3.1.1 Integrability of u

Investigating the transport equation in Section 3.2 below, we need more regularity for u. Therefore, we prove

Lemma 1 ( $\textbf{\textit{L}}^{\textbf{4}}$ -Integrability of the velocity)Let Assumption 3, b) be satisfied, then weak solutions of the flow problem (Model 1) fulfill

(3.6) \begin{equation}\|\mathbb{K}(\theta)^{-\frac{1}{4}}u\|_4\leq\left(3\|\theta^{\frac{1}{2}}p\|_2\|\theta^{\frac{k-1}{2}}\partial_t\theta\|_\infty\right)^{\frac{1}{2}} .\end{equation}

Proof. We consider the weak formulation

(3.7a) \begin{align} \int_\Omega \mathbb{K}(\theta)^{-1} u \cdot\varphi dx &= \int_\Omega p \nabla\cdot\varphi dx , \end{align}

(3.7b) \begin{align} \int_\Omega \nabla\cdot u\psi dx + \int_\Omega\theta p\psi dx &= -\int_\Omega\partial_t \theta \psi dx \end{align}

of the non-transformed flow problem (Model 1). We test (3.7a) with $|u|^{2}u$ . This directly proves the lemma’s statement since

\begin{align*}\|\mathbb{K}(\theta)^{-\frac{1}{4}}u\|_4^4&=\int_\Omega p\nabla\cdot(|u|^2u) dx= 3\int_\Omega p|u|^2\nabla\cdot u dx= -3\int_\Omega p|u|^2(\partial_t\theta+\theta p)dx\\[3pt]&=-3\int_\Omega p|u|^2\partial_t\theta dx - 3\int_\Omega \underbrace{ p^2|u|^2\theta}_{\geq0}dx\\[3pt]&\leq 3\|\theta^{\frac{1}{2}}p\|_2\|\mathbb{K}(\theta)^{-\frac{1}{4}}u\|_4^2\|\theta^{\frac{k-1}{2}}\partial_t\theta\|_\infty .\end{align*}

Here, we used (3.7b), $\mathbb{K}(\theta)=\theta^k$ , cf. Assumption 1, and the regularity of p according to (3.5) (Theorem 3.2).

Theorem 3.3 (Integrability of the velocity)Let the conditions of Theorem 3.5 and Lemma 1 be satisfied. Then there holds for each $a\in(2,4)$ the weighted estimate

(3.8) \begin{equation}\|\mathbb{K}(\theta)^{-\frac{1}{a}}u\|_a^a\leq \|\mathbb{K}(\theta)^{-\frac{1}{4}}u\|_4^{2a-4}\|\mathbb{K}(\theta)^{-\frac{1}{2}}u\|_2^{4-a} <\infty .\end{equation}

Proof. We combine Lemma 1 with the non-transformed energy estimate (3.5) to prove the boundedness of $\|\mathbb{K}(\theta)^{-\frac{1}{a}}u\|_a$ with $a\in(2,4)$ via a weighted estimate of Lyapunov type: To this end, we define

\begin{equation*}A:=\frac{|u|^{2a-4}}{\|\mathbb{K}(\theta)^{-\frac{1}{4}}u\|_4^{2a-4}}\quad\text{and}\quad B:=\frac{|u|^{4-a}}{\|\mathbb{K}(\theta)^{-\frac{1}{2}}u\|_2^{4-a}}\end{equation*}

and apply Young’s inequality (with $\frac{2a-4}{4}+\frac{4-a}{2}=1$ ) to obtain $AB\leq \frac{2a-4}{4}A^{\frac{4}{2a-4}}+\frac{4-a}{2}B^{\frac{2}{4-a}}$ . Multiplying AB with $\mathbb{K}(\theta)^{-1}$ and integrating actually leads to

\begin{align*}&\frac{1}{\|\mathbb{K}(\theta)^{-\frac{1}{4}}u\|_4^{2a-4}\|\mathbb{K}(\theta)^{-\frac{1}{2}}u\|_2^{4-a}}\int_\Omega |u|^a\mathbb{K}(\theta)^{-1}\,dx =\int_\Omega AB\,\mathbb{K}(\theta)^{-1}\,dx\\[3pt]&\qquad \leq \frac{2a-4}{4}\int_\Omega A^{\frac{4}{2a-4}}\mathbb{K}(\theta)^{-1}\,dx+\frac{4-a}{2}\int_\Omega B^{\frac{2}{4-a}}\mathbb{K}(\theta)^{-1}\,dx=1 ,\end{align*}

which directly implies the assertion.

In particular, we are interested in an estimate choosing $a=3$ in (3.8) and hence consider in this special situation the upper bound in more detail. Thereby we obtain with (3.5) and (3.6)

(3.9) \begin{align}\|\mathbb{K}(\theta)^{-\frac{1}{3}}u\|_3^3 &\leq \|\mathbb{K}(\theta)^{-\frac{1}{4}}u\|_4^2\|\mathbb{K}(\theta)^{-\frac{1}{2}}u\|_2\notag\\[3pt]&\leq 3\|\theta^{\frac{1}{2}}p\|_2\|\theta^{\frac{k-1}{2}}\partial_t\theta\|_\infty\cdot C\|\theta^{-\frac{1}{2}}\partial_t \theta\|_2\notag\\[3pt]&\leq 3C^2\|\theta^{-\frac{1}{2}}\partial_t \theta\|_2^2\|\theta^{\frac{k-1}{2}}\partial_t\theta\|_\infty .\end{align}

Since we investigated an elliptic equation during this section, we dropped the time-dependence of $\theta$ and hence of the solution $(\hat{u},\hat{p})$ . The flow model in (2.1) is of stationary type and depends on time only due to $\theta$ , i.e. the integrability of $(\hat{u},\hat{p})$ with respect to time is inherited fromAssumption 3:

\begin{equation*}\hat{u}\in L^\infty(0,T;H_{\theta,div,0}(\Omega)) \quad{and}\quad \hat{p}\in L^\infty(0,T;L^2(\Omega)) .\end{equation*}

For the same reason we have $\mathbb{K}(\theta)^{-\frac{1}{a}}u\in L^\infty(0,T;L^a(\Omega))$ for all $a\in[2,4]$ .

3.2.1 Transformation and function spaces for the transport equation

In this section, we analyse the transport problem, cf. Model 2. Similar to the transformation of the velocity and pressure given in Section 3.1, we define the transformation of the concentration field

\begin{align*}\hat{c}:=\theta^{\frac{1}{2}}c \ \ (\Leftrightarrow c = \theta^{-\frac{1}{2}}\hat{c}) ,\end{align*}

such that we are again able to analytically handle the corresponding problem by the introduction of an appropriate weighted function space.

The scaled transport equation reads

(3.10a) \begin{align} \partial_t(\theta^{\frac{1}{2}} \hat{c}) - \nabla\cdot(\mathbb{D}\nabla (\theta^{-\frac{1}{2}}\hat{c}) -\theta^{-\frac{1}{2}}u\hat{c}) &= -\sigma(\theta)\hat{R}(\hat{c}) - \theta^{\frac{1}{2}}p\hat{c} && \text{in }\Omega_T ,\end{align}

(3.10b) \begin{align} \hat{c}&=0 && \text{on }\partial\Omega \times (0,T) ,\end{align}

(3.10c) \begin{align} \hat{c}(\,.\,,0)&=\hat{c}_0 && \text{in } \Omega ,\end{align}

where $\hat{R}(\hat{c}):=R(\theta^{-\frac{1}{2}}\hat{c})=R(c)$ for the given porosity function $\theta$ and $\hat{c}_0:=\theta(\,.\,,0)^{\frac{1}{2}}c_0$ . Due to the Lipschitz continuity of R and $R(0)=0$ , the transformed reaction rate function $\hat{R}$ satisfies the condition

(3.11) \begin{equation}\hat{R}(\hat{c})\leq L_R\theta^{-\frac{1}{2}}|\hat{c}| .\end{equation}

Inspired by the energy estimate, cf. Theorem 3.3, and [Reference Arbogast and Taicher3], we consider the function space

\begin{align*} V_0(\Omega):=\Big\{\zeta\in L^2(\Omega): \mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}}\zeta)\in (L^2(\Omega))^{3}\ \text{and}\ \zeta=0\ \text{on }\partial\Omega\Big\} .\end{align*}

We remark that due to $\theta$ the function space $V_0(\Omega)$ is time-dependent although it describes the spatial dependence of an unknown. The corresponding inner product is given by

(3.12) \begin{align}(\zeta,\eta)_{V_0}:= (\zeta,\eta)_2 + \left( \mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}}\zeta)\,,\, \mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}}\eta)\right)_2 ,\end{align}

cf. [Reference Arbogast and Taicher3] and we conclude

Proof. Since this is a special case of the situation discussed in [Reference Arbogast and Taicher3], we follow the proof of [Reference Arbogast and Taicher3, Lemma 3.1] and it suffices to verify the completeness of $V_0(\Omega)$ . Thus, let $(u_k)_{k\in\mathbb{N}}\subset V_0(\Omega)$ be a Cauchy sequence, i.e.

\begin{equation*}\|u_k-u_n\|_{V_0}^2=\|u_k-u_n\|_2^2+\|\mathbb{D}(\theta)^{\frac{1}{2}}\nabla(\theta^{-\frac{1}{2}}(u_k-u_n))\|_2^2\ \stackrel{k,n\to\infty}{\longrightarrow}\ 0 .\end{equation*}

The completeness of $L^2(\Omega)$ implies the convergence of the sequences $(u_k)_{k\in\mathbb{N}}$ and $(\mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}} u_k))_{k\in\mathbb{N}}$ in $L^2(\Omega)$ , i.e. $u_k\rightarrow u\in L^2(\Omega)$ and $\mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}} u_k)\rightarrow\psi\in L^2(\Omega)$ as $k\to\infty$ . Testing the sequence $(\mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}}u_k))_{k\in\mathbb{N}}$ with a smooth function $\phi\in C_0^\infty(\Omega)$ yields

\begin{align*}\big(\mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}} u_k)\,,\,\phi\big)_2 &= -\big(u_k\,,\,\theta^{-\frac{1}{2}}\nabla\cdot(\mathbb{D}(\theta)^{\frac{1}{2}}\phi)\big)_2\\&= -\big(u_k\,,\,\frac{1}{2}\theta^{-\frac{1}{2}}\mathbb{D}(\theta)^{-\frac{1}{2}}\mathbb{D}'(\theta)\nabla\theta\phi\big)_2+\big(u_k\,,\,\theta^{-\frac{1}{2}}\mathbb{D}(\theta)^{\frac{1}{2}}\nabla\cdot\phi\big)_2 ,\end{align*}

where $\mathbb{D}(\theta)=\theta$ , cf. Assumption 1, and the condition $\theta^{-1}\nabla\theta\in L^2(\Omega)$ , cf. Assumption 3, ensures that $\theta^{-\frac{1}{2}}\mathbb{D}(\theta)^{-\frac{1}{2}}\mathbb{D}'(\theta)\nabla\theta$ belongs to $L^2(\Omega)$ . Due to the $L^2$ -convergence of $(u_k)_{k\in\mathbb{N}}$ the right-hand side converges to

\begin{align*}&-\big(u\,,\,\frac{1}{2}\theta^{-\frac{1}{2}}\mathbb{D}(\theta)^{-\frac{1}{2}}\mathbb{D}'(\theta)\nabla\theta\phi\big)_2 + \big(u\,,\,\theta^{-\frac{1}{2}}\mathbb{D}(\theta)^{\frac{1}{2}}\nabla\cdot\phi\big)_2\\&\qquad =-\big(u\,,\,\theta^{-\frac{1}{2}}\nabla\cdot(\mathbb{D}(\theta)^{\frac{1}{2}}\phi)\big)_2=\big(\mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}}u)\,,\,\phi\big)_2 .\end{align*}

Finally, we conclude $\psi=\mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}} u)$ which completes the proof.

3.2.2 Weak formulation

In this section, we introduce the weak formulation to the transformed transport equation (3.10) and consider relations between weighted and standard function spaces.

Definition 3.4 We call

(3.13) \begin{equation}\hat{c}\in \mathcal{X}_\theta:=\Big\{\zeta \in L^2(0,T;H_0^1(\Omega))\,|\,\theta^{-\frac{1}{2}}\partial_t (\theta^{\frac{1}{2}}\zeta)\in L^2(0,T;H^{-1}(\Omega))\Big\} ,\end{equation}

a weak solution to (3.10) if the scaled transport equation is fulfilled in the weak sense, i.e.

(3.14) \begin{align}\langle \theta^{-\frac{1}{2}}\partial_t (\theta^{\frac{1}{2}}\hat{c})\,,\,\varphi\rangle_{H^{-1},H^1}&+ \int_\Omega\mathbb{D}(\theta)\nabla (\theta^{-\frac{1}{2}}\hat{c})\nabla(\theta^{-\frac{1}{2}}\varphi) dx+ \int_\Omega\nabla\cdot(\theta^{-\frac{1}{2}}u\hat{c})\theta^{-\frac{1}{2}}\varphi dx \notag \\[3pt]&= -\int_\Omega\theta^{-\frac{1}{2}}\sigma(\theta)\hat{R}(\hat{c})\varphi dx - \int_\Omega p\hat{c}\varphi dx\end{align}

holds for given porosity function $\theta$ , (u,p) from Section 3.1, and test functions $\varphi\in H_0^1(\Omega)$ for a.e. $t\in(0,T)$ . Here $\langle \,.\,,\,.\,\rangle_{H^{-1},H^1}$ denotes the standard dual product of $H^{-1}(\Omega)$ and $H_0^1(\Omega)$ . Moreover the initial value $\hat{c}_0\in L^2(\Omega)$ is taken in the following sense

\begin{equation*}(\hat{c}(t)-\hat{c}_0,\psi)_{2}\stackrel{t\to0}{\longrightarrow}0\qquad \text{for all }\psi\in L^2(\Omega) .\end{equation*}

We now further investigate the relation between the weighted function spaces $V_0(\Omega)$ , $\mathcal{X}_\theta$ defined above and the standard function spaces for general $\mathbb{D}(\theta)=\theta^d$ , $d\geq 0$ :

Lemma 3 For $\mathbb{D}(\theta)^{\frac{1}{2}}\theta^{-\frac{3}{2}}\nabla\theta\in L^3(\Omega)$ , the following embeddings hold:

\begin{align*}H_0^1(\Omega) \hookrightarrow V_0(\Omega)\qquad & \text{for }d\geq 1 ,\\V_0(\Omega)\cap L^6(\Omega) \hookrightarrow H_0^1(\Omega)\qquad & \text{for }d\leq 1 .\end{align*}

Consequently, with Assumptions 1 and 3, c) we have the isomorphism

\begin{align*} V_0(\Omega)\cap L^6(\Omega)\simeq H_0^1(\Omega)\end{align*}

for a.e. $t\in(0,T)$ .

If additionally Assumption 3, d) holds, the solution space $\mathcal{X}_\theta$ is isomorph to the standard parabolic solution space, i.e.

\begin{equation*}\mathcal{X}_\theta\simeq\mathcal{X}:=\Big\{\zeta\in L^2(0,T;H_0^1(\Omega))\,|\,\partial_t\zeta\in L^2(0,T;H^{-1}(\Omega))\Big\} .\end{equation*}

Proof. By the product rule, it holds for an arbitrary function $\xi:\Omega\to\mathbb{R}$

(3.15) \begin{equation} \mathbb{D}(\theta)^{\frac{1}{2}}\nabla(\theta^{-\frac{1}{2}}\xi) = \mathbb{D}(\theta)^{\frac{1}{2}}\theta^{-\frac{1}{2}}\nabla \xi -\frac{1}{2}\mathbb{D}(\theta)^{\frac{1}{2}}\theta^{-\frac{3}{2}}(\nabla \theta) \xi .\end{equation}

Assuming $\xi\in H_0^1(\Omega)$ and applying the Sobolev embedding $H^1(\Omega)\hookrightarrow L^6(\Omega)$ , it suffices that $\mathbb{D}(\theta)^{\frac{1}{2}}\theta^{-\frac{3}{2}}\nabla \theta$ belongs to $L^3(\Omega)$ and that $d\geq 1$ to have the left-hand side in $L^2(\Omega)$ , i.e. $H_0^1(\Omega)\hookrightarrow V_0(\Omega)$ . On the contrary, the embedding $V_0(\Omega)\cap L^6(\Omega)\hookrightarrow H_0^1(\Omega)$ holds for $d\leq1$ again due to $\mathbb{D}(\theta)^{\frac{1}{2}}\theta^{-\frac{3}{2}}\nabla \theta\in L^3(\Omega)$ .

For $\zeta\in L^2(0,T;H_0^1(\Omega))$ , we similarly have

\begin{equation*}\theta^{-\frac{1}{2}}\partial_t(\theta^{\frac{1}{2}}\zeta)=\partial_t\zeta+\frac{1}{2}\theta^{-1}(\partial_t\theta)\zeta ,\end{equation*}

i.e. with $\theta^{-1}(\partial_t\theta)\in L^\infty(0,T;L^{3}(\Omega))$ the term $\theta^{-1}(\partial_t)\zeta$ belongs to $L^2(\Omega_T)\hookrightarrow L^2(0,T;H^{-1}(\Omega))$ and hence the following equivalence holds:

\begin{align*}\theta^{-\frac{1}{2}}\partial_t(\theta^{\frac{1}{2}}\zeta)\in L^2(0,T;H^{-1}(\Omega))\quad\Leftrightarrow\quad \partial_t\zeta\in L^2(0,T;H^{-1}(\Omega)) .\\[-40pt]\end{align*}

3.2.3 Regularisation

In order to prove existence of weak solutions to the degenerating model (3.10) in the sense of Definition 3.4, we consider for a regularisation parameter $\varepsilon>0$ the following non-degenerating regularisation of the transport problem

(3.16a) \begin{align}\partial_t (\theta_\varepsilon^{\frac{1}{2}} \hat{c}_\varepsilon) - \nabla\cdot\left(\mathbb{D}(\theta_\varepsilon)\nabla (\theta_\varepsilon^{-\frac{1}{2}} \hat{c}_\varepsilon) \right.&\left.- u \theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon\right)\notag\\&= -\sigma(\theta)\hat{R}_\varepsilon(\hat{c}_\varepsilon) - \theta p \theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon && \text{in }\Omega\times (0,T) ,\end{align}

(3.16b) \begin{align} \hat{c}_\varepsilon&=0 && \text{on }\partial\Omega \times (0,T) ,\end{align}

(3.16c) \begin{align} \hat{c}_\varepsilon(\,.\,,0)&=\hat{c}_0 && \text{in } \Omega\end{align}

with $\hat{R}_\varepsilon(\hat{c}_\varepsilon):=R(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)$ . Moreover, the given porosity $\theta$ is replaced by

(3.17) \begin{equation}\theta_\varepsilon:=\begin{cases}\theta & \text{for }\theta\geq\varepsilon\\\varepsilon & \text{else}\end{cases}\end{equation}

avoiding the porosity to fall below $\varepsilon>0$ .

In (3.16) the weak solution (u, p) of the degenerating flow model (2.1) is defined as before, i.e. no regularisation is undertaken for these quantities. Furthermore, the argument of $\sigma(\theta)$ is not replaced by $\theta_\varepsilon$ in (3.16) and the stabilisation term is specifically split to ensure the applicability of (3.7b) later on.

Obviously, this system does not degenerate and hence standard parabolic theory [Reference Ladyženskaja, Solonnikov and Ural’ceva16, Chap. III] can be applied with respect to the usual function space $\mathcal{X}$ . Therefore, for each $\varepsilon>0$ we obtain a solution $\hat{c}_\varepsilon\in \mathcal{X}$ .

3.2.4 Regularisation – uniform energy estimates

In this section, we derive energy estimates for solutions to the regularised model (3.16) for $\mathbb{D}(\theta)=\theta$ and the parameter choice $k\geq 3$ , cf. Assumption 1. As a consequence of (3.15) and the homogeneous Dirichlet boundary condition, the Sobolev embedding $H_0^1(\Omega)\hookrightarrow L^6(\Omega)$ ( $C_S>0$ ) and the Poincare’s inequality ( $C_P>0$ ) yield for a solution $\hat{c}_\varepsilon$ to (3.16)

\begin{align*}\|\nabla \hat{c}_\varepsilon\|_2 & \leq \|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2+\frac{1}{2}\|\theta_\varepsilon^{-1}\nabla\theta_\varepsilon\|_3\|\hat{c}_\varepsilon\|_6\notag\\[2pt]& \leq \|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2+C_S\frac{1}{2}\|\theta_\varepsilon^{-1}\nabla\theta_\varepsilon\|_3\|\hat{c}_\varepsilon\|_{H^1}\notag\\[2pt]& \leq \|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2+C_SC_P\|\theta_\varepsilon^{-1}\nabla\theta_\varepsilon\|_3\|\nabla \hat{c}_\varepsilon\|_{2} .\end{align*}

Let us mention that due to the definition of $\theta_\varepsilon$ there holds $\|\theta_\varepsilon^{-1}\nabla\theta_\varepsilon\|_3 \leq \|\theta^{-1}\nabla\theta\|_3$ for all $\varepsilon>0$ . Therefore, in case that the smallness condition

\begin{equation*} \|\theta^{-1}\nabla\theta\|_3<\frac{1}{\tilde{C}},\end{equation*}

is satisfied with $\tilde{C}:=C_SC_P$ , cf. Assumption 3 c), the gradient $\nabla\hat{c}_\varepsilon$ can be estimated uniformly by the weighted gradient $\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)$ as follows:

(3.18) \begin{equation} \|\nabla \hat{c}_\varepsilon\|_2 \leq \Theta(\theta)\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2, \end{equation}

with $\Theta(\theta):= \frac{1}{1-\tilde{C}\|\theta^{-1}\nabla\theta\|_{L^\infty(L^3)}}\in (1,\infty)$ .

Since in the limit $\varepsilon\rightarrow 0$ a degenerating system is considered, it is necessary to derive uniformly bounded estimates in adequately $\theta$ -weighted norms:

Theorem 3.5 Under the Assumptions 1, 2 and 3, the condition $\hat{c}_0\in L^2(\Omega)$ , and the additional smallness assumption

(3.19) \begin{equation}2\tilde{C} \Theta(\theta)\|\mathbb{K}(\theta)^{-\frac{1}{3}}u\|_3\leq 2\tilde{C}\Theta(\theta)\left(3C^2\|\theta^{-\frac{1}{2}}\partial_t \theta\|_2^2\|\theta\partial_t\theta\|_\infty\right)^{\frac{1}{3}} < 1,\end{equation}

for a.e. $t\in(0,T)$ if $k=3$ , the following uniform energy estimate holds for solutions $\hat{c}_\varepsilon$ to the regularised problem (3.16):

\begin{align*}\|\hat{c}_\varepsilon\|^2_{L^\infty(L^2)} + \|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}}_\varepsilon \hat{c}_\varepsilon)\|^2_{L^2(L^2)} + \|\theta^{-\frac{1}{2}}_\varepsilon\partial_t(\theta^{\frac{1}{2}}_\varepsilon \hat{c}_\varepsilon)\|_{L^2(H^{-1})}^2 <\infty .\end{align*}

Proof. We test equation (3.16) with $\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon$ and apply (3.7b) as well as (3.11) to obtain the energy estimate:

\begin{align*} &\frac{1}{2}\frac{d}{dt}\|\hat{c}_\varepsilon\|^2_2 + \|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|^2_2 \\[2pt] &=- \int_\Omega \theta_\varepsilon^{-\frac{1}{2}}\sigma(\theta)\hat{R}_\varepsilon(\hat{c}_\varepsilon)\hat{c}_\varepsilon dx -\int_\Omega \theta p\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx - \int_\Omega \nabla\cdot(u\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx\\[3pt] &\quad - \frac{1}{2}\int_\Omega\theta_\varepsilon^{-1}(\partial_t\theta_\varepsilon) \hat{c}_\varepsilon^2 dx\\ &=- \int_\Omega \theta_\varepsilon^{-\frac{1}{2}}\sigma(\theta)\hat{R}_\varepsilon(\hat{c}_\varepsilon)\hat{c}_\varepsilon dx - \int_\Omega u\cdot\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx + \int_\Omega \underbrace{\theta_\varepsilon^{-1}(\partial_t\theta-\frac{1}{2}\partial_t\theta_\varepsilon) \hat{c}_\varepsilon^2}_{\leq0} dx\\& \leq L_R \|\theta_\varepsilon^{-1}\sigma(\theta)\|_\infty\|\hat{c}_\varepsilon\|_2^2 - \int_\Omega u\cdot\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx . \end{align*}

For the last estimate, we used the fact that $\partial_t\theta-\frac{1}{2}\partial_t\theta_\varepsilon\leq0$ due to the definition of $\theta_\varepsilon$ , cf. (3.17) and Assumption 3. The last term on the right-hand side must be absorbed into the weighted diffusive term, which is possible due to the integrability of the velocity, see Section 3.1.1. More precisely, to manage the integral

(3.20) \begin{equation}\int_\Omega u\cdot\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx,\end{equation}

it is necessary to distinguish the two cases $k>3$ and $k=3$ :

1. (1) If $k>3$ , we set $\kappa:=\min\{k,4\}$ and apply Young’s inequality to obtain with (3.8) and $\frac{1}{\kappa}+\frac{1}{\kappa'}=\frac{1}{2}$ (i.e. $\kappa'<6$ )

   \begin{align*}\int_\Omega u\cdot\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx &= \int_\Omega \theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}(\theta)^{\frac{1}{\kappa}} \mathbb{K}(\theta)^{-\frac{1}{\kappa}}u \hat{c}_\varepsilon \cdot \mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)dx\\ &\leq \delta\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2^2\\[3pt] &\quad + C_\delta \|\theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}(\theta)^{\frac{1}{\kappa}}\|_\infty^2\|\mathbb{K}(\theta)^{-\frac{1}{\kappa}}u\|_\kappa^2\|\hat{c}_\varepsilon\|_{\kappa'}^2 . \end{align*}
   With the assumptions $\mathbb{D}(\theta_\varepsilon)=\theta_\varepsilon \geq \theta$ and $\mathbb{K}(\theta)=\theta^k$ , $k>3$ , it holds $\theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}(\theta)^{\frac{1}{\kappa}}\leq 1$ . Then Gagliardo–Nirenberg interpolation (with $\frac{1}{\kappa'}=(\frac{1}{2}-\frac{1}{3})\alpha+\frac{1-\alpha}{2}\ \Leftrightarrow\ $ exponent $\alpha=3(\frac{1}{2}-\frac{1}{\kappa'})<1$ ), Young’s inequality (with $(1-\alpha)+\alpha=1$ ) and (3.18) yields
   (3.21) \begin{align}C_\delta\|\mathbb{K}(\theta)^{-\frac{1}{\kappa}}u\|_\kappa^2 & \|\hat{c}_\varepsilon\|_{\kappa'}^2\leq C_\delta \|\mathbb{K}(\theta)^{-\frac{1}{\kappa}}u\|_\kappa^2 \left(\|\hat{c}_\varepsilon\|_{2}^{1-\alpha}\|\nabla\hat{c}_\varepsilon\|_2^{\alpha}\right)^2\notag\\[3pt]& \leq C_\delta\Theta(\theta)^{2\alpha} \|\mathbb{K}(\theta)^{-\frac{1}{\kappa}}u\|_\kappa^2 \|\hat{c}_\varepsilon\|_2^{2(1-\alpha)}\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2^{2\alpha}\notag\\[3pt]& \leq (1-\alpha)\left(C_\delta\Theta(\theta)^{2\alpha}\right)^{\frac{1}{1-\alpha}} \|\mathbb{K}(\theta)^{-\frac{1}{\kappa}}u\|_\kappa^{\frac{2}{1-\alpha}} \|\hat{c}_\varepsilon\|_2^{2} + \alpha\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2^{2} . \end{align}
   By choosing $\delta<1-\alpha$ the term $(\delta+\alpha)\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2^2$ may be absorbed in the diffusive term on the left-hand side. Finally, we obtain for $k>3$
   (3.22) \begin{align} \frac{1}{2}\frac{d}{dt}\|\hat{c}_\varepsilon\|^2_2 &+ (1-\delta-\alpha)\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|^2_2 \end{align}
   (3.23) \begin{align} &\leq \left(L_R\|\theta_\varepsilon^{-1}\sigma(\theta)\|_\infty+ (1-\alpha)(C_\delta\Theta(\theta)^{2\alpha})^{\frac{1}{1-\alpha}} \|\mathbb{K}(\theta)^{-\frac{1}{\kappa}}u\|_\kappa^{\frac{2}{1-\alpha}} \right)\|\hat{c}_\varepsilon\|_2^{2} \end{align}

   with $\kappa=\min\{k,4\}$ . Let us remark that the definition of $\theta_\varepsilon$ implies $\|\theta_\varepsilon^{-1}\sigma(\theta)\|_\infty\leq \|\theta^{-1}\sigma(\theta)\|_\infty$ . Then, Gronwall’s lemma ensures $\hat{c}_\varepsilon\in L^\infty(0,T;L^2(\Omega))$ with the uniform estimate

   (3.24) \begin{align}&\sup_{t\in(0,T)}\|\hat{c}_\varepsilon(t)\|_2^2 \leq \|\hat{c}_0\|_2^2\nonumber\\[3pt] &\quad \times \exp\left[L_R\int_0^T\|\theta^{-1}\sigma(\theta)\|_\infty dt+(1-\alpha)(C_\delta\Theta(\theta)^{2\alpha})^{\frac{1}{1-\alpha}}\int_0^T\left(\|\mathbb{K}(\theta)^{-\frac{1}{\kappa}} u\|_\kappa^{\frac{2}{1-\alpha}}\right) dt\right] .\end{align}

2. (2) Contrarily, this approach does not work for the limit case $k=3$ since then $\alpha$ would be equal to 1. However, assuming the additional smallness property (3.19) on the porosity $\theta$ , we are able to absorb the entire integral (3.20) in the diffusive term as follows:

   (3.25) \begin{align}\int_\Omega u\cdot\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx &= \int_\Omega \theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}(\theta)^{\frac{1}{3}} \mathbb{K}(\theta)^{-\frac{1}{3}}u \hat{c}_\varepsilon \cdot \mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)dx\notag\\[3pt] &\leq \|\theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}(\theta)^{\frac{1}{3}}\|_\infty\|\mathbb{K}(\theta)^{-\frac{1}{3}}u\|_3\|\hat{c}_\varepsilon\|_6\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2 . \end{align}
   The Sobolev embedding $H^1(\Omega)\hookrightarrow L^6(\Omega)$ ( $C_S>0$ ), the Poincare inequality ( $C_P>0$ ) and (3.18) lead to
   \begin{align*}\|\hat{c}_\varepsilon\|_6 &\leq C_S\|\hat{c}_\varepsilon\|_{H^1}\leq 2C_SC_P\|\nabla\hat{c}_\varepsilon\|_2\leq 2C_SC_P\Theta(\theta)\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2 .\end{align*}
   Applying this estimate of the $L^6$ -norm of $\hat{c}_\varepsilon$ and the fact that $\theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}(\theta)^{\frac{1}{3}}\leq 1$ (since $\mathbb{D}(\theta_\varepsilon)=\theta_\varepsilon\geq \theta$ and $k=3$ ) to (3.25) yields with $\tilde{C}=C_SC_P$
   \begin{align*}\int_\Omega u\cdot\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon dx&\leq2\tilde{C}\Theta(\theta) \|\mathbb{K}(\theta)^{-\frac{1}{3}}u\|_3 \|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2^{2} .\end{align*}
   This term can be absorbed if the following smallness assumption, cf. (3.19),
   \begin{equation*}2\tilde{C} \Theta(\theta)\|\mathbb{K}(\theta)^{-\frac{1}{3}}u\|_3\leq 2\tilde{C}\Theta(\theta)\left(3C^2\|\theta^{-\frac{1}{2}}\partial_t \theta\|_2^2\|\theta\partial_t\theta\|_\infty\right)^{\frac{1}{3}}\ =: b < 1,\end{equation*}
   is satisfied for a.e. $t\in(0,T)$ . Here, the integrability estimate for the velocity yields the constant C, cf. (3.9) and [Reference Arbogast and Taicher3]. Since $1-b>0$ in summary, for $k=3$ it holds
   \begin{equation*}\frac{1}{2}\frac{d}{dt}\|\hat{c}_\varepsilon\|^2_2 + (1-b)\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|^2_2\leq L_R\|\theta_\varepsilon^{-1}\sigma(\theta)\|_\infty\|\hat{c}_\varepsilon\|_2^2 .\end{equation*}
   Again Gronwall’s lemma ensures $\hat{c}_\varepsilon\in L^\infty(0,T;L^2(\Omega))$ with
   \begin{align*}\sup_{t\in(0,T)}\|\hat{c}_\varepsilon(t)\|_2^2 &\leq \|\hat{c}_0\|_2^2\cdot\exp\left[L_R\int_0^T\|\theta^{-1}\sigma(\theta)\|_\infty dt\right] ,\end{align*}
   i.e. for $\kappa=3$ and $\alpha=1$ this formally coincides with (3.24).

   Applying standard arguments the uniform boundedness of $\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla (\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)$ in $L^2(\Omega_T)$ follows from (3.22) and (3.24) or from the analog estimates in case that $k=3\ (\alpha=1)$ . Moreover, the boundedness of the weighted time derivative in $L^2(0,T;H^{-1}(\Omega))$ can be inferred as follows using (3.7b) (eliminating p):

   \begin{align*}\|\theta_\varepsilon^{-\frac{1}{2}}\partial_t(\theta_\varepsilon^{\frac{1}{2}} \hat{c}_\varepsilon)\|_{L^2(H^{-1})}^2&= \int_0^T\left(\sup_{\|\varphi\|_{H^1}=1}\left|\langle\theta_\varepsilon^{-\frac{1}{2}}\partial_t(\theta_\varepsilon^{\frac{1}{2}} \hat{c}_\varepsilon)\,,\,\varphi\rangle\right|\right)^2 dt\\[3pt] &\leq \int_0^T \sup_{\|\varphi\|_{H^1}=1}\Big(\int_\Omega\left|\mathbb{D}(\theta_\varepsilon)\nabla(\theta_\varepsilon^{-\frac{1}{2}} \hat{c}_\varepsilon)\nabla (\theta_\varepsilon^{-\frac{1}{2}}\varphi)\right|dx\\ &\quad + \Big|\int_\Omega\nabla\cdot(u\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon) \theta_\varepsilon^{-\frac{1}{2}}\varphi dx \\ &+ \int_\Omega \theta_\varepsilon^{-\frac{1}{2}}\theta p\hat{c}_\varepsilon\theta_\varepsilon^{-\frac{1}{2}}\varphi dx\,\Big| + \int_\Omega |\theta_\varepsilon^{-\frac{1}{2}}\sigma(\theta)\hat{R}_\varepsilon(\hat{c}_\varepsilon)\varphi| dx\Big)^2 dt \end{align*}
   \begin{align*}\\[-35pt] &\leq \int_0^T \sup_{\|\varphi\|_{H^1}=1} \Big(\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2 \|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\varphi)\|_2\\[3pt] &\quad+ \|\theta_\varepsilon^{-1}\partial_t\theta\|_{\frac{3}{2}} {\|\hat{c}_\varepsilon\|_6}\|\varphi\|_6 \\[3pt] &+ \int_\Omega |u\cdot\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\varphi| dx + L_R\|\theta_\varepsilon^{-1}\sigma(\theta)\|_\infty\|\hat{c}_\varepsilon\|_2\|\varphi\|_2\Big)^2 dt\end{align*}
   with
   \begin{align*}& \int_\Omega |u\cdot\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\theta_\varepsilon^{-\frac{1}{2}}\varphi| dx\\ &\quad \leq \|\theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}^{\frac{1}{3}}(\theta)\|_\infty\|\mathbb{K}(\theta)^{-\frac{1}{3}}u\|_3\|\mathbb{D}(\theta_\varepsilon)^{\frac{1}{2}}\nabla(\theta_\varepsilon^{-\frac{1}{2}}\hat{c}_\varepsilon)\|_2\|\varphi\|_6 \le \infty.\end{align*}
   Again, since $\mathbb{D}(\theta_\varepsilon)=\theta_\varepsilon\geq\theta$ and $k\geq 3$ , cf. Assumption 1 it holds $\theta_\varepsilon^{-\frac{1}{2}}\mathbb{D}(\theta_\varepsilon)^{-\frac{1}{2}}\mathbb{K}(\theta)^{\frac{1}{3}}\leq 1$ . Owing to the regularity estimate for the velocity (3.9), the weighted time derivative $\theta_\varepsilon^{-\frac{1}{2}}\partial_t(\theta_\varepsilon^{\frac{1}{2}} \hat{c}_\varepsilon)$ is finally bounded in the corresponding norm.

3.2.5 Regularization – Passage to limit

Due to the uniform estimates of Theorem 3.5, we can extract converging subsequences. A compactness argument then yields a limit function $\hat{c}\in\mathcal{X}$ which solves the original degenerating equations (3.10) in a weak sense, cf. Definition 3.4.

Lemma 4 For functions fulfilling the uniform a priori estimates stated in Theorem 3.5, we deduce the following convergences of an appropriate subsequence $(\hat{c}_m)_{m\in\mathbb{N}}$ of $(\hat{c}_\varepsilon)_{\varepsilon>0}$ (and likewise for the other terms) and identify the corresponding limit function as follows:

\begin{equation*}\begin{array}{rr@{\,\,}c@{\,\,}l}\text{a)} & \hat{c}_m & \stackrel{*}{\rightharpoonup} & \hat{c}\ \in\ \mathcal{X} ,\\ \\[-7pt]\text{b)} & \mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m) & \rightharpoonup & \mathbb{D}(\theta)^{\frac{1}{2}}\nabla(\theta^{-\frac{1}{2}}\hat{c})\ \in\ L^2(0,T;L^2(\Omega)) ,\\\\[-7pt] \text{c)} & \theta_m^{-\frac{1}{2}}\partial_t(\theta_m^{\frac{1}{2}}\hat{c}_m) & \stackrel{*}{\rightharpoonup} & \theta^{-\frac{1}{2}}\partial_t(\theta^{\frac{1}{2}}\hat{c})\ \in\ L^2(0,T;H^{-1}(\Omega)) .\end{array}\end{equation*}

Proof. Due to the estimates obtained in Theorem 3.5 and the isomorphism between the spaces $\mathcal{X}_\theta$ and $\mathcal{X}$ , cf. Lemma 3, $(\hat{c}_{\varepsilon})_{\varepsilon>0}$ is uniformly bounded in $\mathcal{X}$ with respect to $\varepsilon>0$ and hence there is a subsequence $(\hat{c}_m)_{m\in\mathbb{N}}$ converging weakly $^*$ to a limit $\hat{c}\in\mathcal{X}$ . Then the Lemma of Aubin-Lions implies strong convergence of a subsequence of $(\hat{c}_m)_{m\in\mathbb{N}}$ in $C(0,T;L^2(\Omega))$ to $\hat{c}$ such that in particular $\partial_t\hat{c}_m\stackrel{*}{\rightharpoonup}\partial_t\hat{c}$ with respect to $L^2(0,T;H^{-1}(\Omega))$ . These properties together with the weak convergence of $(\theta_m^{-\frac{1}{2}}\partial_t\theta_m)_{m\in\mathbb{N}}$ lead to

\begin{align*} \int_0^T & \langle(\partial_t(\theta_m^{\frac{1}{2}}\hat{c}_m) - \partial_t(\theta^{\frac{1}{2}}\hat{c}),\varphi\rangle_{H^{-1},H^1}dt \\ &= \frac{1}{2}\int_0^T\left((\theta_m^{-\frac{1}{2}}\partial_t\theta_m - \theta^{-\frac{1}{2}}\partial_t\theta)\hat{c},\varphi\right)_2dt+ \frac{1}{2}\int_0^T\left(\theta_m^{-\frac{1}{2}}\partial_t\theta_m(\hat{c}_m - \hat{c}),\varphi\right)_2dt \\ &\qquad +\int_0^T\langle(\theta_m^{\frac{1}{2}}-\theta^{\frac{1}{2}})\partial_t\hat{c}_m,\varphi\rangle_{H^{-1},H^1}dt+\int_0^T\langle\theta^{\frac{1}{2}}(\partial_t\hat{c}_m-\partial_t\hat{c}),\varphi\rangle_{H^{-1},H^1}dt\ \stackrel{m\to\infty}{\longrightarrow}\ 0 ,\end{align*}

implying weakly $^*$ convergence of $\partial_t(\theta_m^{\frac{1}{2}}\hat{c}_m)$ to $\partial_t(\theta^{\frac{1}{2}}\hat{c})$ . This property allows us to identify the limit $\xi$ of $\theta_m^{-\frac{1}{2}}\partial_t(\theta_m^{\frac{1}{2}} \hat{c}_m)$ in $L^2(0,T;H^{-1}(\Omega))$ . Due to the strong $L^\infty$ -convergence of $\theta_m^{\frac{1}{2}}$ to $\theta^{\frac{1}{2}}$ we have weak $^*$ convergence of the product $\theta_m^{\frac{1}{2}} \cdot \theta_m^{-\frac{1}{2}}\partial_t(\theta_m^{\frac{1}{2}} \hat{c}_m)=\partial_t(\theta_m^{\frac{1}{2}} \hat{c}_m)$ to $\theta^{\frac{1}{2}}\cdot \xi=\partial_t(\theta^{\frac{1}{2}} \hat{c})$ , i.e. $\xi=\theta^{-\frac{1}{2}}\partial_t(\theta^{\frac{1}{2}}\hat{c})$ .

Similarly, we see that a subsequence of $\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)$ converges weakly to $\mathbb{D}(\theta)^{\frac{1}{2}}\nabla(\theta^{-\frac{1}{2}}\hat{c})$ : The uniform boundedness ensures the existence of a limit $\gamma\in L^2(0,T;L^2(\Omega))$ , cf. Section 3.2.4. On the other hand (3.15) implies with $\mathbb{D}(\theta)=\theta$ the weak $L^2(\Omega_T)$ -convergence

\begin{align*}\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m) &= \nabla\hat{c}_m-\frac{1}{2}(\theta_m^{-1}\nabla\theta_m)\hat{c}_m\\&\rightharpoonup\ \nabla\hat{c}-\frac{1}{2}(\theta^{-1}\nabla\theta)\hat{c}=\mathbb{D}(\theta)^{\frac{1}{2}}\nabla(\theta^{-\frac{1}{2}}\hat{c}) ,\end{align*}

which allows to identify $\gamma$ .

Since the energy estimate stated in Theorem 3.5 is uniform with respect to $\varepsilon>0$ , it is inherited to the limit $\hat{c}$ such that

\begin{align*}\|\hat{c}\|^2_{L^\infty(L^2)} + \|\mathbb{D}(\theta)^{\frac{1}{2}}\nabla (\theta^{-\frac{1}{2}}\hat{c})\|^2_{L^2(L^2)} + \|\theta^{-\frac{1}{2}}\partial_t(\theta^{\frac{1}{2}} \hat{c})\|_{L^2(H^{-1})}^2 < \infty .\end{align*}

Proof. We have for all test functions $\varphi\in L^2(0,T;H_0^1(\Omega))$ the following convergences:

Evolution Term:

\begin{align*}\int_0^T\langle \theta_m^{-\frac{1}{2}}\partial_t(\theta_m^{\frac{1}{2}}\hat{c}_m)-\theta^{-\frac{1}{2}}\partial_t(\theta^{\frac{1}{2}}\hat{c})\,,\,\varphi\rangle_{H^{-1},\,H^1} dt\ \ \stackrel{m\to\infty}{\longrightarrow}\ 0 .\end{align*}

according to the corresponding weak- $^*$ convergence as stated in Lemma 4, c).

Reaction Term: The Lipschitz property (3.11) of $\hat{R}(\hat{c})$ leads to

(3.26) \begin{align*}\int_0^T\Big(\big(\theta_m^{-\frac{1}{2}}\sigma(\theta)\hat{R}_m(\hat{c}_m) &- \theta^{-\frac{1}{2}}\sigma(\theta)\hat{R}(\hat{c})\big),\varphi\Big)_2 dt=\int_0^T\Big(\left(\theta_m^{-1}\sigma(\theta)-\theta^{-1}\sigma(\theta)\right)\theta_m^{\frac{1}{2}}\hat{R}_m(\hat{c}_m),\varphi\Big)_2 dt\notag\\[4pt] & +\int_0^T\Big(\theta^{-1}\sigma(\theta)\left(\theta_m^{\frac{1}{2}}\hat{R}_m(\hat{c}_m)-\theta^{\frac{1}{2}}\hat{R}(\hat{c})\right),\varphi\Big)_2 dt\notag\\[3pt] & \leq \int_0^T\|\theta_m^{-1}\sigma(\theta)-\theta^{-1}\sigma(\theta)\|_{3}\underbrace{\|\theta_m^{\frac{1}{2}}\hat{R}_m(\hat{c}_m)\|_2}_{\leq L_R\|\hat{c}_m\|_2}\|\varphi\|_{6} dt\notag\\[3pt]& +\Big|\int_0^T\Big(\left(\theta_m^{\frac{1}{2}}\hat{R}_m(\hat{c}_m)-\theta^{\frac{1}{2}}\hat{R}(\hat{c})\right)\theta^{-1}\sigma(\theta),\varphi\Big)_2 dt\Big|\ \ \stackrel{m\to\infty}{\longrightarrow}\ 0 ,\notag\end{align*}

where the second summand on the right-hand side can be estimated with Assumption 2 and (3.11) as follows

\begin{align*}\int_0^T & \Big(\left(\theta_m^{\frac{1}{2}}\hat{R}_m(\hat{c}_m)-\theta^{\frac{1}{2}}\hat{R}(\hat{c})\right)\theta^{-1}\sigma(\theta),\varphi\Big)_2 dt\\[2pt]&=\int_0^T\Big((\theta_m^{\frac{1}{2}}-\theta^{\frac{1}{2}})\hat{R}_m(\hat{c}_m)\theta^{-1}\sigma(\theta),\varphi\Big)_2 dt+ \int_0^T\Big(\theta^{\frac{1}{2}}\underbrace{\left(\hat{R}_m(\hat{c}_m)-\hat{R}(\hat{c})\right)}_{\leq L_R|\theta_m^{-\frac{1}{2}}\hat{c}_m-\theta^{-\frac{1}{2}}c|}\theta^{-1}\sigma(\theta),\varphi\Big)_2 dt\\[2pt]&\leq\int_0^T\|\theta^{\rho_1}\left(\frac{\theta^{\frac{1}{2}}}{\theta_m^{\frac{1}{2}}}-1\right)\|_3\underbrace{\|\theta_m^{\frac{1}{2}}\hat{R}_m(\hat{c}_m)\|_2}_{\leq L_R\|\hat{c}_m\|_2}\|\theta^{-(1+\rho_1)}\sigma(\theta)\|_{\infty}\|\varphi\|_6 dt\\[2pt]&\qquad+L_R\int_0^T\underbrace{\|\theta^{\rho_1}\left(\frac{\theta}{\theta_m}\right)^\frac{1}{2}\hat{c}_m-\theta^{\rho_1}\hat{c}\|_{\frac{6}{5}}}_{\leq \|\theta^{\rho_1}\left(\frac{\theta^{\frac{1}{2}}}{\theta_m^{\frac{1}{2}}}-1\right)\|_3\|\hat{c}_m\|_2+\|\hat{c}_m-\hat{c}\|_2}\|\theta^{-(1+\rho_1)}\sigma(\theta)\|_{\infty}\|\varphi\|_6 dt .\end{align*}

Let us define for $m\in\mathbb{N}$ and $t\in(0,T)$ the subdomain $\Omega_m(t):=\{x\in\Omega:\theta(x,t)\geq \varepsilon_m\}$ , where $\varepsilon_m>0$ corresponds to $\theta_m$ via (3.17), i.e. $\theta_m\geq\varepsilon_m$ and $\theta_m=\theta$ for all $x\in\Omega_m(t)$ . Moreover, there holds $\lim_{m\to\infty}\varepsilon_m=0$ . Consequently, the right-hand side of the above estimate converges since $\left(\frac{\theta}{\theta_m}\right)^{\frac{1}{2}}=1$ in $\Omega_m(t)$ and hence

(3.27) \begin{equation} \|\theta^{\rho_1}\left(\frac{\theta^{\frac{1}{2}}}{\theta_m^{\frac{1}{2}}}-1\right)\|_3\leq |\Omega|^{\frac{1}{3}}\|\theta^{\rho_1}\left(\frac{\theta^{\frac{1}{2}}}{\theta_m^{\frac{1}{2}}}-1\right)\|_\infty \leq|\Omega|^{\frac{1}{3}}\,\varepsilon_m^{\rho_1}\ \ \stackrel{m\to\infty}{\longrightarrow}\ 0 .\end{equation}

Similarly, the first summand on the right-hand side of (3.26) can be estimated by applying (3.27) and Assumption 2.

Diffusive Term: Applying the weak convergence as stated in Lemma 4, b), it holds

\begin{align*}\int_0^T & \left(\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla (\theta_m^{-\frac{1}{2}}\hat{c}_m),\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla(\theta_m^{-\frac{1}{2}}\varphi)\right)_2 dt-\int_0^T\left({\mathbb{D}^{\frac{1}{2}}(\theta)\nabla (\theta^{-\frac{1}{2}}\hat{c})},\mathbb{D}^{\frac{1}{2}}(\theta)\nabla(\theta^{-\frac{1}{2}}\varphi)\right)_2 dt\\[2pt]&= \int_0^T\left(\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla (\theta_m^{-\frac{1}{2}}\hat{c}_m)-{\mathbb{D}^{\frac{1}{2}}(\theta)\nabla (\theta^{-\frac{1}{2}}\hat{c})},\mathbb{D}^{\frac{1}{2}}(\theta)\nabla(\theta^{-\frac{1}{2}}\varphi)\right)_2 dt \\[2pt] &\qquad + \int_0^T\left(\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla (\theta_m^{-\frac{1}{2}}\hat{c}_m),\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla(\theta_m^{-\frac{1}{2}}\varphi)-\mathbb{D}^{\frac{1}{2}}(\theta)\nabla(\theta^{-\frac{1}{2}}\varphi)\right)_2 dt\end{align*}

for all $\varphi\in L^2(0,T;H_0^1(\Omega))$ , where the first summand on the right-hand side vanishes. Due to $\mathbb{D}(\theta)=\theta$ , cf. Assumption 1, and the uniform estimate stated in Theorem 3.3, the second summand on the right-hand side can be estimated by

\begin{align*}\int_0^T & \left(\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla (\theta_m^{-\frac{1}{2}}\hat{c}_m),\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla(\theta_m^{-\frac{1}{2}}\varphi)-\mathbb{D}^{\frac{1}{2}}(\theta)\nabla(\theta^{-\frac{1}{2}}\varphi)\right)_2 dt\\[2pt]&\leq \frac{1}{2}\|\mathbb{D}^{\frac{1}{2}}(\theta_m)\nabla (\theta_m^{-\frac{1}{2}}\hat{c}_m)\|_2 \|\theta_m^{-1}\nabla\theta_m-\theta^{-1}\nabla\theta\|_3\|\varphi\|_6 \ \ \stackrel{m\to\infty}{\longrightarrow}\ 0 ,\end{align*}

where the sequence $(\theta_m^{-1}\nabla\theta_m)_{m\in\mathbb{N}}$ even converges in a strong sense since the corresponding $L^3$ -norms converge to $\|\theta^{-1}\nabla\theta\|_3$ .

Advective and stabilisation term: Since $\mathbb{D}(\theta)=\theta$ and $k\geq 3$ , cf. Assumption 1, and applying (3.7b), we obtain

(3.28) \begin{align}-\int_0^T & \left(\theta_m^{-\frac{1}{2}}\nabla\cdot(u\theta_m^{-\frac{1}{2}}\hat{c}_m)-\theta^{-\frac{1}{2}}\nabla\cdot(u\theta^{-\frac{1}{2}}\hat{c}),\varphi\right)_2 dt-\int_0^T\left(\theta_m^{-1}\theta p\hat{c}_m-p\hat{c},\varphi\right)_2 dt \notag\\[2pt]&= -\int_0^T\left((\nabla\cdot u)(\theta_m^{-1}\hat{c}_{m}-\theta^{-1}\hat{c})+\theta_{m}^{-\frac{1}{2}}u\cdot\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)-\theta^{-\frac{1}{2}}u\cdot\nabla(\theta^{-\frac{1}{2}}\hat{c}),\varphi\right)_2 dt \notag\\[2pt]&\qquad +\int_0^T\left((\nabla\cdot u+\partial_t\theta)(\theta_m^{-1}\hat{c}_{m}-\theta^{-1}\hat{c}),\varphi\right)_2 dt \notag\\[2pt]&= -\int_0^T\left((\theta_{m}^{-\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)-\theta^{-\frac{1}{2}}\nabla(\theta^{-\frac{1}{2}}\hat{c}))\cdot u,\varphi\right)_2 dt \notag\\[2pt]&\qquad +\int_0^T\left((\theta_m^{-1}\hat{c}_{m}-\theta^{-1}\hat{c})\partial_t\theta,\varphi\right)_2 dt \notag\\&= -\int_0^T\left(\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)-\mathbb{D}(\theta)^{\frac{1}{2}}\nabla(\theta^{-\frac{1}{2}}\hat{c}),\underbrace{\mathbb{K}(\theta)^{-\frac{1}{k}}u}_{L^3}\underbrace{\varphi}_{L^6}\right)_2 dt \notag\\ &\qquad - \int_0^T\left((\theta_m^{-\frac{1}{2}}-\theta_m^{\frac{1}{2}}\theta^{-1})\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\cdot u,\varphi\right)_2 dt\notag\\&\qquad +\int_0^T\left((\theta_m^{-1}\hat{c}_{m}-\theta^{-1}\hat{c})\partial_t\theta,\varphi\right)_2 dt . \end{align}

Again Lemma 4, b) implies the convergence of the first summand on the right-hand side. If $k>3$ , the second summand can similarly to (3.27) be estimated by

\begin{align*}\int_0^T & \left((\theta_m^{-\frac{1}{2}}-\mathbb{D}(\theta_m)^{\frac{1}{2}}\theta^{\rho_2}\mathbb{K}(\theta)^{-\frac{1}{3}})\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\cdot u,\varphi\right)_2 dt\\&\leq\int_0^T\|\theta^{\rho_2}\left(\frac{\theta}{\theta_m}-1\right)\|_\infty\|\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\|_2\|\mathbb{K}(\theta)^{-\frac{1}{3}}u\|_3\|\varphi\|_6 dt\ \ \stackrel{m\to\infty}{\longrightarrow}\ 0,\end{align*}

with $\rho_2:=\frac{k}{3}-1>0$ .

Contrarily, for the case $k=3$ , the following density argument is needed. First we consider

\begin{align*}&\left|\int_0^T \left((\theta_m^{-\frac{1}{2}}-\mathbb{D}(\theta_m)^{\frac{1}{2}}\theta^{\frac{1}{2}}\mathbb{K}(\theta)^{-\frac{1}{2}})\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\cdot u,\varphi\right)_2 dt\right|\\&\qquad \leq \int_0^T\|\theta^{\frac{1}{2}}\left(\frac{\theta}{\theta_m}-1\right)\|_\infty\|\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\|_2\|\mathbb{K}(\theta)^{-\frac{1}{2}}u\|_2\,\|\varphi\|_\infty dt\ \stackrel{m\to\infty}{\longrightarrow}\ 0,\end{align*}

for all smooth test functions $\varphi\in C_0^\infty(\Omega_T)$ . The density of $C_0^\infty(\Omega_T)$ in $L^2(0,T;L^6(\Omega))$ leads for an arbitrary test function $\varphi\in L^2(0,T;L^6(\Omega))$ and a sequence of smooth functions $\varphi_\ell\in C_0^\infty(\Omega_T)$ converging to $\varphi$ as $\ell\to\infty$ to the following result. Let $\delta>0$ . Since $\|\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla\cdot(\theta_m^{-\frac{1}{2}}\hat{c}_m)\|_{L^2(L^2)}$ is uniformly bounded, choosing $\ell$ sufficiently large such that

\begin{equation*}\sup_{t\in(0,T)}\left(\|\frac{\theta(t)}{\theta_m(t)}-1\|_\infty\|\mathbb{K}(\theta)^{-\frac{1}{3}}u(t)\|_3\right)\|\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\|_{L^2(L^2)}\|\varphi-\varphi_\ell\|_{L^2(L^6)}<\frac{\delta}{2},\end{equation*}

and afterwards choosing m in such a way that

\begin{equation*}\sup_{t\in(0,T)}\left(\|\theta^{\frac{1}{2}}(t)\left(\frac{\theta(t)}{\theta_m(t)}-1\right)\|_\infty\|\mathbb{K}(\theta)^{-\frac{1}{2}}u(t)\|_2\right)\|\mathbb{D}(\theta_m)^{\frac{1}{2}}\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\|_{L^2(L^2)}\|\varphi_\ell\|_{L^2(L^\infty)}<\frac{\delta}{2},\end{equation*}

actually leads for $\varphi\in L^2(0,T;L^6(\Omega))$ to

\begin{equation*}\left|\int_0^T \left((\theta_m^{-\frac{1}{2}}-\mathbb{D}(\theta_m)^{\frac{1}{2}}\theta^{\frac{1}{2}}\mathbb{K}(\theta)^{-\frac{1}{2}})\nabla(\theta_m^{-\frac{1}{2}}\hat{c}_m)\cdot u,\varphi\right)_2 dt\right|<\delta .\end{equation*}

Finally, applying Assumption 3, d) and the weak* convergence of $\frac{\theta}{\theta_m}$ to 1 in $L^\infty(\Omega)$ to the last term on the right-hand side of (3.28) yields

\begin{align*} \int_0^T & \left((\theta_m^{-1}\hat{c}_{m}-\theta^{-1}\hat{c})\partial_t\theta,\varphi\right)_2dt= \int_0^T\left(\left(\frac{\theta}{\theta_m}-1\right)\hat{c}\,\theta^{-1}\partial_t\theta,\varphi\right)_2dt\\[3pt] &\quad + \int_0^T\left(\frac{\theta}{\theta_m}\left(\hat{c}_{m}-\hat{c}\right)\theta^{-1}\partial_t\theta,\varphi\right)_2dt\\&\leq \int_0^T\big\langle\left(\frac{\theta}{\theta_m}-1\right),\,\hat{c}\,\theta^{-1}\partial_t\theta\varphi\big\rangle_{L^\infty,L^1}dt+ \int_0^T\|\frac{\theta}{\theta_m}\|_\infty\|\hat{c}_{m}-\hat{c}\|_2\|\theta^{-1}\partial_t\theta\|_3\|\varphi\|_6 dt\ \stackrel{m\to\infty}{\longrightarrow}\ 0 .\\[-30pt]\end{align*}

3.2.6 Existence and uniqueness

Combining the results of the previous sections, we obtain weak solvability of the original degenerating model. However, the transport equation (2.2) is lost within the area with $\theta=0$ since it trivializes to $0=0$ . In such a situation of vanishing porosity, the behavior of c is not clearly defined and hence a uniqueness assertion needs to be restricted to the subdomain $\Omega_0:=\{(x,t)\in\Omega_T|\theta(x,t)\neq0\}$ .

Theorem 3.7 (Coupled degenerating problem)Under Assumptions 1, 2, and 3, the condition $\hat{c}_0\in L^2(\Omega)$ , and the additional smallness assumption (3.19) if $k=3$ , there exists a weaksolution

\begin{equation*}(\hat{c},\hat{u},\hat{p})\in \mathcal{X}\times L^\infty(0,T;H_{\theta,div,0}(\Omega))\times L^\infty(0,T;L^2(\Omega)),\end{equation*}

to the degenerating transformed problems (3.2) and (3.19) corresponding to the coupled Model 1 and 2. Moreover, this solution is unique and $\hat{c}$ satisfies non-negativity on $\Omega_0$ .

Proof. The Theorems 3.2 and 3.6 directly ensure the existence of a weak solution $(\hat{c},\hat{u},\hat{p})$ to (3.2) and (3.10). The uniqueness of this solution in $\Omega_0$ is verified by a straightforward argument (for the sake of readability it is expressed in the physical non-transformed values): Let $(c_i,u_i,p_i)$ , $i=1,2$ , two weak solutions of (2.1) – (2.2). Then $\tilde{u}:=u_1-u_2$ and $\tilde{p}:=p_1-p_2$ fulfill the equations

\begin{align*} \tilde{u} &= -\mathbb{K}(\theta)\nabla \tilde{p} && \text{in }\Omega_T ,\\ \nabla\cdot \tilde{u} + \theta \tilde{p} &= 0 && \text{in }\Omega_T ,\notag\\ \tilde{u}\cdot \nu &= 0 && \text{on } \partial\Omega\times (0,T),\end{align*}

with the unique solution $\tilde{u}=0$ and $\tilde{p}=0$ , cf. [Reference Arbogast and Taicher3]. Furthermore, the function $\tilde{c}:=c_1-c_2$ satisfies the transport equation

\begin{align*} \partial_t(\theta \tilde{c}) - \nabla\cdot\mathbb{D}(\theta)\nabla \tilde{c} &= -\sigma(\theta)(R(c_1)-R(c_2)) && \text{in }\Omega_T,\end{align*}

with zero initial and boundary conditions. Testing with $\tilde{c}$ leads to

\begin{align*}\frac{1}{2}\|\theta^{\frac{1}{2}}\tilde{c}(t)\|_2^2+\int_0^t\|\mathbb{D}(\theta)^{\frac{1}{2}}\nabla \tilde{c}\|_2^2 ds\leq\int_0^t\|\sigma(\theta)\|_\infty\|R(c_1)-R(c_2)\|_2\|\tilde{c}\|_2 ds+ \frac{1}{2}\int_0^t\|\partial_t\theta\|_\infty\|\tilde{c}\|_2^2 ds .\end{align*}

Finally, the Lipschitz condition of R, cf. Assumption 2 and Gronwall’s Lemma yield

\begin{equation*}\sup_t\|\theta^{\frac{1}{2}}\tilde{c}(t)\|_2^2\leq0 ,\quad\text{i.e. }\tilde{c}=0\text{ within }\Omega_0 .\end{equation*}

The non-negativity follows along the same lines as the energy estimate testing the weak formulation with the negative part $c_-$ of the concentration. Thereby the reaction term is estimated via

\begin{align*} \int_\Omega \sigma(\theta)R(c)c_-dx \leq C_R\|\sigma(\theta)\|_\infty \|c_-\|_2^2,\end{align*}

according to Assumption 2. Again Gronwall’s Lemma guarantees the non-negativity on $\Omega_0$ .