3.1. Setting

For the following, we consider solute transport by diffusion only. As the term $\partial _t (\phi c)$ in (2.1) is difficult to handle analytically, it is shifted to the right-hand side, cf. [Reference Ray and Schulz34]. Therefore, we write

(3.1) \begin{align} \phi \partial _t c - \nabla \cdot (\mathbb{D}\nabla c) = \sigma f(c) - \partial _t \phi c \quad \text{in } (0,T)\times \Omega, \end{align}

rendering the equations for flow and permeability determination (2.4), (2.6) superfluous. Hence, it is sufficient to close the model regarded in this section by (2.3), (2.7) and (2.9). We emphasise that the following considerations assume good-natured conditions such as the smoothly bounded solid geometry being compactly contained within the unit-cell $Y$ . By considering local-in-time estimates, this setting is maintained by an appropriate choice of initial conditions. Using diffeomorphisms to describe solid alteration, we particularly exclude clogging scenarios and degenerating equations.

3.2. Continuous dependence of diffusion tensors

In order to prove existence to the model described in Section 3.1, we make extensive use of existence theory for linear parabolic equations, cf. Theorem A.3 in the appendix. As such, we require moderate regularity for $\mathbb{D}$ as the coefficient of the leading order term in (3.1).

Therefore, this section is concerned with the dependence of the diffusion tensor $\mathbb{D}$ on the evolving geometry. To do so, we first focus on a single unit cell and therefore on a single cell problem posed on the fluid domain $\mathcal{P}$ . While regarding $(t,x)$ as independent parameters for now, we present in Section 3.3 how regularity with respect to the macroscopic variables is established. The method presented consists of three steps: At first, we establish higher regularity for weak solutions to the diffusion cell problem (2.3). Although using standard methods, we state the results in detail due to the uncommon periodic boundary conditions. Based on that, a mapping between the geometry and the elliptic PDE’s solution of desired regularity is constructed using the implicit function theorem following the technique of [Reference Geißert, Heck and Hieber16]. Finally, we extend our smoothness results from the solutions $\zeta _j$ , $j\in \{1,\dots,d\}$ , of (2.3) to the diffusion tensor $\mathbb{D}$ which is given as an affine-linear functional of $\zeta _j$ , cf. (2.2).

For the formulation of problem (2.3) and our regularity result Lemma 3.1, we assume $\partial ^{\text{int}}\mathcal{P}$ to be $C^{2,1}$ -regular. As it becomes apparent in Theorem 1, it is necessary to consider the full class of elliptic PDEs of type (2.3) with general source term and Neumann boundary conditions. In a first step towards a suitable weak formulation of problem (2.3), we introduce periodic Sobolev spaces according to [Reference Chen, Shahgholian and Vazquez8]. Let $\mathcal{P}_{\text{ext}} \subset \mathbb{R}^d$ denote the perforated domain obtained by periodic extension of $\mathcal{P}$ in $\mathbb{R}^d$ . Then we define $H_\#^k (\mathcal{P})$ for $k\in \mathbb{N}$ as the closure of Y-periodic functions in $C^\infty (\mathcal{P}_{\text{ext}})$ with respect to the $H^k$ -norm. For the weak formulation of the diffusion cell problem with Neumann boundary conditions, we introduce the following function spaces:

\begin{align*} H^k_{\#,0}(\mathcal{P}) = \left \lbrace v\in H_\#^k ( \mathcal{P})\,:\, \int _{\mathcal{P}} v \; dy =0 \right \rbrace, \end{align*}

equipped with the norm $||u||_{ H^k_{\#,0}(\mathcal{P})} = ||u_{|\mathcal{P}}||_{ H^k(\mathcal{P})}$ .

Accordingly, the weak problem with general source term $f$ and Neumann boundary condition $g$ reads: Find $u\in H^1_{\#,0}(\mathcal{P})$ such that

(3.2) \begin{align} \int _{\mathcal{P}} \nabla u \cdot \nabla v \; dy - \int _{\partial ^{\text{int}}\mathcal{P}} g v \; d\sigma = \int _{\mathcal{P}} f v \; dy, \quad \forall v\in H^1_{\#}(\mathcal{P}), \end{align}

with $g\in L^2(\partial ^{\text{int}}\mathcal{P})$ , $f\in L^2(\mathcal{P})$ . Apparently, the compatibility condition $\int \limits _{\mathcal{P}} f \; dx =- \int \limits _{\partial ^{\text{int}}\mathcal{P}} g \; d\sigma$ is necessary for solvability.

Next, we establish unique solvability for the above problem and present conditions ensuring solutions to be of higher regularity. Particularly, a $C^{2,1}$ -regular interior boundary $\partial ^{\text{int}}\mathcal{P}$ proves to be sufficient for equations (2.3) to hold in the $L^2$ -sense and therefore point-wise almost everywhere in $\mathcal{P}$ .

Proof. The coercivity of the bilinear form on the Hilbert space $H^1_{\#,0}(\mathcal{P})$ is guaranteed by Poincaré’s inequality. The Lax–Milgram theorem therefore implies the unique existence of a solution $u \in H^1_{\#,0}(\mathcal{P})$ to the weak formulation, cf. [Reference Chen, Shahgholian and Vazquez8].

Let us now suppose that $\partial ^{\text{int}}\mathcal{P}$ is $C^{k+2,1}$ -regular. By the surjectivity of the higher-order trace operator continuously extending the following function, cf. [Reference Molins and Knabner27],

\begin{align*}{\text{Tr}}_k &\,:\, H^{k+2}(\mathcal{P}) \to \prod _{l=0}^{k+1} H^{k-l+\frac{3}{2}} (\partial ^{\text{int}}\mathcal{P}), \quad \forall u \in H^{k+2}(\mathcal{P})\cap C^{k+1}(\bar{\mathcal{P}}), \\{\text{Tr}}_k u &= \left (u|_{\partial ^{\text{int}}\mathcal{P}}, \partial _\nu u|_{\partial ^{\text{int}}\mathcal{P}},\cdots, \partial _\nu ^{k+1} u|_{\partial ^{\text{int}}\mathcal{P}} \right ), \nonumber \end{align*}

we find a function $\Psi \in H^{k+2}_{\#,0}(\mathcal{P})$ with $\dfrac{\partial \Psi }{\partial \nu } = \dfrac{\partial u}{\partial \nu }$ on $\partial ^{\text{int}}\mathcal{P}$ in the trace sense vanishing in a neighbourhood of $\partial Y$ . Exploiting linearity of the problem and carrying out the subsequent argument for $u-\Psi$ and the corresponding source term $\tilde{f}=f-\Delta \Psi \in H^k(\mathcal{P})$ , we can assume homogeneous Neumann boundary conditions. By Theorem 3 of [Reference Meier25], the required interior higher regularity is established. Following the lines of Theorem 4 of [Reference Meier25], higher regularity is also obtained in a neighbourhood of every interior boundary point $y\in \partial ^{\text{int}}\mathcal{P}$ . Using an open covering argument, we obtain $u \in H^{k+2}_{\#,0}(\mathcal{P})$ .

Remark 3.2 enables us to define mappings from the solution space of equation (3.2) to the bulk and boundary data spaces with a point-wise interpretation. More precisely, we consider such mappings which involve the geometry alteration as a parameter. This idea poses the main ingredient in the subsequent investigation of the dependence of solutions to the weak problem (3.2) and functionals thereof on the geometry $\mathcal{P}$ . Following the approach presented by [Reference Geißert, Heck and Hieber16], a Lagrangian description of the initial problem on varying domains is taken. The main step is to rewrite the equation on a fixed domain of reference $\mathcal{P}$ . This technique has also been successfully applied to the homogenisation of PDEs on non-uniformly periodic or evolving domains in the context of porous media [Reference Evans11, Reference Henry17, Reference Olivares, Bringedal and Pop30] or shape optimisation minimising energy functionals depending on a PDE’s solution [Reference Shamir39]. In order to re-define functions mapping from the altered domains as functions on a fixed domain of reference, we make use of the concept of diffeomorphisms and pullbacks:

The action of a diffeomorphism on the set $\mathcal{P}\subset Y$ is illustrated in Figure 3.

Figure 3.

Smooth deformation of domain $\mathcal{P}$ with circular inclusion mediated by a diffeomorphism $h\in{\text{Diff}}_\Box ^{1} (\bar{Y})$ . As $h$ preserves the exterior boundary $\partial Y$ , the image-set $h(\mathcal{P})$ is an admissible unit cell pore-space geometry.

Definition 2. Pullback. Let $h$ be a diffeomorphism $h\in{\text{Diff}}^{1}(Y)\subset C^1(Y, \mathbb{R}^d)$ and $l\,:\,h(Y) \to \mathbb{R}^n$ , $n\in \mathbb{N}$ . We define the pullback $h^*$ by

\begin{align*} h^*(l)\,:\,Y \to \mathbb{R}^n, \quad h^* l(x) \,:\!=\, l(h(x)). \end{align*}

Note that the pullback is a linear operation and its inverse is given as $(h^*)^{-1}(l) = (h^{-1})^*(l)$ . Furthermore, for sufficiently smooth diffeomorphisms, the pullback is a bounded operator between the $H^m$ function spaces on the original and deformed set, cf. [Reference Geißert, Heck and Hieber16, Reference Lee23]. In Example 3.5, we present an explicit construction of a family of diffeomorphisms mapping circular inclusion of different radii to one another and provide illustrations of the associated pullback of a distance function in Figure 4.

Figure 4.

Visualisation of diffeomorphism (3.7) for $r_2=0.3,\; r_1=0.1$ : (a) illustrates the related circles ( $r_2$ -black, $r_1$ -red) posing the interior boundary of the domain. (b) The displacement field is shown, i.e. $h_{r_1}-\text{id}_Y$ . As enforced by the interpolation function $\xi$ in (3.7), the displacement smoothly vanishes close to the exterior boundary $\partial Y$ . (c), The graph of $h_{r_1}$ , cf. (3.7), is shown along the $y_1$ axis illustrating the three different sections (uniform contraction, transition and identity). (d) Displays the pullback $h_{r_1}^*(\Phi )$ with $\Phi (y_1,y_2)=r_1-||(y_1,y_2)||_2$ . Contour lines uniformly spaced in increments of $0.1$ are added in white. The zero-level-set of $h_{r_1}^*(\Phi )$ highlighted in red corresponds to a circle of radius $r_2$ .

Using the tools introduced in Definitions 1 and 2, we are now able to characterise solutions to (3.2) on the domain $h(\mathcal{P})$ as roots of a function $F$ . Taking a Lagrangian point of view, we work on a fixed domain of reference $\mathcal{P}$ . As such, functions first need to be conveyed to the deformed domain $h(\mathcal{P})$ where the differential operators according to (2.3) are applied. Performing a pullback with the inverse deformation, functions are translated back onto the domain of reference. Accordingly, let us consider the following mapping:

(3.3)

where $\nu _{h(\mathcal{P})}$ denotes the outer unit normal with respect to the domain $h(\mathcal{P})$ and ${\text{Tr}}_{h(\mathcal{P})}$ the standard trace operator on $h(\mathcal{P})$ . Note that the normal vectors can be extended within a tubular neighbourhood of $\partial ^{\text{int}}\mathcal{P}$ , cf. Theorem A.1. By the trace theorem and change of variable rule, $F$ is well-defined as a mapping between the stated spaces. Note that in the first component $F_1$ a vanishing mean value is enforced. Thereby, we eliminate an additional degree of freedom to ensure surjectivity of $F$ .

Since the image of the unit-cell under an arbitrary diffeomorphism may not be a unit cell, we must also introduce a restricted class of deformations. Therefore, let ${\text{Diff}}_\Box (\bar{Y})$ denote the set of diffeomorphism preserving the exterior boundary $\partial Y$ defined by:

(3.4) \begin{align} {\text{Diff}}_\Box ^{k,\alpha } (\bar{Y}) = \left \lbrace h \in{\text{Diff}}^{k,\alpha } (\bar{Y},\bar{Y})\,:\, h_{|U} = \text{id}_U \right \rbrace, \end{align}

for some fixed neighbourhood $\partial Y \subset U\subset \bar{Y}$ . In fact, the diffeomorphism illustrated in Figure 3 belongs to this specified class. The construction (3.4) is inspired by the Hanzawa transformation, cf. [Reference Peter32], and requires the diffeomorphism to decay smoothly towards the identity at the exterior boundary. As such, periodic functions with respect to $Y$ admit a periodic pullback for $h\in{\text{Diff}}^{k,\alpha }_\Box (\bar{Y})$ . The importance of the mapping $F$ defined in (3.3) is now reflected in the following characterisation property. For $u\in H^2_{\#,0}(\mathcal{P})$ , it holds

for all $h\in{\text{Diff}}^{2,1}_\Box (\bar{Y})$ . As such, we established a one-on-one relation between the roots of $F$ and the PDE’s solution on the deformed domain $h(\mathcal{P})$ .

We will now apply the implicit function theorem to $F$ to obtain a continuous mapping $h \mapsto u$ in the respective spaces as summarised in Theorem 1 below. As a result, a relation between the deformation of $\mathcal{P}$ and the associated solution to (2.3) is established. Therefore, we check the following properties:

Continuity of F: Let us denote the pullback of $u$ via $h^{-1}$ by $v=h^{*-1}u$ constituting a function on $h(\bar{Y})$ and the image point on $h(\bar{Y})$ by $y=h(x)$ . First, we consider the first term of the first component $F_1$ of the mapping $F$ . By applying the chain rule, we have

(3.5) \begin{align} & h^*\Delta h^{*-1} u (x) = \Delta v(y) = \Delta u(h^{-1}(y)) = \\ &\sum _{i,j=1}^d \frac{\partial ^2 u}{\partial x_i \partial x_j} \left (\sum _{k=1}^d \frac{\partial h^{-1}_i}{\partial y_k} \frac{\partial h^{-1}_j}{\partial y_k} \right ) +\sum _{i=1}^d \frac{\partial u}{\partial x_i} \left (\sum _{k=1}^d \frac{\partial ^2 h^{-1}_i}{\partial y_k \partial y_k} \right ). \nonumber \end{align}

Note that by the inverse function theorem and the representation of a matrix’ inverse via the cofactor matrix, we can rewrite all partial derivatives of $h_i^{-1}(y)$ as a function of derivatives of $h_i(x)$ only involving multiplications and division by $\text{det}(\nabla h(x))$ . By the uniform boundedness of the last expression away from zero, this map is in particular locally Lipschitz continuous. For any sequence $(u_i,h_i)_{i \in \mathbb{N}}$ in $ H^2_{\#,0}(\mathcal{P}) \times{\text{Diff}}^{2,1}(\bar{Y})$ converging to $(u,h)$ in the product topology, we have the convergence of the derivatives of $u_i$ in $L^2$ and the convergence of the derivatives of $h_i$ uniformly. Hence, expression (3.5) is continuous in $(u,h)\in H^2_{\#,0}(\mathcal{P}) \times{\text{Diff}}^{2,1}(\bar{Y})$ . As the integral is a linear and bounded operator $\int \,:\, L^2(\mathcal{P})\to \mathbb{R}$ , we obtain continuity for $F_1$ .

We can apply the same strategy to prove continuity of the second component $F_2$ of $F$ . Noting the representation

\begin{align*} \nu _{h(\mathcal{P})} (y) = (\nabla h)^{-T}(x) \; \nu _{\mathcal{P}}(x) ||(\nabla h)^{-T}(x) \; \nu _{\mathcal{P}}(x)||^{-1}_2 \end{align*}

derived in [Reference Geißert, Heck and Hieber16] using the inverse transposed Jacobian matrix $(\nabla h)^{-T}$ , we compute

\begin{align*} F_{2}(u,h) (x) &= h^{*}{\text{Tr}}_{h(\mathcal{P})}\left ((\nu _{h(\mathcal{P})} \cdot \nabla h^{*-1}u) (y) - \nu _{h(\mathcal{P})}(y) \cdot e_1\right ) \nonumber \\ &={\text{Tr}}_{\mathcal{P}} \left (\left ( (\nabla h)^{-T} \nu _{\mathcal{P}} \right ) \cdot \left [\sum _{i=1}^d\frac{\partial u}{\partial x_i} \frac{\partial h^{-1}_i}{\partial y_j}-\delta _{j,1}\right ]_j \cdot ||(\nabla h)^{-T}(x) \; \nu _{\mathcal{P}}(x)||_2^{-1}\right ) \end{align*}

for $x\in \partial ^{\text{int}}\mathcal{P}$ . As ${\text{Diff}}^{2,1}(\bar{Y}) \subset C^{2,1}(\bar{Y},\mathbb{R}^d)$ is open, we conclude the continuity of $F$ on a neighbourhood $V$ of $(u,\text{id}_{\bar{Y}})$ .

Continuity of $F^{\prime }_u$ : By Definition 2 the pullback operator is linear. Using the linearity of the differential and trace operators involved, we conclude

for all $w\in H^2_{\#,0}(\mathcal{P})$ . Following the arguments from above, we obtain continuous Fréchet differentiability with respect to the first argument on a neighbourhood $V$ of $(u,\text{id}_{\bar{Y}})$ .

Bijectivity of $F^{\prime }_u$ : The bijectivity of $F^{\prime }_u(u,\text{id}_{\bar{Y}})$ onto $L_0^2(\mathcal{P}) \times H^{\frac{1}{2}}(\partial ^{\text{int}}\mathcal{P})$ is equivalent to finding a unique solution to the elliptic problem (2.3) on $\mathcal{P}$ . More precisely, for a given point $(f,g)$ in the image space of $F$ , we search for a weak solution $u$ for the Neumann boundary condition $g$ and source term $f+c$ for a constant $c\in \mathbb{R}$ . By Lemma 3.1, there exists exactly one $c$ such that the problem admits a solution (compatibility condition) in $H^2_{\#,0}(\mathcal{P})$ . In that case, the solution is unique. Note that due to the set of invertible bounded linear operators between Banach spaces being open and the continuity of $F^{\prime }_u$ , the bijectivity property of $F^{\prime }_u$ in fact holds on a neighbourhood of $(u,\text{id}_{\bar{Y}})$ .

Summarising the above arguments, we conclude the following statement.

Theorem 1. Continuous dependence of weak solutions to ( 2.3 ) on diffeomorphisms. Assume a $C^{2,1}$ open set $Y\setminus \bar{\mathcal{P}} \subset Y$ being compactly contained in $Y$ and $u\in H^2_{\#,0}(\mathcal{P})$ such that $F(u,\text{id}_{\bar{Y}})= (0,0)$ , i.e. $u$ is a solution to problem ( 3.2 ). Then there exists a neighbourhood $V\subset C^{2,1}(\bar{Y})$ of $\text{id}_{\bar{Y}}$ and a continuous function $g\,:\,V\to H^2_{\#,0}(\mathcal{P})$ , $g(\text{id}_{\bar{Y}})=u$ , such that

\begin{align*} F(g(h),h)= (0,0), \quad \forall h\in V. \end{align*}

Particularly, $h^{*-1}g(h)$ solves ( 3.2 ) up to an additive constant on $h(\mathcal{P})$ for all $h\in{\text{Diff}}^{2,1}_\Box (\bar{Y})\cap V$ .

Proof. This is an immediate consequence of the implicit function theorem for Banach spaces as given in [Reference Soulaine and Tchelepi41] and the arguments above.

By the previous theorem, we established a continuous relation between the diffeomorphism $h$ describing the alteration of the domain $\mathcal{P}$ and the pullback of the solution $g(h)$ to the associated problem (2.3). In a final step, we show that the continuous dependence carries over to the desired quantity $\mathbb{D}$ defined in (2.3):

Corollary 3.4. Continuous dependence of $\mathbb{D}$ on diffeomorphisms. Under the assumptions of Theorem 1, the mapping

\begin{align*} R\,:\, V \to \mathbb{R},\quad h\mapsto \int _{h(\mathcal{P})} \nabla h^{*-1}g(h) \; dy \end{align*}

is continuous. Therefore, the diffusion tensor depends continuously on ${\text{Diff}}^{2,1}_\Box (\bar{Y})$ -variations of the domain $\mathcal{P}$ .

Proof. By the change of variables theorem and chain rule, we have

(3.6) \begin{align} \int _{h(\mathcal{P})} \nabla h^{*-1}g(h) \; dy &= \int _{\mathcal{P}} \nabla g_h(x) \cdot \nabla h^{-1}(h(x)) \cdot \mid \text{det} (\nabla h(x))\mid \; dx \\ &=\int _{\mathcal{P}} \nabla g_h(x) \cdot \nabla h(x)^{-1} \cdot \mid \text{det} (\nabla h(x))\mid \; dx. \nonumber \end{align}

As $g$ is continuous with respect to the $H^2$ -norm in the image space, continuity of the functional is proven.

By the previous corollary, we established the continuous behaviour of $\mathbb{D}$ on $h\in{\text{Diff}}_\Box ^{2,1}(\bar{Y})$ in the topology of $C^{2,1}(\bar{Y})$ . However, this degree of regularity is insufficient for our later purposes, cf. Theorem A.2. In order to obtain stronger results, we specify the setting more tailored to our later application. Let us now consider a smooth mapping $h\,:\,({-}S,S) \to{\text{Diff}}_\Box ^{2,1}(\bar{Y})$ for some artificial time horizon $S\gt 0$ and $h(0)=\text{id}_{\bar{Y}}$ . This relates to a 1-parametric deformation of the initial geometry.

Example 3.5. Continuous path of diffeomorphisms. Consider inclusions $Y\setminus \bar{\mathcal{P}}$ of circular shape and of different radii. In this case, a smooth diffeomorphism on $Y$ can be easily constructed by radially compressing/expanding annuli within a compact subset of $Y$ . Given two radii $0\lt r_1\leq r_2\lt \frac{1}{2}$ , a deformation mapping a circle of radius $r_2$ to a circle of radius $r_1$ is defined by

(3.7) \begin{align} h_{r_1}(y)=\begin{cases} y, & |y|\gt \frac{1}{2}, \\[4pt] (1-\xi ( |y|))y + \xi (|y|)\left(\frac{r_1}{r_2} y\right), & r_2 \leq |y|\leq \frac{1}{2}, \\[4pt] \frac{r_1}{r_2} y, & |y|\leq r_2 \end{cases} \end{align}

choosing a suitable $\xi \in C^\infty \left ([r_2,\frac{1}{2}]\right )$ with $\xi ^\prime \in C_0^\infty \left ((r_2,\frac{1}{2})\right )$ , $\xi (r_2)=1$ , $\xi (\frac{1}{2}) = 0$ , see [9]. As such, the domain remains unchanged for $|y|\gt \frac{1}{2}$ and is uniformly contracted for $|y|\leq r_2$ with a smooth convex-combination layer in between. We illustrate this procedure for the choice $r_2=0.3,\; r_1=0.1$ in Figure 4. For $r_2$ fixed, we can consider the path

\begin{align*} h\in C^0\left (0,r_2;{\text{Diff}}^{2,1}_\Box (\bar{Y})\right ), \quad h\,:\, s\mapsto h_s. \end{align*}

Then $R\circ h$ with $R$ being defined in Corollary 3.4 is also a continuous mapping. Consequently, the diffusion tensor depends continuously on $s$ .

In the following, we prove differentiability of the diffusion tensors along such 1-parametric curves. More precisely, $C^m$ -mappings $h\,:\,({-}S,S) \to{\text{Diff}}_\Box ^{2,1}(\bar{Y})$ will be considered. Accordingly, we switch the above setting to the following:

tracking along a fixed path of diffeomorphisms in comparison to (3.3). In order to obtain higher differentiability of the resolution function $g$ in Theorem 1, higher regularity of $F$ needs to be established. Revisiting the calculations performed in (3.5) we immediately see a transfer of regularity to $F$ with respect to the second variable. As the mapping is linear with respect to the first variable, we again obtain $C^m$ -regularity for $F$ . More precisely, the following theorem holds extending the smooth dependence results for simple parametric families of shapes as derived in [Reference Cioranescu and Donato9, Reference Ray and Schulz34].

Theorem 2. Smooth dependence of solutions to ( 2.3 ) on geometric parameter. Consider a $C^m$ -curve $s\mapsto h_s$ of ${\text{Diff}}_{\Box }^{2,1}(\bar{Y})$ -embeddings and $h_0=\text{id}_{\bar{Y}}$ for $m\geq 1$ . Assume a $C^{2,1}$ open set $Y\setminus \bar{\mathcal{P}} \subset Y$ being compactly contained in $Y$ and $u\in H^2_{\#,0}(\mathcal{P})$ such that $F(u,0)= (0,0)$ , i.e. $u$ is a solution to problem ( 3.2 ). Then there exists a neighbourhood $V$ of zero and a $C^m$ -function $g\,:\,V\to H^2_{\#,0}(\mathcal{P})$ , $g(0)=u$ , such that

\begin{align*} F(g(s),s)= (0,0), \quad \forall s\in V. \end{align*}

Particularly, $h_s^{*-1}g(s)$ solves ( 3.2 ) up to an additive constant on $h(\mathcal{P})$ for all $s\in V$ .

Proof. This is again an immediate consequence of the implicit function theorem for Banach spaces as given in [Reference Soulaine and Tchelepi41].

Again, we can leverage this regularity to the diffusivity tensors (2.2).

Corollary 3.6. Smooth dependence of $\mathbb{D}$ on geometric parameter. Under the assumptions of Theorem 2, the mapping

\begin{align*} R\,:\, V \to \mathbb{R},\quad s\mapsto \int _{h_s(\mathcal{P})} \nabla h_s^{*-1}g(s) \; dy \end{align*}

is $m$ -times continuously differentiable. Therefore, the diffusion tensor ( 2.2 ) depends $C^m$ -regularly on variations of the domain $\mathcal{P}$ along a path of diffeomorphisms of specified regularity.

Proof. Since the map $s \mapsto h_s$ is $C^m$ -regular, we establish the same degree of Fréchet differentiability in the spatial derivatives $s \mapsto \nabla h_s$ as mappings $({-}S,S) \to C^1(\bar{Y},\bar{Y})$ with respect to the corresponding norms. Revisiting the calculations performed in (3.6) and using the product rule for Fréchet differentiable functions, we conclude the assertion.

3.3. Existence for partial coupling

In this section, we present an existence result for strong local-in-time solutions to (3.1) under the assumption of no back-coupling from the macro- to the micro-scale, cf. Figure 1. That is, we assume the evolution of the underlying pore geometry to be known a-priori. As such, the treatment of the problem is accessible more easily in comparison to the fully coupled system which we will discuss in Section 3.5. In [Reference Ray and Schulz34], a similar fully coupled problem was solved under the assumption of effective parameters being parameterisable by the porosity $\phi$ in a smooth and a-priori known way. As opposed to this restriction, we will introduce a new order parameter $s$ which corresponds to the parametrisation in the path of diffeomorphisms used to describe evolved initial geometries. Particularly, this approach allows for the opposite description of multiple geometries which admit the same porosity. Due to the identical structure of the model, we will use the methods of [Reference Ray and Schulz34] for our following analysis.

To this point, we considered an initial geometry $\mathcal{P}$ of class $C^{2,1}$ and a parameterised path of diffeomorphisms $h\,:\,({-}S,S)\to{\text{Diff}}_\Box ^{2,1}(\bar{Y})$ of class $C^1$ with $S\in \mathbb{R}^+$ and $h_0 = \text{id}_{\bar{Y}}$ . In order to allow for spatial variations of effective parameters, we realise the prescription of the geometry evolution by specifying $s\,:\, \Omega _T \to ({-}S,S)$ . That is, we assume the state of each microscopic unit cell $Y(t,x)$ to be given as the state of a single master unit cell $Y^*$ at time-parameter $s(t,x)$ . The corresponding setup is visualised in Figure 5 illustrating the assignment of initial conditions on $\Omega$ . Accordingly, the effective parameters of (3.1) are given by $\phi (s),\; \sigma (s),\; \mathbb{D} (s)$ . Note that we perform the necessary redefinition of the functions $\phi,\; \sigma,\; \mathbb{D}$ to mappings from an open interval of $\mathbb{R}$ without change of notation. We furthermore restrict to the non-degenerative case, i.e. we assume

(3.8) \begin{align} \forall s\in ({-}S,S)\,:\, 0\lt \phi (s) \lt 1, \quad \sigma (s) \gt 0, \quad \mathbb{D}(s)\gt 0, \end{align}

where the last inequality holds in the sense of matrices (Loewner partial ordering). This assumption is naturally fulfilled for small times $t\gt 0$ by suitably prepared initial conditions. Moreover, we restrict our consideration to locally Lipschitz reaction rates $f(c)$ generalising the linear reaction rates prescribed in [Reference Ray and Schulz34]. Finally, we state the following anisotropic Sobolev spaces:

\begin{align*} \mathcal{X}_1\,:\!=\,W^{1,2}_r(\Omega _T) = L^r(0,T;W^{2,r}(\Omega )) \cap W^{1,r}(0,T;L^r(\Omega )), \end{align*}

with $r\gt d+2$ which play a crucial role in the subsequent existence result. Following the major steps of [Reference Ray and Schulz34], we have

Figure 5.

Left: Master unit cell $Y^*$ with interior boundary $\partial ^{\text{int}}\mathcal{P}^*$ in black corresponding to $s=0$ . Each colour corresponds to the interior boundary of a deformed cell which is reachable from $\mathcal{P}^*$ along a path of diffeomorphisms parameterised by $s$ . Right: Macroscopic domain $\Omega$ coloured according to the initial underlying geometry displayed in the left image. For the two exemplary macroscopic points $x_1,x_2\in \Omega$ the associated microscopic geometry is displayed.

Proof. First, we note that using the results of Theorem 2 the mapping $\mathbb{D}\,:\, ({-}S,S) \to \mathbb{R}^{d,d}$ is of class $C^1$ and accordingly $\mathbb{D}\circ s \in C^1(\overline{\Omega _{T_1}})$ . Similarly, we obtain $\phi \in C^1(({-}S,S))$ as a consequence of the following representation:

\begin{align*} \phi (s) =\int \limits _{h_s(\mathcal{P})} 1\; dy = \int \limits _{\mathcal{P}} \lvert \text{det} \left (\nabla h_s\right )\rvert \; dx \end{align*}

using the change of variables theorem. In order to establish regularity for $\sigma$ , we consider $h(\partial ^{\text{int}}\mathcal{P})$ as an evolving manifold. Let $\varphi \,:\,U\subset \mathbb{R}^{d-1}\to \mathbb{R}^d$ be a suitable local parameterisation of $\partial ^{\text{int}}\mathcal{P}$ . Then, $\hat{\varphi }\,:\, ({-}S,S) \times U \to \mathbb{R}^d$ defined as

\begin{align*} \hat{\varphi }(s,x) \,:\!=\, h_s(\varphi (x)) \end{align*}

is a parameterisation of $h_s(\partial ^{\text{int}}\mathcal{P})$ for each $s\in ({-}S,S)$ . Using the regularity of $\hat{\varphi }$ , we infer continuity of the mapping

\begin{align*} s \mapsto \int \limits _{h_s(\varphi (U))} 1 \; d\sigma = \int \limits _{U} \sqrt{\text{det}\left (\nabla \hat{\varphi } (\nabla \hat{\varphi } )^T \right ) } \; dx \end{align*}

which translates to the continuity of $\sigma$ by using a partition of unity subordinate to the domains of parameterisation, cf. [Reference Ladyzhenskaya, Solonnikov and Ural‘ceva22]. As a result of the continuity of all effective parameters with respect to $s$ , there exists a $\delta \in (0,1)$ such that:

(3.9) \begin{align} \forall s\in ({-}S+\epsilon,S-\epsilon ) \,:\, \delta \lt \phi (s) \lt 1-\delta, \quad \sigma (s) \gt \delta, \quad \mathbb{D}(s)\gt \delta \mathbb{1}_d. \end{align}

Note that the continuity of the eigenvalues is inherited from the continuity of $\mathbb{D}$ , cf. [Reference Jambhekar, Mejri, Schröder, Helmig and Shokri20]. Following the technique presented in [Reference Ray and Schulz34], the proof is now based on Schauder’s fixed-point theorem applied to the set

\begin{align*} \mathcal{K}_1 = \left \lbrace c\in \mathcal{X}_1 \,:\, ||c||_{\mathcal{X}_1}\leq K \right \rbrace \end{align*}

for a constant $K\geq 1$ chosen appropriately later. Apparently, $\mathcal{K}_1$ is a convex, closed and bounded subset of $\mathcal{X}_1$ . In order to apply standard linear solution theory of parabolic equations, we rewrite equation (3.1) in the following fixed-point form:

(3.10) \begin{align} \partial _t c -\nabla \cdot \left ( \frac{\mathbb{D}(s)}{\phi (s)} \nabla c\right ) = -\frac{\partial _t [\phi (s)]}{\phi (s)} \tilde{c} + \frac{\sigma (s) }{\phi (s)} f(\tilde{c}) + \frac{\mathbb{D}(s)}{\phi (s)^2} \nabla [\phi (s)] \cdot \nabla \tilde{c}. \end{align}

Now consider the mapping $\mathcal{F}_1\,:\,\mathcal{K}_1\to L^{r}(\Omega _T)$ which maps a concentration $\tilde{c}\in \mathcal{K}_1$ to the right-hand side of (3.10). Using the compact embeddings

(3.11) \begin{align} \mathcal{X}_1 \hookrightarrow W^{\frac{3}{4},\frac{3}{2}}_{2r}(\Omega _T), \quad \mathcal{X}_1 \hookrightarrow C^{\frac{1}{2},1}(\overline{\Omega _T}), \end{align}

$\mathcal{F}_1$ shows to be compact, cf. [Reference Ray and Schulz34]. Furthermore, the parabolic theory of Theorem A.3 delivers a continuous solution operator $\mathcal{F}_2\,:\, L^{r}(\Omega _T) \to W^{1,2}_r(\Omega _T)$ to (3.10). More precisely, we apply Theorem A.3 to the above-prescribed initial and boundary conditions with coefficients defined as

\begin{align*} a_{i,j}(t,x) = \frac{\mathbb{D}_{i,j}(s(t,x))}{ \phi (s(t,x))},\quad a_i(t,x) = -\sum \limits _{j=1}^d \partial _j \left ( \frac{\mathbb{D}_{i,j}(s(t,x))}{ \phi (s(t,x))}\right ), \quad a(t,x) = 0 \end{align*}

and source term $f$ according to the right-hand side of (3.10).

As such, for a given $\tilde{c}\in \mathcal{X}_1$ , we have

\begin{align*} ||c||_{\mathcal{X}_1} \leq C_p \left (\text{DC}+ \left \lVert -\frac{\partial _t [\phi (s)]}{\phi (s)} \tilde{c} + \frac{\sigma (s)}{\phi (s)} f(\tilde{c}) + \frac{\mathbb{D}(s)}{\phi (s)^2} \nabla [\phi (s)] \cdot \nabla \tilde{c}\right \rVert _{L^r(\Omega _T)} \right ), \end{align*}

abbreviating the contributions from initial and boundary data by

\begin{align*} \text{DC}\,:\!=\,||c_0||_{W^{2-\frac{2}{r},r} (\Omega )} + ||C_0||_{W^{1-\frac{1}{2r}, 2-\frac{1}{r}}_r(\partial \Omega _T)}. \end{align*}

Due to $f$ being locally Lipschitz, there exists a monotone function $\tilde{f}$ satisfying

(3.12) \begin{align} \tilde{f}\,:\, [0,\infty ) \to \mathbb{R}, \quad |f(x)| \leq \tilde{f}(|x|), \end{align}

Similar to the estimates established in [Reference Ray and Schulz34], we obtain using Hölder’s inequality and (3.12)

\begin{align*} ||c||_{\mathcal{X}_1} &\leq C_p \left (\text{DC} + C_s T^{\frac{1}{r}} |\Omega |^{\frac{1}{r}}\left (||\tilde{c}(t)||_{L^\infty (\Omega _T)} + \tilde{f}(||\tilde{c}||_{L^\infty (\Omega )}) \right ) \right ) \\ &+ C_p C_s \left (T^{\frac{1}{2r}} ||\nabla [\phi (s)] ||_{L^\infty (0,T; L^{2r}(\Omega ))} ||\nabla \tilde{c} ||_{L^{2r}(0,T; L^{2r}(\Omega ))} \right ), \nonumber \end{align*}

with constant $C_s$ depending on the bounds of $\sigma, \phi,\partial _t \phi,\mathbb{D}$ , cf. (3.9). By the embeddings (3.11), every appearing norm of $\tilde{c}$ is bounded by a multiple of $||\tilde{c}||_{\mathcal{X}_1}$ and therefore by a multiple of $K$ . As such, we obtain the self-mapping property of $\mathcal{F}=\mathcal{F}_2\circ \mathcal{F}_1\,:\, \mathcal{K}_1 \to \mathcal{K}_1$ for sufficiently large $K$ and sufficiently small $T$ , finishing the existence proof. Uniqueness follows analogously to [Reference Ray and Schulz34] by considering two solutions $c_1,c_2\in \mathcal{X}_1$ as well as their difference $\bar{c}=c_2-c_1$ . Subtracting both associated equations in fixed-point form (3.10), we obtain

(3.13) \begin{align} \partial _t \bar{c} -\nabla \cdot \left ( \frac{\mathbb{D}(s)}{\phi (s)} \nabla \bar{c}\right ) = -\frac{\partial _t [\phi (s)]}{\phi (s)} \bar{c} + \frac{\sigma (s) }{\phi (s)} \left (f(c_2)-f(c_1)\right ) + \frac{\mathbb{D}(s)}{\phi (s)^2} \nabla [\phi (s)] \cdot \nabla \bar{c}. \end{align}

Testing (3.13) with $\bar{c}$ and estimating the right-hand side with Hölder’s and Young’s inequality show

(3.14) \begin{align} \frac{1}{2} \partial _t ||\bar{c}||^2_{L^2(\Omega )}&\leq \left ( \left \lVert \frac{\partial _t [\phi (s)]}{\phi (s)} \right \rVert _{L^\infty (\Omega )} + L \left \lVert \frac{\sigma (s) }{\phi (s)} \right \rVert _{L^\infty (\Omega )} + C(\epsilon )\left \lVert \frac{\mathbb{D}(s)}{\phi (s)^2} \nabla [\phi (s)] \right \rVert _{L^\infty (\Omega )}\right ) ||\bar{c}||^2_{L^2(\Omega )}\nonumber \\ &- \int \limits _\Omega \frac{\mathbb{D}(s)}{\phi (s)}\nabla \bar{c} \cdot \nabla \bar{c} \; dx + \epsilon \left \lVert \frac{\mathbb{D}(s)}{\phi (s)^2} \nabla [\phi (s)] \right \rVert _{L^\infty (\Omega )} ||\nabla \bar{c}||^2_{L^2(\Omega )}, \end{align}

where $L$ denotes the Lipschitz constant of $f$ with respect to the compact interval

\begin{align*} \left [-\max \limits _{i\in \{1,2\}} \left \lbrace ||c_i||_{C^0(\overline{\Omega _T})}\right \rbrace,\; \max \limits _{i\in \{1,2\}} \left \lbrace ||c_i||_{C^0(\overline{\Omega _T})}\right \rbrace \right ]. \end{align*}

By the uniform coercivity of $\mathbb{D}$ , we can absorb the last addend of (3.14) into the diffusion term for sufficiently small $\epsilon \gt 0$ . Uniqueness now follows from Gronwall’s inequality.

3.4. Level-set equation induced diffeomorphisms

The smooth dependence results derived in Section 3.2 are based on geometry deformation by smooth paths of diffeomorphisms. In the following, we investigate under which conditions on the initial geometry and normal velocity field alterations performed by the level-set equation (2.7) induce such paths. To do so, we make use of the method of characteristics. As a result, we can replace the assumption of a prescribed path of diffeomorphisms in Theorem 3 by a prescribed normal velocity field $v_n(t,x,y)$ and let the geometry evolve according to the level-set equation which is a much more natural setup from the viewpoint of applications. At first, we must fix the class of real-valued functions whose level sets are guaranteed to be smooth submanifolds of codimension one:

In order to apply the theory for non-linear first-order PDEs, we rewrite the level-set equation (2.7) in the form $F(\vec{x},\Phi,D\Phi )=0$ with $F\,:\,(\vec{x},z,\vec{p})\mapsto \mathbb{R}$ . Note that in this notation, $\vec{x}=(x,t)$ , $\vec{p}=(p,p_{d+1})$ corresponds to the spatial and temporal variable, and $D=(\nabla _x,\partial _t)$ denotes the related differential operator. Apparently, we have

\begin{align*} F(\vec{x},z,\vec{p}) = p_{d+1}+v_n(\vec{x}) |p|. \end{align*}

with the normal interface velocity $v_n$ of (2.7). Prescribing $v_n$ smooth such that it vanishes in a neighbourhood of $\{|p|=0 \}$ , we have $F\in C^2(\mathbb{R}^{2d+3},\mathbb{R})$ . Switching to the characteristic system of ODEs, the equations read

(3.15) \begin{align} \dot{\vec{p}}(s) &=- \partial _{\vec{x}} F (\vec{x}(s),z(s),\vec{p}(s)) - \partial _{z} F (\vec{x}(s),z(s),\vec{p}(s)) \vec{p}(s) = -Dv_n(\vec{x}) |p|, \nonumber \\ \dot{z}(s) &= \partial _{\vec{p}} F(\vec{x}(s),z(s),\vec{p}(s))\cdot \vec{p}(s) = v_n(\vec{x}) \frac{p}{|p|}\cdot p + p_{d+1} = 0,\\ \dot{\vec{x}}(s) &= \partial _{\vec{p}} F (\vec{x}(s),z(s),\vec{p}(s)) = \left (v_n(\vec{x}) \frac{p}{|p|}, 1\right ), \nonumber \end{align}

with initial conditions

(3.16) \begin{align} \vec{p}(0) = \left (\nabla \Phi (0,y), -v_n(y)|\nabla \Phi (0,y)| \right ), \quad z(0) = \Phi (0,y), \quad \vec{x}(0)=(y,0). \end{align}

According to (3.15), the projected characteristics of a solution to (2.7) move in normal direction to the interface with speed $v_n(\vec{x})$ . Furthermore, the function value of $\Phi$ remains constant along trajectories. As such, the zero-level-set describing the position of the fluid–solid interfaces is transported along $x$ . Note that every admissible point of the characteristic ODE system is non-characteristic, i.e. the characteristics admit a strictly monotone distance to the set of prescribed initial data $\bar{Y}\times \{0\}$ . As such, equation (2.7) admits a unique local-in-time $C^2$ -solution by standard theory [Reference Eden, Nikolopoulos and Muntean10]. Furthermore, the parameterised trajectories associated with $x$ induce a smooth path of diffeomorphisms and the artificial parameter $s$ coincides with the actual physical time $t$ . More precisely, we have the following statement:

Proof. In the following, we investigate the regularity of the trajectories associated with $\vec{x}$ solving the systems of ODEs (3.15). More precisely, we consider the smaller system in the spatial variables $x$ and $p$ which is closed due to the time independence of $v_n$ . As a suitable Banach space for $(x,p)$ , we introduce

\begin{align*} \mathcal{X} = (C^3(\bar{Y}, \mathbb{R}))^{d} \times (C^3(\bar{Y}, \mathbb{R}))^{d}. \end{align*}

Then the reduced initial conditions (3.16) are element of $\mathcal{X}$ . It is straightforward to check that the structure function $f\,:\,\mathcal{X} \supset V \to \mathcal{X}$ of the reduced ODE system

(3.17) \begin{align} f(x,p) = \left (v_n(x) \frac{p}{|p|},\;-\nabla v_n(x) |p| \right ) \end{align}

is well-defined and locally Lipschitz continuous with respect to the norm $|.|_{\mathcal{X}}$ for a small neighbourhood $V$ of $(x(0),p(0))$ , as long as $v_n = 0$ in a neighbourhood of $\{y\in Y \,:\, |\nabla \Phi _0(y)| = 0\}$ . Due to the tubular neighbourhood theorem (Theorem A.1), this can be achieved by modifying $v_n$ appropriately without changing the solution locally at $\partial ^{\text{int}} \mathcal{P}$ . More precisely, we force $v_n$ to zero away from a neighbourhood of $\partial ^{\text{int}} \mathcal{P}$ and within a neighbourhood of $\partial Y$ in a smooth manner. By Picard–Lindelöf theorem, there exists $T\gt 0$ such that the ODE system admits a unique solution $(x,p)\in C^1(0,T; \mathcal{X})$ . Due to the convexity of the domain, we obtain $x\in C^1(0,T; C^{2,1}(\bar{Y},\mathbb{R}^d))$ . By the choice of $v_n$ , we have $\text{im}(x(s)) = \bar{Y}$ for all $s\in (0,T)$ . Since the initial conditions fulfil $x(0)=\text{id}_{\bar{Y}}\in{\text{Diff}}^{2,1}(\bar{Y})$ , the diffeomorphism property for $x(s)$ is obtained in a possibly reduced time interval $(0,T)$ . Consequently, $x\in C^1(0,T;{\text{Diff}}_\Box ^{2,1}(\bar{Y}))$ . Since the value of $\Phi$ is constant along the trajectories associated with $x$ , we finally see $h_t(\bar{\mathcal{P}}) = \left \lbrace \Phi (t,\cdot )\leq 0 \right \rbrace$ .

In order to underline the power of the above Lemma 3.10, let us reconsider the case of contracting and expanding circles as in Example 3.5. Previously, a suitable family of diffeomorphisms had to be constructed explicitly in (3.7) to apply Theorem 2. With the help of Lemma 3.10, it is now sufficient to check that the function

\begin{align*} \Phi _r(y) = r - ||y||_2 \end{align*}

has a zero-level-set of a circle of radius $r\lt 0.5$ and can be smoothed locally around $0$ to fulfil $\Phi _{r,0} \in C^4(\bar{Y})$ . The statement follows by applying Lemma 3.10 to $\Phi _{r,0}$ and $v_n \equiv 1$ . However, the presented framework also applies to much more general geometry deformations as illustrated in Figure 3 where an explicit construction of diffeomorphisms is unsuitable. It especially includes the transition to non-convex shapes as well as the treatment of disconnected components of solid.

3.5. Existence for full coupling

In geoscientific applications of our model, the evolution of the microscopic geometry of the porous medium is not known in advance but depends on the time-dependent macroscopic concentration field. Accordingly, this chapter is dedicated to the local-in-time existence of strong solutions to a model where the geometry evolution is described by the level-set equation (2.7). For clarity, we restate the model equations:

\begin{align*} \phi \partial _t c - \nabla \cdot (\mathbb{D}\nabla c) &= \sigma f(c) - \partial _t \phi c &&\quad \text{in } (0,T)\times \Omega, \\ \frac{\partial \Phi }{\partial t} + v_n |\nabla _y \Phi | &= 0, &&\quad \text{in } (0, T)\times \Omega \times Y, \nonumber \end{align*}

supplemented by boundary and initial conditions as well as diffusion cell problem (2.3), where the level-set normal interface velocity is given by

(3.18) \begin{align} v_n(t,x) = f(c(t,x)), \end{align}

i.e. depending on the solution of the macroscopic concentration. In simplification of (2.9), we consider a uniform velocity within each unit cell to facilitate the establishment of higher regularity for quantities derived from the geometry. Furthermore, we restrict to scenarios where the porosity can be used as the natural order parameter uniquely characterising the geometry state which is the typical case in dissolution/precipitation processes. Moreover, we assume linear reaction rates. By doing so, we retract to the setting investigated in [Reference Ray and Schulz34] enabling us to use respective results stated therein. Additionally, we again consider a single evolving master unit cell $Y^*$ as in Section 3.3 and introduce spatial inhomogeneity in $\Omega$ by choosing different evolution states of $Y^*$ as the initial condition.

In the following, we approach existence of solutions $(c,\Phi )\in (\mathcal{X}_1, \Omega _x\times C^1((0,T)\times Y))$ to the fully coupled model described above, where the solution space with respect to $\Phi$ is defined as

\begin{align*} \Omega _x\times C^1((0,T)\times Y) = \left \lbrace \Phi \,:\, (0, T)\times \Omega \times Y \to \mathbb{R}\,:\, \; \forall x\in \Omega \quad \Phi (\cdot,x,\cdot ) \in C^1((0,T)\times Y) \right \rbrace. \end{align*}

To do so, we rewrite the given problem in the variables $(c,\phi )\in \mathcal{X}_1^2$ . As such, we can apply Theorem A.2 to obtain local-in-time existence in the variables $(c,\phi )$ . In order to do so, the effective parameters $\mathbb{D},\sigma$ must be expressed as functions of $\phi$ . As we will see, this relation is independent of the concentration solution $c$ and is therefore determined by the level-set initial condition alone. Given the solution $(c,\phi )$ of the transformed system, we construct the solution $(c,\Phi )$ to our model of consideration.

Let us consider the geometry evolution being preliminarily parameterised by the time $t$ of the level-set equation. Using the properties of (3.18), we can reparametrise all effective parameters as a function of $\phi$ by setting:

(3.19) \begin{align} \hat{\sigma } ={\sigma } \circ \phi ^{-1}, \quad \hat{\mathbb{D}} ={\mathbb{D}} \circ \phi ^{-1}. \end{align}

The main difficulty in proving existence for the bilaterally coupled system is establishing sufficiently high regularity in the coefficients $\hat{\sigma }, \hat{\mathbb{D}}$ as required to apply Theorem A.2.

First, we notice that the mapping $\phi \mapsto \hat{\sigma }, \hat{\mathbb{D}}$ is independent of the particular choice of a normal velocity $v_n$ in (3.18) in the set of y-independent functions which can be seen as follows: Let $\Phi _1$ solve (2.7) with $v_n \equiv 1$ . Then $\Phi (t,x) = \Phi _1(V(t),x)$ solves (2.7) with $v_n =v$ and $V$ being the primitive of $v$ . As such, solutions are solely rescaled with respect to the time variable but not with respect to space, i.e. the individual geometries relating to $\Phi _1(s,\cdot )$ remain unaffected. It is therefore sufficient to consider the case $v_n\equiv 1$ , where, due to

(3.20) \begin{align} \partial _s \phi (s) = -\sigma (s), \end{align}

$\phi$ is strictly monotonically increasing. This fact justifies the transformations introduced in (3.19). As such, the regularity of $\phi \mapsto \hat{\sigma }, \hat{\mathbb{D}}$ follows immediately from the regularity of $\phi, \sigma, \mathbb{D}$ which we investigate using the same means as in Theorem 3.

Let $\varphi \,:\,U\subset \mathbb{R}^{d-1}\to \mathbb{R}^d$ be a suitable local parameterisation of $\partial ^{\text{int}}\mathcal{P}$ . Then, $\hat{\varphi }\,:\, ({-}S,S)\times U \to \mathbb{R}^d$ defined as

(3.21) \begin{align} \hat{\varphi }(s,x) \,:\!=\, \varphi (x) + s\nu (\varphi (x)) \end{align}

is a local parameterisation of the deformed interface at times $s\in ({-}S,S)$ using the results of Section 3.4. Note that $\hat{\varphi }(s,\cdot )$ is a bijection for $s$ close to zero and in case of $\Phi _0 \in C^4(Y)$ , the map (3.21) is of class $C^3$ , cf. Theorem A.1. Using the regularity of $\hat{\varphi }$ , we infer the mapping

\begin{align*} s \mapsto \int \limits _{U} \sqrt{\text{det}\left (\nabla \hat{\varphi } (\nabla \hat{\varphi } )^T \right ) } \; dx \end{align*}

to be of class $C^2$ , which translates to $\sigma (s)$ by using a partition of unity, cf. [Reference Ladyzhenskaya, Solonnikov and Ural‘ceva22]. By (3.20), we also obtain $\phi \in C^3(({-}S,S))$ . As such, we finally conclude $\hat{\mathbb{D}} \in C^1((\phi _{\text{min}},\phi _{\text{max}}))$ , $\hat{\sigma } \in C^2((\phi _{\text{min}},\phi _{\text{max}}))$ for sufficiently narrow bounds $\phi _{\text{min}}\lt \phi ^0\lt \phi _{\text{max}}$ around the initial condition. Summing up the observations of this section, we obtain the following existence result using Theorem A.2.

Theorem 4. Existence of strong solutions, full coupling, diffusive transport. Let $\Omega \subset \mathbb{R}^d,\; d\in \{2,3\}$ , be a $C^2$ -domain, $r\gt d+2$ with initial conditions $c_0\in W^{2-\frac{2}{r},r}(\Omega )$ , $c_0\geq 0$ , fulfilling $c_0=0$ on $\partial \Omega$ . Furthermore, consider an initial inclusion compactly contained in $Y^*$ given by a regular $C^4(\bar{Y}^*)$ level-set function evolving according to the level-set equation ( 2.7 ) such that solutions for $v_n \equiv \pm 1$ cover $\phi \in (\phi _{\text{min}},\phi _{\text{max}}) \subset \subset (0,1)$ . Let $\phi ^0\in W^{2,r}(\Omega )$ , $\phi ^0(x)\in (\phi _{\text{min}}+\epsilon,\phi _{\text{max}}-\epsilon )$ for some $\epsilon \gt 0$ , $x\in \Omega$ and the reaction rate given as $f(c)=c$ . Then there exists a local-in-time solution $(c,\Phi )\in (\mathcal{X}_1, \Omega _x\times C^1((0,T)\times Y))$ to the bilaterally coupled model. In case of higher regularity, i.e. $c_0\in W_r^{4-\frac{2}{r}}(\Omega )$ , $\phi ^0\in W_r^{3-\frac{2}{r}}(\Omega )$ fulfilling the compatibility condition $(\nabla c_0 \cdot \nabla \phi ^0) \equiv 0$ , $\Delta c_{0|\partial \Omega }\equiv 0$ it either holds for the maximum existence time $T$ :

(3.22) \begin{align} \lim \limits _{t\to T} \inf \limits _{x\in \Omega } \phi (t,x) = \phi _{\text{min}} \quad \quad \text{or} \quad \quad T=\infty. \end{align}

Proof. In the setup presented above, Theorem A.2 ensures the short-time existence of solutions $(c,\phi )\in (\mathcal{X}_1)^2$ to the equations

\begin{align*} \phi \partial _t c -\nabla \cdot (\hat{\mathbb{D}}(\phi )\nabla c) &= \hat{\sigma }(\phi ) c^2 - \hat{\sigma }(\phi ) c,\\ \nonumber \partial _t\phi = -\hat{\sigma }(\phi ) c, \end{align*}

with $\hat{\mathbb{D}},\; \hat{\sigma }$ as constructed in (3.19). Apparently, the second equation corresponds to the level-set function $\Phi$ fulfilling

(3.23) \begin{align} \partial _t \Phi (t,x,y) +c(t,x)|\nabla _y \Phi (t,x,y)| = 0, \quad \forall x\in \Omega, \end{align}

for $y$ sufficiently close to the zero-level-set as required. By the embedding theorems (3.11), $t\mapsto c(t,x)$ is continuous for all $x\in \Omega$ . Furthermore, a solution $\Phi _1$ related to $v_n \equiv 1$ in a neighbourhood of the zero-level-set exists and is of class $C^2$ as discussed in Section 3.4. As such, the transformed solution $\Phi (t,x,y) = \Phi _1(C(t,x),x,y)$ with $C(\cdot,x)$ being a primitive of $c(\cdot,x)$ solving (3.23) is of class $C^1((0,T)\times Y)$ for each $x\in \Omega$ . Finally, we investigate the case of high regularity initial conditions. Note that the structure function f in (3.17) is smooth by the regularity of $v_n$ , leading to a solution $x$ smooth in time. Therefore, we particularly have $\hat{\mathbb{D}} \in C^3((\phi _{\text{min}},\phi _{\text{max}}))$ due to Theorem 2. As such, the assumption follows by Corollary 1 of [Reference Ray and Schulz34].

4.1. Setting

In the following, we consider solute transport by diffusion and advection:

(4.1) \begin{align} \phi \partial _t c +\nabla \cdot (vc)- \nabla \cdot (\mathbb{D}\nabla c) = \sigma f(c) - \partial _t \phi c\quad \text{in } (0,T)\times \Omega \end{align}

which is coupled to Darcy’s equation

(4.2) \begin{align} v &= -\mathbb{K} \nabla p &&\text{in } \Omega,\; t\in (0,T),\\ \nabla \cdot v &=\tilde{f} && \text{in } \Omega,\; t\in (0,T). \nonumber \end{align}

In comparison to the equations stated in Section 2, we allow for a general constant-in-time divergence $\tilde{f}$ . In contrast to the previous analysis, we now consider a partial coupling from the microscopic to the macroscopic scale as in Section 3.3. That is, we suppose the evolution of the underlying geometry is a-priori given by a sufficiently smooth path of diffeomorphisms. Finally, the model regarded in this chapter is closed by the two effective tensors and associated cell problems (2.3), (2.6).

4.2. Continuous dependence of permeability tensors

Using a similar strategy as in Section 3.2, we also show the continuous dependence of the permeability tensor on the geometry evolution.

In order to capture the underlying stationary Stokes equations in (2.6), we introduce the following function spaces for the velocity field $v$ and pressure field $q$ for $k\geq 1$ , cf. [Reference Gahn, Neuss-Radu and Pop12]:

\begin{align*} H^k_v(\mathcal{P}) &= \left \lbrace v \in (H_\#^k(\mathcal{P}))^d\,:\,{\text{Tr}}_{\mathcal{P}}(v) = 0 \right \rbrace,\\[3pt] H^{k-1}_q(\mathcal{P}) &= \left \lbrace q \in H_\#^{k-1}(\mathcal{P})\,:\,\int _{\mathcal{P}} q \; dy = 0 \right \rbrace, \\[3pt] H^k_{v,\Box }(\mathcal{P}) &= \left \lbrace v\in H^k_v(\mathcal{P})\,:\, \nabla \cdot v = 0 \text{ on } U \right \rbrace \end{align*}

for some fixed neighbourhood $\partial Y \subset U \subset \bar{Y}$ , cf. (3.4). In order to formulate the notion of weak solutions stated in [Reference Gahn, Neuss-Radu and Pop12], we furthermore introduce the space

\begin{align*} H^1_{v,\sigma }(\mathcal{P})\,:\!=\,\left \lbrace u\in H^1_{v}(\mathcal{P})\,:\,\nabla \cdot u =0 \right \rbrace \end{align*}

of solenoidal functions in $H^1_{v}(\mathcal{P})$ . As in Section 3.2, we are required to consider the general inhomogeneous class of Stokes equations. Therefore, the weak form of the Stokes problem with general force term $f\in (L^2(\mathcal{P}))^d$ and inhomogeneity $j\in H^1(\mathcal{P})\cap L^2_0(\mathcal{P})$ with $j=0$ in a neighbourhood of $\partial Y$ reads: Find $(u,p)\in H^1_v(\mathcal{P}) \times H^0_q(\mathcal{P})$ such that

(4.3) \begin{align} \int _{\mathcal{P}} \nabla u \,:\, \nabla v \; dy &= \int _{\mathcal{P}} f v \; dy, \nonumber \\ \int _{\mathcal{P}} \nabla u \,:\, \nabla \Psi \; dy - \int _{\mathcal{P}} p \nabla \cdot \Psi \; dy &= \int _{\mathcal{P}} f \Psi \; dy,\\ \nabla \cdot u &= j, \nonumber \end{align}

for all $(v,\Psi )\in (H^1_{v,\sigma }(\mathcal{P}) \times C^\infty _0(\mathcal{P}))$ .

Following the procedure of Section 3.2, we next implement higher regularity of solutions to the Stokes problem given a sufficiently smooth interior boundary $\partial ^{\text{int}}\mathcal{P}$ .

Proof. We start considering the homogeneous case $j\equiv 0$ . By standard Hilbert space arguments, there exists a uniquely determined function $u\in H^1_{v,\sigma }(\mathcal{P})$ such that

\begin{align*} \int _{\mathcal{P}} \nabla u \,:\, \nabla v \; dy = \int _{\mathcal{P}} f v \; dy, \quad \forall v \in H^1_{v,\sigma }(\mathcal{P}). \end{align*}

Due to the smoothness of the domain and source term, there exists a unique pressure field $p\in L^2(\mathcal{P})$ with vanishing average such that

\begin{align*} \int _{\mathcal{P}} \nabla u \,:\, \nabla \Psi \; dy - \int _{\mathcal{P}} p \nabla \cdot \Psi \; dy &= \int _{\mathcal{P}} f \Psi \; dy, \quad \forall \Psi \in (C_0^\infty (\mathcal{P}))^2 \end{align*}

by Lemma IV 1.1 in [Reference Gahn, Neuss-Radu and Pop12]. Furthermore, Theorem IV 4.1 in [Reference Gahn, Neuss-Radu and Pop12] ensures the interior regularity claimed. As the boundary is of class $C^2$ , the regularity can be extended to the full domain by Theorem IV 5.1 in [Reference Gahn, Neuss-Radu and Pop12]. Next, we consider the problem for general inhomogeneities $j$ of the specified class. According to Theorem 3.4 in [Reference Gärttner, Frolkovic, Knabner and Ray15], there exists a function $\beta \in H^2_0(\mathcal{P})$ satisfying $\nabla \cdot \beta = j$ . Solving the homogeneous system for $\tilde{f}=f-\Delta \beta$ and adding $\beta$ to the velocity solution, the full statement is shown. Combining the estimates associated with the previous steps, we conclude boundedness of the solution operator.

In comparison to the case of elliptic equations in Section 3.2, we need to slightly change the setting here in order to accommodate for the additional condition on $\nabla \cdot u$ which is not invariant under pullbacks and cannot be additively compensated for without changing the solution itself. More precisely, we switch to a setting that does not require surjectivity of operators. Serving the analogous purpose as $F$ defined in equation (3.3), let us consider the following mappings:

\begin{align*} G & \,:\, H^2_{v,\Box }(\mathcal{P}) \times H^1_{q}(\mathcal{P}) \times ({-}S,S) \to L^2(\mathcal{P}) \times H^1_{\#}(\mathcal{P}),\\ & (u,p,s) \mapsto \left (h_s^*\Delta h_s^{*-1} u- h_s^*\nabla h_s^{*-1} p,\; h_s^*\nabla \cdot h_s^{*-1} u \right ),\\ \tilde{G} & \,:\, ({-}S,S) \to L^2(\mathcal{P}) \times H^1_{\#}(\mathcal{P}), \\ & (s) \mapsto (h_s^*e_1,\; 0), \end{align*}

with $h$ being a $C^1$ -path of diffeomorphisms in ${\text{Diff}}^2_{\Box }(\bar{Y})$ and $h_0=\text{id}_{\bar{Y}}$ . As the interior boundary conditions are invariant under pullbacks and already implemented in the underlying function spaces, it is sufficient to only map to the function spaces associated with the bulk data $j,\; f$ . Differentiability of this mapping is established using the same reasoning as above. Summing up our considerations, we obtain the following theorem.

Theorem 5. Smooth dependence of weak solutions to ( 2.6 ) on geometric parameter. Assume a $C^{2}$ open set $Y\setminus \bar{\mathcal{P}} \subset Y$ being compactly contained in $Y$ and $(u,p)\in H^2_{v,\Box }(\mathcal{P}) \times H^1_q(\mathcal{P})$ such that $G(u,p,0)= \tilde{G}(0)$ , i.e. $(u,p)$ is a solution to problem ( 4.3 ) with force term $f=e_1$ and heterogeneity $j \equiv 0$ . Furthermore, let $h$ be an $m$ -times differentiable path in ${\text{Diff}}_{\Box }^2(\bar{Y})$ with $h_0=\text{id}_{\bar{Y}}$ . Then there exist a neighbourhood $V$ of zero and a function $g\,:\,V\to H^2_v(\mathcal{P}) \times H^1_{q}(\mathcal{P})$ $m$ -times differentiable in $0\in V$ , $g(0)=(u,p)$ , such that

\begin{align*} G(g(s),s)= \tilde{G}(s), \quad \forall s\in V, \end{align*}

i.e. $h_s^{*-1}g(s)$ solves ( 4.3 ) on $h(\mathcal{P})$ for $s\in V$ .

Proof. By calculations similar to Section 3.2, we find $G,\; \tilde{G}$ to be differentiable in $s=0$ . Furthermore, $G(\cdot,\cdot,s)$ is a bounded linear operator for fixed $s$ . The unique solvability of

\begin{align*} G(u,p,s)= \tilde{G}(s) \end{align*}

is guaranteed for all $s\in V$ by Lemma 4.1, i.e. we can properly define the map $g(s)$ . Finally, we obtain the estimate

\begin{align*} ||G(u,p;0)||_{L^2(\mathcal{P}) \times H^1(\mathcal{P})} \geq C^{-1} ||(u,p)||_{H^2(\mathcal{P}) \times H^1(\mathcal{P})} \quad \forall (u,p) \in H^2_{v,\Box }(\mathcal{P}) \times H^1_q(\mathcal{P}) \end{align*}

using the operator norm $C\lt \infty$ of the Stokes solution operator on $\mathcal{P}$ introduced in Lemma 4.1. The assertion is a consequence of Theorem A.4. This is due to the fact that all values in the second image space of $G$ vanish close to $\partial Y$ by the restrictions on the preimage spaces.

This result immediately translates to the permeability tensor:

Corollary 4.2. Smooth dependence of $\mathbb{K}$ on geometric parameter. Under the assumptions of Theorem 5, the mapping

\begin{align*} R\,:\, V \to \mathbb{R},\quad s\mapsto \int _{h_s(\mathcal{P})} h_s^{*-1}g_u(s) \; dy \end{align*}

is $m$ -times differentiable. Therefore, the permeability tensor, cf. ( 2.5 ), depends differentiably on variations of $s$ .

4.3. Continuous dependence of Darcy velocity and pressure

The Darcy velocity field $v$ enters the first-order term of the transport equation (2.1) as a parameter. Again, in order to apply linear parabolic theory as in Section 3.3, we first need to establish sufficient regularity of $v$ which immediately rises the question of dependence on the permeability $\mathbb{K}$ . A special difficulty arises from the fact that the stationary Darcy equation (2.4) acts on time-slices of the space-time cylinder $\Omega _T$ . In this chapter, we investigate the effect of permeability tensors continuously depending on space and time on the flow field fulfilling Darcy’s equation. The following result establishes local Lipschitz continuity with respect to the $L^2$ -norm in the velocity field. Assume a typical flow-channel scenario with flux boundary conditions on $\partial \Omega _{\text{flux}}$ and an outlet $\partial \Omega _{\text{Dir}}$ with zero Dirichlet data for the pressure with $\partial \Omega = \partial \Omega _{\text{Dir}} \cup \partial \Omega _{\text{flux}}$ and Hausdorff measure $\mathcal{H}^{d-1} (\partial \Omega _{\text{Dir}}) \gt 0$ . First, we consider (2.4) as an elliptic equation for the pressure $p$ in the following weak form with a general source term $\tilde{f}\in L^2(\Omega )$ :

Find $p\in H_{\text{Dir}}^1(\Omega )\,:\!=\,\left \lbrace v\in H^1(\Omega )\,:\, v=0 \text{ on } \Omega _{\text{Dir}}\right \rbrace$ such that for all $q \in H_{\text{Dir}}^1(\Omega )$

(4.4) \begin{align} \int _{\Omega } \mathbb{K} \nabla p \cdot \nabla q \; dx - (g_{\text{flux}},q)_{L^2(\partial \Omega _{\text{flux}}) }= \int _{\Omega } \tilde{f} q \; dx. \end{align}

The associated velocity field is then given as

\begin{align*} v=-\mathbb{K} \nabla p. \end{align*}

As such, we can establish regularity of $v$ by analysing the pressure equation (4.4) and deduce the relevant properties from $p$ . Next, we follow the approach taken in [Reference Čanić6] where continuous dependence on parameters in case of the Brinkman–Forchheimer equation was established. Considering two pressure solutions to Darcy’s equation $(p_1,p_2)$ related to a pair of coefficients $(\mathbb{K}_1, \mathbb{K}_2)$ , we test the difference of the weak formulations with $p_2-p_1$ . Using suitable estimates on the new right-hand side, the assertion is established. More precisely, the following statement holds:

Lemma 4.3. Continuous dependence of Darcy solutions on $\mathbb{K}$

Let $\Omega$ be a Lipschitz bounded and connected domain with flux boundary conditions on $\partial \Omega _{\text{flux}}$ and homogeneous pressure boundary conditions on $\partial \Omega _{\text{Dir}}$ :

\begin{align*} \mathbb{K}\nabla p_{|\partial \Omega _{\text{flux}}} \cdot \nu = g_{\text{flux}} \in L^2(\partial \Omega _{\text{flux}}), \quad p = 0 \in H^{\frac{1}{2}} (\partial \Omega _{\text{Dir}}). \end{align*}

Assume that the Dirichlet boundary part $\partial \Omega _{\text{Dir}}$ is of positive measure. Furthermore, let $\mathbb{K}_1, \mathbb{K}_2 \in \left ( L^\infty \left ({\Omega }\right ) \right )^{d \times d}$ be permeability tensor fields uniformly $\lambda \gt 0$ -coercive and uniformly bounded in the Frobenius norm a.e. by $K_{max}$ . In addition, let $\tilde{f}\in L^2(\Omega )$ . Denoting the respective solutions to Darcy’s equation by $(v_1,p_1), \; (v_2,p_2) \in L^2(\Omega )\times H^1(\Omega )$ the following estimates hold:

\begin{align*} ||p_2-p_1||_{H^1(\Omega )} \leq C ||\mathbb{K}_2-\mathbb{K}_1||_{L^\infty (\Omega )},\\ ||v_2-v_1||_{L^2(\Omega )} \leq C ||\mathbb{K}_2-\mathbb{K}_1||_{L^\infty (\Omega )}, \nonumber \end{align*}

where the constant $C$ only depends on $\lambda$ , $K_{max}$ , $\Omega$ and the Darcy data $g_{\text{flux}},\tilde{f}$ .

Proof. First of all, we note that standard elliptic theory guarantees the existence of solutions $p_1,p_2\in H_{\text{Dir}}^1(\Omega )$ to (4.4). In a first step towards continuous dependence of solutions on $\mathbb{K}$ , we show uniform boundedness of $p$ in the $H^1$ -seminorm by revisiting well-established energy methods. As $\mathbb{K}$ is symmetric positive definite, there exists a unique symmetric root $\mathbb{K}^{\frac{1}{2}}$ . As $p$ itself is an admissible test function, the weak formulation directly yields for an arbitrary $\epsilon \gt 0$ :

\begin{align*} \lambda || \nabla p ||^2_{L^2(\Omega )} &\leq || \mathbb{K}^{\frac{1}{2}} \nabla p ||^2_{L^2(\Omega )} \leq (g_{\text{flux}},p)_{L^2(\partial \Omega _{\text{flux}}) } +||\tilde{f}||_{L^2(\Omega )} ||p||_{H^1(\Omega )} \nonumber \\ &\leq C(\epsilon ) \left (||g_{\text{flux}}||^2_{L^2(\partial \Omega _{\text{flux}})}+ ||\tilde{f}||^2_{L^2(\Omega )} \right ) +\epsilon ||\nabla p||^2_{L^2(\Omega )} \end{align*}

using Young’s and Poincaré’s inequality as well as the trace theorem. The eigenvalues of $\mathbb{K}$ being bounded from below by $\lambda$ and choosing $\epsilon$ small enough, we have

\begin{align*} ||\nabla p ||_{L^2(\Omega )}&\leq C \left ( || g_{\text{flux}}||_{L^2(\partial \Omega _{\text{flux}})}+||\tilde{f}||_{L^2(\Omega )} \right ) \end{align*}

with $C$ depending on Poincaré’s constant and $\lambda$ only.

Next, assume $p_1, \; p_2$ to be pressure solutions to the Darcy problems respective to $\mathbb{K}_1,\mathbb{K}_2$ . Testing the difference of the weak formulations with $p\,:\!=\,p_2-p_1$ leads to

\begin{align*} ||\mathbb{K}_1^{\frac{1}{2}} \nabla p ||^2_{L^2(\Omega )} &= \int _\Omega (\mathbb{K}_1 -\mathbb{K}_2) \nabla p_2 \cdot \nabla p \; dx \\ & \leq \mathop{\textrm{ess sup}}\limits _{x\in \Omega }(||\mathbb{K}_1 -\mathbb{K}_2||_F) \left ( C(\epsilon ) ||\nabla p_2||^2_{L^2(\Omega )} + \epsilon ||\nabla p||^2_{L^2(\Omega )} \right ) \nonumber \\ & \leq \mathop{\textrm{ess sup}}\limits _{x\in \Omega }(||\mathbb{K}_1 -\mathbb{K}_2||_F) C(\epsilon ) ||\nabla p_2||^2_{L^2(\Omega )} + 2K_{max} \epsilon ||\nabla p||^2_{L^2(\Omega )}, \nonumber \end{align*}

where $||.||_F$ denotes the Frobenius norm of the matrices. Using the uniform boundedness in the gradient of $p_2$ established in the first step and choice of $\epsilon$ small enough yields the assertion on the pressure. Calculating

\begin{align*} ||v_1 -v_2||_{L^2(\Omega )} &= ||\mathbb{K}_1 \nabla p_1 -\mathbb{K}_2 \nabla p_2 ||_{L^2(\Omega )} \\ &\leq ||\mathbb{K}_1||_{L^\infty (\Omega )} ||\nabla p ||_{L^2(\Omega )} + ||\mathbb{K}_1-\mathbb{K}_2||_{L^\infty (\Omega )} ||\nabla p_2 ||_{L^2(\Omega )} \nonumber \\ &\leq C ||\mathbb{K}_2-\mathbb{K}_1||_{L^\infty (\Omega )},\nonumber \end{align*}

we obtain the assertion on the velocities applying the same reasoning.

Unfortunately, the last result cannot be generalised to stronger norms in $v$ by demanding higher regularity to the data, coefficient or $\Omega$ due to the mixed boundary conditions. For a counter-example in a setting of maximal smoothness see [Reference Sethian38].

Yet, in the case of pure Dirichlet boundaries, it is straightforward to show that the above estimate indeed can be refined to hold in $L^\infty (\Omega )$ by standard elliptic theory [Reference Eden, Nikolopoulos and Muntean10]. We capture this observation in the following corollary.

Corollary 4.4. Stronger norms. Let the assumptions of Lemma 4.3 hold with $\partial \Omega _{\text{flux}}=\emptyset$ , a $C^3$ -domain $\Omega \subset \mathbb{R}^d$ , $\tilde{f} \in H^1 (\Omega )$ and $d\in \{2,3\}$ . Furthermore, let $\mathbb{K}_1,\mathbb{K}_2 \in (C^2(\bar{\Omega }))^{d\times d}$ be uniformly coercive and uniformly bounded. Then

\begin{align*} ||v_2-v_1||_{L^\infty (\Omega )} \leq C ||\mathbb{K}_2-\mathbb{K}_1||^{1-\frac{d}{4}}_{L^\infty (\Omega )}. \end{align*}

Proof. Using standard elliptic theory, the solutions $p_1,p_2$ to (4.2) are uniformly bounded in $W^{3,2}(\Omega )$ , cf. [Reference Eden, Nikolopoulos and Muntean10]. By the Gagliardo–Nirenberg interpolation theorem [Reference Necas, Simader, Necasová, Tronel and Kufner28] we have for $p=p_2-p_1$

\begin{align*} ||\nabla p||_{L^\infty (\Omega )} \leq C_1 ||D^3p||^{\frac{d}{4}}_{L^2(\Omega )} ||\nabla p||_{L^2(\Omega )}^{1-\frac{d}{4}} + C_2 ||\nabla p||_{L^2(\Omega )}. \end{align*}

Using the uniform boundedness of $||D^3p||_{L^2(\Omega )}$ and Lemma 4.3, we see

\begin{align*} ||\nabla p||_{L^\infty (\Omega )} \leq C ||\mathbb{K}_2-\mathbb{K}_1||_{L^\infty (\Omega )}^{1-\frac{d}{4}} \end{align*}

which translates to the statement claimed.

4.4. Existence for partial coupling

Analogously to Section 3.3, we use our prior results to prove local-in-time existence of solutions to the model specified in Section 4.1. Using the prior regularity results of this section, sufficient smoothness of the Darcy velocity field is established. Again, we re-define the functions $\phi,\; \sigma,\; \mathbb{D},\; \mathbb{K}$ from mappings on $\Omega _T$ , cf. (3.1), to mappings of the real-valued order parameter without change of notation. In order to avoid additional difficulties due to degeneracy, we require

(4.5) \begin{align} \forall s\in ({-}S,S)\,:\, 0\lt \phi (s) \lt 1, \quad \sigma (s) \gt 0, \quad \mathbb{D}(s)\gt 0, \quad \mathbb{K}(s)\gt 0, \end{align}

to hold along the prescribed path $h$ of diffeomorphisms within the master unit cell $Y^*$ .

Proof. The proof of this assertion follows along the lines of the proof of Theorem 3. In addition, we must now also consider the smoothness of $\mathbb{K}(s(t,x))$ as well as of the associated Darcy velocity field $v(t,x)$ . By the assumptions on the geometry alteration and Corollary 4.2, $\mathbb{K}(s(t,\cdot ))\in \left (C^2(\bar{\Omega })\right )^{d\times d}$ for every $t\in [0,T_1)$ . Due to (4.5) and Remark 4.5, we obtain $v \in C(\overline{\Omega _T})$ for some $0\lt T\leq T_1$ . By standard Sobolev embedding theorems, also $\nabla \cdot v\in C(\overline{\Omega _T})$ holds true. As such, we ensure the regularity of $v$ as required for coefficients in Theorem A.3 and may therefore proceed with the fixed-point argument analogously to Theorem 3. As mentioned before, the solutions $(v(t,\cdot ),p(t,\cdot ))$ for Darcy’s equation are uniquely determined for each time-slice $t$ . Repeating the arguments following (3.13), uniqueness for the full solution triplet is obtained.