×Lr0, (3.10) um→u strongly inLs(Ω) for alls∈[1,2r0), (3.11)

um→u a.e. in Ω, (3.12)

θm→θ a.e. in Ω, and a.e. in Γ, (3.13) T(D(umk))→χ weakly inLp0(Ω), (3.14) a(∇θmk )→χ1 weakly inLq0(Ω). (3.15)

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From (3.10), for anyΦ∈C0 (Ω) and lettingm→ ∞ it holds

1 m

Z

|um|t−2um·Φdx ≤ 1

m1/t 1

mkumktt(t−1)/t

kΦkt→0, (3.16) and

Z

θmf·Φdx+ Z

πmdivΦdx→ Z

θf ·Φdx+ Z

πdivΦdx.

On the other hand, since W1,p(Ω) ,→,→ L2(Ω) for p > 2n/(n+ 2), and writing um= (um−u) +u, forΦ∈C0 (Ω) and lettingm→ ∞we obtain

Z

(um⊗um) :D(Φ)dx→ Z

(u⊗u) :D(Φ)dx.

Also, sinceθm→θinL1(Ω) and a.e. in Ω, andµis a Carath´eodory function, then µ(·, θm)→µ(·, θ) a.e. in Ω. Then, collecting the last convergences, we have

Z

µ(·, θ)χ:D(Φ)dx+ Z

(u⊗Φ) :D(u)dx= Z

θf·Φdx+ Z

πdivΦdx, (3.17) for allΦ∈C0 (Ω) and consequently for allΦ∈W01,r(Ω).

As before, since θm → θ in L1(Ω) and a.e. in Ω, and κ is a Carath´eodory function, then κ(·, θm) → κ(·, θ) a.e. in Ω. Then, from the uniform estimates (3.10), (3.13) and (3.15) we also get

Z

κ(·, θ)χ1−θu

· ∇φ dx+γ Z

Γ

θφ dΓ =hg, φi(W1,q(Ω))0+ Z

Γ

hφ dΓ, (3.18) for all φ ∈ C0(Ω) and consequently for all φ ∈ Xq. It remains to prove that χ=T(D(um)) andχ1=a(∇θm). To this end, it is sufficient to prove that

[D(um),∇θm]→[D(u),∇θ] in measure on Ω, (3.19) or almost everywhere convergence on compact subsets of Ω. Having proved (3.19), through a diagonal procedure, we can find a subsequence of ([um, θm])m∈N, still denoted by ([um, θm])m∈N, such that

[D(um),∇θm]→[D(u),∇θ] almost everywhere in Ω. (3.20) Thus, by using Vitali’s theorem we obtain

Z

µ(·, θm)T(D(um)) :D(Φ)dx→ Z

µ(·, θ)T(D(u)) :D(Φ)dx, (3.21) Z

κ(·, θm)a(∇θm)· ∇φ dx→ Z

κ(·, θ)a(∇θ)· ∇φ dx. (3.22) Once we have (3.21) and (3.22) we conclude the proof of Theorem 3.2. In Subsec- tions 3.3 and 3.4, we analyze the convergence of [D(um),∇(θm)] to [D(u), θ] almost everywhere in Ω. This part is closely related to [11, Sections 3 and 4]; however, we expose it with some details for the reader’s convenience.

3.3. Almost everywhere convergence of D(um) to D(u). To prove the con- vergence ofD(um) toD(u) almost everywhere in Ω, we prove that for an arbitrary η1>0, there exists a subsequence of (um)m∈N, still denoted by (um)m∈N, such that for someρ1∈(0,1), it holds that

m→∞lim Z

[(T(D(um))−T(D(u))) :D(um−u)]ρ1dx≤η1. (3.23)

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Following [11, Section 3], we first consider a decomposition of the pressure. Consider the Stokes problems

−∆uIm+∇πIm =HIm in Ω, I = 1,2,3,4,5, div uIm = 0 in Ω,

uIm =0 on∂Ω,

(3.24)

where

H1m =−div (µ(·, θm)T(D(um)))∈W−1,p0(Ω), H2m = div (um⊗(um−u))∈W1,r0(Ω), H3m = div ((um−u)⊗u)∈W1,r0(Ω), H4m = 1

m|um|t−2um∈Lt0(Ω), H5m =−θmf ∈Lq(Ω).

It is well known that there exists a weak solution [uIm, πIm] of (3.24), for I = 1,2,3,4,5; that is, there exist

[u1m,u2m,u3m,u4m,u5m]

∈W1,p0 0(Ω)×W1,r0 0(Ω)×W1,r0 0(Ω)×W2,t0(Ω)×W2,q0(Ω),

1m, π2m, π3m, π4m, π5m]∈Lp0(Ω)×Lr0(Ω)×Lr0(Ω)×W1,t0(Ω)×W1,q0(Ω), satisfying

Z

∇uIm:∇Φdx− Z

πImdivΦdx=hHIm,Φi, ∀Φ∈C0 (Ω), I= 1,2,3,4,5.

(3.25) Moreover, the following estimates hold:

1mkp0 ≤CkH1mk−1,p0 ≤Cµ2kT(D(um))kp0, (3.26) kπ2mkr0 ≤CkH2mk−1,r0 ≤Ckum⊗(um−u)kr0 ≤Ckumk2r0kum−uk2r0, (3.27) kπ3mkr0 ≤CkH3mk−1,r0 ≤Ckum⊗(um−u)kr0 ≤Ckumk2r0kum−uk2r0, (3.28)

k∇π4mkt0 ≤CkH4mkt0 ≤ C m1/t

1

m1/tkumkt

t−1

, (3.29)

k∇π5mkq ≤CkH5mkq ≤ kθmkqkfk. (3.30) Since 2r0=np/(n−p), from (3.11), (3.27) and (3.28), asmgoes to∞, we obtain

2m, π3m]→[0,0] inLs(Ω)×Ls(Ω) for alls∈[1, r0). (3.31) Furthermore, by using (3.5), asmgoes to∞, it holds that

∇π4m →0 inLt0(Ω). (3.32)

Adding the weak formulations (3.25) and using (3.8) we obtain

5

X

I=1

[ Z

∇uIm :∇Φdx− Z

πImdivΦdx]

= Z

(u⊗u) :D(Φ)dx+ Z

πmdivΦdx, ∀Φ∈W1,r0 (Ω).

(3.33)

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TakingΦ∈Vr in (3.33) we obtain

5

X

I=1

Z

∇uIm :∇Φdx= Z

(u⊗u) :D(Φ)dx ∀Φ∈Vr. (3.34) From (3.34) and using thatP5

I=1uIm ∈W1,r0 0(Ω) we obtain

5

X

I=1

uIm =U∈W1,r0 0(Ω), ∀m∈N. (3.35) Finally, taking the term R

πmdivΦdx in (3.33), replacing it in (3.8) and using (3.35) we obtain

Z

µ(·, θm)T(D(um)) :D(Φ)dx+ 1 m

Z

|um|t−2um·Φdx

= Z

(um⊗um) :D(Φ)dx− Z

(u⊗u) :D(Φ)dx +

Z

∇U:∇Φdx−

5

X

I=1

Z

πImdivΦdx+ Z

θmf·Φdx,

(3.36)

for allΦ∈W1,r0 (Ω).

Now we are in a position to prove (3.23). Let us define

Xm:=C(1 +|D(um)|p+|D(u)|p+|π1m|p0). (3.37) Then, from (3.10) and (3.26) we have

Z

Xmdx≤K1, (3.38)

for some positive constantK1independent on m. Fixedp∈(n+22n ,n+12n ], letε1>0 be small enough to be chosen below (see (3.53)). Then, from [11, Proposition 4.1]

there exists a subsequence of (um)m∈N, still denoted by (um)m∈N, and λ1ε1

1

(independent onm), such that Z

Bmλ

1

Xmdx≤ε1, Bλm1 :={x∈Ω :λ1< M(∇(um−u)ext)(x)≤λ21}, (3.39) whereM(∇(um−u)ext) denotes the Hardy-Littlewood maximal function of∇(um− u)ext (cf. [11]), and (um−u)ext ∈W1,p(Rn) is the extension by zero of (um−u).

On the other hand, from [11, Proposition 4.1], there exist a positive constantC= C(Ω, n) and a sequence ((um−u)λ1)m∈N⊂W01,∞(Ω) such that

k(um−u)λ1k1,∞≤Cλ1, (3.40) (um−u)λ1→0, strongly inLs(Ω)∀s∈[1,∞), (3.41) (um−u)λ1 →0, weakly inW1,s0 (Ω)∀s∈[1,∞). (3.42) Moreover, denoting

Amλ1 :={x∈Ω : (um−u)λ1(x)6= (um−u)(x)}, Cλm1:={x∈Ω :M(∇(um−u))(x)> λ21}, it holds

|Amλ

1| ≤ |Bmλ

1|+|Cλm|, (3.43)

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|Amλ1|+|Bλm1| ≤ C

λp1k∇(um−u)kpp, (3.44)

|Cλm1| ≤ C

λ2p1 k∇(um−u)kpp, (3.45) k∇(um−u)λ1kpp≤Ck∇(um−u)kpp≤K1. (3.46) Now, we consider (um−u)λ1 as a test function in (3.36) and add in both sides of the obtained equation the term

− Z

µ(·, θm)T(D(u)) :D((um−u)λ1)dx, (3.47) to obtain

Z

µ(·, θm)[T(D(um))−T(D(u))] :D((um−u)λ1)dx + 1

m Z

|um|t−2um·(um−u)λ1dx

= Z

[(um⊗um)−(u⊗u)] :D((um−u)λ1)dx +

Z

∇U:∇(um−u)λ1dx

5

X

I=1

Z

πImdiv((um−u)λ1)dx+ Z

θmf·(um−u)λ1dx.

(3.48)

Notice thatum−u= (um−u)λ1 on Ω\ Amλ

1, and then, div(um−u)λ1 = 0 almost everywhere on Ω\ Amλ

1. Therefore, from (3.48) we obtain Zm:=

Z

Ω\Amλ

1

µ(·, θm)[T(D(um))−T(D(u))] :D((um−u))dx

=− Z

Amλ

1

µ(·, θm)[T(D(um))−T(D(u))] :D((um−u)λ1)dx

− Z

Amλ

1

π1mdiv((um−u)λ1)dx +

Z

[(um⊗(um−u) + (um−u)⊗u] :D((um−u)λ1)dx +

Z

[∇U−µ(·, θm)T(D(u))] :∇(um−u)λ1dx +

Z

[∇π4m− 1 m

Z

|um|t−2um]·(um−u)λ1dx

− Z

Amλ

1

2m3m5m) div((um−u)λ1)dx+ Z

θmf·(um−u)λ1dx

:=

7

X

i=1

Zim.

From (3.10), (3.11), (3.16), (3.31), (3.32), (3.40), (3.42) we obtain

m→∞lim (Z3m+Z5m+Z6m+Z7m) = 0. (3.49)

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Moreover, from (3.42) and since ∇U ∈ Lr0(Ω), µ1 ≤ µ(x, θ) ≤ µ2, a.e. x ∈ Ω and T(D(u)) ∈ Lp0(Ω), we obtain that limm→∞Z4m = 0. Now we deal with limm→∞Z1m+Z2m. From the H¨older inequality, (3.40) and (3.43)-(3.46) it holds that

|Z1m+Z2m|

≤ Z

Bmλ

1∪Cλm

1

(µ(·, θm)[T(D(um))−T(D(u))] :D((um−u)λ1)

−π1mdiv((um−u)λ1) )dx

≤µ2τ2CZ

Bmλ

1

Xmdx1/p0

k∇(um−u)λ1kp,Bm

λ1

2τ21

Z

Cmλ

1

Xmdx1/p0

|Cmλ1|1/p

≤Cµ2τ21/p1 0K11/p+Cλ1K11/p0(Cλ1−2pK1)1/p)

≤Cµ2τ21/p

0

1 K11/p+K1

λ1

)

≤Cµ2τ21/p

0

1 K11/p1K1).

(3.50)

In (3.50), k∇(um−u)λ1kp,Bm

λ1 denotes the Lp(Bλm

1)-norm of∇(um−u)λ1. Since limm→∞Z4m= 0, from (3.49) and (3.50) we obtain

m→∞lim Zm≤Cµ2τ21/p1 0K11/p1K1). (3.51) Therefore, fixedρ1∈(0,1), by using the H¨older inequality and (3.38) we obtain

Sm:=µρ11 Z

[(T(D(um))−T(D(u))) :D(um−u)]ρ1dx

≤ Z

Ω\Amλ

1

[µ(·, θm)(T(D(um))−T(D(u))) :D(um−u)]ρ1dx +

Z

Amλ

1

[µ(·, θm)(T(D(um))−T(D(u))) :D(um−u)]ρ1dx

≤(Zm)ρ1|Ω\ Amλ1|1−ρ1+C(µ2τ2K1)ρ1|Amλ1|1−ρ1.

(3.52)

Then, takingε1>0 small enough such that C(µ2τ2)ρ1|Ω|1−ρ11/p

0

1 K11/p1K1)ρ1+C(µ2τ2)ρ1K1εp(1−ρ1 1)< ρ1, (3.53) from (3.51)-(3.53) we have

m→∞lim Sm

≤ |Ω|1−ρ1C(µ2τ2)ρ11/p

0

1 K11/p1K1)ρ1+C(µ2τ2K1)ρ1(Cλ−p1 K1)1−ρ1

≤C(µ2τ2)ρ1|Ω|1−ρ11/p1 0K11/p1K1)ρ1+C(µ2τ2)ρ1K1εp(1−ρ1 1)< ρ1.

(3.54)

Thus, we conclude (3.23) and therefore the convergence ofD(um) to D(u) almost everywhere in Ω.

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3.4. Almost everywhere convergence of ∇θm to ∇θ. Let be a fixed value p∈(2n/(n+ 2),2n/(n+ 1)] andq > p(n+2)−n2np . To prove the convergence of∇θm to ∇θ almost everywhere in Ω, we proceed in the same spirit of Subsection 3.3.

We prove that for an arbitraryη2>0, there exists a subsequence of (θm)m∈N, still denoted by (θm)m∈N, such that for someρ2∈(0,1), it holds that

m→∞lim Z

[(a(∇θm)−a(∇θ))· ∇(θm−θ)]ρ2dx≤η2. (3.55) Let us define

Em:=C(1 +|∇θm|q+|∇θ|q). (3.56) Then, from (3.10) we have

Z

Emdx≤K2, (3.57)

for some positive constantK2 independent on m. Let ε2 >0 small enough to be chosen below (see (3.72)). Reasoning as in Subsection 3.3 (see also [11, Proposition 4.1]), there exists a subsequence of (θm)m∈N, still denoted by (θm)m∈N, andλ2ε1

2

(independent onm), such that Z

Dmλ

2

Emdx≤ε2, Dλm2 :={x∈Ω :λ2< M(∇(θm−θ)ext)(x)≤λ22}, (3.58) whereM(∇(θm−θ)ext) denotes the Hardy-Littlewood maximal function of∇(θm− θ)ext, and (θm−θ)ext∈W1,q(Rn) is the extension by zero of (θm−θ). Also, there exist a positive constantC=C(Ω, n) and a sequence ((θm−θ)λ2)m∈N⊂W01,∞(Ω) such that

k(θm−θ)λ2k1,∞≤Cλ2, (3.59) (θm−θ)λ2→0, strongly in Ls(Ω)∀s∈[1,∞), (3.60) (θm−θ)λ2 →0, weakly inW01,s(Ω)∀s∈[1,∞). (3.61) Moreover, denoting

Fλm2 :={x∈Ω : (θm−θ)λ2(x)6= (θm−θ)(x)}, Gλm2 :={x∈Ω :M(∇(θm−θ))(x)> λ22}, it holds

|Fλm2| ≤ |Dmλ2|+|Gλm2|, (3.62)

|Dmλ

2|+|Fλm

2| ≤ C

λq2k∇(θm−θ)kqq, (3.63)

|Gλm2| ≤ C

λ2q2 k∇(θm−θ)kqq, (3.64) k∇(θm−θ)λ2kqq≤Ck∇(θm−θ)kqq ≤K2. (3.65) Now we consider (θm−θ)λ2 as a test function in (3.4), and add in both sides of the obtained equation the term

− Z

κ(·, θm)a(∇θ)· ∇((θm−θ)λ2)dx, (3.66)

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this gives Z

κ(·, θm)[a(∇θm)−a(∇θ)]· ∇((θm−θ)λ2)dx

= Z

mum)· ∇((θm−θ)λ2)dx− Z

κ(·, θm)a(∇θ)· ∇((θm−θ)λ2)dx

−γ Z

Γ

θmm−θ)λ2dΓ +hg,(θm−θ)λ2i(W1,q(Ω))0

+ Z

Γ

h(θm−θ)λ2dΓ.

(3.67)

Notice thatθm−θ= (θm−θ)λ2 on Ω\ Fλm

2. Therefore, from (3.67) we obtain Ym:=

Z

Ω\Fλm

2

κ(·, θm)[a(∇θm)−a(∇θ)]· ∇(θm−θ)dx

=− Z

Fλm

2

κ(·, θm)[a(∇θm)−a(∇θ)]· ∇((θm−θ)λ2)dx +

Z

mum)· ∇((θm−θ)λ2)dx

− Z

κ(·, θm)a(∇θ)· ∇((θm−θ)λ2)dx−γ Z

Γ

θmm−θ)λ2dΓ +hg,(θm−θ)λ2i(W1,q(Ω))0+

Z

Γ

h(θm−θ)λ2dΓ :=

6

X

i=1

Yim.

(3.68)

Using (3.10) we obtain

m→∞lim (Y2m+Y4m+Y5m+Y6m) = 0. (3.69)

Moreover, from (3.61) and sinceκ1≤κ(x, θ)≤κ2, a.e. x∈Ω anda(∇θ)∈Lq0(Ω), we obtain that limm→∞Y3m = 0. Now we deal with limm→∞Y1m. From the properties of (θm−θ)λ2, the H¨older inequality and (3.65) it holds

|Y1m| ≤ Z

Dmλ

2∪Gλm

2

(κ(·, θm)[a(∇θm)−a(∇θ)]· ∇((θm−θ)λ2))dx

≤κ2α2CZ

Dλm

Emdx1/q0

k∇(θm−θ)λ2kq,Dm

λ2

2α22

Z

Gλm

2

Emdx1/q0

|Gλm|1/q

≤Cκ2α21/q2 0K21/q2K21/q0(Cλ2−2qK2)1/q)

≤Cκ2α21/q

0

2 K21/q+K2

λ2)≤Cκ2α21/q

0

2 K21/q2K2).

(3.70)

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Thus, since limm→∞Y3m = 0, from (3.69) and (3.70) we obtain limm→∞Ym ≤ Cκ2α21/q2 0K21/q2K2). Therefore, fixedρ2∈(0,1) we obtain

Lm:=κρ12 Z

[(a(∇θm)−a(∇θ))· ∇(θm−θ)]ρ2dx

≤ Z

Ω\Fλm

2

[κ(·, θm)(a(∇θm)−a(∇θ))· ∇(θm−θ)]ρ2dx +

Z

Fλm

2

[κ(·, θm)(a(∇θm)−a(∇θ))· ∇(θm−θ)]ρ2dx

≤(Ym)ρ2|Ω\ Fλm2|1−ρ2+C(κ2α2K2)ρ2|Fλm2|1−ρ2.

(3.71)

Then, takingε2>0 small enough such that

C|Ω|1−ρ22α2)ρ21/q2 0K21/q2K2)ρ2+C(κ2α2)ρ2K2εq(1−ρ2 2)< ρ2, (3.72) we have

m→∞lim Ym

≤C|Ω|1−ρ22κ2)ρ21/q

0

2 K21/q2K2)ρ2+C(κ2α2K2)ρ2(Cλ2−q)1−ρ2

≤C|Ω|1−ρ22α2)ρ21/q

0

2 K21/q2K2)ρ2+C(κ2α2)ρ2K2εq(1−ρ2 2)< ρ2.

(3.73)

Thus, we conclude (3.55) and therefore the convergence of ∇θm to ∇θ almost everywhere in Ω.

4. Strong solutions

In this section we analyze the existence of a strong solution considering the tensor stress T(η) = T1(η) := 2µ(1 +|η|2)p−22 η or T(η) = T2(η) := 2µ(1 +|η|)p−2η, with p >1. We also simplify the boundary conditions on the temperature θ. In fact, we assume Dirichlet boundary condition for the temperature; however, our approach can be adapted in order to analyze other boundary conditions. Indeed, we want to study the existence of strong solution for the problem

−div(T(Du)) + div(u⊗u) +∇π=θf in Ω, divu= 0 in Ω,

−div(κ(·, θ)∇θ) +u· ∇θ=g in Ω, u= 0 on∂Ω,

θ= 0 on∂Ω,

(4.1)

withTdefined as above. Also, throughout this section we assume thatκ: Ω×R→ Ris a C1-function such that 0< κ1 ≤κ(x, θ)≤κ2 a.e. x∈Ω and for all θ ∈R and, it satisfies|κ0(·, a)−κ0(·, b)| ≤λ0|a−b|, for all a, b∈Rand κ0(·,0) = 0, with κ1, κ2andλ0 are positive constants. Under mild conditions on the dataf ∈Lq, g∈ Lr(Ω), we obtain the existence of strong solution [u, θ]∈W2,q(Ω)×W2,r(Ω), for q, r > n. Our approach is based on regularity results for the Stokes problem and Laplace equation, and a fixed-point argument. Observe that T1 depends on the differentiable term |D(u)|2 while T2 depends on the merely Lipschitz continuous term|D(u)|; thus, in caseT=T1we can use the classical regularity results for the Stokes system to solve the velocity equation for a fixed temperature. However, in the caseT=T2, in order to overcome the difficulty caused by the lack of regularity of

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