\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 46, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/46\hfil Steady-state thermal flow]
{Steady-state thermal Herschel-Bulkley flow with Tresca's
friction law}

\author[F. Messelmi, B. Merouani, F. Bouzeghaya\hfil EJDE-2010/46\hfilneg]
{Farid Messelmi, Boubakeur Merouani, Fouzia Bouzeghaya}

\address{Farid Messelmi \newline
Department de Mathematiques,
Univerisite Zian Achour de Djelfa,
 Djelfa 17000,  Algeria}
 \email{foudimath@yahoo.fr}

\address{Boubakeur Merouani \newline
Department de Mathematiques,
Universite Ferhat-Abbes de Setif,
Setif 19000, Algeria}
\email{mermathsb@hotmail.fr}

\address{Fouzia Bouzeghaya \newline
Department de Mathematiques,
Universite Ferhat-Abbes de Setif,
Setif 19000, Algeria}
\email{bouzeghaya@yahoo.fr}

\thanks{Submitted January 16, 2010. Published April 6, 2010.}
\subjclass[2000]{35J85, 76D03, 80A20}
\keywords{Herschel-Bulkley fluid; thermal friction law;
variational inequality; \hfill\break\indent weak solution}

\begin{abstract}
 We consider a mathematical model which describes the steady-state
 flow of a Herschel-Bulkley fluid whose the consistency and
 the yield limit depend on the temperature and with mixed boundary
 conditions, including a frictional boundary condition.
 We derive a weak formulation of the coupled system of motion and
 energy equations which consists of a variational inequality for
 the velocity field. We prove the existence of weak solutions.
 In the asymptotic limit case of a high thermal conductivity,
 the temperature becomes a constant solving an implicit total energy
 equation involving the consistency function and the yield limit.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

The model of Herschel-Bulkley fluid has been used in various publications
to describe the flow of metals, plastic solids and some polymers. The
literature concerning this topic is extensive; see e.g.
\cite{d3,p1} and
references therein. The new feature in the model is due to a Fourier type
boundary condition, and consists in the appearance of a nonlocal
term on the boundary part where Tresca's thermal friction is taken
into account.

An intrinsic inclusion leads in a natural way to variational
equations which justify the study of problems involving the
incompressible, plastic Herschel-Bulkley fluid using arguments of
the variational analysis. The paper is organized as follows. In
Section 2 we present the mechanical problem of the steady-state
Herschel-Bulkley flow where the consistency and the yield limit
depend on the temperature and with Tresca's thermal friction law.
Moreover, we introduce some notations and preliminaries. In
Section 3 we derive the variational formulation of the problem. We
prove in Section 4 the existence of weak solutions as well as an
existence result to the steady-state Herschel-Bulkley flow with
temperature dependent nonlocal consistency, yield limit and
tresca's friction, which can be obtained as an asymptotic limit
case of a very large thermal conductivity.

\section{Statement of the Problem}

We consider a mathematical problem modelling the steady-state flow
of a thermal Herschel-Bulkley fluid in a bounded domain
$\Omega \subset\mathbb{R}^n$ $(n=2,3)$, with the boundary
$\Gamma $ of class $C^1$, partitioned into two
disjoint measurable parts $\Gamma_0$ and
$\Gamma_1$ such that $\mathop{\rm meas}(\Gamma_0)>0$.
The fluid is supposed to be incompressible, the consistency and
the yield limit depend on the
temperature. The fluid is acted upon by given volume forces of
density $f$. In addition, we admit a possible external
heat source proportional to the temperature. On $\Gamma_0$ we suppose
that the velocity is known. The temperature is
given by a homogeneous Neumann boundary condition on
$\Gamma_0$. We impose on $\Gamma_1$ a
frictional contact described by a Tresca thermal friction law, as
well as a Fourier boundary condition.

We denote by $\mathbb{S}_n$ the space of symmetric tensors on
$\mathbb{R}^n$. We define the inner product and the Euclidean norm on
$\mathbb{R}^n$ and $\mathbb{S}_n$, respectively, by
\begin{gather*}
\mathbf{u}\cdot \mathbf{v}=u_{i}v_{i}\quad \forall \mathbf{u},
\mathbf{v}\in \mathbb{R}^n\quad\text{and}\quad
\boldsymbol{\sigma}\cdot \boldsymbol{\tau} =\sigma_{ij}\tau
_{ij}\quad \forall \boldsymbol{\sigma},\boldsymbol{\tau} \in\mathbb{S}_n.
\\
|\mathbf{u}|=(\mathbf{u}\cdot \mathbf{u})^{1/2}\quad
 \forall \mathbf{u}\in \mathbb{R}^n\quad \text{and}\quad
|\boldsymbol{\sigma}|=(\boldsymbol{\sigma}\cdot \boldsymbol{\sigma})^{1/2}\quad
\forall \boldsymbol{\sigma}\in \mathbb{S}_n.
\end{gather*}

Here and below, the indices $i$ and $j$ run from $1$ to $n$
and the summation convention over repeated indices is used. We
denote by $\tilde{\boldsymbol{\sigma}}$ the deviator of
$\boldsymbol{\sigma}=(\sigma_{ij})$ given by
\[
\tilde{\boldsymbol{\sigma}}=(\tilde{\sigma}_{ij}),\quad
\tilde{ \sigma}_{ij}=\sigma_{ij}-\frac{\sigma_{kk}}{n}\delta _{ij},
\]
 where $\boldsymbol{\delta}=(\delta_{ij})$
denotes the identity tensor.

Let $1<p<2$. We consider the rate of deformation operator defined
for every $\mathbf{u}\in W^{1, p}(\Omega)^n$ by
\[
\varepsilon (\mathbf{u})=(
\varepsilon _{ij}(\mathbf{u}))_{1\leq i,j\leq n},
\quad \mathbf{\varepsilon}_{ij}(\mathbf{u})
=\frac{1}{2}(u_{i,j}+u_{j,i}).
\]

We denote by $\mathbf{\nu }$ the unit outward normal vector on the
boundary $\Gamma $. For every vector field
$\mathbf{v}\in W^{1, p}(\Omega )^n$ we also write
$\mathbf{v}$ for its trace on $\Gamma$. The normal and
the tangential components of $\mathbf{v}$ on the boundary are
\[
\mathbf{v}_{\nu }=\mathbf{v}\cdot \mathbf{\nu },\quad
\mathbf{v}_{\tau}=\mathbf{v-v}_{\nu }\mathbf{\nu }.
\]
Similarly, for a regular tensor field $\boldsymbol{\sigma}$, we
denote by $\boldsymbol{\sigma}_{\nu }$ and
$\boldsymbol{\sigma}_{\tau }$ the normal and
tangential components of $\boldsymbol{\sigma}$ on the boundary given by
\[
\boldsymbol{\sigma}_{\nu }=\boldsymbol{\sigma \nu }\cdot \mathbf{\nu},\quad
 \boldsymbol{\sigma}_{\tau }=\mathbf{\sigma \nu
}-\boldsymbol{\sigma}_{\nu }\mathbf{ \nu }.
\]
We consider now, the following mechanical problem.

\subsection*{Problem 1}
Find  a  velocity  field $\mathbf{u}=(u_{i})
_{i= \overline{1,n}}:\Omega \to \mathbb{R}^n$,
stress field $\boldsymbol{\sigma}=(\sigma_{ij})
_{i,j=\overline{1,n}}:\Omega \to
\mathbb{S}_n$ and a temperature
$\theta:\Omega\to \mathbb{R}$ such that
\begin{gather}
\mathbf{u}\cdot \nabla\mathbf{u}
=\mathop{\rm Div}(\boldsymbol{\sigma})
+\mathbf{f}\quad \text{in }\Omega \label{e2.1}\\
\left.
\begin{gathered}
\tilde{\boldsymbol{\sigma}}=\mu (\theta)|\varepsilon (\mathbf{u})
|^{p-2} \varepsilon (\mathbf{u})+g (\theta )
\frac{\varepsilon (\mathbf{u})}{ |
\varepsilon (\mathbf{u})|}\quad\text{if }
|\varepsilon (\mathbf{u}) |\neq 0 \\
|\tilde{\boldsymbol{\sigma}}|\leq
g (\theta )\quad
\text{if }|\varepsilon (\mathbf{u} )|=0
\end{gathered}
\right\} \quad \text{in } \Omega \label{e2.2}
\\
\mathop{\rm div}(\mathbf{u})=0\quad \text{in } \Omega  \label{e2.3}
 \\
-k\Delta \theta +\mathbf{u}\cdot \nabla\theta
=\boldsymbol{\sigma }\cdot \varepsilon (
\mathbf{u})-\alpha \theta \quad \text{in } \Omega\label{e2.4}
\\
\mathbf{u}=0\quad \text{on }\Gamma_0\label{e2.5} \\
\left. \begin{gathered}
\mathbf{u}_{\nu }=0,\quad |\boldsymbol{\sigma}_{\tau }|\leq
\upsilon (\theta)\\
|\boldsymbol{\sigma}_{\tau }|<\upsilon (
\theta)\implies \mathbf{u}_{\tau }=0  \\
|\boldsymbol{\sigma}_{\tau }|=\upsilon (
\theta)\implies \mathbf{u}_{\tau }=-\lambda \boldsymbol{\sigma}
_{\tau },\quad \lambda \geq 0
\end{gathered}
  \right\}\quad \text{on } \Gamma_1  \label{e2.6}
\\
\frac{\partial \theta }{\partial \nu }=0\quad
\text{on } \Gamma_0 \label{e2.7}
\\
k\frac{\mathbf{\partial \theta }}{\mathbf{\partial \nu }}+\beta
\theta =\upsilon (\theta)|\mathbf{u}_{\tau }|\quad \text{on }\quad  \Gamma_1
\label{e2.8}
\end{gather}
Where $\mathop{\rm Div}(\boldsymbol{\sigma})=(\sigma_{ij,j})$ and
$\mathop{\rm div}(\mathbf{u})=u_{i,i}$.
The flow is given by the equation \eqref{e2.1} where the density
is assumed equal to one. Equation \eqref{e2.2} represents the
constitutive law of a Herschel-Bulkley fluid whose the consistency
$\mu $ and the yield limit $g $ depend on the
temperature, $1<p<2$ is the power
law exponent of the material. \eqref{e2.3} represents the
incompressibility condition. Equation \eqref{e2.4} represents
the energy conservation where the
specific heat is assumed equal to one, $k>0$ is the thermal
conductivity and
the term $-\alpha \theta$ represents the external heat
source with $\alpha >0$. \eqref{e2.5} gives the velocity on $\Gamma_0$.
Condition \eqref{e2.6} represents a Tresca thermal friction law on
$\Gamma_1$ where $\upsilon (\theta)$ is the friction
yield coefficient for liquid-solid interface.
\eqref{e2.7} is a homogeneous Neumann
boundary condition on $\Gamma_0$. Finally, \eqref{e2.8} represents a
Fourier boundary condition on $\Gamma_1$, where $\beta \geq 0$
represents the Robin coefficient.

\begin{remark} \label{rmk2.1} \rm
In the constitutive law \eqref{e2.2} of the Herschel-Bulkley fluid,
 the viscosity is given by the formula
\begin{equation}
\mathbf{\eta (\theta)}=\mu (
\theta )|\varepsilon (
\mathbf{u})|^{p-2}.  \label{e2.9}
\end{equation}
\end{remark}

We define
\[
W=\left\{ \mathbf{v}\in W^{1,p}(
\Omega)^n:\mathop{\rm div}(\mathbf{v})
=0\ \text{in } \Omega ,\ \mathbf{v}=0 \text{ on } \Gamma_0
\text{ and } \mathbf{v}_{\nu }=0\text{ on }\Gamma_1\right\} ,
\]
which  is a Banach space equipped with the norm
\[
\| \mathbf{v}\|_{W}=\|
\mathbf{v} \|_{W^{1,p}(\Omega)^n}.
\]

For the rest of this article, we will denote by $c$ possibly different
positive constants depending only on the data of the problem.
Denote by $p'$ the conjugate of $p$ and by $q'$ the conjugate of
$q$, $q\in [0,+\infty [ $.
We introduce the following functionals
\begin{gather*}
B :W\times W\times W\to\mathbb{R},\quad
B (\mathbf{u}, \mathbf{v}, \mathbf{w} )
=\int_{\Omega }\mathbf{u}\cdot \nabla\mathbf{
v}\cdot \mathbf{w}\,dx
\\
E :W^{1, q}(\Omega )\times W^{1, q'}(\Omega )\times W\to \mathbb{R},\quad
 E (\theta, \tau,  \mathbf{v})=\int_{\Omega }\mathbf{\theta \nabla
\tau }\cdot \mathbf{v}\, dx.
\end{gather*}
We assume
\begin{gather}
\begin{gathered}
\forall x\in \Omega ,\quad
\mu (. ,x)\in C^{0}(\mathbb{R})\quad\text{and}\\
\exists \mu_1, \mu_{2}>0: \mu_1\ \leq \mu (y, x)\leq
\mu_{2}\quad \forall y\in \mathbb{R},\;
\forall x\in \Omega.
\end{gathered}  \label{e2.10}
\\
\begin{gathered}
\forall x\in \Omega ,\quad g (.  ,x)\in C^{0}(
\mathbb{R})\quad\text{and}\\
\exists g_0>0: 0\leq g (y , x)\leq g_0\quad \forall y\in
\mathbb{R},\; \forall x\in \Omega .
\end{gathered}  \label{e2.11}
\\
\begin{gathered}
\forall x\in \Gamma_1, \upsilon(. ,x)\in C^{0}(
\mathbb{R})\quad \text{and}\\
\exists \upsilon_0>0: 0 \leq \upsilon (y , x)\leq
\upsilon_0\quad \forall y\in \mathbb{R},\;
\forall x\in \Gamma_1.
\end{gathered} \label{e2.12}
\end{gather}

\begin{lemma} \label{lem2.2}
Suppose that
\begin{equation}
\frac{3n}{n+2}\leq p<2\quad\text{and}\quad
1<q<\frac{n}{n-1}.  \label{e2.13}
\end{equation}
Then
(1) $B $ is trilinear, continuous on
$W \times W\times W$. Moreover,
for all $(\mathbf{u},  \mathbf{v}, \mathbf{w})\in W\times W \times
W$ we have $B (\mathbf{u}, \mathbf{v},  \mathbf{w})=-B (
\mathbf{u}, \mathbf{w},  \mathbf{v})$.

(2) $E $ is trilinear, continuous on
$W ^{1, q}(\Omega )\times
W^{1,  q'}(\Omega )\times W$ and on
$H^1(\Omega )\times H^1(\Omega )\times W$. Moreover,
$E (\theta, \tau, \mathbf{v})
=-E (\tau, \theta, \mathbf{v})$ for all
$(\theta, \tau, \mathbf{v} )
\in W^{1, q}(\Omega )\times W^{1, q'}(\Omega )\times
W$ and for all $(\theta, \tau,  \mathbf{v})\in
H^1(\Omega )\times H^1(\Omega )\times W$.
\end{lemma}

\begin{proof}
In these two assertions, the trilinearity is evident.

(1) The Soblov imbedding
\[
W^{1, p}(\Omega )\subset L^{\rho }(\Omega )\quad
\forall \rho \in [p,  \frac{np}{n-p}] ,
\]
combined with  \eqref{e2.13}, gives
$W^{1, p}(\Omega )\subset L ^{\rho }(\Omega )$
 for all $\rho \in [p,  \frac{2n}{n-2}[$.
Particularly,
\begin{equation}
W^{1, p}(\Omega )\subset L^{\frac{3n}{n-1}}(\Omega ). \label{e2.14}
\end{equation}
On the other hand, the use of H\"{o}lder's inequality leads to
\[
B (\mathbf{u}, \mathbf{v}, \mathbf{w})\leq \| \mathbf{u}\|
_{L^{\frac{3n}{n-1}}(\Omega )^n}\|
\mathbf{v}\|_{L ^{p}(\Omega )
^n}\| \mathbf{w}\|_{
L^{\frac{3n}{n-1}}(\Omega )^n}.
\]
Consequently, the continuity of $B $ follows from \eqref{e2.14}.

Moreover, the antisymmetry of the convective operator $B $ is valid
by the incompressibility condition \eqref{e2.3} and the boundary
conditions given
by \eqref{e2.5}, \eqref{e2.6}, using an integration by parts.

(2) The continuity of $E $ on
$H^1(\Omega)\times
H^1(\Omega )\times W$ is an
immediate consequence of the Sobolev imbedding
$W \subset L^{3}(\Omega )^n$ and $H
^1(\Omega )\subset L^{3}(\Omega)$.
The proof of the antisymmetry of $E $ is based on the
incompressibility condition \eqref{e2.3} and the boundary
conditions given by
\eqref{e2.5}, \eqref{e2.6}.

Finally, to prove the continuity of $E$ on
$W^{1,  q}(\Omega )\times
W^{1, q'}(\Omega )\times W$, we proceed as follows.
 Sobolev's imbedding asserts that
\begin{equation}
\begin{gathered}
W^{1, q}(\Omega )\subset L^{\rho }(\Omega )\quad \forall \rho \in
]\frac{n}{n-1}, \frac{n}{n-2}[ , \\
W\subset L^{s}(\Omega )^n\quad \forall s\in [ n, \frac{2n}{n-2}[ .
\end{gathered}  \label{e2.15}
\end{equation}

Then, if $\theta \in W^{1, q}(\Omega  )$,
$\tau \in W^{1, q'}(\Omega )$
and $\mathbf{v}\in W$, the
result follows from \eqref{e2.15}, the antisymmetry of $E$ and
the continuity of the injection
$W^{1, q'}(\Omega )\to C(\bar{\Omega})$ for $q'>n$, that is,
$q<\frac{n}{n-1}$, using H\"{o}lder's inequality .
\end{proof}

For the rest of this article, we take $\frac{3n}{n+2}\leq p<2$
and $1<q<\frac{n}{n-1}$.

\section{Variational Formulation}

The aim of this section is to derive a variational formulation to the
problem (P1). To do so we need the following Lemma.

\begin{lemma} \label{lem3.1}
Assume that $f\in W'$. If
$\{\mathbf{u},  \boldsymbol{\sigma}, \theta \} $ are
regular functions satisfying \eqref{e2.1}-\eqref{e2.8}, then
\begin{gather}
\begin{aligned}
&B (\mathbf{u}, \mathbf{u}, \mathbf{v}-\mathbf{u} )
+\int_{\Omega }\mu (\theta)(|\varepsilon (\mathbf{u})|^{p-2}
\varepsilon (\mathbf{u}))\cdot (\varepsilon (\mathbf{v})-\varepsilon (
\mathbf{u}))\,dx
+\phi (\theta, \mathbf{v})
- \phi (\theta, \mathbf{u})\\
&\geq \int_{\Omega}\mathbf{f}\cdot (
\mathbf{v}-\mathbf{u})\,dx \quad \forall
\mathbf{v}\in W,
\end{aligned} \label{e3.1}
\\
\begin{aligned}
&-E (\theta, \tau, \mathbf{u} )
+k\int_{\Omega }\nabla\theta\cdot \nabla \tau \, dx
+\alpha \int_{\Omega }\theta \tau \,dx
+\beta \int_{\Gamma_1}\theta \tau \,ds \\
&=\int_{\Omega} \mu (\theta) |\varepsilon (\mathbf{u})|^{p}
+ g (\theta) |\varepsilon  (\mathbf{u})|) \tau \,dx
+ \int_{\Gamma_1} \upsilon (\theta)|
\mathbf{u}_{\tau }|\tau  \,ds\quad \forall
\tau\in W^{1,  q'}(\Omega),
\end{aligned}  \label{e3.2}
\end{gather}
where
\begin{equation}
\phi (\theta, \mathbf{u})
=\int_{ \Gamma_1}\upsilon (\theta)|\mathbf{u}_{\tau }|
\,ds+\int_{\Omega } g (\theta)|\varepsilon  (\mathbf{u})|\,dx .  \label{e3.3}
\end{equation}
\end{lemma}

\begin{proof}
 Let us start by proving the variational inequality
\eqref{e3.1}. Let
$\{ \mathbf{u}, \boldsymbol{\sigma}, \theta\} $ be regular
 functions satisfying
\eqref{e2.1}-\eqref{e2.8} and let $\mathbf{v}\in W$.
Using Green's formula and \eqref{e2.1}, \eqref{e2.2}, \eqref{e2.3},
\eqref{e2.5} and \eqref{e2.6}, we obtain
\begin{align*}
&\int_{\Omega }\mathbf{u}\cdot \nabla\mathbf{u}\cdot (
\mathbf{v}-\mathbf{u})\,dx
+\int_{\Omega }\mu (\theta)(|\varepsilon
(\mathbf{u})|^{p-2}\varepsilon (\mathbf{u}))\cdot
(\varepsilon (\mathbf{v} )
-\varepsilon (\mathbf{u}))\, dx\\
&+\int_{\Omega }g (\theta)|
\varepsilon (\mathbf{v})|\,dx
-\int_{ \Omega }g (\theta )|\varepsilon (\mathbf{u})|\,dx\\
&\geq \int_{ \Omega }\mathbf{f}\cdot (
\mathbf{v}-\mathbf{u})\,dx+\int_{\Gamma_1}\boldsymbol{\sigma \nu }
\cdot (\mathbf{v}- \mathbf{u})\,ds .
\end{align*}
On the other hand, by \eqref{e2.6},
\[
\int_{\Gamma_1}\boldsymbol{\sigma \nu }\cdot (
\mathbf{v}- \mathbf{u})\,ds
\geq \int_{\Gamma_1}\upsilon (\theta)|
\mathbf{u}_{\tau }|\,ds-\int_{\Gamma_1}
\upsilon (\theta)
|\mathbf{v}_{\tau }|ds.
\]
Then \eqref{e3.1} holds. Now, to prove the variational equation
\eqref{e3.2}, we proceed as follows. Applying Green's formula,
\eqref{e2.4}, \eqref{e2.7}, \eqref{e2.8} and Lemma \ref{lem2.2}, we
obtain, after a simple calculation,
\begin{align*}
&-\int_{\Omega }\theta(\nabla\tau\cdot
\mathbf{v})\,dx +k\int_{\Omega }\nabla\theta\cdot \nabla\tau \, dx
+\alpha \int_{\Omega }\theta \tau \, dx
+\beta \int_{\Gamma_1}\mathbf{\theta \tau ds} \\
&=\int_{\Omega }\boldsymbol{\sigma}\cdot \varepsilon (
\mathbf{u})\tau \, dx
+\int_{\Gamma_1}\upsilon (\theta)|
\mathbf{u}_{\tau }|\tau \,ds\quad
\forall \tau\in W^{1, q'}(\Omega ).
\end{align*}

By definition of $\boldsymbol{\sigma}$, using the incompressibility condition
\eqref{e2.3}, we can infer
\[
\int_{\Omega }\boldsymbol{\sigma}\cdot \varepsilon (
\mathbf{u})\tau \, dx
=\int_{\Omega }(\mathbf{ \mu (\theta)}|\varepsilon (\mathbf{u})|^{p}
+g (\theta)|\varepsilon (\mathbf{u})|)\tau \, dx,
\]
which completes the proof.
\end{proof}

\begin{remark} \label{rmk3.2} \rm
In \eqref{e3.2}, the first term on the right hand side has sense, since
the injection $W^{1, q'}(\Omega)\to C(\bar{\Omega})$
is continuous for $q'>n$, that is, $q<n/(n-1)$.
\end{remark}

Lemma \ref{lem3.1} leads us to consider the following variational system.

\subsection*{Problem P2}
For prescribed data $f\in W'$. Find
$\mathbf{u} \in W$ and $\theta\in W^{1,  q}(\Omega)$,
satisfying the variational system
\begin{equation}
\begin{aligned}
&B (\mathbf{u}, \mathbf{u}, \mathbf{v}-\mathbf{u} )+\int_{\Omega }\mu (
\theta)(|\varepsilon (\mathbf{u})|^{p-2}\varepsilon (\mathbf{u}))\cdot (\varepsilon (\mathbf{v})-\varepsilon (
\mathbf{u}))\,dx
+ \phi (\theta, \mathbf{v})
-\mathbf{ \phi }(\theta, \mathbf{u})\\
&\geq \int_{\Omega}\mathbf{f}\cdot (
\mathbf{v}-\mathbf{u})\,dx \quad \forall \mathbf{v} \in W,
\end{aligned} \label{e3.4}
\end{equation}
and
\begin{equation}
\begin{aligned}
&-E (\theta, \tau, \mathbf{u} )
+k\int_{\Omega }\nabla\theta\cdot
\nabla \tau\, dx
+\alpha \int_{\Omega }\theta \tau  dx
+\beta \int_{\Gamma_1}\theta \tau ds   \\
&=\int_{\Omega }(\mu (\theta)
|\varepsilon (\mathbf{u})
|^{p}+ g (\theta)
|\varepsilon  (\mathbf{u})|)\tau \, dx
 + \int_{\Gamma_1}\upsilon (\theta)|
\mathbf{u}_{\tau }|\tau  ds \quad \forall
\tau\in W^{1,  q'}(\Omega).
\end{aligned} \label{e3.5}
\end{equation}

Now, we consider the weak nonlocal formulation to the mechanical
problem \eqref{e2.1}-\eqref{e2.3} and \eqref{e2.5}-\eqref{e2.6}
corresponding formally to the
limit model $k=\infty $ (modelling the steady-state
Herschel-Bulkley flow with temperature dependent nonlocal
consistency, yield limit and friction).

\subsection*{Problem P3}
For prescribed data $\mathbf{f}\in W'$. Find $\mathbf{u}
\in W$ and $\Theta \in \mathbb{R}_{+}$, satisfying the
variational inequality
\begin{equation}
\begin{aligned}
&B (\mathbf{u}, \mathbf{u}, \mathbf{v}-\mathbf{u} )
+\mu (\Theta ) \int_{\Omega }(|\varepsilon (
\mathbf{u})|^{p-2} \varepsilon (
\mathbf{u}))\cdot (\varepsilon (\mathbf{v})-\varepsilon
(\mathbf{u}))\,dx \\
&+ g (\Theta )\int_{\Omega }(|
\varepsilon (\mathbf{v})|
-|\varepsilon (\mathbf{u})
|)\,dx  +\upsilon (\Theta
)\int_{\Gamma_1}(|
\mathbf{v}_{\tau }|-|\mathbf{u}_{\tau
}|)\,ds   \\
&\geq \int_{\Omega }\mathbf{f}\cdot (\mathbf{v}-\mathbf{u}
)\,dx \quad \forall \mathbf{v}\in W,
\end{aligned}\label{e3.6}
\end{equation}
where $\Theta $ is a solution to the implicit scalar equation
\begin{equation}
\begin{aligned}
&(\alpha \mathop{\rm meas}(\Omega )+\beta
\mathop{\rm meas}(\Gamma_1))\Theta\\
&= \mu (\Theta )\int_{\Omega
}|\varepsilon (\mathbf{u})|^{p}\,dx
+ g (\Theta )\int_{\Omega }|\varepsilon (\mathbf{u})|\,dx
+\upsilon (\Theta ) \int_{\Gamma_1}|\mathbf{u}_{\tau }|\,ds.
\end{aligned}\label{e3.7}
\end{equation}

\section{Existence Results}

In this section we establish two existence theorems for
problems (P2) and (P3).

\begin{theorem} \label{thm4.1}
Problem {\rm (P2)} has a solution $(\mathbf{u}, \theta)$ satisfying
\begin{gather}
\mathbf{u}\in W,  \label{e4.1}\\
\theta\in W^{1, q}(\Omega ). \label{e4.2}
\end{gather}
\end{theorem}

\begin{theorem} \label{thm4.2}
There exists $(\mathbf{u}, \Theta )\in W
\times \mathbb{R}_{+}$ a solution to the nonlocal problem
{\rm (P3)}, which can be obtained as a limit in
$W\times W^{1, q}(\Omega )$
as $k\to \infty $ of solutions $(\mathbf{u}_{k}, \theta_{k})$
of problem {\rm (P2)}.
\end{theorem}

The proof of Theorem \ref{thm4.1} is based on the application of the
Kakutani-Glicksberg fixed point theorem for multivalued mappings,
using two auxiliary existence results. The first one results from
the classical theory for inequalities with monotone operators and
convex functionals. The second one results from the theory of
elliptic equations and $L^1$-Data theory.
Finally, compactness arguments are used to conclude the proofs. For
reader's convenience, let us recall the fixed point theorem \cite{f1}.

\begin{theorem}[Kakutani-Glicksberg] \label{thm4.3}
 Let $X $ be a locally convex
Hausdorff topological vector space and $K$ be a nonempty
convex compact. If $L:K\to P(K)$
is an upper semicontinuous mapping and $L(\mathbf{z})\neq \emptyset $
is a convex and closed subset in $K $ for every $\mathbf{z}\in K $, then
there exists at least one fixed point, $\mathbf{z}\in L (\mathbf{z})$.
\end{theorem}

The first auxiliary existence result is as follows.

\begin{proposition} \label{prop4.4}
For every $\mathbf{w}\in W$ and $\mathbf{\lambda }\in
W ^{1, q}(\Omega )$, there exists a
unique solution $\mathbf{u}=\mathbf{u}(\mathbf{w}, \mathbf{\lambda } )
\in W$ to the problem
\begin{equation}
\begin{aligned}
&B (\mathbf{w}, \mathbf{u}, \mathbf{v}-\mathbf{u} )
+\int_{\Omega }(\mu (\mathbf{\lambda } )|\varepsilon (\mathbf{u})|^{p-2}\varepsilon (\mathbf{u}))\cdot (\varepsilon (\mathbf{v})-\varepsilon (
\mathbf{u}))\,dx +
\phi (\mathbf{\lambda }, \mathbf{v})
-\mathbf{ \phi }(\mathbf{\lambda }, \mathbf{u})\\
&\geq \int_{\Omega}\mathbf{f}\cdot (
\mathbf{v}-\mathbf{u})\,dx \quad \forall \mathbf{v}\in W,
\end{aligned}  \label{e4.3}
\end{equation}
 and it satisfies the estimate
\begin{equation}
\| \mathbf{u}\|_{W}\leq c(\frac{\| \mathbf{f}\|_{W'}}{\mu_1})^{1/(p-1)}.
 \label{e4.4}
\end{equation}
\end{proposition}

\begin{proof}
Introducing the  functional
\[
J :L^{p}(\Omega )_{s}^{n\times
n}\subset \mathbb{S}_n\to \mathbb{R},\quad
 J (\boldsymbol{\sigma})=\int_{\Omega }\frac{\mu }{p}|
\boldsymbol{\sigma} |^{p}\,dx .
\]
This functional is convex, lower semi-continuous on
$L^{p}(\Omega )_{s}^{n\times n}$ and
G\^{a}teaux differentiable. Its G\^{a}teaux derivate at any point
$\boldsymbol{\sigma}\in L ^{p}(\Omega )_{s}^{n\times n}$ is
\[
\langle DJ(\boldsymbol{\sigma}), \boldsymbol{\eta}\rangle_{L^{p'}(
\Omega )_{s}^{n\times n}\times L^{p}(\Omega
)_{s}^{n\times n}}=\int_{\Omega }\mu |
\mathbf{ \sigma }|^{p-2}\boldsymbol{\sigma} \cdot \mathbf{\eta}\, dx \quad
 \forall \boldsymbol{\eta} \in L^{p}(\Omega )_{s}^{n\times n}.
\]
Consequently, $DJ$ is hemi-continuous and monotone. Moreover
$DJ$ is strict monotone and bounded. To this aim, we have
\begin{align*}
&\langle DJ(\boldsymbol{\sigma})
-DJ(\boldsymbol{\eta}), \boldsymbol{\sigma}-\boldsymbol{\eta}\rangle_{L^{p'}(\Omega )_{s}^{n\times
n}\times L^{p}(\Omega )_{s}^{n\times n}} \\
&\geq \int_{\Omega }\mu (|\boldsymbol{\sigma}
|-\mathbf{|\boldsymbol{\eta}|})
(|\boldsymbol{\sigma}|
^{p-1}-\mathbf{|\boldsymbol{\eta}|}^{p-1})\,dx .
\end{align*}
Then if $\boldsymbol{\sigma}\neq \boldsymbol{\eta}$, we get
$\langle DJ(\boldsymbol{\sigma})
-DJ(\boldsymbol{\eta}),  \boldsymbol{\sigma}-\boldsymbol{\eta}\rangle_{L^{p'}( \Omega )_{s}^{n\times n}\times L^{p}(\Omega
)_{s}^{n\times n}}>0$.
It means that $DJ$ is strict monotone. On the other hand, for every
$\boldsymbol{\sigma}\in L^{p}(\Omega )_{s}^{n\times n}$
\[
\int_{\Omega }|\mu |\boldsymbol{\sigma}
|^{p-2}\boldsymbol{\sigma}|^{p'}\,dx
\leq \mu_{2}^{p'}\int_{\Omega }|\boldsymbol{\sigma} |^{p}\,dx ,
\]
which proves that $DJ$ is bounded on $W$.
Now, we consider the differential operator
\begin{equation}
\begin{gathered}
 F _{\mathbf{w}}:W\to W',\quad \mathbf{u}\mapsto
F _{w}\mathbf{u} \quad \forall \mathbf{v}\in W \\
\langle  F _{\mathbf{w}}\mathbf{u}, \mathbf{v} \rangle_{W'\times
W}=B (\mathbf{w}, \mathbf{u}, \mathbf{v})
+\langle DJ(\varepsilon (\mathbf{u} )),
\varepsilon (\mathbf{v})\rangle_{L^{p'}(\Omega )_{s}^{n\times n}\times
L^{p}(\Omega )_{s}^{n\times n}}.
\end{gathered}  \label{e4.5}
\end{equation}
By  Lemma \eqref{e2.2} and the properties of $DJ$, we
deduce that $ F _{\mathbf{w}}$ is hemi-continuous, strict monotone and
bounded on $W$ for every $\mathbf{w}\in W$.
Therefore, for every $\mathbf{u}\in W$ we have
\[
\frac{\langle F_{\mathbf{w}}u, \mathbf{u} \rangle_{W'\times
W}}{\| \mathbf{u} \|_{W}}\geq
\mu_1\frac{\int_{\Omega  }|\varepsilon(\mathbf{u})|^{p} \,dx }{\|
\mathbf{u}\|_{W}}.
\]
Applying the generalized Korn inequality, we obtain
\[
\frac{\langle  F _{\mathbf{w}}\mathbf{u},  \mathbf{u}\rangle
_{W'\times W}}{\|\mathbf{u}\|_{W}}\geq \mu_1c\|
\mathbf{u} \|_{W}^{p-1}.
\]
It follows that the operator $F _{\mathbf{w}}$
is coercive on $W$ for every $\mathbf{w}\in W$.

Furthermore, the functional
$\mathbf{v}\mapsto \phi (\lambda, \mathbf{v})$ is
continuous and convex on $W$, it is then lower
semi-continuous on $W$. Consequently, the existence and
uniqueness of the solution result from the classical theorems (see
\cite{b1}) on variational inequalities with monotone operators and
convex functionals.

To prove the estimate \eqref{e4.4} we proceed as follows,
 by choosing $\mathbf{v}=0$
as test function in \eqref{e4.3}, we get
\[
\int_{\Omega }\mu (\lambda)
|\varepsilon (\mathbf{u})|^{p} \,dx
\leq \| \mathbf{f}\|_{W'}\| \mathbf{u}\|_{W}.
\]
Hence, Korn's inequality permits to conclude the proof.
\end{proof}

The second auxiliary existence result is as follows.

\begin{proposition} \label{prop4.5}
Let $\mathbf{u}=\mathbf{u}(\mathbf{w}, \mathbf{\lambda })$ be the
solution of problem \eqref{e4.3} given
by Proposition \ref{prop4.4}. Then there exists
$\theta=\theta(\mathbf{u}, \lambda )\in W^{1, q}(\Omega )$, a
solution to the problem
\begin{equation}
\begin{aligned}
&-E (\theta, \tau, \mathbf{u} )
+k\int_{\Omega }\nabla\theta\cdot\mathbf{ \nabla \tau}\, dx
+\alpha \int_{\Omega }\theta \tau  \,dx
+\beta \int_{\Gamma_1}\theta \tau \, ds   \\
&=\int_{\Omega }(\mathbf{\mu (\lambda)
} |\varepsilon (\mathbf{u})
|^{p}+ g (\mathbf{\lambda })
|\varepsilon  (\mathbf{u}) |)\tau \, dx
 +\int_{\Gamma_1}\upsilon (\mathbf{\lambda })|
\mathbf{u}_{\tau }|\tau \, ds\quad
\forall \tau\in W^{1,  q'}(\Omega),
\end{aligned} \label{e4.6}
\end{equation}
and satisfies the estimate
\begin{equation}
\alpha \| \theta\|
_{L^{q}(\Omega)}+\beta \|
\theta\|_{L ^{q}(\Gamma )}+\sqrt{k}\| \nabla\theta\|
_{L^{q}(\Omega )^n}
\leq \Re (\upsilon_0, \mu_1, \|\mathbf{f} \|_{W'}),  \label{e4.7}
\end{equation}
where $\Re $ is a positive function.
\end{proposition}

\begin{proof}
 There is a technical difficulty in the resolution of such problem.
To this aim we introduce the following approximate problem
\begin{equation}
\begin{aligned}
&-E (\theta_{m}, \tau,  \mathbf{u})+k\int_{\Omega }\nabla\theta_{m}\cdot \nabla\tau \, dx
+\alpha \int_{\Omega }\theta_{m} \tau \, dx
+\beta \int_{\Gamma_1}\theta_{m} \tau  \,ds   \\
&=\int_{\Omega }F_{m}\tau \, dx+\int_{\mathbf{
\Gamma }_1}\upsilon (\mathbf{\lambda })
|\mathbf{u}_{\tau }|\tau \,ds\quad \forall
 \tau \in H^1(\Omega ),
\end{aligned} \label{e4.8}
\end{equation}
where
\begin{equation}
F_{m}=\frac{m\big[ \mu (\lambda) |\varepsilon (\mathbf{u})|^{p}
+ g ( \mathbf{\lambda })|\varepsilon (\mathbf{u})|\big]}
{m+\mu (\lambda )|\varepsilon (\mathbf{u})|^{p}
+g (\mathbf{\lambda })|\varepsilon ( \mathbf{u})|}
\in L ^{\infty }(\Omega ).  \label{e4.9}
\end{equation}
Let us consider for every $\mathbf{u}\in W$ the form
$G :H^1(\Omega )\times H^1(\Omega )\to \mathbb{R}$,
\begin{equation}
G (\theta, \tau)
=-E  (\theta, \tau, \mathbf{u})+k\int_{\Omega }\nabla\theta\cdot
\nabla\tau  dx
+\alpha \int_{\Omega }\theta \tau  dx
+\beta \int_{ \Gamma_1}\theta \tau  ds.
\label{e4.10}
\end{equation}
Lemma \ref{lem2.2} and the Poincar\'{e} type inequality affirm
that $G $ is bilinear, continuous and coercive on
$H^1(\Omega )\times H^1(
\Omega )$ for every $\mathbf{u}\in W$. Furthermore, by
H\"{o}lder's inequality and Sobolev's trace inequality using the
estimate \eqref{e4.4}, we get
\[
|\int_{\Gamma_1}\upsilon (
\lambda )|\mathbf{u}_{\tau }|\tau \,ds |
\leq c\| \tau \|_{H ^1(\Omega )}.
\]
Consequently, from the Lax-Milgram theorem, there exists a unique
solution
$\theta_{m}\in H^1(\Omega)$ to the
problem \eqref{e4.8}.

Now, we test the apprixamte equation \eqref{e4.8} by the function
\begin{equation}
\tau=\mathop{\rm sign}(\theta_{m})
[ 1-\frac{ 1}{(1+|\theta_{m}|)^{\xi }}] \in
H^1(\Omega )\cap L ^{\infty }(\Omega ),\quad \xi >0.
 \label{e4.11}
\end{equation}
We find by using some integration by parts (see for instance
\cite{d1})
\begin{equation}
\xi k\int_{\Omega }\frac{|\nabla\theta
_{m}|^{2}}{(1+|\theta_{m}|)^{\xi +1}}\,dx
+\beta C(\xi )\int_{\mathbf{ \Gamma }_1}|\theta_{m}|\,ds\leq M,
\label{e4.12}
\end{equation}
where $M=M(\upsilon_0, \mu_1, c, \| f\|_{W'})$
is a positive function.
Particularly
\begin{equation}
\int_{\Omega }\frac{|\nabla\theta
_{m}|^{2}}{(1+|\theta_{m}|)^{\xi +1}}\,dx
\leq \frac{M}{\xi k}.  \label{e4.13}
\end{equation}
Denoting by $\gamma$ the function
\[
\gamma(r)=\int_0^{r}\frac{dt}{(
1+|t|)^{\tfrac{\xi +1}{2}}}.
\]
Then
\[
\nabla\gamma(\theta_{m})
=\frac{\nabla \theta _{m}}
{(1+| \theta_{m}|)^{(\xi +1)/2}}.
\]
We deduce from \eqref{e4.12} that $\nabla\gamma(\theta_{m})$
is bounded in $L^{2}(\Omega )$, hence $\gamma(\theta_{m})$
is bounded in $H^1(\Omega )$. Sobolev's imbedding asserts
 that $H^1(\Omega)\subset L^{\rho }(\Omega )$, where
$\rho =\frac{2n}{n-2}$ if $n\neq 2$ and $2<\rho <+\infty $ if
$n=2$.

Keeping in mind that $\gamma(r)\sim r^{\tfrac{1-\xi }{ 2} }$
as $r\to +\infty $. Then $|\theta_{m}|^{\tfrac{1-\xi}{2}}$ is bounded
in $L^{\rho }(\Omega )$.
Consequently
\begin{equation}
|\theta_{m}|^{\rho(1-\xi )/2}
\quad \text{is bounded in }L^1(\Omega ).  \label{e4.14}
\end{equation}
Moreover, by H\"{o}lder's inequality,
\[
\int_{\Omega }|\nabla\theta_{m}|^{q}\,dx
\leq \Big(\int_{\Omega }\frac{|
 \nabla \theta _{m}|^{2}}{(1+|\theta_{m}|)^{\xi
+1}}\,dx \Big)^{q/2}
\Big(\int_{\Omega }(1+|\theta_{m}|)
^{(\xi +1)q/(2-q)}\,dx \Big)^{(2-q)/2}.
\]
Hence, from \eqref{e4.13}, we obtain
\begin{equation}
\int_{\Omega }|\nabla\theta_{m}|^{q}\,dx
\leq (\frac{M}{k\xi })^{q/2}
\Big(\int_{\Omega }(1+|\theta_{m}|)^{(\xi +1)q/(2-q)}\,dx \Big)^{(2-q)/2}.
\label{e4.15}
\end{equation}
Let us choose the couple $(\xi , q)$ such that
$\frac{ \rho (1-\xi )}{2}=\frac{(\xi+1)q}{2-q}$.
It means that $q=\frac{2\rho (1-\xi)}{\rho (1-\xi )+2(1+\xi )}$.
Then if $1<q<\frac{n}{n-1}$, we can choose $0<\xi <\frac{\rho
-2}{\rho +2}$.
Consequently, by using \eqref{e4.14} and \eqref{e4.15},
the following estimate holds
\begin{equation}
\theta_{m}\quad \text{is bounded in }W^{1, q}(\Omega ).  \label{e4.16}
\end{equation}
Combining this with \eqref{e4.12}, we can extract a subsequence
$(\theta_{\mu })_{\mu }$, satisfying
\begin{gather}
\theta_{\mu}\to \theta \quad \text{in }W ^{1, q}(\Omega )\text{ weakly},
\label{e4.17}\\
\theta_{\mu }\to \theta\quad \text{in }L ^{q}(\Gamma)\text{ weakly}.
\label{e4.18}
\end{gather}

Recall that Rellich-Kondrachof's theorem affirms the compactness of the
imbedding $W^{1, q}(\Omega )\to L^1(\Omega )$. It follows
that we can extract a subsequence of $\theta_{\mu }$,
still denoted by $\theta_{\mu }$ such that
\begin{gather}
\theta_{\mu }\to \theta\quad \text{in }L ^1(\Omega )\text{ strongly},
 \label{e4.19} \\
\theta_{\mu }\to \theta\quad \text{in }L ^1(\Gamma )\text{ strongly}.
\label{e4.20}
\end{gather}
We conclude that problem \eqref{e4.6} admits a solution
$\theta=\theta (\mathbf{u}, \mathbf{\lambda })\in W^{1,  q}(\Omega )$.
Using \eqref{e4.12}, \eqref{e4.14} and \eqref{e4.15},
the estimate \eqref{e4.7} follows immediately.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm4.1}]
To apply the Kakutani-Glicksberg fixed point theorem, let us
consider the closed convex ball
\begin{equation}
K = \{ (\mathbf{w}, \mathbf{\lambda})\in W\times W^{1, p}(\Omega ):
\| \mathbf{w}\|_{W}\leq R_1, \| \mathbf{\lambda }\|_{W^{1, q}(\Omega )}
\leq R_{2}\} ,  \label{e4.21}
\end{equation}
where $R_1\geq c(\frac{\| \mathbf{f}\|_{W '}}{\mu_1})^{\tfrac{1}{p-1}}$ and
$R_{2}$ is given by the estimate \eqref{e4.14}. The ball $K $ is
compact when the topological vector space is provided by the weak
topology. Let us built the mapping
$L :K \to P(K)$, as follows
\[
(\mathbf{w}, \mathbf{\lambda })\mapsto
L (\mathbf{w}, \mathbf{\lambda
})=\{ (\mathbf{u}, \theta)\} \subset K.
\]
For every $(\mathbf{w}, \mathbf{\lambda })\in
K$, equation \eqref{e4.6} is linear with respect to
$\theta $, and the solution $\mathbf{u}$ is unique.
Consequently the set $L(\mathbf{w}, \mathbf{\lambda })$ is convex. To
conclude the proof it remains to prove the closeness in
$K \times K $ of the graph set
\[
G (L )=\{ (
(\mathbf{ w}, \mathbf{\lambda }), (\mathbf{u},  \theta))\in
K \times K :(\mathbf{u}, \theta)\in L (
\mathbf{w}, \mathbf{\lambda })\} .
\]
To do so, we consider a sequence
$(\mathbf{w}_n, \lambda _n)\in K$, such that
$(\mathbf{w}_n, \mathbf{\lambda }_n)
\to (\mathbf{w},  \mathbf{\lambda
})$ in $W\times W^{1,  q}(\Omega )$ weakly and
$(\mathbf{u}_n,\theta_n)\in L (\mathbf{w}_n,  \mathbf{\lambda }_n)$.
Let us remember that $(\mathbf{ u}_n, \theta_n)$ is solution to
the problem
\begin{gather}
\begin{aligned}
&B (\mathbf{w}_n, \mathbf{u}_n, \mathbf{v}- \mathbf{u}_n)
+\int_{\Omega }(\mu (\mathbf{\lambda }_n)|
\varepsilon (\mathbf{u }_n)|
^{p-2}\varepsilon (\mathbf{u}_n))
\cdot (\varepsilon (\mathbf{v} )
-\varepsilon (\mathbf{u}_n))\,dx\\
&+ \phi (\mathbf{\lambda }_n, \mathbf{v})- \phi (\mathbf{\lambda
}_n, \mathbf{u}_n)\\
&\geq \int_{\Omega}\mathbf{f}\cdot (\mathbf{v}-\mathbf{u}_n)
\,dx \quad  \forall \mathbf{v}\in W,
\end{aligned} \label{e4.22}
\\
\begin{aligned}
&-E (\theta_n, \tau,  \mathbf{u}_n)
+k\int_{\Omega }\nabla\theta _n\cdot \nabla\tau \, dx
+\alpha \int_{\Omega }\theta _n\tau \, dx
+\beta \int_{\Gamma_1}\theta _n\tau \,ds   \\
&=\int_{\Omega }(\mu (\mathbf{\lambda }
_n)|\varepsilon (
\mathbf{u}_n)|^{p}+g (
\mathbf{\lambda }_n)|\varepsilon (\mathbf{u}_n)|) \tau \, dx\\
&\quad + \int_{\Gamma_1}\upsilon (\mathbf{\lambda }_n)|(
\mathbf{u}_n)_{\tau }| \tau \,ds\quad
\forall \tau\in W^{1,  q'}(\Omega ).
\end{aligned} \label{e4.23}
\end{gather}
Then, from Propositions \ref{prop4.4} and \ref{prop4.5},
\[
\| \mathbf{u}_n\|_{W}\leq R_1\quad\text{and}\quad
\| \theta_n\|_{W^{1, q}(\Omega )}\leq R_{2}.
\]
Thus, we can extract a subsequences $\mathbf{u}_{\mu }$ and
$\theta_{\mu }\ $such that
\begin{gather}
\mathbf{u}_{\mu }\to \mathbf{u}\quad \text{in }W\text{ weakly},
 \label{e4.24}\\
\theta_{\mu }\to \theta\quad \text{in }W ^{1, q}(\Omega )\text{ weakly}.
\label{e4.25}
\end{gather}
It follows from Rellich-Kondrachof's theorem and Sobolev's
trace theorem, that we can extract a subsequences of
$\lambda_{\mu }$, $\mathbf{u}_{\mu }$ and
$\theta_{\mu }$, still denoted by $\lambda _{\mu }$, $\mathbf{u}_{\mu }$ and $\mathbf{\theta_{\mu }}$, such
that
\begin{gather}
\mathbf{w}_{\mu }\to \mathbf{w}\quad \text{in }L^{s}(\Omega )^n
\text{ strongly and a.e. in }\Omega , \label{e4.26} \\
\mathbf{\lambda }_{\mu }\to \mathbf{\lambda }\quad \text{in }
\mathbf{ L}^1(\Omega )\text{ strongly and a.e. in }\Omega,
 \label{e4.27}
\\
\mathbf{u}_{\mu }\to \mathbf{u}\quad \text{in }L^{s}(\Omega )^n
\text{ strongly and a.e. in }\Omega , \label{e4.28}
\\
\theta_{\mu }\to \theta\quad \text{in }L ^1(\Omega )
\text{ strongly and a.e. in }\Omega,  \label{e4.29}
\\
\mathbf{u}_{\mu }\to \mathbf{u}\quad \text{in }L^{r}(\Gamma )^n
\text{ strongly and a.e. on }\Gamma , \label{e4.30}
\\
\theta_{\mu }\to \theta\quad \text{in }L ^1(\Gamma )\text{ strongly and
a.e. on }\Gamma,  \label{e4.31}
\end{gather}
where $n\leq s<\frac{2n}{n-2}$ and $2\leq r<\frac{2(n-1)}{n-2} $.

Now we prove  that
$\varepsilon (\mathbf{u}_{\mu})\to \varepsilon (\mathbf{u})$ a.e.
in $\Omega $. To do so, we proceed as
follows. Introducing the positive function
\begin{equation}
\begin{aligned}
h_{\mu }(x)
&=[\mu (\lambda _{\mu }(x))
|\varepsilon (\mathbf{u}_{\mu }(
x))|^{p-2}\varepsilon (\mathbf{u}_{\mu }(x))\\
&\quad -   \mu (\mathbf{\lambda }_{\mu }(
x))|\varepsilon (
\mathbf{u}(x))|^{p-2}\varepsilon (\mathbf{u}(
x))]\cdot (\varepsilon (\mathbf{u }_{\mu }(x))
-\varepsilon (\mathbf{u}(x))).
\end{aligned} \label{e4.32}
\end{equation}
Then
\begin{equation}
\begin{aligned}
\int_{\Omega }h_{\mu }(x)\,dx
& \leq f(\mathbf{u}_{\mu}-\mathbf{u})+B (\mathbf{w}_{\mu },
\mathbf{u}_{\mu }, \mathbf{v}-\mathbf{u}_{\mu })
+\phi (\mathbf{\lambda }_{\mu },
\mathbf{u})-\phi (\mathbf{\lambda }_{\mu},
\mathbf{u}_{\mu })\\
&\quad -  \int_{\Omega }\mu (\mathbf{\lambda }_{\mu }(
x))|\varepsilon (\mathbf{u} (x))|
^{p-2}\varepsilon  (\mathbf{u}(
x))\cdot (\varepsilon (\mathbf{u}_{\mu }(x))-
\varepsilon (\mathbf{u}(x)))\,dx .
\end{aligned} \label{e4.33}
\end{equation}
We know from Lemma \ref{lem2.2} that
\[
B (\mathbf{w}_{\mu }, \mathbf{u}_{\mu
}, \mathbf{ v})=-B (\mathbf{w}, \mathbf{v}, \mathbf{u}_{\mu })+B (
\mathbf{w}-\mathbf{w}_{\mu },  \mathbf{v}, \mathbf{u}_{\mu }).
\]
Then,  Lebesgue's dominated convergence theorem applied to
the second term on the right-hand side yields  convergence
\begin{equation}
B (\mathbf{w}_{\mu }, \mathbf{u}_{\mu}, \mathbf{ v})\to B (
\mathbf{w}, \mathbf{u},  \mathbf{v}).
\label{e4.34}
\end{equation}
On the other hand, since $\mathbf{\lambda_{\mu }}\to
\mathbf{\lambda }$ a.e. in $\Omega $ and on $\Gamma$,
the functions $g$ and $\upsilon$ are continuous
and due to the weak lower semicontinuity of the continuous and
convex functional $\mathbf{v}\mapsto \mathbf{\phi (\mathbf{\lambda },
\mathbf{v })}$, combined with convergence result \eqref{e4.24},
we deduce from the Lebesgue dominated convergence theorem that
\begin{gather}
\liminf \phi (\mathbf{\lambda }_{\mu }, \mathbf{u}_{\mu })
\geq \phi (\mathbf{\lambda }, \mathbf{u} ),  \label{e4.35}\\
\lim \phi (\mathbf{\lambda }_{\mu }, \mathbf{u})
=\phi (\mathbf{\lambda }, \mathbf{u}).\label{e4.36}
\end{gather}
Since $\mathbf{\lambda_{\mu }}\to \mathbf{\lambda }$
a.e. in $\Omega $ and on $\Gamma$, the function
$\mu $ is continuous and due to \eqref{e4.24} and the fact that
$|\varepsilon (\mathbf{u}(
x))|^{p-2}\varepsilon (\mathbf{u}(x))$ is bounded in
$L^{p'}(\Omega )_{s}^{n\times n}$,
we obtain by the Lebesgue dominated convergence
theorem that
\begin{equation}
\int_{\Omega }\mu (\mathbf{\lambda }_{\mu }(
x))|\varepsilon (\mathbf{u} (x))|
^{p-2}\varepsilon  (\mathbf{u}(x))(\varepsilon (
\mathbf{u}_{\mu }(x))-\varepsilon (\mathbf{u}(x))
)\,dx \to 0.  \label{e4.37}
\end{equation}
Consequently, \eqref{e4.33}, \eqref{e4.34}, \eqref{e4.35}, \eqref{e4.36}
and \eqref{e4.37} give
\begin{equation}
\lim \| h_{\mu }\|
_{L^1(\Omega)}=0\quad \text{and}\quad
h_{\mu }\to 0\quad \text{a.e.}  \label{e4.38}
\end{equation}
Furthermore, $h_{\mu }(x)$ can be rewritten as
\begin{equation}
\begin{aligned}
h_{\mu }(x)
& =\mu (\lambda _{\mu }(x))
|\varepsilon (\mathbf{u}_{\mu }(
x))|^{p}-\mu (\mathbf{\lambda }_{\mu }(\mathbf{x }))
|\varepsilon (\mathbf{u}_{\mu
}(x))|^{p-2}\varepsilon
(\mathbf{u}_{\mu }(x))\cdot \varepsilon (\mathbf{u}(x)) \\
&\quad -\mu (\mathbf{\lambda }_{\mu }(
x))|\varepsilon (\mathbf{u}(x ))|
^{p-2}\varepsilon (\mathbf{u} (x))\cdot (\varepsilon (
\mathbf{u}_{\mu }(x))
-\varepsilon  (\mathbf{u}(x))),
\end{aligned}  \label{e4.39}
\end{equation}
which proves, using the estimate \eqref{e4.4}, that
\begin{align*}
\mu (\mathbf{\lambda }_{\mu }(x))|\varepsilon (
\mathbf{u}_{\mu }(x ))|^{p}
&\leq h_{\mu }(x )+c\mu (\mathbf{\lambda }_{\mu }(x ))
|\varepsilon (\mathbf{u}_{\mu
}(x))|^{p-1}\\
&\quad + c\mu (\mathbf{\lambda }_{\mu }(x)
)|\varepsilon (\mathbf{u}_{\mu }(x))|+c.
\end{align*}
It follows that $(\varepsilon (\mathbf{u}_{\mu }(x)))_{\mu }$
is bounded in $\mathbb{R}^{n\times n}$, we can then extract a
subsequence still denoted by
$(\varepsilon (\mathbf{u}_{\mu }(x)))_{\mu }$, that
converges to $\mathbf{\xi }\in \mathbb{R}^{n\times n}$. By passage
to the limit in $h_{\mu }$, we deduce that
\[
(\mu (\lambda)|
\mathbf{\xi } |^{p-2}\mathbf{\xi }-\mu (\lambda)|\varepsilon (
\mathbf{u}(x))|^{p-2}\varepsilon (\mathbf{u}(
x)))\cdot (\mathbf{\xi
}-\varepsilon (\mathbf{u}(x)))=0.
\]
Then $\varepsilon (\mathbf{u}(x))=\mathbf{\xi}$.
We conclude that
\begin{equation}
\varepsilon (\mathbf{u}_{\mu })
\to \varepsilon (\mathbf{u})\quad\text{ a.e. in }\Omega . \label{e4.40}
\end{equation}
Therefore, the sequence $(\mu (\mathbf{\lambda}_{\mu }(x))|
\varepsilon (\mathbf{u}_{\mu }(x))|^{p-2}
\varepsilon (\mathbf{u}_{\mu }(x)))_{\mu }$ converges a.e.
in $\Omega $ to $\mu  (\mathbf{\lambda }(x))|
\varepsilon (\mathbf{u}(x))|^{p-2}\varepsilon
(\mathbf{u}(x))$. Moreover, this
sequence is bounded in $L^{p'}(\Omega)_{s}^{n\times n}$,
then the $L^{p}-L^{q}$ compactness theorem (see \cite{d1,l1})
implies the convergence in $L^{r}(\Omega )_{s}^{n\times n}$ for
every $1<r<p'$.
By choosing $\varphi \in D(\Omega )^n$ as test
function in inequality \eqref{e4.22},
\begin{equation}
\begin{aligned}
&B (\mathbf{w}, \mathbf{u}, \boldsymbol{\varphi}
- \mathbf{u})+\int_{\Omega }\mu (\lambda )
|\varepsilon (\mathbf{u})|^{p-2}\varepsilon (\mathbf{u})
\cdot \varepsilon (\boldsymbol{\varphi})\,dx
+\mathbf{ \phi }(\mathbf{\lambda }, \boldsymbol{\varphi})
-\mathbf{f}\cdot (\boldsymbol{\varphi}-\mathbf{u}) \\
&\geq \int_{\Omega }\mu (\mathbf{\lambda }_{\mu
})|\varepsilon (\mathbf{u}_{\mu})|^{p}\,dx +\phi (
\mathbf{\lambda }_{\mu },  \mathbf{u}_{\mu }).
\end{aligned}\label{e4.41}
\end{equation}
Using  \eqref{e4.35}, the fact that $\mathbf{\lambda_{\mu
}}\to \mathbf{\lambda }$ a.e. in $\Omega$,
the continuity of $\mu$ and $g $, the weak
lower semicontinuity of the norm $\|\cdot\|_{W^{1, p}(\Omega)^n}$.
We conclude that $\mathbf{u}$ is solution to \eqref{e4.3}.

Our final goal is to show that $\theta$ is solution of
\eqref{e4.23}. To do so, we proceed as follows. Introducing the function
\begin{equation}
\mathbf{\chi }_{\mu }(x)=\mu (
\lambda _{\mu }(x))
|\varepsilon (\mathbf{u}_{\mu }(
x))|^{p}+g (
\mathbf{\lambda }_{\mu }(x ))
|\varepsilon (\mathbf{u}_{\mu }(
x))|.  \label{e4.42}
\end{equation}
 From \eqref{e4.40}, we remark that $\mathbf{\chi }_{\mu }\to
\mathbf{\chi}$ a.e. in $\Omega $, where
\begin{equation}
\mathbf{\chi }(x)=\mu (
\mathbf{\lambda } (x))|
\varepsilon (\mathbf{u}(x)
)|^{p}+g (\mathbf{\lambda }(
x))|\varepsilon (\mathbf{u}(x))|.
\label{e4.43}
\end{equation}
Substituting in  \eqref{e4.22} and taking $\mathbf{v}=\mathbf{u}$ as
test function, the passage to limit, using Lebesgue's dominated
convergence theorem, gives
\begin{equation}
\lim \big[ \int_{\Omega }\mathbf{\chi }_{\mu }(x
)\,dx +\int_{\Gamma_1}\upsilon (\mathbf{\lambda }_{\mu })|(
\mathbf{u}_{\mu })
_{\tau }|\,ds\big]
\leq \int_{\Omega }\mathbf{\chi }(x)\,dx+\int_{\Gamma_1}\upsilon (
\mathbf{\lambda } )|\mathbf{u}_{\tau }|
\,ds.  \label{e4.44}
\end{equation}
On the other hand, we know from the weak lower semicontinuity of the norm
$\|\cdot\|_{W^{1, p}(\Omega)^n}$ and the functional
$\mathbf{v}\mapsto \mathbf{\phi (\mathbf{\lambda }, \mathbf{v})}$, that
\begin{equation}
\liminf \big[ \int_{\Omega }\mathbf{\chi }_{\mu }(
x)\,dx +\int_{\Gamma_1}\upsilon (\mathbf{\lambda }_{\mu })
|(\mathbf{u}_{\mu }) _{\tau }|\,ds\big]
\geq \int_{\Omega }\mathbf{\chi }(x)\, dx
+\int_{\Gamma_1}\upsilon (\mathbf{\lambda } )|\mathbf{u}_{\tau }|
\,ds.  \label{e4.45}
\end{equation}
We deduce from \eqref{e4.44} and \eqref{e4.45} that
\[
\lim \big[ \int_{\Omega }\mathbf{\chi }_{\mu }(x
)\,dx +\int_{\Gamma_1}\upsilon (\mathbf{\lambda }_{\mu })|(
\mathbf{u}_{\mu })_{\tau }|\,ds\big]
=\int_{\Omega }\mathbf{\chi }(x)
\,dx +\int_{\Gamma_1}\upsilon (\mathbf{\lambda })|\mathbf{u}_{\tau }|
\,ds,
\]
which implies, using the continuity of the injection
$W^{1,  q'}(\Omega )\to C(\bar{\Omega})$ and
the Lebesgue dominated convergence theorem, that for every
$\tau\in W^{1, q'}(\Omega)$,
\begin{equation}
\begin{aligned}
&\lim \big[ \int_{\Omega }(\mu (\lambda _{\mu })|
\varepsilon (\mathbf{u}_{\mu })|^{p}+g (\mathbf{\lambda }_{\mu })|
\varepsilon (\mathbf{u}_{\mu })|
)\tau \, dx+\int_{\Gamma_1}\upsilon (\mathbf{\lambda }_{\mu })|(
\mathbf{u}
_{\mu })_{\tau }|\tau \,ds\big]   \\
&=\int_{\Omega }(\mu (\lambda)
|\varepsilon (\mathbf{u})
|^{p}+ g (\mathbf{\lambda })
|\varepsilon  (\mathbf{u})|)\tau \, dx
+\int_{\Gamma_1}\upsilon (\mathbf{\lambda })|
\mathbf{u}_{\tau }|\tau \,ds.
\end{aligned}\label{e4.46}
\end{equation}
Thus, we conclude that $\theta$ is solution to  \eqref{e4.23}.
Hence, $(\mathbf{u}_n, \theta_n)
\to (\mathbf{u}, \theta)
\in L (\mathbf{w}, \mathbf{\lambda})$ in
$W\times W^{1, q}(\Omega )$ weakly. By virtue of
Kakutani-Glicksberg's fixed point theorem, the mapping
$L $ admits a fixed point
$(\mathbf{u}, \theta)\in L (\mathbf{u}, \theta)$, which solves
 problem (P2).
\end{proof}

\begin{remark} \label{rmk4.6} \rm
This proof permits also to verify the continuous dependence of the
solution $(\mathbf{u}(\mathbf{w}, \mathbf{\lambda }),  \theta(
\mathbf{u}, \mathbf{\lambda }))\in
W\times W^{1, q}(\Omega )$
of problem \eqref{e4.3}, \eqref{e4.6} with respect to the function
$(\mathbf{w},  \mathbf{\lambda })\in W\times W^{1,  q}(\Omega )$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm4.2}]
Let $(\mathbf{u}_{k}, \theta_{k})$
be a solution to the problem (P2), corresponding to each
$k>0$ and let $k\to +\infty $. From the estimates
\eqref{e4.4}, \eqref{e4.7} and using Rellich-Kondrachof's theorem, we can
extract a subsequence of $(\mathbf{u}_{k}, \theta_{k})$, still
denoted by $(\mathbf{u}_{k}, \theta_{k})$, satisfying
\begin{gather*}
\mathbf{u}_{k}\to \mathbf{u}\quad \text{in $W$ weakly},\\
\mathbf{u}_{k}\to \mathbf{u}\quad \text{in $L^{s}(\Omega )^n$ strongly}, \\
\nabla\theta_{k}\to 0\quad \text{in $L^1(\Omega )^n$ strongly}, \\
\theta_{k}\to \Theta =\text{a constant}\quad
\text{in $L ^1(\Omega )$ strongly},
\end{gather*}
where $n\leq s<2n/(n-2)$. We can proceed as in the proof of
Theorem \ref{thm4.1} to get the convergence
\begin{equation}
\begin{aligned}
&\lim \big[ \int_{\Omega }\mu (\theta
_{k})|\varepsilon (\mathbf{u}_{k})|^{p}\tau \, dx
+\int_{\Omega }g (\theta_{k})
|\varepsilon (\mathbf{u}_{k})
|\tau \, dx+\int_{\Gamma_1}\upsilon (\theta_{k})|(
\mathbf{u}_{k})_{\tau }|\tau \,ds\Big]   \\
&=\mu (\Theta )\int_{\Omega }|
\varepsilon (\mathbf{u})|
^{p}\mathbf{\tau dx }+g (\Theta )\int_{\Omega
}|\varepsilon (\mathbf{u})
|\tau \, dx+\upsilon (\Theta
)\int_{\Gamma_1}|\mathbf{u}_{\tau
}|\tau \,ds.
\end{aligned} \label{e4.47}
\end{equation}
Then, we can pass to the limit $k\to +\infty $ in
\eqref{e3.5} and taking $\tau=1$ to obtain the implicit scalar
equation \eqref{e3.7}. Now, taking the limit $k\to +\infty $
in \eqref{e3.4}, it follows that $\mathbf{u}$ solves the nonlocal
inequality \eqref{e3.6}.
Moreover, the scalar equation \eqref{e3.7} asserts that
$\Theta \geq 0$.
\end{proof}


\begin{thebibliography}{00}

\bibitem{b1} H. Brezis;
 \emph{Equations et In\'{e}quations
Non Lin\'{e}aires dans les Espaces en Dualit\'{e}}, Annale de l'Institut
Fourier, Tome 18, No.1, (1968), p. 115-175.

\bibitem{b2} M. Bul\'{\i}\v{c}ek, P. Gwiazda, J. M\'{a}lek
and A. \'{S}wierczewska;
 \emph{On steady Flows of an Incompressible Fluids
with Implicit Power-Law-Like Rheology}, Advances of Calculus of Variation,
(2009)

\bibitem{c1}  L. Consiglieri and J. F. Rodrigues;
 \emph{A Nonlocal Friction Problem for a Class of Non-Newtonian Flows},
Portugaliae Mathematica 60:2 (2003), 237-252.

\bibitem{d1} J. Droniou et C. Imbert;
\emph{Solutions de Viscosit\'{e} et Solutions Variationnelle pour
EDP Non-Lin\'{e}aires}, Cours de DEA, D\'{e}partement de
Math\'{e}matiques, Montpellier II, (2002).

\bibitem{d2} G. Duvaut et J. L. Lions;
 \emph{Transfert de la Chaleur dans un Fluide de Bingham dont
la Viscosit\'{e} D\'{e}pend de la Temp\'{e} rature.}
Journal of Functional Analysis 11 (1972), 85-104.

\bibitem{d3} G. Duvaut et J. L. Lions;
\emph{Les  In \'{e}quations  en  M\'{e}canique et  en Physique},
Dunod (1976).

 \bibitem{f1} Fan, Ky;
 \emph{Fixed Point and Min-max Theorems in Locally Convex Topological
Linear Spaces}, Proc Natl. Acad. Sci. U S A. (1952), 38(2): 121--126.

\bibitem{f2} H. Fellouah, C. Castelain, A. Ould
El Moctar et H. Peerhossaini; \emph{Instabilit\'{e} dans un
\'{e}coulement de fluides complexes en canal courbe}, 16\`{e}me
Congr\`{e}s Fran\c{c}ais de M\'{e} canique, Nice, 1-5 septembre
2003.

\bibitem{g1} P. N. Godbole and O. C. Zienkiewicz;
 \emph{ Flow of Plastic and Viscoplastic Solids with
Special Reference to Extension and Forming Processes}, Int, J,
Num. Eng, 8, (1976), p. 3-16.

\bibitem{k1} S. Kobayashi and N. Robelo;
 \emph{A Coupled Analysis of Viscoplastic Deformation and Heat
Transfert: I Theoretical Consideration, II Applications,}
Int, J. of Mech. Sci, 22, 699-705, 707-718, (1980).

\bibitem{l1}  J. L. Lions; \emph{Quelques M\'{e}thodes de
R\'{e}solution des Probl\`{e}mes Aux Limites Non Lin\'{e}aires},
 Dunod (1969).

\bibitem{l2} J. L. Lions et E. Magenes;
\emph{Probl\`{e}mes aux Limites Non Homog\`{e}nes et
Applications}, Volume I, Dunod (1968).

\bibitem{m1}  J. M\'{a}lek, M. R\r{u}\v{z}i\v{c}ka, and V. V.
Shelukhin; \emph{Herschel-Bulkley fluids,Existence and Regularity of
SteadyFlows}, Math. Models Methods Appl. Sci. 15, 12 (2005), 1845--1861.

\bibitem{m2}  F. Messelmi, B. Merouani and M. Meflah F;
\emph{Nonlinear Thermoelasticity Problem}, Analele of University
Oradea, Fasc. Mathematica, Tome XV (2008), 207-217.

\bibitem{m3} 14] B. Merouani, F. Messelmi and S. Drabla;
\emph{Dynamical Flow of a Bingham Fluid with Subdifferential Boundary Condition},
Analele of University Oradea, Fasc. Mathematica, Tome XVI (2009), 5-30.

\bibitem{p1} Papanastasiou;
\emph{Flow of Materials with Yield}, J. of Rheology. 31(5), 385-404 (1987).

\bibitem{s1}  M. Selmani, B. Merouani and L. Selmani;
\emph{Analysis of a Class of Frictional Contact Problems for the
Bingham Fluid},  Mediterr.  J. Math 1-12, (2004).

\end{thebibliography}

\end{document}