text:
\begin{document}
\title{On a linearized Mullins-Sekerka/Stokes system for two-phase flows}

\author{Helmut Abels\thanks{  \textit{Fakult\"at f\"ur Mathematik,   Universit\"at Regensburg,   93040 Regensburg,   Germany}   \textsf {helmut.abels@ur.de} } \  and Andreas Marquardt\thanks{\textit   {Fakult\"at f\"ur Mathematik,   Universit\"at Regensburg,   93040 Regensburg,   Germany}  } }
\maketitle
\begin{abstract}
   We study a linearized Mullins-Sekerka/Stokes system in a bounded domain with various boundary conditions. This system plays an important role to prove the convergence of a Stokes/Cahn-Hilliard system to its sharp interface limit, which is a Stokes/Mullins-Sekerka system, and to prove solvability of the latter system locally in time. We prove solvability of the linearized system in suitable $L^2$-Sobolev spaces with the aid of a maximal regularity result for non-autonomous abstract linear evolution equations.
\end{abstract}
{\small\noindent
{\textbf {Mathematics Subject Classification (2000):}}
Primary: 76T99; Secondary: 35Q30,
35Q35,
35R35,
76D05,
76D45.}\\
{\textbf {Key words:}} {Two-phase flow, sharp interface limit, Cahn-Hilliard equation, Free boundary problems, Mullins-Sekerka equation,}

\section{Introduction}
We study the following linearized Mullins-Sekerka/Stokes system
\begin{align*}
D_{t,\Gamma}h+\mathbf{b}\cdot\nabla_{\Gamma}h-bh+\tfrac{1}{2}X_{0}^{*}\big((\mathbf{v}^{+}+\mathbf{v}^{-})\cdot\mathbf{n}_{\Gamma_{t}}\big)+\tfrac{1}{2}X_{0}^{*}\big(\big[\partial_{\mathbf{n}_{\Gamma_{t}}}\mu\big]\big) & =g &  & \text{on }\Sigma\times\left(0,T\right),\\
h\left(.,0\right) & =h_{0} &  & \text{in }\Sigma,
\end{align*}
where for every $t\in\left[0,T\right]$, the functions {$\mathbf{v}^{\pm}=\mathbf{v}^{\pm}(x,t)$},
$p^{\pm}=p^{\pm}(x,t)$ and $\mu^{\pm}=\mu^{\pm}(x,t)$
for $(x,t)\in\Omega_{T}^{\pm}$ with $\mathbf{v}^{\pm}\in H^{2}(\Omega^{\pm}(t))^d$,
$p^{\pm}\in H^{1}(\Omega^{\pm}(t))$ and $\mu^{\pm}\in H^{2}(\Omega^{\pm}(t))$
are the unique solutions to
\begin{align}
\Delta\mu^{\pm} & =a_{1} &  & \text{in }\Omega^{\pm}(t),\label{eq:laplmu}\\
  \mu^{\pm} & = X_{0}^{*,-1}\big(\sigma\Delta_{\Gamma}h\pm a_{2}h\big)+a_{3} &  & \text{on }\Gamma_{t},\\
  \no\cdot \nabla \mu^{-} & =a_{4} &  & \text{on }\Gmu{1},\\
  \mu^{-} & =a_{4} &  & \text{on }\Gmu{2},\\
-\Delta\mathbf{v}^{\pm}+\nabla p^{\pm} & =\mathbf{a}_{1} &  & \text{in }\Omega^{\pm}(t),\label{eq:stokes}\\
\operatorname{div}\mathbf{v}^{\pm} & =0 &  & \text{in }\Omega^{\pm}(t),\\
[\mathbf{v}] & =\mathbf{a}_{2} &  & \text{on }\Gamma_{t},\label{eq:stokesjmp}\\
\left[2D_{s}\mathbf{v}-p\mathbf{I}\right]\mathbf{n}_{\Gamma_{t}} & =X_{0}^{*,-1}\big(\mathbf{a}_{3}h+\mathbf{a}_{4}\Delta_{\Gamma}h+a_{5}\nabla_{\Gamma}h+\mathbf{a}_{5}\big) &  & \text{on }\Gamma_{t},\label{eq:stokesbdry}\\
B_j(\mathbf{v}^{-},p^{-})& =\mathbf{a}_{6} &  & \text{on }\Gv{j}, j=1,2,3.\label{eq:stokesoutbdry}
\end{align}
Here $\Omega\subseteq \R^d$, $d=2,3$, is a bounded domain with smooth boundary, which is the disjoint union of $\Omega^+(t)$, $\Omega^-(t)$ and $\Gamma_t$, where $\Gamma_t=\partial\Omega^+(t)$ is a smoothly evolving $(d-1)$-dimensional orientable hypersurface. We assume that $\Gamma_t\subseteq\Omega$ for all $t\in (0,T)$, i.e., there is no boundary contact and contact angle. Moreover, $\Gamma_t$ is given for $t\in [0,T]$ as well as $a_1,\ldots, a_4$, $\mathbf{a}_1,\ldots, \mathbf{a}_6$ are given for some $T>0$, $\sigma>0$ is the surface tension constant and $D_s\mathbf{v}= \frac12(\nabla \mathbf{v}+\nabla \mathbf{v}^T)$. Furthermore,
\begin{align*}
\left[g\right]\left(p,t\right) & :=\lim_{h\searrow0}\left[g^+(p+\mathbf{n}_{\Gamma_{t}}(p)h,t)-g^-(p-\mathbf{n}_{\Gamma_{t}}(p)h,t)\right]\text{ for }p\in\Gamma_{t}
\end{align*}
for suitable functions $g^\pm$ and $X_0\colon \Sigma\times [0,T]\to \Gamma:=\bigcup_{t\in [0,T]} \Gamma_t\times \{t\}$ is a suitable diffeomorphism, which is described in Section~\ref{sec:Prelim} below. For $a\colon \Sigma\times [0,T]\to \R^N$, $N\in\N$, we define $X^{\ast,-1}_0 a\colon \Gamma \to \R^N$ by
\begin{equation*}
  (X^{\ast,-1}_0 a)(p,t)= a(X_0^{-1}(p,t))\qquad \text{for all }(p,t)\in \Gamma
\end{equation*}
and for $b\colon \Gamma\to \R^N$, $N\in\N$, we define $X^{\ast}_0 b\colon \Sigma\times [0,T] \to \R^N$ by
\begin{equation*}
  (X^{\ast}_0 b)(s,t)= b(X_0(s,t))\qquad \text{for all }(s,t)\in \Sigma\times [0,T].
\end{equation*}

This system arises in the construction of approximate solutions in the proof of convergence of a Stokes/Cahn-Hilliard system to its sharp interface limit, which is a Stokes/Mullins-Sekerka system, cf.\ \cite{NSCH2}-\cite{NSCH1}. Here $\mathbf{v}^\pm\colon \bigcup_{t\in [0,T]}\Omega^\pm(t)\times \{t\}\to \R^d$ and $p^\pm\colon \bigcup_{t\in [0,T]}\Omega^\pm(t)\times \{t\}\to \R$ are the velocity and pressure incompressible viscous Newtonian fluids filling the domains $\Omega^\pm(t)$ at time $t$, which are separated by the (fluid) interface $\Gamma_t$. Furthermore, $h\colon \Sigma\times [0,T]\to \R$ is a linearized height function that describes the evolution of the interface at a certain order and $\mu^\pm\colon \bigcup_{t\in [0,T]}\Omega^\pm(t)\times \{t\}\to \R$ is a linearized chemical potential related to the fluids in $\Omega^\pm(t)$. If one neglects the terms related to $\mathbf{v}^\pm, p^\pm$, a similar linearized system arises in the study of the sharp interface limit of the Cahn-Hilliard equation, cf.\ \cite{abc}. Moreover, similar systems arise in the construction of strong solutions for a Navier-Stokes/Mullins-Sekerka system locally in time, cf. \cite{abelswilke}.

We consider different kinds of boundary conditions for $\mathbf{v}^-$ and $\mu^-$ simultaneously. More precisely, we assume that
\begin{equation*}
\partial\Omega= \Gmu{1}\cup\Gmu{2}= \Gv{1}\cup \Gv{2}\cup \Gv{3},
\end{equation*}
where $\Gmu{1}, \Gmu{2}$ and $\Gv{1}, \Gv{2},\Gv{3}$ are disjoint and closed. Moreover, we have
\begin{alignat*}{2}
  B_1(\bv^-,p^-)&= \bv^- &\quad&\text{on }\Gv{1}\\
  (B_2(\bv^-,p^-))_{\btau}&= \left(\left(2D_s \bv^- -p^-\right)\no_{\partial\Omega}\right)_\btau + \alpha_2 \bv_\btau^- &\quad&\text{on }\Gv{2}\\
  \no_{\partial\Omega}\cdot B_2(\bv^-,p^-)&= \no_{\partial\Omega}\cdot \bv^- &\quad&\text{on }\Gv{2}\\
  B_3(\bv^-,p^-)&= \left(2D_s \bv^- -p^-\right)\no_{\partial\Omega} + \alpha_3 \bv^- &\quad&\text{on }\Gv{3},
\end{alignat*}
where $\no_{\partial\Omega}$ denotes the exterior normal on $\partial\Omega$.
To avoid a non-trivial kernel in the following we assume that one of the following cases holds true:
\begin{equation*}
  |\Gv{1}|+ \alpha_2 |\Gv{2}|+\alpha_3|\Gv{3}|>0
\end{equation*}
Then Korn's inequality yields
\begin{equation}\label{eq:Korn}
  \|\bv\|_{H^1(\Omega)}\leq C \left(\|D_s\bv\|_{L^2(\Omega)} + \alpha_2 \|\bv_{\btau}\|_{L^2(\Gv{2})}+ \alpha_3 \|\bv\|_{L^2(\Gv{3})} \right)
\end{equation}
for all $\bv\in H^1(\Omega)^d$ with $\bv|_{\Gv{1}}=0$, $\no_{\partial\Omega}\cdot \bv|_{\Gv{2}}=0$, cf.\ \cite[Corollary 5.9]{korn}.

The structure of this contribution is as follows: In Section~\ref{sec:Prelim} we summarize some preliminaries on the parametrization of the interface $\Gamma_t$ and non-autonomuous evolution equations.
In Section~\ref{sec:Main} we present and prove our main results on existence and smoothness of solutions to the linearized Mullins-Sekerka system. Finally, in the appendix we prove an auxilliary result on the existence of a pressure.

The results of this paper are extensions of results in the second author's PhD Thesis.

\section{Preliminaries}\label{sec:Prelim}
\subsection{Notation}

Throughout this manuscript we denote by $\xi\in C^{\infty}\left(\mathbb{R}\right)$ a
cut-off function such that
\begin{equation}
\xi(s)=1\text{ if }\left|s\right|\leq\delta,\,\xi(s)=0\text{ if }\left|s\right|>2\delta,\text{ and }0\geq s\xi'(s)\geq-4\ \text{if }\delta\leq\left|s\right|\leq2\delta.\label{eq:cut-off}
\end{equation}

\subsection{Coordinates}\label{subsec:Coordinates}

We parametrize $(\Gamma_t)_{t\in[0,T_0]}$ with the aid of a family of smooth diffeomorphisms $X_0\colon \Sigma\times [0,T_0]\to \Gamma= \bigcup_{t\in [0,T_0]}\Gamma_t\times \{t\}$. Here either $\Sigma\subseteq \R^d$ is a smooth $(d-1)$-dimensional compact, orientable manifold without boundary, where $d\geq 2$ is allowed, or $d=2$ and $\Sigma=\mathbb{T}^1$. We have included the latter case to cover the setting in \cite{nsac,NSCH2,NSCH1}. Moreover, $\no_{\Gamma_t}(x)$ denotes the exterior normal of $\Gamma_t$ in $x$ with respect to $\Omega^-(t)$ and we denote
\begin{equation*}
\no (s,t):= \no_{\Gamma_t}((X_0(s,t))_1)\qquad \text{for all}~ s\in \Sigma, t\in [0,T_0],
\end{equation*}
where $(X_0(s,t))_1\in \R^d$ denote the spatial components of $X_0(s,t)$.
In the following we will need a tubular neighborhood of $\Gamma_t$: For $\delta>0$ sufficiently small,  the orthogonal projection $P_{\Gamma_t}(x)$ of all
\begin{equation*}
x\in \Gamma_t(3\delta) =\{y\in \Omega: \dist(y,\Gamma_t)<3\delta\}
\end{equation*}
is well-defined and smooth. Moreover, we choose $\delta$ so small that $\dist(\partial\Omega,\Gamma_t)>3\delta$ for every $t\in [0,T_0]$. Every $x\in\Gamma_t(3\delta)$ has a unique representation
\begin{equation*}
x=P_{\Gamma_t}(x)+r\no_{\Gamma_t}(P_{\Gamma_t}(x))
\end{equation*}
 where $r=\sdist(\Gamma_t,x)$. Here
\begin{equation*}
  d_{\G}(x,t):=\sdist (\Gamma_t,x)=
  \begin{cases}
    \dist(\Omega^-(t),x) &\text{if } x\not \in \Omega^-(t),\\
    -\dist(\Omega^+(t),x) &\text{if } x \in \Omega^-(t).
  \end{cases}
\end{equation*}

For the following we define for $\delta'\in (0,3\delta]$
\begin{equation*}
  \Gamma(\delta') =\bigcup_{t\in [0,T_0]} \Gamma_t(\delta') \times\{t\}.
\end{equation*}

We introduce new coordinates in  $\Gamma(3\delta)$ which we denote by
\begin{equation*}
  X\colon  (-3\delta, 3\delta)\times \Sigma \times [0,T_0]\mapsto \Gamma(3\delta)~\text{by}~  X(r,s,t):= X_0(s,t)+r\no(s,t),
\end{equation*}
where
\begin{equation*}
  r=\sdist(\Gamma_t,x), \qquad s= (X_{0}^{-1}(P_{\Gamma_t}(x),t))_1=: S(x,t),
\end{equation*}
where $(X_{0}^{-1}(P_{\Gamma_t}(x),t))_1$ denote the components in $\Sigma$ of $X_{0}^{-1}(P_{\Gamma_t}(x),t)$.

 In the case that $h$ is twice continuously differentiable with respect to $s$ and continuously differentiable with respect to $t$,
we introduce the notations
 \begin{align}
D_{t,\Gamma}h(s,t) & :=\left.\partial_t\left(h(S(x,t),t)\right)\right|_{x=X_0(s,t)},\quad
\nabla_{\Gamma}h(s,t) :=\left.\nabla\left(h(S(x,t),t)\right)\right|_{x=X_0(s,t)},\nonumber \\
\Delta_{\Gamma}h(s,t)& :=\left.\Delta\left(h(S(x,t),t)\right)\right|_{x=X_0(s,t)},\nonumber
 \end{align}
 where $\nabla$ and $\Delta$ act with respect to $x$.
We note that in the case that $d=2$ and $\Sigma=\mathbb{T}^1$ we have
 \begin{align}
D_{t,\Gamma}h(s,t) & =\left(\partial_{t}+\partial_{t}S(X_0(s,t))\cdot \partial_{s}\right)h(s,t),\nonumber \\
\nabla_{\Gamma}h(s,t)& =\nabla S\left(X_0(s,t)\right)\partial_{s}h(s,t),\nonumber \\
\Delta_{\Gamma}h(s,t)& =\Delta S\left(X_0(s,t)\right) \partial_{s}h(s,t)+|\nabla S\left(X_0(s,t)\right)|^2\partial_{s}^2h(s,t).\nonumber
 \end{align}
 as in \cite{nsac,NSCH2, NSCH1}.

\subsection{Maximal Regularity for Non-autonomous Equations}

In order to prove our main result we use of the theory of maximal
regularity for non-autonomous abstract evolution equations. Therefore, we give a short overview of the basic definitions
and results which we will use. These are taken from \cite{arendt}
and all the proofs of the statements can be found in that article.

In this subsection let $X$ and $D$ be two Banach spaces such that $D$
is continuously and densely embedded in $X$.
\begin{definition}[$L^{p}$-maximal regularity]
 Let $p\in\left(1,\infty\right).$
\begin{enumerate}
\item Let $A\in\mathcal{L}\left(D,X\right)$. Then $A$ has $L^{p}-$\emph{maximal
regularity} and we write $A\in\mathcal{MR}_{p}$ if for some bounded
interval $\left(t_{1},t_{2}\right)\subset\mathbb{R}$ and all $f\in L^{p}\left(t_{1},t_{2};X\right)$
there exists a unique $u\in W^{1,p}\left(t_{1},t_{2};X\right)\cap L^{p}\left(t_{1},t_{2};D\right)$
such that
\begin{align*}
\partial_{t}u+Au & =f\quad\text{a.e. on }\left(t_{1},t_{2}\right),\\
u\left(t_{1}\right) & =0.
\end{align*}
\item Let $T>0$ and $A:\left[0,T\right]\rightarrow\mathcal{L}\left(D,X\right)$
be a bounded and strongly measurable function. Then $A$ has $L^{p}$-\emph{maximal
regularity} and we write $A\in\mathcal{MR}_{p}\left(0,T\right)$ if
for all $f\in L^{p}\left(0,T;X\right)$ there exists a unique $u\in W^{1,p}\left(0,T;X\right)\cap L^{p}\left(0,T;D\right)$
such that
\begin{align*}
\partial_{t}u+A\left(t\right)u & =f\quad\text{a.e. on }\left(0,T\right),\\
u\left(0\right) & =0.
\end{align*}
\end{enumerate}
\end{definition}

It can be shown that if $A\in\mathcal{MR}_{p}$ for some $p\in\left(1,\infty\right)$
then $A\in\mathcal{MR}_{p}$ for all $p\in\left(1,\infty\right)$.
Hence, we often simply write $A\in\mathcal{MR}$.
\begin{definition}[Relative Continuity]~\\
 We say that $A:\left[0,T\right]\rightarrow\mathcal{L}\left(D,X\right)$
is \emph{relatively continuous} if for each $t\in\left[0,T\right]$
and all $\epsilon>0$ there exist $\delta>0$, $\eta\geq0$ such that
for all $x\in D$ and for all $s\in\left[0,T\right]$ with $\left|s-t\right|\leq\delta$
the inequality
\[
\left\Vert A(t)x-A(s)x\right\Vert _{X}\leq\epsilon\left\Vert x\right\Vert _{D}+\eta\left\Vert x\right\Vert _{X}
\]
holds.
\end{definition}

\begin{theorem}
\label{maxres} Let $T>0$ and $A:\left[0,T\right]\rightarrow\mathcal{L}\left(D,X\right)$
be a strongly measurable and relatively continuous function. If $A(t)\in\mathcal{MR}$
for all $t\in\left[0,T\right]$, then $A\in\mathcal{MR}_{p}\left(0,t\right)$
for every $0<t\leq T$ and every $p\in\left(1,\infty\right)$.
\end{theorem}

\begin{proof}
See \cite[Theorem 2.7]{arendt}.
\end{proof}
A very important tool for proving maximal regularity properties of
differential operators are perturbation techniques. Employing these can
often help to show maximal regularity for a variety of operators by
separating them into a main part (for which maximal regularity can
be readily shown) and a perturbation.

In the following we give a perturbation result which is key to many
results in the next chapter.
\begin{definition}[Relatively Close]~\\
\label{relclo} Let $Y$ be a Banach space such that
\[
D\hookrightarrow Y\hookrightarrow X.
\]
We say $Y$ \emph{is close to $X$ compared with $D$}, if for each
$\epsilon>0$ there exists $\eta\geq0$ such that
\[
\left\Vert x\right\Vert _{Y}\leq\epsilon\left\Vert x\right\Vert _{D}+\eta\left\Vert x\right\Vert _{X}\quad\text{for all }x\in D.
\]
\end{definition}

\begin{proposition}
\label{relres} Let $Y$ be as in Definition \ref{relclo} and let
the inclusion $D\hookrightarrow Y$ be compact. Then $Y$
is close to $X$ compared with $D$.
\end{proposition}

\begin{proof}
See \cite[Example 2.9 (d)]{arendt}.
\end{proof}
\begin{theorem}\label{thm:Perturbation}
\label{perres} Let $T>0$ and $Y$ be a Banach space that is close
to $X$ compared with $D$. Furthermore, let $A\colon \left[0,T\right]\rightarrow\mathcal{L}\left(D,X\right)$
be relatively continuous and $B\colon \left[0,T\right]\rightarrow\mathcal{L}\left(Y,X\right)$
be strongly measurable and bounded. If $A\left(t\right)\in\mathcal{MR}$
for every $t\in\left[0,T\right]$ then $A+B\in\mathcal{MR}_{p}\left(0,T\right)$.
\end{theorem}

\begin{proof}
See \cite[Theorem 2.11]{arendt}.
\end{proof}

\section{Main Results}\label{sec:Main}

\subsection{Parabolic Equations on Evolving Surfaces \label{sec:Parabolic-Equations-on}}

We introduce the space
\begin{eqnarray}
X_{T} & = & L^{2}\big(0,T;H^{\frac{7}{2}}(\Sigma)\big)\cap H^{1}\big(0,T;H^{\frac{1}{2}}(\Sigma)\big)\label{eq:XT}
\end{eqnarray}
for $T\in(0,\infty) $, where we equip
$X_{T}$ with the norm
\[
\left\Vert h\right\Vert _{X_{T}}=\left\Vert h\right\Vert _{L^{2}\big(0,T;H^{\frac{7}{2}}(\Sigma)\big)}+\left\Vert h\right\Vert _{H^{1}\big(0,T;H^{\frac{1}{2}}(\Sigma)\big)}+\left\Vert h|_{t=0}\right\Vert _{H^{2}(\Sigma)}.
\]

\begin{theorem}
\label{Max-Reg} Let $T\in\left(0,T_{0}\right]$. Let $\mathbf{b}\colon \Sigma\times[0,T]\rightarrow\mathbb{R}^d$
and $b_{1},b_{2}\colon \Sigma\times[0,T]\rightarrow\mathbb{R}$
be smooth given functions.
 For every $g\in L^{2}\big(0,T;H^{\frac{1}{2}}(\Sigma)\big)$
and $h_{0}\in H^{2}(\Sigma)$, there is a unique
solution $h\in X_{T}$ of
\begin{align}
D_{t,\Gamma}h+\mathbf{b}\cdot\nabla_{\Gamma}h-b_{1}h+X_{0}^{*}\big(\big[\partial_{\mathbf{n}_{\Gamma_{t}}}\mu\big]\big) & =g &  & \text{ on }\Sigma\times(0,T),\label{eq:hmax}\\
h\left(.,0\right) & =h_{0} &  & \text{ on }\Sigma,\nonumber
\end{align}
where $\mu|_{\Omega^{\pm}(t)}\in H^{2}(\Omega^{\pm}(t))$,
for $t\in[0,T]$, is determined by \begin{subequations}\label{eq:musystem}
\begin{align}
\Delta\mu^{\pm} & =0 &  & \text{in }\Omega^{\pm}(t),\label{eq:musystem1}\\
\mu^{\pm} & =X_{0}^{*,-1}(\sigma\Delta_{\Gamma}h\pm b_{2}h) &  & \text{on }\Gamma_{t},\label{eq:musystem2}\\
  \no_{\partial\Omega}\cdot \nabla \mu^{-} & =0 &  & \text{on }\Gmu{1},\\%\label{eq:musystem3}
  \mu^{-} & =0 &  & \text{on }\Gmu{2}.\label{eq:musystem3}
\end{align}
\end{subequations}Furthermore, the estimates
\begin{align}
\sum_{\pm}\left\Vert \mu^{\pm}\right\Vert _{L^{2}\left(0,T;H^{2}\left(\Omega^{\pm}(t)\right)\right)\cap L^{6}\left(0,T;H^{1}\left(\Omega^{\pm}(t)\right)\right)} & \leq C\left\Vert h\right\Vert _{X_{T}},\label{eq:mumaxab}
\end{align}
hold for some constant $C>0$ independent of $\mu$ and $h$.
\end{theorem}

\begin{proof}
We may write (\ref{eq:hmax}) in abstract form as
\begin{alignat*}{2}
\partial_{t}h+\mathcal{A}(t)h & =g &\qquad & \text{in }\Sigma\times [0,T],\\
h\left(.,0\right) & =h_{0} &  & \text{in }\Sigma,
\end{alignat*}
where $\mathcal{A}(t)\colon  H^{\frac{7}{2}}(\Sigma)\to  H^{\frac{1}{2}}(\Sigma)$ depends on  $t\in[0,T]$.
Now we fix $t_{0}\in [0,T]$ and analyze the operator $\mathcal{A}(t_{0})$,
where we replace $t$ with the fixed $t_{0}$ in all time dependent
coefficients.

In order to understand this operator we define
\begin{align*}
\mathfrak{D}_{t_{0}}\colon & H^{\frac{7}{2}}(\Sigma)\rightarrow H^{\frac{3}{2}}(\Gamma_{t_{0}})\colon  h\mapsto\big(X_{0}^{*,-1}(\sigma\Delta_{\Gamma}h)\big)(.,t_{0}),\\
S_{t_{0}}^{N}\colon & H^{\frac{3}{2}}(\Gamma_{t_{0}})\rightarrow H^{2}(\Omega^{+}(t_{0}))\times H^{2}(\Omega^{-}(t_{0}))\colon f\mapsto(\Delta_{N})^{-1}f,\\
B_{t_{0}}\colon & H^{2}(\Omega^{+}(t_{0}))\times H^{2}(\Omega^{-}(t_{0}))\rightarrow H^{\frac{1}{2}}(\Sigma)\colon (\mu^{+},\mu^{-})\mapsto \big(X_{0}^{*}([\nabla\mu\cdot\mathbf{n}_{\Gamma_{t_{0}}}])\big)(.,t_{0}),
\end{align*}
where $(\Delta_{N})^{-1}f$ is the
unique solution $(\mu_{N}^{+},\mu_{N}^{-})$ to \begin{subequations}\label{eq:neumannelliptic}
\begin{align}
\Delta\mu_{N}^{\pm} & =0 &  & \text{in }\Omega^{\pm}(t_{0}),\label{eq:neumannelliptic1}\\
\mu_{N}^{\pm} & =f &  & \text{on }\Gamma_{t_{0}},\label{eq:neumannelliptic2}\\
\nabla\mu_{N}^{-}\cdot\mathbf{n}_{\partial\Omega} & =0 &  & \text{on }\partial\Omega.\label{eq:neumannelliptic3}
\end{align}
\end{subequations}
In the literature the concatenation $B_{t_{0}}\circ S_{t_{0}}^{N}$
is often referred to as the \emph{Dirichlet-to-Neumann} operator and
$A_{0}\left(t_{0}\right):=B_{t_{0}}\circ S_{t_{0}}^{N}\circ\mathfrak{D}_{t_{0}}$
is called the \emph{Mullins-Sekerka} operator. It can be shown that
\[
A_{0}\colon [0,T]\to\mathcal{L}\big(H^{\frac{7}{2}}(\Sigma),H^{\frac{1}{2}}(\Sigma)\big)
\]
has $L^{p}$-maximal regularity, i.e., $A_{0}\in\mathcal{MR}_{p}(0,T)$.
We will not prove this in detail but just give a short sketch describing
the essential ideas: first, a reference surface $\tilde\Sigma\subset\subset\Omega$
is fixed such that $\Gamma_{t}$ can be expressed as a graph over
$\tilde\Sigma$ for $t$ in some time interval $\left[\tilde{t},\tilde{t}+\epsilon\right]\subset\left[0,T\right]$.
e.g.\ one may choose $\tilde\Sigma:=\Gamma_{0}$ and then determine $\epsilon_{0}>0$
such that $\Gamma_{t}$ may be written as graph over $\Gamma_{0}$
for all $t\in\left[0,\epsilon_{0}\right]$, which is possible since
$\Gamma$ is a smoothly evolving hypersurface. Next, a Hanzawa transformation
is applied, enabling us to consider (\ref{eq:neumannelliptic3}) as
a system on fixed domains $\Omega^{\pm}$ and $\tilde\Sigma$, but with
time dependent coefficients (see e.g.\ \cite[Chapter 2.2]{abelswilke} or and \cite[Chapter 4]{stefan}). Here, $\Omega^{+}$, $\Omega^{-}$
and $\tilde\Sigma$ denote disjoint sets such that $\partial\Omega^{+}=\tilde\Sigma$
and $\Omega=\Omega^{+}\cup\Omega^{-}\cup\tilde\Sigma$ holds and we assume
in the following that $t_{0}\in\left[0,\epsilon_{0}\right]$. To be
more specific, the Hanzawa transformation results in a system of the
form
\begin{align*}
a\left(x,t,\nabla_{x}\right)\bar{\mu}^{\pm} & =0 &  & \text{in }\Omega^{\pm},\\
\bar{\mu}^{\pm} & =\tilde{f} &  & \text{on }\tilde\Sigma,\\
\nabla\bar{\mu}^{-}\cdot\mathbf{n}_{\partial\Omega} & =0 &  & \text{on }\partial\Omega,
\end{align*}
where $a$ is the transformed Laplacian, depending smoothly on $t$
and $\tilde{f}$ is the transformation of $f$. Applying the Hanzawa
transformation (and the diffeomorphism $X_{0}$)
also to the operators $\mathfrak{D}_{t_{0}}$ and $B_{t_{0}}$, we
end up with a transformed operator $\tilde{A}_{0}(t_{0})\in\mathcal{L}\big(H^{\frac{7}{2}}(\Sigma),H^{\frac{1}{2}}(\Sigma)\big)$
and \cite[Corollary 6.6.5]{pruess} implies that $\tilde{A}_{0}(t_{0})$
has $L^{p}$-maximal regularity. As all involved differential operators
and coefficients depend smoothly on $t$, it is possible to show that
$\tilde{A}_{0}\colon \left[0,\epsilon_{0}\right]\to\mathcal{L}\big(H^{\frac{7}{2}}(\Sigma),H^{\frac{1}{2}}(\Sigma)\big)$
is relatively continuous. Therefore Theorem \ref{maxres} implies
$\tilde{A}_{0}\in\mathcal{MR}_{p}(0,\epsilon_{0})$ and,
transforming back, also $A_{0}\in\mathcal{MR}_{p}(0,\epsilon_{0})$.
Repeating this procedure with a new reference surface $\Sigma:=\Gamma_{\epsilon_{0}}$
and iteratively continuing the argumentation, we end up with $A_{0}\in\mathcal{MR}_{p}(0,T)$.

We proceed by showing that $\mathcal{A}(t_{0})=A_{0}(t_{0})+\mathcal{B}(t_{0})$
holds for some lower order perturbation $\mathcal{B}$. We introduce
\[
S_{t_{0}}^{DN}\colon H^{\frac{3}{2}}(\Gamma_{t_{0}})\rightarrow H^{2}(\Omega^{+}(t_{0}))\times H^{2}(\Omega^{-}(t_{0}))\colon f\mapsto (\Delta_{DN})^{-1}f,
\]
where $(\mu_{DN}^{+},\mu_{DN}^{-}):=(\Delta_{DN})^{-1} f$
is the unique solution to (\ref{eq:neumannelliptic}), replacing
$\nabla\mu_{N}^{-}\cdot\mathbf{n}_{\partial\Omega}=0$ on $\Gmu{2}$ by $\mu_{D}^{-}=0$
on $\Gmu{2}$. Moreover, we write $S_{t_{0}}^{\Delta}:=S_{t_{0}}^{DN}-S_{t_{0}}^{N}$
and observe that the equality
\begin{equation}
B_{t_{0}}\circ S_{t_{0}}^{DN}\circ\mathfrak{D}_{t_{0}}=A_{0}(t_{0})+\mathcal{B}_{0}(t_{0})\label{eq:neumanndirichlet}
\end{equation}
is satisfied, where $\mathcal{B}_{0}(t_{0}):=B_{t_{0}}\circ S_{t_{0}}^{\Delta}\circ\mathfrak{D}_{t_{0}}$.
Let $f\in H^{\frac{3}{2}}(\Gamma_{t_{0}})$ be fixed, $(\mu_{DN}^{+},\mu_{DN}^{-}):=S_{t_{0}}^{DN}f$,
$(\mu_{N}^{+},\mu_{N}^{-}):=S_{t_{0}}^{N}f$
and $\tilde{\mu}^{\pm}:=\mu_{DN}^{\pm}-\mu_{N}^{\pm}$, implying $(\tilde{\mu}^{+},\tilde{\mu}^{-})=S_{t_{0}}^{\Delta}f$.
Then $\tilde{\mu}^{\pm}\in H^{2}(\Omega^{\pm}(t_{0}))$
solves
\begin{alignat*}{2}
\Delta\tilde{\mu}^{\pm} & =0 & \qquad & \text{in }\Omega^{\pm}\left(t_{0}\right),\\
\tilde{\mu}^{\pm} & =0 &  & \text{on }\Gamma_{t_{0}},\\
\no_{\partial\Omega}\cdot \nabla \tilde{\mu}^{-} & =0 &  & \text{on }\Gmu{1}\\
\tilde{\mu}^{-} & =\mu_{N}^{-} &  & \text{on }\Gmu{2}
\end{alignat*}
and elliptic regularity theory implies
\begin{equation}
\left\Vert \tilde{\mu}^{-}\right\Vert _{H^{2}(\Omega^{-}(t_{0}))}\leq C\Vert \mu_{N}^{-}\Vert _{H^{\frac{3}{2}}(\Gmu{2})}\label{eq:mutildeelliptic}
\end{equation}
 and $\tilde{\mu}^{+}\equiv0$ in $\Omega^{+}(t_{0})$.
For the further argumentation, we show
\begin{equation}
\left\Vert \mu_{N}^{-}\right\Vert _{H^{\frac{3}{2}}(\partial\Omega)}\leq C\Vert \mu_{N}^{-}\Vert _{H^{\frac{1}{2}}(\Gamma_{t_{0}})}.\label{eq:zwischenstand}
\end{equation}
To this end let $\gamma(x):=\xi(4d_{\mathbf{B}}(x))$
for all $x\in\Omega$, where $\xi$ is a cut-off function satisfying
(\ref{eq:cut-off}). In particular $\text{supp}\gamma\cap\Gamma_{t}=\emptyset$
for all $t\in[0,T_{0}]$ by our assumptions and $\gamma\equiv1$
in $\partial\Omega(\frac{\delta}{4})$. Denoting $\hat{\mu}:=\gamma\mu_{N}^{-}\in H^{2}\left(\Omega^{-}(t_{0})\right)$,
we compute using $\Delta\mu_{N}^{-}=0$ in $\Omega^{-}(t_{0})$
that $\hat{\mu}$ is a solution to
\begin{alignat*}{2}
\Delta\hat{\mu} & =2\nabla\gamma\cdot\nabla\mu_{N}^{-}+(\Delta\gamma)\mu_{N}^{-} &\qquad  & \text{in }\Omega^{-}(t_{0}),\\
\hat{\mu} & =0 &  & \text{on }\Gamma_{t_{0}},\\
\nabla\hat{\mu}\cdot\mathbf{n}_{\partial\Omega} & =0 &  & \text{on }\partial\Omega,
\end{alignat*}
which, again regarding elliptic regularity theory, implies $\left\Vert \hat{\mu}\right\Vert _{H^{2}(\Omega^{-}(t_{0}))}\leq C\Vert \mu_{N}^{-}\Vert _{H^{1}(\Omega^{-}(t_{0}))}$.
This is essential in view of (\ref{eq:zwischenstand}) as it leads
to
\begin{align*}
  \Vert \mu_{N}^{-}\Vert _{H^{\frac{3}{2}}(\partial\Omega)} & =\Vert \hat{\mu}\Vert _{H^{\frac{3}{2}}(\partial\Omega)}\leq C\Vert \hat{\mu}\Vert _{H^{2}(\Omega^{-}(t_{0}))}\\
  &\leq C\Vert \mu_{N}^{-}\Vert _{H^{1}(\Omega^{-}\left(t_{0})\right)} \leq \tilde{C}\Vert \mu_{N}^{-}\Vert _{H^{\frac{1}{2}}(\Gamma_{t_{0}})},
\end{align*}
where we used the continuity of the trace operator $\operatorname{tr}\colon H^{2}(\Omega^{-}(t_{0}))\rightarrow H^{\frac{3}{2}}(\partial\Omega^{-}(t_{0}))$
in the first inequality (cf.\ \cite[Theorem 3.37]{mclean})
and standard estimates for elliptic equations in the second and third
inequality.

Let now $h\in H^{\frac{7}{2}}(\Sigma)$ and $(\tilde{\mu}^{+},\tilde{\mu}^{-}):=S_{t_{0}}^{\Delta}\circ\mathfrak{D}_{t_{0}} h$.
Our prior considerations enable us to estimate
\begin{align*}
\left\Vert B_{t_{0}}\circ S_{t_{0}}^{\Delta}\circ\mathfrak{D}_{t_{0}}h\right\Vert _{H^{\frac{1}{2}}(\Sigma)} & \leq C\left\Vert \tilde{\mu}^{-}\right\Vert _{H^{2}(\Omega^{-}(t_{0}))}\le C\Vert \mu_{N}^{-}\Vert _{H^{\frac{3}{2}}(\partial\Omega)}\\
 & \leq C\Vert \mu_{N}^{-}\Vert _{H^{\frac{1}{2}}(\Gamma_{t_{0}})}\leq C\Vert \sigma\Delta_{\Gamma}h\Vert _{H^{\frac{1}{2}}(\Sigma)} \leq C\Vert h\Vert_{H^{\frac{5}{2}}(\Sigma)},
\end{align*}
where we employed the continuity of the trace in the first line, (\ref{eq:mutildeelliptic})
in the second, (\ref{eq:zwischenstand}) in the third and the definition
of $\mu_{N}^{-}$ in the fourth. As $H^{\frac{7}{2}}(\Sigma)$
is dense in $H^{\frac{5}{2}}(\Sigma)$, we may
extend $\mathcal{B}_{0}(t_{0})$ to an operator
\begin{equation}
\mathcal{B}_{0}(t_{0})\colon H^{\frac{5}{2}}(\Sigma)\rightarrow H^{\frac{1}{2}}(\Sigma),\label{eq:B0}
\end{equation}
which shows in regard to (\ref{eq:neumanndirichlet}) that we may
view $B_{t_{0}}\circ S_{t_{0}}^{\Delta}\circ\mathfrak{D}_{t_{0}}$ as a lower order
perturbation of $A_{0}(t_0)$.

Next we take care of the term involving $b_{2}$ in (\ref{eq:musystem2}).
For this we consider the operator
\[
\mathcal{B}_{1}(t_0)\colon H^{\frac{7}{2}}(\Sigma)\rightarrow H^{\frac{1}{2}}(\Sigma)\colon h\mapsto X_{0}^{*}\big(\big[\partial_{\mathbf{n}_{\Gamma_{t_{0}}}}\mu_{1}\big]\big),
\]
where $\mu_{1}^{\pm}\in H^{2}(\Omega^{\pm}(t_0))$
is the solution to
\begin{alignat*}{2}
\Delta\mu_{1}^{\pm} & =0 &\qquad  & \text{in }\Omega^{\pm}(t_0),\\
\mu_{1}^{\pm} & =\pm b_{2}h &  & \text{on }\Gamma_{t_{0}},\\
\no_{\partial\Omega}\cdot \nabla \mu_{1}^{-} & =0 &  & \text{on }\Gmu{1},\\
\mu_{1}^{-} & =0 &  & \text{on }\Gmu{2}.
\end{alignat*}
We estimate
\begin{align}
\left\Vert \mathcal{B}_{1}(t_0)h\right\Vert _{H^{\frac{1}{2}}(\Sigma)} & \leq C\big\Vert \big[\partial_{\mathbf{n}_{\Gamma_{t_{0}}}}\mu_{1}\big]\big\Vert _{H^{\frac{1}{2}}\left(\Gamma_{t_{0}}\right)}\nonumber\\&\leq C\left(\Vert \mu_{1}^{+}\Vert _{H^{2}(\Omega^{+}(t_0))}+\Vert \mu_{1}^{-}\Vert _{H^{2}(\Omega^{-}(t_0))}\right)
  \leq C\Vert h\Vert _{H^{\frac{3}{2}}(\Sigma)},\label{eq:max-reg-per}
\end{align}
where $C>0$ can be chosen independent of $h$ and $t_{0}\in\left[0,T\right]$.
Here we again employed the continuity of the trace operator and elliptic
theory.

Defining
\[
\mathcal{B}(t_0)\colon H^{\frac{7}{2}}(\Sigma)\to H^{\frac{1}{2}}(\Sigma)\colon h\mapsto\mathcal{B}(t_0)h:=\tilde{\mathbf{b}}(.,t_{0})\cdot \nabla_{\Gamma}h-b_{1}(.,t_{0})h+(\mathcal{B}_{0}(t_0)+\mathcal{B}_{1}(t_0))h,
\]
where $\tilde{\mathbf{b}}$ is chosen such that $\partial_th+\tilde{\mathbf{b}}\cdot \nabla_\Gamma h= D_{t,\Gamma}h+{\mathbf{b}}\cdot \nabla_\Gamma h $, and using (\ref{eq:max-reg-per})
and (\ref{eq:B0}), we find that
\[
\left\Vert \mathcal{B}(t_0)h\right\Vert _{H^{\frac{1}{2}}(\Sigma)}\leq C\Vert h\Vert _{H^{\frac{5}{2}}(\Sigma)}.
\]
Thus, we can extend $\mathcal{B}(t_0)$ to a bounded
operator $\mathcal{B}(t_0)\colon H^{\frac{5}{2}}(\Sigma)\rightarrow H^{\frac{1}{2}}(\Sigma)$.
Since $H^{\frac{5}{2}}(\Sigma)$ is close to $H^{\frac{1}{2}}(\Sigma)$
compared to $H^{\frac{7}{2}}(\Sigma)$ as the embedding $H^{\frac{7}{2}}(\Sigma)\hookrightarrow H^{\frac{5}{2}}(\Sigma)$
is compact, we get due to Theorem~\ref{thm:Perturbation}, that $\text{\ensuremath{\mathcal{A}}}=A_{0}+\mathcal{B}$
has $L^{p}$-maximal regularity for all $t\in\left[0,T\right]$.

By elliptic theory
\begin{align*}
\left\Vert \mu^{\pm}\right\Vert _{H^{1}\left(\Omega^{\pm}(t)\right)} & \leq C\left\Vert X_{0}^{*,-1}\left(\sigma\Delta_{\Gamma}h+b_{2}h\right)\right\Vert _{H^{\frac{1}{2}}\left(\Gamma_{t}\right)}\leq C\left\Vert h\right\Vert _{H^{\frac{5}{2}}(\Sigma)}
\end{align*}
for almost all $t\in\left[0,T\right]$ and thus
\[
\left\Vert \mu^{\pm}\right\Vert _{L^{6}\left(0,T;H^{1}\left(\Omega^{\pm}(t)\right)\right)}\leq C\left\Vert h\right\Vert _{L^{6}\left(0,T;H^{\frac{5}{2}}(\Sigma)\right)}\leq C\left\Vert h\right\Vert _{X_{T}}.
\]
\end{proof}
\begin{theorem}
\label{stokesthe} Let $T\in\left(0,T_{0}\right]$ and $t\in\left[0,T\right]$.
For every $\mathbf{f}\in L^{2}(\Omega)^d$, $\mathbf{s}\in H^{\frac{3}{2}}(\Gamma_{t})^d$, $\mathbf{a}\in H^{\frac{1}{2}}(\Gamma_{t})^d$
and $\mathbf{g}\colon \partial \Omega\to \R^d$ such that  $\mathbf{g}|_{\Gv{1}}\in H^{\frac{3}{2}}(\Gv{1})^d$, $\no_{\partial\Omega}\cdot \mathbf{g}|_{\Gv{2}}\in H^{\frac{3}{2}}(\Gv{2})$, $(I-\no_{\partial\Omega}\otimes\no_{\partial\Omega})\mathbf{g}|_{\Gv{2}}\in H^{\frac{1}{2}}(\Gv{2})^d$, $\mathbf{g}|_{\Gv{3}}\in H^{\frac{1}{2}}(\Gv{3})^d$ satisfying the compatibility condition
\begin{equation}
  \label{eq:compStokes}
  \int_{\Gamma_t} \no_{\Gamma_t}\cdot \mathbf{s} \, d\mathcal{H}^{d-1} + \int_{\partial\Omega} \no_{\partial\Omega}\cdot \mathbf{g} \, d\mathcal{H}^{d-1} =0\qquad \text{if }\Gv{3}=\emptyset
\end{equation}
the system
\begin{align}
-\Delta\mathbf{v}^{\pm}+\nabla p^{\pm} & =\mathbf{f} &  & \text{in }\Omega^{\pm}(t),\label{eq:zweiphasen1}\\
\mathrm{div}\mathbf{v}^{\pm} & =0 &  & \text{in }\Omega^{\pm}(t),\\
B_j(\mathbf{v}^{-},p^{-}) & =\mathbf{g}|_{\Gv{j}}=: \mathbf{g}_j &  & \text{on }\Gv{j}, j=1,2,3,\\
\left[\mathbf{v}\right] & =\mathbf{s} &  & \text{on }\Gamma_{t},\\
\left[2D_{s}\mathbf{v}-p\mathbf{I}\right]\mathbf{n}_{\Gamma_{t}} & =\mathbf{a} &  & \text{on }\Gamma_{t}\label{eq:zweiphasen5}
\end{align}
has a unique solution $\left(\mathbf{v}^{\pm},p^{\pm}\right)\in H^{2}(\Omega^{\pm}(t))^d\times H^{1}(\Omega^{\pm}(t))$ satisfying $\int_\Omega p\, dx=0$ if $\Gv{3}=\emptyset$.
Moreover, there is a constant $C>0$ independent of $t\in\left[0,T_{0}\right]$
such that
\begin{align}\nonumber
  &\left\Vert \left(\mathbf{v},p\right)\right\Vert _{H^{2}(\Omega^{\pm}(t))\times H^{1}(\Omega^{\pm}(t))}\leq C \left(\| \mathbf{f}\| _{L^{2}(\Omega)}+\left\Vert \mathbf{s}\right\Vert _{H^{\frac{3}{2}}(\Gamma_{t})}+\left\Vert \mathbf{a}\right\Vert _{H^{\frac{1}{2}}(\Gamma_{t})}\right. \\
  &\qquad \left.+\left\Vert \mathbf{g}_1\right\Vert _{H^{\frac{3}{2}}(\Gv{1})}+ \left\Vert \mathbf{g}_{2,\no}\right\Vert _{H^{\frac{3}{2}}(\Gv{2})} + \left\Vert \mathbf{g}_{2,\btau}\right\Vert _{H^{\frac{1}{2}}(\Gv{2})} +\left\Vert \mathbf{g}_3\right\Vert _{H^{\frac{1}{2}}(\Gv{3})}\right)\label{eq:instoab}
\end{align}
holds.
\end{theorem}
\begin{proof}
 We can assume for simplicity that $\mathbf{g}=0$ on $\Gv{1}$ and $\no_{\partial\Omega}\cdot \mathbf{g}=0$ on $\Gv{2}$. Otherwise we substract a suitable extension of $\mathbf{g}$.
As a first step, we reduce the system (\ref{eq:zweiphasen1})\textendash (\ref{eq:zweiphasen5})
to the case $\mathbf{s}=0$. Elliptic theory implies that the equation
\begin{align*}
\Delta q & =0 &  & \text{in }\Omega^{-}(t),\\
\nabla q\cdot\mathbf{n}_{\Gamma_{t}} & =\mathbf{s}\cdot\mathbf{n}_{\Gamma_{t}} &  & \text{on }\Gamma_{t},\\
  \no_{\partial\Omega} \cdot \nabla q & =\no_{\partial\Omega}\cdot \mathbf{g}_{2} &  & \text{on }\Gv{1}\cup \Gv{2},\\
  q & =0 &  & \text{on }\Gv{3}
\end{align*}
has a unique solution $q\in H^{3}\left(\Omega^{-}(t)\right)$ with $\int_{\Omega^-(t)} q\, dx =0$ if $\Gv{3}=\emptyset$
since $\mathbf{s}\in H^{\frac{3}{2}}\left(\Gamma_{t}\right)^d$ and $\no_{\partial\Omega}\cdot \mathbf{g}_{2}\in H^{\frac32}(\Gv{1}\cup \Gv{2})$. Here, if $\Gv{3}=\emptyset$, the necessary compatibility condition is satisfied because of \eqref{eq:compStokes}. Moreover,
we have the estimate
\[
\left\Vert q\right\Vert _{H^{3}\left(\Omega^{-}(t)\right)}\leq C\left(\left\Vert \mathbf{s}\right\Vert _{H^{\frac{3}{2}}(\Gamma_{t})}+ \|\no_{\partial\Omega}\cdot \mathbf{g}_{2}\|_{H^{\frac32}(\Gv{1}\cup \Gv{2})}\right).
\]
Regarding the tangential part of $\mathbf{s}$, we may solve the stationary
Stokes system
\begin{align*}
-\Delta\mathbf{w}+\nabla\tilde{p} & =0 &  & \text{in }\Omega^{-}(t),\\
\operatorname{div}\mathbf{w} & =0 &  & \text{in }\Omega^{-}(t),\\
  \mathbf{w} & =(I-\no_{\Gamma_t}\otimes \no_{\Gamma_t})\left(\mathbf{s}-\nabla q\right) &  & \text{on }\Gamma_{t},\\
  \mathbf{w} & =0 &  & \text{on }\partial\Omega.
\end{align*}
We may find a solution $\left(\mathbf{w},\tilde{p}\right)\in H^{2}(\Omega^{-}(t))^d\times H^{1}(\Omega^{-}(t))$
(made unique by the normalization $\int_{\Omega^{-}(t)}\tilde{p}\sd x=0$)
and also get the estimate
\[
\left\Vert \mathbf{w}\right\Vert _{H^{2}(\Omega^{-}(t))}+\left\Vert \tilde{p}\right\Vert _{H^{1}(\Omega^{-}(t))}\leq C\left(\left\Vert \mathbf{s}\right\Vert _{H^{\frac{3}{2}}(\Gamma_{t})}+ \|\no_{\partial\Omega}\cdot \mathbf{g}_{2}\|_{H^{\frac32}(\Gv{1}\cup \Gv{2})}\right).
\]
 Thus, defining
$
\tilde{\mathbf{w}}:=\mathbf{w}+\nabla q,
$
the couple $\left(\tilde{\mathbf{w}},\tilde{p}\right)$ solves
\begin{align*}
-\Delta\tilde{\mathbf{w}}+\nabla\tilde{p} & =0 &  & \text{in }\Omega^{-}(t),\\
\operatorname{div}\tilde{\mathbf{w}} & =0 &  & \text{in }\Omega^{-}(t),\\
\tilde{\mathbf{w}} & =\mathbf{s} &  & \text{on }\Gamma_{t},\\
\tilde{\mathbf{w}} & =0 &  & \text{on }\partial\Omega,
\end{align*}
and may be estimated by $\mathbf{s}$ in strong norms. Next, let
\[
  \tilde{\mathbf{g}}:=\mathbf{g}_j+B_j(\tilde{\mathbf{w}},\tilde{p})\qquad \text{on }\Gv{j}, j=1,2,3
\]
and $\tilde{\mathbf{a}}:=\mathbf{a}-\left(2D_{s}\tilde{\mathbf{w}}-\tilde{p}\mathbf{I}\right)\mathbf{n}_{\Gamma_t}\in H^{\frac{1}{2}}(\Gamma_{t})^d$,
where the regularity is due to the properties of the trace operator.
Then, for every strong solution $\left(\hat{\mathbf{v}}^{\pm},\hat{p}^{\pm}\right)$
of \eqref{eq:zweiphasen1}-\eqref{eq:zweiphasen5}, with
$\mathbf{s\equiv}0$ and $\mathbf{g},\mathbf{a}$ substituted by $\tilde{\mathbf{g}},\tilde{\mathbf{a}}$,
the functions
\[
\left(\mathbf{v}^{+},p^{+}\right):=\left(\hat{\mathbf{v}}^{+},\hat{p}^{+}\right)\text{ and }\left(\mathbf{v}^{-},p^{-}\right):=\left(\hat{\mathbf{v}}^{-}-\tilde{\mathbf{w}},\hat{p}^{-}-\tilde{p}\right)
\]
are solutions to the original system \eqref{eq:zweiphasen1}-\eqref{eq:zweiphasen5}.
So, we will consider $\mathbf{s}\equiv0$ in the following and show
existence of strong solutions in that case.

As a starting point, we construct a solution $\left(\mathbf{v},p\right)\in V(\Omega)\times L^{2}(\Omega)$
to the weak formulation
\begin{align}
  &\int_{\Omega}2D_{s}\mathbf{v}:D_{s}\bpsi\, dx -\int_\Omega p\operatorname{div}\bpsi\sd x+\int_{\Gv{2}}\alpha_{2}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)+\int_{\Gv{3}}\alpha_{3}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)\nonumber\\
  &\qquad =\int_{\Omega}\mathbf{f}\cdot\bpsi\sd x+\int_{\Gamma_{t}}\mathbf{a}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)-\int_{\partial\Omega}\mathbf{g}\cdot\bpsi\sd\mathcal{H}^{d-1}(s),\label{eq:weak2phase}
\end{align}
for all $\bpsi\in H^{1}(\Omega)^d$ with $\bpsi|_{\Gv{1}}=0$, $\no \cdot \bpsi|_{\Gv{2}}=0$, where
\begin{equation*}
  V(\Omega)= \{\mathbf{u}\in H^1(\Omega)^d: \operatorname{div} \mathbf{u}=0, \mathbf{u}|_{\Gv{1}}=0, \no \cdot \mathbf{u}|_{\Gv{2}}=0\}.
\end{equation*}
Considering first $\bpsi\in V(\Omega)$
and the right hand side as a functional $\mathbf{F}\in\left(V(\Omega)\right)'$,
the Lemma of Lax-Milgram implies the existence of a unique $\mathbf{v}\in V(\Omega)$
solving (\ref{eq:weak2phase}) for all $\bpsi\in V(\Omega)$,
where the coercivity of the involved bilinear form is a consequence
of \eqref{eq:Korn}.

Next consider the functional
\begin{align*}
 F(\bpsi):= &-\int_{\Omega}2D_{s}\mathbf{v}:D_{s}\bpsi\, dx -\int_{\Gv{2}}\alpha_{2}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)-\int_{\Gv{3}}\alpha_{3}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)\\
  &\qquad +\int_{\Omega}\mathbf{f}\cdot\bpsi\sd x+\int_{\Gamma_{t}}\mathbf{a}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)-\int_{\partial\Omega}\mathbf{g}\cdot\bpsi\sd\mathcal{H}^{d-1}(s),
\end{align*}
for all $\bpsi\in H^1(\Omega)^d$ with $\bpsi|_{\Gv{1}}=0$, $\no \cdot \bpsi|_{\Gv{2}}=0$. Then $F$ vanishes on $V(\Omega)$ and by Lemma~\ref{lem:pressure} in  Appendix A
there is a unique $p\in L^2 (\Omega)$ with  $\int_\Omega p\, dx =0$ if $\Gv{3}=\emptyset$ such that
\begin{equation*}
  F(\bpsi)= -\int_\Omega p \operatorname{div} \bpsi \, dx \quad \text{for all }\ \bpsi\in H^1(\Omega)^d\text{ with }\bpsi|_{\Gv{1}}=0, \no \cdot \bpsi|_{\Gv{2}}=0.
\end{equation*}
Hence $(\mathbf{v},p)$ solve \eqref{eq:weak2phase}.

Moreover, we obtain the estimate
\begin{equation}
\left\Vert (\mathbf{v},p)\right\Vert _{H^{1}(\Omega)\times L^{2}(\Omega)}\leq C\left(\left\Vert \mathbf{f}\right\Vert _{L^{2}(\Omega)}+\left\Vert \mathbf{a}\right\Vert _{H^{\frac{1}{2}}(\Gamma_{t})}+\left\Vert \mathbf{g}\right\Vert _{H^{\frac{1}{2}}(\partial\Omega)}\right).\label{eq:2weak2estimate}
\end{equation}
We now show higher regularity of $\left(\mathbf{v},p\right)$ by localization.

Let $\eta^{\pm}\in C^{\infty}\left(\overline{\Omega}\right)$ be a
partition of unity of $\Omega$, such that the inclusions $\Omega^{+}(t)\cup\Gamma_{t}\left(\delta\right)\subset\left\{ \left.x\in\Omega\right|\eta^{+}(x)=1\right\} $
and $\partial\Omega\left(\delta\right)\subset\left\{ \left.x\in\Omega\right|\eta^{-}(x)=1\right\} $
hold. We choose $\eta^\pm$ such that $\{x\in\Omega: \eta^\pm(x)=1\}$ has smooth boundary and define $U^{\pm}:=\text{supp}\left(\eta^{\pm}\right)$, $\partial U_{0}^{-}:=\partial U^{-}\backslash\partial\Omega$
and
\[
\dot{U}:=\left\{ \left.x\in\Omega\right|\eta^{+}(x)\in\left(0,1\right)\right\} =\left\{ \left.x\in\Omega\right|\eta^{-}(x)\in\left(0,1\right)\right\} .
\]
Moreover, we set $\dot{p}^{-}:=p\eta^{-}$ and $\tilde{\mathbf{v}}^{-}:=\mathbf{v}\eta^{-}$
in $\Omega$ and we correct the divergence of $\tilde{\mathbf{v}}^{-}$
with the help of the Bogovskii-operator: Let $\varphi\in C_{c}^{\infty}(\Omega)$
with $\text{supp}\left(\varphi\right)\subset U^{+}\backslash\dot{U}$
and $\int_{\Omega}\varphi\sd x=1$ and set
\[
\hat{g}:=\operatorname{div}\left(\tilde{\mathbf{v}}^{-}\right)-\varphi\int_{U^{+}}\operatorname{div}\left(\tilde{\mathbf{v}}^{-}\right)\sd x
\]
in $U^{+}$. As $\mathbf{v}\in V(\Omega)$,
we have $\operatorname{div}\left(\tilde{\mathbf{v}}^{-}\right)=\mathbf{v}\cdot\nabla\eta^{-}$
and thus $\hat{g}\in H_{0}^{1}\left(U^{+}\right)$, $\int_{U^{+}}\hat{g}\sd x=0$.
Consequently, \cite[Theorem III.3.3]{galdi} implies that
there is $\hat{\mathbf{v}}^{-}\in H_{0}^{2}\left(U^{+}\right)$, which
we extend onto $\Omega$ by $0$, satisfying
\begin{align}
\operatorname{div}\hat{\mathbf{v}}^{-} & =\hat{g}\text{ in }U^{+},\nonumber \\
\left\Vert \hat{\mathbf{v}}^{-}\right\Vert _{H^{2}(\Omega)} & \leq C\left\Vert \mathbf{v}\right\Vert _{H^{1}(\Omega)}.\label{eq:vhut-}
\end{align}
Therefore, $\dot{\mathbf{v}}^{-}:=\tilde{\mathbf{v}}^{-}-\hat{\mathbf{v}}^{-}$
fulfills $\operatorname{div}\dot{\mathbf{v}}^{-}=0$ in $U^{-}$ since $\varphi\equiv0$
in that domain. Let now
$$
\bpsi\in\left\{ \mathbf{w}\in H^{1}(U^{-})^d: \mathbf{w}=0\text{ on }\partial U_{0}^{-}, \mathbf{w}|_{\Gv{1}}=0, \no\cdot \mathbf{w}|_{\Gv{2}}=0\right\},
$$
then
\begin{align*}
\int_{U^{-}} & 2D_{s}\dot{\mathbf{v}}^{-}:D_{s}\bpsi-\dot{p}^{-}\operatorname{div}\bpsi\sd x+\int_{\Gv{2}}\alpha_{2}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)+\int_{\Gv{3}}\alpha_{3}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)\\
 & =\int_{U^{-}}2D_{s}\tilde{\mathbf{v}}^{-}:D_{s}\bpsi-p\operatorname{div}\left(\bpsi\eta^{-}\right)+\left(p\nabla\eta^{-}\right)\cdot\bpsi\sd x+\int_{\Gv{2}}\alpha_{2}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)\\
 & \quad+\int_{\Gv{3}}\alpha_{3}\mathbf{v}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)-\int_{U^{-}}2D_{s}\hat{\mathbf{v}}^{-}:D_{s}\bpsi\sd x\\
 & =\int_{U^{-}}\mathbf{f}\cdot\bpsi\eta^{-}\sd x-\int_{\partial\Omega}\mathbf{g}\cdot\bpsi\sd\mathcal{H}^{d-1}(s)+\left(p\nabla\eta^{-}\right)\cdot\bpsi\sd x\\
 & \quad+\int_{U^{-}}2\operatorname{div}\left(D_{s}\hat{\mathbf{v}}\right)\cdot\bpsi+\left(2D_{s}\mathbf{v}\nabla\eta^{-}-\operatorname{div}\left(\mathbf{v}\otimes\nabla\eta^{-}+\nabla\eta^{-}\otimes\mathbf{v}\right)\right)\cdot\bpsi\sd x,
\end{align*}
where we used the definition of $\dot{\mathbf{v}}^{-}$ and $\dot{p}^{-}$
in the first equality and integration by parts together with $\hat{\mathbf{v}}^{-}\in H_{0}^{2}\left(U^{+}\right)$
and $\nabla\eta^{-}=0$ on $U^{-}$ in the second equality. Additionally,
we employed the fact that $\left(\mathbf{v},p\right)$ is the weak
solution to (\ref{eq:weak2phase}). Hence, $\left(\dot{\mathbf{v}}^{-},\dot{p}^{-}\right)$
are a weak solution to the system
\begin{align}
-\Delta\dot{\mathbf{v}}^{-}+\nabla\dot{p}^{-} & =\tilde{\mathbf{f}} &  & \text{in }U^{-},\nonumber \\
\operatorname{div}\dot{\mathbf{v}}^{-} & =0 &  & \text{in }U^{-},\nonumber \\
\dot{\mathbf{v}}^{-} & =\hat{\mathbf{v}}^{-} &  & \text{on }\partial U_{0}^{-},\nonumber \\
  B_j(\mathbf{v}^{-},\dot{p}^{-}) & =\mathbf{g} &  & \text{on }\Gv{j}, j=1,2,3,\label{eq:onephase outer}
\end{align}
where
\[
  \tilde{\mathbf{f}}:=\mathbf{f}\eta^-
  + p\nabla\eta^{-}+2\operatorname{div}(D_{s}\hat{\mathbf{v}})+2D_{s}\mathbf{v}\nabla\eta^{-}-\operatorname{div}\left(\mathbf{v}\otimes\nabla\eta^{-}+\nabla\eta^{-}\otimes\mathbf{v}\right)\in L^{2}(U^{-})
\]
and $\hat{\mathbf{v}}^{-}\in H^{\frac{3}{2}}(\partial U_{0}^{-})$
by the properties of the trace operator. Writing
\begin{equation*}
  \tilde{\mathbf{g}}:=
  \begin{cases}
\mathbf{g} &\text{on }\Gv{1},\\
\alpha_{j}\dot{\mathbf{v}}^{-}+\mathbf{g} &\text{on }\Gv{j},j=2,3,
  \end{cases}
\end{equation*}
using localization techniques and results for strong solutions of
the stationary Stokes equation in one phase with inhomogeneous do-nothing
boundary condition (cf.\ Theorem 3.1 in \cite{ShimizuStokes}),
with Dirichlet boundary condition (cf.\ \cite{galdi}) and slip-boundary conditions (cf.\  Solonnikov and \v{S}\v{c}adilov~\cite[Theorem~2]{SolonnikovNavier}),
we find that
$\left(\dot{\mathbf{v}}^{-},\dot{p}^{-}\right)\in H^{2}(U^{-})\times H^{1}(U^{-})$.
Moreover, regarding (\ref{eq:vhut-}), (\ref{eq:2weak2estimate})
and the definition of $\tilde{\mathbf{f}}$, we get
\[
\left\Vert \left(\dot{\mathbf{v}}^{-},\dot{p}^{-}\right)\right\Vert _{H^{2}\left(U^{-}\right)\times H^{1}\left(U^{-}\right)}\leq C\left(\left\Vert \mathbf{f}\right\Vert _{L^{2}(\Omega)}+\left\Vert \mathbf{a}\right\Vert _{H^{\frac{1}{2}}\left(\Gamma_{t}\right)}+\left\Vert \mathbf{g}\right\Vert _{H^{\frac{1}{2}}\left(\partial\Omega\right)}\right).
\]
Analogously, we define $\tilde{\mathbf{v}}^{+}:=\mathbf{v}\eta^{+}$
and $\hat{\mathbf{v}}^{+}\in H_{0}^{2}(\dot{U})$ as a
solution to $\operatorname{div}\mathbf{\hat{\mathbf{v}}}^{+}=\operatorname{div}\tilde{\mathbf{v}}^{+}$.
Here, we do not need to correct the mean value, since
\[
\int_{\dot{U}}\operatorname{div}\tilde{\mathbf{v}}^{+}\sd x=\int_{\partial\dot{U}}\mathbf{v}\cdot\mathbf{n}_{\partial\dot{U}}\eta^{+}\sd\mathcal{H}^{d-1}(s)=-\int_{\left\{ \eta^{+}=1\right\} }\operatorname{div}\mathbf{v}\sd x=0.
\]
We set $\dot{\mathbf{v}}^{+}:=\tilde{\mathbf{v}}^{+}-\hat{\mathbf{v}}^{+}$
and $\dot{p}^{+}:=p\eta^{+}$ and get after similar calculations as
before that $\left(\dot{\mathbf{v}}^{+},\dot{p}^{+}\right)$ is a
weak solution to the two phase stationary Stokes system
\begin{align}
-\Delta\dot{\mathbf{v}}^{+}+\nabla\dot{p}^{+} & =\hat{\mathbf{f}} &  & \text{in }U^{+},\label{eq:zweiphasen1-1}\\
\operatorname{div}\dot{\mathbf{v}}^{+} & =0 &  & \text{in }U^{+},\\
\dot{\mathbf{v}}^{+} & =0 &  & \text{on }\partial U^{+},\\
\left[\dot{\mathbf{v}}^{+}\right] & =0 &  & \text{on }\Gamma_{t},\\
\left[2D_{s}\dot{\mathbf{v}}^{+}-\dot{p}^{+}\mathbf{I}\right]\mathbf{n}_{\Gamma_{t}} & =\mathbf{a} &  & \text{on }\Gamma_{t},\label{eq:zweiphasen5-1}
\end{align}
where $\hat{\mathbf{f}}\in L^{2}\left(U^{+}\right)$. Then \cite[Theorem 1.1]{shibatastokesdirichlet}
 implies $\dot{\mathbf{v}}^{+}|_{\Omega^{+}(t)}\in H^{2}(\Omega^{+}(t))$
and $\dot{\mathbf{v}}^{+}|_{U^{+}\backslash\Omega^{+}(t)}\in H^{2}(U^{+}\backslash\Omega^{+}(t))$,
and also that the pressure satisfies $\dot{p}^{+}|_{\Omega^{+}(t)}\in H^{1}(\Omega^{+}(t))$
and $\dot{p}^{+}|_{U^{+}\backslash\Omega^{+}(t)}\in H^{1}(U^{+}\backslash\Omega^{+}(t))$
with estimates in the associated norms. In particular, $\mathbf{v}=\dot{\mathbf{v}}^{+}$
in $\Omega^{+}(t)$ and $\mathbf{v}=\dot{\mathbf{v}}^{+}+\dot{\mathbf{v}}^{-}+\hat{\mathbf{v}}^{+}+\text{\ensuremath{\hat{\mathbf{v}}}}^{-}$
in $\Omega^{-}(t)$, yielding the desired regularity and
(\ref{eq:instoab}). To show that $C>0$ may be chosen independently
of $t\in\left[0,T_{0}\right]$, one may make use of extension arguments,
see e.g.\ the proof of  \cite[Lemma 2.10]{nsac}.
\end{proof}
\begin{theorem}
\label{scharfthe} Let $T\in\left(0,T_{0}\right]$. Let $\mathbf{b}\colon \Sigma\times\left[0,T\right]\rightarrow\mathbb{R}^d$,
$b\colon \Sigma\times\left[0,T\right]\rightarrow\mathbb{R}$, $a_{1}\colon \Omega\times\left[0,T\right]\rightarrow\mathbb{R}$,
$a_{2},a_{3},a_{5}\colon \Gamma\rightarrow\mathbb{R}$, $a_{4}\colon \partial\Omega\times[0,T]\rightarrow\mathbb{R}$,
$\mathbf{a}_{1}\colon \Omega\times\left[0,T\right]\rightarrow\mathbb{R}^d$,
$\mathbf{a}_{2},\mathbf{a}_{3},\mathbf{a}_{4},\mathbf{a}_{5}\colon\Gamma\rightarrow\mathbb{R}^d$
and $\mathbf{a}_{6}\colon \partial\Omega\times [0,T]\rightarrow\mathbb{R}^d$
be smooth given functions such that
\begin{equation*}
    \int_{\Gamma_t} \no_{\Gamma_t}\cdot \mathbf{a}_2 \, d\mathcal{H}^{d-1} + \int_{\partial\Omega} \no_{\partial\Omega}\cdot\mathbf{a}_6 \, d\mathcal{H}^{d-1} =0\qquad \text{if }\Gv{3}=\emptyset.
\end{equation*}
For every $g\in L^{2}\big(0,T;H^{\frac{1}{2}}(\Sigma)\big)$
and $h_{0}\in H^{2}(\Sigma)$ there exists a unique
solution $h\in X_{T}$ of
\begin{align*}
D_{t,\Gamma}h+\mathbf{b}\cdot\nabla_{\Gamma}h-bh+\tfrac{1}{2}X_{0}^{*}\big((\mathbf{v}^{+}+\mathbf{v}^{-})\cdot\mathbf{n}_{\Gamma_{t}}\big)+\tfrac{1}{2}X_{0}^{*}\big(\big[\partial_{\mathbf{n}_{\Gamma_{t}}}\mu\big]\big) & =g &  & \text{in }\Sigma\times\left(0,T\right),\\
h\left(.,0\right) & =h_{0} &  & \text{in }\Sigma,
\end{align*}
where for every $t\in\left[0,T\right]$, the functions \textbf{$\mathbf{v}^{\pm}=\mathbf{v}^{\pm}(x,t)$},
$p^{\pm}=p^{\pm}(x,t)$ and $\mu^{\pm}=\mu^{\pm}(x,t)$
for $(x,t)\in\Omega_{T}^{\pm}$ with $\mathbf{v}^{\pm}\in H^{2}(\Omega^{\pm}(t))$,
$p^{\pm}\in H^{1}(\Omega^{\pm}(t))$ and $\mu^{\pm}\in H^{2}(\Omega^{\pm}(t))$
are the unique solutions to
\begin{align}
\Delta\mu^{\pm} & =a_{1} &  & \text{in }\Omega^{\pm}(t),\label{eq:laplmu}\\
  \mu^{\pm} & = X_{0}^{*,-1}\big(\sigma\Delta_{\Gamma}h\pm a_{2}h\big)+a_{3} &  & \text{on }\Gamma_{t},\\
  \no_{\partial\Omega}\cdot \nabla \mu^{-} & =a_{4} &  & \text{on }\Gmu{1},\\
  \mu^{-} & =a_{4} &  & \text{on }\Gmu{2},\\
-\Delta\mathbf{v}^{\pm}+\nabla p^{\pm} & =\mathbf{a}_{1} &  & \text{in }\Omega^{\pm}(t),\label{eq:stokes}\\
\operatorname{div}\mathbf{v}^{\pm} & =0 &  & \text{in }\Omega^{\pm}(t),\\
[\mathbf{v}] & =\mathbf{a}_{2} &  & \text{on }\Gamma_{t},\label{eq:stokesjmp}\\
\left[2D_{s}\mathbf{v}-p\mathbf{I}\right]\mathbf{n}_{\Gamma_{t}} & =X_{0}^{*,-1}\big(\mathbf{a}_{3}h+\mathbf{a}_{4}\Delta_{\Gamma}h+a_{5}\nabla_{\Gamma}h+\mathbf{a}_{5}\big) &  & \text{on }\Gamma_{t},\label{eq:stokesbdry}\\
B_j(\mathbf{v}^{-},p^{-})& =\mathbf{a}_{6} &  & \text{on }\Gv{j}, j=1,2,3.\label{eq:stokesoutbdry}
\end{align}
Moreover, if $g$, $h_{0}$ and $b$, $\mathbf{b}$, $a_{i}$, and
$\mathbf{a}_{j}$ are smooth on their respective domains for $i\in\left\{ 1,\ldots,5\right\}$, $j\in\left\{ 1,\ldots,6\right\}$,
then $h$ is smooth and $p^{\pm}$, $\mathbf{v}^{\pm}$ and $\mu^{\pm}$
are smooth on $\Omega^{\pm}(t)$.
\end{theorem}
\begin{proof}
We show this by a perturbation argument. First of all note that we
may without loss of generality assume that $a_{1}$, $a_{3}$, $a_{4}$,
$\mathbf{a}_{1}$, $\mathbf{a}_{2}$, $\mathbf{a}_{5}$, $\mathbf{a}_{6}=0$
on their respective domains. The above system may be reduced to this
case by solving
\begin{align*}
\Delta\hat{\mu}^{\pm} & =a_{1} &  & \text{in }\Omega^{\pm}(t),\\
  \hat{\mu}^{\pm} & =a_{3} &  & \text{on }\Gamma_{t},\\
    \no\cdot \nabla \hat{\mu}^{-} & =a_{4} &  & \text{on }\Gmu{1},\\
  \hat{\mu}^{-} & =a_{4} &  & \text{on }\Gmu{2}
\end{align*}
with the help of standard elliptic theory and
\begin{align*}
-\Delta\hat{\mathbf{v}}^{\pm}+\nabla\hat{p}^{\pm} & =\mathbf{a}_{1} &  & \text{in }\Omega^{\pm}(t),\\
\operatorname{div}\mathbf{\hat{v}}^{\pm} & =0 &  & \text{in }\Omega^{\pm}(t),\\
[\mathbf{\hat{\mathbf{v}}}] & =\mathbf{a}_{2} &  & \text{on }\Gamma_{t},\\
\left[2D_{s}\mathbf{\hat{v}}-p\right]\mathbf{n} & =\mathbf{a}_{5} &  & \text{on }\Gamma_{t},\\
B_j(\mathbf{\hat{v}}^{-},\hat{p}^{-}) & =\mathbf{a}_{6} &  & \text{on }\Gv{j}, j=1,2,3,
\end{align*}
with the help of Theorem \ref{stokesthe} and setting
\[
\hat{g}=g-\tfrac{1}{2}X_{0}^{*}\left([\partial_{\mathbf{n}_{\Gamma_{t}}}\hat{\mu}]+(\hat{\mathbf{v}}^{+}+\hat{\mathbf{v}}^{-})\cdot\mathbf{n}_{\Gamma_{t}}\right).
\]
Now let $t\in\left[0,T\right]$, $h\in H^{\frac{7}{2}}(\Sigma)$
and let $\mathbf{v}_{h}^{\pm}\in H^{2}(\Omega^{\pm}(t))^d$,
$p_{h}^{\pm}\in H^{1}(\Omega^{\pm}(t))$ be
the solution to (\ref{eq:stokes})\textendash (\ref{eq:stokesoutbdry}).
Multiplying (\ref{eq:stokes}) by $\mathbf{v}_{h}^{\pm}$ and integrating
in $\Omega^{\pm}(t)$ together with integration by parts
and the consideration of the boundary values (\ref{eq:stokesbdry})
and (\ref{eq:stokesoutbdry}) allows us to deduce
\begin{align}
\int_{\Omega^{+}(t)} & 2|D_{s}\mathbf{v}_{h}^{+}|^{2}\sd x+\int_{\Omega^{-}(t)}2|D_{s}\mathbf{v}_{h}^{-}|^{2}\sd x+\sum_{j=2,3}\int_{\Gv{j}}\alpha_j|\mathbf{v}_{h}^{-}|^{2}\sd\mathcal{H}^{d-1}(s)\nonumber \\
 & =\int_{\Gamma_{t}}X_{0}^{*,-1}\big(\mathbf{a}_{3}h+\mathbf{a}_{4}\Delta_{\Gamma}h+a_{5}\nabla_{\Gamma}h\big)\cdot\mathbf{v}_{h}^{-}\sd\mathcal{H}^{d-1}(s).\label{eq:v+-test}
\end{align}
Hence, by \cite[Corollary 5.8]{korn}  and the continuity of the trace
we find
\begin{equation}
\Vert \mathbf{v}_{h}^{-}\Vert _{H^{1}(\Omega^{-}(t))}\leq C\Vert h\Vert _{H^{2}(\Sigma)}\label{eq:v-est}
\end{equation}
for $C$ independent of $h$ and $t$. \cite[Corollary~5.8]{korn},
also implies
\[
\int_{\Omega^{+}(t)}2|D_{s}\mathbf{v}_{h}^{+}|^{2}\sd x+\int_{\Gamma_{t}}|\mathbf{v}_{h}^{+}|^{2}\sd\mathcal{H}^{d-1}(s)\geq C\Vert \mathbf{v}_{h}^{+}\Vert _{H^{1}\left(\Omega^{+}(t)\right)}^{2},
\]
 leading to
\begin{equation}
\Vert \mathbf{v}_{h}^{+}\Vert _{H^{1}(\Omega^{-}(t))}\leq C\Vert h\Vert _{H^{2}(\Sigma)}\label{eq:v+est}
\end{equation}
due to $\mathbf{v}_{h}^{+}=\mathbf{v}_{h}^{-}$ on $\Gamma_{t}$,
(\ref{eq:v-est}) and (\ref{eq:v+-test}). Defining
\[
\mathcal{B}(t)\colon H^{\frac{7}{2}}(\Sigma)\rightarrow H^{\frac{1}{2}}(\Sigma)\colon h\mapsto\mathcal{B}(t)h=\tfrac{1}{2}X_{0}^{*}\big((\mathbf{v}_{h}^{+}+\mathbf{v}_{h}^{-})\cdot\mathbf{n}_{\Gamma_{t}}\big),
\]
we may use (\ref{eq:v-est}) and (\ref{eq:v+est}) to confirm
\[
\Vert \mathcal{B}(t)h\Vert _{H^{\frac{1}{2}}(\Sigma)}\le C\Vert h\Vert _{H^{2}(\Sigma)}
\]
for $C>0$ independent of $h$ and $t$. As $H^{\frac{7}{2}}(\Sigma)$
is dense in $H^{2}(\Sigma)$ we can extend $\mathcal{B}(t)$
to an operator $\mathcal{B}(t):H^{2}(\Sigma)\rightarrow H^{\frac{1}{2}}(\Sigma)$
and $H^{2}(\Sigma)$ is close to $H^{\frac{1}{2}}(\Sigma)$
compared with $H^{\frac{7}{2}}(\Sigma)$.

The existence of a unique solution $h\in X_{T}$ with the properties
stated in the theorem is now a consequence of Theorem~\ref{thm:Perturbation}.
Higher regularity may be shown by localization and e.g.\ the
usage of difference quotients.
\end{proof}

\section*{Acknowledgments}
 The authors acknowledge support by the SPP 1506 "Transport Processes
at Fluidic Interfaces" of the German Science Foundation (DFG) through the grant AB285/4-2. Moreover, we are grateful to the anonymous referee for the careful reading a previous version of the manuscript and many helpful comments.

\appendix
\section{Existence of a Pressure}
\begin{lemma}\label{lem:pressure}
  Let $F\in \{\bpsi\in H^1(\Omega)^d: \bpsi|_{\Gv{1}}=0, \no_{\partial\Omega}\cdot \bpsi|_{\Gv{2}}=0 \}\to \R$ be linear and bounded such that
  \begin{equation*}
    F(\bpsi)= 0 \quad \text{for all } \bpsi\in V(\Omega)=\{ \bpsi\in H^1(\Omega)^d: \operatorname{div} \bpsi=0, \bpsi|_{\Gv{1}}=0, \no_{\partial\Omega}\cdot \bpsi|_{\Gv{2}}=0\}.
  \end{equation*}
  Then there is a unique $p\in L^2(\Omega)$ with $\int_\Omega p\,dx =0$ if $\Gv{3}=\emptyset$ such that
   \begin{equation*}
    F(\bpsi)= -\int_\Omega p\operatorname{div}\bpsi \,dx \qquad \text{for all } \bpsi\in H^1(\Omega)^d\text{ with }\bpsi|_{\Gv{1}}=0, \no_{\partial\Omega}\cdot \bpsi|_{\Gv{2}}=0.
  \end{equation*}
\end{lemma}
\begin{proof}
  We will apply the closed range theorem. To this end let
  \begin{align*}
    X&= \{\bpsi\in H^1(\Omega)^d: \bpsi|_{\Gv{1}}=0, \no_{\partial\Omega}\cdot \bpsi|_{\Gv{2}}=0 \},\\
    Y&= \left\{g\in L^2(\Omega): \int_\Omega g(x)\,dx =0 \text{ if }\Gv{3}=\emptyset\right\}
  \end{align*}
  and consider
  \begin{equation*}
    T \colon X\to Y \colon \bpsi\mapsto -\operatorname{div} \bpsi.
  \end{equation*}
  Then $T$ is onto, which can be seen as follows: Let $g\in Y$.

  If $\Gv{3}\neq \emptyset$, then there is a unique solution $q\in H^1(\Omega)$ of
  \begin{alignat*}{2}
    \Delta q &= g &\qquad & \text{in }\Omega,\\
    q|_{\Gv{3}} &= 0 && \text{on }\Gv{3},\\
    \no_{\partial\Omega} \cdot \nabla q|_{\Gv{1}\cup\Gv{2}}& =0 &&\text{on }\Gv{1}\cup \Gv{2}.
  \end{alignat*}
  Moreover, using the solvability of the stationary Stokes equation with non-homogenoues Dirichlet boundary conditions, we find some $\mathbf{w}\in H^1(\Omega)^d$ with $\operatorname{div} \mathbf{w}=0$ and
  \begin{equation*}
    \mathbf{w}|_{\Gv{1}} = \nabla q|_{\Gv{1}\cup \Gv{2}},\qquad \mathbf{w}|_{\Gv{3}}=0.
  \end{equation*}
  Then $\bpsi = \mathbf{w}-\nabla q\in X$ with $-\operatorname{div}\bpsi= g$.

  If $\Gv{3}= \emptyset$, we have $\int_\Omega g(x)\, dx=0$ and  can use the well-known Bogovskii operator to obtain some $\bpsi\in H^1_0(\Omega)$ with $-\operatorname{div} \bpsi=g$.

  Now the closed range theorem implies that $T' \colon Y'\to X'$ is injective and
  \begin{equation*}
    \mathcal{R}(T') =\mathcal{N}(T)^\circ=\{F\in X':  F(\bpsi) =0 \text{ for all }\bpsi\in V(\Omega)\}.
  \end{equation*}
  This proves the statement of the lemma.
\end{proof}

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\providecommand{\MRhref}[2]{
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text:
\section{Empirical Evaluation: Synthetic Data}\label{section:numerical}

\subsection{Inference from testing data}

\rev{We first illustrate the effectiveness of the proposed inference and parameter identification methods. Recall that the hyper-parameter $\alpha$, defined as the ratio of the probability of an infected individual undergoing testing and the probability of a healthy individual undergoing testing, plays a key role in our inference scheme. We now show how it can be learned from testing data following the procedure discussed in Section \ref{sec:alpha_tuning}.}

\begin{figure}
\centering
    \includegraphics[scale=0.6]{optcost_vs_alpha.pdf}
    \caption{Variation of optimal least squares cost \eqref{eq:ls_opt_new} for different values of $\alpha$ for $n = 5$ and $\alpha_{tr} = 50$.}
    \label{fig:opt_alpha}
\end{figure}

\rev{We generate random network topologies on $n$ nodes where there is an edge between any two nodes with probability $0.25$. For each such network, we choose infection rates to be uniformly distributed between $0.03$ and $0.05$ and the recovery rate for each node to be uniformly distributed between $0.01$ to $0.03$. We let the SIR epidemic state trajectory evolve following \eqref{eq:sir_si} with an initial infected proportion of $0.01$ at a few randomly chosen nodes. We then generate synthetic testing data according to our observer model defined in \eqref{eq:cik_bin} and \eqref{eq:nonlin_obs_d} with the number of daily tests being uniformly distributed between $2000$ and $2050$ at each node and the (true) value of $\alpha_{tr}$ being $10$, $50$, and $100$.}

\begin{figure*}
\centering
    \includegraphics[scale=0.6]{testing_data.pdf}
    ~\includegraphics[scale=0.6]{inf_learntalpha_n10.pdf}
    ~\includegraphics[scale=0.6]{suc_learntalpha_n10.pdf}
    \caption{\small Synthetic testing data and comparison of true and predicted infected and susceptible proportions at two different nodes.}
    \label{fig:learned_vs_predicted}
\end{figure*}

\rev{We now use the synthetic testing data generated above to learn the (true) value of $\alpha_{tr}$ as well as the parameters of the epidemic model. We use testing data from $T_1 = 30$ and $T_2 = 60$, i.e., when the testing data is available once the disease has spread significantly among the population. We solve the (non-convex) least squares problem formulated in \eqref{eq:ls_opt_new} using optimization package YALMIP \cite{lofberg2004yalmip} with `fmincon' solver in MATLAB environment on a Desktop computer with an Intel Xeon 3.4GHz processor and 32GB of RAM for different integer values of $\alpha$ over the range $0.5\alpha_{tr}$ to $2\alpha_{tr}$ and examine the optimal cost value.}

\rev{In Table \ref{table:learn_alpha}, we report the mean and standard deviation of the values of $\alpha$ that achieve the smallest optimal cost for different values of $\alpha_{tr}$ over $10$ independent runs; each run with a different randomly chosen network topology and epidemic parameters. We consider networks with $n = 5$ and $n=10$.\footnote{\rev{We note that computation time was significantly higher (more than one hour) for values of $n \geq 15$ due to the non-convexity of the problem.}} The result shows that the optimal least square cost is smallest when $\alpha$ is approximately close to $\alpha_{tr}$ with a relatively low variance. In Fig. \ref{fig:opt_alpha}, we show how the optimal cost varies as a function of $\alpha$ for three different instances with $n = 5$ and $\alpha_{tr} = 50$. The variation is qualitatively similar in all other instances we have considered.}

\begin{table}[htb]
\centering
\caption{\small \rev{Learned value of $\alpha$ for different network sizes and $\alpha_{tr}$ for $10$ independent runs.}}
\label{tab:case3}
\begin{tabular}{l c  c  c c}
{} & {} & $\alpha_{tr} = 10$ & $\alpha_{tr}  = 50$ & $\alpha_{tr}  = 100$\\
\toprule
& Mean & $10.25$& $47$& $95.7$\\
$n = 5$ & Std. & $1.03$ &  $1.41$ & $2.87$\\
& Worst Dev. &$12$ & $45$ & $91$\\
\midrule
& Mean & $9.11$& $46.8$& $94.6$\\
$n = 10$ & Std. & $0.78$&  $2.25$& $3.17$\\
& Worst Dev. & $8$ &$44$ & 87 \\
\bottomrule
\end{tabular}
\label{table:learn_alpha}
\end{table}

\rev{We now illustrate how the inferred or predicted epidemic states compare with the true states. We consider a randomly generated network on $10$ nodes. The plot in the left panel of Fig. \ref{fig:learned_vs_predicted} shows the synthetic testing data at two nodes between $T_1 = 30$ to $T_2 = 60$ with $\alpha_{tr} = 10$. For ease of exposition, we assume $\tau = 0$. We first solve the problem in \eqref{eq:ls_opt_new} for different values of $\alpha$ and learn that the optimal least squares cost is smallest for $\alpha = 9$. We then learn the infection and recovery rates as well as the proportion of susceptible and infected nodes at time $T_1-1$ by solving \eqref{eq:ls_opt_new}. Finally, we predict the SIR epidemic states using the learned values following \eqref{eq:for}.}

\rev{The evolution of the true and predicted proportions of infected and susceptible individuals for the two different nodes ($v_2$ and $v_3$) are shown in the middle and right panels of Fig. \ref{fig:learned_vs_predicted}. As the figure shows, for node $3$, the epidemic has spread significantly by time $T_1$. However, our estimate of the initial proportion of infected and susceptible individuals lead to a very accurate predicted state trajectory. We also note that while our estimates of initial states and recovery rates tend to be quite accurate, the learned values of infection rates ($\beta_{ij}$ values) tend to be very different from their true values; nevertheless, the predicted states closely align with the true states, consistent with the observations in \cite{prasse2020network} (also see Remark \ref{remark:prasse}).}

\subsection{Performance of optimal NPIs}

\begin{figure}
\begin{subfigure}{.45\linewidth}
  \centering

  \begin{tikzpicture}[-,>=stealth',shorten >=1pt,auto,thick,main node/.style={circle,draw,minimum size = 0.6cm,inner sep=0pt}]

  \node[main node] (1) {$\mathtt{CH}$};
  \node[main node] (2) [below = 0.7cm of 1] {$\mathtt{IT}$};
  \node[main node] (3) [above = 0.7cm of 1] {$\mathtt{DE}$};
  \node[main node] (4) [right = 0.8cm of 1] {$\mathtt{AT}$};
  \node[main node] (5) [left = 0.8cm of 1] {$\mathtt{FR}$};

  \path[every node/.style={font=\sffamily\small}]
    (1) edge (3)
    (1) edge (5)
    (1) edge (4)
    (1) edge (2)
    (3) edge (5)

    (2) edge (5)
    (2) edge (4)
    (3) edge (4);
\end{tikzpicture}

\label{fig:graph}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
  \centering

  \includegraphics[width=\linewidth]{evolution.pdf}

    \label{fig:dynamics}
\end{subfigure}
\caption{Network topology and evolution of proportion of infected subpopulations without NPIs at nodes DE and IT.}
\label{fig:baseline}
\end{figure}

We now evaluate the effectiveness of the proposed online NPI allocation scheme via simulations. We consider a network with $5$ nodes whose topology is given in Fig.~\ref{fig:baseline}. The nodes are roughly modeled after five countries in continental Europe: France (FR), Germany (DE), Italy (IT), Austria (AT) and Switzerland (CH). Two nodes are neighbors if they share a border. \rev{The chosen infection and recovery rates are shown in Table \ref{tab:beta_gamma_baseline}, which satisfy Assumption \ref{assm:1} with $h = 1$.} The evolution of the infected proportions under these parameters at two nodes (DE and IT) with the initially infected proportions being $0.02$ at node IT and $0$ elsewhere is shown in Fig. \ref{fig:baseline}.

\begin{table}[htb]
\centering
\caption{\small \rev{Baseline infection and recovery rate parameters}}
\begin{tabular}{lllllll}
 & \multicolumn{5}{c}{$\beta$}         & $\gamma$ \\ \cline{2-6}
                  & DE   & FR   & AT   & IT   & CH   &                        \\ \hline
DE                & 0.05 & 0.05 & 0.05 & 0    & 0.05 & 0.03                   \\
FR                & 0.05 & 0.2  & 0    & 0.03 & 0.05 & 0.03                   \\
AT                & 0.05 & 0    & 0.2  & 0.05 & 0.04 & 0.03                   \\
IT                & 0    & 0.03 & 0.05 & 0.2  & 0.05 & 0.03                   \\
CH                & 0.05 & 0.05 & 0.04 & 0.05 & 0.2  & 0.03                   \\ \hline
\end{tabular}
\label{tab:beta_gamma_baseline}
\end{table}

We solve the problems stated in \eqref{eq:GP_SIR} and \eqref{eq:GP_SIR2} \rev{using the Python solver GPkit \cite{gpkit}}, and evaluate their performance in controlling the spread of the SIR epidemic. Following \cite{preciado2014optimal}, we consider the following cost functions for NPIs:
\begin{equation}\label{eq:NPI_cost_fn}
f_{ij}(\beta_{ij}) = \frac{\beta_{ij}^{-1} - \beta^{-1}_{u, ij}}{\beta^{-1}_{l, ij} - \beta^{-1}_{u, ij}}, \quad \text{and} \quad g_{i}(\bar{\gamma}_{i}) =  \frac{\bar{\gamma}_{i}^{-1} - \bar{\gamma}_{u,i}^{-1}}{\bar{\gamma}_{l,i}^{-1} - \bar{\gamma}_{u,i}^{-1}};
\end{equation}
recall that $\bar{\gamma}_{i} = 1-h\gamma_{i}$.

\begin{figure*}
\centering
    \includegraphics[scale=0.6]{rate_minimizer.pdf}
    ~\includegraphics[scale=0.6]{rate_minimizer2.pdf}
    ~\includegraphics[scale=0.6]{optimal_gamma_rate_min.pdf}
    \caption{\small Evolution of the infected proportions at node DE, optimal growth rate ($\lambda^*_k$) and optimal recovery rates ($\gamma^*$) under optimal budget-constrained NPIs obtained by solving \eqref{eq:GP_SIR}. Baseline scenario refers to the case without NPIs with parameters shown in Table \ref{tab:beta_gamma_baseline}.}
    \label{fig:curve_flattening}
\end{figure*}

\begin{figure*}
\centering
    \includegraphics[scale=0.6]{cost_minimization.pdf}
    ~\includegraphics[scale=0.6]{costmin2.pdf}
    ~\includegraphics[scale=0.6]{optimal_continuous_cost_min.pdf}
    \caption{\small Evolution of the infected proportion and the optimal cost of NPIs obtained by solving \eqref{eq:GP_SIR2} for different constraints on $\lambda^{A_k}_{\max}$.}
    \label{fig:secondwave}
\end{figure*}

Note that $f_{ij}$ is monotonically decreasing (i.e., it is costly to reduce contact rates), \rev{with $f_{ij}(\beta_{l,ij}) = 1$, $f_{ij}(\beta_{u,ij}) = 0$, and $f_{ij}(\beta_{ij}) \in [0,1]$ for $\beta_{ij} \in [\beta_{l, ij},\beta_{u, ij}]$. In other words, as $\beta_{ij}$ decreases from $\beta_{u,ij}$ to $\beta_{l,ij}$, the corresponding cost increases from $0$ to $1$.} The function $g_i$ has analogous properties.

\rev{In our evaluation, we choose the bounds such that $\bar{\gamma}_i \in [0.91,0.97]$, $\beta_{ii} \in [0.02,0.2]$ for $i \in [n]$ and $\beta_{ij} \in [0.005,0.05]$ when $i \neq j$.} The upper and lower limits on $\beta_{ii}$ and $\beta_{ij}$ are chosen differently to reflect the fact it is easier to reduce contact between individuals from different subpopulations compared to individuals within a subpopulation.

In order to isolate the performance of the online optimization approaches described in Section \ref{section:optimization}, we assume that $s(k)$ is exactly known, and report the results under NPIs obtained by solving \eqref{eq:GP_SIR} at every instance for different budget combinations. We assume that the initial fraction of infected nodes is $0.01$ in DE, and $0$ for all other nodes.

Fig.~\ref{fig:curve_flattening} shows the evolution of the proportions of the infected individuals at node DE, and the instantaneous growth rate $\lambda^{A_k}_{\max}$ at the optimal NPIs. The figure shows that higher budgets lead to ``flattening" of the curve of the infected proportion. The peak of the infected proportion is smaller and occurs much later compared to the baseline setting without NPIs. In addition, we observe that the cumulative number of people who become infected shows significant reduction depending on the budget. When the budget is sufficiently high, not many people become infected in the first place. Consequently, the proportion of susceptible individuals remains high, which (perhaps counter-intuitively) results in a larger value of $\lambda^*_k$ compared to the cases with smaller budgets for NPIs. Nevertheless, the infected proportion eventually decreases to $0$. \rev{The evolution of the optimal recovery rates with time for different nodes when $C_1 = 3.537$ and $C_2 = 3$ is shown in the right panel of Fig.~\ref{fig:curve_flattening}. Node CH receives a significantly higher share of the budget-constrained recovery rates, potentially due to its central location. These results highlight the importance of optimal allocation of NPIs exploiting the structure of the network.}

We then focus on computing cost optimal NPIs subject to constraints on the growth rate or the reproduction number. We solve the optimization problem in \eqref{eq:GP_SIR2} starting from $T_1 = 15$ with constraint $\lambda_k = 0.99$ for $k\in[16,100]$ and $\lambda_k = 1.05$ for $k\in[101,200]$. The evolution of the infected proportion at node DE and the optimal cost obtained by solving \eqref{eq:GP_SIR2} at every time instance are shown as solid (blue) curves in Fig.~\ref{fig:secondwave}. The dashed (black) curves show the evolution when the NPIs are computed only at $k=15$ and at $k=100$ and maintained over the intervals $k \in [16,100]$ and $k \in [101,200]$. \rev{The evolution of the infection rate parameters $\beta_{DE,j}$ for $j \in \{DE, FR, AT, CH\}$ are shown in the right panel of Fig.~\ref{fig:secondwave}. The figure shows that around $k = 150$, the proportion of susceptible individuals become negligible leading to the growth rate being inherently smaller than the desired value, and as a result, the NPIs are gradually withdrawn.}

Fig.~\ref{fig:secondwave} shows that early interventions reduced the growth of the epidemic, albeit at a high cost of NPIs. As a result, the proportion of susceptible individuals remains high for $k \in [16,100]$. Consequently, the baseline reproduction number remains high and it remains costly to maintain the growth-rate below $1$. Furthermore, if NPIs are withdrawn before the infected proportion becomes negligible, we observe an exponential increase in the infected proportion (i.e., a ``second wave" of infections).

Fig.~\ref{fig:secondwave} also highlights the importance of regularly adapting the NPI strategy using feedback from testing data. Recall from Theorem \ref{prop:main} that the baseline reproduction number is monotonically decreasing. Accordingly, if optimal NPIs are computed at every time step, then the optimal cost shows a gradual decrease and the optimal solution pertains to a gradual easing of NPIs while maintaining the desired growth rate. In contrast, when updated sporadically (as is the case with the dashed black curve), the NPIs are more stringent. This achieves better performance in reducing the spread of the epidemic, but at a much higher cost.

config: arxiv


TITLE: Convergence with distance $d$ implies convergence with distance $d'$
QUESTION [0 upvotes]: Let $E$ be a space and $d$ and $d'$ be two distances on $E$ with the property that $U$ open with $d$ implies that $U$ is open with $d'$. Show that if $x_n$ converges to $x$ with $d'$, then $x_n$ converges to $x$ with $d$.
Just to clarify, the property means that for a ball of radius $r$ centered at $x$ contained in $U$, if $d(x,x_0) < r$ implies $d'(x,x_0) < r$ for any $x_0$ in the ball, correct? Would this imply that $d'(x,x_0) < d(x,x_0)$ for all $x$ and $x_0$? Aside from that, I'm not sure where to go with this, so any pointers in the right direction are greatly appreciated.
Thank you very much!

REPLY [1 votes]: So your assumption says that $\mathcal{T}_d$ (the topology generated by the $d$-balls) is a subset of $\mathcal{T}_{d'}$ (ditto for $d'$). Now if $x_n \to x$ under $d'$, we have $x_n \to x$ under $d$: Let $r>0$ and consider $B(x,r) \in \mathcal{T}_d$, then $B_d(x,r)$ is thus open in the $d'$-topology and a $d'$-neighbourhood of $x$ and so, by convergence under $d'$, from some $N$ onwards we must have $x_n \in B_d(x,r)$, and so $\forall n \ge N: d(x_n, x) < r$, and we are done.
Having more open sets makes convergence a stronger property.


TITLE: How Do I solve The Below System Of Equations By Using Substitution?
QUESTION [0 upvotes]: $$a^2+b^2-c^2=0$$
$$5a-b=c$$
$$a+b+c=30$$
I have tried everything I could think of and I still don't know how to solve this.

REPLY [2 votes]: $$a^2+b^2-c^2=0\tag 1$$
$$5a-b=c\tag 2$$
$$a+b+c=30\tag 3$$
Using $(2)$, $b=5a-c$. Plug in $(3)$ to get $6a=30$ that is to say $a=5$ and $b=25-c$. Plug in $(1)$ to get $650-50 c=0$.


TITLE: Matlab, anonymous functions, making a piecewise function, how?
QUESTION [0 upvotes]: If I want to make a piecewise function where lets say $f(x) = x^2 +1$ at $1 \leq t \leq 2$ and zero elsewhere.  How can I make this possible using an anonymous function?  I wanted to invent y = @(t) ((t.^2 + 1)*(t>=1)&&(t<=2))
But that obviously did not work out at all.
Thanks

REPLY [2 votes]: The function you have works for scalar input, but I guess you want to make it for vectors, in that case you need to adjust it a bit:
y = @(t) ((t.^2 + 1).*(t>=1)&(t<=2))


TITLE: Motion equation given the following function, is this a potential function?
QUESTION [0 upvotes]: I'm a bit rusty in this stuff.
I'm trying to design an energy function that would allow me to derive a motion equation.
I'm considering the vector field
$$
F(x,y,z) = k\left( \frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}},0 \right)^T
$$
($k$ positive constant). I've checked the condition
$$
curl(F) = 0
$$
for all $x,y,z$, and indeed I get 0. However the components aren't differentiable in $(x,y) = 0$ which basically confuses me a little bit.
Is there a slightly modification I could do maybe to get something more sensible? But I'm not sure if this matters though.
The other question is I have the potential function
$$
U(x,y,z) = k \sqrt{x^2 + y^2}
$$
and I clearly have
$$
\nabla U = F
$$
How do I derive the motion equation? My intuition is that since
$$
F = ma = m \frac{d^2x}{dt^2}
$$
I have to relate both $U$ and the motion equation, is there some Euler lagrange equation (in general not for this specific case) I should use?

REPLY [2 votes]: You should use the cylindric coordinates $\phi$, $r$, $z$, where
$$[x,y,z]= [r\cos(\phi),r\sin(\phi),z].$$
We then have for the kinetic energy
$$
T = \frac{m}{2}\left(r^2\dot{\phi}^2 + \dot{r}^2 + \dot{z}^2\right)
$$
and for the potential simply
$$
U = k r.
$$
Now apply Euler-Lagrange:
$$
L = T - V
$$
and
$$
d_t\partial_{\dot{q}}L = \partial_q L$$
where $q\in \lbrace\phi,r,z\rbrace$. This will give you your equations of motion:
$$
\ddot{z} = 0,
$$
conservation of momentum in $z$-direction,
$$
r\ddot{\phi} + 2\dot{r}\dot{\phi} = 0 = \dot{J},
$$
conservation of angluar momentum $J = mr^2\dot{\phi}$, and finally
$$
m\ddot{r} = mr\dot{\phi}^2-k = \frac{J^2}{mr^3} - k.
$$
$J$ in the last equation is constant because of the conservation of angluar momentum. So now you are left with a decoupled equation for $r$.


TITLE: Expanding products of type $\prod_{k=1}^N (1 + \lambda_k x)$
QUESTION [0 upvotes]: Is there any closed form for the expansion of products of type
$\prod_{k=1}^N  (1 + \lambda_k x)$ for some sequence $\{\lambda_k\}_{k=1}^N$?
In particular, I'm interested in the case where $\lambda_k = k^{-t}$ for some fixed $t > 1$.

REPLY [1 votes]: Does the following help you (collecting coefficients of $x^n$)
$$\prod_{k=1}^N(1+\lambda_kx)=1+\sum_{n=1}^N\sum_{1\leq k_1<\dots<k_n\leq N}\lambda_{k_1}\dots\lambda_{k_n}x^n.$$
Substituting $\lambda_k=k^{-t}$, this becomes
$$\prod_{k=1}^N(1+\lambda_kx)=1+\sum_{n=1}^N\sum_{1\leq k_1<\dots<k_n\leq N}\left(k_1\dots k_n\right)^{-t}x^n.$$


TITLE: Is the quotient of a tensor product by a group a free module
QUESTION [1 upvotes]: Let $C_{2}$ be the group $\left\langle \tau|\tau^{2}=1\right\rangle $
let $X$ be a finite $C_{2}$-set. Let $\mathbb{Z}^{X}$ be the free
$\mathbb{Z}$-module whose generators are $X$. We naturally extend
the action of $C_{2}$ on $X$ to an action on $\mathbb{Z}^{X}$.
Let $x,y,z\in X$. we define an action of $C_{2}$ on $\mathbb{Z}^{X}\otimes\mathbb{Z}^{X}\otimes\mathbb{Z}^{X}$
by $\tau\left(x\otimes y\otimes z\right)=\tau\left(z\right)\otimes\tau\left(y\right)\otimes\tau\left(x\right)$.
Let
$$
\mathbb{Z}^{X}\otimes\mathbb{Z}^{X}\otimes\mathbb{Z}^{X}/\left\langle \tau\left(x\otimes y\otimes z\right)-x\otimes y\otimes z\right\rangle
$$
 What is its rank?

REPLY [1 votes]: If $Y$ is a $C_2$-set then $C_2$ acts on the free $\mathbb{Z}$-module $\mathbb{Z}Y$. The quotient map $Y\to Y/C_2$ extends linearly to a module homomorphism $\mathbb{Z}Y\to\mathbb{Z}Y/C_2$ whose kernel is generated by all elements of the form $\sigma y-y$, where $y\in Y$ and $\sigma\in C_2$. Thus, $\mathbb{Z}Y/C_2$ is isomorphic to the module of coinvariants $\mathbb{Z}Y_{C_2}=\mathbb{Z}Y/\langle \sigma y-y\rangle$.
In your case, $\mathbb{Z}X\otimes\mathbb{Z}X\otimes \mathbb{Z}X\cong \mathbb{Z}(X\times X\times X)$, which respects the $C_2$ action, so we are treating the set as $Y=X^3$. To find the rank, we simply need the number of orbits in $X^3$.
Let $n=|X|$ and let $f$ be the number of fixed points of $C_2\curvearrowright X$. Then the number of fixed points in $X^3$ is $f^3$, and the number of other orbits is the number of non-fixed-points halved, i.e. the rank is
$$ f^3+\frac{n^3-f^3}{2}. $$


TITLE: Hausdorff measure of an embedded submanifold
QUESTION [1 upvotes]: Let $ M $ be a $ C^1 $- embedded n-submanifold (without boundary) of $ R^{n+k} $. Is it true that for every $ K $ compact set in $ R^{n+k} $ the n-dimensional Haussdorf measure of $ M \cap K $ is finite?
Thanks

REPLY [0 votes]: Not in general, as the example $y=\sin(1/x)$, $0<x<1$, shows (given by Ilya as a comment).
But the answer is yes if $M$ is a closed subset of $\mathbb R^{n+k}$. Indeed, by the usual compactness argument it suffices to show that every point $x\in \mathbb R^{n+k}$ has a neighborhood $U$ such that $\mathcal H^n(M\cap U)<\infty$.

If $x\notin M$ then there is a neighborhood $U$ such that $M\cap U=\varnothing$.
If $x\in M$ then there is a neighborhood $V$ such that $M\cap V$ is diffeomorphic to a hyperplane with a diffeomorphism of $V$. This diffeomorphism is Lipschitz on any  compactly contained neighborhood $U\Subset V$. This $U$  satisfies $\mathcal H^n(M\cap U)<\infty$.