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+2}^{2n} ,q > _{p(n+1)−n}^{np} , 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 R^{n},
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 → R^{n}, 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:M^{n×n}sym →M^{n×n}sym 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) = ^{1}_{2}(∇u+∇^{T}u); 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 inM^{n×n}sym and
R^{n}respectively, and satisfy the following conditions for somep, q >1 (see notation
in Section 2):

i) (Coercivity) There existτ_{1}, α_{1}>0 such that
T(η) :η≥τ_{1}|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

for allη∈M^{n×n}sym, χ∈R^{n}.

(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∈M^{n×n}sym, χ∈R^{n}.

(iii) (Strict monotonicity)

(T(η)−T(ξ)) : (η−ξ)>0, ∀η,ξ∈M^{n×n}sym,η6=ξ,

(a(ς)−a(χ))·(ς−χ)>0, ∀ς,χ∈R^{n}, 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)/2}η and T2(η) = 2µ(1 +|η|)^{p−2}η 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< κ_{1}≤κ(x, θ)≤κ_{2} 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 inW^{1,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(u^{m}) toD(u), where u^{m} corresponds to
a sequence of approximated solutions u^{m} of the original problem. However, this
method only works for p≥2n/(n+ 1) because of the required L^{1}-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

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−2}^{2} η
or T(η) = T2(η) := 2µ(1 +|η|)^{p−2}η, where µ > 0 is a viscosity coefficient and
p >1. He proved the existence of strong solutionsu∈W^{2,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=T_{1} orT=T_{2}. 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
T_{1}, and with the Lipschitz continuous stress tensorT_{2}. 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,C_{0}^{∞}(Ω) denotes the set of allC^{∞}-functions with compact support
in Ω, while C^{∞}_{0,σ}(Ω) consists of functions Φ ∈ C_{0}^{∞}(Ω) such that divΦ = 0. For
p, q >1 we set

Hq :=C^{∞}_{0,σ}(Ω)^{k·k}^{q} ={u∈L^{q}(Ω) : divu= 0, u·n= 0 on∂Ω},
Vp:=C^{∞}_{0,σ}(Ω)^{k∇·k}^{p}={u∈W^{1,p}_{0} (Ω) : divu= 0},

Xq :={θ∈W^{1,q}(Ω)∩L^{2}(Γ) :θ= 0 on Γ0}.

HereVpandXq are Banach spaces with the normskD(u)kp andkθkX_{q} =k∇θkq+
kθk2,Γ. As usual,k · kpdenotes theL^{p}-norm. Notice that, due to the trace theorem,
X_{q} = {θ ∈ W^{1,q}(Ω) : θ = 0 on Γ_{0}} 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

use the notationh·,·iX^{0} (or simplyh·,·iif there is no ambiguity) to represent the
duality product between X^{0} and X, for the Banach space X. We also introduce
the constants

2r_{p}= 1 + (p−3)^{+}−(p−4)^{+}, γ_{p}= [(p,3)^{+}−2]^{(p,3)}^{+}^{−2}
[(p,3)^{+}−1]^{(p,3)}^{+}^{−1}.

Form∈Nand 1< p <∞, the standard Sobolev Spaces are denoted byW^{m,p}(Ω)
and their norms byk · km,p. In particular,W^{−1,p}(Ω) denotes the dual ofW_{0}^{1,p}(Ω).

We also consider the space

V_{m,p}={v∈W^{1,p}_{0} (Ω)∩W^{m,p}(Ω) : divv= 0 in Ω},

equipped with the usual normk · km,p:=k · kW^{m,p}(Ω). Notice thatV_{1,p}=V_{p}. Also,
forr, q > nandδ >0, let us denote byB_{δ} the convex set defined by

B_{δ} ={[ξ, ω]∈V_{2,q}×W^{2,r}(Ω) :C_{E}k∇ξk1,q≤δ, CE˜k∇ωk1,r ≤δ}, (2.1)
where CE is the norm of the embedding of W^{1,q}(Ω) into L^{∞}(Ω) and CE˜ is the
norm of the embedding ofW^{1,r}(Ω) intoL^{∞}(Ω). Also, we consider the spaceV2,q×
(W^{2,r}(Ω)∩W_{0}^{1,r}(Ω)) endowed with the norm

k[ξ, ω]k1,q,r:= max{k∇ξk1,q,k∇ωk1,r}.

Throughout the paper,M^{n×n}denotes the space of all realn×nmatrices andM^{n×n}sym

its subspace of all symmetric n×n matrices. We use the following summation
convention on repeated indices: η:ξ:=η_{ij}ξ_{ij} forη:ξ∈M^{n×n}, (u⊗v)ij:=u^{i}v^{j}
foru,v∈R^{n} andu·v:=u^{i}v^{i}. Also we set|u|:= (u·u)^{1/2} and|η|:= (η:η)^{1/2}
foru∈R^{n}, η∈M^{n×n}. Finally, the letter C stands for several positive constants
that may change line by line; alsoC_{P} =C_{P}(n, s,Ω) denotes the Poincar´e constant
corresponding to the general Poincar´e inequalityk · k_{s}≤C_{P}k∇(·)k_{s}.

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+2}^{2n} < p ≤ _{n+1}^{2n} , q > _{p(n+1)−n}^{np} . The existence of
weak solutions for p > _{n+1}^{2n} was analyzed in [6]. We assume that f ∈L^{∞}(Ω), g∈
(W^{1,q}(Ω))^{0}, h∈L^{2}(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

Ω

µ(·, θ)T(D(u)) :D(Φ)dx− Z

Ω

(u⊗u) :D(Φ)dx= Z

Ω

θf·Φdx,

∀Φ∈C^{∞}_{0,σ}(Ω),
Z

Ω

κ(·, θ)a(∇θ)· ∇φ dx+ Z

Ω

φu· ∇θ dx+γ Z

Γ

θφ dΓ =hg, φi_{(W}1,q(Ω))^{0}+
Z

Γ

hφ dΓ,

∀φ∈C_{0}^{∞}(Ω).

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

Theorem 3.2. Letp > _{n+2}^{2n} ,q > _{p(n+1)−n}^{np} ,f ∈L^{∞}(Ω),g∈(W^{1,q}(Ω))^{0}, h∈L^{2}(Γ).

There exists a weak solution[u, θ]∈V_{p}×X_{q} to problem (1.1)-(1.6).

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(u^{m}),∇θ^{m}]→[D(u),∇θ] through the Lipschitz
truncation method.

3.1. Approximate solutions. Form∈N andt >max{_{p−1}^{2p} ,_{q(n+1)−2n}^{nq} }, we de-
fine the approximated problem: Find a weak solution [u^{m}, θ^{m}] of the system

−div (µ(·, θ^{m})T(D(u^{m}))) + div(u^{m}⊗u^{m}) + 1

m|u^{m}|^{t−2}u^{m}+∇π=θ^{m}f in Ω,
div u^{m}= 0 in Ω,

−div(κ(·, θ^{m})a(∇θ^{m})) +u^{m}· ∇θ^{m}=g in Ω,

(3.1) with the boundary conditions

u^{m}= 0 on∂Ω,

θ^{m}= 0 on Γ0, κ(·, θ^{m})a(∇θ^{m})·n+γθ^{m}=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 [u^{m}, θ^{m}] of (3.1)-
(3.2). A weak solution of the system (3.1)-(3.2) is a pair [u^{m}, θ^{m}] ∈ V_{p} ×X_{q}
satisfying

Z

Ω

µ(·, θ^{m})T(D(u^{m})) :D(Φ)dx−
Z

Ω

(u^{m}⊗u^{m}) :D(Φ)dx
+ 1

m Z

Ω

|u^{m}|^{t−2}u^{m}·Φdx

= Z

Ω

θ^{m}f·Φdx,

(3.3)

Z

Ω

κ(·, θ^{m})a(∇θ^{m})· ∇φ dx−
Z

Ω

θ^{m}u^{m}· ∇φ dx+γ
Z

Γ

θ^{m}φ dΓ

=hg, φi_{(W}1,q(Ω))^{0}+
Z

Γ

hφ dΓ,

(3.4)

for allΦ∈C^{∞}_{0,σ}(Ω), φ∈C_{0}^{∞}(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≥2p^{0},q > _{p(n+1)−n}^{np} . Assume thatf ∈L^{∞}(Ω),
g ∈(W^{1,q}(Ω))^{0}, h∈L^{2}(Γ). Then, there exists a unique weak solution [u^{m}, θ^{m}]∈
(V_{p}∩H_{t})×X_{q} of (3.3)-(3.4). Moreover, the following uniform estimates hold

τ_{1}µ_{1}

2 ku^{m}k^{p}_{1,p}+ 1

mku^{m}k^{t}_{t}≤C_{1}kfk^{p}_{∞}^{0}

kgk^{q}_{(W}^{0} 1,q)^{0}+khk^{2}_{2,Γ}^{p}^{0}^{/q}

, (3.5) α1κ1

2 k∇θ^{m}k^{q}_{q}+γ

2kθ^{m}k^{2}_{2,Γ}≤C2

kgk^{q}_{(W}^{0} 1,q)^{0}+khk^{2}_{2,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 = u^{m} and

φ=θ^{m}in (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 [u^{m}, θ^{m}] 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 pressuresπ^{m}, observing that in (3.3) we can consider test functions Φ from
V_{p}∩V_{r} =V_{r} with r =np/[(n+ 2)p−2n]. Notice that for this value of r and
2n/(n+ 2)< p≤2n/(n+ 1), it holds thatV_{r} ,→,→L^{y} for all y ∈[1,∞). Then,
defining

hF^{m},Φi_{(W}1,r
0 (Ω))^{0} :=

Z

Ω

µ(·, θ^{m})T(D(u^{m})) :D(Φ)dx−
Z

Ω

(u^{m}⊗u^{m}) :D(Φ)dx
+ 1

m Z

Ω

|u^{m}|^{t−2}u^{m}·Φdx−
Z

Ω

θ^{m}f·Φdx,
it holds that hF^{m},Φi_{(W}^{1,r}

0 (Ω))^{0} = 0, for all Φ ∈ C^{∞}_{0,σ}(Ω). Furthermore, F^{m} ∈
W^{−1,r}^{0}(Ω). Thus, because of the De Rham Theorem (cf. [1]), there existsπ^{m} ∈
L^{r}^{0}(Ω) such that

hF^{m},Φi_{(W}1,r

0 (Ω))^{0} =h−∇π^{m},Φi_{(W}1,r
0 (Ω))^{0} =

Z

Ω

π^{m}divΦdx andkπ^{m}kr^{0} ≤C.

(3.7)
Therefore, we obtain the following weak formulation (for the velocityu^{m}) equivalent
to (3.3):

Z

Ω

µ(·, θ^{m})T(D(u^{m})) :D(Φ)dx−
Z

Ω

(u^{m}⊗u^{m}) :D(Φ)dx
+ 1

m Z

Ω

|u^{m}|^{t−2}u^{m}·Φdx

= Z

Ω

θ^{m}f·Φdx+
Z

Ω

π^{m}divΦdx,

(3.8)

for allΦ∈W^{1,r}_{0} (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 ([u^{m}, π^{m}, θ^{m}])m∈N⊆
Vp×L^{r}^{0}(Ω)×Xq, still denoted by ([u^{m}, π^{m}, θ^{m}])_{m∈N}, and [u, π, θ,χ,χ_{1}]∈Vp×
L^{r}^{0}(Ω)×Xq×L^{p}^{0}(Ω)×L^{q}^{0}(Ω) such that asm→ ∞the following holds

D(u^{m})→D(u) weakly inL^{p}, (3.9)
[u^{m}, θ^{m}, π^{m}]→[u, θ, π] weakly inVp×Xq×L^{r}^{0}, (3.10)
u^{m}→u strongly inL^{s}(Ω) for alls∈[1,2r^{0}), (3.11)

u^{m}→u a.e. in Ω, (3.12)

θ^{m}→θ a.e. in Ω, and a.e. in Γ, (3.13)
T(D(u^{m}_{k}))→χ weakly inL^{p}^{0}(Ω), (3.14)
a(∇θ^{m}_{k} )→χ_{1} weakly inL^{q}^{0}(Ω). (3.15)

From (3.10), for anyΦ∈C^{∞}_{0} (Ω) and lettingm→ ∞ it holds

1 m

Z

Ω

|u^{m}|^{t−2}u^{m}·Φdx
≤ 1

m^{1/t}
1

mku^{m}k^{t}_{t}^{(t−1)/t}

kΦkt→0, (3.16) and

Z

Ω

θ^{m}f·Φdx+
Z

Ω

π^{m}divΦdx→
Z

Ω

θf ·Φdx+ Z

Ω

πdivΦdx.

On the other hand, since W^{1,p}(Ω) ,→,→ L^{2}(Ω) for p > 2n/(n+ 2), and writing
u^{m}= (u^{m}−u) +u, forΦ∈C^{∞}_{0} (Ω) and lettingm→ ∞we obtain

Z

Ω

(u^{m}⊗u^{m}) :D(Φ)dx→
Z

Ω

(u⊗u) :D(Φ)dx.

Also, sinceθ^{m}→θinL^{1}(Ω) and a.e. in Ω, andµis a Carath´eodory function, then
µ(·, θ^{m})→µ(·, θ) a.e. in Ω. Then, collecting the last convergences, we have

Z

Ω

µ(·, θ)χ:D(Φ)dx+ Z

Ω

(u⊗Φ) :D(u)dx= Z

Ω

θf·Φdx+ Z

Ω

πdivΦdx, (3.17)
for allΦ∈C^{∞}_{0} (Ω) and consequently for allΦ∈W_{0}^{1,r}(Ω).

As before, since θ^{m} → θ in L^{1}(Ω) and a.e. in Ω, and κ is a Carath´eodory
function, then κ(·, θ^{m}) → κ(·, θ) a.e. in Ω. Then, from the uniform estimates
(3.10), (3.13) and (3.15) we also get

Z

Ω

κ(·, θ)χ_{1}−θu

· ∇φ dx+γ Z

Γ

θφ dΓ =hg, φi(W^{1,q}(Ω))^{0}+
Z

Γ

hφ dΓ, (3.18)
for all φ ∈ C_{0}^{∞}(Ω) and consequently for all φ ∈ X_{q}. It remains to prove that
χ=T(D(u^{m})) andχ_{1}=a(∇θ^{m}). To this end, it is sufficient to prove that

[D(u^{m}),∇θ^{m}]→[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 ([u^{m}, θ^{m}])m∈N, still
denoted by ([u^{m}, θ^{m}])m∈N, such that

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

Z

Ω

µ(·, θ^{m})T(D(u^{m})) :D(Φ)dx→
Z

Ω

µ(·, θ)T(D(u)) :D(Φ)dx, (3.21) Z

Ω

κ(·, θ^{m})a(∇θ^{m})· ∇φ dx→
Z

Ω

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

m→∞lim Z

Ω

[(T(D(u^{m}))−T(D(u))) :D(u^{m}−u)]^{ρ}^{1}dx≤η1. (3.23)

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

−∆u^{I}^{m}+∇π^{I}^{m} =H^{I}^{m} in Ω, I = 1,2,3,4,5,
div u^{I}^{m} = 0 in Ω,

u^{I}^{m} =0 on∂Ω,

(3.24)

where

H^{1}^{m} =−div (µ(·, θ^{m})T(D(u^{m})))∈W^{−1,p}^{0}(Ω),
H^{2}^{m} = div (u^{m}⊗(u^{m}−u))∈W^{1,r}^{0}(Ω),
H^{3}^{m} = div ((u^{m}−u)⊗u)∈W^{1,r}^{0}(Ω), H^{4}^{m} = 1

m|u^{m}|^{t−2}u^{m}∈L^{t}^{0}(Ω),
H^{5}^{m} =−θ^{m}f ∈L^{q}(Ω).

It is well known that there exists a weak solution [u^{I}^{m}, π^{I}^{m}] of (3.24), for I =
1,2,3,4,5; that is, there exist

[u^{1}^{m},u^{2}^{m},u^{3}^{m},u^{4}^{m},u^{5}^{m}]

∈W^{1,p}_{0} ^{0}(Ω)×W^{1,r}_{0} ^{0}(Ω)×W^{1,r}_{0} ^{0}(Ω)×W^{2,t}^{0}(Ω)×W^{2,q}^{0}(Ω),

[π^{1}^{m}, π^{2}^{m}, π^{3}^{m}, π^{4}^{m}, π^{5}^{m}]∈L^{p}^{0}(Ω)×L^{r}^{0}(Ω)×L^{r}^{0}(Ω)×W^{1,t}^{0}(Ω)×W^{1,q}^{0}(Ω),
satisfying

Z

Ω

∇u^{I}^{m}:∇Φdx−
Z

Ω

π^{I}^{m}divΦdx=hH^{I}^{m},Φi, ∀Φ∈C^{∞}_{0} (Ω), I= 1,2,3,4,5.

(3.25) Moreover, the following estimates hold:

kπ^{1}^{m}kp^{0} ≤CkH^{1}^{m}k−1,p^{0} ≤Cµ_{2}kT(D(u^{m}))kp^{0}, (3.26)
kπ^{2}^{m}kr^{0} ≤CkH^{2}^{m}k−1,r^{0} ≤Cku^{m}⊗(u^{m}−u)kr^{0} ≤Cku^{m}k2r^{0}ku^{m}−uk2r^{0}, (3.27)
kπ^{3}^{m}kr^{0} ≤CkH^{3}^{m}k−1,r^{0} ≤Cku^{m}⊗(u^{m}−u)kr^{0} ≤Cku^{m}k2r^{0}ku^{m}−uk2r^{0}, (3.28)

k∇π^{4}^{m}kt^{0} ≤CkH^{4}^{m}kt^{0} ≤ C
m^{1/t}

1

m^{1/t}ku^{m}kt

t−1

, (3.29)

k∇π^{5}^{m}k_{q} ≤CkH^{5}^{m}k_{q} ≤ kθ^{m}k_{q}kfk_{∞}. (3.30)
Since 2r^{0}=np/(n−p), from (3.11), (3.27) and (3.28), asmgoes to∞, we obtain

[π^{2}^{m}, π^{3}^{m}]→[0,0] inL^{s}(Ω)×L^{s}(Ω) for alls∈[1, r^{0}). (3.31)
Furthermore, by using (3.5), asmgoes to∞, it holds that

∇π^{4}^{m} →0 inL^{t}^{0}(Ω). (3.32)

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

5

X

I=1

[ Z

Ω

∇u^{I}^{m} :∇Φdx−
Z

Ω

π^{I}^{m}divΦdx]

= Z

Ω

(u⊗u) :D(Φ)dx+ Z

Ω

π^{m}divΦdx, ∀Φ∈W^{1,r}_{0} (Ω).

(3.33)

TakingΦ∈Vr in (3.33) we obtain

5

X

I=1

Z

Ω

∇u^{I}^{m} :∇Φdx=
Z

Ω

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

I=1u^{I}^{m} ∈W^{1,r}_{0} ^{0}(Ω) we obtain

5

X

I=1

u^{I}^{m} =U∈W^{1,r}_{0} ^{0}(Ω), ∀m∈N. (3.35)
Finally, taking the term R

Ωπ^{m}divΦdx in (3.33), replacing it in (3.8) and using
(3.35) we obtain

Z

Ω

µ(·, θ^{m})T(D(u^{m})) :D(Φ)dx+ 1
m

Z

Ω

|u^{m}|^{t−2}u^{m}·Φdx

= Z

Ω

(u^{m}⊗u^{m}) :D(Φ)dx−
Z

Ω

(u⊗u) :D(Φ)dx +

Z

Ω

∇U:∇Φdx−

5

X

I=1

Z

Ω

π^{I}^{m}divΦdx+
Z

Ω

θ^{m}f·Φdx,

(3.36)

for allΦ∈W^{1,r}_{0} (Ω).

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

X^{m}:=C(1 +|D(u^{m})|^{p}+|D(u)|^{p}+|π^{1}^{m}|^{p}^{0}). (3.37)
Then, from (3.10) and (3.26) we have

Z

Ω

X^{m}dx≤K1, (3.38)

for some positive constantK_{1}independent on m. Fixedp∈(_{n+2}^{2n} ,_{n+1}^{2n} ], letε_{1}>0
be small enough to be chosen below (see (3.53)). Then, from [11, Proposition 4.1]

there exists a subsequence of (u^{m})_{m∈}_{N}, still denoted by (u^{m})_{m∈}_{N}, and λ_{1} ≥ _{ε}^{1}

1

(independent onm), such that Z

B^{m}_{λ}

1

X^{m}dx≤ε1, B_{λ}^{m}_{1} :={x∈Ω :λ1< M(∇(u^{m}−u)ext)(x)≤λ^{2}_{1}}, (3.39)
whereM(∇(u^{m}−u)ext) denotes the Hardy-Littlewood maximal function of∇(u^{m}−
u)ext (cf. [11]), and (u^{m}−u)ext ∈W^{1,p}(R^{n}) is the extension by zero of (u^{m}−u).

On the other hand, from [11, Proposition 4.1], there exist a positive constantC=
C(Ω, n) and a sequence ((u^{m}−u)λ_{1})_{m∈}_{N}⊂W_{0}^{1,∞}(Ω) such that

k(u^{m}−u)λ_{1}k_{1,∞}≤Cλ1, (3.40)
(u^{m}−u)_{λ}_{1}→0, strongly inL^{s}(Ω)∀s∈[1,∞), (3.41)
(u^{m}−u)_{λ}_{1} →0, weakly inW^{1,s}_{0} (Ω)∀s∈[1,∞). (3.42)
Moreover, denoting

A^{m}_{λ}_{1} :={x∈Ω : (u^{m}−u)λ_{1}(x)6= (u^{m}−u)(x)},
C_{λ}^{m}_{1}:={x∈Ω :M(∇(u^{m}−u))(x)> λ^{2}_{1}},
it holds

|A^{m}_{λ}

1| ≤ |B^{m}_{λ}

1|+|C_{λ}^{m}|, (3.43)

|A^{m}_{λ}_{1}|+|B_{λ}^{m}_{1}| ≤ C

λ^{p}_{1}k∇(u^{m}−u)k^{p}_{p}, (3.44)

|C_{λ}^{m}_{1}| ≤ C

λ^{2p}_{1} k∇(u^{m}−u)k^{p}_{p}, (3.45)
k∇(u^{m}−u)λ_{1}k^{p}_{p}≤Ck∇(u^{m}−u)k^{p}_{p}≤K1. (3.46)
Now, we consider (u^{m}−u)λ1 as a test function in (3.36) and add in both sides of
the obtained equation the term

− Z

Ω

µ(·, θ^{m})T(D(u)) :D((u^{m}−u)λ_{1})dx, (3.47)
to obtain

Z

Ω

µ(·, θ^{m})[T(D(u^{m}))−T(D(u))] :D((u^{m}−u)λ1)dx
+ 1

m Z

Ω

|u^{m}|^{t−2}u^{m}·(u^{m}−u)_{λ}_{1}dx

= Z

Ω

[(u^{m}⊗u^{m})−(u⊗u)] :D((u^{m}−u)_{λ}_{1})dx
+

Z

Ω

∇U:∇(u^{m}−u)_{λ}_{1}dx

−

5

X

I=1

Z

Ω

π^{I}^{m}div((u^{m}−u)λ_{1})dx+
Z

Ω

θ^{m}f·(u^{m}−u)λ_{1}dx.

(3.48)

Notice thatu^{m}−u= (u^{m}−u)λ_{1} on Ω\ A^{m}_{λ}

1, and then, div(u^{m}−u)λ_{1} = 0 almost
everywhere on Ω\ A^{m}_{λ}

1. Therefore, from (3.48) we obtain
Z^{m}:=

Z

Ω\A^{m}_{λ}

1

µ(·, θ^{m})[T(D(u^{m}))−T(D(u))] :D((u^{m}−u))dx

=− Z

A^{m}_{λ}

1

µ(·, θ^{m})[T(D(u^{m}))−T(D(u))] :D((u^{m}−u)λ_{1})dx

− Z

A^{m}_{λ}

1

π^{1}^{m}div((u^{m}−u)λ_{1})dx
+

Z

Ω

[(u^{m}⊗(u^{m}−u) + (u^{m}−u)⊗u] :D((u^{m}−u)λ_{1})dx
+

Z

Ω

[∇U−µ(·, θ^{m})T(D(u))] :∇(u^{m}−u)λ_{1}dx
+

Z

Ω

[∇π^{4}^{m}− 1
m

Z

Ω

|u^{m}|^{t−2}u^{m}]·(u^{m}−u)λ1dx

− Z

A^{m}_{λ}

1

(π^{2}^{m}+π^{3}^{m}+π^{5}^{m}) div((u^{m}−u)_{λ}_{1})dx+
Z

Ω

θ^{m}f·(u^{m}−u)_{λ}_{1}dx

:=

7

X

i=1

Z_{i}^{m}.

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

m→∞lim (Z_{3}^{m}+Z_{5}^{m}+Z_{6}^{m}+Z_{7}^{m}) = 0. (3.49)

Moreover, from (3.42) and since ∇U ∈ L^{r}^{0}(Ω), µ1 ≤ µ(x, θ) ≤ µ2, a.e. x ∈ Ω
and T(D(u)) ∈ L^{p}^{0}(Ω), we obtain that lim_{m→∞}Z_{4}^{m} = 0. Now we deal with
lim_{m→∞}Z_{1}^{m}+Z_{2}^{m}. From the H¨older inequality, (3.40) and (3.43)-(3.46) it holds
that

|Z_{1}^{m}+Z_{2}^{m}|

≤ Z

B^{m}_{λ}

1∪C_{λ}^{m}

1

(µ(·, θ^{m})[T(D(u^{m}))−T(D(u))] :D((u^{m}−u)λ1)

−π^{1}^{m}div((u^{m}−u)λ_{1}) )dx

≤µ2τ2CZ

B^{m}_{λ}

1

X^{m}dx1/p^{0}

k∇(u^{m}−u)λ_{1}k_{p,B}^{m}

λ1

+µ2τ2Cλ1

Z

C^{m}_{λ}

1

X^{m}dx^{1/p}^{0}

|C^{m}_{λ}_{1}|^{1/p}

≤Cµ_{2}τ_{2}(ε^{1/p}_{1} ^{0}K_{1}^{1/p}+Cλ_{1}K_{1}^{1/p}^{0}(Cλ_{1}^{−2p}K_{1})^{1/p})

≤Cµ2τ2(ε^{1/p}

0

1 K_{1}^{1/p}+K1

λ1

)

≤Cµ2τ2(ε^{1/p}

0

1 K_{1}^{1/p}+ε1K1).

(3.50)

In (3.50), k∇(u^{m}−u)λ_{1}k_{p,B}^{m}

λ1 denotes the L^{p}(B_{λ}^{m}

1)-norm of∇(u^{m}−u)λ_{1}. Since
lim_{m→∞}Z_{4}^{m}= 0, from (3.49) and (3.50) we obtain

m→∞lim Z^{m}≤Cµ2τ2(ε^{1/p}_{1} ^{0}K_{1}^{1/p}+ε1K1). (3.51)
Therefore, fixedρ1∈(0,1), by using the H¨older inequality and (3.38) we obtain

S^{m}:=µ^{ρ}_{1}^{1}
Z

Ω

[(T(D(u^{m}))−T(D(u))) :D(u^{m}−u)]^{ρ}^{1}dx

≤ Z

Ω\A^{m}_{λ}

1

[µ(·, θ^{m})(T(D(u^{m}))−T(D(u))) :D(u^{m}−u)]^{ρ}^{1}dx
+

Z

A^{m}_{λ}

1

[µ(·, θ^{m})(T(D(u^{m}))−T(D(u))) :D(u^{m}−u)]^{ρ}^{1}dx

≤(Z^{m})^{ρ}^{1}|Ω\ A^{m}_{λ}_{1}|^{1−ρ}^{1}+C(µ2τ2K1)^{ρ}^{1}|A^{m}_{λ}_{1}|^{1−ρ}^{1}.

(3.52)

Then, takingε1>0 small enough such that
C(µ2τ2)^{ρ}^{1}|Ω|^{1−ρ}^{1}(ε^{1/p}

0

1 K_{1}^{1/p}+ε1K1)^{ρ}^{1}+C(µ2τ2)^{ρ}^{1}K1ε^{p(1−ρ}_{1} ^{1}^{)}< ρ1, (3.53)
from (3.51)-(3.53) we have

m→∞lim S^{m}

≤ |Ω|^{1−ρ}^{1}C(µ2τ2)^{ρ}^{1}(ε^{1/p}

0

1 K_{1}^{1/p}+ε1K1)^{ρ}^{1}+C(µ2τ2K1)^{ρ}^{1}(Cλ^{−p}_{1} K1)^{1−ρ}^{1}

≤C(µ2τ2)^{ρ}^{1}|Ω|^{1−ρ}^{1}(ε^{1/p}_{1} ^{0}K_{1}^{1/p}+ε1K1)^{ρ}^{1}+C(µ2τ2)^{ρ}^{1}K1ε^{p(1−ρ}_{1} ^{1}^{)}< ρ1.

(3.54)

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

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

We prove that for an arbitraryη_{2}>0, there exists a subsequence of (θ^{m})_{m∈}_{N}, still
denoted by (θ^{m})_{m∈}_{N}, such that for someρ2∈(0,1), it holds that

m→∞lim Z

Ω

[(a(∇θ^{m})−a(∇θ))· ∇(θ^{m}−θ)]^{ρ}^{2}dx≤η_{2}. (3.55)
Let us define

E^{m}:=C(1 +|∇θ^{m}|^{q}+|∇θ|^{q}). (3.56)
Then, from (3.10) we have

Z

Ω

E^{m}dx≤K_{2}, (3.57)

for some positive constantK_{2} independent on m. Let ε_{2} >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 (θ^{m})_{m∈}_{N}, still denoted by (θ^{m})_{m∈}_{N}, andλ2≥ _{ε}^{1}

2

(independent onm), such that Z

D^{m}_{λ}

2

E^{m}dx≤ε2, D_{λ}^{m}_{2} :={x∈Ω :λ2< M(∇(θ^{m}−θ)ext)(x)≤λ^{2}_{2}}, (3.58)
whereM(∇(θ^{m}−θ)_{ext}) denotes the Hardy-Littlewood maximal function of∇(θ^{m}−
θ)ext, and (θ^{m}−θ)ext∈W^{1,q}(R^{n}) is the extension by zero of (θ^{m}−θ). Also, there
exist a positive constantC=C(Ω, n) and a sequence ((θ^{m}−θ)_{λ}_{2})_{m∈}_{N}⊂W_{0}^{1,∞}(Ω)
such that

k(θ^{m}−θ)λ_{2}k1,∞≤Cλ2, (3.59)
(θ^{m}−θ)λ_{2}→0, strongly in L^{s}(Ω)∀s∈[1,∞), (3.60)
(θ^{m}−θ)λ_{2} →0, weakly inW_{0}^{1,s}(Ω)∀s∈[1,∞). (3.61)
Moreover, denoting

F_{λ}^{m}_{2} :={x∈Ω : (θ^{m}−θ)λ_{2}(x)6= (θ^{m}−θ)(x)},
G_{λ}^{m}_{2} :={x∈Ω :M(∇(θ^{m}−θ))(x)> λ^{2}_{2}},
it holds

|F_{λ}^{m}_{2}| ≤ |D^{m}_{λ}_{2}|+|G_{λ}^{m}_{2}|, (3.62)

|D^{m}_{λ}

2|+|F_{λ}^{m}

2| ≤ C

λ^{q}_{2}k∇(θ^{m}−θ)k^{q}_{q}, (3.63)

|G_{λ}^{m}_{2}| ≤ C

λ^{2q}_{2} k∇(θ^{m}−θ)k^{q}_{q}, (3.64)
k∇(θ^{m}−θ)_{λ}_{2}k^{q}_{q}≤Ck∇(θ^{m}−θ)k^{q}_{q} ≤K_{2}. (3.65)
Now we consider (θ^{m}−θ)λ_{2} as a test function in (3.4), and add in both sides of the
obtained equation the term

− Z

Ω

κ(·, θ^{m})a(∇θ)· ∇((θ^{m}−θ)λ_{2})dx, (3.66)

this gives Z

Ω

κ(·, θ^{m})[a(∇θ^{m})−a(∇θ)]· ∇((θ^{m}−θ)λ_{2})dx

= Z

Ω

(θ^{m}u^{m})· ∇((θ^{m}−θ)λ_{2})dx−
Z

Ω

κ(·, θ^{m})a(∇θ)· ∇((θ^{m}−θ)λ_{2})dx

−γ Z

Γ

θ^{m}(θ^{m}−θ)λ_{2}dΓ +hg,(θ^{m}−θ)λ_{2}i(W^{1,q}(Ω))^{0}

+ Z

Γ

h(θ^{m}−θ)λ2dΓ.

(3.67)

Notice thatθ^{m}−θ= (θ^{m}−θ)_{λ}_{2} on Ω\ F_{λ}^{m}

2. Therefore, from (3.67) we obtain
Y^{m}:=

Z

Ω\F_{λ}^{m}

2

κ(·, θ^{m})[a(∇θ^{m})−a(∇θ)]· ∇(θ^{m}−θ)dx

=− Z

F_{λ}^{m}

2

κ(·, θ^{m})[a(∇θ^{m})−a(∇θ)]· ∇((θ^{m}−θ)λ_{2})dx
+

Z

Ω

(θ^{m}u^{m})· ∇((θ^{m}−θ)λ_{2})dx

− Z

Ω

κ(·, θ^{m})a(∇θ)· ∇((θ^{m}−θ)λ_{2})dx−γ
Z

Γ

θ^{m}(θ^{m}−θ)λ_{2}dΓ
+hg,(θ^{m}−θ)λ_{2}i_{(W}1,q(Ω))^{0}+

Z

Γ

h(θ^{m}−θ)λ_{2}dΓ :=

6

X

i=1

Y_{i}^{m}.

(3.68)

Using (3.10) we obtain

m→∞lim (Y_{2}^{m}+Y_{4}^{m}+Y_{5}^{m}+Y_{6}^{m}) = 0. (3.69)

Moreover, from (3.61) and sinceκ1≤κ(x, θ)≤κ2, a.e. x∈Ω anda(∇θ)∈L^{q}^{0}(Ω),
we obtain that lim_{m→∞}Y_{3}^{m} = 0. Now we deal with lim_{m→∞}Y_{1}^{m}. From the
properties of (θ^{m}−θ)λ_{2}, the H¨older inequality and (3.65) it holds

|Y_{1}^{m}| ≤
Z

D^{m}_{λ}

2∪G_{λ}^{m}

2

(κ(·, θ^{m})[a(∇θ^{m})−a(∇θ)]· ∇((θ^{m}−θ)λ_{2}))dx

≤κ_{2}α_{2}CZ

D_{λ}^{m}

E^{m}dx1/q^{0}

k∇(θ^{m}−θ)_{λ}_{2}k_{q,D}^{m}

λ2

+κ2α2Cλ2

Z

G_{λ}^{m}

2

E^{m}dx^{1/q}^{0}

|G_{λ}^{m}|^{1/q}

≤Cκ_{2}α_{2}(ε^{1/q}_{2} ^{0}K_{2}^{1/q}+λ_{2}K_{2}^{1/q}^{0}(Cλ_{2}^{−2q}K_{2})^{1/q})

≤Cκ2α2(ε^{1/q}

0

2 K_{2}^{1/q}+K2

λ_{2})≤Cκ2α2(ε^{1/q}

0

2 K_{2}^{1/q}+ε2K2).

(3.70)

Thus, since limm→∞Y_{3}^{m} = 0, from (3.69) and (3.70) we obtain limm→∞Y^{m} ≤
Cκ_{2}α_{2}(ε^{1/q}_{2} ^{0}K_{2}^{1/q}+ε_{2}K_{2}). Therefore, fixedρ_{2}∈(0,1) we obtain

L^{m}:=κ^{ρ}_{1}^{2}
Z

Ω

[(a(∇θ^{m})−a(∇θ))· ∇(θ^{m}−θ)]^{ρ}^{2}dx

≤ Z

Ω\F_{λ}^{m}

2

[κ(·, θ^{m})(a(∇θ^{m})−a(∇θ))· ∇(θ^{m}−θ)]^{ρ}^{2}dx
+

Z

F_{λ}^{m}

2

[κ(·, θ^{m})(a(∇θ^{m})−a(∇θ))· ∇(θ^{m}−θ)]^{ρ}^{2}dx

≤(Y^{m})^{ρ}^{2}|Ω\ F_{λ}^{m}_{2}|^{1−ρ}^{2}+C(κ2α2K2)^{ρ}^{2}|F_{λ}^{m}_{2}|^{1−ρ}^{2}.

(3.71)

Then, takingε2>0 small enough such that

C|Ω|^{1−ρ}^{2}(κ_{2}α_{2})^{ρ}^{2}(ε^{1/q}_{2} ^{0}K_{2}^{1/q}+ε_{2}K_{2})^{ρ}^{2}+C(κ_{2}α_{2})^{ρ}^{2}K_{2}ε^{q(1−ρ}_{2} ^{2}^{)}< ρ_{2}, (3.72)
we have

m→∞lim Y^{m}

≤C|Ω|^{1−ρ}^{2}(α2κ2)^{ρ}^{2}(ε^{1/q}

0

2 K_{2}^{1/q}+ε2K2)^{ρ}^{2}+C(κ2α2K2)^{ρ}^{2}(Cλ2−q)^{1−ρ}^{2}

≤C|Ω|^{1−ρ}^{2}(κ2α2)^{ρ}^{2}(ε^{1/q}

0

2 K_{2}^{1/q}+ε2K2)^{ρ}^{2}+C(κ2α2)^{ρ}^{2}K2ε^{q(1−ρ}_{2} ^{2}^{)}< ρ2.

(3.73)

Thus, we conclude (3.55) and therefore the convergence of ∇θ^{m} 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) = T_{1}(η) := 2µ(1 +|η|^{2})^{p−2}^{2} η or T(η) = T_{2}(η) := 2µ(1 +|η|)^{p−2}η,
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 C^{1}-function such that 0< κ1 ≤κ(x, θ)≤κ2 a.e. x∈Ω and for all θ ∈R
and, it satisfies|κ^{0}(·, a)−κ^{0}(·, b)| ≤λ^{0}|a−b|, for all a, b∈Rand κ^{0}(·,0) = 0, with
κ1, κ2andλ^{0} are positive constants. Under mild conditions on the dataf ∈L^{q}, g∈
L^{r}(Ω), we obtain the existence of strong solution [u, θ]∈W^{2,q}(Ω)×W^{2,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=T_{2}, in order to overcome the difficulty caused by the lack of regularity of