Homogenization is a mathematical tool that allows changing the scale in problems containing several characteristic scales. Typical examples of its utilization are finding effective models for composite materials, in optimal shape design, etc. Another important example, which we are interested in, is the fluid mechanics of the flow through porous medium.

In porous medium, there are at least two length scales: microscopic scale and macroscopic scale. The partial differential equations describing a physical phenomenon are posed at the microscopic level whereas only macroscopic quantities are of interest for the engineers or the physicists. Therefore, effective or homogenized equations should be derived from the microscopic ones by an asymptotic analysis. To this end, it is convenient to assume that the porous medium has a periodic structure.

A number of known laws from the dynamics of fluids in porous media were derived using homogenization. The most well-known example is Darcy's law, being the homogenized equation for one-phase flow through a rigid porous medium. Its formal derivation by two-scale expansion goes back to the classical paper by Sanchez- Palencia [1], Keller [2] and the classical book Bensoussan [3]. It was rigorously derived by using oscillating functions by Tartar [4]. In other cases of periodic porous media, we refer the readers to the papers by Allaire [5-8] and Mikelic [9, 10]. Other works can be seen in [11-13] and the references therein. Besides the Darcy law, Brinkman [14] introduced a new set of equations, which is called the Brinkman law, an intermediate law between the Darcy and Stokes equations. The so-called Brinkman law is obtained from the Stokes equations by adding to the momentum equation a term proportional to the velocity (see [7]).

Inspired by the work from Feireisl [11], we consider the asymptotic behavior of a compressible fluid in a periodic medium. Before stating the system, let us recall the domain we consider. A porous medium is defined as the periodic repetition of an elementary cell of size $ \varepsilon $ (we assume that $ \frac{1}{\varepsilon} $ to be an integral) in a bounded domain $ \Omega $ of $ R^n $ with $ n = 2,3 $. The solid part of the porous medium is also taken of size $ \varepsilon $. The domain $ \Omega_\varepsilon $ is then defined as the intersection of $ \Omega $ with the fluid part. We consider the density dependent fluid governed by the full compressible Navier-Stokes equations. So, we have the following equations

$ \begin{equation} \left\{\begin{aligned} &\varepsilon^2\frac{\partial \rho_\varepsilon}{\partial t}+ \rm{div}(\rho_\varepsilon u_\varepsilon) = 0\; \; \; \; \rm{in}\; \; \Omega_\varepsilon\times(0,\mathit{T}),\\ &\varepsilon^2\frac{\partial( \rho_\varepsilon u_\varepsilon)}{\partial t}+ \rm{div}(\rho_\varepsilon u_\varepsilon\otimes u_\varepsilon)- \rm{div}(\mu \nabla u_\varepsilon)+\nabla p_\varepsilon = \rho_\varepsilon f\; \; \rm{ in}\; \Omega_\varepsilon\times(0,\mathit{T}),\\ &\varepsilon^2 \frac{\partial }{\partial t}\{\rho_\varepsilon(\frac{|u_\varepsilon|^2}{2}+e_\varepsilon)\}+ \rm{div}\{u_\varepsilon(\rho_\varepsilon\frac{|u_\varepsilon|^2}{2} +\rho_\varepsilon e_\varepsilon+p_\varepsilon)\} \\ = &\; \; \rm{div}(\kappa \nabla \mathbb{T}_\varepsilon) + \rm{div}(\mu \nabla u_\varepsilon\cdot u_\varepsilon)+\rho_\varepsilon u_\varepsilon f \; \; \rm{in}\; \; \Omega_\varepsilon\times(0,\mathit{T}), \end{aligned} \right. \end{equation} $

(1.1)

where $ u_\varepsilon, \rho_\varepsilon, \mathbb{T}_\varepsilon $ are the unknown quantities velocity, density and temperature. $ p_\varepsilon = p_\varepsilon(\rho_\varepsilon,\mathbb{T}_\varepsilon) $ is the pressure, $ e_\varepsilon = e_\varepsilon(\rho_\varepsilon,\mathbb{T}_\varepsilon) $ is the internal energy, $ f $ (the exterior force) is given on $ \Omega\times(0,T) $. We assume that $ f $ is smooth enough. $ T\in (0,\infty) $ is fixed; $ \kappa = \kappa(\mathbb{T}_\varepsilon), \mu = \mu(\mathbb{T}_\varepsilon) $ are positive for $ \mathbb{T}_\varepsilon\geq0 $ and $ \kappa,\mu\in W^{1,\infty}([0,\infty)) $.

We also assume that $ u_\varepsilon $ satisfies

$ \begin{equation} u_\varepsilon = 0\; \; \; \; \rm{on}\; \; \partial\Omega_\varepsilon\times(0,\mathit{T}), \end{equation} $

(1.2)

and in order to fix ideas we impose Neumann boundary conditions on $ \mathbb{T}_\varepsilon $ namely

$ \begin{equation} \frac{\partial\mathbb{T}_\varepsilon}{\partial n} = 0 \; \; \; \; \rm{on}\; \; \partial\Omega_\varepsilon\times(0,\mathit{T}), \end{equation} $

(1.3)

where $ n $, as usual, the unit outward normal to $ \partial\Omega_\varepsilon $.

In this paper, we assume that the initial conditions

$ \begin{equation} \rho_\varepsilon|_{t = 0} = \rho_{0,\varepsilon},\; \; \; u_\varepsilon|_{t = 0} = u_{0,\varepsilon}, \; \; \mathbb{T}_\varepsilon|_{t = 0} = \mathbb{T}_{0,\varepsilon},\; \; e_\varepsilon|_{t = 0} = e_{0,\varepsilon} \end{equation} $

(1.4)

are bounded in $ L^\infty(\Omega_\varepsilon) $.

In this paper, we also assume that the transport coefficients $ \mu(\mathbb{T}_{\varepsilon}) $ and $ \kappa(\mathbb{T}_{\varepsilon}) $ satisfying the following conditions

$ \begin{equation} \begin{aligned} &\kappa,\; \mu\in W^{1,\infty}[0,+\infty),\; \; 0<\underline{\mu}(1+\mathbb{T}_{\varepsilon}) \leq \mu(\mathbb{T}_{\varepsilon}),\\ &0<\underline{\kappa}(1+\mathbb{T}_{\varepsilon}^3)\leq \kappa(\mathbb{T}_{\varepsilon})\leq \overline{\kappa}(1+\mathbb{T}_{\varepsilon}^3) \end{aligned} \end{equation} $

(1.5)

for all $ \mathbb{T}_{\varepsilon}\geq 0 $. $ \underline{\mu},\underline{\kappa}, \overline{\kappa} $ are positive constants.

Let us recall that the equation with temperature in (1.1), it is equivalent (at least formally) to

$ \begin{equation} \begin{aligned} &\varepsilon^2 \frac{\partial }{\partial t}(\rho_\varepsilon e_\varepsilon)+ \rm{div}(\rho_\varepsilon u_\varepsilon e_\varepsilon) - \rm{div}(\kappa \nabla \mathbb{T}_\varepsilon)-\mu| \nabla u_\varepsilon|^2 +p_\varepsilon \rm{div}u_\varepsilon = 0. \end{aligned} \end{equation} $

(1.6)

For simplicity, in this paper, we consider the models in astrophysics and the state equation for the pressure $ p_\varepsilon $ and the internal energy $ e_\varepsilon $ satisfying the Joule's law (see [15])

$ \begin{equation*} \begin{aligned} p_\varepsilon = a \rho_\varepsilon^\gamma+b\rho_\varepsilon^\beta \mathbb{T}_{\varepsilon},\; \; \; \; \; e_\varepsilon = \frac{a}{\gamma-1} \rho_\varepsilon^{\gamma-1}+c\mathbb{T}_{\varepsilon}^{\frac{3}{2}}, \end{aligned} \end{equation*} $

where $ \gamma\geq n $ for $ n = 2,3 $, $ \frac{3}{2}<\beta<\gamma $, $ a,b,c $ are positive constants. Then the specific entropy reads,

$ \begin{equation*} \begin{aligned} s_\varepsilon( \rho_\varepsilon,\mathbb{T}_{\varepsilon}) = 3c\sqrt{\mathbb{T}_{\varepsilon}}-\frac{b}{\beta-1}\rho_\varepsilon^{\beta-1}. \end{aligned} \end{equation*} $

We assume that the initial condition

$ \begin{equation*} \begin{aligned} \rho_\varepsilon s_\epsilon^+(\rho_\varepsilon,\mathbb{T}_\varepsilon)\in C([0,+\infty)^2),\; \; \rho_\varepsilon (s_\varepsilon)^-(x,0)\; \; \rm{is\; bounded\; in}\; \; L^1(\Omega_\varepsilon). \end{aligned} \end{equation*} $

Let us also recall that, at least formally, the following identity holds

$ \begin{equation} \begin{aligned} &\varepsilon^2 \frac{\partial }{\partial t}(\rho_\varepsilon s_\varepsilon)+ \rm{div}(\rho_\varepsilon u_\varepsilon s_\varepsilon) - \rm{div}(\kappa\frac{\nabla \mathbb{T}_\varepsilon}{\mathbb{T}_\varepsilon}) = \mu(\mathbb{T}_{\varepsilon})\frac{| \nabla u_\varepsilon|^2}{\mathbb{T}_\varepsilon} +\kappa(\mathbb{T}_{\varepsilon}) \frac{|\nabla \mathbb{T}_\varepsilon|^2}{\mathbb{T}_\varepsilon^2}. \end{aligned} \end{equation} $

(1.7)

Our aim here is to investigate the asymptotic behaviors of $ \rho_\varepsilon,\; u_\varepsilon $ and $ \mathbb{T}_{\varepsilon} $ as $ \varepsilon\rightarrow 0 $ under the assumptions mentioned above. The main difficulty in this paper is how to pass the limit in the momentum and energy equations. To overcome this obstacle, we have to regularize the system both in time and in space before we can pass the limit. In this paper, we exert the conditions on the entropy to get the estimates. Moreover, we rigorously proved that the low boundary of $ \gamma $ would be $ n $ when passing the limit to the convection term. At the limit process, we fall back on the two-scale convergence method to obtain the homogenized model. Those are quite different from [11].

Let $ \Omega $ be an open bounded subset of $ R^n $ with $ n = 2 $ or $ 3 $ and defined $ \mathcal{Y} = [0,1]^n $ to be the unit open cube of $ R^n $. Let $ \mathcal{Y}_s $ be a closed smooth subset of $ \mathcal{Y} $ with a strictly positive measure. The fluid part is then defined by $ \mathcal{Y}_f = \mathcal{Y}-\mathcal{Y}_s $. Let $ \theta = |\mathcal{Y}_f| $. The constant $ \theta $ is called the porosity of the porous medium. We assume that $ 0<\theta<1. $

Repeating the domain $ \mathcal{Y}_f $ by $ \mathcal{Y} $-periodicity, we get the whole fluid domain $ D_f $, we can write it as

$ \begin{equation*} \begin{aligned} D_f = \{x\in R^n|\exists k\in Z^n\; \rm{such\; that\; }x-k\in \mathcal{Y}_\mathit{f}\}. \end{aligned} \end{equation*} $

Then the solid part is defined by $ D_s = R^n-D_f $. It is easy to see that $ D_f $ is a connected domain, while $ D_s $ is formed by separated smooth subsets. In the sequel, we denote for all $ k\in Z^n $, $ \mathcal{Y}^k = \mathcal{Y}+k $ and then $ \mathcal{Y}_f^k = \mathcal{Y}_f+k $. For all $ \varepsilon $, we define the domain $ \Omega_\varepsilon $ as the intersection of $ \Omega $ with the fluid domain scaled by $ \varepsilon $, namely, $ \Omega_\varepsilon = \Omega\cap \varepsilon D_f $. To get a smooth connected domain, we will not remove the solid part of the cells which intersect with the boundary of $ \Omega $. Now, the fluid domain can be also defined by

$ \begin{equation*} \begin{aligned} \Omega_\varepsilon = \Omega-\cup\{\varepsilon \mathcal{Y}_s^k, k\in Z^n, \varepsilon\mathcal{Y}^k\subset \Omega\}. \end{aligned} \end{equation*} $

Throughout this paper, we denote $ L^p(0,T;L^q(X)) $, the time-space Lebesgue spaces, where $ X $ would be $ \Omega $ or $ \Omega_\varepsilon $; $ W^{s,p}(X) $ is the classical Sobolev space with all functions, whose all derivatives up to order $ s $ belong to $ L^p $ and $ H^s(X) = W^{s,2}(X) $; $ W_0^{1,p}(X) $ is the subset of $ W^{1,p}(X) $ with trace 0 on $ X $. We also denote $ W^{-s,p'}(X) $, the dual space of $ W_0^{s,p}(X) $, where $ p' $ is the conjugate exponent of $ p $; $ C $ is a constant that may differ from one place to another. Throughout this paper, we use $ ||\cdot||_X $ to denote the modules for all vectors and matrixes if there is no confusion.

Due to the presence of the holes, the domain $ \Omega_\varepsilon $ depends on $ \varepsilon $ and hence to study the convergence of $ \{u_\varepsilon,\rho_\varepsilon,p_\varepsilon\} $, we have to extend the functions defined in $ \Omega_\varepsilon $ to the whole domain. This can be done in two different possible ways.

Definition 1.1   For any fixed $ \varepsilon\in L^1(\Omega_\varepsilon) $, we define

$ \begin{equation*} \widetilde{\varphi} = \left\{\begin{aligned} &\varphi\; \; \; \; {\rm in} \; \; \Omega,\\ &0\; \; \; \; {\rm in} \; \; \Omega-\Omega_\varepsilon \end{aligned} \right. \end{equation*} $

the null extension and

$ \begin{equation*} \widehat{\varphi} = \left\{\begin{aligned} &\varphi\; \; \; \; {\rm in} \; \; \Omega_\varepsilon,\\ &\frac{1}{|\varepsilon\mathcal{Y}_f^k|}\int_{\varepsilon\mathcal{Y}_f^k} \varphi(x)dx\; \; \; \; {\rm in} \; \; \Omega\cap\varepsilon\mathcal{Y}_s^k \end{aligned} \right. \end{equation*} $

the mean value extension.

The relation between the weak limits of both types of extensions is given by the following lemma (see [13]).

Lemma 1.1  For all $ \omega_\varepsilon\in L^1(\Omega_\varepsilon) $, the following two assertions are equivalent

1. $ \widehat{\omega}_\varepsilon\rightharpoonup \omega $ in $ L^1(\Omega) $; 2. $ \widetilde{\omega}_\varepsilon\rightharpoonup \theta\omega $ in $ L^1(\Omega) $.

A very important property of the porous media is a variant of the Poincare's inequality. Due to the presence of the holes in $ \Omega_\varepsilon $, the Poincare's inequality reads in [12].

Lemma 1.2  Let $ 1\leq p,q<\infty $ and $ u\in W^{1,p}_0(\Omega_\varepsilon) $, then

$ \begin{equation*} ||u||_{L^{q}(\Omega_\varepsilon)}\leq C\varepsilon^{1+n(\frac{1}{q}-\frac{1}{p})}||\nabla u||_{L^p(\Omega_\varepsilon)}, \end{equation*} $

where $ C $ depends only on $ \mathcal{Y}_f $ and $ p,q $ satisfies (1) $ 1\leq p <n, p\leq q\leq p^* = \frac{np}{n-p} $; (2) $ p = n, p\leq q<\infty $; (3) $ p>n, p\leq q\leq\infty $.

We also need the restriction operator constructed by Tartar (see [4]).

Lemma 1.3  There exists an operator $ \mathcal{R}_\varepsilon $ with the following properties

1. $ \mathcal{R}_\varepsilon $ is a bounded linear operator on $ W^{1,p}_0(\Omega) $ ranging in $ W^{1,p}_0(\Omega_\varepsilon),\; \; p\geq 2 $;

2. $ \mathcal{R}_\varepsilon[\varphi] = \varphi|_{\Omega_\varepsilon} $ provide $ \varphi = 0 $ in $ \Omega-\Omega_\varepsilon $;

3. $ \rm{div}_x\varphi = 0 $ in $ \Omega $ implies $ \rm{div}_x\mathcal{R}_\varepsilon[\varphi] = 0 $ in $ \Omega_\varepsilon $;

4. $ ||\mathcal{R}_\varepsilon[\varphi]||_{L^p(\Omega_\varepsilon)}+\varepsilon||\nabla\mathcal{R}_\varepsilon[\varphi]||_{L^p(\Omega_\varepsilon)} \leq C(||\varphi||_{L^p(\Omega)}+\varepsilon||\nabla\varphi||_{L^p(\Omega)}) $.

In addition, we can find the restriction operator $ \mathcal{R}_\varepsilon $ satisfies a compatibility relation with the extension operator introduces in Definition 1.1, namely,

$ \begin{equation*} \langle \nabla \widehat{\omega}, \varphi\rangle = -\int_{\Omega}\widehat{\omega}\; \rm{div}\varphi dx = -\int_{\Omega_\varepsilon}\omega \; \rm{div}R_\varepsilon[\varphi] dx,\; \; \; \forall \varphi\in C_0^\infty(\Omega). \end{equation*} $

Lemma 1.4  (Bogovskii operator) There exists a linear operator $ \mathcal{B}_\varepsilon $ with the properties: if $ f\in L^p(\Omega_\varepsilon) $, then $ \phi = \mathcal{B}_\varepsilon[f] $ such that

$ \begin{equation*} \phi\in W^{1,p}_0(\Omega_\varepsilon),\; \; \; \; \rm{div}\phi = f-\frac{1}{|\Omega_\varepsilon|}\int_{\Omega_\varepsilon}\mathit{fdx}. \end{equation*} $

Moreover, the following estimates is satisfied

$ \begin{equation*} ||\mathcal{B}_\varepsilon[f]||_{W^{1,p}_0(\Omega_\varepsilon)}\leq C \varepsilon^{-1}||f||_{L^p(\Omega_\varepsilon)},\; \; \; 1<p<\infty. \end{equation*} $

In addition, if $ f = \rm{div}g $ with $ g\in L^q(\Omega_\varepsilon),\; g\cdot n|_{\partial\Omega_\varepsilon} = 0 $, then

$ \begin{equation*} ||\mathcal{B}_\varepsilon[f]||_{L^{q}(\Omega_\varepsilon)}\leq C||g||_{L^q(\Omega_\varepsilon)},\; \; \; 1<q<\infty. \end{equation*} $

There are many ways to construct $ \mathcal{B}_\varepsilon $. An explicit formula was proposed by Bogovskii [16] on Lipschitz domains. Some properties of $ \mathcal{B}_\varepsilon $ were discussed by Galdi [17]. In the domain with porous medium, the relevant estimates were obtained by Masmoudi [13].

Finally, we define the permeability matrix $ \mathcal{A} $. For $ 1\leq i\leq n $, let $ (\omega_i,\pi_i)\in H^1(\mathcal{Y}_f)\times L^2(\mathcal{Y}_f)/R $ be the unique solution of the following system

$ \begin{equation*} \left\{\begin{aligned} &-\triangle \omega_i+\nabla \pi_i = e_i\; \; \; \rm{in}\; \mathcal{Y}_\mathit{f},\\ & \rm{div} \omega_i = 0\; \; \; \rm{in}\; \mathcal{Y}_\mathit{f},\\ &\omega_i = 0\; \; \rm{on}\; \partial\mathcal{Y}_s,\; \; \omega_i, \pi_i\; \; \rm{are}\; \; \mathcal{Y} \rm{-periodic}, \end{aligned} \right. \end{equation*} $

where $ e_i $ is the standard basis of $ R^n $. Set $ \omega_i^\varepsilon = \omega_i(\frac{x}{\varepsilon}),\; \pi_i^\varepsilon = \pi_i(\frac{x}{\varepsilon}) $. Then we get the cell problem

$ \begin{equation*} \left\{\begin{aligned} &-\varepsilon^2\triangle \omega_i^\varepsilon+\varepsilon\nabla \pi_i^\varepsilon = e_i\; \; \; \rm{in}\; \varepsilon\mathcal{Y}_\mathit{f},\\ & \rm{div} \omega_i^\varepsilon = 0\; \; \; \rm{in}\; \varepsilon\mathcal{Y}_\mathit{f},\\ &\omega_i^\varepsilon = 0\; \; \rm{on}\; \partial(\varepsilon\mathcal{Y}_s),\; \; \omega_i^\varepsilon, \pi_i^\varepsilon\; \; \rm{are}\; \; \mathcal{\varepsilon Y} \rm{-periodic}. \end{aligned} \right. \end{equation*} $

Lemma 1.5  Let $ \omega_i^\varepsilon, \pi_i^\varepsilon $ be the solution to the cell problem and be extended to zero outside $ \Omega_\varepsilon $. Then the following estimates hold

$ \begin{equation*} \begin{aligned} ||\omega_i^\varepsilon||_{[L^{q}(\Omega_{ \varepsilon})]^{n}}\leq C,\; \; ||\pi_i^\varepsilon||_{L^{q}(\Omega_{\varepsilon})/R} \leq C,\; \; \varepsilon||\nabla \omega_i^\varepsilon||_{[L^{q}(\Omega_{ \varepsilon})]^{n}}\leq C \end{aligned} \end{equation*} $

for any $ 1\leq q\leq +\infty $, $ C $ only depends on $ q $ and $ \mathcal{Y}_f $.

The construction of $ \omega_i^\varepsilon $ and the properties of $ \omega_i^\varepsilon,\; \pi_i^\varepsilon $ stated in Lemma 1.5 can be found in Tartar [4] or Masmoudi [13]. Let us define

$ \begin{equation*} \begin{aligned} \mathcal{A} = (\mathcal{A}_{i,j})_{i,j = 1}^n, \; \; \; \; \mathcal{A}_{i,j} = \int_{\mathcal{Y}_f}(\omega_i)_jdx. \end{aligned} \end{equation*} $

It is easy to see that $ \mathcal{A} $ is a symmetric positive defined matrix. The form of the permeability matrix has different form if $ \mathcal{Y}_s $ has different forms. For more information about $ \mathcal{A} $, we refer the interested readers to Allaire [7] for detail.

Now we introduce the definition of weak solution to the systems (1.1)–(1.4)

Definition 1.2  We shall say that a trio $ \{u_\varepsilon,\rho_\varepsilon,p_\varepsilon\} $ is a weak solution of (1.1), supplemented with the boundary and initial conditions (1.2)–(1.4) if only if

1. $ \rho_\varepsilon\in L^\infty(0,T;L^\gamma(\Omega_\varepsilon)),\; \; \gamma\geq n $ and $ u_\varepsilon \in L^2(0,T;H^1_ 0(\Omega_\varepsilon)) $, and the integral identity

$ \begin{equation*} \int_0^T\int_{\Omega_\varepsilon}(\varepsilon^2\rho_\varepsilon \varphi_t+\rho_\varepsilon u_\varepsilon \cdot\nabla\varphi)dxdt = -\int_{\Omega_\varepsilon}\varepsilon^2\rho_{0,\varepsilon} \varphi(x,0) dx \end{equation*} $

holds for any test function $ \varphi \in C_0^\infty([0,T)\times\overline{\Omega}_\varepsilon) $.

2. $ p_\varepsilon \in L^q(\Omega_\varepsilon\times (0,T)) $ for some $ q>1 $, and

$ \begin{equation*} \begin{aligned} &\int_0^T\int_{\Omega_\varepsilon}(\varepsilon^2\rho_\varepsilon u_\varepsilon\varphi_t +\rho_\varepsilon u_\varepsilon\otimes u_\varepsilon :\nabla\varphi+p_\varepsilon {\hbox{div}} \varphi)dxdt\\ = &\int_0^T\int_{\Omega_\varepsilon}\mu \nabla u_\varepsilon :\nabla\varphi dxdt -\int_{\Omega_\varepsilon}\varepsilon^2m_{0,\varepsilon} \varphi(x,0) dx-\int_0^T\int_{\Omega_\varepsilon}\rho_\varepsilon f\varphi dxdt. \end{aligned} \end{equation*} $

3. $ \mathbb{T}_\varepsilon \in L^2(0,T; L^6(\Omega_\varepsilon)) \cap L^2(0,T; H^1(\Omega_\varepsilon)) $, $ \mathbb{T}_\varepsilon >0 $ a.a in $ \Omega_\varepsilon\times (0,T) $ and the integral identity

$ \begin{equation*} \begin{aligned} &\int_0^T\int_{\Omega_\varepsilon}(\varepsilon^2\rho_\varepsilon s_\varepsilon\varphi_t+\rho_\varepsilon u_\varepsilon s_\varepsilon\cdot\nabla\varphi-\kappa(\mathbb{T}_\varepsilon) \frac{\nabla\mathbb{T}_\varepsilon}{\mathbb{T}_\varepsilon}\nabla\varphi )dxdt\\ = &-\int_0^T\int_{\Omega_\varepsilon}(\mu (\mathbb{T}_\varepsilon) \frac{|\nabla u_\varepsilon|^2}{\mathbb{T}_\varepsilon}+ \kappa (\mathbb{T}_\varepsilon) \frac{|\nabla \mathbb{T}_\varepsilon|^2}{\mathbb{T}_\varepsilon^2})\varphi dxdt +\int_{\Omega_\varepsilon}\varepsilon^2\rho_{\varepsilon0} s_\varepsilon(x,0)\varphi(x,0) dx \end{aligned} \end{equation*} $

holds for any $ \varphi\in C_0^\infty([0,T)\times\bar{\Omega}_\varepsilon) $.

With all the preparation above, we are now in the position to state our main result in this paper.

Theorem 1.1  Let $ \{u_\varepsilon, \rho_\varepsilon, \mathbb{T}_\varepsilon\}_{\varepsilon>0} $ be a family of weak solutions to system (1.1). We assume that $ \gamma\geq n $ for $ n = 2,3 $ and

$ \begin{equation*} \begin{aligned} \widehat{\rho_\varepsilon(x,0)}\rightarrow \rho_0(x),\; \; e_\varepsilon(x,0)\rightarrow e_0(x)\; \; \rm{strongly\; in}\; \; L^1(\Omega),\; \; \rm{respectively}. \end{aligned} \end{equation*} $

Then, there exist three functions $ u, \rho, \Xi $ such that

$ \begin{equation*} \left\{\begin{aligned} &\widehat{\rho_\varepsilon}\rightarrow \rho\; \; \; \; \; \rm{in}\; \; L^p(0,T;L^{q}(\Omega)),\; \; 1<p<\infty,\; 1<q<\gamma+1,\\ &\widehat{\mathbb{T}_\varepsilon}\rightarrow \Xi\; \; \; \; \; \rm{in}\; \; L^2(\Omega\times(0,T)),\\ &\frac{u_\varepsilon}{\varepsilon^2}\rightharpoonup u\; \; \; \; \; \rm{weakly\; in}\; \; L^2(\Omega\times(0,T)), \end{aligned} \right. \end{equation*} $

where $ \Xi = \Xi(t) $ is a spatially homogeneous function, $ \rho_(x,0) = \rho_0(x) $. Moveover, $ \{u, \rho, \Xi\} $ satisfies the following homogenized system

$ \begin{equation*} \left\{\begin{aligned} &|\theta|\partial_t \rho+ \rm{div}(\rho u) = 0\; \; \rm{in}\; \; \Omega\times(0,T),\\ &\mu(\Xi)u = \mathcal{A}(-\nabla p(\rho,\Xi) +\rho f)\; \; \; \; \rm{on\; the\; set\; }\; \{\rho>0\}\; \; \Omega\times(0,T),\\ &\frac{\partial(\rho e)}{\partial t} = \rho u\cdot f, \end{aligned} \right. \end{equation*} $

where $ p,e $ are given by

$ \begin{equation*} \begin{aligned} p = a\rho^\gamma+b\rho^\beta \Xi,\; \; e = \frac{a}{\gamma-1}\rho^{\gamma-1}+c\Xi^{\frac{3}{2}}, \end{aligned} \end{equation*} $

and $ \mathcal{A} $ is the so-called permeability matrix. The specific homogenized entropy $ s $ related to the homogenized pressure $ p $ and the homogenized inner energy $ e $ through Gibbs' equation

$ \begin{equation*} \begin{aligned} \Xi D s(\rho,\Xi) = D e(\rho,\Xi)+p(\rho,\Xi) D (\frac{1}{\rho})\; \; \; \; \rm{on\; the\; set}\; \{\rho>0\}, \end{aligned} \end{equation*} $

where $ s $ is given by $ s = 3c\sqrt{\Xi}-\frac{b}{\beta-1}\rho^{\beta-1}. $

In this section, we collect all available bounds on the family $ \{u_\varepsilon, \rho_\varepsilon, \mathbb{T}_\varepsilon\} $. Let us begin with the basic estimates

In this subsection, we obtain some estimates for the solutions to system (1.1) which are independent of $ \varepsilon $. First, from the conservation of mass, we have $ \rho_\varepsilon\in L^\infty(0,T;L^1(\Omega_\varepsilon)) $. We set

$ \begin{equation} \begin{aligned} \int_{\Omega_\varepsilon} \rho_\varepsilon dx = \int_{\Omega_\varepsilon} \rho_{0,\varepsilon} dx = M. \end{aligned} \end{equation} $

(2.1)

Next, integrating (1.7) over $ \Omega_\varepsilon\times(0,t) $ for any $ t\in [0,T] $, we have

$ \begin{equation} \begin{aligned} \varepsilon^{-1}||\nabla \mathbb{T}_\varepsilon||_{L^2(\Omega_\varepsilon\times(0,\mathit{T}))} \leq C,\; \; \varepsilon^{-1}||\nabla u_\varepsilon||_{L^2(\Omega_\varepsilon\times(0,\mathit{T}))}\leq C. \end{aligned} \end{equation} $

(2.2)

We then deduce the uniform bound of $ \rho_\varepsilon |s_\varepsilon| $ in $ L^\infty(0,T;L^1(\Omega_\varepsilon)) $ and $ \rho_\varepsilon $ is uniformly bounded in $ L^\infty(0,T;L^\gamma(\Omega_\varepsilon)) $. By Lemma 1.2 and the special $ s_\varepsilon $, $ e_\varepsilon $, we also have

$ \begin{equation} \begin{aligned} &\varepsilon^{-2}||u_\varepsilon||_{L^2(\Omega_\varepsilon\times(0,\mathit{T}))}\leq C;\\ &\rho_\varepsilon|u_\varepsilon|^2,\; \rho_\varepsilon e_\varepsilon,\; \rho_\varepsilon \mathbb{T}_\varepsilon^{\frac{3}{2}} \; \rm{are\; uniformly\; bounded\; in}\; L^\infty(0,T;L^\gamma(\Omega_\varepsilon)). \end{aligned} \end{equation} $

(2.3)

In this subsection, we want to deduce the uniform bounds on $ \mathbb{T}_\varepsilon $. Note that we only have the bounds of $ \nabla\mathbb{T}_\varepsilon $ and $ \rho_\varepsilon \mathbb{T}_\varepsilon^{\frac{3}{2}} $. The estimate on $ \mathbb{T}_\varepsilon $ itself isn't included. To fill this gap, we fall back on the Ne\u{c}as' lemma [18] and the Sobolev embedding theorem in the porous medium

Lemma 2.1  Let $ \Omega $ be a bounded Lipschitz domain in $ R^2 $ or $ R^3 $. Let $ M,K $ be two positive real numbers and $ \rho $ a non-negative function such that

$ \begin{equation*} \begin{aligned} 0<M\leq M_\rho = \int_\Omega \rho dx,\; \; \; \; \int_\Omega \rho^\gamma dx\leq K\; \; \; \rm{for\; a\; certain}\; \gamma\geq 2. \end{aligned} \end{equation*} $

Then there exists a constant $ C = C(M,K) $ such that

$ \begin{equation*} \begin{aligned} ||\omega-\frac{1}{M_\rho}\int_\Omega \omega \rho dx||_{L^2(\Omega)}\leq C(M,K) ||\nabla \omega||_{H^{-1}(\Omega)} \end{aligned} \end{equation*} $

for any $ \omega\in L^2(\Omega) $.

Proof see [11].

Following the idea in [11] and [12], we prove the Sobolev embedding theorem in the porous medium has the form.

Lemma 2.2  Let $ v\in W^{1,p}(\Omega_\varepsilon) $. Then we have

$ \begin{equation*} \begin{aligned} ||v||_{L^{q}(\Omega_\varepsilon)}\leq C(|\int_{\Omega_\varepsilon} v dx| +\varepsilon^{n(\frac{1}{q}-\frac{1}{p})}||\nabla v||_{L^p(\Omega_\varepsilon)}) \end{aligned} \end{equation*} $

with $ C $ independent of $ \varepsilon $, $ 1<p\leq q \leq \frac{np}{n-p} $ for $ n\geq 3 $ and $ 1<p\leq q <\infty $ for $ n = 2 $.

Proof Obviously, it is enough to show

$ \begin{equation*} \begin{aligned} ||v||_{L^{q}(\Omega_\varepsilon)}\leq C\varepsilon^{n(\frac{1}{q}-\frac{1}{p})} ||\nabla v||_{L^p(\Omega_\varepsilon)},\; \; \; v\in W^{1,p}(\Omega_\varepsilon),\; \int_{\Omega_\varepsilon}v dx = 0. \end{aligned} \end{equation*} $

By the definition of the module of $ L^p $ and Lemma 1.2, Lemma 1.3, we have

$ \begin{equation*} \begin{aligned} ||v||_{L^{q}(\Omega_\varepsilon)} = &\sup\limits_{||\varphi||_{L^{q'}}\leq1, \int_{\Omega_\varepsilon}\varphi dx = 0} \int_{\Omega_\varepsilon} \nabla v\mathcal{B}_\varepsilon[\varphi]dx\\ \leq&\sup\limits_{||\varphi||_{L^{q'}}\leq1, \int_{\Omega_\varepsilon}\varphi dx = 0} ||\nabla v||_{L^p(\Omega_\varepsilon)}\varepsilon^{1+n(\frac{1}{p'}-\frac{1}{q'})} ||\nabla\mathcal{B}_\varepsilon[\varphi]||_{L^{q'}(\Omega_\varepsilon)}\\ \leq&C\varepsilon^{n(\frac{1}{q}-\frac{1}{p})}||\nabla v||_{L^p(\Omega_\varepsilon)}, \end{aligned} \end{equation*} $

where $ \frac{1}{q}+\frac{1}{q'} = 1 $, $ \frac{1}{p}+\frac{1}{p'} = 1 $.

By Lemma 1.3 and (2.2), we obtain

$ \begin{equation*} \begin{aligned} |\langle \nabla \widehat{\mathbb{T}_\varepsilon},\varphi\rangle|\leq C||\nabla\mathbb{T}_\varepsilon||_{L^2(\Omega_\varepsilon)}(||\varphi||_{L^2(\Omega_\varepsilon)} +\varepsilon||\nabla\varphi||_{L^2(\Omega_\varepsilon)}) \leq C\varepsilon||\varphi||_{H_0^{1}(\Omega_\varepsilon)}, \end{aligned} \end{equation*} $

which implies that

$ \begin{equation} \begin{aligned} ||\nabla \widehat{\mathbb{T}_\varepsilon}||_{L^2(0,T;H^{-1}(\Omega_\varepsilon))}\leq C\varepsilon. \end{aligned} \end{equation} $

(2.4)

Next, we write

$ \begin{equation*} \begin{aligned} \mathbb{T}_\varepsilon = \mathbb{T}_\varepsilon-\frac{1}{M_\varepsilon}\int_{\Omega_\varepsilon} \rho_\varepsilon\mathbb{T}_\varepsilon dx +\frac{1}{M_\varepsilon}\int_{\Omega_\varepsilon}\rho_\varepsilon\mathbb{T}_\varepsilon dx. \end{aligned} \end{equation*} $

In accordance with Lemma 2.1 and (2.2)–(2.4), we have

$ \begin{equation*} \begin{aligned} ||\mathbb{T}_\varepsilon-\frac{1}{M_\varepsilon}\!\! \int_{\Omega_\varepsilon}\!\rho_\varepsilon\mathbb{T}_\varepsilon dx||_{L^2(\Omega_\varepsilon)}\leq C\varepsilon,\; \Xi_\varepsilon = \frac{1}{M_\varepsilon}\!\!\int_{\Omega_\varepsilon} \!\rho_\varepsilon\mathbb{T}_\varepsilon dx \rm{\; is\; bounded\; in\; } L^\infty(0,T). \end{aligned} \end{equation*} $

We deduce that $ ||\mathbb{T}_\varepsilon||_{L^2(\Omega_\varepsilon\times (0,T))}\leq C $. By Lemma 2.2, we obtain

$ \begin{equation} \begin{aligned} ||\mathbb{T}_\varepsilon||_{L^2(0,T;L^6(\Omega_\varepsilon))}\leq C \end{aligned} \end{equation} $

(2.5)

with $ C $ independent of $ \varepsilon $.

We get the uniform bound on $ \rho_\varepsilon $ in $ L^\infty(0,T;L^\gamma(\Omega_\varepsilon)) $. This is not enough when we consider the strong convergence of the density. In this part, we want to obtain some more delicate estimates. To this end, we choose the test function $ v_\varepsilon = \psi(t)\mathcal{B}_\varepsilon(\rho_\varepsilon- \frac{1}{|\Omega_\varepsilon|} \int_{\Omega_\varepsilon}\rho_\varepsilon dx) $, where $ \psi(t)\in C_0^\infty(0,T),\; \psi(t)>0 $, $ \mathcal{B}_\varepsilon $ denotes Bogovskii's operator introduced in Lemma 1.3.

For any $ t\in (0,T) $, we have

$ \begin{equation} \begin{aligned} ||v||_{L^2(\Omega_\varepsilon)}\leq C\varepsilon||\nabla v||_{L^2(\Omega_\varepsilon)} \leq C||\rho_\varepsilon||_{L^{2}(\Omega_\varepsilon)},\\ ||v||_{L^\gamma(\Omega_\varepsilon)}\leq C\varepsilon||\nabla v||_{L^\gamma(\Omega_\varepsilon)} \leq C||\rho_\varepsilon||_{L^\gamma(\Omega_\varepsilon)}. \end{aligned} \end{equation} $

(2.6)

Multiplying the second equation of (1.1) by $ v_\varepsilon $, we get (we drop the $ dxdt $)

$ \begin{equation} \begin{aligned} \int_0^T\!\!\psi(t) \int_{\Omega_\varepsilon}\!\! a\rho_\varepsilon^{\gamma+1} = & \int_0^T \int_{\Omega_\varepsilon}\varepsilon^2 \rho_\varepsilon u_\varepsilon\cdot \partial_tv_\varepsilon+\frac{a}{|\Omega_\varepsilon|} \int_0^T\psi(t) \int_{\Omega_\varepsilon} \rho_\varepsilon^\gamma \int_{\Omega_\varepsilon}\rho_\varepsilon\\ &+ \int_0^T \int_{\Omega_\varepsilon}\rho_\varepsilon f\cdot v_\varepsilon + \int_0^T \int_{\Omega_\varepsilon}\mu\nabla u_\varepsilon: \nabla v_\varepsilon - \int_0^T\psi(t) \int_{\Omega_\varepsilon}b\rho_\varepsilon^{\beta+1}\mathbb{T}_\varepsilon\\ &+\frac{b}{|\Omega_\varepsilon|} \int_0^T\psi(t) \int_{\Omega_\varepsilon}b \rho_\varepsilon^\beta\mathbb{T}_\varepsilon \int_{\Omega_\varepsilon}\rho_\varepsilon - \int_0^T \int_{\Omega_\varepsilon} \rho_\varepsilon u_\varepsilon\otimes u_\varepsilon:\nabla v_\varepsilon \\ = &\sum\limits_{i = 1}^7 I_i. \end{aligned} \end{equation} $

(2.7)

The first term is the most technical and requires some spatial regularization of $ v_\varepsilon $ (see [13]). Let us explain how the difficulty related to $ I_1 $ can be solved. The estimation of rest terms are the same as [13]. Set $ \chi(x)\in C_0^\infty(R^n) $ such that $ \chi(x)\geq0 $, $ \int_{R^n}\chi(x) dx = 0 $. For all $ \delta\in(0,1), $ we denote $ \chi_\delta(x) = \frac{1}{\delta^n}\chi(\frac{x}{\delta}) $. Next, we denote $ \widetilde{\rho}_{\varepsilon,\delta} = \widetilde{\rho}_{\varepsilon}\ast\chi_\delta(x) $, where $ \widetilde{\rho}_{\varepsilon} $ is defined in Definition 1.1. Then we have the following relation

$ \begin{equation} \begin{aligned} \varepsilon^2\frac{\partial \widetilde{\rho}_{\varepsilon,\delta}}{\partial t}+{\hbox{div}}(\widetilde{\rho}_{\varepsilon,\delta}\widetilde{u}_\varepsilon) = r_{\varepsilon,\delta}, \end{aligned} \end{equation} $

(2.8)

where $ r_{\varepsilon,\delta} $ is nothing but a commutator, $ r_{\varepsilon,\delta}\rightarrow 0 $ in $ L^2(0,T;L^{\frac{2\gamma}{\gamma+2}}(\Omega)) $ as $ \delta\rightarrow 0 $. Now we take $ v_{\varepsilon,\delta} = \psi(t) \mathcal{B}_\varepsilon(\widetilde{\rho}_{\varepsilon,\delta}-\frac{1}{|\Omega_\varepsilon|} \int_{\Omega_\varepsilon}\widetilde{\rho}_{\varepsilon,\delta} dx) $ in stead of $ v_{\varepsilon} $. Taking the time derivative of $ t $ of $ v_{\varepsilon,\delta} $, we have

$ \begin{equation*} \begin{aligned} \frac{\partial v_{\varepsilon,\delta}}{\partial t} = &\psi(t)\mathcal{B}(\frac{\partial \widetilde{\rho}_{\varepsilon,\delta}}{\partial t} -\frac{1}{|\Omega_\varepsilon|} \int_{\Omega_\varepsilon} \frac{\partial \widetilde{\rho}_{\varepsilon,\delta}}{\partial t})+\psi'(t) \mathcal{B}_\varepsilon(\widetilde{\rho}_{\varepsilon,\delta}-\frac{1}{|\Omega_\varepsilon|} \int_{\Omega_\varepsilon}\widetilde{\rho}_{\varepsilon,\delta} dx)\\ = &\psi(t)I_{11}+\psi'(t)I_{12}. \end{aligned} \end{equation*} $

It is easy to check that

$ | \int_0^T \int_{\Omega_\varepsilon}\varepsilon^2 \rho_\varepsilon u_\varepsilon\cdot \psi'(t)I_{12}| \leq C\varepsilon||\rho_\varepsilon||_{L^\infty(0,T;L^{\gamma}(\Omega_\varepsilon))}^{2} ||u_\varepsilon||_{L^2(0,T;L^{p}(\Omega_\varepsilon))} \leq C\varepsilon^{3+n(\frac{1}{p}-\frac{1}{2})}, $

where $ 3+n(\frac{1}{p}-\frac{1}{2})\geq 2 $ for $ n = 3 $ and $ 3+n(\frac{1}{p}-\frac{1}{2})>2 $ for $ n = 2 $.

Next, after a straightforward manipulation we have

$ \begin{equation*} \begin{aligned} I_{11} = &-\varepsilon^{-2}\mathcal{B}({\hbox{div}}(\widetilde{\rho}_{\varepsilon,\delta}\widetilde{u}_\varepsilon)) +\mathcal{B}_\varepsilon(r_{\varepsilon,\delta}-\frac{1}{|\Omega_\varepsilon|} \int_{\Omega_\varepsilon} r_{\varepsilon,\delta} dx) = I_{111}+I_{112}. \end{aligned} \end{equation*} $

By Lemma 1.2 and Lemma 1.3, we deduce that

$ \left | \int_0^T \int_{\Omega_\varepsilon}\varepsilon^2 \rho_\varepsilon u_\varepsilon\cdot \psi(t)I_{111}\right| \leq C ||\rho_\varepsilon||_{L^\infty(0,T;L^{\gamma}(\Omega_\varepsilon))}^{2} ||u_\varepsilon||_{L^2(0,T;L^{p}(\Omega_\varepsilon))}^2 \leq C\varepsilon^{4+\frac{2n}{p}-n}. $

To $ I_{112} $, we have

$ \left| \int_0^T \int_{\Omega_\varepsilon}\varepsilon^2 \rho_\varepsilon u_\varepsilon\cdot \psi(t)I_{112}\right| \leq C\varepsilon^{\frac{2\gamma-n}{\gamma}}||\rho_\varepsilon u_\varepsilon||_{L^2_T(\Omega_\varepsilon)}||r_{\varepsilon,\delta} ||_{L^2(0,T;L^{\frac{2\gamma}{\gamma+2}}(\Omega_\varepsilon))}, $

where $ L^2_T(\Omega_\varepsilon) = L^2(0,T;L^{2}(\Omega_\varepsilon)) $. Then we let $ \delta\rightarrow0 $ and deduce that $ \int_0^T \int_{\Omega_\varepsilon}\varepsilon^2 \rho_\varepsilon u_\varepsilon\cdot \psi(t)I_{112} $ also tends to 0.

From above, we get the estimate

$ \begin{equation} \begin{aligned} \rho_\varepsilon \rm{\; is\; bounded\; uniformly\; in\; }L^{\gamma+1}(\Omega_\varepsilon\times(0,\mathit{T})). \end{aligned} \end{equation} $

(2.9)

With those estimations, we can also obtain $ \widehat{p_\varepsilon} $ converges strongly to $ p $ in $ L^2(0,T;L^s(\Omega)) $ for some $ s>1 $, see [12].

We divide three steps to finish the proof.

Step 1    Passing the limit in the continuum equation. As in [13], we can prove that

$ \begin{equation} \begin{aligned} \frac{\partial\widetilde{\rho}_\varepsilon}{\partial t}+\varepsilon^{-2} \rm{div}(\widehat{\rho}_\varepsilon \widetilde{u}_\varepsilon) = 0. \end{aligned} \end{equation} $

(3.1)

Multiplying above equation by $ \varphi \in C_0^\infty(\Omega\times(0,T]) $ and integrating over $ \Omega\times(0,T) $, we have

$ \begin{equation*} \begin{aligned} - \int_0^T \int_{\Omega} \widetilde{\rho}_\varepsilon \partial_t\varphi dxdt- \int_0^T \int_{\Omega}\widehat{\rho}_\varepsilon \frac{\widetilde{u}_\varepsilon}{\varepsilon^2}\cdot \nabla \varphi dxdt = \int_{\Omega} \widetilde{\rho}_\varepsilon(x,0) \varphi(x,0) dx. \end{aligned} \end{equation*} $

Passing the limit and using the strong convergence of $ \widehat{\rho}_\varepsilon $ and $ \widehat{\rho}_{0,\varepsilon} $, we obtain

$ \begin{equation*} \begin{aligned} -|\theta| \int_0^T \int_{\Omega} \rho \partial_t\varphi dxdt- \int_0^T \int_{\Omega}\rho u\cdot \nabla \varphi dxdt = |\theta| \int_{\Omega} \rho_0(x) \varphi(x,0) dx. \end{aligned} \end{equation*} $

We then recover the homogenized equation and the initial condition

$ \begin{equation} \begin{aligned} |\theta|\partial_t\rho+ \rm{div}(\rho u) = 0, \; \; \; \; \; \rho(x,0) = \rho_0(x). \end{aligned} \end{equation} $

(3.2)

Step 2  Passing the limit in the momentum equations.

To pass the limit, we have to regularize the second equation of (1.1) both $ x $ and $ t $. To this end, we set $ \chi_1(x)\in C_0^\infty(R^n) $ such that $ \chi_1(x)\geq 0 $, $ \int_{R^n}\chi_1(x) dx = 1 $. Let $ \eta_1\in (0,1) $ and set $ \chi_{1\eta_1}(x) = \frac{1}{\eta_1^n}\chi_1(\frac{x}{\eta_1}) $. We also set $ \chi_2(t)\in C_0^\infty(R^+) $ such that $ \chi_2(t)\geq 0 $, $ \int_{R^+}\chi_2(t) dt = 1 $. Let $ \eta_2\in (0,1) $ and set $ \chi_{2\eta_2}(t) = \frac{1}{\eta_2}\chi_2(\frac{t}{\eta_2}) $. Take $ T_m(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\omega_n^\varepsilon \varphi $ as the test function, where $ T_m $ is the truncated function by integer $ m $ and $ T_m(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2} $ is prolonged by zero outside $ \Omega\times(0,T) $, $ \omega_n^\varepsilon $ be the solution of the cell problem and $ \varphi \in C_0^\infty(\Omega\times(0,T)) $. We have

$ \begin{equation} \begin{aligned} & \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)\nabla u_\varepsilon:\nabla(\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\omega_n^\varepsilon \varphi)dxdt\\ = & \int_0^T \int_{\Omega}\rho_\varepsilon u_\varepsilon\otimes u_\varepsilon:\nabla(\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\omega_n^\varepsilon \varphi)dxdt\\ &+\varepsilon^2 \int_0^T \int_{\Omega}\rho_\varepsilon u_\varepsilon\cdot \omega_n^\varepsilon\; \partial_t(\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2} \varphi) dxdt\\ &+ \int_0^T \int_{\Omega}\rho_\varepsilon f\cdot \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\omega_n^\varepsilon \varphi dxdt\\ &+ \int_0^T \int_{\Omega}p_\varepsilon(\rho_\varepsilon,\mathbb{T}_\varepsilon) \rm{div}(\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\omega_n^\varepsilon \varphi)dxdt\\ = &I_1+I_2+I_3+I_4. \end{aligned} \end{equation} $

(3.3)

To pass the limit in $ I_1 $ and $ I_2 $, we can easily deduce that $ I_1\rightarrow 0, $ $ I_2\rightarrow 0 $ as $ \varepsilon\rightarrow 0. $ Now, we can pass the limit in $ I_3 $ and $ I_4 $ since $ \widehat{\rho}_\varepsilon $ converges strongly,

$ \begin{equation} \begin{aligned} I_3\rightarrow \mathcal{A}_n \int_0^T \int_{\Omega}\rho f\cdot \mathbb{T}_\varepsilon(\rho)\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\varphi dxdt \end{aligned} \end{equation} $

(3.4)

and

$ \begin{equation} \begin{aligned} I_4\rightarrow&\mathcal{A}_n \int_0^T \int_{\Omega}p(\rho,\mathbb{T}) \rm{div}(\mathbb{T}_\varepsilon(\rho)\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\varphi)dxdt\\ = &-\mathcal{A}_n \int_0^T \int_{\Omega}\nabla p(\rho,\mathbb{T}) \mathbb{T}_\varepsilon(\rho)\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\varphi dxdt. \end{aligned} \end{equation} $

(3.5)

Finally, we consider the limit on the left-hand side. We write it as

$ \begin{equation*} \begin{aligned} & \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)\nabla u_\varepsilon:\nabla(\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\omega_n^\varepsilon \varphi)dxdt\\ = & \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)\nabla u_\varepsilon:\nabla \omega_n^\varepsilon \; \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\; \varphi dxdt\\ &+ \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)\nabla u_\varepsilon:\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon}) \ast\nabla\chi_{1\eta_1}\ast\chi_{2\eta_2}\otimes \omega_n^\varepsilon\; \varphi dxdt\\ &+ \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)\nabla u_\varepsilon:\omega_n^\varepsilon\otimes\nabla\varphi\; \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}dxdt\\ = &L_1+L_2+L_3. \end{aligned} \end{equation*} $

It is easy to check that

$ \begin{equation*} \begin{aligned} |L_2+L_3|\leq C\varepsilon\rightarrow 0\; \; \rm{as}\; \; \varepsilon\rightarrow 0. \end{aligned} \end{equation*} $

On the other hand,

$ \begin{equation*} \begin{aligned} L_1 = &- \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)u_\varepsilon \cdot\triangle \omega_n^\varepsilon\; \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\; \varphi dxdt\\ &- \int_0^T \int_{\Omega}\mu'(\mathbb{T}_\varepsilon)\nabla \mathbb{T}_\varepsilon\otimes u_\varepsilon:\nabla \omega_n^\varepsilon \; \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\; \varphi dxdt\\ &- \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)u_\varepsilon\otimes \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\nabla\chi_{1\eta_1}\ast\chi_{2\eta_2} :\nabla \omega_n^\varepsilon\; \varphi dxdt\\ &- \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)u_\varepsilon\otimes\nabla\varphi:\nabla \omega_n^\varepsilon\; \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}dxdt\\ = &L_{11}+L_{12}+L_{13}+L_{14}. \end{aligned} \end{equation*} $

We also have

$ \begin{equation*} \begin{aligned} |L_{12}+L_{13}+L_{14}|\leq C\varepsilon\rightarrow 0\; \; \rm{as}\; \; \varepsilon\rightarrow 0. \end{aligned} \end{equation*} $

To $ L_{11} $, by virtue of the cell problem, we have

$ \begin{equation*} \begin{aligned} L_{11} = & \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)\varepsilon^{-2}u_\varepsilon \cdot e_n\; \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\; \varphi dxdt\\ &- \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)u_\varepsilon \cdot\varepsilon^{-1} \nabla\pi^\varepsilon\; [\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1} \ast\chi_{2\eta_2}-\mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})]\; \varphi dxdt\\ &- \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)u_\varepsilon \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon}) \cdot\varepsilon^{-1} \nabla\pi^\varepsilon\; \varphi dxdt\\ = &L_{111}+L_{112}+L_{113}. \end{aligned} \end{equation*} $

We continue discussing the convergence of those three terms.

$ \begin{equation*} \begin{aligned} |L_{112}+L_{113}|\rightarrow 0\; \; \; \rm{as}\; \; \; \varepsilon\rightarrow 0. \end{aligned} \end{equation*} $

Finally, passing the limit with $ \varepsilon\rightarrow 0 $, the limit of $ L_{111} $ is given by

$ \begin{equation} \begin{aligned} & \int_0^T \int_{\Omega}\mu(\mathbb{T}_\varepsilon)\varepsilon^{-2}u_\varepsilon \cdot e_n\; \mathbb{T}_\varepsilon(\widehat{\rho_\varepsilon})\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\; \varphi dxdt\\ \rightarrow& \int_0^T \int_{\Omega}\mu(\Xi)u\cdot e_n T_m(\rho)\ast\chi_{1\eta_1}\ast\chi_{2\eta_2}\varphi dxdt. \end{aligned} \end{equation} $

(3.6)

Let $ \eta_1,\eta_2\rightarrow 0 $ and $ m\rightarrow +\infty $ and by using the arbitrary property of $ \varphi $, (3.4), (3.5) and (3.6) lead to the homogenized system of the momentum. That is

$ \begin{equation} \begin{aligned} \mu(\Xi)u = \mathcal{A}(-\nabla p(\rho,\Xi) +\rho f)\; \; \rm{on\; the\; set\; }\; \{\rho>0\}. \end{aligned} \end{equation} $

(3.7)

Step 3  Passing the limit in the energy equation.

Integrating the energy equation in $ \Omega\times(0,t) $ and passing the limit, we have

$ \begin{equation} \begin{aligned} \frac{\partial(\rho e)}{\partial t} = \rho u\cdot f \; \; \; \; \; \rm{a.e. in}\; \Omega\times(0,T). \end{aligned} \end{equation} $

(3.8)

We also have

$ \begin{equation*} \begin{aligned} p_\varepsilon \rightarrow p = a\rho^\gamma+b\rho^\beta \Xi, e_\varepsilon \rightarrow e = \frac{a}{\gamma-1}\rho^{\gamma-1}+c\Xi^{\frac{3}{2}}, s_\varepsilon \rightarrow s = 3c\sqrt{\Xi}-\frac{b}{\beta-1}\rho^{\beta-1}. \end{aligned} \end{equation*} $

Then the following Gibbs' equation holds

$ \begin{equation} \begin{aligned} \Xi Ds(\rho,\Xi) = De(\rho,\Xi)+p(\rho,\Xi)D(\frac{1}{\rho})\; \; \; \; \rm{on\; the\; set}\; \{\rho>0\}. \end{aligned} \end{equation} $

(3.9)

Now we finish Theorem 1.2.