C\|\left(f(t),g(t)\right)\|_{\mathcal{L}^{2}}^{2}\\\ &\leq
C\Big{(}\|\mu(t)\|_{L^{2}(\Omega)}^{2}+\|\theta(t)\|_{L^{2}(\Gamma)}^{2}+\|F^{\prime}(\phi(t))\|_{L^{2}(\Omega)}^{2}+\|G^{\prime}(\psi(t))\|_{L^{2}(\Gamma)}^{2}\Big{)}\\\
&\leq
C+C\|\mu(t)\|_{L^{2}(\Omega)}^{2}+C\|\theta(t)\|_{L^{2}(\Gamma)}^{2}\end{split}$

for almost all $t\in[0,T]$. Integrating this inequality in time from $0$ to
$T$, we use the regularity of $\left(\mu,\theta\right)$ to infer that
$\left(\phi,\psi\right)\in L^{2}(0,T;\mathcal{H}^{2})$. In the case
$K\in(0,\infty)$, we even obtain $\left(\phi,\psi\right)\in
L^{4}(0,T;\mathcal{H}^{2})$ since $\left(\mu,\theta\right)\in
L^{4}(0,T;\mathcal{L}^{2})$.
If $K=\infty$, we may choose
$\left(\eta,\vartheta\right)=\left(1,0\right)\in\mathcal{H}^{1}$ as a test
function (6.7), which yields

$\displaystyle\int_{\Omega}f(t)\;\mathrm{d}x=\int_{\Omega}\mu(t)-F^{\prime}(\phi(t))\;\mathrm{d}x=0$

for almost all $t\in[0,T]$. This allows us to use regularity theory for
Poisson’s equations with homogeneous Neumann boundary condition (see, e.g.
[40, s.5, Proposition 7.7]) to infer that $\phi(t)\in H^{2}(\Omega)$ with

$\displaystyle\|\phi(t)\|_{H^{2}(\Omega)}\leq
C\|f(t)\|_{L^{2}(\Omega)}+C\|\phi(t)\|_{H^{1}(\Omega)}$

for almost all $t\in[0,T]$. In combination with (6.13) we deduce that

$\displaystyle\|\phi(t)\|_{H^{2}(\Omega)}\leq C+C\|\mu(t)\|_{L^{2}(\Omega)}$
(6.15)

for almost all $t\in[0,T]$. Next, due to (6.14), we can apply regularity
theory for elliptic equations on submanifolds (see, e.g., [40, s.5, Theorem
1.3]) for (6.10) and infer that $\psi(t)\in H^{2}(\Gamma)$ with

$\displaystyle\|\psi(t)\|_{H^{2}(\Gamma)}\leq
C\|g(t)\|_{L^{2}(\Gamma)}+C\|\psi(t)\|_{H^{1}(\Gamma)}\leq
C+C\|\theta(t)\|_{L^{2}(\Gamma)}$ (6.16)

for almost all $t\in[0,T]$. Squaring the equations (6.15) and (6.16) and
integrating in time over $[0,T]$ yields $\left(\phi,\psi\right)\in
L^{2}(0,T;\mathcal{H}^{2})$ which proves the assertion in the case $K=\infty$.
Additionally, as $\left(\mu,\theta\right)\in L^{4}(0,T;\mathcal{L}^{2})$, we
obtain $\left(\phi,\psi\right)\in L^{4}(0,T;\mathcal{H}^{2})$.

Let us now consider the case $k=3$. Since $p\leq 4$, we use Hölder’s
inequality and the Sobolev embedding $H^{1}(\Omega)\hookrightarrow
L^{6}(\Omega)$ to derive the estimate

$\displaystyle\begin{split}\|F^{\prime\prime}(\phi(t))\nabla\phi(t)\|_{L^{2}(\Omega)}&\leq
C\|\nabla\phi(t)\|_{L^{2}(\Omega)}+C\|\left|\phi(t)\right|^{4}\nabla\phi(t)\|_{L^{2}(\Omega)}\\\
&\leq
C\|\nabla\phi(t)\|_{L^{2}(\Omega)}+C\|\phi(t)\|_{L^{6}(\Omega)}^{2}\|\nabla\phi(t)\|_{L^{6}(\Omega)}\\\
&\leq C+C\|\phi(t)\|_{H^{2}(\Omega)}.\end{split}$

for almost all $t\in[0,T]$. In combination with (6.11) this yields

$\displaystyle\|F^{\prime}(\phi(t))\|_{H^{1}(\Omega)}\leq\|F^{\prime}(\phi(t))\|_{L^{2}(\Omega)}+\|F^{\prime\prime}(\phi(t))\nabla\phi(t)\|_{L^{2}(\Omega)}\leq
C+C\|\phi(t)\|_{H^{2}(\Omega)}$ (6.17)

for almost all $t\in[0,T]$. Proceeding similarly, we derive the estimate

$\displaystyle\|G^{\prime\prime}(\psi(t))\nabla_{\Gamma}\psi(t)\|_{L^{2}(\Gamma)}\leq
C+C\|\psi(t)\|_{H^{2}(\Gamma)},$

which leads to

$\displaystyle\|G^{\prime}(\psi(t))\|_{H^{1}(\Gamma)}\leq
C+C\|\psi(t)\|_{H^{2}(\Gamma)}$ (6.18)

for almost all $t\in[0,T]$. We thus obtain
$\left(f(t),g(t)\right)\in\mathcal{H}^{1}$. If $K\in[0,\infty)$ we may use
again regularity theory for elliptic problems with bulk-surface coupling (see
[30, Theorem 3.3]) to infer $\left(\phi(t),\psi(t)\right)\in\mathcal{H}^{3}$
with

$\displaystyle\begin{split}\|\left(\phi(t),\psi(t)\right)\|_{\mathcal{H}^{3}}^{2}&\leq
C\|\left(f(t),g(t)\right)\|_{\mathcal{H}^{1}}^{2}\\\ &\leq
C\|\mu(t)\|_{H^{1}(\Omega)}^{2}+C\|\theta(t)\|_{H^{1}(\Gamma)}^{2}+C\|F^{\prime}(\phi(t))\|_{H^{1}(\Omega)}^{2}+C\|G^{\prime}(\psi(t))\|_{H^{1}(\Gamma)}^{2}\\\
&\leq
C+C\|\mu(t)\|_{H^{1}(\Omega)}^{2}+C\|\theta(t)\|_{H^{1}(\Gamma)}^{2}+C\|\phi(t)\|_{H^{2}(\Omega)}^{2}+C\|\psi(t)\|_{H^{2}(\Gamma)}^{2}\end{split}$

for almost all $t\in[0,T]$. Integrating this inequality in time over $[0,T]$
yields $\left(\phi,\psi\right)\in L^{2}(0,T;\mathcal{H}^{3})$.
If $K=\infty$, the result can be established analogously to the case $k=2$ as
the regularity results cited above hold true for any integer $k\geq 0$. We
again find that

$\displaystyle\|\left(\phi(t),\psi(t)\right)\|_{\mathcal{H}^{3}}\leq
C\|\left(f(t),g(t)\right)\|_{\mathcal{H}^{1}}$

for almost all $t\in[0,T]$, where the right-hand side is bounded due to (6.17)
and (6.18).

This means that (6.1) and (6.2) are established. If
$(K,L)\in[0,\infty]\times(0,\infty]$ and $k=3$, the additional continuity
property (6.3) follows from Proposition A.1. Therefore, the proof is complete.
∎

######  Remark 6.2.

In the other cases, where the boundary conditions involving $K$ and $L$ are of
the same type (i.e., $K=L=0$ or $K=L=\infty$), it should also be possible to
obtain the regularities (6.1) and (6.2) even for $d=3$ and $p\in(4,6)$. This
is because in these cases, the system (1.1) can be discretized by a
Faedo–Galerkin scheme (cf. Remark 4.2) and therefore, the higher order
regularity estimates can be performed on the level of the approximate
solutions. In the case $K=L=0$, we refer the reader to the proof of [23,
Theorem 3.3], where this line of argument is carried out in detail.

However, it should also be possible to achieve the same regularity for
$p\in(4,6)$ even if the boundary conditions involving $K$ and $L$ are not of
the same type. This could be done by approximating the potential $F$ by a
suitable sequence $(F_{m})_{m\in\mathbb{N}}$ of potentials satisfying (A4)
with $p\leq 4$. Then, proceeding similarly to the proof of Theorem 4.1,
additional uniform bounds on the corresponding weak solutions
$(\phi_{m},\psi_{m},\mu_{m},\theta_{m})$ need to be established to obtain the
convergences $\left(\phi_{m},\psi_{m}\right)\rightarrow\left(\phi,\psi\right)$
weakly in $L^{2}(0,T;\mathcal{H}^{k})$ as $m\rightarrow\infty$ for
$k\in\\{2,3\\}$ as well as
$\left(\phi_{m},\psi_{m}\right)\rightarrow\left(\phi,\psi\right)$ weakly in
$L^{4}(0,T;\mathcal{H}^{2})$ as $m\rightarrow\infty$ in the case
$k\in\\{2,3\\}$ and $K\in(0,\infty]$.

### 6.2 Continuous dependence and uniqueness

  * Proof of Theorem 3.5.

As the functions $m_{\Omega}$ and $m_{\Gamma}$ are constant, we assume,
without loss of generality, that $m_{\Omega}\equiv 1$ and $m_{\Gamma}\equiv
1$. In the following, we use the letter $C$ to denote generic positive
constants depending only on $\Omega$, the parameters of the system (3.1), and
the initial data and the prescribed velocity field.

We consider two weak solutions $(\phi_{1},\psi_{1},\mu_{1},\theta_{1})$ and
$(\phi_{2},\psi_{2},\mu_{2},\theta_{2})$ corresponding to the initial data
$\left(\phi_{0}^{1},\psi_{0}^{1}\right),\left(\phi_{0}^{2},\psi_{0}^{2}\right)\in\mathcal{H}^{1}_{K,\alpha}$
and the velocity fields $\boldsymbol{v}_{1},\boldsymbol{v}_{2}\in
L^{\infty}(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}_{\mathbf{n}}^{1})$,
respectively, and set

$\displaystyle(\phi,\psi,\mu,\theta,\boldsymbol{v})\coloneqq(\phi_{2}-\phi_{1},\psi_{2}-\psi_{1},\mu_{2}-\mu_{1},\theta_{2}-\theta_{1},\boldsymbol{v}_{2}-\boldsymbol{v}_{1}).$

Then, due to their respective weak formulations (3.4), the quadruplet
$(\phi,\psi,\mu,\theta)$ satisfies

$\displaystyle\langle\left(\partial_{t}\phi,\partial_{t}\psi\right),\left(\zeta,\xi\right)\rangle_{\mathcal{H}_{L,\beta}^{1}}-\int_{\Omega}\phi_{2}\boldsymbol{v}\cdot\nabla\zeta\;\mathrm{d}x-\int_{\Omega}\phi\boldsymbol{v}_{1}\cdot\nabla\zeta\;\mathrm{d}x$
$\displaystyle\qquad-\int_{\Gamma}\psi_{2}\boldsymbol{v}_{\tau}\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma-\int_{\Gamma}\psi\boldsymbol{v}_{1,\tau}\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma$
(6.19a)
$\displaystyle\quad=-\int_{\Omega}\nabla\mu\cdot\nabla\zeta\;\mathrm{d}x-\int_{\Gamma}\nabla_{\Gamma}\theta\cdot\nabla_{\Gamma}\xi\;\mathrm{d}\Gamma-\mathds{h}(L)\int_{\Gamma}(\beta\theta-\mu)(\beta\xi-\zeta)\;\mathrm{d}\Gamma,$
$\displaystyle\begin{split}&\int_{\Omega}\mu\,\eta\;\mathrm{d}x+\int_{\Gamma}\theta\,\vartheta\;\mathrm{d}\Gamma-\int_{\Omega}\big{[}F^{\prime}(\phi_{2})-F^{\prime}(\phi_{1})\big{]}\eta\;\mathrm{d}x-\int_{\Gamma}\big{[}G^{\prime}(\psi_{2})-G^{\prime}(\psi_{1})\big{]}\vartheta\;\mathrm{d}\Gamma\\\
&\quad=\int_{\Omega}\nabla\phi\cdot\nabla\eta\;\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}\psi\cdot\nabla_{\Gamma}\vartheta\;\mathrm{d}\Gamma+\mathds{h}(K)\int_{\Gamma}(\alpha\psi-\phi)(\alpha\vartheta-\eta)\;\mathrm{d}\Gamma,\end{split}$
(6.19b)

a.e. on $[0,T]$ for all $\left(\zeta,\xi\right)\in\mathcal{H}^{1}_{L,\beta}$
and $\left(\eta,\vartheta\right)\in\mathcal{H}^{1}_{K,\alpha}$. Now, due to
(3.11), we have

$\displaystyle\beta\left|\Omega\right|{\langle\phi\rangle}_{\Omega}+\left|\Gamma\right|{\langle\psi\rangle}_{\Gamma}=0\quad\text{a.e.
on\leavevmode\nobreak\ }[0,T].$

This shows that the pair $\left(\phi(t),\psi(t)\right)$ is an admissible
right-hand side in the elliptic problem with bulk-surface coupling (2.9) for
almost all $t\in[0,T]$. Further, testing the weak formulation (6.19) with
$\left(\zeta,\xi\right)=\left(\beta,1\right)\in\mathcal{H}^{1}_{L,\beta}$, we
infer that

$\displaystyle\beta\left|\Omega\right|{\langle\partial_{t}\phi\rangle}_{\Omega}+\left|\Gamma\right|{\langle\partial_{t}\psi\rangle}_{\Gamma}=0\quad\text{a.e.
on\leavevmode\nobreak\ }[0,T].$

Hence, the pair $\left(\partial_{t}\phi(t),\partial_{t}\psi(t)\right)$ is an
admissible right-hand side in (2.9) for almost all $t\in[0,T]$ as well. Due to
the linearity of the operator $\mathcal{S}_{L,\beta}$, we deduce that
$\partial_{t}\mathcal{S}_{L,\beta}(\phi,\psi)=\mathcal{S}_{L,\beta}(\partial_{t}\phi,\partial_{t}\psi)$
in $\mathcal{H}^{1}_{L,\beta}$ for almost all $[0,T]$. Thus, choosing
$\left(\zeta,\xi\right)=\mathcal{S}_{L,\beta}(\phi,\psi)\in\mathcal{H}^{1}_{L,\beta}$
in (6.19), we derive the identity

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}=\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\|\mathcal{S}_{L,\beta}(\phi,\psi)\|_{L,\beta}^{2}$
$\displaystyle\quad=\left(\mathcal{S}_{L,\beta}(\partial_{t}\phi,\partial_{t}\psi),\mathcal{S}_{L,\beta}(\phi,\psi)\right)_{L,\beta}$
$\displaystyle\quad=-\langle\left(\partial_{t}\phi,\partial_{t}\psi\right),\mathcal{S}_{L,\beta}(\phi,\psi)\rangle_{\mathcal{H}^{1}_{L,\beta}}$
$\displaystyle\quad=\big{(}\left(\mu,\theta\right),\mathcal{S}_{L,\beta}(\phi,\psi)\big{)}_{L,\beta}$
(6.20)
$\displaystyle\qquad-\big{(}\left(\phi_{2}\boldsymbol{v}+\phi\boldsymbol{v}_{1},\psi_{2}\boldsymbol{v}_{\tau}+\psi\boldsymbol{v}_{1,\tau}\right),\left(\nabla\mathcal{S}_{L,\beta}^{\Omega}(\phi,\psi),\nabla_{\Gamma}\mathcal{S}_{L,\beta}^{\Gamma}(\phi,\psi)\right)\big{)}_{\mathcal{L}^{2}}$
$\displaystyle\quad=-\big{(}\left(\mu,\theta\right),\left(\phi,\psi\right)\big{)}_{\mathcal{L}^{2}}$
$\displaystyle\qquad-\big{(}\left(\phi_{2}\boldsymbol{v}+\phi\boldsymbol{v}_{1},\psi_{2}\boldsymbol{v}_{\tau}+\psi\boldsymbol{v}_{1,\tau}\right),\left(\nabla\mathcal{S}_{L,\beta}^{\Omega}(\phi,\psi),\nabla_{\Gamma}\mathcal{S}_{L,\beta}^{\Gamma}(\phi,\psi)\right)\big{)}_{\mathcal{L}^{2}}$

a.e. on $[0,T]$. Here, the second term on the right-hand side can be bounded
by

$\displaystyle\begin{split}&\big{(}\left(\phi_{2}\boldsymbol{v}+\phi\boldsymbol{v}_{1},\psi_{2}\boldsymbol{v}_{\tau}+\psi\boldsymbol{v}_{1,\tau}\right),\left(\nabla\mathcal{S}_{L,\beta}^{\Omega}(\phi,\psi),\nabla_{\Gamma}\mathcal{S}_{L,\beta}^{\Gamma}(\phi,\psi)\right)\big{)}_{\mathcal{L}^{2}}\\\
&\leq\|\left(\phi_{2}\boldsymbol{v}+\phi\boldsymbol{v}_{1},\psi_{2}\boldsymbol{v}_{\tau}+\psi\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{2}}\|\left(\nabla\mathcal{S}_{L,\beta}^{\Omega}(\phi,\psi),\nabla_{\Gamma}\mathcal{S}_{L,\beta}^{\Gamma}(\phi,\psi)\right)\|_{\mathcal{L}^{2}}\\\
&\leq\Big{(}\|\left(\phi_{2}\boldsymbol{v},\psi_{2}\boldsymbol{v}_{\tau}\right)\|_{\mathcal{L}^{2}}+\|\left(\phi\boldsymbol{v}_{1},\psi\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{2}}\Big{)}\|\mathcal{S}_{L,\beta}(\phi,\psi)\|_{L,\beta}\\\
&\leq\Big{(}C\|\left(\boldsymbol{v},\boldsymbol{v}_{\tau}\right)\|_{\mathcal{L}^{3}}+\|\left(\phi,\psi\right)\|_{\mathcal{L}^{6}}\|\left(\boldsymbol{v}_{1},\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{3}}\Big{)}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}\end{split}$
(6.21)

a.e. on $[0,T]$. Further, due to Ehrling’s lemma, for all $\varepsilon>0$
there exists a constant $C_{\varepsilon}$ such that

$\displaystyle\begin{split}\|\left(\phi,\psi\right)\|_{\mathcal{L}^{6}}&\leq\varepsilon\|\left(\phi,\psi\right)\|_{K,\alpha}+C_{\varepsilon}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}\end{split}$
(6.22)

a.e. on $[0,T]$. Using Young’s inequality, we obtain

$\displaystyle\begin{split}&\|\left(\phi,\psi\right)\|_{\mathcal{L}^{6}}\|\left(\boldsymbol{v}_{1},\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{3}}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}\\\
&\leq\Big{(}\varepsilon\|\left(\phi,\psi\right)\|_{K,\alpha}+C\|\left(\phi,\psi\right)\|_{L,\beta,\ast}\Big{)}\|\left(\boldsymbol{v}_{1},\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{3}}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}\\\
&\leq\varepsilon^{2}\|\left(\phi,\psi\right)\|^{2}_{K,\alpha}+C\Big{(}1+\|\left(\boldsymbol{v}_{1},\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{3}}^{2}\Big{)}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}\end{split}$
(6.23)

a.e. on $[0,T]$. Another application of Young’s inequality yields

$\displaystyle\begin{split}C\|\left(\boldsymbol{v},\boldsymbol{v}_{\tau}\right)\|_{\mathcal{L}^{3}}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}\leq\frac{1}{2}\|\left(\boldsymbol{v},\boldsymbol{v}_{\tau}\right)\|_{\mathcal{L}^{3}}^{2}+C\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}\end{split}$
(6.24)

a.e. on $[0,T]$. Plugging (6.23) and (6.24) back into (6.21), we infer

$\displaystyle\begin{split}&\big{(}\left(\phi_{2}\boldsymbol{v}+\phi\boldsymbol{v}_{1},\psi_{2}\boldsymbol{v}_{\tau}+\psi\boldsymbol{v}_{1,\tau}\right),\left(\nabla\mathcal{S}_{L,\beta}^{\Omega}(\phi,\psi),\nabla_{\Gamma}\mathcal{S}_{L,\beta}^{\Gamma}(\phi,\psi)\right)\big{)}_{\mathcal{L}^{2}}\\\
&\leq\frac{1}{2}\|\left(\boldsymbol{v},\boldsymbol{v}_{\tau}\right)\|_{\mathcal{L}^{3}}^{2}+\varepsilon^{2}\|\left(\phi,\psi\right)\|_{K,\alpha}^{2}+C\Big{(}1+\|\left(\boldsymbol{v}_{1},\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{3}}^{2}\Big{)}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}\end{split}$
(6.25)

a.e. on $[0,T]$. For the first term on the right-hand side of (6.20) we find

$\displaystyle\begin{split}-\big{(}\left(\mu,\theta\right),\left(\phi,\psi\right)\big{)}_{\mathcal{L}^{2}}&=-\|\left(\phi,\psi\right)\|_{K,\alpha}^{2}-\int_{\Omega}[F^{\prime}(\phi_{2})-F^{\prime}(\phi_{1})]\phi\;\mathrm{d}x\\\
&\qquad-\int_{\Gamma}[G^{\prime}(\psi_{2})-G^{\prime}(\psi_{1})]\psi\;\mathrm{d}\Gamma\end{split}$
(6.26)

a.e. on $[0,T]$. Now, exploiting the growth condition on $F^{\prime\prime}$
(see (A4)) and using the fundamental theorem of calculus as well as Hölder’s
inequality, we compute

$\displaystyle\begin{split}&\left|\int_{\Omega}[F^{\prime}(\phi_{2})-F^{\prime}(\phi_{1})]\phi\;\mathrm{d}x\right|=\left|\int_{\Omega}\int_{0}^{1}F^{\prime\prime}\big{(}\tau\phi_{2}+(1-\tau)\phi_{1}\big{)}\;\mathrm{d}\tau(\phi_{2}-\phi_{1})\phi\;\mathrm{d}x\right|\\\
&\quad\leq
C\int_{\Omega}\left(1+\left|\phi_{1}\right|^{4}+\left|\phi_{2}\right|^{4}\right)\left|\phi\right|^{2}\;\mathrm{d}x\leq
C\|\phi\|^{2}_{L^{6}(\Omega)}\end{split}$ (6.27)

a.e. on $[0,T]$. Recalling $H^{1}(\Gamma)\hookrightarrow L^{q}(\Gamma)$, we
analogously infer

$\displaystyle\left|\int_{\Gamma}[G^{\prime}(\psi_{2})-G^{\prime}(\psi_{1})]\psi\;\mathrm{d}\Gamma\right|$
$\displaystyle\leq C\|\psi\|_{L^{6}(\Gamma)}^{2}$ (6.28)

a.e. on $[0,T]$. Combining (6.27)–(6.28) yields

$\displaystyle\left|\int_{\Omega}[F^{\prime}(\phi_{2})-F^{\prime}(\phi_{1})]\phi\;\mathrm{d}x\right|+\left|\int_{\Gamma}[G^{\prime}(\psi_{2})-G^{\prime}(\psi_{1})]\psi\;\mathrm{d}\Gamma\right|\leq
C\|\left(\phi,\psi\right)\|_{\mathcal{L}^{6}}^{2}$ (6.29)

a.e. on $[0,T]$. Therefore,

$\displaystyle\begin{split}&\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}+(1-C\varepsilon^{2})\|\left(\phi,\psi\right)\|_{K,\alpha}^{2}\leq\frac{1}{2}\|\left(\boldsymbol{v},\boldsymbol{v}_{\tau}\right)\|_{\mathcal{L}^{3}}^{2}+C\mathcal{F}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}\end{split}$

a.e. on $[0,T]$, where $\mathcal{F}\coloneqq
1+\|\left(\boldsymbol{v}_{1},\boldsymbol{v}_{1,\tau}\right)\|_{\mathcal{L}^{3}}^{2}$.
Choosing $\varepsilon>0$ sufficiently small, we conclude

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{2}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}\leq\frac{1}{2}\|\left(\boldsymbol{v},\boldsymbol{v}_{\tau}\right)\|_{\mathcal{L}^{3}}^{2}+C\mathcal{F}\|\left(\phi,\psi\right)\|_{L,\beta,\ast}^{2}.$

As $\mathcal{F}\in L^{1}(0,T)$, Gronwall’s lemma directly implies

$\displaystyle\|\left(\phi(t),\psi(t)\right)\|^{2}_{L,\beta,\ast}$
$\displaystyle\leq\|\left(\phi(0),\psi(0)\right)\|^{2}_{L,\beta,\ast}\exp\left(C\int_{0}^{t}\mathcal{F}(\tau)\;\mathrm{d}\tau\right)$
$\displaystyle\quad+\int_{0}^{t}\|\left(\boldsymbol{v}(s),\boldsymbol{v}_{\tau}(s)\right)\|^{2}_{\mathcal{L}^{3}}\exp\left(C\int_{s}^{t}\mathcal{F}(\tau)\;\mathrm{d}\tau\right)\;\mathrm{d}s$

for almost all $t\in[0,T]$, which proves (3.12). As a consequence, if
$(\phi_{0}^{1},\psi_{0}^{1})=(\phi_{0}^{2},\psi_{0}^{2})$ a.e. in
$\Omega\times\Gamma$ and $\boldsymbol{v}_{1}=\boldsymbol{v}_{2}$ a.e. in $Q$,
the uniqueness of the corresponding weak solution immediately follows. Thus,
the proof is complete. ∎

## Appendix: Some calculus for bulk-surface function spaces

######  Proposition A.1.

Let $\Omega\subset\mathbb{R}^{d}$ with $d\in\\{2,3\\}$ be a non-empty, bounded
domain with Lipschitz boundary $\Gamma\coloneqq\partial\Omega$. Moreover, let
$T>0$, $K,L\in[0,\infty]$ and $\alpha,\beta\in\mathbb{R}$ be arbitrary. Let
$\left(u,v\right)\in C([0,T];\mathcal{L}^{2})\cap L^{2}(0,T;\mathcal{H}^{3})$
with $\left(-\Delta
u,-\Delta_{\Gamma}v+\alpha\partial_{\mathbf{n}}u\right)\in\mathcal{H}^{1}_{L,\beta}$
and

$\displaystyle\begin{cases}K\partial_{\mathbf{n}}u=\alpha v-u&\text{if}\
K\in[0,\infty),\\\ \partial_{\mathbf{n}}u=0&\text{if}\
K=\infty\end{cases}\quad\text{a.e.\leavevmode\nobreak\ on }\Sigma.$ (A.1)

We further suppose that their weak time derivative satisfies
$\left(\partial_{t}u,\partial_{t}v\right)\in
L^{2}(0,T;(\mathcal{H}^{1}_{L,\beta})^{\prime})$. Then, the continuity
property $\left(u,v\right)\in C([0,T];\mathcal{H}^{1})$ holds, the mapping

$\displaystyle t\mapsto\|\left(u(t),v(t)\right)\|^{2}_{K,\alpha}$

is absolutely continuous on $[0,T]$, and the chain rule formula

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\left(u(t),v(t)\right)\|^{2}_{K,\alpha}=2\big{<}\left(\partial_{t}u,\partial_{t}v\right)(t),\left(-\Delta
u,-\Delta_{\Gamma}v+\alpha\partial_{\mathbf{n}}u\right)(t)\big{>}_{\mathcal{H}^{1}_{L,\beta}}$
(A.2)

holds for almost all $t\in[0,T]$.

  * Proof of Proposition A.1..

In the cases $(K,L)\in[0,\infty)\times\\{\infty\\}$, a proof was already given
in [12, Proposition A.1.(b)]. There, the claim was established by
approximating the occurring functions by a sequence of time-mollified
functions, and eventually passing to the limit. For all other choices of
$(K,L)$, the proof can be carried out analogously. ∎

## Acknowledgement

This research was funded by the Deutsche Forschungsgemeinschaft (DFG,
GermanResearch Foundation) – Project 52469428. Their support is gratefully
acknowledged.

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