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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 57, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/57\hfil Corrector estimates for a thermoelasticity problem]
{Corrector estimates for the homogenization of a two-scale thermoelasticity
problem with a priori known phase transformations}

\author[M. Eden, A. Muntean \hfil EJDE-2017/57\hfilneg]
{Michael Eden, Adrian Muntean}

\address{Michael Eden \newline
Center for Industrial Mathematics, FB 3,
University of Bremen,
Bibliotheksstr. 1, 28359, Bremen, Germany}
\email{leachim@math.uni-bremen.de}

\address{Adrian Muntean \newline
Department of Mathematics and Computer Science,
University of Karlstad,
Universitetsgatan 2, 651 88 Karlstad,  Sweden}
\email{adrian.muntean@kau.se}

\dedicatory{Communicated by Ralph Showalter}

\thanks{Submitted February 12, 2017. Published February 23, 2017.}
\subjclass[2010]{35B27, 35B40, 74F05}
\keywords{Homogenization; two-phase thermoelasticity; corrector estimates; 
\hfill\break\indent time-dependent domains; distributed microstructures}

\begin{abstract}
 We investigate corrector estimates for the solutions of a thermoelasticity
 problem posed in a highly heterogeneous two-phase medium  and its
 corresponding two-scale thermoelasticity model which was derived in \cite{EM17}
 by two-scale convergence arguments.
 The medium in question consists of a connected matrix with disconnected,
 initially periodically distributed inclusions separated by a sharp interface
 undergoing a priori known phase transformations.
 While such estimates seem not to be obtainable in the fully coupled setting,
 we show that for some simplified scenarios optimal convergence rates can be
 proven rigorously.
 The main technique for the proofs are energy estimates using special
 reconstructions of two-scale functions and particular operator estimates for
 periodic functions with zero average.
 Here, additional regularity results for the involved functions are necessary.
\end{abstract}

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

\section{Introduction}

We aim to derive quantitative estimates that show the quality of the upscaling 
process of a coupled linear  thermoelasticity system with a priori known phase 
transformations posed in a high-contrast media to its corresponding 
two-scale thermoelasticity system.

The problem we have in mind is posed in a medium where the two building components, 
initially assumed to be periodically distributed, are different solid phases of 
the same material in which phase transformations that are a priori known occur.
As a main effect, the presence of phase transformations leads to evolution 
problems in time-dependent domains that are not necessarily periodic anymore.

In our earlier paper \cite{EM17}, we rigorously studied the well-posedness of
such a thermoelasticity problem and conducted a homogenization procedure via 
the two-scale convergence technique (cf. \cite{Al92} for details).
Those results were obtained after transforming the problem to a fixed reference 
geometry. In this work, our goal is to further investigate the connection of
 those problems and to derive an upper bound for the convergence rate (in some 
yet to be defined sense) of their solutions.
While historically a tool to also justify the homogenization (via asymptotic 
expansions) in the first place, such estimates, which in the homogenization 
literature are usually called \emph{error} and \emph{corrector estimates}, 
provide a means to evaluate the accuracy of the upscaled model.
Also, such estimates are especially interesting from a computational point of view.
In the context of \emph{Multiscale FEM}, for example, they are needed to
 ensure/control the convergence of the method, we refer to, e.g., \cite{AB05,HW97}.

The basic idea is to estimate the $L^2-$ and $H^1$-errors of the solutions of 
the problems using energy techniques, additional regularity results, and special 
operator estimates for functions with zero average. Since, in general, 
the solutions of the $\varepsilon$-problem do not even have the same domain as the 
solutions of the two-scale problem, we additionally rely on so-called 
\emph{macroscopic reconstructions} (we refer to Section~\ref{sec:preliminaries}).
The difficulty in getting such estimates in our specific scenario is twofold:
 First, the coupling between the quasi-stationary momentum equation and the heat 
equation and, second, the interface movement which (after transforming to a 
reference domain) leads to additional terms as well as time-dependent and 
non-periodic coefficients functions.

As typical for a corrector estimate result, the main goal is to show that there 
is a constant $C>0$ which is independent on the particular choice of $\varepsilon$ such that
\begin{equation}\label{goal_est}
\begin{aligned}
&\|\Theta_\mathrm{err}^\varepsilon\|_{L^\infty(S\times\Omega)}
 +\|U_{\mathrm{err}}^\varepsilon\|_{L^\infty(S;L^2(\Omega))^{3}}
 +\|\nabla\Theta_\mathrm{cor}^\varepsilon\|_{L^2(S\times\Omega_A^\varepsilon)^3}\\
&+\|\nabla U_{\mathrm{cor}}^\varepsilon\|_{L^\infty(S;L^2(\Omega_A^\varepsilon))^{3\times3}}
+\varepsilon\|\nabla\Theta_\mathrm{err}\|_{L^2(S\times\Omega_B^\varepsilon)^3} \\
&+\varepsilon\|\nabla U_\mathrm{cor}^\varepsilon\|_{L^\infty(S;L^2(\Omega_B^\varepsilon))^{3\times3}}
\leq C (\sqrt{\varepsilon}+\varepsilon).
\end{aligned}
\end{equation}
Here, $\Omega$ represents the full medium, $\Omega_A^\varepsilon$ the fast-heat-conducting 
connected matrix and $\Omega_B^\varepsilon$ the slow-heat-conducting inclusions.
For the definitions of the (error and corrector) functions, we refer the 
reader to the beginning of Section~\ref{sec:corrector}.

Unfortunately, in the general setting of a fully-coupled thermoelasticity 
problem with moving interface, such corrector estimates as stated in
\eqref{goal_est} seem not to be obtainable; in Section~\ref{subsec:overall}, 
we point out where and why the usual strategy for establishing such estimates 
is bound to fail.

Instead, we show that there are a couple of possible simplifications of the 
full model in which \eqref{goal_est} holds:
\begin{itemize}
\item[(a)] \emph{Weakly coupled problem}: If we assume either the 
\emph{mechanical dissipation} or the \emph{thermal stresses} to be negligible,
 we are led to weakly coupled problems, where the desired estimates can be 
established successively, see Theorem~\ref{cor1}.
We note that the regularity requirements are higher in the case of no thermal 
stress compared to the case of mechanical dissipation.

\item[(b)] \emph{Microscale coupling}: 
If \emph{mechanical dissipation} and \emph{thermal stress} are only really
 significant in the slow-conducting component and negligible in the connected 
matrix part, the estimates hold, see Theorem~\ref{cor2}.
\end{itemize}
As pointed out in \cite{W99}, neglecting the effect of mechanical dissipation
is a step that is quite usual in modeling thermoelasticity problems.

Convergence rates for specific one-phase problems with periodic constants 
(some of them posed in perforated domains) were investigated in, e.g.,
\cite{BP13,BPC98,CP98}.
In \cite{E04}, convergence rates for a complex nonlinear problem modeling 
liquid-solid phase transitions via a phase-field approach were derived.
A homogenization result including corrector estimates for a two-scale diffusion 
problem posed in a locally-periodic geometry was proven in \cite{MV13,VM11}.
Here, similar to our scenario, the microstructures are non-uniform, non-periodic, 
and assumed to be \emph{a priori} known; the microstructures are however 
time independent and there are no coupling effects.
For some corrector estimate results in the context of thermo and elasticity problems,
 we refer to \cite{BP13,STV13}, e.g.
We also want to point out to the newer and different philosophy in which the 
solutions are compared in the two-scale spaces (e.g., $L^2(\Omega;H_\#^1(Y_B)$) 
as opposed to the, possibly $\varepsilon$-dependent, spaces for the $\varepsilon$-problem 
(e.g., $L^2(\Omega_B^\varepsilon)$),\footnote{Here, we have used notation and 
domains as introduced in Section~\ref{sec:setting}.} 
a method which requires considerably less regularity on part of the solutions 
of the homogenized problems, we refer to \cite{G04,MR16,R15}.

The paper is organized as follows: In Section~\ref{sec:setting},
 we introduce the $\varepsilon$-microscopic geometry and formulate the thermoelasticity 
micro-problem in the moving geometry and transformed to a fixed reference domain 
and also state the homogenized two-scale problem.
The assumptions on our data, some regularity statements, and auxiliary estimate 
results are then collected in Section~\ref{sec:preliminaries}.
Finally, in Section~\ref{sec:corrector}, we focus on establishing convenient 
$\varepsilon$-control for the terms arising in the \emph{error formulation} 
(see \eqref{corrector_eq}).
Based on these estimates, the corrector estimate~\eqref{goal_est} is then 
shown to hold for the above described cases $(a)$ and $(b)$.


\section{Setting}\label{sec:setting}

\subsection{Interface movement}
The following notation is taken from \cite{EM17}.
Let $S=(0,T)$, $T>0$, be a time interval.
Let $\Omega$ be the interior of a union of a finite number of closed cubes $Q_j$, 
$1\leq j\leq n$, $n\in\mathbb{N}$, whose vertices are in $\mathbb{Q}^3$ such that, 
in addition, $\Omega$ is a Lipschitz domain.

In addition, we denote the outer normal vector of $\Omega$ with $\nu=\nu(x)$.
Let $Y=(0,1)^3$ be the open unit cell in $\mathbb{R}^3$.
Take $Y_A,$ $Y_B\subset Y$  two disjoint open sets, such that $Y_A$ is connected, 
such that $\Gamma:=\overline{Y_A}\cap\overline{Y_B}$ is a $C^3$ interface,
 $\Gamma=\partial Y_B$, $\overline{Y_B}\subset Y$, and $Y=Y_A\cup Y_B\cup \Gamma$.
With $n=n(y)$, $y\in\Gamma$, we denote the normal vector of $\Gamma$ 
pointing outwards of $Y_B$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth, clip=true, trim = 120 525 120 120]{fig1}
\end{center}
\caption{Reference geometry and the resulting $\varepsilon$-periodic initial configuration. 
Note that for $t\neq0$, these domains typically loose their periodicity.}
\label{s:fig}
\end{figure}

For $\varepsilon>0$, we introduce the $\varepsilon Y$-periodic, initial domains 
$\Omega_A^\varepsilon$ and $\Omega_B^\varepsilon$ and interface $\Gamma^\varepsilon$ representing 
the two phases and the phase boundary, respectively, via ($i\in\{A,B\}$)
\[
 \Omega^\varepsilon_i=\Omega\cap\big(\cup_{k\in\mathbb{Z}^3}\varepsilon(Y_i+k)\big),\quad
 \Gamma^\varepsilon=\Omega\cap\big(\cup_{k\in\mathbb{Z}^3}\varepsilon(\Gamma+k)\big).
\]
Here, for a set $M\subset\mathbb{R}^3$, $k\in\mathbb{Z}^3$, and $\varepsilon>0$, we employ the notation
%
$$
 \varepsilon(M+k):=\big\{x\in\mathbb{R}^3 : \frac{x}{\varepsilon}-k\in M\big\}.
$$
From now on, we take $\varepsilon=(\varepsilon_n)_{n\in\mathbb{N}}$ to be a sequence of monotonically 
decreasing positive numbers converging to zero such that $\Omega$ can be 
represented as the union of cubes of size $\varepsilon$.
Note that this is possible due to the assumed structure of $\Omega$.

Here $n^\varepsilon=n(\frac{x}{\varepsilon})$, $x\in\Gamma^\varepsilon$, denotes the unit normal 
vector (extended by periodicity) pointing outwards $\Omega_B^\varepsilon$ into $\Omega_A^\varepsilon$.
The above construction ensures that  $\Omega_A^\varepsilon$ is connected and that 
$\Omega_B^\varepsilon$ is disconnected.
We also have that $\partial\Omega_B^\varepsilon\cap\partial\Omega=\emptyset$.

We assume that $s\colon\overline{S}\times\overline\Omega\times\mathbb{R}^3\to\overline{Y}$ is a function such that
%
\begin{enumerate}
 \item $s\in C^2(\overline{S};C^2(\overline{\Omega})\times C^2_\#(Y))
$,\footnote{Here, and in the following, the $\#$ subscript denotes periodicity, i.e., 
for $k\in\mathbb{N}_0$, we have $C^k_\#(Y)=\{f\in C^k(\mathbb{R}^3): f(x+e_i)=f(x) \text{ for all } 
 x\in\mathbb{R}^3\}$, $e_i$ basis vector of $\mathbb{R}^3$.}

 \item $s(t,x,\cdot)_{|\overline{Y}}\colon\overline{Y}\to\overline{Y}$ 
is bijective for every $(t,x)\in\overline{S}\times\overline{\Omega}$,

 \item $s^{-1}\in C^2(\overline{S};C^2(\overline{\Omega})\times C^2_\#(Y))
$,\footnote{Here, $s^{-1}\colon\overline{S}\times\overline\Omega\times\mathbb{R}^3
\to\overline{Y}$ is the unique function such that $s(t,x,s^{-1}(t,x,y))=y$ 
for all $(t,x,y)\in\overline{S}\times\overline{\Omega}\in\overline{Y}$ extended 
by periodicity to all $y\in\mathbb{R}^3$.}

 \item $s(0,x,y)=y$ for all $x\in\overline{\Omega}$ and all $y\in\overline Y$,

 \item $s(t,x,y)=y$ for all $(t,x)\in\overline{S}\times\overline{\Omega}$ 
and all $y\in\partial Y$,

 \item there is a constant $c>0$ such that $\mathrm{dist}(\partial Y,\gamma)>c$ 
for all $\gamma\in s(t,x,\Gamma)$ and $(t,x)\in\overline{S}\times\overline{\Omega}$,

 \item $s(t,x,y)=y$ for all $(t,x)\in\overline{S}\times\overline{\Omega}$ and 
for all $y\in Y$ such that $\mathrm{dist}(\partial Y,y)<\frac{c}{2}$,

 \item there are constants $c_s,C_s>0$ such that
       $$
     c_s\leq\det(\nabla s(t,x,y))\leq C_s,\quad (t,x,y)\in\overline{S}\times\overline\Omega\times\mathbb{R}^3
    $$
\end{enumerate}
and set the $(t,x)$-parametrized sets
\begin{align*}
 Y_A(t,x)=s(t,x,Y_A),\quad
 Y_B(t,x)=s(t,x,Y_B),\quad
 \Gamma(t,x)=s(t,x,\Gamma).
\end{align*}

We introduce the operations
%
\begin{gather*}
 [\cdot]\colon\mathbb{R}^3\to\mathbb{Z}^3,\quad [x]=k \ \text{such that}\  x-[x]\in Y,\\
 \{\cdot\}\colon\mathbb{R}^3\to Y,\quad \{x\}=x-[x]
\end{gather*}
and define the $\varepsilon$-dependent function\footnote{This is the typical 
notation in the context of homogenization via the \emph{periodic unfolding method}, 
see, e.g., \cite{CDG08, D12}.}
$$
 s^\varepsilon\colon\overline{S}\times\overline{\Omega}\to\mathbb{R}^3,\quad s^\varepsilon(t,x)
:=\varepsilon\big[\frac{x}{\varepsilon}\big]+\varepsilon s\Big(t,\varepsilon\big[\frac{x}{\varepsilon}\big],\frac{x}{\varepsilon}\Big).
$$
For $i\in\{A,B\}$ and $t\in\overline{S}$, we set the time dependent sets 
$\Omega_i^\varepsilon(t)$ and $\Gamma^\varepsilon(t)$ and the corresponding non-cylindrical 
space-time domains $Q_i^\varepsilon$ and space-time phase boundary $\Sigma^\varepsilon$ via
\begin{gather*}
 \Omega_i^\varepsilon(t)=s^\varepsilon(t,\Omega_i^\varepsilon), \quad
 Q_i^\varepsilon=\cup_{t\in S}\left(\{t\}\times\Omega_i^\varepsilon(t)\right),\\
 \Gamma^\varepsilon(t)=s^\varepsilon(t,\Gamma^\varepsilon),\quad
\Sigma^\varepsilon=\cup_{t\in S}\left(\{t\}\times\Gamma^\varepsilon(t)\right),
\end{gather*}
and denote by $n^\varepsilon=n^\varepsilon(t,x)$, $t\in S$, $x\in\Gamma^\varepsilon(t)$, the unit normal 
vector pointing outwards $\Omega_B^\varepsilon(t)$ into $\Omega_A^\varepsilon(t)$.
The time-dependent domains $\Omega_i^\varepsilon$ host the phases at time $t\in\overline{S}$ 
and model the movement of the interface $\Gamma^\varepsilon$.
We emphasize that, for any $t\neq0$, the sets $\Omega_A^\varepsilon(t)$, $\Omega_B^\varepsilon(t)$, 
and $\Gamma^\varepsilon(t)$ are, in general, not periodic.

We introduce the transformation-related functions (here, $W_\Gamma^\varepsilon$ 
is the normal velocity and $H_\Gamma^\varepsilon$ the mean curvature of the interface) via
\begin{gather*}
 F^\varepsilon\colon\overline{S}\times\overline{\Omega}\to\mathbb{R}^{3\times3},\quad 
  F^\varepsilon(t,x):=\nabla s^\varepsilon(t,x),\\
 J^\varepsilon\colon\overline{S}\times\overline{\Omega}\to\mathbb{R},\quad
  J^\varepsilon(t,x):=\det\left(\nabla s^\varepsilon(t,x)\right),\\
 v^\varepsilon\colon\overline{S}\times\overline{\Omega}\to\mathbb{R}^{3},\quad
  v^\varepsilon(t,x):=\partial_ts^\varepsilon(t,x),\\
 W_\Gamma^\varepsilon\colon\overline{S}\times\Gamma^\varepsilon\to\mathbb{R},\quad
  W_\Gamma^\varepsilon(t,x):=v^\varepsilon(t,x)\cdot n^\varepsilon(t,s^\varepsilon(t,x)),\\
 H_\Gamma^\varepsilon\colon\overline{S}\times\Gamma^\varepsilon\to\mathbb{R},\quad
  H_\Gamma^\varepsilon(t,x):=-\operatorname{div}\left((F^{\varepsilon})^{-1}(t,x)n^\varepsilon(t,s^\varepsilon(t,x))\right)
\end{gather*}
for which we have the following estimates
\begin{equation}\label{s:estimate_movement}
\begin{aligned}
& \|F^\varepsilon\|_{L^\infty(S\times\Omega)^{3\times3}}
+\|(F^\varepsilon)^{-1}\|_{L^\infty(S\times\Omega)^{3\times3}}
+\|J^\varepsilon\|_{L^\infty(S\times\Omega)}\\
&+\varepsilon^{-1}\|v^\varepsilon\|_{L^\infty(S\times\Omega)^3}
 +\varepsilon^{-1}\|W_\Gamma^\varepsilon\|_{L^\infty(S\times\Gamma^\varepsilon)}
+\varepsilon\|H_\Gamma^\varepsilon\|_{L^\infty(S\times\Gamma^\varepsilon)}\leq C.
\end{aligned}
\end{equation}
By design, the constant $C$ entering~\eqref{s:estimate_movement} 
is independent on the choice of $\varepsilon$.
Note that the same estimates also hold for the time derivatives of 
these functions.

\subsection{$\varepsilon$-problem and homogenization result}

The bulk equations of the coupled thermoelasticity problem are given as 
(we refer to \cite{B56,EM17,K79})
\begin{subequations}\label{p:full_problem_moving}
\begin{gather}
 -\operatorname{div}(\mathcal{C}_A^\varepsilon e(u_A^\varepsilon)-\alpha_A^\varepsilon\theta_A\mathds{I}_3)
=f_{u_A}^\varepsilon \quad\text{in } Q_A^\varepsilon,\label{p:full_problem_moving:1} \\
 -\operatorname{div}(\mathcal{C}_B^\varepsilon e(u_B^\varepsilon)-\alpha_B^\varepsilon\theta_B\mathds{I}_3)
=f_{u_B}^\varepsilon \quad \text{in } Q_B^\varepsilon,\label{p:full_problem_moving:2}\\
 \partial_t\left(\rho_Ac_{dA}\theta_A^\varepsilon+\gamma_A^\varepsilon\operatorname{div} u_A^\varepsilon\right)
-\operatorname{div}(K_A^\varepsilon\nabla\theta_A^\varepsilon)=f_{\theta_A}^\varepsilon \quad \text{in }
  Q_A^\varepsilon,\label{p:full_problem_moving:3}\\
 \partial_t\left(\rho_Bc_{dB}\theta_B^\varepsilon+\gamma_B^\varepsilon\operatorname{div} u_B^\varepsilon\right)
-\operatorname{div}(K_B^\varepsilon\nabla\theta_B^\varepsilon)=f_{\theta_B}^\varepsilon \quad \text{in }
 Q_B^\varepsilon.\label{p:full_problem_moving:4}
\end{gather}
Here, $\mathcal{C}_i^\varepsilon\in\mathbb{R}^{3\times3\times3\times3}$ are the \emph{stiffness} tensors, 
$\alpha_i^\varepsilon>0$ the \emph{thermal expansion} coefficients, $\rho_i>0$ the 
\emph{mass densities}, $c_{di}>0$ the \emph{heat capacities}, $\gamma_i^\varepsilon>0$ are
 the \emph{dissipation coefficients}, $K_i^\varepsilon\in\mathbb{R}^{3\times3}$ the 
\emph{thermal conductivities}, and $f_{u_i}^\varepsilon$, $f_{\theta_i}^\varepsilon$ are volume 
densities.
In addition, $e(v)=1/2(\nabla v+\nabla v^T)$ denotes the linearized strain
 tensor and $\mathds{I}_3$ the identity matrix.

At the interface between the phases, we assume continuity of both the temperature 
and deformation and the fluxes of force and heat densities to be given via 
the mean curvature and the interface velocity, 
resp.:\footnote{Here, the scaling via $\varepsilon^2$ counters the effects of both 
the interface surface area, note that $\varepsilon|\Gamma^\varepsilon|\in\mathcal{O}(1)$, 
and the curvature itself, note that $\varepsilon|\widetilde{H_\Gamma^\varepsilon}|\in\mathcal{O}(1)$.}
\begin{gather}
 \llbracket u^\varepsilon\rrbracket=0,\quad \llbracket \theta^\varepsilon 
\rrbracket
=0\quad\text{on }\Sigma^\varepsilon,\label{p:full_problem_moving:5}\\
 \llbracket \mathcal{C}^\varepsilon\varepsilon(u^\varepsilon)-\alpha^\varepsilon\theta^\varepsilon\mathds{I}_3 \rrbracket n^\varepsilon
=-\varepsilon^2H_\Gamma^\varepsilon n^\varepsilon \quad \text{on } \Sigma^\varepsilon,\\
 \llbracket \rho c_d\rrbracket\theta^\varepsilon W_{\Gamma}^\varepsilon
+\llbracket \gamma^\varepsilon\operatorname{div} u^\varepsilon\rrbracket W_\Gamma^\varepsilon
-\llbracket K^\varepsilon\nabla\theta^\varepsilon \rrbracket\cdot n^\varepsilon
= L_{AB}W_\Gamma^\varepsilon \quad \text{in } \Sigma^\varepsilon.
\end{gather}
Here, $\llbracket v\rrbracket:=v_A-v_B$ denotes the jump across the boundary
separating phase $A$ from phase $B$, $\sigma_0>0$ is the coefficient of
 surface tension, and $L_{AB}\in\mathbb{R}$ is the latent heat

Finally, at the boundary of $\Omega$ and for the initial condition, we pose
\begin{gather}
 u_A^\varepsilon=0\quad \text{on } S\times\partial\Omega_A^\varepsilon,\\
 \theta_A^\varepsilon =0\quad \text{on } S\times\partial\Omega_A^\varepsilon,\\
 \theta^\varepsilon(0)=\theta_{0}^\varepsilon \quad \text{on } \Omega,\label{p:full_problem_moving:9}\\
 u^\varepsilon(0)=0\quad \text{on }\Omega,
\end{gather}
where $\theta_{0}^\varepsilon$ is some (possibly highly heterogeneous) initial 
temperature distribution.
The scaling of the coefficients is chosen as
\begin{gather*}
 \mathcal{C}_A^\varepsilon=\mathcal{C}_A,\quad K_A^\varepsilon=K_A,\quad\alpha_A^\varepsilon=\alpha_A,\quad
\gamma_A^\varepsilon=\gamma_A,\\
 \mathcal{C}_B^\varepsilon=\varepsilon^2\mathcal{C}_B,\quad K_B^\varepsilon=\varepsilon^2K_B,\quad\alpha_B^\varepsilon=\varepsilon\alpha_B,
\quad\gamma_B^\varepsilon=\varepsilon\gamma_B.
\end{gather*}
\end{subequations}

\begin{remark} \label{rmk1} \rm
The simplified models described in the introduction (for which corrector estimates 
can be established) correspond to $\alpha_A=\alpha_B=0$ or $\gamma_A=\gamma_B=0$ 
(case (1), \emph{weakly coupled problem}) and $\alpha_A=\gamma_A=0$
(case (2), \emph{mirco coupled} problem).
\end{remark}

Now, taking the back-transformed quantities (defined on the initial, periodic 
domains $\Omega_i^\varepsilon$) $U_i^\varepsilon\colon S\times\Omega_i^\varepsilon\to\mathbb{R}^3$ and 
$\Theta_i^\varepsilon\colon S\times\Omega_i^\varepsilon\to\mathbb{R}^3$ given via 
$U_i^\varepsilon(t,x)=u_i^\varepsilon(t,s^{-1,\varepsilon}(t,x))$ and 
$\Theta_i^\varepsilon(t,x)=\theta_i^\varepsilon(t,s^{-1,\varepsilon}(t,x))$,\footnote{%
Here, $s^{-1,\varepsilon}\colon \overline{S}\times\overline{\Omega}\to\overline{\Omega}$ 
is the inverse function of $s^\varepsilon$.} we obtain the following problem in fixed
coordinates (for more details regarding the transformation to a fixed domain, 
we refer to \cite{D12,M08, PSZ13}):\footnote{Here, the superscript $r,\varepsilon$ 
denotes the transformed quantities (w.r.t~$s^\varepsilon$), for example 
$K_A^{r,\varepsilon}=J^\varepsilon(F^\varepsilon)^{-1}K_A(F^\varepsilon)^{-T}$ (cf. \cite{EM17}).}
\begin{subequations}\label{p:ref}
\begin{gather}
 -\operatorname{div}\left(\mathcal{C}_A^{r,\varepsilon}(U_A^\varepsilon)-\Theta_A^\varepsilon\alpha_A^{r,\varepsilon}\right)
=f_{u_A}^{r,\varepsilon} \quad \text{in }S\times\Omega_A^\varepsilon,\\
 -\operatorname{div}\left(\varepsilon^2\mathcal{C}_B^{r,\varepsilon} e(U_B^\varepsilon)-\varepsilon\Theta_B^\varepsilon\alpha_B^{r,\varepsilon}\right)
=f_{u_B}^{r,\varepsilon} \quad \text{in }S\times\Omega_B^\varepsilon,\\
 \begin{split}
&\partial_t\left(c_A^{r,\varepsilon}\Theta_A^\varepsilon+\gamma_A^{r,\varepsilon}:\nabla U_A^\varepsilon\right)
-\operatorname{div}\left(K_A^{r,\varepsilon}\nabla\Theta_A^\varepsilon\right)\\
& -\operatorname{div}\left(\left(c_A^{r,\varepsilon}\Theta_A^\varepsilon+\gamma_A^{r,\varepsilon}:
\nabla U_A^\varepsilon\right)v^{r,\varepsilon}\right)\\
&=f_{\theta_A}^{r,\varepsilon}
\quad\text{in } S\times\Omega_A^\varepsilon, \end{split}
\\
 \begin{split}
&\partial_t\left(c_B^{r,\varepsilon}\Theta_B^\varepsilon
 +\varepsilon\gamma_B^{r,\varepsilon}:\nabla U_B^\varepsilon\right)
 -\operatorname{div}\left(\varepsilon^2K_B^{r,\varepsilon}
 \nabla\Theta_B^\varepsilon\right) \\
&-\operatorname{div}\left(\left(c_B^{r,\varepsilon}\Theta_B^\varepsilon+\varepsilon\gamma_B^{r,\varepsilon}:
 \nabla U_B^\varepsilon\right)v^{r,\varepsilon}\right) \\
&=f_{\theta_B}^{r,\varepsilon}
\quad\text{in } S\times\Omega_B^\varepsilon,
 \end{split}
\end{gather}
\end{subequations}
complemented with interface transmission, boundary, and initial conditions.


Now, for $j,k\in\{1,2,3\}$ and $y\in Y$, set
 $d_{jk}=(y_j\delta_{1k},y_j\delta_{2k},y_j\delta_{3k})^T$.
For $t\in S$, $x\in\Omega$, let $\tau_j^\theta(t,x,\cdot)\in H^1_{\overline{\#}}(Y_A)$,
 $\tau^u_{jk}(t,x,\cdot)$, $\tau^u(t,x,\cdot)\in H^1_{\overline{\#}}(Y_A)^3$ be the 
solutions to the following variational cell 
problems\footnote{Here, and in the following, the superscript $r$ denotes 
the transformed quantities (w.r.t~$s$), e.g., $K_A^{r}=J(F)^{-1}K_A(F)^{-T}$.}
\begin{subequations}\label{cell}
\begin{gather}
 0=\int_{Y_A}K_A^{r}\left(\nabla_y\tau_j^\theta+e_j\right)
\cdot\nabla_yv\,\mathrm{d}{y}\quad\text{for all } v\in H^1_\#(Y_A),\label{cell:1}\\
 0=\int_{Y_A}\mathcal{C}_A^{r}e_y(\tau^u_{jk}+d_{jk}):e_y(v)\,\mathrm{d}{y}\quad\text{for all } v\in H^1_\#(Y_A)^3,\label{cell:2}\\
 0=\int_{Y_A}\mathcal{C}_A^{r}e_y(\tau^u):e_y(v_A)\,\mathrm{d}{y}-\int_{Y_A}\alpha_A^{r}:
\nabla_yv\,\mathrm{d}{y}\quad\text{for all }v\in H^1_\#(Y_A)^3\label{cell:3}.
\end{gather}
\end{subequations}
Using these functions, we introduce the fourth rank tensor $\mathcal{C}$ and the matrix 
$K$ via
\begin{gather*}
\left(\mathcal{C}\right)_{i_1i_2j_1j_2}=\mathcal{C}_A^{r}e_y\left(\tau_{i_1i_2}^u+d_{i_1i_2}\right)
:e_y\left(\tau_{j_1j_2}^u+d_{j_1j_2}\right),\\
\left(K\right)_{ij}=K_A^{r}\left(\nabla_Y\tau_j^\theta+e_j\right)
\cdot\left(\nabla_Y\tau_i^\theta+e_i\right).
\end{gather*}
Furthermore, we define the following set of averaged coefficients
\begin{alignat*}{2}
C^h&=\int_{Y_A}C\,\mathrm{d}{y},\quad&
 \alpha_A^{h}&=\int_{Y_A}\left(\alpha_A^{r}C_A^re_y(\tau^{m})\right)\,
\mathrm{d}{y},\\
H_\Gamma^h&=\int_\Gamma H_\Gamma n\,\mathrm{d}{s},\quad&
 f_u^{h}&=\int_{Y_A}f_{u_A}^{r}\,\mathrm{d}{y}
+\int_{Y_B}f_{u_B}^{r}\,\mathrm{d}{y},\\
c^{h}&=\rho_Ac_{dA}\left|Y_A\right|+\int_{Y_A}\gamma_A^{r}:
\nabla_y\tau^u\,\mathrm{d}{y},\quad&
 W_{\Gamma}^{h}&=\int_{\Gamma}W_\Gamma^{r}\,\mathrm{d}{s},\\
K_A^{h}&=\int_{Y_A}K\,\mathrm{d}{y},\quad&
 f_\theta^{h}&=\int_{Y_A}f_{\theta_A}^{r}\,\mathrm{d}{y}
 +\int_{Y_B}f_{\theta_B}^{r}\,\mathrm{d}{y},\\
A^{h}(\Theta_B,U_B)&=\int_{Y_B}\left(c_B^{r}\Theta_B
+\gamma_B^{r}U_B\right)\,\mathrm{d}{y},\quad&
\gamma_A^{h}&=\int_{Y_A}\left(\gamma_A^{r}+\gamma_A^{r}\nabla_y\tau_{jk}^{m}\right)
\,\mathrm{d}{y}.
\end{alignat*}
After a homogenization procedure (the details of which are presented 
in \cite{EM17}), we obtain the following upscaled two-scale model
\begin{subequations}\label{p:refhom}
\begin{gather}
-\operatorname{div}\left(\mathcal{C}_A^{h}e(u_A)-\alpha_A^{h}\theta_A\right)
=f_u^{h}+H_\Gamma^{h}\quad \text{in }S\times\Omega,\label{h:hom_prob:1}\\[4pt]
\begin{gathered}
\partial_t\left(c^{h}\theta_A+\gamma_A^{h}:\nabla u_A+A^{h}(\Theta_B,U_B)\right)
-\operatorname{div}\left(K_A^{h}\nabla\theta_A\right)
=f_\theta^{h}-W_{\Gamma}^{h}\\
 \text{in }S\times\Omega,
\end{gathered}\label{h:hom_prob:2}\\[4pt]
-\operatorname{div}_y\left(\mathcal{C}_B^r e_y(U_B)-\alpha_B^r\Theta_B\right)
=f_{u_B}^r\quad \text{in }S\times\Omega\times Y_B\label{h:hom_prob:3},\\[4pt]
\begin{split}
&\partial_t\left(c_B^r\Theta_B+\gamma_B^r:\nabla_y U_B\right)
 -\operatorname{div}_y\left(K_B^r\nabla_Y\Theta_B\right) \\
&-\operatorname{div}_y\left(\left(c_B^r\Theta_B^r
 +\gamma_B^r:\nabla_y U_B\right)v^r\right) \\
&=f_{\theta_B}^r \quad\text{in }S\times\Omega\times Y_B,
\end{split} \label{h:hom_prob:4}\\[4pt]
U_B=u_A,\quad \Theta_B =\theta_A\quad \text{on }
 S\times\Omega\times\Gamma,\label{h:hom_prob:5}
\end{gather}
\end{subequations}
again, complemented with corresponding initial and boundary values.

\section{Preliminaries}\label{sec:preliminaries}

In this section, we lay the groundwork for the corrector estimations that 
are done in Section~\ref{sec:corrector} in stating the existence and 
regularity results for the solutions and also giving some auxiliary estimates.
We introduce the spaces
$$
H^1(\Omega_A^\varepsilon;\partial\Omega)=\big\{u\in H^1(\Omega_A^\varepsilon)  : u=0  \text{ on } 
\partial\Omega\big\}
$$
and, for a Banach space $X$,
$$
\mathcal{W}(S;X)=\left\{u\in L^2(S;X)  :  \text{such that } 
 \partial_tu\in L^2(S;X')\right\}.
$$
In general, we will not differentiate (in the notation) between a function 
defined on $\Omega$ and its restriction to $\Omega_A^\varepsilon$ or 
$\Omega_B^\varepsilon$ or between a function defined on one of those subdomains and 
its trivial extension to the whole of $\Omega$.
Here, and in the following, $C$, $C_1$, $C_2$ denote generic constants which
 are independent of $\varepsilon$ but whose values might change even from line to line.

For a function $f=f(x,y)$, we introduce the so called macroscopic reconstruction 
$[f]_\varepsilon=[f]_\varepsilon(x)=f(x,x/\varepsilon)$.
Note that, for general $f\in L^\infty(\Omega;H^1(Y)$, $[f]_\varepsilon$ 
may not even be measurable (see \cite{Al92}); continuity in one variable,
 e.g., $f\in L^\infty(\Omega;C_\#(Y))$ is sufficient, though.
Applying the chain rule leads to 
$D[f]_\varepsilon=[D_xf]_\varepsilon+1/\varepsilon[D_yf]_\varepsilon$, 
where $D=\nabla, e(\cdot),\operatorname{div}(\cdot)$, for sufficiently smooth 
functions $f$.

\subsection*{Assumption (A1)}
We assume that $\theta_0^\varepsilon\in H^1(\Omega)$, 
$f_{u_i}^{r,\varepsilon}\in C^1(S;L^2(\Omega_i^\varepsilon))$, 
$f_{\theta_i}^{r,\varepsilon}\in L^2(S\times\Omega_i^\varepsilon)$.
Furthermore, let $\theta_{A_0}\in H^1(\Omega)$, 
$f_{u_A}^h\in C^1(S;L^2(\Omega))$, $f_{\theta_A}^h\in L^2(S\times\Omega)$ 
for the macroscopic homogenized part and $\theta_{B_0}\in C(\Omega;H^1(Y_B))$, 
$f_{u_B}^r\in C^1(S;L^2(\Omega))$, $f_{\theta_B}^r\in L^2(S\times\Omega)$ 
for the two-scale part.
Moreover, we expect the following convergence rates to hold for our data
\begin{gather*}
\|\mathds{1}_{\Omega_A^\varepsilon}\theta_0^\varepsilon-\theta_{A_0}\|_{L^2(\Omega_A^\varepsilon)}
\leq C\sqrt{\varepsilon},\\
\|\mathds{1}_{\Omega_B^\varepsilon}\theta_0^\varepsilon-\big[\theta_{B_0}\big]_\varepsilon\|_{L^2(\Omega_B^\varepsilon)}
+\|f_{u_B}^{r,\varepsilon}-\big[f_{u_B}^r\big]_\varepsilon\|_{L^2(\Omega_B^\varepsilon)^3}
+\|f_{\theta_B}^{r,\varepsilon}-\big[f_{\theta_B}^r\big]_\varepsilon\|_{L^2(\Omega_B^\varepsilon)}
\leq C\varepsilon
\end{gather*}
and, in addition, to have the following estimates
\begin{gather*}
\int_{\Omega_A^\varepsilon}\left|(f_{u_A}^{r,\varepsilon}-f_{u_A}^h)\varphi(x)\right|\,\mathrm{d}{x}
\leq C\varepsilon\|\varphi\|_{H^1(\Omega_A^\varepsilon)}\quad\text{for all } 
 \varphi\in H^1(\Omega_A^\varepsilon;\partial\Omega)^3,\\
\int_{\Omega_A^\varepsilon}\left|(f_{\theta_A}^{r,\varepsilon}-f_{\theta_A}^h)\varphi(x)\right|\,\mathrm{d}{x}
\leq C\varepsilon\|\varphi\|_{H^1(\Omega_A^\varepsilon)}\quad\text{for all } 
 \varphi\in H^1(\Omega_A^\varepsilon;\partial\Omega).
\end{gather*}
If we are also interested in developing estimates for the time derivatives, 
we need stronger regularity assumptions.

\subsection*{Assumption (A2)}
Additionally to Assumptions (A1), we also expect the following convergence 
rates to hold:
\begin{gather*}
\|\mathds{1}_{\Omega_A^\varepsilon}\theta_0^\varepsilon-\theta_{A_0}\|_{H^1(\Omega_A^\varepsilon)}
\leq C\sqrt{\varepsilon},\\
\begin{aligned}
&\|\mathds{1}_{\Omega_B^\varepsilon}\theta_0^\varepsilon
 -\big[\theta_{B_0}\big]_\varepsilon\|_{H^1(\Omega_B^\varepsilon)}
+\|\partial_t(f_{u_B}^{r,\varepsilon}-\big[f_{u_B}^r\big]_\varepsilon)
\|_{L^2(\Omega_B^\varepsilon)^3}
+\|\partial_t(f_{\theta_B}^{r,\varepsilon}
-\big[f_{\theta_B}^r\big]_\varepsilon)\|_{L^2(\Omega_B^\varepsilon)}\\
&\leq C\varepsilon.
\end{aligned}
\end{gather*}
Moreover, we assume
\begin{gather*}
\int_{\Omega_A^\varepsilon}|\partial_t(f_{u_A}^{r,\varepsilon}-f_{u_A}^h)\varphi(x)|\,\mathrm{d}{x}
 \leq C\varepsilon\|\varphi\|_{H^1(\Omega_A^\varepsilon)}\quad\text{for all } 
 \varphi\in H^1(\Omega_A^\varepsilon;\partial\Omega)^3,\\
\int_{\Omega_A^\varepsilon}|\partial_t(f_{\theta_A}^{r,\varepsilon}-f_{\theta_A}^h)\varphi(x)|\,\mathrm{d}{x}
\leq C\varepsilon\|\varphi\|_{H^1(\Omega_A^\varepsilon)}\quad\text{for all }  
\varphi\in H^1(\Omega_A^\varepsilon;\partial\Omega).
\end{gather*}

\subsection{Regularity results}
To be able to justify the steps in the estimates in Section~\ref{sec:corrector},
 some of the involved functions need to be of higher regularity than is guaranteed 
via the standard $H^1$-theory for elliptic/parabolic problems.
In the following lemmas, we will collect the appropriate regularity results.

\begin{lemma}[Regularity of cell problem solutions]\label{reg_cell}
The solutions of the problems \eqref{cell:1}-\eqref{cell:3} possess the
 regularity $\tau_j^\theta\in C^2(\overline{S};C^1(\overline{\Omega};W^{2,p}(Y_A)))$
 and $\tau^u_{jk}$, $\tau^u\in C^2(\overline{S};C^1(\overline{\Omega};W^{2,p}(Y_A)^3))$
for some $p>3$.
\end{lemma}

\begin{proof}
The regularity w.r.t.~$y\in Y_A$ can be derived from standard elliptic 
regularity theory (we refer to \cite{GT13} for the general results and \cite{E04} 
for the application to our case of the cell problems).
The rest is a direct consequence of the regularity (w.r.t.~$t\in S$ and 
$x\in\Omega$) of the involved coefficients.
\end{proof}

Note that this implies, in particular, that the cell problem functions and 
their gradients (w.r.t.~$y\in Y$) are bounded and that their macroscopic 
reconstructions are well-defined measurable functions.

In the following, we denote $U^\varepsilon=(U_A^\varepsilon,U_B^\varepsilon)$ and 
$\Theta^\varepsilon=(\Theta_A^\varepsilon,\Theta_B^\varepsilon)$.

\begin{lemma}[Existence and regularity theorem for the $\varepsilon$-problem]
\label{reg_epsilon}
There is a \newline
unique $(U^\varepsilon,\Theta^\varepsilon)\in \mathcal{W}
(S;H_0^1(\Omega)^3\times H_0^1(\Omega))$ solving the variational system~\eqref{p:ref} 
for which standard energy estimates hold independently of the parameter $\varepsilon$.
Furthermore, this solution possesses the regularity 
$(U^\varepsilon,\Theta^\varepsilon)\in C^1(S;H^2(\Omega_A^\varepsilon)^3\times H^2(\Omega_B^\varepsilon)^3)
\times L^2(S;H^2(\Omega_A^\varepsilon)\times H^2(\Omega_B^\varepsilon))$ such that
 $\partial_t\Theta^\varepsilon\in L^2(S;H^1(\Omega))$.
\end{lemma}

\begin{proof}
The proof of the existence of a unique solution and of the energy estimates 
is given in \cite[Theorem 3.7, Theorem 3.8]{EM17}.
As a linear transmission problem (with sufficiently regular coefficients), 
regularity results apply (we refer to, e.g., \cite{E10}; see, also, \cite{SM02}
 for a similar coupling problem).
\end{proof}

Since $s^\varepsilon$ is a diffeomorphism, this leads to a unique solution to the 
moving interface problem, also.
However, while the solution has $H^2$-regularity, its second derivatives are
 not necessarily bounded (and in general will not be) independently of $\varepsilon>0$.

\begin{lemma}[Existence and regularity theorem for the homogenized problem]
\label{reg_homo} \ \newline
There is a unique 
$$
(u_A,\theta_A,U_B,\Theta_B)\in\mathcal{W}(S;H_0^1(\Omega)^3
\times H_0^1(\Omega)\times L^2(\Omega;W^{1,2}(Y_B)^3)
\times L^2(\Omega;W^{1,2}(Y_B)))
$$ 
solving the variational system~\eqref{p:refhom}.
Furthermore, 
\[
(u_A,\theta_A)\in C^1(S;H^2(\Omega)^3)\times L^2(S;H^2(\Omega))
\]
such that $\partial_t\theta_A\in L^2(S;H^1(\Omega))$ and
\[
(U_B,\Theta_B)\in C^1(S;H^2(\Omega;H^2(Y_B)^3))
\times L^2(S;H^2(\Omega;H^2(Y_B)))
\] 
such that $\partial_t\Theta_B\in L^2(S;H^2(\Omega;H^1(Y_B)))$.
\end{lemma}

\begin{proof}
Existence of a solution is given via the two-scale homogenization procedure 
outlined in \cite{EM17} and uniqueness for this linear coupled transmission 
problem can then be shown using energy estimates.
As to the higher regularity, this follows, again, via the regularity of domain, 
coefficients, and data, we refer to results outlined in \cite{E10, SM02, VM11}.
\end{proof}

\subsection{Auxiliary estimates}

For the transformation related quantities, we have the following estimates 
available as stated in Lemma~\ref{lemma:estimates_trafo} and 
Lemma~\ref{lemma:average}.

\begin{lemma}\label{lemma:estimates_trafo}
There is a constant $C>0$ independent of the parameter $\varepsilon$ such that
\begin{align*}
&\|F^\varepsilon-[F]_\varepsilon\|_{L^\infty(S\times\Omega)^{3\times3}}
+\|J^\varepsilon-[J]_\varepsilon\|_{L^\infty(S\times\Omega)}
+\|v^\varepsilon-\varepsilon[v]_\varepsilon\|_{L^\infty(S\times\Omega)}\\
& +\|W_\Gamma^\varepsilon-\varepsilon\big[W_\Gamma\big]_\varepsilon\|_{L^\infty(S\times\Gamma)}
+\|H_\Gamma^\varepsilon-\big[H_\Gamma\big]_\varepsilon\|_{L^\infty(S\times\Gamma)}
\leq C\varepsilon
\end{align*}
The same estimates hold for the time derivatives of those functions.
\end{lemma}

\begin{proof}
We show this, by way of example, only for $F^\varepsilon$, the other estimates 
follow in the same way:
\[
\|F^\varepsilon-[f]_\varepsilon \|_{L^\infty(S\times\Omega)^{3\times3}}
=\operatorname{ess\,sup}_{(t,x)\in S\times\Omega}
\big|\nabla_y s\big(t,\varepsilon \big[\frac{x}{\varepsilon}\big],{\frac{x}{\varepsilon}}\big)
 -\nabla_y s\big(t,x,{\frac{x}{\varepsilon}}\big)\big|
\leq L\frac{\varepsilon}{\sqrt{2}}.
\]
Here, $L$ is the Lipschitz constant of $F^\varepsilon$ w.r.t.~$x\in\Omega$ 
(uniform in $S\times Y$).
\end{proof}
Based on these estimates and due to the fact that all material parameters 
are assumed to be constant in the moving geometry, we obtain the same estimates
for the material parameters ($K^{r,e}$, $\alpha^{r,\varepsilon}$ and so on) in the 
reference configuration.

The following lemma is concerned with $\varepsilon$-independent estimates for the 
macroscopic reconstruction of periodic functions with zero average.
There are several different but similar theorems that can be found in the 
literature regarding corrector estimates in the context of homogenization,
 we refer to, e.g., \cite{CPS07, CP98, E04, MV13}, but for our purposes the 
following version suffices:

\begin{lemma}\label{lemma:average}
Let $f\in L^2(S\times\Omega_A^\varepsilon;C_\#(Y)$ such that
%
$$
\int_{Y_A}f(t,x,y)\,\mathrm{d}{y}=0\quad \text{a.e.~in}\ \ S\times\Omega_A^\varepsilon.
$$
Then, there is a constant $c>0$ such that, independently of $\varepsilon$,
$$
\int_{\Omega_A^\varepsilon}\left|[f]_\varepsilon(t,x)\varphi(x)\right|\,\mathrm{d}{x}
\leq C\varepsilon\|\varphi\|_{H^1(\Omega_A^\varepsilon)}.
$$
for all $\varphi\in H^1(\Omega_A^\varepsilon;\partial\Omega)$.
\end{lemma}

The above lemma can be proven similarly to the corresponding statements 
in \cite[Lemma 3]{CP98} and \cite[Lemma 5.2]{MV13}.

\section{Corrector estimates}\label{sec:corrector}

In this section, we are concerned with the actual corrector estimates.
Reconstructing micro-solutions from the homogenized functions via the 
$[\cdot]_\varepsilon$-operation and subtracting the heat 
equations~\eqref{p:full_problem_moving:1}, \eqref{p:full_problem_moving:2},
\eqref{h:hom_prob:2}, and \eqref{h:hom_prob:4} and momentum 
equations~\eqref{p:full_problem_moving:1}, \eqref{p:full_problem_moving:1},
\eqref{h:hom_prob:1}, and \eqref{h:hom_prob:3} we obtain
\begin{subequations}\label{corrector_eq}
\begin{gather}
\begin{aligned}
&\partial_t\left(c_A^{r,\varepsilon}\Theta_A^\varepsilon
 -\kappa_Ac^{h}\theta_A\right)
+\partial_t\left(\gamma_A^{r,\varepsilon}:\nabla U_A^\varepsilon
 -\kappa_A\gamma_A^{h}:
\nabla u_A\right) \\
&-\operatorname{div}\left(K_A^{r,\varepsilon}\nabla\Theta_A^\varepsilon
-\kappa_AK_A^{h}\nabla\theta_A\right)
-\kappa_AW_{\Gamma}^{h} \\
 &=f_{\theta_A}^{r,\varepsilon}-\kappa_Af_\theta^{h},
\end{aligned}\label{corrector_eq:ha}
\\[4pt]
\begin{aligned}
&-\operatorname{div}\left(\mathcal{C}_A^{r,\varepsilon}(U_A^\varepsilon)-\kappa_A\mathcal{C}_A^{h}e(u_A)-\alpha_A^{r,\varepsilon}
\Theta_A^\varepsilon+\kappa_A\alpha_A^{h}\theta_A\right)+H_\Gamma^{h} \\
&=f_{u_A}^{r,\varepsilon} -\kappa_Af_u^{h},\label{corrector_eq:ma}
\end{aligned}\\[4pt]
\begin{aligned}
&\partial_t\left(c_B^{r,\varepsilon}\Theta_B^\varepsilon-\big[c_B^{r}\Theta_B\big]_\varepsilon\right)
+\partial_t\left(\varepsilon\gamma_B^{r,\varepsilon}:\nabla U_B^\varepsilon-\big[\gamma_B^{r}:
\nabla_Y U_B\big]_\varepsilon\right) \\
& -\operatorname{div}\left(\varepsilon^2K_B^{r,\varepsilon}\nabla\Theta_B^\varepsilon\right)
+\big[\operatorname{div}_Y\left(K_B^{r}\nabla_Y\Theta_B\right)\big]_\varepsilon\\
&=f_{\theta_B}^{r,\varepsilon}-\left[f_{\theta_B}^{r}\right]_\varepsilon,
\end{aligned}\label{corrector_eq:hb}
\\[4pt]
\begin{aligned}
&-\operatorname{div}\left(\varepsilon^2\mathcal{C}_B^{r,\varepsilon} 
e(U_B^\varepsilon)-\varepsilon\alpha_B^{r,\varepsilon}\Theta_B^\varepsilon\right)\\
&+\Big[\operatorname{div}_Y\left(\mathcal{C}_B^{r} e_y(U_B)
-\alpha_B^{r}\Theta_B\mathds{I}_3\right)\Big]_\varepsilon
=f_{u_B}^{r,\varepsilon}-\big[f_{u_B}^r\big]_\varepsilon .
\end{aligned}\label{corrector_eq:mb}
\end{gather}
\end{subequations}
These equations hold in $\Omega_A^\varepsilon$ and $\Omega_B^\varepsilon$, respectively.
Using the interface and boundary conditions for both the $\varepsilon$-problem and 
the homogenized problem and then doing an integration by parts, these equations 
correspond to a variational problem in $H^{-1}(\Omega)$.

Our strategy in establishing the estimates is as follows: After introducing 
error and corrector functions and doing some further preliminary estimates, 
we first, in Section~\ref{subsec:momentum}, concentrate on the momentum part, 
i.e., equations~\eqref{corrector_eq:ma} and~\eqref{corrector_eq:mb}.
Here, we take the different terms arising in the weak formulation and estimate 
them individually using the results from Section~\ref{sec:preliminaries} 
and usual energy estimation techniques.
Combining those estimates, it is shown that the \emph{mechanical error} 
can be controlled by the \emph{heat error}, see Remark~\ref{rem:mech}.
Then, in Section~\ref{subsec:heat}, we basically do the same for the heat 
conduction part, i.e., equations~\eqref{corrector_eq:ha} and~\eqref{corrector_eq:hb}, 
thereby arriving at the corresponding result that the \emph{heat error} 
is controlled by the \emph{mechanical error} (and the time derivative 
of the \emph{mechanical error}), see Remark~\ref{rem:heat}.

Finally, in Section~\ref{subsec:overall}, we go about combining those 
individual estimates.
Here, we show that for the scenarios $(a)$ (Theorem~\ref{cor1}) and $(b)$ 
(Theorem~\ref{cor2}) (as described in the introduction), we obtain the desired
estimates, i.e., equation~\eqref{goal_est}.
Moreover, we point out why the same strategy will not work for the full problem.

Now, we introduce the functions the subscripts ``$\mathrm{err}$'' and 
``$\mathrm{cor}$'' for \emph{error} and \emph{corrector}, respectively.
\begin{gather*}
U_\mathrm{err}^\varepsilon
=\begin{cases} 
U_A^\varepsilon-u_A & \text{in } S\times\Omega_A^\varepsilon\\
 U_B^\varepsilon-[U_B]_\varepsilon & \text{in } S\times\Omega_B^\varepsilon
\end{cases}, \quad
\Theta_\mathrm{err}^\varepsilon=\begin{cases}
\Theta_A^\varepsilon-\theta_A & \text{in } S\times\Omega_A^\varepsilon\\ 
\Theta_B^\varepsilon-[\Theta_B]_\varepsilon & \text{in } S\times\Omega_B^\varepsilon
\end{cases},
\\
U_\mathrm{cor}^\varepsilon=\begin{cases}
U_\mathrm{err}^\varepsilon-\varepsilon\big[\widetilde{U}\big]_\varepsilon& \text{in } S\times\Omega_A^\varepsilon\\
U_\mathrm{err}^\varepsilon & \text{in } S\times\Omega_B^\varepsilon
\end{cases},\quad
\Theta_\mathrm{cor}^\varepsilon=\begin{cases}
\Theta_\mathrm{err}^\varepsilon-\varepsilon\big[\widetilde{\Theta}\big]_\varepsilon & \text{in }
 S\times\Omega_A^\varepsilon\\
 \Theta_\mathrm{err}^\varepsilon & \text{in } S\times\Omega_B^\varepsilon
\end{cases}.
\end{gather*}
The functions $\widetilde{U}$ and $\widetilde{\theta}$ are exactly the 
functions arising in the two-scale limits of the gradients of $U_A^\varepsilon$ and
 $\Theta_A^\varepsilon$, respectively, and are given as (cf. \cite{EM17})
\begin{align*}
\widetilde{U}=\sum_{j,k=1}^3\tau_{jk}^ue(u_A)_{jk}+\tau^u\theta_A,\quad
\widetilde{\Theta}=\sum_{j=1}^3\tau^\theta\partial_j\theta_A.
\end{align*}
Because of the corrector part (namely, $\varepsilon\big[\widetilde{\Theta}\big]_\varepsilon$ 
and
$\varepsilon\big[\widetilde{U}\big]_\varepsilon$, respectively), our corrector functions will,
 in general, not vanish at $\partial\Omega$ and are therefore not valid 
choices of test functions for a weak variational formulation of the system 
given via equations~\eqref{corrector_eq:ha}-\eqref{corrector_eq:mb}.
Because of that, we introduce a smooth cut-off function
 $m^\varepsilon\colon\Omega\to[0,1]$ 
such that $m^\varepsilon(x)=0$ for all $x\in\Omega_A^\varepsilon$ such that 
$\operatorname{dist}(x,\partial\Omega)\leq\varepsilon c/2$ and $m^\varepsilon(x)=1$ for all $x\in\Omega_A^\varepsilon$ 
such that $\operatorname{dist}(x,\partial\Omega)\geq\varepsilon c$.
Furthermore, we require the estimate
\begin{align}\label{estimate_cut}
\sqrt{\varepsilon}\|\nabla m^\varepsilon\|_{L^2(\Omega)}
+\sqrt{\varepsilon^3}\|\Delta m^\varepsilon\|_{L^2(\Omega)}+\frac{1}{\sqrt{\varepsilon}}
\|1-m^\varepsilon\|_{L^2(\Omega)}\leq C
\end{align}
to hold independently of the parameter $\varepsilon$.
For this cut-off function $m^\varepsilon$, we set
\begin{gather*}
U_{\mathrm{cor}0}^\varepsilon=\begin{cases}
U_\mathrm{cor}^\varepsilon+(1-m^\varepsilon)\varepsilon\big[\widetilde{U}\big]_\varepsilon &\text{in } S\times\Omega_A^\varepsilon\\
 U_\mathrm{cor}^\varepsilon-\varepsilon\big[\widetilde{U}\big]_\varepsilon & \text{in } S\times\Omega_B^\varepsilon
\end{cases},\\
\Theta_{\mathrm{cor}0}^\varepsilon=\begin{cases}
\Theta_\mathrm{cor}^\varepsilon+(1-m^\varepsilon)\varepsilon\big[\widetilde{\Theta}\big]_\varepsilon& \text{in }S\times\Omega_A^\varepsilon\\
\Theta_\mathrm{cor}^\varepsilon-\varepsilon\big[\widetilde{\Theta}\big]_\varepsilon & \text{in } S\times\Omega_B^\varepsilon
\end{cases}.
\end{gather*}
Obviously, we then have $\Theta_{\mathrm{cor}0}^\varepsilon\in H_0^1(\Omega)$ and 
$U_{\mathrm{cor}0}^\varepsilon\in H_0^1(\Omega)^3$.
Owing to the regularity of $\widetilde{U}$ and $\widetilde{\Theta}$ 
(Lemma~\ref{reg_cell}) and the estimate~\eqref{estimate_cut} for $m^\varepsilon$, 
these modified correctors admit the following $\varepsilon$-independent estimates
 (for $i=A,B$)\footnote{The same estimates hold when replacing the 
linearized strain tensor with the gradient operator.}
\begin{gather*}
\|U_{\mathrm{err}}^\varepsilon\|_{L^2(\Omega_i^\varepsilon)^3}
 \leq\|U_{\mathrm{cor}0}^\varepsilon\|_{L^2(\Omega_i^\varepsilon)^3}+C\varepsilon\leq\|U_{\mathrm{err}}^\varepsilon
\|_{L^2(\Omega_i^\varepsilon)^3}+2C\varepsilon,\\
\begin{aligned}
\|e(U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}
&\leq\|e( U_{\mathrm{cor}0}^\varepsilon)\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}+C(\sqrt{\varepsilon}+\varepsilon) \\
&\leq\|e(U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}+2C(\sqrt{\varepsilon}+\varepsilon),
\end{aligned}\\
\begin{aligned}
\varepsilon\|e( U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}
&\leq\varepsilon\|e(U_{\mathrm{cor}0}^\varepsilon)\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}+C(\sqrt{\varepsilon}+\varepsilon) \\
&\leq\varepsilon\|e(U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}+2C(\sqrt{\varepsilon}+\varepsilon),
\end{aligned}\\
\|\Theta_{\mathrm{err}}^\varepsilon\|_{L^2(\Omega_i^\varepsilon)}
\leq\|\Theta_{\mathrm{cor}0}^\varepsilon\|_{L^2(\Omega_i^\varepsilon)}
 +C\varepsilon\leq\|\Theta_{\mathrm{err}}^\varepsilon\|_{L^2(\Omega_i^\varepsilon)}+2C\varepsilon,\\
\begin{aligned}
\|\nabla \Theta_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^{3}}
&\leq\|\nabla \Theta_{\mathrm{cor}0}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^{3}}+C(\sqrt{\varepsilon}+\varepsilon) \\
&\leq\|\nabla \Theta_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^{3}}+2C(\sqrt{\varepsilon}+\varepsilon),
\end{aligned}\\
\begin{aligned}
\varepsilon\|\nabla \Theta_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^{3}}
&\leq\varepsilon\|\nabla \Theta_{\mathrm{cor}0}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^{3}}+C(\sqrt{\varepsilon}+\varepsilon) \\
&\leq\varepsilon\|\nabla \Theta_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^{3}}+2C(\sqrt{\varepsilon}+\varepsilon).
\end{aligned}
\end{gather*}
Applying Korns inequality to $U_{\mathrm{cor}0}$ (see \cite[Lemma 3.1]{EM17}) 
and using the above estimates, we then get
\begin{equation}\label{estimate_ucor}
\begin{aligned}
&\|U_\mathrm{err}^\varepsilon\|_{L^2(\Omega)}+\|\nabla U_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^3}
+\varepsilon\|\nabla U_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}\\ 
&\leq \|e(U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_A^\varepsilon)^3}
 +\varepsilon\|e(U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}+C(\sqrt{\varepsilon}+\varepsilon).
\end{aligned}
\end{equation}

\subsection{Estimates for the momentum equations}\label{subsec:momentum}

Let us first concentrate on the mechanical part of the corrector equations, 
namely equations~\eqref{corrector_eq:ma} and~\eqref{corrector_eq:mb}.
After integrating over $\Omega$, multiplying with a test function 
$\varphi\in H_0^1(\Omega)^3$, and integrating by parts, we obtain
\begin{align*}
&\underbrace{\int_{\Omega_A^\varepsilon}\left(\mathcal{C}_A^{r,\varepsilon} e(U_A^\varepsilon)
-\kappa_A\mathcal{C}_A^{h}e(u_A)\right):e(\varphi)\,\mathrm{d}{x}
-\int_{\Gamma^\varepsilon}\kappa_A\mathcal{C}_A^{h}e(u_A)n^\varepsilon\cdot\varphi\,\mathrm{d}{s}}_{=:I_1^\varepsilon(t,\varphi)}\\
&+\underbrace{\varepsilon^2\int_{\Omega_B^\varepsilon}\left(\mathcal{C}_B^{r,\varepsilon} e(U_B^\varepsilon)
-\big[\mathcal{C}_B^{r}\big]_\varepsilon e([U_B]_\varepsilon )\right):e(\varphi)\,\mathrm{d}{x}}_{=:I_2^\varepsilon(t,\varphi)}\\
&-\underbrace{\int_{\Omega_A^\varepsilon}\left(\alpha_A^{r,\varepsilon}\Theta_A^\varepsilon
-\kappa_A\alpha_A^{h}\theta_A\right):\nabla\varphi\,\mathrm{d}{x}
+\int_{\Gamma^\varepsilon}\kappa_A\alpha_A^{h}\theta_An^\varepsilon\cdot\varphi\,\mathrm{d}{s}}_{=:I_3^\varepsilon(t,\varphi)}\\
&+\underbrace{\varepsilon\int_{\Omega_B^\varepsilon}\left(\alpha_B^{r,\varepsilon}
\Theta_B^\varepsilon-\big[\alpha_B^{r}\Theta_B\big]_\varepsilon \right):\nabla\varphi\,\mathrm{d}{x}}_{=:I_4^\varepsilon(t,\varphi)}\\
&=\underbrace{\int_{\Omega_A^\varepsilon}\left(f_{u_A}^{r,\varepsilon}
 -\kappa_A\int_{Y_A}f_{u_A}^{r}\,\mathrm{d}{y}\right)\cdot\varphi\,\mathrm{d}{x}}_{=:I_5^\varepsilon(t,\varphi)} \\
&\quad +\underbrace{\varepsilon^2\int_{\Gamma^\varepsilon}[\mathcal{C}_B^r]_\varepsilon e([U_B]_\varepsilon )n^\varepsilon\cdot\varphi\,\mathrm{d}{s}
-\int_{\Omega_A^\varepsilon}\kappa_A\int_{Y_B}f_{u_B}^{r}\,\mathrm{d}{y}\cdot\varphi\,\mathrm{d}{x}}_{=:I_6^\varepsilon(t,\varphi)}\\
&\quad +\underbrace{\varepsilon^2\int_{\Gamma^\varepsilon}H_{\Gamma}^{r,\varepsilon}n^\varepsilon\cdot\varphi\,\mathrm{d}{s}
-\int_{\Omega_A^\varepsilon}\kappa_AH_\Gamma^{h}\cdot\varphi\,\mathrm{d}{x}}_{=:I_7^\varepsilon(t,\varphi)}
+\underbrace{\int_{\Omega_B^\varepsilon}\left(f_{u_B}^{r,\varepsilon}-\big[f_{u_B}^r\big]_\varepsilon\right)
\cdot\varphi\,\mathrm{d}{x}}_{=:I_8^\varepsilon(t,\varphi)}
+I_9^\varepsilon(t,\varphi),
\end{align*}
where
\begin{align*}
&I_9^\varepsilon(t,\varphi) \\
&:=\int_{\Omega_B^\varepsilon}
\left(\varepsilon^2\operatorname{div}\left([\mathcal{C}_B^{r}e_x(U_B)]_\varepsilon\right)
+\varepsilon\big[\operatorname{div}_x\left(\mathcal{C}_B^{r}e_y(U_B)\right)\big]_\varepsilon
+\varepsilon\big[\operatorname{div}_x\left(\alpha_B^{r}\Theta_B\right)\big]_\varepsilon\right)
\cdot\varphi\,\mathrm{d}{x}.
\end{align*}
We now go on estimating these terms individually and then combine the resulting 
estimates. We proceed for both $U_{\mathrm{cor}0}^\varepsilon$ and $\partial_tU_{\mathrm{cor}0}$ 
as test function choices.
While the first choice is the natural one for energy estimates, 
the second choice is needed in order to merge those estimates with the 
heat equation estimates.

Taking a look at $I_1^\varepsilon$, we calculate
\begin{align*}
 I_1^\varepsilon(t,\varphi)
&=\int_{\Omega_A^\varepsilon}\mathcal{C}_A^{r,\varepsilon} e(U_{\mathrm{cor}}^\varepsilon):e(\varphi)\,\mathrm{d}{x}\\
&\quad +\int_{\Omega_A^\varepsilon}\left(\mathcal{C}_A^{r,\varepsilon} e(U_A^\varepsilon-U_{\mathrm{cor}}^\varepsilon)
-\kappa_A\mathcal{C}_A^{h}e(u_A)\right):e(\varphi)\,\mathrm{d}{x} \\
&\quad -\int_{\Gamma^\varepsilon}\kappa_A\mathcal{C}_A^{h}e(u_A)n^\varepsilon\cdot\varphi\,\mathrm{d}{s}
\end{align*}
We also see (via the definition of $\mathcal{C}$ and $\widetilde{U}$)
%
\begin{align*}
\mathcal{C}_A^{r,\varepsilon}e(U_A^\varepsilon-U_{\mathrm{cor}}^\varepsilon)
&=[\mathcal{C}]_\varepsilon e(u_A)+\big[\mathcal{C}_A^{r}e_y(\tau^u)\big]_\varepsilon\theta_A\\
&\quad +\underbrace{\left(\mathcal{C}_A^{r,\varepsilon}-\big[\mathcal{C}_A^{r}\big]_\varepsilon\right)
\left(e(u_A)+\big[e_y(\widetilde{U})\big]_\varepsilon\right)
+\varepsilon\mathcal{C}_A^{r,\varepsilon}\big[e_x\left(\widetilde{U}\right)\big]_\varepsilon}_{R_1^\varepsilon},
\end{align*}
where, using Lemma~\ref{lemma:average}, the estimate
$$
\big|\int_{\Omega_A^\varepsilon}R_1^\varepsilon\varphi\,\mathrm{d}{x}\big|\leq C\varepsilon\|p\|_{H^1(\Omega_A^\varepsilon)}
$$
holds.
Now, seeing that $\int_\Gamma \mathcal{C} e(u_A)n\,\mathrm{d}{s}=0$ and
 $\operatorname{div}_y(\mathcal{C} e(u_A))=0$ a.e.~in $S\times\Omega$ and using Lemma~\ref{lemma:average}, 
we obtain
\begin{align*}
&\big|\int_{\Omega_A^\varepsilon}\left([\mathcal{C}]_\varepsilon
-\kappa_A(0)\mathcal{C}_A^{h}\right)e(U_A):e(\varphi)\,\mathrm{d}{x}
 -\int_{\Gamma^\varepsilon}\kappa_A(0)\mathcal{C} e(u_A) n^\varepsilon\cdot\varphi\,\mathrm{d}{x}\big|\\
&=\big|\int_{\Omega_A^\varepsilon}\operatorname{div}\left(\left([\mathcal{C}]_\varepsilon 
-\kappa_A(0)\mathcal{C}_A^{h}\right)e(U_A)\right)\cdot\varphi\,\mathrm{d}{x}\big| \\
&\leq C\varepsilon\|\varphi\|_{H^1(\Omega_A^\varepsilon)}
\end{align*}
for all $\varphi\in H^1(\Omega_A^\varepsilon;\partial\Omega)$.
Also,
\begin{align*}
&\int_{\Omega_A^\varepsilon}\mathcal{C}_A^{r,\varepsilon} e(U_{\mathrm{cor}}^\varepsilon):e(U_{\mathrm{cor}0}^\varepsilon)\,\mathrm{d}{x} \\
&\geq C_1\|e(U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}^2
+\varepsilon\int_{\Omega_A^\varepsilon}\mathcal{C}_A^{r,\varepsilon}e(U_\mathrm{cor}^\varepsilon):
e\big((1-m^\varepsilon)\big[\widetilde{U}\big]_\varepsilon\big)\,\mathrm{d}{x}\\
&\geq \frac{C_1}{2}\|e(U_{\mathrm{cor}}^\varepsilon)\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}^2
-C_2(\varepsilon+\varepsilon^2).
\end{align*}
Here, the second inequality can be shown using the assumptions on $m^\varepsilon$ 
and the known estimates of the involved functions.
In summary, for $I_1^\varepsilon$, we arrive at
\begin{equation}\label{estimate_i1}
\begin{aligned}
&I_1^\varepsilon(t,U_{\mathrm{cor}0}^\varepsilon)-\int_{\Omega_A^\varepsilon}\big[\mathcal{C}_A^{r}e_y(\tau^u)\big]_\varepsilon
\theta_A :e(U_{\mathrm{cor}0}^\varepsilon)\,\mathrm{d}{x} \\
&\geq C_1\|e(U_{\mathrm{cor}}^\varepsilon)\|^2_{L^2(\Omega_A^\varepsilon)^{3\times3}}
-C_2(\varepsilon+\varepsilon^2).
\end{aligned}
\end{equation}
Now, going on with $I_2^\varepsilon$, it is easy to see that
\begin{equation} \label{estimate_i2}
\begin{aligned}
I_2^\varepsilon(t,U_{\mathrm{cor}0}^\varepsilon)
&=\varepsilon^2\int_{\Omega_B^\varepsilon}\big[\mathcal{C}_B^{r}\big]_\varepsilon\left(e(U_B^\varepsilon)
 -e([U_B]_\varepsilon )\right):e(U_\mathrm{cor})\,\mathrm{d}{x} \\
&\quad +\varepsilon^2\int_{\Omega_B^\varepsilon}\left(\mathcal{C}_B^{r,\varepsilon}-\big[\mathcal{C}_B^{r}\big]_\varepsilon
\right)e(U_B^\varepsilon): e(U_\mathrm{cor})\,\mathrm{d}{x} \\
&\geq C_1\varepsilon^2\|e(U_\mathrm{cor}^\varepsilon)\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}^2-C_2\varepsilon^2.
\end{aligned}
\end{equation}
Going forward with term $I_3^\varepsilon$, we decompose
\begin{equation}\label{decompose}
\alpha_A^{r,\varepsilon}\Theta_A^\varepsilon-\kappa_A\alpha_A^{h}\theta_A
=\left(\alpha_A^{r,\varepsilon}-\big[\alpha_A^{r}\big]_\varepsilon\right)\Theta_A^\varepsilon
+\big[\alpha_A^{r}\big]_\varepsilon\Theta_\mathrm{err}^\varepsilon+\left(\big[\alpha_A^{r}
\big]_\varepsilon
-\kappa_A\alpha_A^{h}\right)\theta_A
\end{equation}
from which we can estimate the first two terms as
\[
\big|\int_{\Omega_A^\varepsilon}\left(\left(\alpha_A^{r,\varepsilon}-\big[\alpha_A^{r}\big]_\varepsilon\right)
\Theta_A^\varepsilon+\big[\alpha_A^r\big]_\varepsilon \Theta_\mathrm{err}^\varepsilon\right):
\nabla\varphi\big|\leq C\left(\varepsilon+\|\Theta_\mathrm{err}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)}\right)
\|\nabla\varphi\|_{L^2(\Omega_A^\varepsilon)}.
\]
For the remaining term of equation~\eqref{decompose} combined with
the interface integral part of $I_3^\varepsilon$, we obtain
\begin{align*}
&\int_{\Omega_A^\varepsilon}\left(\big[\alpha_A^r\big]_\varepsilon
 -\kappa_A\alpha_A^{h}\right)\theta_A:\nabla\varphi\,\mathrm{d}{x}
 +\int_{\Gamma^\varepsilon}\kappa_A\alpha_A^{h}\theta_An^\varepsilon\cdot\varphi\,\mathrm{d}{s}\\
&=-\int_{\Omega_A^\varepsilon}\Big[\operatorname{div}_x\Big(\Big(\alpha_A^{r}
 -\kappa_A\int_{Y_A}\alpha_A^{r}\,\mathrm{d}{y}\Big)\theta_A\Big)\Big]_\varepsilon
 \cdot\varphi\,\mathrm{d}{x} \\
&\quad -\int_{\Omega_A^\varepsilon}\kappa_A\operatorname{div}
\Big(\int_{Y_A}C_A^re_y(\tau^u)\,\mathrm{d}{y} \theta_A\Big)\cdot\varphi\,\mathrm{d}{x}\\
&\quad -\int_{\Gamma^\varepsilon}\big[\alpha_A^r\big]_\varepsilon \theta_An^\varepsilon\cdot\varphi\,\mathrm{d}{s}
-\frac{1}{\varepsilon}\int_{\Omega_A^\varepsilon}\Big[\operatorname{div}_y\left(\alpha_A^{r}\theta_A\right)\Big]_\varepsilon
\cdot\varphi\,\mathrm{d}{x}
\end{align*}
We apply Lemma~\ref{lemma:average} to
\begin{gather*}
f_1=\operatorname{div}_x\Big(\Big(\alpha_A^{r}-\kappa_A\int_{Y_A}\alpha_A^{r}\,\mathrm{d}{y}
 \Big)\theta_A\Big),\\
f_2=\operatorname{div}_x\Big(\Big(\mathcal{C}_A^{r}e_y(\tau^u))
 -\kappa_A\int_{Y_A}C_A^re_y(\tau^u)\,\mathrm{d}{y}\Big)\theta_A\Big)
\end{gather*}
and recall that $\tau^u$ is a solution of the cell problem~\eqref{cell:3}
(because of this the ``$\frac{1}{\varepsilon}\big[\operatorname{div}_y\big]_\varepsilon$''-terms vanish)
which leads to
\begin{equation}\label{estimate_i3}
\big|I_3^\varepsilon(t,\varphi)+\int_{\Omega_A^\varepsilon}
\big[\mathcal{C}_A^{r}e_y(\tau^u)\big]_\varepsilon\theta_A:e(\varphi)\,\mathrm{d}{x}\big|
\leq C\left(\varepsilon+\|\Theta_\mathrm{err}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)}\right)
\|\varphi\|_{H^1(\Omega_A^\varepsilon)}.
\end{equation}
Similarly as with $I_2^\varepsilon$, for the thermo-elasticity term $I_4^\varepsilon$, we obtain
\begin{align}\label{estimate_i4}
\left|I_4^\varepsilon(t,\varphi)\right|
\leq C\|\Theta_\mathrm{err}^\varepsilon\|^2_{L^2(\Omega_B^\varepsilon)}
+\varepsilon^2\|\nabla\varphi\|^2_{L^2(\Omega_B^\varepsilon)^{3\times3}}.
\end{align}
Next, we take the mean curvature error term $I_7^\varepsilon$
\[
|I_7^\varepsilon(t,\varphi)|\leq \varepsilon^2\big|\int_{\Gamma^\varepsilon}\left(H_{\Gamma}^{r,\varepsilon}
-\big[H_{\Gamma}^r\big]_\varepsilon\right)n^\varepsilon\cdot\varphi\,\mathrm{d}{s}\big|
+\big|\varepsilon^2\int_{\Gamma^\varepsilon}\big[H_{\Gamma}^r\big]_\varepsilon n^\varepsilon\cdot\varphi\,\mathrm{d}{s}
-\int_{\Omega_A^\varepsilon}\kappa_AH_\Gamma^{h}\cdot\varphi\,\mathrm{d}{x}\big|.
\]
Now, in view of the assumptions on our data and for the source density errors
 (stated in Assumptions (A1)) and the curvature estimate, using
 Lemma~\ref{lemma:estimates_trafo} (for the functional $I_6^\varepsilon(t,\varphi)$),
and the boundedness of the functions involved in $I_9^\varepsilon$, we estimate
\begin{equation} \label{estimate_i5}
\begin{aligned}
&\left|I_5^\varepsilon(t,\varphi)\right|+|I_6^\varepsilon(t,\varphi)|+\left|I_7^\varepsilon(t,\varphi)\right|
+\left|I_8^\varepsilon(t,\varphi)\right|+\left|I_9^\varepsilon(t,\varphi)\right| \\
&\leq C\varepsilon\left(\|\varphi\|_{H^1(\Omega_A^\varepsilon)}+\|\varphi\|_{L^2(\Omega_B^\varepsilon}\right).
\end{aligned}
\end{equation}
Finally, merging the individual estimates for the error terms (namely,
estimates \eqref{estimate_i1},
\eqref{estimate_i2}, \eqref{estimate_i3}, \eqref{estimate_i4},
\eqref{estimate_i5}) and the estimate~\eqref{estimate_ucor}, we conclude
\begin{equation}\label{estimate_mech}
\begin{aligned}
&\|U_{\mathrm{err}}^\varepsilon\|_{L^2(\Omega)^{3}}
 +\|\nabla U_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}
+\varepsilon\|\nabla U_\mathrm{cor}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^{3\times3}} \\
&\leq C\Big(\sqrt{\varepsilon}
+\varepsilon+\|\Theta_{\mathrm{err}}\|_{L^2(\Omega)}\Big).
\end{aligned}
\end{equation}

\begin{remark}\label{rem:mech} \rm
With estimate~\eqref{estimate_mech} it is clear that the error in the mechanical
 part inherits the convergence rate from the heat-error (at least if it is not 
faster than $\sqrt{\varepsilon}$).
\end{remark}

If, additionally, Assumptions (A2) are fulfilled, it is also possible to first 
differentiate equations~\eqref{corrector_eq:ma} and~\eqref{corrector_eq:mb} with respect to time and to then choose the test function $\partial_tU_\mathrm{cor}$ for the variational formulation of the arising system.
This way, we can get the estimate
\begin{equation}\label{estimate_mechtime1}
\begin{aligned}
&\|\partial_tU_{\mathrm{err}}^\varepsilon\|_{L^2(\Omega)^{3}}
 +\|\nabla\partial_t U_{\mathrm{cor}}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}
 +\varepsilon\|\nabla\partial_t U_\mathrm{cor}^\varepsilon\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}\\
&\leq C\left(\sqrt{\varepsilon}+\varepsilon+\|\Theta_{\mathrm{err}}\|_{L^2(\Omega)}
 +\|\partial_t\Theta_{\mathrm{err}}\|_{L^2(\Omega)}\right).
\end{aligned}
\end{equation}
Since this is done quite analogously to the estimates leading to 
inequality~\eqref{estimate_mech}, except for a few terms arising due to 
the time differentiation, the details of this are omitted.

If we take $\partial_t U_{\mathrm{cor}^\varepsilon}$ as a test function and follow the
 same strategy as in~\eqref{estimate_i1}, we can estimate
\begin{equation}\label{estimate_i1time}
\begin{aligned}
&\int_0^tI_1^\varepsilon(\tau,\partial_tU_{\mathrm{cor}0}^\varepsilon)\,\mathrm{d}{\tau}
-\int_0^t\int_{\Omega_A^\varepsilon}\big[\mathcal{C}_A^{r}e_y(\tau^u)\big]_\varepsilon\theta_A
:e(\partial_tU_{\mathrm{cor}0}^\varepsilon)\,\mathrm{d}{x}\,\mathrm{d}{\tau}\\
&\geq C_1\left(\|e(U_\mathrm{cor}^\varepsilon)(t)\|^2_{L^2(\Omega_A^\varepsilon)}
-\|e(U_\mathrm{cor}^\varepsilon)(0)\|^2_{L^2(\Omega_A^\varepsilon)}\right) \\
&\quad -C_2\int_0^t\|U_{\mathrm{cor}}^\varepsilon\|^2_{H^1(\Omega_A^\varepsilon)^3}\,\mathrm{d}{\tau}-C_3(\varepsilon+\varepsilon^2),
\end{aligned}
\end{equation}
where integration by parts w.r.t.~time was done.
Similarly, we obtain
\begin{equation}\label{estimate_i2time}
\begin{aligned}
&\int_0^tI_2^\varepsilon(\tau,\partial_tU_{\mathrm{cor}0}^\varepsilon)\,\mathrm{d}{\tau}\\
&\geq C_1\varepsilon^2\left(\|e(U_\mathrm{cor}^\varepsilon)(t)\|^2_{L^2(\Omega_B^\varepsilon)^{3\times3}}
-\|e(U_\mathrm{cor}^\varepsilon)(0)\|^2_{L^2(\Omega_B^\varepsilon)^{3\times3}}\right)\\
&\quad -C_2\varepsilon^2\int_0^t\|e(U_{\mathrm{cor}}^\varepsilon)\|^2_{L^2(\Omega_B^\varepsilon)^{3\times3}}
\,\mathrm{d}{\tau}-C_3\varepsilon^2.
\end{aligned}
\end{equation}
Moreover, 
\begin{equation} \label{estimate_i5time}
\sum_{i=5}^9\left|I_i^\varepsilon(t,\partial_tU_{\mathrm{cor}0}^\varepsilon)\right|
\leq C\varepsilon\left(\|U_\mathrm{cor}^\varepsilon\|_{H^1(\Omega_A^\varepsilon)^3}
+\|U_\mathrm{cor}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^3}\right).
\end{equation}

\subsection{Estimates for the heat conduction equations}\label{subsec:heat}
Now we go on with establishing some control on the error terms in the heat
 conduction equations.
Multiplying equations~\eqref{corrector_eq:ha} and~\eqref{corrector_eq:hb}
with test functions $\varphi\in H_0^1(\Omega)$, integrating over $\Omega$, 
and then integrating by parts while using the interface conditions, we are lead to
\begin{align*}
&\underbrace{\int_{\Omega_A^\varepsilon}\partial_t\left(c_A^{r,\varepsilon}\Theta_A^\varepsilon
-\kappa_A\int_{Y_A}c^{r}\,\mathrm{d}{y}\,\theta_A\right)\varphi\,\mathrm{d}{x}}_{=:E_1^\varepsilon(t,\varphi)}
+\underbrace{\int_{\Omega_A^\varepsilon}\left(K_A^{r,\varepsilon}\nabla\Theta_A^\varepsilon
 -\kappa_AK_A^{h}\nabla\theta_A\right)\cdot\nabla\varphi\,\mathrm{d}{x}}_{=:E_2^\varepsilon(t,\varphi)}\\
&+\underbrace{\int_{\Omega_A^\varepsilon}\left(\left(c_A^{r,\varepsilon}\Theta_A^\varepsilon
 +\gamma_A^{r,\varepsilon}:\nabla U_A^\varepsilon\right)v^{r,\varepsilon}\right)
 \cdot\nabla\varphi\,\mathrm{d}{x}}_{=:E_3^\varepsilon(t,\varphi)}\\
&+\underbrace{\int_{\Omega_A^\varepsilon}\left(K_A^{r,\varepsilon}\nabla\Theta_A^\varepsilon
 -\kappa_AK_A^{h}\nabla\theta_A\right)\cdot\nabla\varphi\,\mathrm{d}{x}
-\int_{\Gamma^\varepsilon}\kappa_AK_A^{h}\nabla\theta_A\cdot n^\varepsilon\varphi\,\mathrm{d}{x}}_{=:E_4^\varepsilon(t,\varphi)}\\
&+\underbrace{\int_{\Omega_B^\varepsilon}\left(\partial_t\left(c_B^{r,\varepsilon}\Theta_B^\varepsilon\right)
 -\big[\partial_t(c_B^{r}\Theta_B)\big]_\varepsilon\right)\varphi\,\mathrm{d}{x}}_{=:E_6^\varepsilon(t,\varphi)}\\
&+\underbrace{\int_{\Omega_B^\varepsilon}\left(\partial_t\left(\varepsilon\gamma_B^{r,\varepsilon}
 :\nabla U_B^\varepsilon\right)-\Big[\partial_t(\gamma_B^{r}:\nabla_y U_B)\Big]_\varepsilon\right)
\varphi\,\mathrm{d}{x}}_{=:E_6^\varepsilon(t,\varphi)}\\
&+\underbrace{\int_{\Omega_B^\varepsilon}\left(c_B^{r,\varepsilon}\Theta_B^\varepsilon v^{r,\varepsilon}
 -\big[c_B^{r}\Theta_B\big]_\varepsilon\big[v^{r}\big]_\varepsilon\right)\cdot\nabla\varphi\,\mathrm{d}{x}}_{=:E_7^\varepsilon(t,\varphi)}\\
&+\underbrace{\varepsilon\int_{\Omega_B^\varepsilon}\left(\gamma_B^{r,\varepsilon}:\nabla U_B^\varepsilon v^{r,\varepsilon}
 -\big[\gamma^{r}_B\big]_\varepsilon:\nabla[U_B]_\varepsilon [v^{r}]_\varepsilon\right)
 \cdot\nabla\varphi\,\mathrm{d}{x}}_{=:E_8^\varepsilon(t,\varphi)}\\
&+\underbrace{\varepsilon^2\int_{\Omega_B^\varepsilon}\left(K_B^{r,\varepsilon}\nabla\Theta_B^\varepsilon
 -\big[K_B^{r}\big]_\varepsilon\nabla\big[\Theta_B\big]_\varepsilon\right)\cdot\nabla\varphi\,\mathrm{d}{x}}
 _{=:E_9^\varepsilon(t,\varphi)}\\
&=\underbrace{-\int_{\Gamma^\varepsilon}W_\Gamma^{r,\varepsilon}\varphi\,\mathrm{d}{\sigma}
 +\int_{\Omega_A^\varepsilon}\kappa_AW_{\Gamma}^{h}\varphi\,\mathrm{d}{x}}_{=:E_{10}^\varepsilon(t,\varphi)} \\
&\quad +\underbrace{\int_{\Omega_A^\varepsilon}\kappa_A\int_{\Gamma}K_B^{r}
 \nabla_y\Theta_B\cdot n\,\mathrm{d}{s}\,\varphi\,\mathrm{d}{x}
 -\varepsilon^2\int_{\Gamma^\varepsilon}[K_B^{r}]_\varepsilon\nabla[\Theta_B]_\varepsilon\cdot
  n^\varepsilon\varphi\,\mathrm{d}{\sigma}}_{=:E_{11}^\varepsilon(t,\varphi)}\\
&\quad +\underbrace{\int_{\Omega_A^\varepsilon}\left(f_{\theta_A}^{r,\varepsilon}
 -\kappa_A\int_{Y_A}f_{\theta_A}^{r}\,\mathrm{d}{y}\right)\varphi\,\mathrm{d}{x}
 +\int_{\Omega_B^\varepsilon}\left(f_{\theta_B}^{r,\varepsilon}
 -\big[f_{\theta_B}^{r}\big]_\varepsilon\right)
\varphi\,\mathrm{d}{x}}_{=:E_{12}^\varepsilon(t,\varphi)}+E_{13}^\varepsilon(t,\varphi)
\end{align*}
where
$$
E_{13}^\varepsilon(t,\varphi)=\varepsilon\int_{\Omega_B^\varepsilon}\left(\Big[\operatorname{div}_x\left(K_B^{r}
\nabla\Theta_B\right)\Big]_\varepsilon+\operatorname{div}\big[K_B^{r}\nabla_x\Theta_B
\big]_\varepsilon\right)\varphi\,\mathrm{d}{x}.
$$
For the first term, we see that
\begin{align*}
E_1^\varepsilon(t,\Theta_{\mathrm{cor}0}^\varepsilon)
&=\int_{\Omega_A^\varepsilon}\rho_Ac_{A}\partial_t\Big(\Theta_A^\varepsilon(J^\varepsilon
 -\frac{1}{|Y_A(0)|}\left|Y_A\right|)\Big)\Theta_{\mathrm{cor}0}^\varepsilon\,\mathrm{d}{x}\\
&\quad  +\int_{\Omega_A^\varepsilon}\rho_Ac_{A}\frac{1}{|Y_A(0)|}
 \partial_t\left(\left|Y_A\right|\Theta_\mathrm{err}^\varepsilon\right)\Theta_\mathrm{err}^\varepsilon\,\mathrm{d}{x}
 +R_2^\varepsilon(t),
\end{align*}
where
\begin{equation*}
R_2^\varepsilon(t)=\varepsilon\int_{\Omega_A^\varepsilon}\rho_Ac_{A}\frac{1}{|Y_A(0)|}\partial_t
 \left(\left|Y_A\right|\Theta_\mathrm{err}^\varepsilon\right)m^\varepsilon\big[\widetilde{\Theta}\big]_\varepsilon.
\end{equation*}
Using the regularity estimates for $\widetilde{\Theta}$ and $\Theta_A^\varepsilon$, 
it is easy to see that there is a constant $c>0$ independent of $\varepsilon$ such 
that (for every $\delta>0$)
$$
\big|\int_0^t R_2^\varepsilon(\tau)\,\mathrm{d}{\tau}\big|
\leq \int_0^t\|\Theta_\mathrm{err}^\varepsilon(\tau)\|^2\,\mathrm{d}{\tau}
+\delta\left(\|\Theta_\mathrm{err}^\varepsilon(t)\|^2-\|\Theta_\mathrm{err}^\varepsilon(0)\|^2\right)+C_\delta t\varepsilon^2.
$$
With this estimate and Lemma~\ref{lemma:average}, we then get
\begin{equation}\label{estimate:E1}
\begin{aligned}
&\int_0^tE_1^\varepsilon(\tau,\Theta_{\mathrm{cor}0}^\varepsilon)\,\mathrm{d}{\tau}\\
& \geq C_1\left(\|\Theta_\mathrm{err}^\varepsilon(t)\|_{\Omega_A^\varepsilon}^2
 -\|\Theta_\mathrm{err}^\varepsilon(0)\|_{\Omega_A^\varepsilon}^2\right) \\
&\quad -C_2\Big(\int_0^t\|\Theta_\mathrm{err}^\varepsilon(\tau)\|_{\Omega_A^\varepsilon}^2
 +\varepsilon\|\Theta_{\mathrm{cor}}^\varepsilon(\tau)\|_{H^1(\Omega_A^\varepsilon)}\,\mathrm{d}{\tau}+t\varepsilon^2\Big).
\end{aligned}
\end{equation}
For the dissipation term of $\Omega_A^\varepsilon$, namely $E_2^\varepsilon$, we start by noticing 
that
\[
\kappa_A\gamma_A^{h}:\nabla u_A+\kappa_A\int_{Y_A}\gamma_A^{r}
:\nabla_y\tau^u\,\mathrm{d}{y}\,\theta_A=\kappa_A\int_{Y_A}\gamma_A^{r}
:\left(\nabla u_A+\nabla_y\widetilde{U}\right)\,\mathrm{d}{y}
\]
and decompose
\begin{align*}
E_2^\varepsilon(t,\varphi)
&=\int_{\Omega_A^\varepsilon}\partial_t\left(\gamma_A^{r,\varepsilon}:\nabla U_\mathrm{cor}^\varepsilon\right)\varphi\,\mathrm{d}{x}
 +\int_{\Omega_A^\varepsilon}\partial_t\left(\left(\gamma_A^{r,\varepsilon}
-\big[\gamma_A^{r}\big]_\varepsilon\right):\nabla(U_A^\varepsilon-U_\mathrm{cor}^\varepsilon)\right)\varphi\,\mathrm{d}{x}\\
&\quad+\int_{\Omega_A^\varepsilon}\partial_t\Big(\big[\gamma_A^{r}\big]_\varepsilon:
\nabla(U_A^\varepsilon-U_\mathrm{cor}^\varepsilon)-\kappa_A\int_{Y_A}\gamma_A^{r}:
\left(\nabla u_A+\nabla_y\widetilde{U}\right)\,\mathrm{d}{y}\Big)\varphi\,\mathrm{d}{x}.
\end{align*}
Applying Lemma~\ref{lemma:average} to
$$
f=\partial_t\Big(\gamma_A^{r}:\left(\nabla U_A+\nabla_y\widetilde{U}\right)
-\kappa_A\int_{Y_A}\gamma_A^{r}:\left(\nabla u_A+\nabla_y\widetilde{U}\right)
\,\mathrm{d}{y}\Big),
$$
leads to
\[
|E_2^\varepsilon(t,\varphi)|\leq C\Big(\Big(\|\nabla U_\mathrm{cor}\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}
+\|\nabla\partial_tU_\mathrm{cor}\|_{L^2(\Omega_A^\varepsilon)^{3\times3}}
\Big)\|\varphi\|_{L^2(\Omega_A^\varepsilon)}+\varepsilon\|\varphi\|_{H^1(\Omega_A^\varepsilon)}\big).
\]
In the case of $E_3^\varepsilon$, the estimate  $\varepsilon^{-1}\|v^\varepsilon\|_{L^\infty}(\Omega)\leq C$ 
(see \eqref{s:estimate_movement}) implies
\begin{align}\label{estimate:E3}
\left|E_3^\varepsilon(\tau,\varphi)\right|\leq C\varepsilon\|\nabla\varphi\|_{L^2(\Omega_A^\varepsilon)}
\quad\text{for all } \varphi\in H^1(\Omega).
\end{align}
For handling the heat conduction functional,
\begin{align*}
E_4^\varepsilon(t,\varphi)
&=\int_{\Omega_A^\varepsilon}K_A^{r,\varepsilon}\nabla\Theta_\mathrm{cor}^\varepsilon\cdot\nabla\varphi\,\mathrm{d}{x}\\
&\quad +\int_{\Omega_A^\varepsilon}\Big(K_A^{r,\varepsilon}\nabla(\Theta_A^\varepsilon-\Theta_\mathrm{cor}^\varepsilon)
-\frac{1}{|Y_A(0)|}K_A^{h}\nabla\Theta_A\Big)\cdot\nabla\varphi\,\mathrm{d}{x}\\
&\quad -\int_{\Gamma^\varepsilon}\frac{1}{|Y_A(0)|}K_A^{h}\nabla\Theta_A\cdot n^\varepsilon\varphi\,\mathrm{d}{x},
\end{align*}
the strategy is exactly the same as with dealing with the $I_1^\varepsilon$-estimate 
of the mechanical part, see estimate~\eqref{estimate_i1}, which then leads to
\begin{equation}\label{estimate:E4}
E_4^\varepsilon(t,\Theta_{\mathrm{cor}0}^\varepsilon)\geq C_1\|\nabla\Theta_\mathrm{cor}^\varepsilon\|^2_{L^2(\Omega_A^\varepsilon)}
-C_2\varepsilon(\|\Theta_{\mathrm{cor}0}^\varepsilon\|_{H^1(\Omega_A^\varepsilon)}+1+\varepsilon).
\end{equation}
Now, turning our attention to the next to functionals, $E_5^\varepsilon$, it follows 
easily from Lemma~\ref{lemma:estimates_trafo} that
\begin{equation} \label{estimate:E5}
\begin{aligned}
\int_0^tE_5^\varepsilon(\tau,\Theta_{\mathrm{cor}0}^\varepsilon)\,\mathrm{d}{\tau}
&\geq C_1\Big(\|\Theta_\mathrm{err}^\varepsilon(t)\|^2_{L^2(\Omega_B^\varepsilon)}
 -\|\Theta_\mathrm{err}^\varepsilon(0)\|_{L^2(\Omega_B^\varepsilon)}^2\Big) \\
&\quad -\int_0^t\|\Theta_\mathrm{err}^\varepsilon(\tau)\|_{L^2(\Omega_B^\varepsilon)}^2\,\mathrm{d}{\tau}-\varepsilon^2.
\end{aligned}
\end{equation}
Estimates for the dissipation error terms, $E_5^\varepsilon$-$E_7^\varepsilon$, are given by
\begin{gather}
|E_6^\varepsilon(t,\varphi)|\leq C\varepsilon\|\varphi\|_{L^2(\Omega_B^\varepsilon)}
 \Big(\|\nabla U_\mathrm{err}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}
 +\|\nabla\partial_tU_\mathrm{err}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}+\varepsilon\Big),
 \label{estimate:E6}\\
|E_7^\varepsilon(t,\varphi)|\leq C\varepsilon\|\nabla\varphi\|_{L^2(\Omega_B^\varepsilon)}
 \Big(\|\Theta_\mathrm{err}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)}+\varepsilon\Big),\label{estimate:E7}\\
|E_8^\varepsilon(t,\varphi)|\leq C\varepsilon^2\|\nabla\varphi\|_{L^2(\Omega_B^\varepsilon)}
 \Big(\|\nabla U_\mathrm{err}^\varepsilon\|_{L^2(\Omega_B^\varepsilon)^{3\times3}}+\varepsilon\Big)
\label{estimate:E8}.
\end{gather}
Similarly, we obtain
\begin{equation} \label{estimate:E9}
E_9^\varepsilon(t,\Theta_{\mathrm{cor}0}^\varepsilon)
\geq C_1\varepsilon^2\|\nabla\Theta_\mathrm{err}^\varepsilon\|^2_{L^2(\Omega_B^\varepsilon)}-C_2\varepsilon^2.
\end{equation}
Taking a look at the interface velocity terms, we obtain
\[
|E_{10}^\varepsilon(t,\varphi)|
\leq\Big|\int_{\Gamma^\varepsilon}\left(W_\Gamma^{r,\varepsilon}
 -\varepsilon\big[W_\Gamma^{r}\big]_\varepsilon\varphi\,\mathrm{d}{\sigma}\right)\Big|
+\Big|\int_{\Gamma^\varepsilon}\varepsilon\big[W_\Gamma^{r}\big]_\varepsilon\varphi\,\mathrm{d}{\sigma}
+\int_{\Omega_A^\varepsilon}\kappa_AW_{\Gamma}^{h}\varphi\,\mathrm{d}{x}\Big|.
\]
Using Lemma~\ref{lemma:estimates_trafo} for the functional $E_{11}^\varepsilon$,
 cf. \cite{MV13}, the estimates on the functions that are involved, and our
assumptions on the data, it is straightforward to show
\begin{equation} \label{estimate:E10}
|E_{10}^\varepsilon(t,\varphi)|+|E_{11}^\varepsilon(t,\varphi)|+|E_{12}^\varepsilon(t,\varphi)|
\leq C\varepsilon\big(\|p\|_{H^1(\Omega_A^\varepsilon)}+\|\varphi\|_{L^2(\Omega_B^\varepsilon)}\big).
\end{equation}
Finally, for the functional $E_{13}^\varepsilon$ catching some of the terms arising in
the elliptic part for $\Theta_{\mathrm{err}}$, we obtain
\begin{equation} \label{estimate:E12}
|E_{13}^\varepsilon(t,\Theta_{\mathrm{cor}0})|\leq C\varepsilon\left(\|\Theta_{\mathrm{err}}\|+\varepsilon\right).
\end{equation}
Summarizing those estimates~\eqref{estimate:E1}-\eqref{estimate:E12}
 and using Young's and Gronwall's inequalities, we arrive at
\begin{equation}\label{estimate_heat}
\begin{aligned}
&\|\Theta_\mathrm{err}^\varepsilon\|_{L^\infty(S\times\Omega)}
 +\|\nabla\Theta_\mathrm{cor}^\varepsilon\|_{L^2(S\times\Omega_A^\varepsilon)^3}
 +\varepsilon\|\nabla\Theta_\mathrm{err}\|_{L^2(S\times\Omega_B^\varepsilon)^3}\\
&\leq C\big(\|\nabla U_\mathrm{cor}\|_{L^2(S\times\Omega_A^\varepsilon)^{3\times3}}
 +\varepsilon\|\nabla U_\mathrm{err}\|_{L^2(S\times\Omega_B^\varepsilon)^{3\times3}}\\
&\quad +\|\nabla \partial_tU_\mathrm{cor}\|_{L^2(S\times\Omega_A^\varepsilon)^{3\times3}}
+\varepsilon\|\nabla \partial_tU_\mathrm{err}\|_{L^2(S\times\Omega_B^\varepsilon)^{3\times3}}
+\sqrt{\varepsilon}+\varepsilon\big).
\end{aligned}
\end{equation}

\begin{remark}\label{rem:heat} \rm
With~\eqref{estimate_heat} at hand, we conclude that estimates for 
$\nabla U_\mathrm{cor}^\varepsilon$ and $\nabla\partial_tU_\mathrm{cor}^\varepsilon$ will also lead to
 corresponding corrector estimates for the heat part.
\end{remark}

\subsection{Overall estimates}\label{subsec:overall}
Here, we combine the estimates from the preceding sections, 
Section~\ref{subsec:momentum} and Section~\ref{subsec:heat}.
It is clear that the following statement now follows directly from 
estimates~\eqref{estimate_mech} and~\eqref{estimate_heat}.

\begin{theorem}[Corrector for Weakly Coupled Problem]\label{cor1}
If we reduce our problem to a weakly coupled problem, that is, if 
we assume either $\alpha_A=\alpha_B=0$ (together with {\rm (A1)}) or 
$\gamma_A=\gamma_B=0$ (together with {\rm (A2)}), we have the following 
corrector estimate
\begin{align*}
&\|\Theta_\mathrm{err}^\varepsilon\|_{L^\infty(S\times\Omega)}
 +\|U_{\mathrm{err}}^\varepsilon\|_{L^\infty(S;L^2(\Omega)^{3}}
 +\|\nabla\Theta_\mathrm{cor}^\varepsilon\|_{L^2(S\times\Omega_A^\varepsilon)^3}\\
&+\|\nabla U_{\mathrm{cor}}^\varepsilon\|_{L^\infty(S;L^2(\Omega_A^\varepsilon))^{3\times3}}
 +\varepsilon\|\nabla\Theta_\mathrm{err}\|_{L^2(S\times\Omega_B^\varepsilon)^3}
 +\varepsilon\|\nabla U_\mathrm{cor}^\varepsilon\|_{L^\infty(S;L^2(\Omega_B^\varepsilon))^{3\times3}} \\
&\leq C (\sqrt{\varepsilon}+\varepsilon).
\end{align*}
\end{theorem}

Moreover, for the heat part, we take $\Theta_{\mathrm{cor}0}$ and for the mechanical 
part $\partial_t U_{\mathrm{cor}0}^\varepsilon$ as a test function, sum the weak formulations, 
integrate over $(0,t)$ and obtain
\begin{equation}
\begin{aligned}
&\int_0^t\Big(\sum_{i=1}^6I_i^\varepsilon(\tau,\partial_tU_{\mathrm{cor}0}^\varepsilon)
 +\sum_{i=1}^9E_i^\varepsilon(\tau,\Theta_{\mathrm{cor}0}^\varepsilon) \Big)\,\mathrm{d}{\tau}\\
&=\int_0^t\Big(\sum_{i=7}^9I_i^\varepsilon(\tau,\partial_tU_{\mathrm{cor}0}^\varepsilon)
 +\sum_{i=10}^{13}E_i^\varepsilon(\tau,\Theta_{\mathrm{cor}0}^\varepsilon) \Big)\,\mathrm{d}{\tau}.
\end{aligned}
\end{equation}
Now, we first take a view on the error terms corresponding to the coupling 
terms for the $\Omega_B^\varepsilon$ part for both the mechanical and the heat part, 
namely $I_4^\varepsilon$, $E_6^\varepsilon$, and $E_8^\varepsilon$.
While $E_8^\varepsilon$ can be controlled in terms of $\varepsilon\nabla U_\mathrm{err}$ and 
$\varepsilon\nabla\Theta_\mathrm{err}$ (see inequality~\eqref{estimate:E8}), this is not 
possible for either $I_4^\varepsilon$ or $E_6^\varepsilon$ due to the involved time derivatives.
If we take a look at the sum of those (appropriately 
scaled, assuming $\alpha_B\neq0$) two terms, however, we see 
that they counterbalance each other leading to
\[
\big|\frac{\gamma_B}{\alpha_B}I_4^\varepsilon(\tau,\partial_tU_{\mathrm{cor}0}^\varepsilon)
+E_6^\varepsilon(\tau,\Theta_{\mathrm{cor}0}^\varepsilon)\big|
\leq C\|\Theta_\mathrm{err}^\varepsilon\|^2_{L^2(\Omega_B^\varepsilon)}
+\varepsilon^2\|\nabla U_\mathrm{cor}\|^2_{L^2(\Omega_B^\varepsilon)}+\varepsilon^2.
\]
Note that with estimate~\eqref{estimate_i2time}, the 
$\varepsilon^2\|\nabla U_\mathrm{cor}\|^2_{L^2(\Omega_B^\varepsilon)}$ is resolvable via Gronwall's inequality.

This, unfortunately, does not work for the coupling parts in $\Omega_A^\varepsilon$:
 Here, we would have to apply Lemma~\ref{lemma:average} at the cost 
of additional derivatives (we only get control in $H^1$ and not in $L^2$),
 which, in general, can not be compensated without structural assumptions.

As a result of this observation and the estimates collected in the previous 
sections, we obtain:

\begin{theorem}[Corrector for microscale coupled problem]\label{cor2}
If we simplify our pro\-blem so that there is only coupling in the
 $\Omega_B^\varepsilon$ part, that is, if we assume $\alpha_A=\gamma_A=0$, 
we have the  corrector estimate
\begin{align*}
&\|\Theta_\mathrm{err}^\varepsilon\|_{L^\infty(S\times\Omega)}
 +\|U_{\mathrm{err}}^\varepsilon\|_{L^\infty(S;L^2(\Omega)^{3}}
 +\|\nabla\Theta_\mathrm{cor}^\varepsilon\|_{L^2(S\times\Omega_A^\varepsilon)^3}
 +\|\nabla U_{\mathrm{cor}}^\varepsilon\|_{L^\infty(S;L^2(\Omega_A^\varepsilon))^{3\times3}}\\
&+\varepsilon\|\nabla\Theta_\mathrm{err}\|_{L^2(S\times\Omega_B^\varepsilon)^3}
+\varepsilon\|\nabla U_\mathrm{cor}^\varepsilon\|_{L^\infty(S;L^2(\Omega_B^\varepsilon))^{3\times3}}
\leq C (\sqrt{\varepsilon}+\varepsilon).
\end{align*}
\end{theorem}


\subsection*{Acknowledgments}
The authors are indebted to Michael B\"ohm (Bremen) for initiating and 
supporting this research.
AM thanks NWO MPE ``Theoretical estimates of heat losses in geothermal wells'' 
(grant nr. 657.014.004) for funding.

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\end{document}