text:
\section{Analytical Preliminaries} \label{Sec:Preliminaries}

Notationally, we equip the function spaces $L^p(\Omega)$, $L^p(\Lambda)$, $W^{m,p}(\Omega)$, $W^{m,p}(\Lambda)$ with the norms $|\cdot|_{L^p(\Omega)}$, $|\cdot|_{L^p(\Lambda)}$, $|\cdot|_{W^{m,p}(\Omega)}$, $|\cdot|_{W^{m,p}(\Lambda)}$. In the case of $d$-dimensional vector functions, we write $L^p(\Omega;\R^d)$ and in the same way for the other Banach spaces, but we do not make this distinction in the notation of norms, scalar products and applications with its dual.

Throughout this paper, $C<\infty$ stands for a generic constant, which may change from line to line. For brevity, we write $x \lesssim y$ for $x \leq Cy$. We recall the Poincar\'e--Wirtinger and Sobolev inequalities, see \cite{brezis2010functional,roubicek,evans2010partial},
\begin{equation}\begin{aligned}
	|f-f_\Omega|_{L^p(\Omega)} & \lesssim |\nabla f|_{L^p(\Omega)} && \text{for all } f \in W^{1,p}(\Omega),   \\
	|f|_{L^p(\Omega)}              & \lesssim  |\nabla f|_{L^p(\Omega)} && \text{for all } f \in W^{1,p}_0(\Omega), \\[-.18cm]
	 |f|_{W^{m,q}(\Omega)} & \lesssim |f|_{W^{k,p}(\Omega)} && \text{for all } f \in W^{k,p}(\Omega), \quad k-\frac{d}{p}  \geq m-\frac{d}{q} , \quad k\geq m,
\end{aligned} \label{Eq:SobolevInequality} \end{equation}
where $p,q \in [1,\infty)$ and $f_\Omega=\frac{1}{|\Omega|}\int_\Omega f(x) \,\textup{d} x$ denotes the mean of $f$ with respect to $\Omega$. Also, the last inequality yields the continuous embedding $W^{k,p}(\Omega) \hookrightarrow W^{m,q}(\Omega)$.

For a given Banach space $X$, we define the Bochner space, see e.g.,\cite{diestel1977vector},
$$L^p(0,T;X)=\{ u :(0,T) \to X: u \text{ is strongly measurable, } \int_0^T |u(t)|_X^p \,\text{d$t$} < \infty \},$$
where $1 \leq p < \infty$, with the norm $\|u\|_{L^p X}^p = \int_0^T |u(t)|_X^p \,\textup{d} t$. For $p=\infty$, we equip $L^\infty(0,T;X)$ with the norm $\|u\|_{L^\infty X} = \text{ess\,sup}_{t\in (0,T)} |u(t)|_X$. Moreover, we introduce the Sobolev--Bochner space,
$$W^{1,p}(0,T;X)=\{ u \in L^p(0,T;X) : \pt u \in L^p(0,T;X) \}.$$

Let $X$, $Y$, $Z$ be Banach spaces such that $X$ is compactly embedded in $Y$, and $Y$ is continuously embedded in $Z$, i.e.,  $X\com Y \con Z$. In the proof of the existence theorem below, we make use of the Aubin--Lions--Simon compactness lemma, see \cite[Corollary 4]{simon1986compact},
\begin{equation}\begin{alignedat}{2} L^p(0,T;X) \cap W^{1,1}(0,T;Z) &\com L^p(0,T;Y), &&\quad 1\leq p<\infty, \\
		L^\infty(0,T;X) \cap W^{1,r}(0,T;Z) &\com C^0([0,T];Y), &&\quad r >1,
	\end{alignedat}
	\label{Eq:EmbeddingComp}
\end{equation}
where we equip an intersection space $X \cap Y$ with the norm $\|\cdot\|_{X\cap Y}=\max\{\|\cdot\|_X,\|\cdot\|_Y\}$. Further, we make use of the following continuous embeddings, see \cite[Theorem 3.1, Chapter 1]{lions2012non},
\begin{equation}\begin{alignedat}{2} L^2(0,T;Y)  & \cap H^1(0,T;Z)   &  & \con C^0([0,T];[Y,Z]_{1/2}),  \\
	L^\infty(0,T;Y) & \cap C_w([0,T];Z) &  & \con C_w([0,T];Y),
\end{alignedat}
\label{Eq:EmbeddingCont}
\end{equation}
where $[Y,Z]_{1/2}$ denotes the interpolation space between $Y$ and $Z$, see \cite[Definition 2.1, Chapter 1]{lions2012non} for more details. Also, $C_w([0,T];Y)$ denotes the space of the weakly continuous functions on the interval $[0,T]$ with values in $Y$.

We note the following special case of the Gagliardo--Nirenberg inequality, see \cite[Lemma II.2.33]{boyer2012mathematical},
$$|f|_{L^p(\Omega)} \lesssim |f|_{H^1(\Omega)}^\alpha |f|_{L^2(\Omega)}^{1-\alpha} \quad  \text{for all } f \in H^1(\Omega), \quad \frac{1}{p} = \frac12 - \frac{\alpha}{3}, \quad \alpha \in [0,1],$$
which gives in a time-dependent setting, choosing $\alpha=2/q$ with $q \geq 2$,
\begin{equation} \label{Eq:GagliardoTime}
\begin{aligned}\|u\|_{L^q(0,T;L^p(\Omega))}^q = \int_0^T |u(t)|_{L^p(\Omega)}^q \dt &\lesssim \int_0^T |u(t)|_{H^1(\Omega)}^{q \alpha} |u(t)|_{L^2(\Omega)}^{q(1-\alpha)} \dt  \\ &= \int_0^T | u(t)|_{H^1(\Omega)}^2 |u(t)|_{L^2(\Omega)}^{q-2} \dt  \\ &\leq \| u\|_{L^2(0,T; H^1(\Omega))}^2 \|u\|_{L^\infty(0,T;L^2(\Omega))}^{q-2}
\\ &\leq (\max \{ \|u\|_{L^\infty(0,T;L^2(\Omega))}, \| u\|_{L^2(0,T; H^1(\Omega))} \})^q
.\end{aligned}
\end{equation}
In particular, it yields the continuous embedding
$$L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega)) \con L^q(0,T;L^p(\Omega)), \quad \frac{1}{p} + \frac{2}{3q} = \frac12. $$

We also make use of the classical Gr\"onwall--Bellman lemma in the energy estimates to absorb solution-dependent terms on the right hand side of the energy inequalities.
\begin{lemma}[Gr\"onwall--Bellman, cf. {\cite[Lemma II.4.10]{boyer2012mathematical}}] Let $u \in L^\infty(0,T)$, $g \in L^1(0,T;\R_{\geq 0})$ and $u_0 \in \mathbb{R}$. If we have
	$$u(t) \leq u_0 +  \int_0^t g(s) u(s) \, \dd s \quad \text{ for a.e. } t \in (0,T),$$
	then it holds $u(t) \leq u_0 \exp(\int_0^t g(s) \, \dd s)$ for almost every $t \in (0,T)$.
	\label{Lem_Gronwall}
\end{lemma}

\section{Existence of Solutions} \label{Sec:Theorem}

In this section, we lay down some general assumptions on the model that are in force throughout this paper. Under these assumptions, we state the definition of a weak solution, and we then state a theorem, which provides the existence of a weak solution.

For simplicity, we write
$$\begin{gathered}
			S_\alpha = S_\alpha( \phib), \quad
			m_\beta = m_\beta( \phib), \quad
			\Psi = \Psi(\phi_P,\phi_H,\phi_N), \\  J_{p v} = J_{p v}( \ov p, p_v), \quad J_{p v,\Gamma} = J_{p v}(  p,\Pi_\Gamma p_v), \quad J_{\sigma v} = J_{\sigma v}(\ov \phi_\sigma, \ov p, \phi_v,p_v), \quad J_{\sigma v,\Gamma} = J_{\sigma v}(\phi_\sigma, p, \Pi_\Gamma \phi_v,\Pi_\Gamma p_v),
\end{gathered} $$
where $\alpha \in \A$ and $\beta \in \A \backslash \{N,\ECM\}$. \blue{We introduce the scaled parameters $\wdr=2\pi R_i$ and $\wdk=R_i^2 \pi K_{v,i}$ in order to express the 1D model \cref{Eq:Model1D} in a shorter way. Moreover, we define the cut-off operator
	\begin{equation} \mathcal{C}(x)=\max\{0,\min\{1,x\}\}.\label{Eq:Cutoff}
	\end{equation} }
	Moreover, we introduce the following abbreviations for frequently appearing function spaces,
$$\begin{alignedat}{4}
		&V = H^1(\Omega) &&\con \quad H = L^2(\Omega) \quad&&\con\quad  V'=(H^1(\Omega))' ,\\
			&V_0 = H^1_D(\Omega) &&\con \quad H = L^2(\Omega) &&\con\quad  V_0'=(H^1_D(\Omega))' ,\\
		&W = W^{1,3/2}(\Omega) ~&&\con \quad H=L^2(\Omega) &&\con\quad  W'=(W^{1,3/2} (\Omega))', \\
		&X= H^1(\Lambda) &&\con \quad Y = L^2(\Lambda) &&\con\quad  X'=(H^1(\Lambda))', \\
		&X_0 = H^1_D(\Lambda) &&\con \quad Y = L^2(\Lambda) &&\con\quad  X_0'=(H^1_D(\Lambda))',

	\end{alignedat}$$
where we have denoted the Sobolev space of vanishing trace on $\p \Omega_D \subset \p \Omega$ by $H_D^1(\Omega) = \{u \in H^1(\Omega) : u|_{\partial \Omega_D} =0 \}$  and in the same way $H_D^1(\Lambda) = \{u \in H^1(\Lambda) : u|_{\partial \Lambda_D} =0 \}$. We equip these spaces of vanishing trace with the norms $|\cdot|_{V_0}=|\nabla \cdot|_H$ and $|\cdot|_{X_0} = |\nablala \cdot|_Y$, respectively. \blue{Here, we use the notation $\nablala$ for the space derivative of the 1D fields.}

The space $W$ with the Lebesgue order $3/2$ becomes useful in the application of the H\"older inequality. Indeed, we have the relation $\frac23 = \frac16 + \frac12$, and therefore, we obtain
	$$|u \varphi|_{L^{3/2}(\Omega)} \leq |u|_{L^6(\Omega)} |\varphi|_H \lesssim |u|_V |\varphi|_H \quad \text{for all } u \in V, \varphi \in H,$$
	where we also applied the Sobolev embedding theorem $V\con L^6(\Omega)$ in the three-dimensional domain $\Omega$. Hence, we have for all $u , \varphi \in V$,
	\begin{equation} \label{Eq:Holder32}
		|u\varphi|_W = \Big(|u\varphi|_{L^{3/2}(\Omega)}^{3/2} + |\nabla(u\varphi)|_{L^{3/2}(\Omega)}^{3/2} \Big)^{2/3} \leq |u\varphi|_{L^{3/2}(\Omega)} + |\nabla(u\varphi)|_{L^{3/2}(\Omega)} \lesssim |u|_V |\varphi|_V,
	\end{equation}
	where we used the Bernoulli inequality to obtain $(a+b)^r \leq a^r + b^r$ with $a,b \geq 0$, $r \in [0,1]$.

\begin{assumption} \, ~
	\begin{enumerate}[label=(A\arabic*), ref=A\arabic*] \itemsep.25em
		\item $\Omega\subset \R^3$ is a bounded domain with $C^{1,1}$-boundary, \blue{$\Lambda$ is a 1D structure as depicted in \cref{fig:1DNetworkc}, $\Gamma$ is the 2D associated cylindrical surface, see \cref{fig:1DNetworkd},} and $T>0$ denotes a finite time horizon,
		\item $\phi_{\alpha,0} \in V$ for all $\alpha \in \CH \cup \{\ECM\}$, $\phi_{\beta,0} \in H$ for all $\beta \in \RD$, $\phi_{v,0} \in Y$, $\phi_{v,\infty}, p_{v,\infty} \in H^1(0,T) \subset C([0,T])$ and $p_\infty \in H^1(0,T;H) \cap L^2(0,T;V) \subset C([0;T];H)$, \label{Ass:Initial}
		\item $\chi_c,\chi_h \geq 0$ and $\eps_\alpha, D_\beta, \blue{C_\sigma,} \wdk, \wdr >0$ for $\alpha \in \{P,H\}$, $\beta \in \RD$, \label{Ass:Parameters}
		\item $S_\alpha$ are of the form

		      $$\begin{aligned}
				      S_\alpha(\phib) & =  \sum_{\gamma \in \A} \phi_\gamma f_{\alpha,\gamma} (\phib) ,  &&\alpha \in  \A \backslash \{ N, \ECM\}, \\
				      S_\beta(\phib)  & = f_\beta( \phib), &&\beta \in \{N,\ECM\}, \\
				      \hspace{-5cm}\!\! S_p(\phib,\mu_P,\mu_H) &= \rlap{$\displaystyle \begin{multlined}[t] -\mathcal{C}(\phi_P) (\nabla \mu_P+\chi_c \nabla \phi_\sigma+\chi_h \nabla \ecm) \\ \qquad -\mathcal{C}(\phi_H) (\nabla \mu_H + \chi_c \nabla \phi_\sigma+\chi_h \nabla \ecm), \end{multlined}$}
			      \end{aligned}$$
			  where $f_{\alpha,\gamma} \in C_b(\R^{|\A|})$, $f_\beta \in \Lip(\R^{|\A|}) \cap PC^1(\R^{| \A|})$, such that $|f_{\alpha,\gamma}|, |f_\beta|, |\p_{\phi_\gamma} f_\beta| \leq f_\infty$ for all $\alpha \in \A\backslash\{N,\ECM\}$, $\beta \in \{N,\ECM\}$, $\gamma \in \A$, \label{Ass:Source}
				\item $J_{pv}$ and $J_{\sigma v}$ are of the form $$\begin{aligned} J_{pv}(y_1,y_2)&=L_p (y_2-y_1), \\ J_{\sigma v}(x_1,y_1,x_2,y_2) &=f_{\sigma,v} (x_1,x_2) J_{pv}(y_1,y_2) + L_\sigma (x_2-x_1),\end{aligned}$$ where $f_{\sigma,v} \in C_b(\mathbb{R}^2)$ such that $|f_{\sigma,v}(x)|\leq f_\infty$ \blue{for all $x \in \R^2$} and $L_p, L_\sigma, K \geq 0$ are sufficiently small in the sense that the prefactors in \cref{Eq:Collect} are positive,

		\label{Ass:Jv}
		\item $m_\alpha \in C_b(\R^{| \A|})$ such that  $0<m_0 \leq m_\alpha(x) \leq m_\infty$ \blue{for all $x \in \R^{|\A|}$} for all $\alpha \in \A \backslash \{N,\ECM\},$ \label{Ass:Mobility}
		\item \label{Ass:Psi} $\Psi \in C^{1}(\R^3)$ non-negative such that $\Psi(0,0,0)=\Psi'(0,0,0)=0$, and there are constants $C_{\Psi_j}$, $j \in \{1,\dots,3\}$, such that \blue{for all $(x,y,z) \in \R^3$ it holds} $$\begin{aligned} \Psi(x,y,z) &\geq C_{\Psi_1} (|x|^2+|y|^2+|z|^2) - C_{\Psi_2},

		\\ |\p_x \Psi(x,y,z)|, |\p_y \Psi(x,y,z)|, |\p_z \Psi(x,y,z)| &\leq C_{\Psi_3}(1+|x|+|y|+|z|) .\end{aligned}$$
	\end{enumerate}
	\label{Ass:Assumption}
\end{assumption}

Remarks on the assumptions:
	\begin{itemize} \itemsep1em
	\item[\cref{Ass:Source}] After a suitable reformulation of the source functions \cref{Eq:DerivationSourceCH} and \cref{Eq:DerivationSourceRD} with the cut-off operator $\C$, see \cref{Eq:Cutoff}, and replacing the Heaviside functions by the continuous Sigmoid function, the source functions can be brought into the form as stated in assumption \cref{Ass:Source}. Further, the assumption  $f_\beta \in \Lip(\R^{|\A|}) \cap PC^1(\R^{| \A|})$, $\beta \in \{N,\ECM\}$, ensures the validity of the chain rule if $f_\beta$ is composed with a vector-valued Sobolev function; see \cite{murat2003chain,leoni2007necessary}. In particular, we have for all $\alpha \in \A$,
	$$(\nabla f_\beta(\phib), \nabla \phi_\alpha)_H = \sum_{\gamma \in \A} ( \p_{\phi_\gamma} f_\beta(\phib) \nabla \phi_\gamma, \nabla \phi_\alpha)_H \leq f_\infty \sum_{\gamma \in \A} |\nabla \phi_\gamma|_H |\nabla \phi_\alpha|_H .$$
	\item[\cref{Ass:Jv}] We consider the unique, linear and continuous trace operator, see \cite{grosse2013sobolev},
	$$\tr_\Gamma: W \to W^{1/3,3/2}(\Gamma)\text{ such that }\tr_\Gamma u = u_{|\Gamma} \text{ for } u \in C^\infty(\Omega), $$ onto the two dimensional \blue{associated cylindrical} surface $\Gamma$ of the one-dimensional network $\Lambda$, \blue{see \cref{fig:1DNetwork}}. In two dimensions, we can apply the Sobolev embedding theorem to obtain $W^{1/3,3/2}(\Gamma) \con L^2(\Gamma)$, see \cref{Eq:SobolevInequality}. Note that this embedding does not hold in three dimensions. Consequently, we have
	\begin{align*}  \| \delta_\Gamma \|_{W'} = \sup_{|\varphi|_W \leq 1}| \langle \delta_\Gamma, \varphi \rangle_W |= \sup_{|\varphi|_W \leq 1} \Big|\int_{\Gamma} \tr_\Gamma\varphi(s) \ds \Big| &\leq \sup_{|\varphi|_W \leq 1}   |\tr_\Gamma\varphi|_{L^1(\Gamma)} \\ &\leq C_{W^{1/3,3/2}(\Gamma)}^{L^1(\Gamma)} |\tr_\Gamma|_{\mathcal{L}(W;W^{1/3,3/2}(\Gamma))},
	\end{align*}
	where $C_{W^{1/3,3/2}(\Gamma)}^{L^1(\Gamma)}$ denotes the embedding constant from $W^{1/3,3/2}(\Gamma) \hookrightarrow L^1(\Gamma)$. Therefore, we have $\delta_\Gamma \in W'$  and  in the following existence proof we often apply the estimate for $\varphi \in W$
	\begin{equation}\begin{aligned} \langle \delta_\Gamma, J_{\alpha v,\Gamma}\varphi \rangle_W = \int_\Gamma J_{\alpha v,\Gamma} \tr_\Gamma\varphi(s) \ds &\leq |J_{\alpha v,\Gamma}|_{L^2(\Gamma)} |\tr_\Gamma \varphi|_{L^2(\Gamma)} \leq C_\Gamma |J_{\alpha v,\Gamma}|_{L^2(\Gamma)} |\varphi|_W,
	\end{aligned} \label{Eq:EstimateDelta}
	\end{equation}
	for $\alpha \in \{\sigma,p\}$, where $$C_\Gamma= C_{W^{1/3,3/2}(\Gamma)}^{L^2(\Gamma)}|\tr_\Gamma|_{\mathcal{L}(W;W^{1/3,3/2}(\Gamma))}.$$
	Further, we can estimate the fluxes by
	\begin{equation} \label{Eq:EstimateJ}
	\begin{aligned}
	|J_{p v,\Gamma}|_{L^2(\Gamma)} &\leq L_p  (C_\Gamma |p|_W+ |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))} |p_v|_Y), \\
	 |J_{\sigma v,\Gamma}|_{L^2(\Gamma)} &\leq f_\infty L_p  (C_\Gamma |p|_W+ |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))} |p_v|_Y) +  L_\sigma (C_\Gamma|\phi_\sigma|_W+|\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))}|\phi_v|_Y).
 	\end{aligned}
	\end{equation}
	The assumption of smallness of $L_p$ and $L_\sigma$ is generally accepted in the analysis of very weak solution of the stationary Navier--Stokes equation. There, one also considers a distributional divergence, which should be sufficiently small, see \cite{kim2009existence}. Additionally, in \cite{dangelo2008coupling} the authors have shown well-posedness of an abstract stationary 3D-1D model if the prefactor of the Dirac delta functional is sufficiently small.

	\item[\cref{Ass:Psi}] The assumption on the potential $\Psi$ is quite typical in the analysis of Cahn--Hilliard equations, see also \cite{fritz2018unsteady,fritz2019local}.
	In order to take the fourth order polynomial $(x+y+z)^2(1-x-y-z)^2$, we have to extend it by a quadratic function outside of the interval $[0,1]$, i.e.,
	$$\Psi(x,y,z)=\begin{cases} (x+y+z)^2, &x+y+z <0, \\ (x+y+z)^2(1-x-y-z)^2, &x+y+z \in [0,1], \\(1-x-y-z)^2, & x+y+z >1, \end{cases}  $$
	and one can show that $\Psi \in C^2(\R^3;\R)$.

	We invoke from \cref{Ass:Psi} and the fundamental lemma of calculus the upper estimate
	\begin{equation}  \label{Eq:PsiGrowth}
	\begin{aligned}
	\Psi(x,y,z) &= \Psi(0,y,z)+ \int_0^x \partial_x \Psi(\tilde x,y,z) \,\dd \tilde x \\
	&= \Psi(0,0,0) + \int_0^x \p_x \Psi(\tilde x,y,z) \, \dd \tilde x + \int_0^y \p_y \Psi(0,\tilde y, z) \, \dd \tilde y + \int_0^z \p_z \Psi(0,0,\tilde z) \, \dd \tilde z \\ &\lesssim 1+|x|^2+|y|^2+|z|^2.
	\end{aligned}
	\end{equation}
\end{itemize}

We define a weak solution of the coupled 3D-1D system, see \cref{Eq:Model3D} and \cref{Eq:Model1D}, in the following way.

\begin{definition}[Weak solution] \label{Def:Weak} We call the tuple $( \phib, \mu_P, \mu_H, v, p, \phi_v, v_v, p_v)$ a weak solution of \cref{Eq:Model3D} and \cref{Eq:Model1D} with boundary data \cref{Eq:Initial3D} and \cref{Eq:Initial1D} if the functions $ \phib:(0,T) \times \Omega \to \R^{|\A|}$, $\mu_P, \mu_H, v, p, \phi_v, v_v, p_v: (0,T) \times \Omega \to \R$ have the regularity
	\begin{equation} \begin{aligned}
		  \phi_\alpha  &\in H^1(0,T;V') \cap L^\infty(0,T;V),      &&\alpha \in \{P,H\},              \\
		  \mu_\alpha    & \in L^2(0,T;V),            &&\alpha \in \{P,H\} ,       \\
		  \phi_\beta  &\in H^1(0,T;H) \cap L^\infty(0,T;V),      &&\beta \in \{N,\ECM\},                               \\
		 \phi_\gamma    & \in H^1(0,T;V') \cap L^\infty(0,T;H) \cap L^2(0,T;V) , && \gamma \in \RD,  \\

		  (v,p-p_\infty) &\in L^2((0,T)\times \Omega;\R^3)\times L^2(0,T;V_0),  \\
		  \phi_v-\phi_{v,\infty}  &   \in H^1(0,T;X_0') \cap L^\infty(0,T;Y) \cap L^2(0,T;X_0) ,   \\
		  (v_v,p_v-p_{v,\infty}) & \in L^2(0,T;Y) \times L^2(0,T;X_0),
	\end{aligned} \label{Eq:DefSolutionRegularity} \end{equation}
	fulfill the initial data $\phi_\alpha(0)=\phi_{\alpha,0}$, $\alpha \in \A$, $\phi_v(0)=\phi_{v,0}$, and satisfy the following variational form of \cref{Eq:Model3D},
	\begin{equation}  \label{Eq:ModelWeak3D}\begin{aligned}
			\langle \pt \phi_P , \varphi_1 \rangle_{W^{1,3}(\Omega)} -  (\mathcal{C}(\phi_P) v,\nabla \varphi_1)_H +(m_P \nabla \mu_P,\nabla \varphi_1)_H               & = (S_P,\varphi_1)_H,
			\\
			-(\mu_P,\varphi_2)_H + (\p_{\phi_P} \Psi,\varphi_2)_H + \eps^2_P (\nabla \phi_P,\nabla \varphi_2)_H                                         & = \chi_c (\phi_\sigma,\varphi_2)_H+\chi_h (\ecm,\varphi_2)_H,
			\\
			\langle \pt \phi_H , \varphi_3 \rangle_{W^{1,3}(\Omega)} - (\C(\phi_H) v,\nabla \varphi_3)_H +(m_H \nabla \mu_H,\nabla \varphi_3)_H                          & = (S_H,\varphi_3)_H       ,        \\
			-(\mu_H,\varphi_4)_H + (\p_{\phi_H}\Psi,\varphi_4)_H + \eps^2_H (\nabla \phi_H,\nabla \varphi_4)_H                                         & = \chi_c (\phi_\sigma,\varphi_4)_H+\chi_h (\ecm,\varphi_4)_H,
			\\
			(\pt \phi_N , \varphi_5)_H                                                                                & = (S_N,\varphi_5)_H,
			\\
			\langle \pt \phi_\sigma,\varphi_6 \rangle_{W^{1,3}(\Omega)}  - ( \C(\phi_\sigma)v,\nabla \varphi_6)_H +  D_\sigma(m_\sigma \nabla \phi_\sigma, \nabla \varphi_6)_H & = (S_\sigma,\varphi_6)_H + \langle \delta_\Gamma, J_{\sigma v,\Gamma} \varphi_6\rangle_W            \\ &\qquad -\chi_c (m_\sigma \nabla (\phi_P +\phi_H  ),\nabla \varphi_6)_H,
			\\
			\langle \pt \mde,\varphi_7 \rangle_V +  D_{\MDE} (m_{\MDE} \nabla \mde,\nabla \varphi_7)_H                                                   & = (S_{\MDE},\varphi_7)_H,
			\\
			\langle \pt \taf,\varphi_8 \rangle_V + D_{\TAF} (m_{\TAF} \nabla \taf,\nabla \varphi_8)_H                                                    & = (S_{\TAF},\varphi_8)_H,
			\\
			\langle \pt \ecm,\varphi_9 \rangle_V                                                                                & = (S_{\ECM},\varphi_9)_H,
			\\
			(v,\varphi_{10})_H &= -K (\nabla p, \varphi_{10})_H + K (S_p,\varphi_{10})_H,
			\\
			K(\nabla  p,\nabla \varphi_{11})_H                                                                                & = \langle \delta_\Gamma, J_{pv,\Gamma} \varphi_{11} \rangle_{W}  +K(S_p,\nabla \varphi_{11})_H,

		\end{aligned}
	\end{equation}
	for all $\varphi_j \in V$, $j \in \{1,\dots,9\}$, $\varphi_{10} \in L^2(\Omega;\mathbb{R}^3)$, $\varphi_{11} \in V_0$, and the variational form of \cref{Eq:Model1D},
	\begin{equation}  \begin{aligned}
			\langle \pt \phi_v, \varphi_{12} \rangle_X -  (\C(\phi_v) v_v ,\nablala \varphi_{12} )_Y + D_v (m_v \nablala \phi_v,\nablala \varphi_{12})_Y & = -\wdr(J_{\sigma v},\varphi_{12})_Y  ,   \\
			(v_v, \varphi_{13})_Y &= - \wdk (\nablala p_v,\varphi_{13})_Y, \\
			\wdk( \nablala p_v, \nablala \varphi_{14} )_Y                                                                                & =  -\wdr(J_{pv},\varphi_{14})_Y,
		\end{aligned}
		\label{Eq:ModelWeak1D}
	\end{equation}
	for all $\varphi_j \in X_0$, $j\in \{12,14\}$, $\varphi_{13} \in Y$.
\end{definition}

The initial data $\phi_\alpha(0)=\phi_{\alpha,0}$, $\alpha \in \A$, are well-defined with assumption \cref{Ass:Initial} on the regularity of the initial data. Indeed, from the regularity given in \cref{Eq:DefSolutionRegularity}, we achieve, by the embeddings \cref{Eq:EmbeddingCont}, the continuity-in-time regularity
	$$\begin{aligned}
		\phi_\alpha   &\in C^0([0,T];H) \cap C_w([0,T];V),  && \alpha \in \CH \cup \{\ECM\}, \\
		\phi_\beta &\in C^0([0,T];V') \cap C_w([0,T];H),  &&\beta \in \RD, \\
		\phi_v &\in C^0([0,T];X_0') \cap C_w([0,T];Y),  &&
   \end{aligned}$$
and therefore, $\phi_\alpha(0)$ is well-defined in $H$, $\phi_\beta(0)$ in $V'$ and $\phi_v(0)$ in $X_0'$.

We use a mixed boundary approach for $p, \phi_v, p_v$, e.g., for the pressure $p$ we define $\tilde p= p-p_\infty$ with $\tilde p|_{\p \Omega_D}=0$ and $(\p_n \tilde p+ \p_n p_\infty)_{\p \Omega \backslash \p \Omega_D}=0$. Hence, we consider the partial differential equation
$$-\div (K \nabla \tilde p) - \div (K \nabla p_\infty) = \delta_\Gamma J_{pv,\Gamma} - \div S_p,$$
with the weak form with the test function $q \in V_0$
$$K (\nabla \tilde p + \nabla p_\infty, \nabla q)_H - K (\p_n \tilde p + \p_n p_\infty ,q)_{L^2(\p \Omega)} = \langle \delta_\Gamma, J_{pv,\Gamma} q \rangle_W + (K S_p, \nabla q)_H - (K S_p \cdot n,q)_{L^2(\p \Omega)},   $$
or, after the cancellation of the boundary terms,
$$K (\nabla p , \nabla q)_H  = \langle \delta_\Gamma, J_{pv,\Gamma} q \rangle_W + (K S_p, \nabla q)_H.   $$

The main result of this paper involves stating the existence of a weak solution of the 3D-1D model, see \cref{Eq:Model3D} and \cref{Eq:Model1D},  in the sense of \cref{Def:Weak}.

\begin{theorem}[Existence of a global weak solution] Let \cref{Ass:Assumption} hold.
	Then there exists a weak solution tuple $( \phib, \mu_P, \mu_H, p, \phi_v, p_v)$ to the 3D-1D model in the sense of \cref{Def:Weak}, which additionally satisfies the energy inequality
	\begin{equation} \label{Eq:EnergySolution}
		\begin{aligned}
				& \quad \|\Psi\|_{L^\infty(0,T;L^1(\Omega))} +\sum_{\mathclap{\alpha \in \CH \cup \{\ECM\}}} ~ \| \phi_\alpha\|^2_{L^\infty(0,T;V)} + ~\sum_{\mathclap{\beta \in \{P,H\}}} ~ \| \mu_\beta \|_{L^2(0,T;V)}^2 + \sum_{\gamma \in \RD}
			   \|\phi_\gamma\|^2_{L^\infty(0,T;H) \cap L^2(0,T;V)}   \\[-.1cm] &\qquad +\|v\|_{L^2(0,T;H)}^2 + \|p\|_{L^2(0,T;V)}^2 +\|\phi_v\|^2_{L^\infty(0,T;Y) \cap L^2(0,T;X)}    +\|v_v\|_{L^2(0,T;Y)}^2+ \|p_v\|_{L^2(0,T;X)}^2  \\ &\quad \lesssim  1+ |\phi_{v,0}|_{Y}^2+  \sum_{\mathclap{\alpha \in \CH \cup \{\ECM\}}} ~ | \phi_{\alpha,0}|^2_{V} + \sum_{\beta \in \RD}
			   |\phi_{\beta,0}|^2_{H}  + \|p_\infty \|^2_{L^2(0,T;V)}  + |\phi_{v,\infty}|^2_{H^1(0,T)} + |p_{v,\infty}|^2_{L^2(0,T)}  .
		   \end{aligned} \end{equation}
	\label{Thm:Existence}
\end{theorem}

\section{Proof of Theorem 1} \label{Sec:Proof}

To prove the existence of a weak solution, we use the Faedo--Galerkin method \cite{evans2010partial} and semi-discretize the original problem in space. The discretized model can be formulated as an ordinary differential equation system and by the Cauchy--Peano theorem \cite{walter1998ordinary}, we conclude the existence of a discrete solution, see \cref{Sec:Discrete}.  Having derived energy estimates in \cref{Sec:Energy}, we deduce from the Banach--Alaoglu theorem the existence of limit functions which eventually form a weak solution, see \cref{Sec:Limit}. This method is by now standard in the analysis of tumor growth models, e.g.,see \cite{frigeri2015diffuse,jiang2015well,lowengrub2013analysis, frigeri2018on}. Nevertheless, the novel nonlinear coupling of the equations requires a thorough proof of the existence of a solution to the system.

\subsection{Faedo--Galerkin discretization} \label{Sec:Discrete}
We introduce the discrete spaces
\begin{align*}
	H_k & =\text{span}\{ h_1,\dots,h_k\}, \\
	H_k^0 &= \text{span}\{ h_1^0,\dots,h_k^0\}, \\
	Y_k & =\text{span}\{ y_1,\dots,y_k\},

\end{align*}
where $h_j: \Omega \to \R$, $h_j^0: \Omega \to \R$, $y_j : \Lambda \to \R$, $j \in \{1,\dots,k\}$, are the eigenfunctions to the eigenvalues $\lambda_{h,j},  \lambda_{h^0,j},\lambda_{y,j} \in \R$ of the following respective problems
$$\begin{aligned}
(\nabla h_j,\nabla v)_H &= \lambda_{h,j} (h_j,v)_H &&\forall  v \in V,  \\
(\nabla h_j^0,\nabla v)_H &= \lambda_{h^0,j} (h_j^0,v)_H &&\forall  v \in V_0,  \\
(\nabla y_j,\nabla v)_Y &= \lambda_{y,j} (y_j,v)_Y &&\forall  v \in X_0.
\end{aligned}$$

Since the inverse Neumann--Laplace operator is a compact, self-adjoint, injective, positive operator on $L_0^2(\Omega)$, we conclude by the spectral theorem, see e.g.,\cite[12.12 and 12.13]{alt2016linear}, that
\begin{alignat*}{3}
	 & \{h_j\}_{j \in \mathbb{N}} &  & \text{ is an orthonormal basis in } H &  & \text{ and orthogonal in } V.
\end{alignat*}
Therefore, $\blue{\cup_{k\in\mathbb{N}}} H_k$ is dense in $V$. Additionally, $\{h_j\}_{j \in \N}$ is a basis in $H^2_N(\Omega)=\{u \in H^2(\Omega) : \partial_n u=0 \text{ on } \p\Omega\}$, see \cite{garcke2016global}.

Next, we investigate the inverse Dirichlet--Neumann Laplacian $(-\Delta)^{-1}|_H : H \to H$, see, e.g., \cite{saedpanah2014well} for the consideration of the Dirichlet--Neumann Laplacian in a Faedo--Galerkin approach. According to the Lax--Milgram theorem, for all $f \in H$ there exists a unique solution $u_f\in V_0$ to the problem
$$(\nabla u_f,\nabla v)_H = (u_f,f)_H \quad \forall v \in V_0.$$
Additionally, it holds $|u_f|_{V_0} \lesssim |f|_H$ for all $f\in H$ and we can construct an operator $T \in \L(H;V_0)$ with $Tf=u_f$. Since $V_0$ is compactly embedded in $H$, we conclude the compactness of $T \in \L(H;H)$. Taking the test function $v=Tg$ for an arbitrary element $g \in H$, we obtain the self-adjointness of $T$,
$$(Tg,f)_H=(\nabla Tf,\nabla Tg)_H=(g,Tf)_H,$$
and taking $g=f$ yields the positivity of $T$,
$$(Tf,f)_H=|\nabla Tf|_H^2 \geq 0.$$
Additionally, $T$ is injective, since $Tf=0$ yields $(f,v)_H=0$ for all $v \in H$ and hence, $f=0$ almost everywhere. Similarly, we can derive the same results for an operator $\widetilde T \in \L(Y;Y)$ corresponding to the eigenvalue problem on $Y$. Hence, by the spectral theorem we conclude
\begin{alignat*}{3}
& \{h_j^0\}_{j \in \mathbb{N}} &  & \text{ is an orthonormal basis in } H &  & \text{ and orthogonal in } V_0, \\
& \{y_j\}_{j \in \mathbb{N}} &  & \text{ is an orthonormal basis in } Y &  & \text{ and orthogonal in } X_0.
\end{alignat*}
Additionally, we deduce that $\blue{\cup_{k\in\mathbb{N}}}H_k^0$ is dense in $V_0$ and $\blue{\cup_{k\in\mathbb{N}}}Y_k$ is dense in $X_0$.

We consider the Faedo--Galerkin approximations, $\alpha \in \A$, $\beta \in \{P,H\}$,
\begin{equation}\begin{gathered}
		\phi_\alpha^k (t) = \sum_{j=1}^k \xi_{\alpha,j}(t) h_j,
		\quad \mu_\beta^k (t) = \sum_{j=1}^k \zeta_{\beta,j}(t) h_j,
		\quad \phi_v^k(t) = \phi_{v,\infty}(t)+ \sum_{j=1}^k \xi_{v,j}(t) y_j,  \\
		p^k(t) =p_\infty(t) + \sum_{j=1}^k \zeta_{p,j}(t) h^0_j, \quad p_v^k(t) =  p_{v,\infty}(t) + \sum_{j=1}^k \zeta_{p_v,j}(t) y_j,
	\end{gathered}
	\label{Eq:GalerkinAnsatzFunctions}
\end{equation}
where $(\xi_{\alpha,j})_{\alpha \in \A}:(0,T) \to \R^{|\A|}$,  $(\zeta_{\beta,j})_{\beta \in \{P,H\}}:(0,T) \to \R^{2}$ and

$\xi_{v,j},\, \zeta_{p,j}, \, \zeta_{p_v,j} : (0,T) \to \R$
 are coefficient functions for all $j \in \{1,\dots,k\}$.
To simplify the notation, we set $\phib^k = (\phi_\alpha^k)_{\alpha \in \A}$, and
$$\begin{gathered}
S_\alpha^k = S_\alpha( \phib^k), \quad
m_\beta^k = m_\beta( \phib^k), \quad
\Psi^k = \Psi(\phi_P^k,\phi_H^k,\phi_N^k), \\  J_{p v}^k = J_{p v}( \ov p^k, p_v^k), \quad J_{p v,\Gamma}^k = J_{p v}(  p^k,\Pi_\Gamma p_v^k), \quad J_{\sigma v}^k = J_{\sigma v}(\ov \phi_\sigma^k, \ov p^k, \phi_v^k,p_v^k), \quad J_{\sigma v,\Gamma}^k = J_{\sigma v}(\phi_\sigma^k, p^k, \Pi_\Gamma \phi_v^k,\Pi_\Gamma p_v^k),
\end{gathered} $$
where $\alpha \in \A$ and $\beta \in \A \backslash \{N,\ECM\}$. The Faedo--Galerkin system of the model then reads
\begin{equation}  \label{Eq:ModelGalerkin3D}\begin{aligned}
		(\pt \phi_P^k , \varphi_1)_H -  (\C(\phi_P^k) v^k,\nabla \varphi_1)_H +(m_P^k \nabla \mu_P^k,\nabla \varphi_1)_H                             & = (S_P^k,\varphi_1)_H,
		\\
		-(\mu_P^k,\varphi_2)_H + (\p_{\phi_P^k}\Psi^k,\varphi_2)_H + \eps^2_P (\nabla \phi_P^k,\nabla \varphi_2)_H                                           & = \chi_c (\phi_\sigma^k,\varphi_2)_H +\chi_h (  \ecm^k,\varphi_2)_H,
		\\
		( \pt \phi_H^k , \varphi_3 )_H -  (\C(\phi_H^k) v^k,\nabla \varphi_3)_H +(m_H^k \nabla \mu_H^k,\nabla \varphi_3)_H                             & = (S_H^k,\varphi_3)_H           ,           \\
		-(\mu_H^k,\varphi_4)_H + (\p_{\phi_H^k}\Psi^k,\varphi_4)_H + \eps^2_H (\nabla \phi_H^k,\nabla \varphi_4)_H                                           & = \chi_c (\phi_\sigma^k,\varphi_4)_H+ \chi_h ( \ecm^k,\varphi_4)_H,
		\\
		(\pt \phi_N^k , \varphi_5)_H                                                                                & = (S_N^k,\varphi_5)_H,
		\\
		(\pt \phi_\sigma^k,\varphi_6)_H - (\C(\phi_\sigma^k) v^k,\nabla \varphi_6)_H+  D_\sigma(m_\sigma^k \nabla \phi_\sigma^k, \nabla \varphi_6)_H & = (S_\sigma^k,\varphi_6)_H + \langle \delta_\Gamma, J_{\sigma v,\Gamma}^k \varphi_6\rangle_W             \\ &\qquad -\chi_c (m_\sigma^k \nabla (\phi_P^k +\phi_H^k  ),\nabla \varphi_6)_H,
		\\
		(\pt \mde^k,\varphi_7 )_H +  D_{\MDE} (m_{\MDE}^k \nabla \mde^k,\nabla \varphi_7)_H                                                       & = (S_{\MDE}^k,\varphi_7)_H,
		\\
		( \pt \taf^k,\varphi_8)_H + D_{\TAF} (m_{\TAF}^k \nabla \taf^k,\nabla \varphi_8)_H                                                        & = (S_{\TAF}^k,\varphi_8)_H,
		\\
		(\pt \ecm^k,\varphi_9 )_H                                                                                & = (S_{\ECM}^k,\varphi_9)_H,

		\\
		K(\nabla p^k,\nabla \varphi_{10})_H                                                                                & =   \langle \delta_\Gamma, J_{pv,\Gamma}^k\varphi_{10} \rangle_W + K(S_p^k,\nabla \varphi_{10})_H,
	\end{aligned}
\end{equation}
for all $\varphi_i \in H_k$, $i \in \{1,\dots,9\}$, $\varphi_{10} \in H_k^0$, and
\begin{equation} \label{Eq:ModelGalerkin1D}  \begin{aligned}
		(\pt \phi_v^k, \varphi_{11})_Y + D_v (m_v^k \nablala \phi_v^k,\nablala \varphi_{11})_Y & = (\C( \phi_v^k) v_v^k ,\nablala \varphi_{11} )_Y   -\wdr(J_{\sigma v}^k,\varphi_{11})_Y,
		\\
		\wdk( \nablala p_v^k, \nablala \varphi_{12} )_Y                                                                                & = -\wdr (J_{pv}^k,\varphi_{12})_Y ,
	\end{aligned}
\end{equation}
for all $\varphi_j \in Y_k$, $j\in \{11,12\}$, where we define the Faedo--Galerkin ansatz for the velocities $v^k, v_v^k$ by
\begin{equation} \label{Eq:ModelGalerkinVelocity}
	\begin{aligned}
		v^k &= -K ( \nabla p^k - S_p^k), \\

		v_v^k &= - \wdk \nablala p_v^k.

	\end{aligned}
\end{equation}
We equip the system with the initial data,
\begin{equation} \begin{aligned}
	\phi_\alpha^k(0) &= \Pi_{H_k} \phi_{\alpha,0}, &&\alpha \in \A, \\
	\phi_v^k(0) &=\phi_{v,\infty}(0)  + \Pi_{Y_k} \phi_{v,0},
\end{aligned}
\label{Eq:GalerkinInitial}\end{equation}
where  $\Pi_{H_k}: H \to H_k$ and $\Pi_{Y_k} : Y \to Y_k$ are the orthogonal projections onto the finite dimensional spaces, which can be written as
$$\begin{aligned}
	\Pi_{H_k} h = \sum_{j=1}^k (h,h_j)_H h_j, ~\text{ and }~
	\Pi_{Y_k} y = \sum_{j=1}^k (y,y_j)_Y y_j.
\end{aligned}$$

After inserting the Faedo--Galerkin ansatz functions \cref{Eq:GalerkinAnsatzFunctions} into the system \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D}, one can see that the Faedo--Galerkin system is equivalent to a system of nonlinear ordinary differential equations in the unknowns $( (\xi_{\alpha,j})_{\alpha \in \A \cup \{ v\}}, (\zeta_{\beta,j})_{\beta \in \CH \cup \{p,p_v\}})_{1\leq j \leq k}$ with the initial data,
$$\begin{aligned} \xi_{\alpha,j}(0)  & = (\phi_{\alpha,0},h_j)_H, &&\alpha \in \A, \\
		\xi_{v,j}(0)       & = (\phi_{v,0},y_j)_Y.
\end{aligned}$$
Due to the continuity of the involved nonlinear functions the existence of solutions to \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D} with the initial data \cref{Eq:GalerkinInitial} follows from the standard theory of ordinary differential equations, according to the Cauchy--Peano theorem \cite{walter1998ordinary}. We thus have local-in-time existence of a continuously differentiable solution,  $$\begin{aligned} (\phib^k,\mu_P^k,\mu_H^k,p^k-p_\infty,\phi_v^k-\phi_{v,\infty},p_v^k-p_{v,\infty}) &\in [C^1([0,T_k];H_k)]^{|\A|} \times [C^0([0,T_k];H_k)]^{2} \times C^0([0,T_k];H_k^0) \\ & \qquad  \times C^1([0,T_k];Y_k)  \times C^0([0,T_k];Y_k), \end{aligned}$$ to the Faedo--Galerkin problem \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D} on some sufficiently short time interval $[0, T_k]$. Further,  we obtain $\div S_p^k \in H$ \blue{by the representation of $S_p^k$, see \cref{Ass:Source},} and therefore, $v^k \in H$ with $\div v^k = -K (\Delta p^k-\div S_p^k)=J_{pv,\Gamma}^k \delta_\Gamma$. Similarly, $v_v^k \in Y$ with $\div v_v^k = - RJ_{pv}^k$.

\subsection{Energy estimates} \label{Sec:Energy}
Next, we extend the existence interval to $[0,T]$ by deriving $T_k$-independent estimates. In particular, these estimates allow us to deduce that the solution sequences converge to some limit functions as $k \to \infty$. It will turn out that exactly these limit functions will form a weak solution to our 3D-1D model \cref{Eq:Model3D}--\cref{Eq:Model1D} in the sense of \cref{Def:Weak}.

\subsubsection*{Step 1 (Testing)}
We derive energy estimates of the model \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D} by choosing suitable test functions in the variational form. For the Cahn--Hilliard type equations, we choose $\varphi_1 = \mu_P^k+\chi_c \phi_\sigma^k+\chi_h \ecm^k$, $\varphi_2 = \partial_t \phi_P^k-\mu_P^k$, $\varphi_3 = \mu_H^k+\chi_c \phi_\sigma^k+\chi_h \ecm^k$, $\varphi_4 = \partial_t \phi_H^k-\mu_H^k$, $\varphi_5 =\Pi_{H_k} \p_{\phi_N^k}\Psi^k - \eps_N^2 \Delta \phi_N^k$, and we arrive at the system of equations,

\begin{equation} \label{Eq:TestingCH}  \begin{aligned}
		(\pt \phi_P^k,\mu_P^k+\chi_c \phi_\sigma^k+\chi_h \ecm^k)_H  + \Big|\sqrt{m_P^k}\nabla \mu_P^k\Big|_H^2
		                                                                              & =  (\C(\phi_P^k) v^k, \nabla \mu_P^k+\chi_c \nabla \phi_\sigma^k+\chi_h \nabla \ecm^k)_H \\[-.15cm] &\qquad -(m_P^k \nabla \mu_P^k,\chi_c \nabla \phi_\sigma^k+\chi_h \nabla \ecm^k)_H \\[-.05cm]  &\qquad + (S_P^k,\mu_P^k + \chi_c\phi_\sigma^k+\chi_h \ecm^k)_H,
		\\
		(\p_{\phi_P^k}\Psi^k,\pt \phi_P^k)_H + \frac{\eps^2_P}{2} \ddt |\nabla \phi_P^k|_H^2 + |\mu_P^k|_H^2
		                                                                              & = (\mu_P^k+\chi_c \phi_\sigma^k + \chi_h \ecm^k,\pt \phi_P^k)_H+ (\p_{\phi_P^k}\Psi^k,\mu_P^k)_H \\[-.1cm] &\qquad -  (\chi_c \phi_\sigma^k+\chi_h \ecm^k,\mu_P^k)_H  + \eps_P^2 (\nabla \phi_P^k,\nabla \mu_P^k)_H,
		\\
		(\pt \phi_H^k,\mu_H^k+\chi_c \phi_\sigma^k+\chi_h \ecm^k)_H +  \Big|\sqrt{m_H^k}\nabla \mu_H^k\Big|_H^2
		                                                                              & =(\C(\phi_H^k) v^k,\nabla \mu_H^k+\chi_c \nabla \phi_\sigma^k+\chi \nabla \ecm^k )_H  \\[-.15cm] &\qquad -(m_H^k \nabla \mu_H^k,\chi_c\nabla \phi_\sigma^k+\chi_h \nabla \ecm^k)_H  \\&\qquad + (S_H^k,\mu_H^k + \chi_c\phi_\sigma^k+\chi_h \ecm^k)_H,
		\\
		(\p_{\phi_H^k}\Psi^k,\pt \phi_H^k)_H + \frac{\eps^2_H}{2} \ddt |\nabla \phi_H^k|_H^2   + |\mu_H^k|_H^2
		                                                                              & = (\mu_H^k+\chi_c \phi_\sigma^k+\chi_h \ecm^k,\pt \phi_H^k)_H + (\p_{\phi_H^k}\Psi^k,\mu_H^k)_H \\[-.1cm] &\qquad -(\chi_c\phi_\sigma^k+\chi_h \ecm^k,\mu_H^k)_H + \eps_H^2 (\nabla \phi_H^k,\nabla \mu_H^k)_H,
		\\
		(\pt \phi_N^k,\Pi_{H_k}\p_{\phi_N^k}\Psi^k)_H  + \frac{\eps_N^2}{2} \ddt |\nabla \phi_N^k|_H^2  & =    (S_N^k,\Pi_{H_k}\p_{\phi_N^k}\Psi^k)_H  + \eps_N^2 (\nabla S_N^k, \nabla \phi_N^k)_H.
	\end{aligned} \end{equation}
We exploit that the time derivative operator is invariant under the adjoint of the orthogonal projection.

Further, for the reaction-diffusion type equations, we choose $\varphi_6 = C_\sigma \phi_\sigma^k$, $C_\sigma>0$ to be determined, $\varphi_7 = \mde^k$, $\varphi_8 = \taf^k$, $\varphi_9 = \ecm^k -\Delta \ecm^k$, which yields the system,
\begin{equation} \label{Eq:TestingRD}  \begin{aligned}
		\frac{C_\sigma}{2} \ddt |\phi_\sigma^k|_H^2 + C_\sigma D_\sigma \Big|\sqrt{m_\sigma^k} \nabla \phi_\sigma^k \Big|_H^2
		 & =  \chi_c C_\sigma  (m_\sigma^k  \nabla (\phi_P^k +\phi_H^k  ),\nabla \phi_\sigma^k )_H+C_\sigma (S_\sigma^k,\phi_\sigma^k)_H  \\[-.1cm] &\qquad +C_\sigma  (\C(\phi_\sigma^k)  v^k , \nabla \phi_\sigma^k)_H  + C_\sigma\langle \delta_\Gamma,J_{\sigma v,\Gamma}^k\phi_\sigma^k \rangle _W,
		\\
		 \frac12 \ddt |\mde^k|_H^2 + D_{\MDE} \Big|\sqrt{m_{\MDE}^k} \nabla \mde^k\Big|_H^2
		 & = (S_{\MDE}^k,\mde^k)_H,
		\\
		\frac12 \ddt |\taf^k|_H^2  + D_{\TAF} \Big|\sqrt{m_{\TAF}^k} \nabla \taf^k\Big|_H^2
		 & = (S_{\TAF}^k,\taf^k )_H,
		\\
		\frac12 \ddt |\ecm^k|_H^2 +\frac12 \ddt |\nabla \ecm^k|_H^2
		 & =(S_{\ECM}^k,\ecm^k)_H+ (\nabla S_{\ECM}^k,\nabla \ecm^k)_H,
	\end{aligned} \end{equation}
and for the equations in $Y_k$, we choose $\varphi_{12}=C_v(\phi_v^k-\phi_{v,\infty})$, $C_v>0$ to be determined, giving
\begin{equation} \label{Eq:Testing1D}  \begin{aligned}
		\frac{C_v}{2} \ddt |\phi_v^k-\phi_{v,\infty}|_Y^2   +  C_vD_v \Big|\sqrt{m_v^k} \nablala \phi_v^k\Big|^2_Y
		                                  & =  C_v(\C(\phi_v^k)  v_v^k,\nablala \phi_v^k )_Y
		                                  \! -\! C_v(RJ_{\sigma v}^k+\phi_{v,\infty}',\phi_v^k-\phi_{v,\infty})_Y,

	\end{aligned} \end{equation}

		Similarly, we test the equations \cref{Eq:ModelGalerkinVelocity}$_2$ by $\frac{1}{\wdk}v_v^k$ to obtain
				\begin{align}
		\frac{1}{\wdk} |v_v^k|_H^2 &= - (\nablala (p_v^k-p_{v,\infty}),v_v^k)_Y  =   -\wdr(J_{pv}^k, p_v^k-p_{v,\infty})_Y .

		\label{Eq:TestingVel1D}
		\end{align}
		We test \cref{Eq:ModelGalerkinVelocity}$_1$ by $\frac{1}{K} v^k$ and simplify the first term on the right hand side by comparing it with the equation \cref{Eq:ModelGalerkinVelocity}$_1$ for the velocity $v^k$ and the pressure equation \cref{Eq:ModelGalerkin3D}, which is tested by $\varphi_{10}=p^k-p_\infty$. This procedure yields
		\begin{equation} \begin{aligned}
		\frac{1}{K} |v^k|_H^2  &= - (\nabla (p^k-p_\infty),v^k)_H + (\nabla p_\infty,v^k)_H + (S_p^k,v^k)_H
		\\  &= K (\nabla p^k-S_p^k,\nabla (p^k-p_\infty))_H  + (\nabla p_\infty,v^k)_H + (S_p^k,v^k)_H
		 \\ & =    \langle \delta_\Gamma, J_{pv,\Gamma}^k (p^k-p_\infty) \rangle_W - (\nabla p_\infty,v^k)_H+(S_p^k,v^k)_H .

		\end{aligned} \label{Eq:TestingVel3D} \end{equation}

	\subsubsection*{Step 2 (Estimates)} We separate this step into three sub-steps by deriving the energy estimates separately for \cref{Eq:TestingCH}, \cref{Eq:TestingRD} and \cref{Eq:Testing1D}.

	\subsubsection*{Step 2.1 (Estimate for \cref{Eq:TestingCH})}

	Adding the equations in \cref{Eq:TestingCH} and \cref{Eq:TestingVel3D} gives
	\begin{equation} \label{Eq:AddingCH} \begin{aligned}
			 &     \ddt  |\Psi^k|_{L^1(\Omega)}   +  \frac{\eps^2_P}{2} \ddt |\nabla \phi_P^k|_H^2
			+\frac{\eps^2_H}{2} \ddt |\nabla \phi_H^k|_H^2 +  \frac{\eps_N^2}{2} \ddt |\nabla \phi_N^k|_H^2 +\Big|\sqrt{m_P^k}\nabla \mu_P^k\Big|_H^2 +   \Big|\sqrt{m_H^k}\nabla \mu_H^k\Big|_H^2  \\
			& \qquad +|\mu_P^k|_H^2 + |\mu_H^k|_H^2  + \frac{1}{K} |v^k|_H^2 \\
			 & = (\C(\phi_P^k)v^k,\nabla \mu_P^k + \chi_c \nabla \phi_\sigma^k + \chi_h \nabla \ecm^k)_H -(m_P^k \nabla \mu_P^k,\chi_c\nabla \phi_\sigma^k+\chi_h \nabla \ecm^k)_H
			 \\ &\qquad  + (S_P^k,\mu_P^k + \chi_c\phi_\sigma^k+\chi_h \ecm^k)_H +   (\p_{\phi_P^k}\Psi^k-\chi_c \phi_\sigma^k-\chi_h \ecm^k,\mu_P^k)_H + \eps_P^2 (\nabla \phi_P^k,\nabla \mu_P^k)_H
			 \\ &\qquad+ (\C(\phi_H^k)v^k,\nabla \mu_H^k + \chi_c \nabla \phi_\sigma^k + \chi_h \nabla \ecm^k)_H-(m_H^k \nabla \mu_H^k,\chi_c\nabla \phi_\sigma^k + \chi_h \nabla \ecm^k)_H
			\\ &\qquad+(S_H^k,\mu_H^k + \chi_c\phi_\sigma^k+ \chi_h \ecm^k)_H + (\p_{\phi_H^k}\Psi^k-\chi_c \phi_\sigma^k - \chi_h \ecm^k,\mu_H^k)_H    + \eps_H^2 (\nabla \phi_H^k,\nabla \mu_H^k)_H
	   \\ &\qquad 	+ (S_N^k,\Pi_{H_k}\p_{\phi_N^k}\Psi^k)_H + \eps_N^2 (\nabla S_N^k,\nabla \phi_N^k )_H
	   +\langle \delta_\Gamma, J_{pv,\Gamma}^k (p^k-p_\infty) \rangle_W - (\nabla p_\infty,v^k)_H+(S_p^k,v^k)_H \\
			& = \RHS_{\CH}.
		\end{aligned} \end{equation}
		We note that the two convection terms cancel together with the the last term $(S_p^k,v^k)_H$ on the right hand side.

		We apply the H\"older inequality on the terms on the right hand side, and use the assumptions \cref{Ass:Source} and \cref{Ass:Mobility}, which gives
	\begin{equation} \label{Eq:HolderCH} \begin{aligned}
		\RHS_\CH & \leq m_\infty  | \nabla \mu_P^k|_H (\chi_c |\nabla \phi_\sigma^k|_H + \chi_h |\nabla \ecm^k|_H)  + |S_P^k|_H ( |\mu_P^k|_H +  \chi_c |\phi_\sigma^k|_H+  \chi_h |\ecm^k|_H)
		\\ &\qquad  +|\mu_P^k|_H (|\p_{\phi_P^k}\Psi^k|_H+ \chi_c |\phi_\sigma^k|_H + \chi_H |\ecm^k|_H)  + \eps_P^2 |\nabla \phi_P^k|_H |\nabla \mu_P^k|_H
	    \\ &\qquad + m_\infty  |\nabla \mu_H^k|_H (\chi_c |\nabla \phi_\sigma^k|_H+ \chi_h  |\nabla \ecm^k|_H) + |S_H^k|_H ( |\mu_H^k|_H + \chi_c |\phi_\sigma^k|_H+ \chi_h |\ecm^k|_H  )
		  \\ &\qquad +  |\mu_H^k|_H (|\p_{\phi_H^k}\Psi^k|_H+\chi_c |\phi_\sigma^k|_H + \chi_h |\ecm^k|_H) + \eps_H^2 |\nabla \phi_H^k|_H |\nabla \mu_H^k|_H
		  \\ &\qquad + |S_N^k|_{H} |\Pi_{H_k}\p_{\phi_N^k}\Psi^k|_H + \eps_N^2 |\nabla S_N^k|_H |\nabla \phi_N^k |_H + \langle \delta_\Gamma, J_{pv,\Gamma}^k (p^k-p_\infty) \rangle_W + |\nabla p_\infty|_H |v^k|_H.
	\end{aligned} \end{equation}
	We note that the norm of the orthogonal projection is bounded by $1$. We use a similar argument as in \cref{Eq:EstimateDelta} and \cref{Eq:EstimateJ} to estimate the term involving the Dirac delta functional $\delta_\Gamma$, i.e., with the assumption on the form of $J_{pv,\Gamma}^k$, see \cref{Ass:Jv}, we obtain
	$$\begin{aligned} &\langle \delta_\Gamma, J_{pv,\Gamma}^k (p^k-p_\infty) \rangle_W\\ &\leq  C_\Gamma |p^k-p_\infty|_W|J_{pv,\Gamma}^k|_{L^2(\Gamma)}  \\ &\leq C_\Gamma L_p  |p^k-p_\infty|_W (C_\Gamma |p^k-p_\infty|_W + C_\Gamma |p_\infty|_W + |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))} |p_v^k-p_{v,\infty}|_Y +|\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))} |p_{v,\infty}|_Y) \\
	&\leq C_1 L_p ( |\nabla p^k|_H^2 + |\nablala p_v^k|_Y^2 + |p_\infty|^2_V + |p_{v,\infty}|^2),
	\end{aligned} $$
	where we also applied the Poincar\'e inequality on $p^k-p_\infty \in V_0$ and $p_v^k-p_{v,\infty} \in X_0$ with the Poincar\'e constants $C_{P,\Omega}$ and $C_{P,\Lambda}$, giving the constant
	$$C_1 =  \max\{2C_\Gamma^2 (C_{W}^{V})^2 (C_{P,\Omega}^2+1);|\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))}^2 C_{P,\Lambda}^2 ;|\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))}^2 |\Lambda|  \}.$$
	 Further, using the form on $v^k$ and $v_v^k$ gives
		$$\begin{aligned} \langle \delta_\Gamma, J_{pv,\Gamma}^k (p^k-p_\infty) \rangle_W \leq C_1 L_p (K^{-2} |v^k|_H^2 +|S_p^k|_H^2  + \wdk^{-2} |v_v^k|_Y^2 + |p_\infty|^2_V + |p_{v,\infty}|^2).
	\end{aligned} $$

	We apply Young's inequality on the norm products to separate the terms. Here, the goal is to make the terms involving $|\mu_P^k|_V$, $|\mu_H^k|_V$, $|\nabla \phi_\sigma^k|_H$ small, since we cannot absorb them with the Gr\"onwall--Bellman lemma later on. We only track the important constants, which are used to absorb the terms on the right hand side with the left hand side, the other ones we simply denote by the generic constant $C$. We have
	\begin{equation} \label{Eq:YoungCH} \begin{aligned}
		\RHS_\CH & \leq \frac{m_0}{4} | \nabla \mu_P^k|_H^2 + \frac{2m_\infty^2 \chi_c^2}{m_0} |\nabla \phi_\sigma^k|_H^2 + \frac{2m_\infty^2 \chi_h^2}{m_0} |\nabla \ecm^k|_H^2  + 3 |S_P^k|_H^2  +\frac14 \big( |\mu_P^k|_H^2 + \chi_c^2|\phi_\sigma^k|_H^2
		\\ &\qquad + \chi_h^2|\ecm^k|_H^2\big)
		  + \frac14 |\mu_P^k|_H^2 + 3 ( |\p_{\phi_P^k}\Psi^k|_H^2+ \chi_c^2 |\phi_\sigma^k|_H^2 +\chi_h^2 |\ecm^k|_H^2) + \frac{\eps_P^4}{m_0} |\nabla \phi_P^k|_H^2
		    \\ &\qquad		 + \frac{m_0}{4} |\nabla \mu_P^k|_H^2  + \frac{m_0}{4} |\nabla \mu_H^k|_H^2    + \frac{2m_\infty^2 \chi_c^2}{m_0} |\nabla \phi_\sigma^k|_H^2 + \frac{2m_\infty^2 \chi_h^2}{m_0} |\nabla \ecm^k|_H^2   + 3|S_H^k|_H^2
		 \\ &\qquad+\frac14 ( |\mu_H^k|_H^2 + \chi_c^2|\phi_\sigma^k|_H^2 + \chi_h^2|\ecm^k|_H^2) + \frac14 |\mu_H^k|_H^2    + 3 ( |\p_{\phi_H^k}\Psi^k|_H^2+ \chi_c^2 |\phi_\sigma^k|_H^2 +\chi_h^2 |\ecm^k|_H^2)
		  \\ &\qquad + \frac{\eps_H^4}{m_0} |\nabla \phi_H^k|_H^2 + \frac{m_0}{4} |\nabla \mu_H^k|_H^2+ \frac12 |S_N^k|_H^2 + \frac12 |\p_{\phi_N^k}\Psi^k|_H^2 + \eps|\nabla S_N^k|_H^2 + \frac{\eps_N^4}{4\eps} |\nabla \phi_N^k |_H^2
		  \\ &\qquad + C_1 L_p\big(K^{-2} |v^k|_H^2 +|S_p^k|_H^2 + \wdk^{-2} |v_v^k|_Y^2 + |p_\infty|^2_V + |p_{v,\infty}|^2\big) + \frac{K}{2} |\nabla p_\infty|_H^2 + \frac{1}{2K} |v^k|_H^2,
	\end{aligned} \end{equation}
	where $\eps>0$ is a constant, which will be determined later on, see \cref{Eq:Epsilon} below for more details. We estimate the terms involving the potential $\Psi$ via assumption \cref{Ass:Psi} and afterwards, we collect the terms with the same norms, which yields
	\begin{equation} \label{Eq:CollectCH} \begin{aligned}
		\RHS_\CH & \leq \frac{m_0}{2} \big( | \nabla \mu_P^k|_H^2 +  | \nabla \mu_H^k|_H^2 \big) +\frac12 \big( |\mu_P^k|_H^2 + |\mu_H^k|_H^2 \big) + \frac{4m_\infty^2 \chi_c^2}{m_0}  |\nabla \phi_\sigma^k|_H^2    + \eps|\nabla S_N^k|_H^2
				\\ &\qquad+ \frac{1}{2K} |v^k|_H^2 + C_1 L_p \big(K^{-2} |v^k|^2_H  + |S_p^k|_H^2 + \wdk^{-2} | v_v^k|_Y^2+ |p_\infty|_V^2 + |p_{v,\infty}|^2 \big)
				\\ &\qquad+ C \big( |\phi_\sigma^k|_H^2 +  |\phi_P|_V^2 + |\phi_H|_V^2 + |\phi_N|_V^2 +  |\ecm^k|_V^2   +  |p_\infty|_V^2+ |S_P^k|_H^2 + |S_H^k|_H^2 +  |S_N^k|_H^2 \big) .
	\end{aligned} \end{equation}
	We insert this estimate into \cref{Eq:AddingCH}, and absorb the terms involving the chemical potentials, and arrive at the upper bound
	\begin{equation} \label{Eq:FinalCH} \begin{aligned}
		&    \ddt \bigg[  |\Psi^k|_{L^1(\Omega)}   +  \frac{\eps^2_P}{2} |\nabla \phi_P^k|_H^2
	   +\frac{\eps^2_H}{2}  |\nabla \phi_H^k|_H^2 +  \frac{\eps_N^2}{2}  |\nabla \phi_N^k|_H^2 \bigg] +\frac{m_0}{2} |\nabla \mu_P^k|_H^2 +   \frac{m_0}{2} |\nabla \mu_H^k|_H^2 +\frac12 |\mu_P^k|_H^2   \\
	   &\qquad + \frac12 |\mu_H^k|_H^2+\left(  \frac{1}{2K}- \frac{C_1 L_p}{K^2} \right) |v_k|_H^2 \\
	   &\leq  \frac{4m_\infty^2 \chi_c^2}{m_0} |\nabla \phi_\sigma^k|_H
	   +  \eps|\nabla S_N^k|_H^2 + C_1 L_p \big(|S_p^k|_H^2 + \wdk^{-2} |v_v^k|_Y^2  \big)
	    + C \big( |\phi_\sigma^k|_H^2 +  |\phi_P^k|_V^2 + |\phi_H^k|_V^2  \\ &\qquad+ |\phi_N^k|_V^2+  |\ecm^k|_V^2 + |\phi_v^k|_Y^2  +  |p_\infty|_V^2 + |p_{v,\infty}|^2 + |S_P^k|_H^2 + |S_H^k|_H^2 +  |S_N^k|_H^2 \big).
   \end{aligned} \end{equation}

   \subsubsection*{Step 2.2 (Estimate for \cref{Eq:TestingRD})}

   Adding the equations in \cref{Eq:TestingRD} gives
	\begin{equation} \label{Eq:AddingRD} \begin{aligned}
			 & \frac{C_\sigma }{2} \ddt |\phi_\sigma^k|_H^2
			+ C_\sigma D_\sigma \Big|\sqrt{m_\sigma^k} \nabla \phi_\sigma^k\Big|_H^2
			+  \frac12 \ddt |\mde^k|_H^2 + D_{\MDE} \Big|\sqrt{m_{\MDE}^k} \nabla \mde^k\Big|_H^2+  \frac12 \ddt |\taf^k|_H^2
			\\ &\qquad  + D_{\TAF} \Big|\sqrt{m_{\TAF}^k} \nabla \taf^k\Big|_H^2  +\frac12 \ddt |\ecm^k|_H^2 +\frac12 \ddt |\nabla \ecm^k|_H^2
			\\
			 & =  \chi_c C_\sigma  (m_\sigma^k  \nabla (\phi_P^k +\phi_H^k ),\nabla \phi_\sigma^k)_H
			+C_\sigma (S_\sigma^k,\phi_\sigma^k)_H + C_\sigma (\C(\phi_\sigma^k) v^k,\nabla \phi_\sigma^k)_H
			+C_\sigma\langle \delta_\Gamma, J_{\sigma v,\Gamma}^k \phi_\sigma^k \rangle_W
			\\ &\qquad + (S_{\MDE}^k,\mde^k)_H + (S_{\TAF}^k,\taf^k)_H +(S_{\ECM}^k,\ecm^k)_H + (\nabla S_{\ECM}^k,\nabla \ecm^k)_H
			\\ & = \RHS_\RD.
		\end{aligned} \end{equation}

		We estimate the term involving the Dirac delta functional as before, i.e., we use assumption \cref{Ass:Jv} and the inequalities \cref{Eq:EstimateDelta} and \cref{Eq:EstimateJ} to obtain
		$$\begin{aligned}&C_\sigma\langle \delta_\Gamma, J_{\sigma v,\Gamma}^k \phi_\sigma^k \rangle_W \\ &\leq C_\sigma C_\Gamma |\phi_\sigma^k|_W \Big(f_\infty L_p (C_\Gamma |p^k|_W + |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))} |p_v^k|_Y) + L_\sigma (C_\Gamma |\phi_\sigma^k|_W + |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))} |\phi_v^k|_Y  ) \Big) \\
		&\leq C_2 C_\sigma \max\{L_p;L_\sigma\} \big( |\phi_\sigma^k|_V^2 +|\phi_v^k|_Y^2 +  K^{-2} |v^k|_H^2 + |S_p^k|_H^2 + \wdk^{-2} |v_v^k|_Y^2 + |p_\infty|_V^2 + |p_{v,\infty}|^2   \big),
		\end{aligned}$$
		where
		$$C_2 = \max\{2^4 C_\Gamma^2 (C_W^V)^2; f_\infty^2 C_\Gamma^2 (C_W^{V})^2 (C_{P,\Omega}^2+1); f_\infty^2 |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))}^2 C_{P,\Lambda}^2 ; f_\infty^2 |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))}^2 |\Lambda|; |\Pi_\Gamma|_{\mathcal{L}(Y;L^2(\Gamma))}^2 \}. $$
		In a similar way to the estimates before, we apply H\"older's and Young's inequalities on the terms on the right hand side, which results in
\begin{equation} \label{Eq:HolderRD} \begin{aligned}
	\RHS_\RD & \leq  \frac{C_\sigma m_\infty^2 \chi_c^2}{D_\sigma m_0} |\nabla (\phi_P^k+\phi_H^k)|_H^2 + \frac{C_\sigma D_\sigma m_0}{4} |\nabla \phi_\sigma^k|_H^2 + C( |S_\sigma^k|_H^2 +  |\phi_\sigma^k|_H^2 )
	\\ &\qquad  + \frac{C_\sigma D_\sigma m_0}{4} |\nabla \phi_\sigma^k|_H^2+ \frac{C_\sigma}{D_\sigma m_0} |v^k|_H^2 +C_2 C_\sigma  \max\{L_p;L_\sigma\} \big(|\phi_\sigma^k|_V^2 +|\phi_v^k|_Y^2 +  K^{-2} |v^k|_H^2 \\ &\qquad + |S_p^k|_H^2 + \wdk^{-2} |v_v^k|_Y^2 + |p_\infty|_V^2 + |p_{v,\infty}|^2  \big)     +  C \big( |S_{\MDE}^k|_H^2 +  |\mde^k|_H^2  +  |S_{\TAF}^k|_H^2
	\\ &\qquad+  |\taf^k|_H^2   +  |S_{\ECM}^k|_H^2 + |\ecm^k|_H^2 \big)+ \eps |\nabla S_{\ECM}^k|_H^2 + \frac{1}{4 \eps} |\nabla \ecm^k|_H^2 ,
\end{aligned} \end{equation}
where we used the same constant $\eps$ as before in \cref{Eq:AddingCH} and applied the assumption on the form of $J_{\sigma v,\Gamma}^k$, see \cref{Ass:Jv}. Again, collecting the terms on the right hand side and absorbing the terms with their counterparts, we have
\begin{equation} \label{Eq:FinalRD} \begin{aligned}
	& \frac12 \ddt \bigg[ C_\sigma |\phi_\sigma^k|_H^2  +|\mde^k|_H^2+|\taf^k|_H^2 + |\ecm^k|_H^2 + |\nabla \ecm^k|_H^2 \bigg]
   + D_{\MDE} m_0| \nabla \mde^k|_H^2
   \\ &\qquad+ D_{\TAF} m_0 |\nabla \taf^k|_H^2+  \frac{C_\sigma}{2} \big(  D_\sigma m_0 - 2C_2 \max\{L_p;L_\sigma\}\big) |\nabla \phi_\sigma^k|_H^2
    \\	& \leq  \left(\frac{C_\sigma}{D_\sigma m_0} + \frac{C_2 C_\sigma \max\{L_p;L_\sigma\} }{K^2}  \right) |v^k|_H^2 + C_2 C_\sigma \max\{L_p;L_\sigma\} \big(|S_p^k|_H^2 + \wdk^{-2} |v_v^k|_Y^2  \big)+ \eps |\nabla S_{\ECM}^k|_H^2
	\\ &\qquad   + C \big( |\phi_P|_V^2+ |\phi_H|_V^2  + |\phi_\sigma^k|_H^2  + |\mde^k|_H^2 + |\taf^k|_H^2 + |\ecm^k|_V^2 + |\phi_v^k|_Y^2 + |p_\infty|_V^2
	\\ &\qquad  + |p_{v,\infty}|^2+|S_\sigma^k|_H^2 + |S_{\MDE}^k|_H^2
	 + |S_{\TAF}^k|_H^2 + |S_{\ECM}^k|_H^2 \big).
\end{aligned} \end{equation}

\subsubsection*{Step 2.3 (Estimate for \cref{Eq:Testing1D})}
		Lastly, adding the equations in \cref{Eq:Testing1D} and \cref{Eq:TestingVel1D} gives
		\begin{equation} \label{Eq:Adding1D} \begin{aligned}
			 &\frac{C_v}{2} \ddt |\phi_v^k-\phi_{v,\infty}|_Y^2
		   + C_vD_v \Big|\sqrt{m_v^k} \nablala \phi_v^k\Big|_Y^2 + \frac{1}{\wdk} |v_v^k|_Y^2 \\
			&=C_v (\C(\phi_v^k) v_v^k,\nablala \phi_v^k)_Y -C_v(RJ_{\sigma v}^k+\phi_{v,\infty}',\phi_v^k-\phi_{v,\infty})_Y  -  \wdr (J_{pv}^k,p_v^k-p_{v,\infty})_Y. \end{aligned}
	  \end{equation}

We estimate the last term on the right hand side with the Poincar\'e--Wirtinger inequality \cref{Eq:SobolevInequality} with constant $C_P$, the Darcy law \cref{Eq:ModelGalerkinVelocity} and the Young inequality as follows
$$-\wdr (J_{pv}^k,p_v^k-p_{v,\infty})_Y \leq \wdr C_P |J_{pv}^k|_Y |\nabla p_v^k|_Y = \frac{R^2 C_P^2}{\wdk} |J_{pv}^k|_Y |v_v^k|_Y \leq \frac{R^2 C_P^2}{\wdk} |J_{pv}^k|_Y^2 + \frac{1}{4\wdk} |v_v^k|_Y^2  .  $$
Additionally, repeating the steps from before and using the assumption on the forms on $J_{pv}^k$ and $J_{\sigma v}^k$, see \cref{Ass:Jv}, we arrive at
\begin{equation*} \begin{aligned} &\frac{C_v}{2} \ddt |\phi_v^k-\phi_{v,\infty}|_Y^2
   + C_vD_v m_0 |\nablala \phi_v^k|_Y^2 + \frac{1}{\wdk} |v_v^k|_Y^2
     \\ &\leq \frac{C_v}{D_v m_0} |v_v^k|_Y^2 + \frac{C_vD_vm_0}{4} |\nablala \phi_v^k|_Y^2 +  R^2|J_{\sigma v}^k|_Y^2 + C\big( |\phi_{v,\infty}'|^2 +  |\phi_v^k-\phi_{v,\infty}|^2_Y\big)  + \frac{R^2C_P^2}{\wdk} |J_{pv}^k|_Y^2   + \frac{1}{4\wdk} |v_v^k|_Y^2
          \\ &\leq \frac{C_v}{D_v m_0} |v_v^k|_Y^2 + \frac{C_vD_vm_0}{4} |\nablala \phi_v^k|_Y^2 + C\big(|\phi_{v,\infty}'|^2 +  |\phi_v^k-\phi_{v,\infty}|^2_Y \big)    + \frac{1}{4\wdk} |v_v^k|_Y^2
          \\ &\qquad + (1+C_P^2\wdk^{-1})  R^2C_2 \max\{L_p;L_\sigma\}   \big( |\phi_\sigma^k|_V^2 +|\phi_v^k|_Y^2 +  K^{-2} |v^k|_H^2 + |S_p^k|_H^2 + \wdk^{-2} |v_v^k|_Y^2 + |p_\infty|_V^2 + |p_{v,\infty}|^2   \big),
\end{aligned}\end{equation*}
which gives after choosing $C_v > \frac{4 \wdk}{D_v m_0}$ and absorbing
\begin{equation}
\label{Eq:Final1D}
\begin{aligned} &\frac{C_v}{2} \ddt |\phi_v^k-\phi_{v,\infty}|_Y^2
   + \frac{3C_v D_v m_0}{4} |\nablala \phi_v^k|_Y^2 + \left(\frac{1}{2\wdk}  - \frac{(1+C_P^2\wdk^{-1})R^2 C_2 \max\{L_p;L_\sigma\}}{\wdk^2} \right) |v_v^k|_Y^2\\
    &\leq (1+C_P^2\wdk^{-1})R^2C_2 \max\{L_p;L_\sigma\} \big( |\phi_\sigma^k|_V^2 +|\phi_v^k-\phi_{v,\infty}|_Y^2 + |\phi_{v,\infty}|^2 +  K^{-2} |v^k|_H^2 + |S_p^k|_H^2
\\ &\qquad + |p_\infty|_V^2 + |p_{v,\infty}|^2  \big) +C\big( |\phi_{v,\infty}'|^2 +  |\phi_v^k-\phi_{v,\infty}|^2_Y    \big).
    \end{aligned}
\end{equation}

	\subsubsection*{Step 3 (Adding)}

We add the equations \cref{Eq:FinalCH}, \cref{Eq:FinalRD} and \cref{Eq:Final1D} to arrive at
\begin{equation} \label{Eq:Adding} \begin{aligned}
	&   \frac12 \ddt \bigg[2  |\Psi^k|_{L^1(\Omega)}   +  \eps^2_P |\nabla \phi_P^k|_H^2
   +\eps^2_H  |\nabla \phi_H^k|_H^2 + \eps_N^2 |\nabla \phi_N^k|_H^2 + C_\sigma |\phi_\sigma^k|_H^2  +|\mde^k|_H^2+|\taf^k|_H^2 \\
   &\qquad + |\ecm^k|_V^2 + C_v|\phi_v^k-\phi_{v,\infty}|_Y^2  \bigg]  +\frac{m_0}{2} |\nabla \mu_P^k|_H^2 +   \frac{m_0}{2} |\nabla \mu_H^k|_H^2 +\frac12 |\mu_P^k|_H^2 + \frac12 |\mu_H^k|_H^2   \\
   &\qquad +  \left( \frac{1}{2K} - \frac{C_1L_p}{K^2}  - \frac{C_\sigma}{D_\sigma m_0} - \frac{ (C_\sigma+R^2+R^2C_P^2 \wdk^{-1}) C_2 \max\{L_p;L_\sigma\}}{K^2} \right) |v^k|_H^2
   \\ &\qquad  + \left(\frac{C_\sigma D_\sigma m_0}{2} - \frac{4m_\infty^2 \chi_c^2}{m_0} - (C_\sigma+\wdk+1) C_2 \max \{L_p;L_\sigma\}  \right)  |\nabla \phi_\sigma^k|_H^2
  \\ &\qquad  + D_{\MDE} m_0| \nabla \mde^k|_H^2  + D_{\TAF} m_0 |\nabla \taf^k|_H^2  + \frac{3C_vD_v m_0}{4} |\nablala \phi_v^k|_Y^2   \\ &\qquad
   +  \left( \frac{1}{2\wdk} - \frac{C_1L_p}{\wdk^2} - \frac{(C_\sigma +R^2+R^2C_P^2\wdk^{-1}) C_2 \max\{L_p;L_\sigma\}}{\wdk^2}  \right) |v_v^k|_Y^2
   \\ &\leq
    \eps \big(|\nabla S_N^k|_H^2+  |\nabla S_{\ECM}^k|_H^2 \big) + (C_1 + (C_\sigma +R^2+R^2C_P^2 \wdk^{-1}) C_2) \max\{L_p;L_\sigma\} |S_p^k|_H^2
	+ C \big( 1+|\phi_P|_V^2\\ &\qquad  + |\phi_H^k|_V^2 + |\phi_N^k|_V^2 + \blue{|\phi_v^k|_Y^2}+ |\phi_\sigma^k|_H^2  + |\mde^k|_H^2
	  + |\taf^k|_H^2
   + |\ecm^k|_V^2 + |\phi_v - \phi_{v,\infty}|_Y^2 + |p_\infty|_V^2 + |p_{v,\infty}|^2  \\ &\qquad  + |\phi_{v,\infty}|^2+|\phi_{v,\infty}'|^2+|S_P^k|_H^2 + |S_H^k|_H^2+  |S_N^k|_H^2 +|S_\sigma^k|_H^2 + |S_{\MDE}^k|_H^2
	+ |S_{\TAF}^k|_H^2 + |S_{\ECM}^k|_H^2 \big).
\end{aligned} \end{equation}
By assumption \cref{Ass:Source} on the source functions, we have the three estimates
\begin{itemize} \itemsep1em
	\item $\displaystyle \sum_{\alpha \in \A} |S_\alpha^k|_H^2
	\lesssim  \sum_{\alpha \in \A} |\phi_\alpha|_H^2$,
	\item $\displaystyle|\nabla S_N^k|_H^2 +  |\nabla S_{\ECM}^k|_H^2 \leq 2^{|\A|} f_\infty^2 |\A|^2 \sum_{\alpha \in \A} |\nabla \phi_\alpha|_H^2$,
	\item $\displaystyle|S_p^k|_H^2 \leq 8 \big( |\nabla \mu^k_P|^2_H +  |\nabla \mu^k_H|^2_H+  2\chi_c^2 |\nabla \phi_\sigma^k|_H^2 + 2\chi_h^2 |\nabla \ecm^k|_H^2  \big)$,
	\end{itemize}
and insert these estimates into \cref{Eq:Adding}. Further, in order to treat the factor $|\nabla S_N^k|_H^2 +  |\nabla S_{\ECM}^k|_H^2$, we choose the constant
\begin{equation} \label{Eq:Epsilon} \eps=\frac{m_0}{2^{|\A|+2} f_\infty^2 |\A|^2} \min\{C_\sigma D_\sigma,D_{\MDE},D_{\TAF}\}, \end{equation}
so that we can conclude
\begin{equation*}
	\begin{aligned}
	\eps \big(|\nabla S_N^k|_H^2 +  |\nabla S_{\ECM}^k|_H^2 \big)&\leq \frac{m_0}{4}  \min\{C_\sigma D_\sigma,D_{\MDE},D_{\TAF}\} \sum_{\alpha \in \A} |\nabla \phi_\alpha|_H^2 \\ &\leq \frac{C_\sigma D_\sigma m_0}{4} |\nabla \phi_\sigma^k|_H^2 + \frac{D_{\MDE} m_0}{4} |\nabla \mde^k|_H^2 + \frac{D_{\TAF} m_0}{4} |\nabla \taf^k|_H^2 \\
	&\qquad + C (|\phi_P^k|_V^2+|\phi_H^k|^2+|\phi_N^k|^2+|\ecm^k|_V^2).
	\end{aligned}
\end{equation*}
We absorb, collect and summarize the constants, giving
\begin{equation} \label{Eq:Collect} \begin{aligned}
	&  \frac12  \ddt \bigg[ 2 |\Psi^k|_{L^1(\Omega)}   +  \eps^2_P |\nabla \phi_P^k|_H^2
   + \eps^2_H  |\nabla \phi_H^k|_H^2 +  \eps_N^2  |\nabla \phi_N^k|_H^2 +C_\sigma |\phi_\sigma^k|_H^2  +|\mde^k|_H^2+|\taf^k|_H^2
   \\ &\qquad + |\ecm^k|_V^2 + |\phi_v^k-\phi_{v,\infty}|_Y^2 \bigg]+\frac12 |\mu_P^k|_H^2 + \frac12 |\mu_H^k|_H^2
   \\ &\qquad  +\left( \frac{m_0}{2} - 8(C_1 + (C_\sigma +R^2+R^2 C_P^2 \wdk^{-1}) C_2) \max\{L_p;L_\sigma\} \right) \big(|\nabla \mu_P^k|_H^2 +   |\nabla \mu_H^k|_H^2 \big)
   \\  &\qquad +  \left( \frac{1}{2K} - \frac{C_1L_p}{K^2}  - \frac{C_\sigma}{D_\sigma m_0} - \frac{(C_\sigma +R^2+R^2 C_P^2 \wdk^{-1}) C_2 \max\{L_p;L_\sigma\}}{K^2} \right) |v^k|_H^2
      \\ &\qquad  + \left(\frac{C_\sigma D_\sigma m_0}{2} - \frac{4m_\infty^2 \chi_c^2}{m_0} - (C_\sigma +R^2+R^2 C_P^2 \wdk^{-1}) C_2 \max \{L_p;L_\sigma\}  \right)  |\nabla \phi_\sigma^k|_H^2
 \\ &\qquad + \frac{3m_0}{4}\big(D_{\MDE}  | \nabla \mde^k|_H^2  + D_{\TAF} |\nabla \taf^k|_H^2  + D_v  |\nablala \phi_v^k|_Y^2\big)
 \\ &\qquad +\left( \frac{1}{2\wdk} - \frac{C_1L_p}{\wdk^2} - \frac{(C_\sigma +R^2+R^2 C_P^2 \wdk^{-1}) C_2 \max\{L_p;L_\sigma\}}{\wdk^2}  \right) |v_v^k|_Y^2
   \\ &\leq  C \big( 1+    |\phi_P^k|_V^2 + |\phi_H^k|_V^2 + |\phi_N^k|_V^2 +\blue{|\phi_v^k|_Y^2}+ |\phi_\sigma^k|_H^2   +|\mde^k|_H^2+|\taf^k|_H^2 + |\ecm^k|_V^2  +|\phi_v^k-\phi_{v,\infty}|_Y^2
   \\ &\qquad +  |p_\infty|_V^2 + |p_{v,\infty}|^2 + |\phi_{v,\infty}|^2 + |\phi_{v,\infty}'|^2 \big),
\end{aligned} \end{equation}
and we choose $C_\sigma$ and $L_p$, $L_\sigma$, $K$ such that the prefactors are positive, see also assumption \cref{Ass:Jv}. In particular, we have to ensure the condition
$$\frac{8m_\infty^2 \chi_c^2}{m_0^2 D_\sigma} < C_\sigma < \frac{D_\sigma m_0}{2K}.$$

	\subsubsection*{Step 4 (Gr\"onwall--Bellman lemma)}
\blue{First, we eliminate the prefactors on the left hand side of the energy inequality \cref{Eq:Collect}  by estimating it with the minimum of all prefactors and bringing it to the right hand side to the generic constant $C$.} Afterwards, we integrate the inequality over the time interval $(0,t)$ with $t \in (0,T_k)$, apply the growth assumption \cref{Ass:Psi}, and obtain
\begin{equation*} \begin{aligned}
	&    |\Psi^k(t)|_{L^1(\Omega)}   +  |\phi_P^k(t)|_V^2
	+  |\phi_H^k(t)|_V^2 +    |\phi_N^k(t)|_V^2 + |\phi_\sigma^k(t)|_H^2  +|\mde^k(t)|_H^2+|\taf^k(t)|_H^2+ |\ecm^k(t)|_V^2
	\\ &\qquad  + |\phi_v^k(t)-\phi_{v,\infty}|_Y^2  +  \|\phi_\sigma^k\|_{L^2(0,T_k;V)}^2 + \| \mu_P^k\|_{L^2(0,T_k;V)}^2 +   \|\mu_H^k\|_{L^2(0,T_k;V)}^2  +  \| \mde^k\|_{L^2(0,T_k;V)}^2   \\ & \qquad + \| \taf^k\|_{L^2(0,T_k;V)}^2
	+  \|\phi_v^k - \phi_{v,\infty} \|_{L^2(0,T_k;X_0)}^2
	+ \|v^k\|_{L^2(0,T_k;H)}^2 + \|v_v^k\|_{L^2(0,T_k;Y)}^2 \\ & \qquad - C \big( \|\phi_P^k\|_{L^2(0,T_k;V)}^2 + \|\phi_H^k\|_{L^2(0,T_k;V)}^2 + \|\phi_N^k\|_{L^2(0,T_k;V)}^2 +\blue{\|\phi_v^k\|_{L^2(0,T_k;Y)}^2}+ \|\phi_\sigma^k\|_{L^2(0,T_k;H)}^2 \\ &\qquad  +\|\mde^k\|_{L^2(0,T_k;H)}^2 +\|\taf^k\|_{L^2(0,T_k;H)}^2   +\|\ecm^k\|_{L^2(0,T_k;V)}^2  +\|\phi_v^k-\phi_{v,\infty}\|_{L^2(0,T_k;Y)}^2   \big)
	\\ &\leq C(T_k) \cdot \big (  1+   |\Psi^k(0)|_{L^1(\Omega)}   +  |\nabla \phi_{P,0}^k|_H^2
	+   |\nabla \phi_{H,0}^k|_H^2 +    |\nabla \phi_{N,0}^k|_H^2 + |\phi_{\sigma,0}^k|_H^2  +|\phi_{\MDE,0}^k|_H^2+|\phi_{\TAF,0}^k|_H^2
	  \\ &\qquad + |\phi_{\ECM,0}^k|_V^2+ |\phi_{v,0}^k|_Y^2 + \|p_\infty\|_{L^2(0,T;V)}^2 + |p_{v,\infty}|_{L^2(0,T)}^2+ |\phi_{v,\infty}|_{H^1(0,T)}^2  \big)  .
\end{aligned} \end{equation*}

By applying the Gr\"onwall--Bellman lemma, see \cref{Lem_Gronwall}, we obtain
\begin{equation} \label{Eq:Gronwall} \begin{aligned}
		 & \|\Psi^k\|_{L^\infty(0,T_k;L^1(\Omega))} +  \sum_{\mathclap{\alpha \in \CH \cup \{\ECM\}}} ~ \| \phi_\alpha^k\|^2_{L^\infty(0,T_k;V)} + ~\sum_{\mathclap{\alpha \in \{P,H\}}} ~ \| \mu_\alpha^k\|^2_{L^2(0,T_k;V)} + \sum_{\beta \in \RD}
		\|\phi_\beta^k\|^2_{L^\infty(0,T_k;H) \cap L^2(0,T_k;V)}
		\\ &\qquad + \|v^k\|_{L^2(0,T_k;H)}^2  +\|\phi_v^k-\phi_{v,\infty}\|^2_{L^\infty(0,T_k;Y) \cap L^2(0,T_k;X_0)}  + \|v_v^k\|_{L^2(0,T_k;Y)}^2  \\ &\leq C(T_k) \cdot \Big( 1+|\phi_{v,0}^k|^2_{Y}+ \sum_{\mathclap{\alpha \in \CH \cup \{\ECM\}}} ~ | \phi_{\alpha,0}^k|^2_{V} + \sum_{\beta \in \RD}
		|\phi_{\beta,0}^k|^2_{H}   +  |\Psi(\phi_{P,0}^k,\phi_{H,0}^k,\phi_{N,0}^k)|_{L^1(\Omega)} + \|p_\infty\|_{L^2(0,T;V)}^2
		\\[-.2cm] &\qquad  + |p_{v,\infty}|_{L^2(0,T)}^2 + |\phi_{v,\infty}|_{H^1(0,T)}^2  \Big).
	\end{aligned} \end{equation}

We have chosen the initial values of the Faedo--Galerkin approximations as the orthogonal projections of the initial values of their counterpart, see \cref{Eq:GalerkinInitial}. The operator norm of an orthogonal projection is bounded by $1$ and, therefore, uniform estimates are obtained in \cref{Eq:Gronwall}; for example
$$|\phi_{P,0}^k|_V^2 = |\Pi_{H_k} \phi_{P,0}|_V^2 \leq |\phi_{P,0}|_V^2.$$
Using the upper bound \cref{Eq:PsiGrowth} of $\Psi$, we treat the term involving the potential function on the right hand side in the following way:
$$\begin{aligned}
	|\Psi(\phi_{P,0}^k,\phi_{H,0}^k,\phi_{N,0}^k)|_{L^1(\Omega)} &\lesssim 1+  |\phi_{P,0}^k|_H^2+|\phi_{H,0}^k|_H^2+|\phi_{N,0}^k|_H^2 \\  &= 1+  |\Pi_{H_k} \phi_{P,0}|_H^2|+|\Pi_{H_k} \phi_{H,0}|_H^2+|\Pi_{H_k} \phi_{N,0}|_H^2 \\ &\leq 1+  |\phi_{P,0}|_H^2+|\phi_{H,0}|_H^2+|\phi_{N,0}|_H^2.
\end{aligned}$$
Now, the $k$-independent right hand side in the estimate allows us to extend the time interval by setting $T_k=T$ for all $k \in \mathbb{N}$.
Therefore, we have the final uniform energy estimate,
\begin{equation}\label{Eq:FinalEnergy}
	\begin{aligned}
			& \|\Psi^k\|_{L^\infty(0,T;L^1(\Omega))} + \sum_{\mathclap{\alpha \in \CH \cup \{\ECM\}}} ~ \| \phi_\alpha^k\|^2_{L^\infty(0,T;V)} + ~\sum_{\mathclap{\alpha \in \{P,H\}}} ~ \| \mu_\alpha^k\|^2_{L^2(0,T;V)} + \sum_{\mathclap{\beta \in \RD}}
		   \|\phi_\beta^k\|^2_{L^\infty H \cap L^2(0,T;V)}   \\ &\qquad + \|v^k\|_{L^2(0,T;H)}^2+\|\phi_v^k-\phi_{v,\infty}\|^2_{L^\infty(0,T;Y) \cap L^2(0,T;X_0)}  + \|v_v^k\|_{L^2(0,T;Y)}^2  \\ &\leq C(T)\! \cdot \! \Big(1+|\phi_{v,0}|^2_{Y}+ \sum_{\mathclap{\alpha \in \CH \cup \{\ECM\}}} ~ | \phi_{\alpha,0}|^2_{V} + \sum_{\mathclap{\beta \in \RD}}
		   |\phi_{\beta,0}|^2_{H} +\|p_\infty\|_{L^2(0,T;V)}^2
		     + |p_{v,\infty}|_{L^2(0,T)}^2 + |\phi_{v,\infty}|_{H^1(0,T)}^2   \Big).
	   \end{aligned} \end{equation}
	   From this energy inequality and \cref{Eq:ModelGalerkinVelocity} we also get bounds for the pressures $p^k$ and $p_v^k$ in the following way $$\|p^k-p_\infty\|_{L^2(0,T;V_0)}  + \|p_v^k - p_{v,\infty} \|_{L^2(0,T;X_0)}  \leq C.$$

\subsection{Limit process} \label{Sec:Limit}

\subsubsection*{Weak convergence}
Next, we prove that there are subsequences of $\phib^k,\mu_P^k,\mu_H^k,p^k,\phi_v^k,p_v^k$, which converge to a weak solution of our model \cref{Eq:Model3D}--\cref{Eq:Model1D} in the sense of \cref{Def:Weak}. From the energy estimate \cref{Eq:FinalEnergy} we deduce that
\begin{equation}\begin{aligned}
		\{\phi_\alpha^k\}_{k \in \mathbb{N}}      & \text{ is bounded in } L^\infty(0,T;V),        &&\alpha \in \CH \cup \{\ECM\}            ,            \\
		\{\mu_\alpha^k\}_{k \in \mathbb{N}}         & \text{ is bounded in } L^2(0,T;V),       &&\alpha \in \{P,H\},                                    \\
		\{\phi_\beta^k\}_{k \in \mathbb{N}} & \text{ is bounded in } L^\infty(0,T;H) \cap L^2(0,T;V), &&\beta \in \RD,     \\
		\{v^k\}_{k \in \mathbb{N}}           & \text{ is bounded in }  L^2(0,T;L^2(\Omega;\R^3)), \\
				\{p^k\}_{k \in \mathbb{N}}           & \text{ is bounded in }  (p_\infty + L^2(0,T;V_0)), \\
		\{\phi_v^k\}_{k \in \mathbb{N}} & \text{ is bounded in } L^\infty(0,T;Y) \cap (\phi_{v,\infty} + L^2(0,T;X_0)),     \\
				\{v_v^k\}_{k \in \mathbb{N}}           & \text{ is bounded in }  L^2(0,T;Y), \\
		\{p_v^k\}_{k \in \mathbb{N}}           & \text{ is bounded in }  (p_{v,\infty} + L^2(0,T;X_0)),
	\end{aligned} \label{Eq:Bounds1} \end{equation}
and, by the Banach--Alaoglu theorem, these bounded sequences have weakly/weakly-$*$ convergent subsequences. By a standard abuse of notation, we drop the subsequence index. Consequently, there are functions $\phib:(0,T)\times \Omega \to \R^{|\A|}$, $\mu_P, \mu_H, p :(0,T)\times \Omega \to \R$, $v:(0,T) \times \Omega \to \R^3$, $ \phi_v, v_v,p_v:(0,T)\times \Lambda \to \R$ such that, for $k \to \infty$,
\begin{equation}
	\begin{alignedat}{3}
		\phi_\alpha^k &\longweak \phi_\alpha &&\text{ weakly-$*$ in } L^\infty(0,T;V), &&\quad \alpha \in \CH \cup \{\ECM\},
		\\
		\mu_\alpha^k &\longweak \mu_\alpha &&\text{ weakly\phantom{-*} in } L^2(0,T;V), &&\quad \alpha \in \CH \backslash \{N\},      \\
		\phi_\beta^k &\longweak \phi_\beta &&\text{ weakly-$*$ in } L^\infty(0,T;H) \cap L^2(0,T;V), &&\quad \beta \in \RD, \\
		v^k &\longweak v &&\text{ weakly\phantom{-*} in } L^2(0,T;L^2(\Omega;\R^3)) , \\
		p^k &\longweak p &&\text{ weakly\phantom{-*} in } (p_\infty +L^2(0,T;V_0)) , \\
		\phi_v^k &\longweak \phi_v &&\text{ weakly-$*$ in } L^\infty(0,T;Y) \cap (\phi_{v,\infty} + L^2(0,T;X_0)), \\
				v_v^k &\longweak v_v &&\text{ weakly\phantom{-*} in }  L^2(0,T;Y), \\
		p_v^k &\longweak p_v &&\text{ weakly\phantom{-*} in } (p_{v,\infty} + L^2(0,T;X_0)).
	\end{alignedat}
	\label{Eq:WeakConv}
\end{equation}

\subsubsection*{Strong convergence} We now consider taking the limit $k \to \infty$ in the Faedo--Galerkin system \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D} in the hope to attain the initial variational system \cref{Eq:ModelWeak3D}--\cref{Eq:ModelWeak1D}. Since the equations in \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D} are nonlinear in $\phib^k$ and $\phi_v^k$, we want to achieve strong convergence of these sequences before we take the limit in \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D}. Therefore, our goal is to bound their time derivatives and to apply the Aubin--Lions--Simon compactness lemma \cref{Eq:EmbeddingComp}.

Let $(\varphi,\hat \varphi, \tilde \varphi)$ be such that $\varphi \in L^2(0,T;V)$, $\hat \varphi \in L^2(0,T;H)$, $\tilde \varphi  \in L^2(0,T;X_0)$, and
$$\Pi_{H_k}\varphi = \sum_{j=1}^k \varphi_j^k h_j, \quad \Pi_{H_k}\hat \varphi = \sum_{j=1}^k \hat \varphi_j^k h_j, \quad \Pi_{Y_k} \tilde \varphi = \sum_{j=1}^k \tilde \varphi_j^k y_j,$$
with time-dependent coefficient functions $\varphi_j^k, \hat \varphi_j^k, \tilde \varphi_j^k : (0,T) \to \R$, $j \in \{1,\dots,k\}$.  We multiply the equations \cref{Eq:ModelGalerkin3D} and \cref{Eq:ModelGalerkin1D} by $\tilde \varphi_j^k$ by the appropriate coefficient functions, sum up each equation from $j=1$ to $k$ and integrate in time over $(0,T)$, to obtain the equation system,
\begin{equation} \label{Eq:Derivative}  \begin{aligned}
	\int_0^T \langle \pt \phi_\alpha^k , \varphi \rangle_V \dt &= \int_0^T (\C(\phi_\alpha^k) v^k,\nabla \Pi_{H_k} \varphi)_H -(m_\alpha^k \nabla \mu_\alpha^k,\nabla \Pi_{H_k} \varphi)_H                             + (S_\alpha^k,\Pi_{H_k} \varphi)_H \dt,
	\\
	\int_0^T \langle \pt \phi_\beta^k , \hat \varphi \rangle_V     \dt                                                                                & = \int_0^T (S_\beta^k,\Pi_{H_k} \hat \varphi)_H \dt,
	\\
	\int_0^T \langle \pt \phi_\sigma^k,\varphi \rangle_V \dt &=\int_0^T (\C(\phi_\sigma^k)v^k,\nabla\Pi_{H_k}\varphi)_H -  D_\sigma(m_\sigma^k \nabla \phi_\sigma^k, \nabla \Pi_{H_k}\varphi)_H + (S_\sigma^k,\Pi_{H_k}\varphi)_H \\ &\qquad+ \langle \delta_\Gamma, J_{\sigma v,\Gamma}^k \Pi_{H_k}\varphi\rangle_W              -\chi_c (m_\sigma^k \nabla (\phi_P^k +\phi_H^k +\phi_N^k ),\nabla \Pi_{H_k} \varphi)_H \dt,
	\\
	\int_0^T \langle \pt \phi_\gamma^k,\varphi\rangle_V \dt &= \int_0^T -  D_\gamma (m_\gamma^k \nabla \phi_\gamma^k,\nabla \Pi_{H_k} \varphi)_H                                                       + (S_\gamma^k,\Pi_{H_k} \varphi)_H \dt,
	\\
	\int_0^T \langle \pt \phi_v^k, \tilde \varphi\rangle_X \dt &= \int_0^T  (\C(\phi_v^k) v_v^k ,\blue{\nablala} \Pi_{Y_k}\tilde \varphi )_Y - D_v (m_v^k \nablala \phi_v^k,\nablala \Pi_{Y_k} \tilde \varphi)_Y  -\wdr(J_{\sigma v}^k,\Pi_{Y_k} \tilde \varphi)_Y \dt,
\end{aligned}
\end{equation}
where $\alpha \in \{P,H\}$, $\beta \in \{N,\ECM\}$, $\gamma \in \{\MDE,\TAF\}$. Each equation in \cref{Eq:Derivative} can be treated using standard inequalities \blue{and the estimate involving the trace operator, see \cref{Eq:EstimateDelta},} the boundedness of the orthogonal projection and the energy estimate \cref{Eq:FinalEnergy}, e.g., we find
\begin{equation*} \begin{aligned} \int_0^T \langle \pt \phi_\sigma^k , \varphi \rangle_V \dt &\lesssim \int_0^T  |v^k|_H |\varphi|_V +  | \phi_\sigma^k|_V |\varphi|_V + \sum_{\alpha \in \A} |\phi_\alpha^k|_H |\varphi|_H + |J_{\sigma v,\Gamma}^k|_{L^2(\Gamma)} |\varphi|_V + \sum_{\beta \in \CH } |\phi_\beta^k|_V |\varphi|_V  \dt
	\\ & \lesssim  \|\varphi\|_{L^2(0,T;V) }.
	\end{aligned} \label{Eq:DerivativeBoundCH}
\end{equation*}
From this inequality and the bounds derived earlier, see \cref{Eq:Bounds1}, we conclude that
\begin{alignat*}{3}
	 & \{\phi_\alpha^k\}_{k \in \mathbb{N}}          &  & \text{ is bounded in }  H^1(0,T;V') \cap L^\infty(0,T;V) ,                          && \qquad \alpha \in \{P,H\},                                     \\
	 & \{\phi_\beta^k\}_{k \in \mathbb{N}}          &  & \text{ is bounded in } H^1(0,T;H) \cap L^\infty(0,T;V),                          && \qquad \beta \in \{N,\ECM\},                                     \\
	 & \{\phi_\gamma^k\}_{k \in \mathbb{N}}     &  & \text{ is bounded in } H^1(0,T;V') \cap L^\infty(0,T;H) \cap L^2(0,T;V) ,     && \qquad \gamma \in \RD,              \\
	 & \{\phi_v^k\}_{k \in \mathbb{N}}     &  & \text{ is bounded in } H^1(0,T;X_0') \cap L^\infty(0,T;Y) \cap (\phi_{v,\infty} + L^2(0,T;X_0)) .
\end{alignat*}
We apply the Aubin--Lions--Simon compactness lemma \cref{Eq:EmbeddingComp},
yielding the strong convergences as $k \to \infty$
\begin{equation} \label{Eq:StrongConv}
	\begin{alignedat}{3}
		\phi_\alpha^k &\longrightarrow \phi_\alpha &&\text{ strongly in } C^0([0,T];H), && \qquad \alpha \in \CH \cup \{\ECM\},\\
		\phi_\beta^k &\longrightarrow \phi_\beta &&\text{ strongly in } L^2(0,T;H)\cap C^0([0,T];V'),&& \qquad \beta \in \RD, \\
		\phi_v^k &\longrightarrow \phi_v &&\text{ strongly in } L^2(0,T;Y)\cap C^0([0,T];X_0').
	\end{alignedat}
\end{equation}
The strong convergence $\phi_\alpha^k \to \phi_\alpha$ in $C^0([0,T];H)$ implies $\phi_\alpha(0)=\phi_{\alpha,0}$ in $H$ and similarly $\phi_\beta(0)=\phi_{\beta,0}$ in $V'$ and $\phi_v(0)=\phi_{v,0}$ in $X'_0$. Therefore, the limit functions $(\phib,\phi_v)$ of the Faedo--Galerkin approximations fulfill the initial conditions for the system \cref{Eq:Model3D}--\cref{Eq:Initial1D}.

\subsubsection*{Limit process} We show that the limit functions also satisfy the variational form \cref{Eq:ModelWeak3D}--\cref{Eq:ModelWeak1D}, as defined in \cref{Def:Weak}. Multiplying the Faedo--Galerkin system \cref{Eq:ModelGalerkin3D}--\cref{Eq:ModelGalerkin1D} by $\eta \in C_c^\infty(0,T)$ and integrating from $0$ to $T$, gives
\begin{equation}\begin{aligned}
	\int_0^T \langle \pt \phi_\alpha^k , h_j \rangle_V \eta(t) \dt &= \int_0^T \big( (\C(\phi_\alpha^k) v^k,\nabla h_j)_H -(m_\alpha^k \nabla \mu_\alpha^k,\nabla  h_j)_H                             + (S_\alpha^k,h_j)_H\big) \eta(t) \dt,
	\\
	\int_0^T (\mu_\alpha^k,h_j)_H \eta(t) \dt &= \int_0^T \big( (\p_{\phi_\alpha} \Psi^k - \chi_c \phi_\sigma^k - \chi_h \ecm^k,h_j)_H + \eps_\alpha^2 (\nabla \phi_\alpha^k, \nabla h_j)_H \big) \eta(t) \dt,
	\\
	\int_0^T ( \pt \phi_\beta^k , h_j )_H   \eta(t) \dt                                                                                & = \int_0^T (S_\beta^k,h_j)_H \eta(t) \dt,	\end{aligned}
\end{equation}
 and
\begin{equation}  \begin{aligned}
	\int_0^T \langle \pt \phi_\sigma^k,h_j \rangle_V \dt &=\int_0^T \big( (\C(\phi_\sigma^k) v^k,\nabla h_j)_H -  D_\sigma(m_\sigma^k \nabla \phi_\sigma^k, \nabla h_j)_H + (S_\sigma^k,h_j)_H \\ &\qquad+ \langle \delta_\Gamma, J_{\sigma v,\Gamma}^k h_j\rangle_W              -\chi_c (m_\sigma^k \nabla (\phi_P^k +\phi_H^k +\phi_N^k ),\nabla h_j)_H \big) \eta(t) \dt,
	\\
	\int_0^T \langle \pt \phi_\gamma^k,h_j\rangle_V  \eta(t) \dt &= \int_0^T \big( -  D_{\gamma} (m_{\gamma}^k \nabla \phi_\gamma^k,\nabla h_j)_H                                                       + (S_{\gamma}^k,h_j)_H \big) \eta(t) \dt,
	\\
		\int_0^T (v^k,h_j)_H \eta(t) \dt &= \int_0^T \big(-K(\nabla p,h_j)_H + (S_p,\nabla h_j)_H \big) \eta(t) \dt,  \\
	\int_0^T K(\nabla p^k,\nabla h_j)_H \eta(t) \dt &= \int_0^T \big( \langle \delta_\Gamma, J_{pv,\Gamma}^k h_j \rangle_W + (S_p,\nabla h_j)_H \big) \eta(t) \dt,
\end{aligned}
\end{equation}
and
\begin{equation}  \begin{aligned}
	\int_0^T \langle \pt \phi_v^k, y_j\rangle_X \eta(t)  \dt &= \int_0^T \big(   (\C( \phi_v^k) v_v^k ,\nabla y_j )_Y - D_v (m_v^k \nablala \phi_v^k,\nablala y_j)_Y  -\wdr(J_{\sigma v}^k,y_j)_Y \big) \eta(t) \dt, \\
		\int_0^T  (v_v^k, y_j)_Y \eta(t) \dt &= \int_0^T -  \wdk (\nablala p_v^k,y_j)_Y \eta(t) \dt, \\
	\int_0^T \wdk (\nablala p_v^k, \nablala y_j)_Y \eta(t) \dt &= \int_0^T -  \wdr(J_{pv}^k,y_j)_Y \eta(t) \dt,
\end{aligned}
\end{equation}
for each $j\in \{1,\dots,k\}$, $\alpha \in \{P,H\}$, $\beta \in \{N,\ECM\}$, $\gamma \in \{\MDE,\TAF\}$. We take the limit $k \to \infty$ in each equation. The linear terms can be treated directly in the limit process since they can be justified via the weak convergences \cref{Eq:WeakConv}, e.g.,the functional
$$\mu_P^k \mapsto \int_0^T (\mu_P^k,h_j)_H \eta(t) \, \dt \leq \|\mu_P^k\|_{L^2(0,T;H)} |h_j|_{H} |\eta|_{L^2(0,T)}, $$
is linear and continuous on $L^2(0,T;H)$ and hence, as $k\to \infty$,
$$\int_0^T (\mu_P^k,h_j)_H \eta(t) \, \dt  \longrightarrow \int_0^T (\mu_P,h_j) \eta(t) \, \dt.$$
Thus, it remains to examine the nonlinear terms. We do so in the steps (i)--(v) as follows. \\[0.1cm]
\begin{enumerate}[label=(\roman*), wide, labelwidth=!, labelindent=0pt] \itemsep1em
\item We have derived the convergence, see \cref{Eq:StrongConv}, $$\begin{aligned}\phi_\alpha^k \longrightarrow \phi_\alpha &\text{ in } L^2(0,T;H) \cong L^2((0,T)\times \Omega), &&\alpha \in \A,\end{aligned}$$ for $k \to \infty$ and, consequently, we have by the continuity and boundedness of $m_\alpha$,
	$$m_\alpha^k=m_\alpha\big(\phib^k(t,x)\big) \longrightarrow m_\alpha\big(\phib(t,x)\big)=:m_\alpha \text{ a.e. in } (0,T)\times \Omega \text{ for } k \to \infty.$$
Applying the Lebesgue dominated convergence theorem, gives for $k \to \infty$
$$m_\alpha^k  \nabla h_j \eta \longrightarrow m_\alpha \nabla h_j \eta \text{ in } L^2((0,T) \times \Omega;\R^d),$$
and, together with $\nabla \mu_\alpha^k \weak \nabla \mu_\alpha$ weakly in $L^2((0,T)\times \Omega;\R^d)$ as $k\to\infty$, we have for $k \to \infty$
$$m_\alpha(\phib^k) \eta \nabla h_j \cdot \nabla \mu_\alpha^k \longrightarrow m_\alpha(\phib) \eta \nabla h_j \cdot \nabla \mu_\alpha \text{ in } L^1((0,T)\times \Omega).$$
We use here the fact that the product of a strongly and a weakly converging sequence in $L^2$ converges strongly in $L^1$.  \\[0.1cm]
\item By \cref{Eq:StrongConv}, we have $\phi_\alpha^k \to \phi_\alpha$ in $L^2((0,T)\times \Omega)$ and $v^k \weak v$ in $L^2((0,T)\times \Omega;\R^d)$ as $k\to \infty$, hence for $k \to \infty$ $$\C(\phi_\alpha^k) v^k  \cdot \nabla h_j \eta \longrightarrow \C(\phi_\alpha) v \cdot \nabla h_j \eta \text{ in } L^1((0,T)\times \Omega).$$
\item By the continuity and the growth assumptions on $\p_{\phi_\alpha} \Psi$, we have for $k \to \infty$
\begin{gather*} \p_{\phi_\alpha} \Psi\big(\phi_P^k(t,x),\phi_H^k(t,x),\phi_N^k(t,x)\big) \longrightarrow \p_{\phi_\alpha}\Psi\big(\phi_P(t,x),\phi_H(t,x),\phi_N(t,x)\big) \text{ a.e. in } (0,T)\times \Omega, \\
	|\p_{\phi_\alpha}\Psi(\phi_P^k,\phi_H^k,\phi_N^k) \eta h_j| \leq C(1+|\phi^k_P|+|\phi^k_H|+|\phi^k_N|)|\eta h_j|,
\end{gather*}
and the Lebesgue dominated convergence theorem yields for $k \to \infty$
$$\p_{\phi_\alpha}\Psi(\phi_P^k,\phi_H^k,\phi_N^k) \eta h_j \longrightarrow \p_{\phi_\alpha}\Psi(\phi_P,\phi_H,\phi_N) \eta h_j \text{ in } L^1((0,T)\times \Omega).$$
\item  We have the strong convergence of $\phi_P^k$ and $\phi_H^k$ in $L^2((0,T)\times \Omega)$ and the continuity and boundedness of $\C$. Together with the weak convergence of $\nabla \phi_\sigma^k$ and $\nabla \mu^k$ in $L^2((0,T)\times \Omega;\R^d)$ it is enough to conclude the convergence of the term involving $$S_p^k=-\C(\phi_P^k) (\nabla \mu_P^k+\chi_c \nabla \phi_\sigma^k)-\C(\phi_H^k) (\nabla \mu_H^k+\chi_c \nabla \phi_\sigma^k).$$
\item We have $\phi_v^k \to \phi_v$ in $L^2((0,T)\times \Lambda)$ and $\phi_\sigma^k \to \phi_\sigma$ in $L^2((0,T) \times \Omega)$ as $k \to \infty$ and therefore, also $\Pi_\Gamma \phi_v^k \to \Pi_\Gamma \phi_v$ in $L^2((0,T)\times \Gamma)$. Since $f_{\sigma,v}$ is a continuous and bounded function, we conclude
$$\int_0^T \int_\Gamma |f_{\sigma,v}(\phi_v^k,\Pi_\Gamma \phi_\sigma^k) \tr_\Gamma h_j \eta(t) |^2 \dS \dt \lesssim  \|f_{\sigma,v}(\Pi_\Gamma \phi_v^k,\phi_\sigma^k)\|_{L^\infty((0,T)\times \Gamma)}^2 |h_j|_{V}^2 |\eta|_{L^2(0,T)}^2,$$
and the Lebesgue dominated convergence theorem gives for $k\to \infty$
$$f_{\sigma,v}(\phi_v^k,\Pi_\Gamma \phi_\sigma^k) \tr_\Gamma h_j \eta(t) \longrightarrow f_{\sigma,v}(\phi_v,\Pi_\Gamma \phi_\sigma) \tr_\Gamma h_j \eta(t) \text{ in } L^2((0,T) \times \Gamma).  $$
Together with the weak convergence of $\Pi_\Gamma p_v^k$ and $p^k$ we have for $k \to \infty$
$$\begin{aligned} \int_0^T \langle \delta_\Gamma,J_{\sigma v,\Gamma}^k h_j \rangle_W \eta(t) \dt &=\int_0^T \int_\Gamma \Big( f_{\sigma,v}(\phi_\sigma^k,\Pi_\Gamma \phi_v^k) L_p (\Pi_\Gamma p_v^k-p^k) + L_\sigma (\Pi_\Gamma \phi_v^k - \phi_\sigma^k) \Big) \tr_\Gamma h_j  \eta(t) \dS \dt \\ &\to \int_0^T \langle \delta_\Gamma,J_{\sigma v,\Gamma} h_j \rangle_W \eta(t) \dt .
\end{aligned}$$
\end{enumerate}

Using the densities of $\cup_{k \in \mathbb{N}} H_k$ in $V$, $\cup_{k \in \mathbb{N}} H_k^0$ in $V_0$ and $\cup_{k \in \mathbb{N}} Y_k$ in $X$, and the fundamental lemma of the calculus of variations, we obtain a solution tuple $(\phib,\mu_P,\mu_H,v,p,\phi_v,v_v,p_v)$ to our model \cref{Eq:Model3D} and \cref{Eq:Model1D} in the weak sense as defined in \cref{Def:Weak}.

\subsubsection*{Energy inequality} It remains to prove that found solution tuple satisfies the energy inequality \cref{Eq:EnergySolution}. First, we note that norms are weakly/weakly-$*$ lower semicontinuous, e.g.,we have $\mu_P^k \weak \mu_P$ in $L^2(0,T;V)$ and therefore, we infer
$$\| \mu_P \|_{L^2(0,T;V)} \leq \liminf_{k \to \infty} \|\mu_P^k\|_{L^2(0;T;V)}.$$
We apply the Fatou lemma on the continuous and non-negative function $\Psi$ to obtain
$$\int_\Omega \Psi(\phi_P,\phi_H,\phi_N) \, \dd x  \leq \liminf_{k \to \infty} \int_\Omega \Psi(\phi_P^k,\phi_H^k,\phi_N^k) \, \dd x.$$
Consequently, passing the limit $k \to \infty$ in the discrete energy inequality \cref{Eq:FinalEnergy} leads to  \cref{Eq:EnergySolution}.
\qed


text:
\paragraph{Proof of \propref{p:to_2r_bnd}}
First, we make the following claim:
\begin{equation}\label{eq:bold_claim}
e^{-G v^\frac{1}{y\log v-1}} \leq e^{-G 2^{1/y}} \, \left( 1 - e^{-A(v-2^{1/y})} \right),\;\text{for all $v\geq 2^{1/y}$},
\end{equation}
for some $A \equiv A(y,\,G) \geq 0$ to be determined. We proceed to derive the bound on $\prob{Y_1+Y_2\leq y}$ starting with \eqref{eq:to_2r}.
\begin{align}
\prob{Y_1+Y_2\leq y}
&= G \int\limits_{2^{1/y}}^\pinf
\expof{ -G\left(v+v^{1/y\log v - 1}-2\right) } \, \diff v \\

&\leq G e^{2G}e^{-G 2^{1/y}} \int\limits_{2^{1/y}}^\pinf
e^{-G v}\,\left(1 - e^{-A(v-2^{1/y})}\right) \,\diff v \\

&= G e^{2G}e^{-G 2^{1/y}} \int\limits_{2^{1/y}}^\pinf e^{-G v} \,\diff v
\;\;+\;\; G e^{2G}e^{(A-G)2^{1/y}} \int\limits_{2^{1/y}}^\pinf e^{-(A+G)v} \,\diff v \\

&= e^{2G}e^{-2G 2^{1/y}} - \left(\frac{G}{A+G}\right) e^{2G} e^{(A-G)2^{1/y}} e^{-(A+G)2^{1/y}} \\

&= \left(1-\frac{G}{A+G}\right) e^{2G} e^{-2G 2^{1/y}}.
\end{align}
Finally,
\begin{equation}
\prob{Y_1+Y_2\leq y}  \leq \left(\frac{A}{A+G}\right) e^{-2G(2^{1/y}-1)}.
\end{equation}

Now that we have tightened our bound on the probability, we prove the claim that we have made. We first rewrite the inequality in \eqref{eq:bold_claim} as
\begin{align}
\expof{-2^\frac{\ln v}{y\ln v - \ln 2}}
&\leq e^{-G 2^{1/y}} - \expof{-v\left(A-\frac{(A-G)2^{1/y}}{v}\right)}   \\

\expof{-v\left(A-\frac{(A-G)2^{1/y}}{v}\right)}
&\leq  e^{-G 2^{1/y}} - \expof{-2^\frac{\ln v}{y\ln v - \ln 2}} \\

\expof{-v\,\Psi(v)} &\leq \expof{-v\,\Phi(v)}.
\end{align}
Now, we determine $\Phi(v)$.
\begin{align}
\expof{-v\,\Phi(v)}
&= \expof{\lnof{e^{-G 2^{1/y}} - \expof{-2^\frac{\ln v}{y\ln v - \ln 2}}}} \\

&= \expof{-v\cdot \frac{-1}{v}\lnof{e^{-G 2^{1/y}} - \expof{-2^\frac{\ln v}{y\ln v - \ln 2}}}}.
\end{align}
Since the exponential function is increasing, a necessary condition for \eqref{eq:bold_claim} is $\Psi(v)\geq \Phi(v)$, for all $v\geq 2^{1/y}$. After a few steps of calculation, we obtain the following condition:
\begin{align}
A \geq \left(\frac{1}{2^{1/y}-v}\right)\,\lnof{1- \expof{G 2^\frac{1}{y}-G 2^\frac{\ln v}{y\ln v - \ln 2}}}.
\end{align}

\noindent We still need to prove that such an $A$ does exist. We write the \gls{rhs} as a product of two functions $f$ and $g$, where $f$ is negative and monotonically increasing towards $0$, and $g$ is also negative but monotonically decreasing towards $0$. It is clear that $\lim_{v\rightarrow 2^{1/y}}=0$. To show that $\lim_{v\rightarrow \,+\infty}=0$, we replace $g$ with a function $h$ that goes faster to $\ninf$ as its argument goes to $0$. One candidate is $h(x)=-1/x$ because $\lim_{x\rightarrow 0} \ln(x)/(-1/x) = 0$. Hence,
\begin{align}
\lim_{v\rightarrow\,\pinf} f(v)g(v)
&=\lim_{v\rightarrow\pinf} \left(\frac{1}{2^{1/y}-v}\right)\,\lnof{1- \expof{G 2^\frac{1}{y}-G 2^\frac{\ln v}{y\ln v - \ln 2}}} \\

&\leq
\lim_{v\rightarrow\,\pinf} \frac{1}{G 2^{1/y}-v}\cdot\frac{-1}{{1- \expof{G 2^\frac{1}{y}-G 2^\frac{\ln v}{y\ln v - \ln 2}}}} \\

&=
\lim_{v\rightarrow\,\pinf} \frac{1}{v}\cdot\frac{e^{-G 2^{1/y}}}{e^{-G 2^{1/y}} - \expof{-G 2^\frac{\ln v}{y\ln v - \ln 2}}} \\

&=
\lim_{v\rightarrow\,\pinf} \frac{1}{v}\cdot\frac{1}{e^{-G 2^{1/y}} - \expof{-G 2^\frac{\ln v}{y\ln v - \ln 2}}} \\

&=
\lim_{x\rightarrow\,\pinf} \frac{e^{-x}}{e^{-G 2^{1/y}} - \expof{-G 2^\frac{x}{yx - \ln 2}}}.
\end{align}
By using l'Hopital's rule, we get
\begin{align}
\lim_{v\rightarrow\,\pinf} f(v)g(v)
&=
\lim_{x\rightarrow\,\pinf} \frac{e^{-x}\;(yx-\ln  2)^2}{ G(\ln 2)^2 \, 2^{\frac{x}{yx-\ln 2}}\, e^{-G 2^{\frac{x}{yx-\ln 2}}}} \phantom{\;\;\;\;\;\;\;}\\

&=\frac{\lim_{x\rightarrow\,\pinf} e^{-x}\;(yx-\ln  2)^2}{G(\ln 2)^2 \, 2^\frac{1}{y} \, e^{-G 2^{1/y}}} \; = 0.
\end{align}

\noindent Since $fg$ does not admit any singularities on $(2^{1/y},\pinf)$, $fg$ is differentiable on that interval. Since, additionally, $\lim_{v\rightarrow 2^{1/y}} f(v)g(v)$  $=\lim_{v\rightarrow\,\pinf} f(v)g(v)=0$, Rolle's theorem says that $fg$ must have a critical point on $(2^{1/y},\pinf)$. Finally, since $fg\geq 0$, we conclude that that critical point is in fact a maximum; we choose $A$ to be that.

config: arxiv


TITLE: Differentiable atlas for infinite Mobius strip
QUESTION [2 upvotes]: I want to construct a differentiable atlas for the abstract Mobius strip $M:=X/\sim$ equipped with the quotient topology, where $X:=[0,1]\times\mathbb{R}\subset\mathbb{R}^2$ is a subset equipped with the subspace topology and $\sim$ is the equivalence relation, that identifies any $x\in X^\circ$ with itself and any $x\in\partial X$ with itself and its reflection $\bar{x}=\overline{(x_1,x_2)}:=(1-x_1,-x_2)$. I have already shown that $M$ is a Hausdorff space. It is obvious, that it is a non-empty topological space.

1.) Is there any way to show that $M$ is second-countable? My initial idea was to show that $M$ has a global chart. A global chart is a homeomorphism $M\cong U\subset\mathbb{R}^n$. Because second-countability is a topological invariant this would mean that $M$ is second-countable because $U\subset\mathbb{R}^n$ with the subset topology is itself second-countable. But finding such a global chart seems to be impossible.

Let $\pi:X\rightarrow M, x\mapsto [x]$ be the quotient map. Given $\Box:=[0,1]\times (0,1)\subset X$ and
$$\Phi:\pi(\Box)\rightarrow\Psi\circ\pi(\Box),\quad(\varphi,r)\mapsto(x,y,z),$$
where $$
x:=\cos 2\pi\varphi\left(1+\frac{r}{2}\cos\pi\varphi\right),\\
y:=\sin 2\pi\varphi\left(1+\frac{r}{2}\cos\pi\varphi\right),\\
z:=\frac{r}{2}\sin\pi\varphi,$$
I want to show that $\Phi$ is a chart of $\pi(\Box)\subset M$.
That means I have to show that $\Phi$ is a continuous bijection with an inverse map $\Phi^{-1}$ that is also continuous.
My thoughts on this are the following. To show that $\Phi$ is continuous I want to use that compositions of continuous maps are continuous themselves. However, there is my second problem.

2.) Since we are dealing with topological spaces, it only makes sense to speak of continuity with regards to the topologies at hand. Does this mean I have to check continuity for all elemental functions contained in $\Phi$?

I have found the following candidate as an inverse function:
$$g: \Phi\circ\pi(\Box)\rightarrow\pi(\Box),\quad(x,y,z)\mapsto(\varphi,r),$$
where $$r:=2\gamma, \quad\varphi:=\frac{1}{\pi}\arccos\frac{R-1}{\gamma},\\
R:=\sqrt{x^2+y^2}, \quad\gamma:=\sqrt{z^2+\left(R-1\right)^2}.$$

3.) Now, to show that, in fact, $g=\Phi^{-1}$, I must show that $\Phi\circ g=id_{\Phi\circ\pi(\Box)}$ and $g\circ\Phi=id_{\pi(\Box)}$. However, upon trying to compute both equalities I never reobtain the input. Is there maybe another way to show that $g=\Phi^{-1}$?

After having shown that $\Psi$ is actually a chart for $\pi(\Box)\subset M$, I want to look at shifted segments $\Box_k:=[0,1]\times(\frac{k}{2},1+\frac{k}{2})$ for $k\in\mathbb{Z}$. Then we obviously have $\Box=\Box_0$ and each adjacent box intersects: $\Box_k\cap\Box_{k+1}=[0,1]\times[\frac{k+1}{2},1+\frac{k}{2}]\neq\emptyset$. It also obviously follows that $\bigcup_{k\in\mathbb{Z}}\Box_k=X$.

4.) The next step would be to define analogous charts
$$\Phi_k:\pi(\Box_k)\rightarrow\Psi\circ\pi(\Box_k),\quad(\varphi,r)\mapsto(x,y,z)$$
for each neighbourhood $\pi(\Box_k)$. And prove that the transition $\tau_{k,k+1}:=\Phi_{k+1}\circ\Phi^{-1}_k$ is differentiable. Since each $\Phi_k$ is differentiable as a composition of differentiable maps,  $\tau_{k,k+1}$ is itself differentiable.
But how do I argue that $\Phi^{-1}_k$ is differentiable?

Given all of the steps above I would then argue that we have an atlas $A_0:=\{(\pi(\Box_k),\Phi_k)|k\in\mathbb{Z}\}$ and therefore a differentiable structure $A=[A_0]$ on $M$. Hence, it is a differentiable manifold.
I am sorry to ask so many questions at once. They all belong to that one exercise that I have been pondering about for the past five days and while I cannot spend much more time pondering so intensely about it, because I have new exercises waiting to be solved, maybe someone can give me some hints, tips or even propose a solution to one of the questions. I find this exercise really good since it involves really revising the definitions and putting the theorems to work.

REPLY [4 votes]: You are working too hard. Since $M$ is given as a quotient of a nice, rectangular subset of of the plane, we don't need to introduce trigonometric functions. Think of charts as ways of flattening subsets of manifolds. It is intuitively clear that we only need two charts to flatten the whole Möbius strip: cut across the strip in two places, and you are left with two rectangular pieces which you can lay down on the plane. Formally, let $U = \pi(X \setminus \partial X)$, and let $V = \pi(X \setminus \{ x  = 1/2 \})$. Define a chart $\phi:U \to \mathbb R^2$ via $$\phi([x,y]) = (x,y),$$ and a chart $\psi:V \to \mathbb R^2$ via $$\psi([x,y]) = \begin{cases} (x,y) &\text{if } x < 1/2  \\ (x-1,-y) &\text{if } x > 1/2 \end{cases}.$$
You can check that these are well-defined homeomorphisms onto their images, with $\phi^{-1}(x,y) = [x,y]$ and $\psi^{-1}(x,y) = [x,y]$. Moreover, $$\phi \psi^{-1}(x,y) = \phi([x,y]) = (x,y)$$ wherever it is defined and $$\psi \phi^{-1}(x,y) = \psi([x,y]) = \begin{cases} (x,y) &\text{if } 0 < x < 1/2  \\ (x-1,-y) &\text{if } 1 > x > 1/2 \end{cases}; $$ both of these maps are obviously smooth on their respective domains.

Regarding second-countability, we can take open balls or rectangles around points $[x,y] \in M$ with rational $x,y$ as a countable base.


TITLE: Can quarks and antiquarks exist in an atom without annihilating?
QUESTION [0 upvotes]: Elaborating on antimatter, where atoms consist of antiprotons, antineutrons, and positrons
Is it possible to have something in between? with antiprotons, positrons, and regular neutrons?
Can such an atom exist in a stable mode without the anti-upquarks annihilating the upquarks and
likewise for the antidowns and downs?
Taking, for simplicity, an example analogous to deuterium, an atom consisting of an antiproton, a neutron and a positron. Is then the reaction$$\bar{u}\bar{u}\bar{d}+udd\rightarrow\bar{u}d+Energy$$inevitable? Or can these quarks/antiquarks exist safely confined within their respective particle in the nucleus to make a stable atom?

REPLY [2 votes]: I don't think it's annihilation you have to worry about. It's my understanding that at typical energies adjacent (anti)baryons in a cluster confine their respective valence (anti)quarks too tightly for them to annihilate those in another (anti)baryon.
The real stability issue for a neutron/antiproton nucleus is that beta-decay of one or more neutrons to protons will reduce the electrostatic potential. Having said that, it's possible some specific numbers of neutrons & antiprotons are especially stable; you'd just need to deduce their equivalent of the whole of neutron-proton physics: magic numbers, drip lines etc. That's well beyond the scope of this question.
(A thought on terminology: if a cluster of nucleons leads to nuclear physics, we may as well call the antinucleon case antinuclear physics, while your mixed case would be sesquinuclear physics.)


TITLE: Question on why Lee chooses to define the standard smooth structure on a f.d. vector space in a certain way
QUESTION [0 upvotes]: In Lee's Introduction to Smooth Manifolds, Lee defines the standard smooth structure of a finite-dimensional vector space $V$ to be the smooth structure induced by the collection of charts $$\{(V,E^{-1}) : E : \mathbf{R}^n \to V \text{ is a basis coefficient assignment function}\}$$
That is, one chart for each choice of basis of $V$. He further shows that these charts are smoothly compatible; i.e., the transition function $\tilde{E} \circ E^{-1}$ is a diffeomorphism of $\mathbf{R}^n$.

Here is my question: why did he go to the effort of showing that this collection of charts are smoothly compatible? Is it not sufficient to observe that for a chosen basis $(E_1,\dots,E_n)$ of $V$, the basis coefficient assignment function $E(\mathbf{x}) = \sum x^iE_i$ is a diffeomorphism, hence $(V,E^{-1})$ is a smooth chart, and is in fact a smooth atlas itself, contained within a smooth structure which we define as the standard smooth structure of $V$.
I may just be confusing myself here-- perhaps Lee decided that more rather than less information about the standard structure is helpful. On the other hand, I may be missing something.

REPLY [2 votes]: Is it not sufficient to observe that for a chosen basis $(E_1,\dots,E_n)$ of $V$, the basis coefficient assignment function $E(\mathbf{x}) = \sum x^iE_i$ is a diffeomorphism, hence $(V,E^{-1})$ is a smooth chart, and is in fact a smooth atlas itself, contained within a smooth structure which we define as the standard smooth structure of $V$.

Of course each choice of a basis $\mathcal E$ induces a smooth structure $\mathfrak S_\mathcal E$ on $V$. But there are many choices of a basis, and none of these choices has priority. Theoretically it could be possible that the $\mathfrak S_\mathcal E$ are distinct for distinct bases of $V$. Wouldn't that be a disaster?
Lee proves that all bases $\mathcal E$ induce the same smooth structure on $V$, that is, the choice of $\mathcal E$ is irrelevant. In other words, $V$ has a canonical smooth structure. With this smooth structure all linear isomorphisms $\phi : \mathbb R^n \to V$ become diffeomorphisms which is not a priori obvious.