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Asymptotic behavior of solutions to a boundary value problem with mixed boundary conditions and friction law

Boundary Value Problems20172017:55

https://doi.org/10.1186/s13661-017-0791-z

©  The Author(s) 2017

Received: 20 December 2016

Accepted: 9 April 2017

Published: 20 April 2017

Abstract

In this paper, we consider a non-linear problem in a stationary regime in a three-dimensional thin domain \(\Omega^{\varepsilon}\) with Fourier and Tresca boundary conditions. In the first step, we derive a variational formulation of the mechanical problem. We then study the asymptotic behavior in the one dimension case when the domain parameter tends to zero. In the latter case, a specific Reynolds equation associated with variational inequalities is obtained and the uniqueness of the limit velocity and pressure are proved.

Keywords

asymptotic approach Bingham fluid coupled problem Fourier law Tresca law variational inequality weak solution

MSC

35R35 76F10 78M35

1 Introduction

The asymptotic behavior of Bingham fluid has been studied by many authors, see for instance [1], where the analysis of the Bingham fluid flow variational inequality was carried out, and where the authors investigated the existence, uniqueness and regularity of the solution for the steady and in-stationary flows in a reservoir. Existence and extra regularity results for the d-dimensional Bingham fluid flow problem with Dirichlet boundary conditions were studied in [2, 3].

In [4], the author has addressed the asymptotic behavior of a Bingham fluid in a thin domain. Unfortunately in this work a complete characterization was not given because of the difficulty encountered due to the choice of the test functions and of the imposed boundary conditions.

The authors in [5], studied the same problem, in which only the Dirichlet conditions on the boundary have been considered.

Along the same lines, the authors in [6] have proved the asymptotic analysis of an isothermal Bingham fluid in a thin domain with non-linear Tresca boundary conditions.

The asymptotic behavior of a coupled system involving an incompressible Bingham fluid and the equation of the heat energy, in a three-dimensional bounded domain with Tresca free boundary friction conditions were investigated in [7].

The numerical solution of the stationary Bingham fluid flow problem were studied in [8–10]. A variety of work has been done on the mechanical contact with the various laws of behavior and various friction boundary conditions close to our problem; however, these papers were restricted only to the results of existence and uniqueness of the weak solution under several assumptions. Let us mention for example the work by [11] in which the authors worked the contact problems with friction and adhesion for electro-elastic-visco-plastic materials. The existence and stability results of solitary-wave solutions to coupled non-linear Schrödinger equations with power-type non-linearities arising in several models of modern physics were studied in [12]. The authors in [13] have studied a symmetric, non-linear eigenvalue problem arising in earthquake initiation, and proved the existence of infinitely many solutions. The new variational principles for solving extended Dirichlet-Neumann problems is given in [14]. The dynamic evolution with frictional contact of an elastic body was studied in [15]. Other similar problems can be found in monographs such as [16–21], and the literature quoted there.

In our present work, we further the research of [6, 7] on the asymptotic behavior of a stationary problem for the isothermal Bingham fluid. However, this time our operator will be perturbed by a term \(( u\cdot \nabla u ) \) in a three-dimensional thin domain \(\Omega^{\varepsilon}\) with Fourier and Tresca boundary conditions. This source term plays an essential role in quantum mechanics, where this non-linear term is used to perturb a linear operator to obtain a non-linear operator which gives applicable results; see [22].

We consider the Dirichlet boundary conditions on \(\overline{\Gamma }_{1}^{\varepsilon }\cup\overline{\Gamma}_{L}^{\varepsilon}\), where \(\overline{\Gamma}_{L}^{\varepsilon}\) is the lateral one, the Fourier boundary condition at the top surface \(\overline{\Gamma}_{1}^{\varepsilon}\), finally, a non-linear Tresca interface condition at the bottom one ω. The weak form of the problem is a variational inequality and the main difficulty here is to estimate the velocity, the pressure and its gradient.

Our work is organized as follows. In Section 2, the related weak formulation is given and the existence theorem of weak solutions is proved. In Section 3, we use the change of variable \(z={x_{3}}/{\varepsilon}\), to transform the initial problem posed in the domain \(\Omega^{\varepsilon}\) into a new problem posed on a fixed domain Ω independent of the parameter ε. We find some estimates on the velocity and pressure which are independent of the parameter and prove the convergence theorem by using different inequalities. The limit problem with a specific weak form of the Reynolds equation, the uniqueness of the limit velocity and pressure are given in Section 4.

2 Problem statement and variational formulation

We consider a mathematical problem governed to the stationary equations for Bingham fluid in a three-dimensional bounded domain \(\Omega^{\varepsilon }\subset \mathbb{R} ^{3}\) with boundary \(\Gamma^{\varepsilon}\). This boundary of the domain is assumed to be composed of three portions: ω, the bottom of the domain, \(\Gamma_{1}^{\varepsilon}\), the upper surface, and \(\Gamma_{L}^{\varepsilon}\), the lateral surface. The fluid is supposed to be incompressible. We impose the Fourier boundary condition at the top surface, a non-linear Tresca interface condition at the bottom one and Dirichlet boundary conditions on the top and the lateral parts. The fluid is acted upon by given body forces of density f.

Let ω be a fixed bounded domain of \(\mathbb{R} ^{3}\) of equation \(x_{3}=0\). We suppose that ω has a Lipschitz continuous boundary and is the bottom of the fluid. The upper surface \(\overline{\Gamma}_{1}^{\varepsilon}\) is defined by \(x_{3}=\varepsilon h(x^{\prime})\), \((x^{\prime}=(x_{1},x_{2}))\). We introduce a small parameter ε, which will tend to zero, and h a smooth bounded function such that \(0< h_{\star}\leq h ( x^{\prime} ) \leq h^{\star}\), for all \(( x^{\prime},0 ) \in\omega\).

We denote by \(\Omega^{\varepsilon}\) the domain of the flow: \(\Omega ^{\varepsilon}= \{ ( x^{\prime},x_{3} ) \in \mathbb{R} ^{3}: ( x^{\prime},0 ) \in\omega, 0< x_{3}<\varepsilon h ( x^{\prime} ) \}\), by \(\sigma^{\varepsilon}\) the deviatoric part and \(p^{\varepsilon}\) the pressure. The fluid is supposed to be visco-plastic, and the relation between \(\sigma^{\varepsilon}\) and \(D(u^{\varepsilon})\) is given:
$$\textstyle\begin{cases} \sigma_{ij}^{\varepsilon}=\widetilde{\sigma}_{ij}^{\varepsilon }-p^{\varepsilon }\delta_{ij},\\ \widetilde{\sigma}^{\varepsilon}=2\mu D(u^{\varepsilon})+\alpha ^{\varepsilon }\frac{D(u^{\varepsilon})}{ \vert D(u^{\varepsilon}) \vert }\quad\text{if }D(u^{\varepsilon})\neq0,\\ \vert \widetilde{\sigma}^{\varepsilon} \vert \leq\alpha ^{\varepsilon}\quad\text{if }D(u^{\varepsilon})\neq0, \end{cases} $$
here \(\alpha^{\varepsilon}\geq0\) is the yield stress, \(\mu>0\) is the constant viscosity, \(u^{\varepsilon}\) is the velocity field, \(\delta_{ij}\) is the Kronecker symbol, and \(D(u^{\varepsilon})=\frac{1}{2} ( \nabla u^{\varepsilon}+ ( \nabla u^{\varepsilon} ) ^{T} ) \).
For any tensor \(D=(d_{ij})\), the notation \(\vert D \vert \) represents the matrix norm: \(\vert D \vert =\sqrt{{ \sum_{i,j}} d_{ij}d_{ij}}\). We denote by \(n= ( n_{1},n_{2},n_{3} ) \) the unit outward normal vector on the boundary \(\Gamma^{\varepsilon}\). The normal and the tangential velocity on the boundary ω are
$$u_{n}^{\varepsilon}=u^{\varepsilon}\cdot n=u_{i}^{\varepsilon} \cdot n_{i}\quad\text{and}\quad u_{T_{i}}^{\varepsilon}=u_{i}^{\varepsilon}-u_{n}^{\varepsilon} \cdot n_{i}.$$
Similarly, for a regular stress tensor field \(\sigma^{\varepsilon}\), we denote by \(\sigma_{n}^{\varepsilon}\) and \(\sigma_{T}^{\varepsilon}\) the normal and tangential components of \(\sigma^{\varepsilon}\) on the boundary ω given by
$$\sigma_{n}^{\varepsilon}= \bigl( \sigma^{\varepsilon}\cdot n \bigr) \cdot n=\sigma _{ij}^{\varepsilon}\cdot n_{i}\cdot n_{j}\quad\text{and}\quad\sigma_{T_{i}}^{\varepsilon }= \sigma_{ij}^{\varepsilon}\cdot n_{j}-\sigma_{n}^{\varepsilon} \cdot n_{i}.$$
We consider now the following mechanical problem.

Problem 1

Find a velocity field \(u^{\varepsilon}:\Omega^{\varepsilon}\rightarrow \mathbb{R} ^{3}\) and the pressure \(p^{\varepsilon}\) such that
$$\begin{aligned} &{-}\operatorname{div} \bigl( \sigma^{\varepsilon} \bigr) +u\cdot \nabla u=f^{\varepsilon} \quad \text{in }\Omega^{\varepsilon}, \end{aligned}$$
(2.1)
$$\begin{aligned} & \textstyle\begin{array} [l]{l}\sigma_{ij}^{\varepsilon}=\widetilde{\sigma}_{ij}^{\varepsilon }-p^{\varepsilon }\delta_{ij},\\ \widetilde{\sigma}^{\varepsilon}=2\mu D(u^{\varepsilon})+\alpha ^{\varepsilon }\frac{D(u^{\varepsilon})}{ \vert D(u^{\varepsilon}) \vert }\quad\text{if }D(u^{\varepsilon})\neq0,\\ \vert \widetilde{\sigma}^{\varepsilon} \vert \leq\alpha ^{\varepsilon}\quad\text{if }D(u^{\varepsilon})\neq0 \end{array}\displaystyle \Biggr\} \quad\text{in } \Omega^{\varepsilon}, \end{aligned}$$
(2.2)
$$\begin{aligned} & \operatorname{div}\bigl(u^{\varepsilon}\bigr)=0 \quad \text{in } \Omega^{\varepsilon}, \end{aligned}$$
(2.3)
$$\begin{aligned} & \textstyle\begin{array} [l]{l}\sigma_{T}^{\varepsilon} ( u^{\varepsilon} ) +l^{\varepsilon }u^{\varepsilon}=0,\\ u^{\varepsilon}\cdot n=0 \end{array}\displaystyle \biggr\} \quad \text{on $ \Gamma_{1}^{\varepsilon}$,} \end{aligned}$$
(2.4)
$$\begin{aligned} & u^{\varepsilon}=0 \quad \text{on }\Gamma_{L}^{\varepsilon}, \end{aligned}$$
(2.5)
$$\begin{aligned} & u^{\varepsilon}\cdot n=0 \quad \text{on } \omega, \end{aligned}$$
(2.6)
$$\begin{aligned} & \textstyle\begin{array} [l]{l} \vert \sigma_{\tau}^{\varepsilon} \vert < k^{\varepsilon }\quad\Rightarrow\quad u_{\tau}^{\varepsilon}=s, \\ \vert \sigma_{\tau}^{\varepsilon} \vert =k^{\varepsilon}\quad\Rightarrow\quad\exists\lambda\geq0:u_{\tau}^{\varepsilon}=s-\lambda \sigma_{\tau }^{\varepsilon},\end{array}\displaystyle \biggr\} \quad \text{on }\omega . \end{aligned}$$
(2.7)

The law of conservation of momentum is given by equation (2.1). The relation between \(\sigma^{\varepsilon}\) and \(D(u^{\varepsilon})\) is given by (2.2), equation (2.3) represents the incompressibility equation, the boundary condition (2.4) is of Fourier type where \(l^{\varepsilon}>0\) is a given scalar, the Dirichlet boundary conditions on \(\overline{\Gamma}_{L}^{\varepsilon}\) is given by (2.5). Since there is a no-flux condition across ω we have equation (2.6). Condition (2.7) represents a Tresca thermal friction law on ω where \(k^{\varepsilon}\) is the friction coefficient.

To get a weak formulation, we introduce the closed convex,
$$\begin{aligned} &V^{\varepsilon} = \bigl\{ v\in \bigl( H^{1} \bigl( \Omega^{\varepsilon } \bigr) \bigr) ^{3}:v=0\text{ on } \Gamma_{L}^{\varepsilon}; v\cdot n=0\text{ on }\omega\cup \Gamma_{1}^{\varepsilon} \bigr\} , \\ &V_{\operatorname{div}}^{\varepsilon} = \bigl\{ v\in V^{\varepsilon }: \operatorname{div}v=0 \bigr\} , \\ &L_{0}^{2} \bigl( \Omega^{\varepsilon} \bigr) = \biggl\{ q \in L^{2} \bigl( \Omega^{\varepsilon} \bigr) :{ \int _{\Omega^{\varepsilon}}} q\,dx=0 \biggr\} . \end{aligned}$$
By standard calculations, the variational formulation of the problem \((Pb_{1})\) is given by: Find \(u^{\varepsilon}\in V_{\operatorname{div}}^{\varepsilon}\) and \(p^{\varepsilon}\in L_{0}^{2} ( \Omega ^{\varepsilon } ) \) such that, for all \(\varphi\in V^{\varepsilon}\), we have
$$\begin{aligned} &a \bigl(u^{\varepsilon},\varphi-u^{\varepsilon} \bigr)- \bigl( p^{\varepsilon }, \operatorname{div} ( \varphi ) \bigr) +l^{\varepsilon}{ \int _{\Gamma_{1}^{\varepsilon}}} u^{\varepsilon} \bigl( \varphi-u^{\varepsilon} \bigr) \,d\tau +b \bigl(u^{\varepsilon }, \varphi-u^{\varepsilon} \bigr)+j^{\varepsilon}(\varphi)-j^{\varepsilon } \bigl(u^{\varepsilon} \bigr) \\ &\quad\geq \bigl(f^{\varepsilon},\varphi-u^{\varepsilon} \bigr), \end{aligned}$$
(2.8)
where
$$\begin{aligned} &a \bigl(u^{\varepsilon},\varphi \bigr) =2\mu \int_{\Omega^{\varepsilon}}d_{ij} \bigl(u^{\varepsilon} \bigr)d_{ij}(\varphi)\,dx, \qquad \bigl( p^{\varepsilon }, \operatorname{div} ( \varphi ) \bigr) = \int_{\Omega ^{\varepsilon}}p^{\varepsilon}\operatorname{div} ( \varphi ) \,dx, \\ &b \bigl(u^{\varepsilon},\varphi \bigr) = \int_{\Omega^{\varepsilon }}u_{i}^{\varepsilon }\frac{\partial u_{j}^{\varepsilon}}{\partial x_{i}} \varphi_{j}\,dx, \qquad \bigl(f^{\varepsilon},\varphi \bigr)= \int_{\Omega^{\varepsilon}}f_{i}^{\varepsilon }\varphi_{i} \,dx, \\ &j^{\varepsilon} ( \varphi ) = \int_{\omega}k^{\varepsilon } \vert \phi \vert dx^{\prime}+ \alpha^{\varepsilon} \int _{\Omega^{\varepsilon}} \bigl\vert D(\varphi) \bigr\vert \,dx. \end{aligned}$$
We introduce some results which will be used in the next. The detailed description can be found in [23]. We have
$$\begin{aligned} &{ \int_{\Omega^{\varepsilon}}} \bigl\vert \nabla \bigl(u^{\varepsilon} \bigr) \bigr\vert ^{2} \leq2{ \int_{\Omega^{\varepsilon}}} \bigl\vert D \bigl( u^{\varepsilon} \bigr) \bigr\vert ^{2}\,dx+\mathcal {C} \bigl( \Gamma_{1}^{\varepsilon} \bigr) \int_{\Gamma_{1}^{\varepsilon }} \bigl\vert u^{\varepsilon} \bigr\vert ^{2}\,d\tau,\quad\text{Korn inequality,} \end{aligned}$$
(2.9)
$$\begin{aligned} & \int_{\Omega^{\varepsilon}} \bigl\vert u^{\varepsilon} \bigr\vert ^{2} \leq2h^{\star}\varepsilon \int_{\Gamma_{1}^{\varepsilon}} \bigl\vert u^{\varepsilon } \bigr\vert ^{2}+2h^{\star 2} \varepsilon^{2} \int_{\Omega^{\varepsilon}} \biggl\vert {\frac{\partial u^{\varepsilon}}{\partial x_{3}}} \biggr\vert ^{2},\quad \text{Poincar\'{e} inequality,} \end{aligned}$$
(2.10)
$$\begin{aligned} &ab \leq\eta^{2}\frac{a^{2}}{2}+\eta^{-2} \frac{b^{2}}{2},\quad\forall ( a,b ) \in\mathbb{R}^{2},\forall\eta,\quad \text{Young inequality,} \end{aligned}$$
(2.11)
where \(C(\Gamma_{1}^{\varepsilon})=2 \Vert D_{2}h^{\varepsilon } \Vert _{C ( \overline{\omega} ) } ( 1+ \Vert D_{1}h^{\varepsilon} \Vert _{C ( \overline{\omega} ) }^{2} ) \).

Theorem 2.1

Let \(u^{\varepsilon}\) be a solution of the variational problem (2.8), then
$$\begin{aligned} & a \bigl(u^{\varepsilon},u^{\varepsilon} \bigr)+\frac{l^{\varepsilon}}{2}{ \int _{\Gamma_{1}^{\varepsilon}}} \bigl\vert u^{\varepsilon} \bigr\vert ^{2}\,d\tau+ \int_{\omega }k^{\varepsilon } \bigl\vert u^{\varepsilon} \bigr\vert \,dx^{\prime}+\alpha ^{\varepsilon}\int_{\Omega^{\varepsilon}} \bigl\vert D \bigl(u^{\varepsilon} \bigr) \bigr\vert \,dx \\ & \quad\leq\frac{\mu}{16} \bigl\Vert \nabla u^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega^{\varepsilon} ) }^{2}+ \biggl[ \frac {8\varepsilon ^{2}h^{\star2}}{\mu}+\frac{\varepsilon h^{\star}}{l^{\varepsilon }} \biggr] \bigl\Vert f^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon } ) }^{2} . \end{aligned}$$
(2.12)

Proof

Choosing \(\varphi=0\) as test function in inequality (2.8), we get
$$ a \bigl(u^{\varepsilon},u^{\varepsilon} \bigr)+l^{\varepsilon}{ \int _{\Gamma_{1}^{\varepsilon}}} \bigl\vert u^{\varepsilon} \bigr\vert ^{2}\,d\tau+b \bigl(u^{\varepsilon },u^{\varepsilon} \bigr)+ \int_{\omega}k^{\varepsilon} \bigl\vert u^{\varepsilon } \bigr\vert \,dx^{\prime}+\alpha^{\varepsilon} \int_{\Omega^{\varepsilon} } \bigl\vert D \bigl(u^{\varepsilon} \bigr) \bigr\vert \,dx\leq \bigl(f^{\varepsilon},u^{\varepsilon} \bigr) $$
(2.13)
as \((f^{\varepsilon},u^{\varepsilon})\leq \Vert f^{\varepsilon } \Vert _{L^{2} ( \Omega^{\varepsilon} ) } \Vert u^{\varepsilon} \Vert _{L^{2} ( \Omega^{\varepsilon} ) }\) and by using the Poincaré inequality (2.10), we obtain
$$\bigl(f^{\varepsilon},u^{\varepsilon} \bigr)\leq\sqrt{2\varepsilon h^{\star }} \bigl\Vert f^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega^{\varepsilon} ) } \biggl( \int_{\Gamma_{1}^{\varepsilon}} \bigl\vert u^{\varepsilon } \bigr\vert ^{2}\,d\tau \biggr) ^{\frac{1}{2}}+\sqrt{2} \bigl( \varepsilon h^{\star } \bigr) \bigl\Vert f^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon } ) } \bigl\Vert \nabla u^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega^{\varepsilon} ) } .$$
By the inequality (2.11), we have
$$ \bigl\vert \bigl(f^{\varepsilon},u^{\varepsilon} \bigr) \bigr\vert \leq \frac {\mu}{16} \bigl\Vert \nabla u^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon} ) }^{2}+ \biggl[ \frac{8\varepsilon^{2}h^{\star 2}}{\mu }+\frac{\varepsilon h^{\star}}{l^{\varepsilon}} \biggr] \bigl\Vert f^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega^{\varepsilon} ) } ^{2}+\frac{l^{\varepsilon}}{2} \int_{\Gamma_{1}^{\varepsilon}} \bigl\vert u^{\varepsilon} \bigr\vert ^{2}\,d\tau. $$
(2.14)
On the other hand
$$b \bigl(u^{\varepsilon},u^{\varepsilon} \bigr)= \int_{\Omega^{\varepsilon}}u_{i}^{\varepsilon}\frac{\partial u_{j}^{\varepsilon}}{\partial x_{i}} u_{j}^{\varepsilon}\,dx.$$
Using the fact that \(2u_{j}^{\varepsilon}\frac{\partial u_{j}^{\varepsilon}}{\partial x_{i}}=\frac{\partial}{\partial x_{i}} ( u_{j}^{\varepsilon} ) ^{2}\) and by the Green formula, we find
$$b \bigl(u^{\varepsilon},u^{\varepsilon} \bigr)=\frac{1}{2} \int_{\Gamma^{\varepsilon} } \bigl( u_{j}^{\varepsilon} \bigr) ^{2}u_{i}^{\varepsilon}\cdot n_{i}\,d\tau - \int_{\Omega^{\varepsilon}} \bigl( u_{j}^{\varepsilon} \bigr) ^{2}\frac{\partial u_{i}^{\varepsilon}}{\partial x_{i}}\,dx.$$
As \(u^{\varepsilon}\in V_{\operatorname{div}}^{\varepsilon}\), we obtain \(b(u^{\varepsilon},u^{\varepsilon})=0\), and from (2.13)-(2.14), we deduce (2.12). □

Theorem 2.2

Let \(u^{\varepsilon}\) be a solution of the variational inequality (2.8). We assume that \(\varepsilon l^{\varepsilon}=\hat{l}\) and
$$ \frac{C(\Gamma_{1}^{\varepsilon})}{l^{\varepsilon}}\leq\frac{1}{\mu }, $$
(2.15)
then
$$ \bigl\Vert \nabla u^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon} ) }^{2} \leq \biggl( \frac{384\varepsilon^{2}h^{\star 2}}{{\mu}^{2}}+\frac {48\varepsilon h^{\star}}{l^{\varepsilon}{\mu}} \biggr) \bigl\Vert f^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon} ) }^{2}. $$
(2.16)

Proof

We estimate \(a(u^{\varepsilon },u^{\varepsilon})\) and \(\int_{\Gamma_{1}^{\varepsilon}} \vert u^{\varepsilon} \vert ^{2}\,d\tau\). According to (2.12), we obtain
$$\begin{aligned} \bigl\Vert \nabla u^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon} ) }^{2} \leq{}&\frac{4}{\mu}a \bigl(u^{\varepsilon },u^{\varepsilon} \bigr)+4C \bigl( \Gamma_{1}^{\varepsilon} \bigr) \int_{\Gamma _{1}^{\varepsilon}} \bigl\vert u^{\varepsilon} \bigr\vert ^{2}\,d\tau \\ \leq{}&\frac{1}{4} \bigl\Vert \nabla u^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega^{\varepsilon} ) }^{2}+ \biggl[ \frac{32\varepsilon ^{2}h^{\star2}}{\mu^{2}}+\frac{4\varepsilon h^{\star}}{\mu l^{\varepsilon}} \biggr] \bigl\Vert f^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon } ) }^{2} \\ &{} +\frac{1}{2}\frac{\mu C(\Gamma_{1}^{\varepsilon})}{l^{\varepsilon}} \bigl\Vert \nabla u^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon} ) }^{2}+ \biggl[ \frac{64C(\Gamma_{1}^{\varepsilon })}{\mu l^{\varepsilon}} \varepsilon^{2}h^{\star2}+\frac{8C(\Gamma _{1}^{\varepsilon})}{ ( l^{\varepsilon} ) ^{2}}\varepsilon h^{\star} \biggr] \bigl\Vert f^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ^{\varepsilon } ) }^{2}. \end{aligned}$$

Now, using the inequality (2.15) we deduce (2.16). □

3 Change of the domain and some estimates

In this section, we will use the technique of scaling in \(\Omega ^{\varepsilon }\) on the coordinate \(x_{3}\), by introducing the change of the variables \(z=\frac{x_{3}}{\varepsilon}\). We obtain a fixed domain Ω which is independent of ε:
$$\Omega= \bigl\{ \bigl( x^{\prime},z \bigr) \in \mathbb{R} ^{3}: \bigl( x^{\prime},0 \bigr) \in\omega, 0< z< h \bigl( x^{\prime } \bigr) \bigr\} .$$
We denote by \(\Gamma=\overline{\omega}\cup\overline{\Gamma}_{1}\cup \overline{\Gamma}_{L}\) its boundary, then we define the following functions in Ω:
$$ \textstyle\begin{cases} \hat{u}_{i}^{\varepsilon}(x^{\prime},z)=u_{i}^{\varepsilon}(x^{\prime},x_{3})\quad\text{for }i=1,2;\\ \hat{u}_{3}^{\varepsilon}(x^{\prime},z)=\varepsilon ^{-1}u_{3}^{\varepsilon }(x^{\prime},x_{3});\\ \widehat{p}^{\varepsilon}(x^{\prime},z)=\varepsilon^{2}p^{\varepsilon }(x^{\prime},x_{3}). \end{cases} $$
(3.1)
Let us assume the following:
$$\hat{k}=\varepsilon k^{\varepsilon}; \qquad\hat{f} \bigl( x^{\prime },z \bigr) = \varepsilon^{2}f^{\varepsilon} \bigl( x^{\prime},x_{3} \bigr) \quad\text{and}\quad\widehat{l}=\varepsilon l^{\varepsilon};\qquad \widehat { \alpha }= \varepsilon\alpha^{\varepsilon}. $$
Let
$$\begin{aligned} &V = \bigl\{ \widehat{\varphi}\in \bigl( H^{1} ( \Omega ) \bigr) ^{3}:\varphi=0\text{ on }\Gamma_{L}; \varphi\cdot n=0 \text{ on }\omega \cup\Gamma_{1} \bigr\} , \\ &V_{\operatorname{div}} = \bigl\{ \widehat{\varphi}\in V:\operatorname {div} ( \widehat{\varphi} ) =0 \bigr\} , \\ &\Pi(V) = \bigl\{ \varphi^{\star}= \bigl(\varphi_{1}^{\star}, \varphi_{2}^{\star} \bigr)\in \bigl( H^{1} ( \Omega ) \bigr) ^{2}:\varphi^{\star }=0\text{ on } \Gamma_{L} \bigr\} , \\ &V_{z} = \biggl\{ \varphi^{\star}= \bigl(\varphi_{1}^{\star}, \varphi_{2}^{\star } \bigr)\in \bigl( L^{2} ( \Omega ) \bigr) ^{2}:\frac{\partial\varphi _{i}^{\star}}{\partial z}\in L^{2} ( \Omega ), i=1,2\text{ and }\varphi^{\star}=0\text{ on }\Gamma_{L} \biggr\} , \\ &\sum ( V ) = \bigl\{ \phi\in\Pi(V):\text{satisfies the condition } \bigl( D^{\prime} \bigr) \bigr\} , \end{aligned}$$
where the condition \(( D^{\prime} ) \) is given by
$$\int _{\Omega} \biggl( \varphi_{1}^{\star} \bigl(x^{\prime},z \bigr)\frac {\partial\theta}{\partial x_{1}} \bigl(x^{\prime} \bigr)+ \varphi_{2}^{\star } \bigl(x^{\prime },z \bigr) \frac{\partial\theta}{\partial x_{2}} \bigl(x^{\prime} \bigr) \biggr) \,dx^{\prime }\,dz=0, \quad \forall\theta\in C_{0}^{1} ( \omega ) . $$
Injecting the new data and unknown in (2.8), and after multiplication by ε we deduce that \((\hat{u}^{\varepsilon},\widehat {p}^{\varepsilon })\) satisfies the following problem:
$$\begin{aligned} & \sum_{1\leq i,j\leq2} \int _{\Omega} \biggl[ \mu\varepsilon ^{2} \biggl( \frac{\partial\hat{u}_{i}^{\varepsilon}}{\partial x_{j}}+\frac{\partial\hat{u}_{j}^{\varepsilon}}{\partial x_{i}} \biggr) -\delta_{ij} \widehat{p}^{\varepsilon} \biggr] \frac{\partial}{\partial x_{j}} \bigl(\hat{ \phi}_{i}-\hat{u}_{i}^{\varepsilon} \bigr)\,dx^{\prime} \,dz \\ &\qquad{}+\mu\sum_{i=1}^{{2} } \int _{\Omega} \biggl( \frac {\partial\hat{u}_{i}^{\varepsilon}}{\partial z}+\varepsilon^{2} \frac {\partial\hat{u}_{3}^{\varepsilon}}{\partial x_{i}} \biggr) \frac {\partial }{\partial z} \bigl(\hat{\phi}_{i}- \hat{u}_{i}^{\varepsilon} \bigr)\,dx^{\prime}\,dz \\ &\qquad{}+\sum_{j=1}^{{2} } \int _{\Omega}\mu\varepsilon ^{2} \biggl( \varepsilon^{2}\frac{\partial\hat{u}_{3}^{\varepsilon }}{\partial x_{j}}+\frac{\partial\hat{u}_{j}^{\varepsilon}}{\partial z} \biggr) \frac{\partial}{\partial x_{j}} \bigl(\hat{\phi}_{3}-\hat {u}_{3}^{\varepsilon } \bigr)\,dx^{\prime}\,dz \\ &\qquad{}+ \int _{\Omega} \biggl( 2\mu\varepsilon^{2} \frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial z}-\widehat{p}^{\varepsilon} \biggr) \frac{\partial}{\partial z} \bigl( \hat{ \phi}_{3}-\hat{u}_{3}^{\varepsilon } \bigr) \,dx^{\prime}\,dz \\ &\qquad{}+\sum_{1\leq i,j\leq2} \int _{\Omega}\varepsilon^{2}\hat{u}_{i}^{\varepsilon} \frac{\partial\hat{u}_{j}^{\varepsilon}}{\partial x_{i}} \bigl(\hat{\phi}_{j}-\hat{u}_{j}^{\varepsilon} \bigr)\,dx^{\prime}\,dz+\overset{2}{\sum _{i=1}} \int _{\Omega}\varepsilon^{4}\hat{u}_{i}^{\varepsilon}\frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial x_{i}} \bigl(\hat{\phi }_{3}- \hat{u}_{3}^{\varepsilon} \bigr)\,dx^{\prime}\,dz \\ &\qquad{}+\overset{2}{\sum_{i=1}} \int _{\Omega}\varepsilon^{2}\hat{u}_{3}^{\varepsilon} \frac{\partial\hat{u}_{i}^{\varepsilon}}{\partial z} \bigl(\hat{\phi}_{i}- \hat{u}_{i}^{\varepsilon} \bigr)\,dx^{\prime}\,dz+ \int _{\Omega }\varepsilon^{4}\hat{u}_{3}^{\varepsilon} \frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial z} \bigl(\hat{\phi}_{3}-\hat{u}_{3}^{\varepsilon } \bigr)\,dx^{\prime}\,dz \\ &\qquad{}+\overset{2}{\sum_{i=1}} \int_{\omega}\widehat{l}\hat{u}_{i}^{\varepsilon }(x^{\prime},h \bigl(x^{\prime} \bigr) \bigl( \hat{\phi}_{i} \bigl(x^{\prime},h \bigl(x^{\prime } \bigr) \bigr)-\hat{u}_{i}^{\varepsilon} \bigl(x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigr) \sqrt {1+ \bigl\vert \nabla h^{\varepsilon} \bigl(x^{\prime} \bigr) \bigr\vert ^{2}}\,dx^{\prime} \\ &\qquad{}+ \int_{\omega}\widehat{l}\varepsilon^{2} \hat{u}_{3}^{\varepsilon }(x^{\prime },h \bigl(x^{\prime} \bigr) \bigl( \hat{\phi}_{3} \bigl(x^{\prime},h \bigl(x^{\prime} \bigr) \bigr)-\hat{u}_{3}^{\varepsilon} \bigl(x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigr) \sqrt{1+ \bigl\vert \nabla h^{\varepsilon} \bigl(x^{\prime} \bigr) \bigr\vert ^{2}}\,dx^{\prime} \\ &\qquad{}+ \int_{\omega}\hat{k} \bigl( \vert \hat{\phi} \vert - \bigl\vert \hat{u}^{\varepsilon} \bigr\vert \bigr) \,dx^{\prime}+\widehat{ \alpha}\int_{\Omega} \bigl\vert D(\hat{\phi}) \bigr\vert \,dx- \widehat{ \alpha}\int_{\Omega} \bigl\vert D \bigl(\hat{u}^{\varepsilon} \bigr) \bigr\vert \,dx^{\prime }\,dz \\ &\quad \geq\sum_{i=1}^{2} \int_{\Omega}\hat{f}_{i} \bigl(\hat{\phi}_{i} -\hat{u}_{i}^{\varepsilon} \bigr)\,dx^{\prime}\,dz+ \int_{\Omega}\varepsilon\hat {f}_{3} \bigl( \hat{\phi}_{3}-\hat{u}_{3}^{\varepsilon} \bigr) \,dx^{\prime}\,dz, \quad\forall \hat{\phi}\in V, \end{aligned}$$
(3.2)
where
$$\begin{aligned} &\bigl\vert D(v) \bigr\vert = \Biggl[ \frac{1}{4}\sum _{1\leq i,j \leq 2} \varepsilon^{2} \biggl( \frac{\partial v_{i}}{\partial x_{j}}+ \frac {\partial v_{j}}{\partial x_{i}} \biggr) ^{2}+\frac{1}{2}\sum _{i=1}^{2} \biggl( \frac{\partial v_{i}}{\partial z}+ \varepsilon^{2}\frac{\partial v_{3}}{\partial x_{i}} \biggr) ^{2}+ \varepsilon^{2} \biggl( \frac{\partial v_{3}}{\partial z} \biggr) ^{2} \Biggr] ^{\frac{1}{2} } , \\ &\quad\forall v\in V. \end{aligned}$$
(3.3)

Now we establish the estimates and convergences for the velocity field \(\hat{u}^{\varepsilon}\) and the pressure \(\widehat{p}^{\varepsilon}\) in the domain Ω.

Theorem 3.1

Assuming that \(f^{\varepsilon}\) in \(( L^{2} ( \Omega^{\varepsilon} ) ) ^{3}\), the friction coefficient \(k^{\varepsilon}\) is a non-negative function in \(L^{\infty}(\omega)\), and under the assumption of (2.15), there exists a constant \(c_{l}>0\), \(l=0\) to 4 independent of ε such that
$$\begin{aligned} &\sum_{1\leq i,j\leq2} \biggl\Vert \varepsilon \frac{\partial \hat {u}_{i}^{\varepsilon}}{\partial x_{j}} \biggr\Vert _{L^{2} ( \Omega ) }^{2}+ \biggl\Vert \varepsilon\frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial z} \biggr\Vert _{L^{2} ( \Omega ) }^{2}+ \sum_{i=1}^{2} \biggl( \biggl\Vert \frac {\partial \hat{u}_{i}^{\varepsilon}}{\partial z} \biggr\Vert _{L^{2} ( \Omega ) }^{2}+ \biggl\Vert \varepsilon^{2}\frac{\partial\hat{u} _{3}^{\varepsilon}}{\partial x_{i}} \biggr\Vert _{L^{2} ( \Omega ) }^{2} \biggr) \leq c_{0}, \end{aligned}$$
(3.4)
$$\begin{aligned} & \bigl\Vert \hat{u}_{i}^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ) } \leq c_{1},\quad\textit{for }i=1,2, \end{aligned}$$
(3.5)
$$\begin{aligned} & \bigl\Vert \varepsilon\hat{u}_{3}^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ) } \leq c_{2}, \end{aligned}$$
(3.6)
$$\begin{aligned} & \biggl\Vert \frac{\partial\widehat{p}^{\varepsilon}}{\partial x_{i}} \biggr\Vert _{H^{-1} ( \Omega ) } \leq c_{3},\quad \textit{for }i=1,2, \end{aligned}$$
(3.7)
$$\begin{aligned} & \biggl\Vert \frac{\partial\widehat{p}^{\varepsilon}}{\partial z} \biggr\Vert _{H^{-1} ( \Omega ) } \leq\varepsilon c_{4}. \end{aligned}$$
(3.8)

Proof

As \(\varepsilon^{3} \Vert f^{\varepsilon } \Vert _{L^{2} ( \Omega^{\varepsilon} ) }^{2}= \Vert \hat{f} \Vert _{L^{2} ( \Omega ) }^{2}\) and passing to the fixed domain Ω in the right member of the inequality (2.16), we obtain
$$ \bigl\Vert \nabla\hat{u}^{\varepsilon} \bigr\Vert _{L^{2} ( \Omega ) }^{2} \leq \biggl[ \frac{384h^{\star2}}{\mu^{2}}+\frac {48h^{\star}}{\mu\widehat{l}} \biggr] \Vert \hat{f} \Vert _{L^{2} ( \Omega ) }^{2} ; $$
(3.9)
from (3.9), we deduce (3.4) with \(c_{0}= [ \frac{384h^{\star2}}{\mu^{2}}+\frac{48h^{\star}}{\mu\widehat{l}} ] \Vert \hat {f} \Vert _{L^{2} ( \Omega ) }^{2}\).
For getting the estimates (3.5) and (3.6), using the Poincaré inequality in the domain Ω:
$$ \int_{\Omega} \bigl\vert \hat{u}_{i}^{\varepsilon} \bigr\vert ^{2}\,dx\leq2h^{\star} \int_{\Gamma_{1}} \bigl\vert \hat{u}_{i}^{\varepsilon } \bigr\vert ^{2}\,d\tau^{\prime}+2h^{\star2} \int_{\Omega} \biggl\vert \frac{\partial\hat{u}_{i}^{\varepsilon}}{\partial z} \biggr\vert ^{2}\,dx^{\prime}\,dz,\quad i=1,2,3, $$
(3.10)
from (3.4), we have
$$2h^{\star2} \int_{\Omega} \biggl\vert \frac{\partial\hat {u}_{i}^{\varepsilon}}{\partial z} \biggr\vert ^{2}\,dx^{\prime}\,dz\leq c_{0}\quad\text{and}\quad 2 \varepsilon^{2}h^{\star2} \int_{\Omega} \biggl\vert \frac{\partial\hat {u}_{3}^{\varepsilon}}{\partial z} \biggr\vert ^{2}\,dx^{\prime}\,dz\leq c_{0},\quad i=1,2 .$$
On the other hand
$$\begin{aligned} & \int_{\Gamma_{1}} \bigl\vert \hat{u}_{i}^{\varepsilon} \bigr\vert ^{2}\,d\tau^{\prime}\leq\max_{x^{\prime}\in\omega} \sqrt{1+ \bigl\vert \nabla h \bigl(x^{\prime} \bigr) \bigr\vert ^{2}} \int_{\Gamma_{1}^{\varepsilon}} \bigl\vert u_{i}^{\varepsilon} \bigr\vert ^{2}\,d\tau,\quad\text{for }i=1,2, \end{aligned}$$
(3.11)
$$\begin{aligned} & \int_{\Gamma_{1}} \bigl\vert \hat{u}_{3}^{\varepsilon} \bigr\vert ^{2} \,d\tau^{\prime}\leq\frac{1}{\varepsilon^{2}} \max_{x^{\prime}\in\omega} \sqrt{1+ \bigl\vert \nabla h \bigl(x^{\prime} \bigr) \bigr\vert ^{2}} \int_{\Gamma _{1}^{\varepsilon}} \bigl\vert u_{3}^{\varepsilon} \bigr\vert ^{2}\,d\tau \end{aligned}$$
(3.12)
from (2.12) and (3.4), we have \({ \int _{\Gamma_{1}^{\varepsilon}}} \vert u^{\varepsilon} \vert ^{2}\,d\tau\leq\frac{\mu c_{0}}{8\widehat{l}}+ [ \frac{16h^{\star2}}{\mu\widehat{l}}+\frac {2h^{\star}}{\widehat{l}^{2}} ] \Vert \hat{f} \Vert _{L^{2} ( \Omega ) }^{2}\).
Then
$$ { \int _{\Gamma_{1}^{\varepsilon}}} \bigl\vert u^{\varepsilon} \bigr\vert ^{2}\,d\tau\leq C^{\prime } \bigl(\Omega, \widehat{l}, \mu, \hat{f}, h^{\star} \bigr). $$
(3.13)
From (3.10)-(3.13), we deduce \(\Vert \hat{u}_{i}^{\varepsilon} \Vert _{L^{2} ( \Omega ) }\leq c_{1},i=1,2\) and \(\Vert \varepsilon\hat{u}_{3}^{\varepsilon} \Vert _{L^{2} ( \Omega ) }\leq c_{2}\).
Choosing \(\hat{\phi}=(\hat{u}_{1}^{\varepsilon}+\psi, \hat {u}_{2}^{\varepsilon},\hat{u}_{3}^{\varepsilon})\) in (3.2) and by Cauchy-Schwarz’s inequality, we have
$$\begin{aligned} & \int_{\omega}\widehat{l}\hat{u}_{1}^{\varepsilon} \bigl(x^{\prime },h \bigl(x^{\prime } \bigr) \bigr)\psi \bigl(x^{\prime},h \bigl(x^{\prime} \bigr) \bigr)\sqrt{ \bigl|1+\nabla h \bigl(x^{\prime } \bigr) \bigr|^{2}}\,dx^{\prime } \\ &\quad\leq\widehat{l}\max_{x^{\prime}\in\omega}\sqrt{ \bigl|1+\nabla h \bigl(x^{\prime } \bigr)\bigr|^{2}} \biggl[ \int_{\omega} \bigl\vert \hat{u}_{1}^{\varepsilon} \bigl(x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) \bigr\vert ^{2}\sqrt{ \bigl|1+\nabla h \bigl(x^{\prime } \bigr) \bigr|^{2}} \,dx^{\prime } \biggr] ^{\frac{1}{2}}\\ &\qquad{}\times \biggl[ \int_{\omega} \bigl\vert \psi \bigl(x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) \bigr\vert ^{2}\sqrt{ \bigl|1+\nabla h \bigl(x^{\prime } \bigr) \bigr|^{2}}\,dx^{\prime } \biggr] ^{\frac{1}{2}} \\ &\quad \leq\widehat{l}\max_{x^{\prime}\in\omega}\sqrt{\bigl|1+\nabla h \bigl(x^{\prime } \bigr)\bigr|^{2}} \biggl( { \int _{\Gamma_{1}}} \bigl\vert \hat{u}_{1}^{\varepsilon} \bigr\vert ^{2}\,d\tau^{\prime } \biggr) ^{\frac{1}{2}} \biggl( { \int _{\Gamma_{1}}} \vert \psi \vert ^{2}\,d \tau^{\prime} \biggr) ^{\frac {1}{2}}. \end{aligned}$$

As \(( { \int _{\Gamma_{1}}} \vert \hat{u}_{1}^{\varepsilon} \vert ^{2}\,d\tau^{\prime } ) ^{\frac{1}{2}}\leq C^{\prime}\) and according to the continuity of the trace application from \(H^{1}(\Omega)\) in \(L^{2}(\Gamma_{1})\), there exists a constant \(C^{\prime\prime}\) independent of ε such that \(( { \int _{\Gamma_{1}}} \vert \psi \vert ^{2}\,d\tau^{\prime} ) ^{\frac {1}{2}}\leq C^{\prime\prime} \Vert \psi \Vert _{H^{1} ( \Omega ) }\).

Therefore
$$\int_{\omega}\widehat{l}\hat{u}_{1}^{\varepsilon} \bigl(x^{\prime},h \bigl(x^{\prime } \bigr) \bigr)\psi \bigl(x^{\prime},h \bigl(x^{\prime} \bigr) \bigr)\sqrt{\bigl|1+\nabla h \bigl(x^{\prime } \bigr)\bigr|^{2}}\,dx^{\prime }\leq C^{\prime\prime\prime} \Vert \psi \Vert _{H^{1} ( \Omega ) }, $$
where \(C^{\prime\prime\prime}=C^{\prime}C^{\prime\prime }\widehat {l}\max_{x^{\prime}\in\omega}\sqrt{|1+\nabla h(x^{\prime})|^{2}}\).
By the Hölder inequality, we get
$$\biggl\vert \varepsilon^{2} \int _{\Omega}\hat{u}_{1}^{\varepsilon}\frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial x_{1}}\psi \,dx^{\prime }\,dz \biggr\vert \leq \varepsilon^{2} \bigl\Vert \hat{u}_{1}^{\varepsilon } \bigr\Vert _{L^{4}(\Omega)} \biggl\Vert \frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial x_{1}} \biggr\Vert _{L^{2}(\Omega)} \Vert \psi \Vert _{L^{4}(\Omega)}.$$
Using a Sobolev imbedding \(\Vert v \Vert _{L^{4}(\Omega )}\leq c \Vert v \Vert _{H^{1} ( \Omega ) }\), we obtain
$$\biggl\vert \varepsilon^{2} \int _{\Omega}\hat{u}_{1}^{\varepsilon}\frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial x_{1}}\psi \,dx^{\prime }\,dz \biggr\vert \leq c \varepsilon^{2} \bigl\Vert \hat {u}_{1}^{\varepsilon } \bigr\Vert _{H^{1} ( \Omega ) } \biggl\Vert \frac{\partial \hat {u}_{1}^{\varepsilon}}{\partial x_{1}} \biggr\Vert _{L^{2}(\Omega )} \Vert \psi \Vert _{H^{1} ( \Omega ) }.$$
In the same way,
$$\begin{aligned} & \biggl\vert \varepsilon^{2} \int _{\Omega}\hat{u}_{2}^{\varepsilon}\frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial x_{2}}\psi \,dx^{\prime }\,dz \biggr\vert \leq c \varepsilon^{2} \bigl\Vert \hat {u}_{2}^{\varepsilon } \bigr\Vert _{H^{1} ( \Omega ) } \biggl\Vert \frac{\partial \hat {u}_{1}^{\varepsilon}}{\partial x_{2}} \biggr\Vert _{L^{2}(\Omega )} \Vert \psi \Vert _{H^{1} ( \Omega ) }, \\ & \biggl\vert \varepsilon^{2} \int _{\Omega}\hat{u}_{3}^{\varepsilon}\frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial z}\psi \,dx^{\prime }\,dz \biggr\vert \leq c \varepsilon^{2} \bigl\Vert \hat {u}_{3}^{\varepsilon } \bigr\Vert _{H^{1} ( \Omega ) } \biggl\Vert \frac{\partial \hat {u}_{1}^{\varepsilon}}{\partial z} \biggr\Vert _{L^{2}(\Omega)} \Vert \psi \Vert _{H^{1} ( \Omega ) }. \end{aligned}$$
The Cauchy-Schwarz inequality gives
$$\begin{aligned} &\int _{\Omega}\frac{\partial\widehat{p}^{\varepsilon}}{\partial x_{1}}\psi \,dx^{\prime}\,dz \\ &\quad \leq \mu \varepsilon^{2}\sum^{2}_{j=1} \biggl( \biggl\Vert \frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial x_{j}} \biggr\Vert _{L^{2} ( \Omega ) }+ \biggl\Vert \frac {\partial\hat {u}_{j}^{\varepsilon}}{\partial x_{1}} \biggr\Vert _{L^{2} ( \Omega ) } \biggr) \biggl\Vert \frac{\partial\psi}{\partial x_{j}} \biggr\Vert _{L^{2} ( \Omega ) } \\ &\qquad{} +\mu \biggl( \biggl\Vert \frac{\partial\hat{u}_{1}^{\varepsilon }}{\partial z} \biggr\Vert _{L^{2} ( \Omega ) }+ \varepsilon^{2} \biggl\Vert \frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial x_{1}} \biggr\Vert _{L^{2} ( \Omega ) } \biggr) \biggl\Vert \frac{\partial\psi }{\partial z} \biggr\Vert _{L^{2} ( \Omega ) }\\ &\qquad{}+c\varepsilon ^{2} \bigl\Vert \hat{u}_{1}^{\varepsilon} \bigr\Vert _{H^{1} ( \Omega ) } \biggl\Vert \frac{\partial\hat{u}_{1}^{\varepsilon }}{\partial x_{1}} \biggr\Vert _{L^{2}(\Omega)} \Vert \psi \Vert _{H^{1} ( \Omega ) } \\ &\qquad {}+c\varepsilon^{2} \bigl\Vert \hat{u}_{2}^{\varepsilon} \bigr\Vert _{H^{1} ( \Omega ) } \biggl\Vert \frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial x_{2}} \biggr\Vert _{L^{2}(\Omega)} \Vert \psi \Vert _{H^{1} ( \Omega ) }+c\varepsilon^{2} \bigl\Vert \hat{u}_{3}^{\varepsilon} \bigr\Vert _{H^{1} ( \Omega ) } \biggl\Vert \frac{\partial\hat{u}_{1}^{\varepsilon}}{\partial z} \biggr\Vert _{L^{2}(\Omega)} \Vert \psi \Vert _{H^{1} ( \Omega ) } \\ &\qquad{} +C^{\prime\prime\prime} \Vert \psi \Vert _{H^{1} ( \Omega ) }+\widehat{\alpha} \sqrt{ \vert \Omega \vert } \bigl\Vert D(\psi) \bigr\Vert _{L^{2}(\Omega)}+ \Vert \hat {f} \Vert _{L^{2} ( \Omega ) } \Vert \psi \Vert _{L^{2} ( \Omega ) } \end{aligned}$$
as \(\Vert D(\psi) \Vert _{L^{2}(\Omega)}\leq \Vert \psi \Vert _{H^{1} ( \Omega ) }\) and from (3.4)-(3.6), we deduce
$$\begin{aligned} &\int _{\Omega}\frac{\partial\widehat{p}^{\varepsilon}}{\partial x_{1}}\psi \,dx^{\prime}\,dz \\ &\quad \leq \mu C \Biggl( \sum^{2}_{j=1} \biggl\Vert \frac{\partial\psi}{\partial x_{j}} \biggr\Vert _{L^{2} ( \Omega ) }+ \biggl\Vert \frac{\partial\psi}{\partial z} \biggr\Vert _{L^{2} ( \Omega ) } \Biggr) \\ &\qquad{} + \bigl( 2cc_{1}C+cc_{2}C+C^{\prime\prime\prime} \bigr) \Vert \psi \Vert _{H^{1} ( \Omega ) }+\widehat{\alpha}\sqrt{ \vert \Omega \vert } \Vert \psi \Vert _{H^{1} ( \Omega ) }+ \Vert \hat{f} \Vert _{L^{2} ( \Omega ) } \Vert \psi \Vert _{H^{1} ( \Omega ) }. \end{aligned}$$
(3.14)
In the same way, if we choose \(\hat{\phi}=(\hat{u}_{1}^{\varepsilon }-\psi, \hat{u}_{2}^{\varepsilon}, \hat{u}_{3}^{\varepsilon})\) in (3.14), we obtain
$$\begin{aligned} &{-} \int _{\Omega}\frac{\partial\widehat{p}^{\varepsilon}}{\partial x_{1}}\psi \,dx^{\prime}\,dz \\ &\quad \leq \mu C \Biggl( \sum^{2}_{j=1} \biggl\Vert \frac{\partial\psi}{\partial x_{j}} \biggr\Vert _{L^{2} ( \Omega ) }+ \biggl\Vert \frac{\partial\psi}{\partial z} \biggr\Vert _{L^{2} ( \Omega ) } \Biggr) \\ &\qquad{} + \bigl( 2cc_{1}C+cc_{2}C+C^{\prime\prime\prime} \bigr) \Vert \psi \Vert _{H^{1} ( \Omega ) }+\widehat{\alpha}\sqrt{ \vert \Omega \vert } \Vert \psi \Vert _{H^{1} ( \Omega ) }\\ &\qquad{}+ \Vert \hat{f} \Vert _{L^{2} ( \Omega ) } \Vert \psi \Vert _{H^{1} ( \Omega ) }. \end{aligned}$$
Then we get (3.7) for \(i=1\).

Choosing now \(\hat{\phi}=(\hat{u}_{1}^{\varepsilon}, \hat{u} _{2}^{\varepsilon}\pm\psi, \hat{u}_{3}^{\varepsilon})\) in (3.14), we obtain (3.7) for \(i=2\), and we choose \(\hat{\phi}=(\hat {u}_{1}^{\varepsilon }, \hat{u}_{2}^{\varepsilon},\hat{u}_{3}^{\varepsilon}\pm\psi)\) for getting (3.8). □

4 Convergence results and limit problem

Theorem 4.1

Under the same assumptions as in Theorem  3.1, there exist \(u^{\star}=(u_{1}^{\star},u_{2}^{\star })\in V_{z}\) and \(p^{\star}\in L_{0}^{2}(\Omega)\), such that
$$\begin{aligned} &\hat{u}_{i}^{\varepsilon} \rightharpoonup u_{i}^{\star} \quad( 1\leq i\leq2 ) \textit{ weakly in} V_{z}, \end{aligned}$$
(4.1)
$$\begin{aligned} &\varepsilon\frac{\partial\hat{u}_{i}^{\varepsilon}}{\partial x_{j}} \rightharpoonup0 \quad( 1\leq i,j\leq2 ) \textit{ weakly in }L^{2} ( \Omega ) , \end{aligned}$$
(4.2)
$$\begin{aligned} &\varepsilon\frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial z} \rightharpoonup0 \quad\textit{weakly in }L^{2} ( \Omega ) , \end{aligned}$$
(4.3)
$$\begin{aligned} &\varepsilon^{2}\frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial x_{i}} \rightharpoonup0\quad ( 1\leq i\leq2 ) \textit{ weakly in }L^{2} ( \Omega ) , \end{aligned}$$
(4.4)
$$\begin{aligned} &\varepsilon\hat{u}_{3}^{\varepsilon} \rightharpoonup 0\quad \textit{weakly in }L^{2} ( \Omega ) , \end{aligned}$$
(4.5)
$$\begin{aligned} &\widehat{p}^{\varepsilon} \rightharpoonup p^{\star} \quad\textit{weakly in }L^{2} ( \Omega ) , \textit{ depend only of }x^{\prime }. \end{aligned}$$
(4.6)
For the proof of this theorem, we follow the same steps as in [6].

Theorem 4.2

With the same assumptions of Theorem  3.1, the solution \((u^{\star},p^{\star})\) satisfies the following relations:
$$\begin{aligned} &\sum_{i=1} ^{{2} } \int _{\Omega}\frac{\partial u_{i}^{\star}}{\partial z}\frac{\partial}{\partial z} \bigl(\hat{ \phi}_{i}-u_{i}^{\star} \bigr) \,dx^{\prime}\,dz- \int _{\Omega}p^{\star} \bigl( x^{\prime } \bigr) \biggl( \frac{\partial\hat{\phi}_{1}}{\partial x_{1}}+\frac{\partial\hat{\phi}_{2}}{\partial x_{2}} \biggr) \,dx^{\prime }\,dz \\ &\qquad{}- \int _{\omega}p^{\star} \bigl( x^{\prime} \bigr) \biggl[ \hat{\phi }_{1} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \frac{\partial h ( x^{\prime} ) }{\partial x_{1}}+\hat{\phi}_{2} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) \frac{\partial h ( x^{\prime} ) }{\partial x_{2}} \biggr] \,dx^{\prime} \\ &\qquad{}+ \sum_{i=1}^{{2}}\widehat{l} \int _{\omega}u_{i}^{\star } \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigl[ \hat{ \phi}_{i} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) -u_{i}^{\star} \bigl( x^{\prime },h \bigl(x^{\prime } \bigr) \bigr) \bigr] \,dx^{\prime} \\ &\qquad{}+ \int _{\omega}\hat{k} \bigl( \vert \hat{\phi} \vert - \bigl\vert u^{\star} \bigr\vert \bigr) \,dx^{\prime}+\widehat{ \alpha}\int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2}} \biggl( \frac{\partial\hat{\phi}_{i}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\qquad{}-\widehat{\alpha} \int _{\Omega} \Biggl(\frac{1}{2}\sum _{i=1} ^{{2}} \biggl( \frac{\partial u_{i}^{\star}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\quad \geq\sum _{i=1}^{{2}} \bigl(\hat{f}_{i},\hat{ \phi}_{i}-u_{i}^{\star} \bigr), \quad\forall\hat { \phi}\in \Pi(V). \end{aligned}$$
(4.7)

Proof

As \(\int_{\omega}\widehat{l}\varepsilon ^{2}\hat {u}_{3}^{\varepsilon}(x^{\prime},h(x^{\prime})\hat{u}_{3}^{\varepsilon }(x^{\prime},h(x^{\prime})))\sqrt{1+ \vert \nabla h(x^{\prime}) \vert ^{2}}\,dx^{\prime }\geq0\), then applying \(\lim_{\varepsilon\rightarrow0}\inf\) in the left part of (3.2) and the \(\lim_{\varepsilon \rightarrow0}\) in the right part of (3.2) and from the convergence results of Theorem 4.1, we obtain
$$\begin{aligned} &\sum_{i=1}^{{2}} \int _{\Omega}\frac{\partial u_{i}^{\star}}{\partial z}\frac{\partial u_{i}^{\star}}{\partial z}\,dx^{\prime}\,dz+ \sum_{i=1}^{2} \widehat{l} \int _{\omega }u_{i}^{\star} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) u_{i}^{\star} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \,dx^{\prime} \\ &\qquad{}+ \int _{\omega}\hat{k} \bigl\vert u^{\star} \bigr\vert \,dx^{\prime }+\widehat{\alpha} \int _{\Omega} \Biggl(\frac{1}{2}\sum _{i=1}^{{2}} \biggl( \frac{\partial u_{i}^{\star}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\quad \leq\sum _{i=1}^{ {2} } \int _{\Omega}\frac{\partial u_{i}^{\star}}{\partial z}\frac {\partial\hat{\phi}_{i}}{\partial z} \,dx^{\prime}\,dz \\ &\qquad{}+\sum_{i=1} ^{{2} }\widehat{l} \int _{\omega}u_{i}^{\star } \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \hat{\phi}_{i} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) \,dx^{\prime}+ \int _{\Omega}p^{\star} \bigl( x^{\prime} \bigr) \frac{\partial\hat{\phi}_{3}}{\partial z}\,dx^{\prime }\,dz \\ &\qquad{}+ \int _{\omega}\hat{k} \vert \hat{\phi} \vert \,dx^{\prime }+ \widehat{\alpha} \int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2}} \biggl( \frac{\partial\hat{\phi}_{i}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\qquad{}+ \int _{\Omega}p^{\star} \bigl( x^{\prime} \bigr) \biggl( \frac {\partial\hat{\phi}_{1}}{\partial x_{1}}+\frac{\partial\hat{\phi}_{2} }{\partial x_{2}} \biggr) \,dx^{\prime}\,dz+ \sum _{i=1}^{{2}} \int _{\Omega}\hat{f}_{i} \bigl( \hat{ \phi}_{i}-u_{i}^{\star} \bigr) \,dx^{\prime}\,dz \end{aligned}$$
(4.8)
as \(\int _{\Omega}p^{\star} ( x^{\prime} ) \frac {\partial\hat{\phi}_{3}}{\partial z}\,dx^{\prime}\,dz=\int _{\omega}p^{\star} ( x^{\prime} ) \hat{\phi}_{3} ( x^{\prime },h(x^{\prime}) ) \,dx^{\prime}\) and \(\hat{\phi}_{3}=\hat{\phi}_{1}\frac{\partial h}{\partial x_{1}}+\hat{\phi}_{2}\frac{\partial h}{\partial x_{2}}\), we have
$$ \int _{\Omega}p^{\star} \bigl( x^{\prime} \bigr) \frac{\partial \hat{\phi}_{3}}{\partial z}\,dx^{\prime}\,dz= \int _{\omega}p^{\star} \bigl( x^{\prime} \bigr) \biggl( \hat{\phi}_{1}\frac{\partial h}{\partial x_{1}}+\hat{\phi}_{2} \frac{\partial h}{\partial x_{2}} \biggr) \,dx^{\prime } $$
(4.9)
and from (4.8)-(4.9), we deduce (4.7). □

Remark 4.1

If ϕ̂ satisfies the condition \(( D^{\prime} ) \), the inequality (4.7) is reduced as follows:
$$\begin{aligned} &\sum_{i=1}^{2} \int _{\Omega}\frac{\partial u_{i}^{\star}}{\partial z} \biggl( \frac{\partial\hat{\phi }_{i}}{\partial z}- \frac{\partial u_{i}^{\star}}{\partial z} \biggr) \,dx^{\prime}\,dz+\sum _{i=1}^{{2} }\widehat{l} \int _{\omega}u_{i}^{\star} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigl( \hat{ \phi}_{i} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) -u_{i}^{\star} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) \bigr) \,dx^{\prime} \\ &\qquad{}+ \int _{\omega}\hat{k} \bigl( \vert \hat{\phi} \vert - \bigl\vert u^{\star} \bigr\vert \bigr) \,dx^{\prime}+\widehat{\alpha} \int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac{\partial\hat{\phi}_{i}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\qquad -\widehat{\alpha} \int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2}} \biggl( \frac{\partial u_{i}^{\star}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime }\,dz \\ &\quad{}\geq\sum^{2}_{i=1} \int _{\Omega}\hat{f}_{i} \bigl( \hat{\phi }_{i}-u_{i}^{\star} \bigr) \,dx^{\prime}\,dz,\quad \forall\hat{\phi }\in \sum ( V ) . \end{aligned}$$
(4.10)

Theorem 4.3

The limit functions \(u^{\star}\) and \(p^{\star}\) satisfy
$$\begin{aligned} &p^{\star} ( x_{1},x_{2},z ) =p^{\star} ( x_{1},x_{2} ) \quad\textit{in }\Omega\quad\textit{and}\quad p^{\star }\in H^{1} ( \omega ) , \end{aligned}$$
(4.11)
$$\begin{aligned} &{-}\mu\frac{\partial^{2}u_{i}^{\star}}{\partial z^{2}}+\frac{\partial p^{\star}}{\partial x_{i}} =\hat{f}_{i},\quad i=1,2\textit{ in }L^{2} ( \Omega ) . \end{aligned}$$
(4.12)

Proof

Choosing in (3.2) \(\hat{\phi}_{i}=\hat{u}_{i}^{\varepsilon}\) for \(i=1,2\), \(\hat{\phi}_{3}=\hat {u}_{3}^{\varepsilon}\pm\psi\) with \(\psi\in H_{0}^{1} ( \Omega ) \), we obtain
$$\begin{aligned} &\sum_{j=1} ^{{2} } \int _{\Omega}\mu\varepsilon ^{2} \biggl( \varepsilon^{2}\frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial x_{j}}+\frac{\partial\hat{u}_{j}^{\varepsilon}}{\partial z} \biggr) \frac {\partial\psi}{\partial x_{j}}\,dx^{\prime}\,dz+ \int _{\Omega } \biggl( 2\mu\varepsilon^{2} \frac{\partial\hat{u}_{3}^{\varepsilon }}{\partial z}-\widehat{p}^{\varepsilon} \biggr) \frac{\partial\psi}{\partial z} \,dx^{\prime}\,dz \\ &\qquad{}+\overset{2}{\sum_{i=1}} \int _{\Omega}\varepsilon^{4}\hat{u}_{i}^{\varepsilon} \frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial x_{i}}\psi \,dx^{\prime}\,dz+ \int _{\Omega}\varepsilon^{4}\hat{u}_{3}^{\varepsilon} \frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial z}\psi \,dx^{\prime}\,dz\\ &\quad = \int_{\Omega}\varepsilon\hat{f}_{3}\psi \,dx^{\prime}\,dz. \end{aligned}$$
By the results of convergence (4.1)-(4.7), we deduce
$$ \int _{\Omega}p^{\star}\frac{\partial\psi}{\partial z}\,dx^{\prime } \,dz=0, \quad\forall\psi\in H_{0}^{1} ( \Omega ) , $$
(4.13)
then
$$ \frac{\partial p^{\star}}{\partial z}=0\quad\text{in }H^{-1} ( \Omega ) . $$
(4.14)
Choosing now \(\hat{\phi}_{i}=\hat{u}_{i}^{\varepsilon}\pm\psi_{i}\) for \(i=1,2\), with \(\psi_{i}\in H_{0}^{1} ( \Omega ) \) and \(\hat {\phi}_{3}=\hat{u}_{3}^{\varepsilon}\) in (3.2) we obtain
$$\begin{aligned} &\sum_{1\leq i,j\leq2} \int _{\Omega} \biggl[ \mu\varepsilon ^{2} \biggl( \frac{\partial\hat{u}_{i}^{\varepsilon}}{\partial x_{j}}+\frac{\partial\hat{u}_{j}^{\varepsilon}}{\partial x_{i}} \biggr) -\delta_{ij} \widehat{p}^{\varepsilon} \biggr] \frac{\partial\psi_{i}}{\partial x_{j}}\,dx^{\prime}\,dz \\ &\qquad{}+\mu\sum_{i=1}^{{2}} \int _{\Omega} \biggl( \frac {\partial\hat{u}_{i}^{\varepsilon}}{\partial z}+\varepsilon^{2} \frac {\partial\hat{u}_{3}^{\varepsilon}}{\partial x_{i}} \biggr) \frac {\partial\psi_{i}}{\partial z}\,dx^{\prime}\,dz+\sum _{1 \leq i,j\leq 2} \int _{\Omega}\varepsilon^{2}\hat{u}_{i}^{\varepsilon} \frac {\partial\hat{u}_{j}^{\varepsilon}}{\partial x_{i}}\psi_{j}\,dx^{\prime }\,dz \\ &\qquad{}+\overset{2}{\sum_{i=1}} \int _{\Omega}\varepsilon^{2}\hat{u}_{3}^{\varepsilon} \frac{\partial\hat{u}_{i}^{\varepsilon}}{\partial z}\psi_{i}\,dx^{\prime}\,dz\\ &\quad = \sum _{i=1}^{{2} } \int_{\Omega }\hat {f}_{i}\psi_{i} \,dx^{\prime}\,dz. \end{aligned}$$
Using (4.1)-(4.7), and choosing \(\psi_{1}=0\) and \(\psi_{2}\in H_{0}^{1} ( \Omega ) \), then \(\psi_{2}=0\) and \(\psi_{1}\in H_{0}^{1} ( \Omega ) \), we obtain
$$ \sum_{i=1}^{-{2} } \int _{\Omega}p^{\star}\frac {\partial\psi_{i}}{\partial x_{i}}\,dx^{\prime} \,dz+ \mu\sum_{i=1}^{{2}} \int _{\Omega}\frac{\partial u_{i}^{\star}}{\partial z}\frac{\partial\psi_{i}}{\partial z} \,dx^{\prime}\,dz= \sum_{i=1}^{{2} } \int_{\Omega}\hat{f}_{i}\psi_{i} \,dx^{\prime}\,dz. $$
(4.15)
By utilizing the Green formula, we obtain (4.12) but in \(H^{-1} ( \Omega ) \).
To prove \(p^{\star}\in H^{1} ( \omega ) \), we see from (4.14) that \(p^{\star}\) does not depend on z, then following [23], we choose \(\psi_{i}\) in (4.15) such that \(\psi_{i} ( x^{\prime},z ) =z ( z-h(x^{\prime}) ) \theta ( x^{\prime} ) \) with \(\theta\in H_{0}^{1} ( \omega ) \), and using the Green formula, we obtain
$$\frac{1}{6} \int _{\omega}p^{\star}\frac{\partial(h^{3}\theta )}{\partial x_{i}} \,dx^{\prime}-2 \int _{\omega}h\tilde{u}_{i}^{\star}\theta \,dx^{\prime}+ \int _{\omega}h \bigl(x^{\prime} \bigr) \bigl[ u_{i}^{\star} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) +u_{i}^{\star} \bigl( x^{\prime},0 \bigr) \bigr] \theta \,dx^{\prime}= \int _{\omega}\widetilde{f_{i}}\theta \,dx^{\prime}, $$
where
$$\tilde{u}_{i}^{\star}=\frac{1}{h ( x^{\prime} ) }{ \int _{0}^{h ( x^{\prime} ) }} u_{i}^{\star} \bigl( x^{\prime},z \bigr) \,dz\quad\text{and}\quad\widetilde {f}_{i}={ \int _{0}^{h ( x^{\prime} ) }} z \bigl( z-h \bigl(x^{\prime} \bigr) \bigr) \hat{f}_{i} \bigl( x^{\prime},z \bigr) \,dz; $$
so
$$ h \bigl(x^{\prime} \bigr) \bigl[ u_{i}^{\star} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) +u_{i}^{\star} \bigl( x^{\prime},0 \bigr) \bigr] -2h\tilde {u}_{i}^{\star }- \frac{1}{6}h^{3}\frac{\partial p^{\star}}{\partial x_{i}}=\widetilde {f}_{i},\quad i=1,2\text{ in }H^{-1} ( \omega ) $$
(4.16)
as \(\hat{f}_{i}\in L^{2} ( \Omega ) \), \(u_{i}^{\star}\in V_{z}\) in particular in \(L^{2} ( \omega ) \), therefore \(\tilde {u}_{i}^{\star }\) and \(\widetilde{f}_{i}\) are in \(L^{2} ( \omega ) \). Then from (4.16) we get (4.11). As \(\hat{f}_{i}\in L^{2} ( \omega ) \), from (4.16) we have \(\frac{\partial^{2}u_{i}^{\star}}{\partial z^{2}}\) in \(L^{2} ( \Omega ) \); then (4.12) holds. □

Theorem 4.4

Under the assumptions of the preceding theorems, the traces \(s^{\star}\), \(\tau^{\star}\) satisfy the following equality:
$$ \int _{\omega} \biggl( \frac{h^{3}}{12\mu}\nabla p^{\star}- \frac{h}{2}s^{\star}-\frac{h}{2}s_{h}^{\star}+ \widetilde{F} \biggr) \nabla\hat {\phi }\,dx^{\prime}+ \int _{\partial\omega}h\widetilde{u}^{\star}\phi \cdot n\,d\sigma=0, \quad \forall\hat{\phi}\in H^{1} ( \omega ) , $$
(4.17)
where
$$\begin{aligned} &s^{\star} \bigl( x^{\prime} \bigr) =u^{\star} \bigl( x^{\prime},0 \bigr) ,\qquad s_{h}^{\star} \bigl( x^{\prime} \bigr) =u^{\star} \bigl( x^{\prime },h \bigl( x^{\prime} \bigr) \bigr) ,\\ & \tau^{\star}=\frac {\partial u^{\star}}{\partial z} \bigl( x^{\prime},0 \bigr) ,\qquad \tau _{h}^{\star}= \frac{\partial u^{\star}}{\partial z} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr), \\ &\widetilde{g} \bigl( x^{\prime} \bigr) ={ \int _{0}^{h ( x^{\prime} ) }} \widehat{g} \bigl( x^{\prime},z \bigr) \,dz, \quad\forall x^{\prime}\in \partial \omega, \\ &F_{i} \bigl( x^{\prime},z \bigr) ={ \int _{0}^{z}} { \int _{0}^{\zeta}} \hat{f}_{i} \bigl( x^{\prime},z \bigr) \,dt\,d\zeta,\qquad \widetilde{F}_{i} \bigl( x^{\prime} \bigr) =\frac{1}{\mu}{ \int _{0}^{h ( x^{\prime} ) }} F_{i} \bigl( x^{\prime},z \bigr) \,dz-\frac{h}{2\mu}F \bigl(x^{\prime },h \bigl(x^{\prime } \bigr) \bigr). \end{aligned}$$

Proof

Integrating twice (4.12) between 0 and z, we obtain
$$ \mu u^{\star} \bigl( x^{\prime},z \bigr) =\mu s^{\star} \bigl( x^{\prime } \bigr) +\frac{z^{2}}{2}\nabla p^{\star}+\mu z \tau^{\star}-F \bigl( x^{\prime},z \bigr) . $$
(4.18)
Replacing z by h, we obtain
$$\mu u^{\star} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) = \mu s^{\star} \bigl( x^{\prime} \bigr) +\frac{h^{2}}{2}\nabla p^{\star}+\mu h\tau^{\star }-F \bigl( x^{\prime},h \bigr) $$
so
$$ h\tau^{\star}=\frac{1}{\mu}F \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) +s_{h}^{\star} \bigl( x^{\prime} \bigr) -s^{\star} \bigl( x^{\prime } \bigr) - \frac{h^{2}}{2\mu}\nabla p^{\star}; $$
(4.19)
integrating (4.18) with respect to z, in the interval \((0,h(x^{\prime}))\), we obtain
$$ h\widetilde{u}^{\star} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) =hs^{\star } \bigl( x^{\prime} \bigr) + \frac{h^{3}}{6\mu} \nabla p^{\star}+\frac {h^{2}}{2}\tau^{\star}- \frac{1}{\mu}{ \int _{0}^{z}} F \bigl( x^{\prime},z \bigr) \,dz $$
(4.20)
we set for any function φ
$$\widetilde{\varphi} \bigl( x^{\prime} \bigr) =\frac{1}{h(x^{\prime})}{ \int _{0}^{h ( x^{\prime} ) }} \varphi \bigl( x^{\prime},z \bigr) \,dz, \quad\forall x^{\prime} \in \omega, $$
on the other hand, \(\forall\phi\in H^{1}(\omega)\),
$$\begin{aligned} { \int _{\Omega}} \phi\operatorname{div} \bigl( \hat{u}^{\varepsilon} \bigr)\,dx^{\prime}\,dz & =0={ \int _{\omega}} \phi \bigl( x^{\prime} \bigr) { \int _{0}^{h(x^{\prime})}} { \sum_{i=1}^{2}} \biggl( \frac{\partial\hat{u}_{i}^{\varepsilon}}{\partial x_{i}}+\frac{\partial\hat{u}_{3}^{\varepsilon}}{\partial z} \biggr) \,dz\,dx^{\prime} \\ & ={ \int _{\omega}} \phi \bigl( x^{\prime} \bigr) \Biggl[ { \sum_{i=1}^{2}} \frac{\partial ( h\widetilde{\hat{u}}_{i}^{\varepsilon} ) }{\partial x_{i}}+\hat{u}_{3}^{\varepsilon} \bigl( x^{\prime },h \bigl(x^{\prime } \bigr) \bigr) -\hat{u}_{3}^{\varepsilon} \bigl(x^{\prime},0 \bigr) \Biggr] \,dx^{\prime}, \end{aligned}$$
we have \(\hat{u}_{3}^{\varepsilon}(x^{\prime},0)=0\) in ∂Ω. Then
$${ \int _{\omega}} \phi \bigl( x^{\prime} \bigr) { \sum_{i=1}^{2}} \frac{\partial ( h\widetilde{\hat{u}}_{i}^{\varepsilon} ) }{\partial x_{i}}\,dx^{\prime}=0.$$
Using the Green formula, we get
$${ \sum_{i=1}^{2}} { \int _{\omega}} \frac{\partial\phi}{\partial x_{i}}\widetilde{ \hat{u}}_{i}^{\varepsilon }h\,dx^{\prime}={ \sum_{i=1}^{2}} { \int _{\partial\omega}} h\widetilde{\hat{u}}_{i}^{\varepsilon} \phi\cdot n_{i}\,d\sigma.$$
As \(\hat{u}_{i}^{\varepsilon}\rightharpoonup u^{\star}\) in \(V_{z}\) and consequently in \(L^{2} ( \omega ) \), therefore \(\widetilde {\hat {u}_{i}^{\varepsilon}}\rightharpoonup\widetilde{u}_{i}^{\star}\) in \(L^{2} ( \omega ) \), we deduce
$$ { \sum_{i=1}^{2}} { \int _{\omega}} \frac{\partial\phi}{\partial x_{i}}\widetilde{u}_{i}^{\star }h \,dx^{\prime}={ \sum _{i=1}^{2}} { \int _{\partial\omega}} h\widetilde{u}_{i}^{\star} \phi\cdot n_{i}\,d\sigma,\quad \forall\phi \in H^{1}( \omega), $$
(4.21)
using (4.19) to eliminate the term containing \(\tau^{\star}\) from (4.20). Multiplying now (4.20) by ∇ϕ, then integrating it in ω and using (4.21), we obtain (4.16). □

Theorem 4.5

The solution \((u^{\star },p^{\star})\) of our limit problem is unique.

Proof

Let \((U^{1},p^{1})\), \((U^{2},p^{2})\) be two solutions of the limit problem (4.10), then
$$\begin{aligned} &\sum_{i=1} ^{{2} } \int _{\Omega}\frac{\partial U_{i}^{1}}{\partial z} \biggl( \frac{\partial\hat{\phi}_{i}}{\partial z}- \frac{\partial U_{i}^{1}}{\partial z} \biggr) \,dx^{\prime}\,dz \\ &\qquad{}+\sum _{i=1}^{{2} }\widehat{l} \int _{\omega}U_{i}^{1} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigl( \hat{ \phi}_{i} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) -U_{i}^{1} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigr) \,dx^{\prime} \\ &\qquad{}+ \int _{\omega}\hat{k} \bigl( \vert \hat{\phi} \vert - \bigl\vert U^{1} \bigr\vert \bigr) \,dx^{\prime}+\widehat{ \alpha}\int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1} ^{{2} } \biggl( \frac{\partial\hat{\phi}_{i}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\qquad{}-\widehat{\alpha} \int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac{\partial U_{i}^{1}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\quad\geq\sum_{i=1} ^{{2} } \int _{\Omega}\hat{f}_{i} \bigl( \hat{ \phi}_{i}-U_{i}^{1} \bigr) \,dx^{\prime}\,dz, \quad \forall\hat{\phi}\in\sum ( V ) , \end{aligned}$$
(4.22)
and
$$\begin{aligned} &\sum_{i=1}^{{2} } \int _{\Omega}\frac{\partial U_{i}^{2}}{\partial z} \biggl( \frac{\partial\hat{\phi}_{i}}{\partial z}- \frac{\partial U_{i}^{2}}{\partial z} \biggr) \,dx^{\prime}\,dz \\ &\qquad{}+\sum _{i=1}^{{2}}\widehat{l} \int _{\omega}U_{i}^{2} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigl( \hat{ \phi}_{i} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) -U_{i}^{2} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigr) \,dx^{\prime} \\ &\qquad{}+ \int _{\omega}\hat{k} \bigl( \vert \hat{\phi} \vert - \bigl\vert U^{2} \bigr\vert \bigr) \,dx^{\prime}+\widehat{ \alpha}\int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac{\partial\hat{\phi}_{i}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\qquad{}-\widehat{\alpha} \int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac{\partial U_{i}^{2}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\quad\geq\sum_{i=1}^{{2}} \int _{\Omega}\hat{f}_{i} \bigl( \hat{ \phi}_{i}-U_{i}^{2} \bigr) \,dx^{\prime}\,dz,\quad \forall\hat{\phi}\in\sum ( V ) . \end{aligned}$$
(4.23)
Taking \(\hat{\phi}=U^{2}\) in (4.22) and \(\hat{\phi}=U^{1}\) in (4.23), we obtain
$$\begin{aligned} &\sum_{i=1}^{{2}} \int _{\Omega}\frac{\partial U_{i}^{1}}{\partial z} \biggl( \frac{\partial U_{i}^{2}}{\partial z}- \frac {\partial U_{i}^{1}}{\partial z} \biggr) \,dx^{\prime}\,dz \\ &\qquad{}+\sum _{i=1}^{{2} }\widehat{l} \int _{\omega}U_{i}^{1} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) \bigl( U_{i}^{2} \bigl( x^{\prime},h \bigl(x^{\prime } \bigr) \bigr) -U_{i}^{1} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigr) \,dx^{\prime} \\ &\qquad{}+ \int _{\omega}\hat{k} \bigl( \bigl\vert U^{2} \bigr\vert - \bigl\vert U^{1} \bigr\vert \bigr) \,dx^{\prime}+ \widehat{\alpha} \int _{\Omega } \Biggl( \frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac {\partial U_{i}^{2}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime }\,dz \\ &\qquad{}-\widehat{\alpha} \int _{\Omega} \Biggl( \frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac{\partial U_{i}^{1}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\quad\geq\sum_{i=1}^{{2} } \int _{\Omega}\hat{f}_{i} \bigl( U_{i}^{2}-U_{i}^{1} \bigr) \,dx^{\prime}\,dz , \end{aligned}$$
(4.24)
$$\begin{aligned} &\sum_{i=1}^{{2} } \int _{\Omega}\frac{\partial U_{i}^{2}}{\partial z} \biggl( \frac{\partial U_{i}^{1}}{\partial z}- \frac {\partial U_{i}^{2}}{\partial z} \biggr) \,dx^{\prime}\,dz \\ &\qquad{}+\sum _{i=1}^{{2} }\widehat{l} \int _{\omega}U_{i}^{2} \bigl( x^{\prime },h \bigl(x^{\prime} \bigr) \bigr) \bigl( U_{i}^{1} \bigl( x^{\prime},h \bigl(x^{\prime } \bigr) \bigr) -U_{i}^{2} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigr) \,dx^{\prime} \\ &\qquad{}+ \int _{\omega}\hat{k} \bigl( \bigl\vert U^{1} \bigr\vert - \bigl\vert U^{2} \bigr\vert \bigr) \,dx^{\prime}+ \widehat{\alpha} \int _{\Omega } \Biggl( \frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac {\partial U_{i}^{1}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime }\,dz \\ &\qquad{}-\widehat{\alpha} \int _{\Omega} \Biggl(\frac{1}{2}\sum _{i=1}^{{2} } \biggl( \frac{\partial U_{i}^{2}}{\partial z} \biggr) ^{2} \Biggr) ^{\frac{1}{2}}\,dx^{\prime}\,dz \\ &\quad\geq\sum_{i=1} ^{{2} } \int _{\Omega}\hat{f}_{i} \bigl( U_{i}^{1}-U_{i}^{2} \bigr) \,dx^{\prime}\,dz. \end{aligned}$$
(4.25)
By adding the two inequalities (4.24), (4.25), it follows that
$$\begin{aligned} & \sum_{i=1}^{{2} } \int _{\Omega}\frac{\partial U_{i}^{1}}{\partial z} \biggl( \frac{\partial U_{i}^{2}}{\partial z}- \frac{\partial U_{i}^{1}}{\partial z} \biggr) \,dx^{\prime}\,dz+\sum _{i=1} ^{{2} } \int _{\Omega}\frac{\partial U_{i}^{2}}{\partial z} \biggl( \frac{\partial U_{i}^{1}}{\partial z}- \frac {\partial U_{i}^{2}}{\partial z} \biggr) \,dx^{\prime}\,dz \\ & \quad\geq\sum_{i=1}^{2}\widehat{l} \int _{\omega } \bigl\vert U_{i}^{2} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) -U_{i}^{1} \bigl( x^{\prime},h \bigl(x^{\prime} \bigr) \bigr) \bigr\vert ^{2}\,dx^{\prime }, \end{aligned}$$
then \(\sum_{i=1}^{{2}}\int _{\Omega } \vert \frac{\partial}{\partial z} ( U_{i}^{2}-U_{i}^{1} ) \vert ^{2}\,dx^{\prime}\,dz\leq0\).
Using the Poincaré inequality, we deduce that
$$ \bigl\Vert U_{i}^{2}-U_{i}^{1} \bigr\Vert _{V_{z}}=0, $$
(4.26)
so \(u^{\star}\) is unique.
The uniqueness of \(p^{\star}\) in \(L_{0}^{2}(\omega )\cap H^{1}(\omega)\) follows then from the specific weak Reynolds equation (4.17), indeed we have
$$\begin{aligned} & \int _{\omega} \biggl( \frac{h^{3}}{12}\nabla p^{1}- \frac{h}{2}U^{1} \bigl(x^{\prime},0 \bigr)- \frac{h}{2}U^{1} \bigl(x^{\prime},h \bigl(x^{\prime } \bigr) \bigr)+\widetilde {F} \biggr) \nabla\hat{\phi}\,dx^{\prime}+ \int _{\partial\omega }h\widetilde{U^{1}}\hat{\phi}\cdot n\,d \sigma =0 \\ &\quad\forall\hat{\phi}\in H^{1} ( \omega ) , \end{aligned}$$
(4.27)
$$\begin{aligned} & \int _{\omega} \biggl( \frac{h^{3}}{12}\nabla p^{2}- \frac{h}{2}U^{2} \bigl(x^{\prime},0 \bigr)- \frac{h}{2}U^{2} \bigl(x^{\prime},h \bigl(x^{\prime } \bigr) \bigr)+\widetilde {F} \biggr) \nabla\hat{\phi}\,dx^{\prime}+ \int _{\partial\omega }h\widetilde{U^{2}}\hat{\phi}\cdot n\,d \sigma =0 \\ &\quad\quad\forall\hat{\phi}\in H^{1} ( \omega ). \end{aligned}$$
(4.28)
Subtracting (4.28) and (4.27) and using (4.26), we obtain
$$\int _{\omega}\frac{h^{3}}{12} \bigl( \nabla p^{1}- \nabla p^{2} \bigr) \nabla\hat{\phi}\,dx^{\prime}=0, $$
taking \(\hat{\phi}=p^{1}-p^{2}\) and by Poincaré’s inequality, we get \(\Vert p^{1}-p^{2} \Vert _{L^{2} ( \omega ) }=0\).

This ends the proof of the uniqueness. □

5 Conclusions

In this research, we studied the asymptotic analysis of a non-linear problem in a three-dimensional thin domain \(\Omega^{\varepsilon}\) with Fourier and Tresca boundary conditions. Firstly, the problem statement and variational formulation were formulated. We then have studied the asymptotic behavior in the one-dimensional case when the domain parameter tends to zero. In the latter case, the main results concerning the limit of weak problem and its uniqueness have been obtained.

Declarations

Acknowledgements

The authors would like to thank very much the referees for their expert and constructive comments, which have further enriched our article.

This work is supported by the MESRS of Algeria (CNEPRU Project C00L03UN190120150002).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Bordj Bou Arreridj University
(2)
Applied Math Lab, Department of Mathematics, Setif 1 University

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