- \int_{\Omega_\tau} \beta \dv (d \nabla v) \cdot v \abs{v}^{q-2} \dd(t,\vec x)
\\
&= \left[ \frac{1}{q} \int_\Omega \abs{v}^q \dd \vec x \right]_{t=0}^{t=\tau}
- \frac{1}{q} \int_{\Omega_\tau} \dv \vec v \cdot \abs{v}^q \dd(t,\vec x)
+ \frac{1}{q} \int_{\Sigma_\tau} (\vec v \cdot \vec \nu) \abs{v}^q \dd(t,\sigma(\vec x))
\\
& \quad
+ \int_{\Omega_\tau} \beta d \nabla v \cdot \nabla (v \cdot \abs{v}^{q-2}) \dd(t,\vec x)
- \int_{\Sigma_\tau} \beta \partial_{\vec \nu} v \cdot v \abs{v}^{q-2} \dd(t,\sigma(\vec x))
\\
&= \frac{1}{q} \big[ \norm{v(\tau,\cdot)}_{\LL_q(\Omega)}^q - \norm{v(0,\cdot)}_{\LL_q(\Omega)}^q \big]
+ \int_{\Omega_\tau} \beta d (q-1) \abs{\nabla v}^2 \abs{v}^{q-2} \dd(t,\vec x)
\\ &\qquad
+ \int_{\Sigma_\tau} g^\Sigma \, v \abs{v}^{q-2} \dd(t,\sigma(\vec x)),
\end{align*}

provided these integrals exist.
The latter will follow for general $q \in (1, \infty)$ from Lemma <ref> below.
Hence, after rearrangement of the terms, we obtain the estimate

\begin{align*}
&\frac{1}{q} \norm{v(\tau,\cdot)}_{\LL_q(\Omega)}^q
+ \int_{\Omega_\tau} \beta d (q-1) \abs{\nabla v}^2 \abs{v}^{q-2} \dd(t,\vec x)
\\
&= \frac{1}{q} \norm{v(0,\cdot)}_{\LL_q(\Omega)}^q
+ \int_{\Omega_\tau} f v \abs{v}^{q-2} \dd(t,\vec x)
- \int_{\Sigma_\tau} g^\Sigma \, v \abs{v}^{q-2} \dd(t,\sigma(\vec x))
\\
&\leq \frac{1}{q} \norm{v(0,\cdot)}_{\LL_q(\Omega)}^q
+ \int_{\Omega_\tau} \abs{f} \abs{v}^{q-1} \dd(t,\vec x)
+ \int_{\Sigma_\tau} \abs{g^\Sigma} \abs{v}^{q-1} \dd(t,\sigma(\vec x)).
\end{align*}

The desired estimate follows from this, if we can ensure that for every $\varepsilon > 0$ there is $C = C(q,\varepsilon) > 0$ (independent of $v$) such that

\begin{equation}
\varepsilon \norm{v|_{\Sigma_\tau}}_{\LL_q(\Sigma)}^q
\leq \int_{\Omega_\tau} \abs{\nabla v}^2 \abs{v}^{q-2} \dd (t, \vec x)
+ C(q,\varepsilon) \norm{v}_{\LL_q(\Omega_\tau)}^q,
\label{eqn:dual-estimate-2}
\end{equation}

as we may then employ the generalized Hölder inequality to establish

\begin{align*}
\int_{\Omega_\tau} \abs{f} \abs{v}^{q-1} \dd(t,x)
&\leq \norm{f}_{\LL_q(\Omega_\tau)} \norm{v}_{\LL_q(\Omega_\tau)}^{q-1},
\\
\int_{\Sigma_\tau} \abs{g^\Sigma} \abs{v}^{q-1} \dd(t,\sigma(x))
&\leq \norm{g^\Sigma}_{\LL_q(\Sigma_\tau)} \norm{v}_{\LL_q(\Sigma_\tau)}^{q-1}
\end{align*}

and the statement follows by Young's inequality for products.
To get estimate (<ref>), we may employ the subsequent Lemma <ref>.
Finally, since the constant $C > 0$ is independent of the particular choice of the data, by approximation these estimates are also valid for less smooth data $(f, g^\Sigma, v_0)$, i.e. the estimate is valid for all such strong $\WW_p^{(1,2)}$-solutions.

Let $q \in (1, \infty)$, $\tau > 0$ and $\Omega \subsetneq \R^d$ be a bounded $\CC^{1,1}$-domain with boundary $\Sigma = \partial \Omega$.
Denote by $\cdot|_\Sigma: \WW^{(1,2)}_q(\Omega_\tau) \rightarrow \LL_q(\Sigma_\tau)$ the (continuous) spatial boundary trace operator.
Then, for every $s \in (\frac{1}{2}, 1)$, there is a constant $C = C(q,s,\Omega) > 0$ such that

\begin{align*}
\norm{v|_\Sigma}_{\LL_q(\Sigma_\tau)}^q
&\leq C \norm{ \abs{v}^{\nicefrac{q}{2}} }_{\LL_2((0,\tau);\HH^1(\Omega))}^{2s} \norm{v}_{\LL_q(\Omega_\tau)}^{(1-s)q}
\\
&\leq C \big( \big( \int_{\Omega_\tau} \abs{v}^{q-2} \abs{\nabla v}^2 \dd (t,\vec x) \big)^{1/2} + \norm{v}_{\LL_q(\Omega_\tau)}^{q/2} \big)^{2s} \norm{v}_{\LL_q(\Omega_\tau)}^{(1-s)q},
\quad
v \in \WW^{(1,2)}_q(\Omega_\tau).
\end{align*}

As a preliminary step, we will derive the following identity, which is valid for every $u \in \CC^2(\overline{\Omega})$:

\begin{equation}
\int_{\Omega_\tau} \Delta u \cdot u \abs{u}^{q-2} \dd(t,\vec x)
= - \int_{\Omega_\tau} \nabla u \cdot \nabla \big( u \abs{u}^{q-2} \big) \dd(t, \vec x)
+ \int_{\Sigma_\tau} \partial_{\vec \nu} u \cdot u \abs{u}^{q-2} \dd(t,\sigma(\vec x)).
\label{eqn:test_function_CC^2}
\end{equation}

This identity does not immediately follow from the Gauß divergence theorem, since for $q \in (1,2)$ the term $u \abs{u}^{q-2}$ will in general not be classically differentiable, nor can we ensure that its weak derivative lies in the space $\LL_{q'}(\Omega_\tau)$.
Therefore, we will approximate the integrands by a regularised version and, instead of $\int_{\Omega_\tau} \nabla u \cdot \nabla (u \abs{u}^{q-2}) \dd (t,\vec x)$, we consider the integral

\begin{align*}
\int_{\Omega_\tau} \nabla u \cdot \nabla \big( u (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \big) \dd (t,\vec x)
&= \int_{\Omega_\tau} \abs{\nabla u}^2 (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \big[ 1 + (q-2) \abs{u}^2 (u^2 + \varepsilon^2)^{-1} \big] \dd (t,\vec x).
\end{align*}

The terms in the integrand on the right-hand side are all positive for each $\varepsilon > 0$ and decrease in $\varepsilon > 0$, thus, for the critical case $q \in (1,2)$ we may use the Beppo–Levi theorem on montone convergence to conclude that

\begin{align*}
&\int_{\Omega_\tau} \abs{\nabla u}^2 (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \big[ 1 + (q-2) \abs{u}^2 (u^2 + \varepsilon^2)^{-1} \big] \dd (t,\vec x)
\\ &\qquad
\xrightarrow{\varepsilon \rightarrow 0+} \int_{\Omega_\tau} \nabla u \cdot \nabla (u \abs{u}^{q-2}) \dd (t,\vec x)
\in [0, \infty].
\end{align*}

For the regularised integral, we may use the Gauß divergence theorem, and obtain that

\begin{align*}
&\int_{\Omega_\tau} \nabla u \cdot \nabla \big( u (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \big) \dd (t,\vec x)
\\
&= - \int_{\Omega_\tau} \Delta u \cdot u (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \dd(t,\vec x)
+ \int_{\Sigma_\tau} \partial_{\vec \nu} u \cdot u (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \dd(t,\sigma(\vec x)).
\end{align*}

Since we may estimate

\begin{align*}
\norm{u (u^2 + \varepsilon^2)^{\frac{q-2}{2}}}_{\LL_{q'}(\Omega_\tau)}^{q'}
= \int_{\Omega_\tau} \big( \frac{\abs{u}}{\sqrt{u^2 + \varepsilon^2}} \big)^{q'} (u^2 + \varepsilon^2)^{\frac{q}{2}} \dd (t, \vec x)
\leq \int_{\Omega_\tau} (u^2 + \varepsilon^2)^{\frac{q}{2}} \dd (t, \vec x)
\end{align*}

and, using that $q' = \frac{q}{q-1}$, so that $\frac{q-1}{2} \cdot q' = \frac{q}{2}$,

\begin{align*}
\norm{u (u^2 + \varepsilon^2)^{\frac{q-2}{2}}}_{\LL_{q'}(\Sigma_\tau)}^{q'}
= \int_{\Sigma_\tau} \big( \frac{\abs{u}}{\sqrt{u^2 + \varepsilon^2}} \big)^{q'} (u^2 + \varepsilon^2)^{\frac{q}{2}} \dd(t,\sigma(\vec x))
\leq \int_{\Sigma_\tau} (u^2 + \varepsilon^2)^{\frac{q}{2}} \dd(t,\sigma(\vec x)),
\end{align*}

we may conclude by Lebesgue's theorem on dominated convergence that

\begin{align*}
\int_{\Omega_\tau} \Delta u \cdot u (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \dd(t,\vec x)
&\xrightarrow{\varepsilon \rightarrow 0+}
\int_{\Omega_\tau} \Delta u \cdot u \abs{u}^{q-2} \dd(t, \vec x),
\\
\int_{\Sigma_\tau} \partial_{\vec \nu} u \cdot u (u^2 + \varepsilon^2)^{\frac{q-2}{2}} \dd(t,\sigma(\vec x))
&\xrightarrow{\varepsilon \rightarrow 0+}
\int_{\Sigma_\tau} \partial_{\vec \nu} u \abs{u}^{q-2} \dd(t,\sigma(\vec x))
\end{align*}

and identity (<ref>) follows.
Having identity (<ref>) at hand, we may proceed as follows: Via the boundary trace map, the space $\WW^{(1,2)}_q(\Omega_\tau)$ embeds continuously into $\LL_q((0,\tau);\WW_q^1(\Sigma)) \subseteq \WW_q^{(1/2,1)}(\Sigma_\tau)$ by comparing the parabolic Sobolev indices $2 - \frac{d+2}{q} \geq 1 - \frac{d+1}{q}$ (which is true for all $q \geq 1$) and, therefore, as $q' = \frac{q}{q-1}$ and $\abs{u \abs{u}^{q-2}}^{q'} = \abs{u}^q$ it follows that $u \abs{u}^{q-2} \in \LL_{q'}(\Sigma_\tau)$ for every $u \in \WW_q^{(1,2)}(\Omega_\tau)$ with $\norm{u \abs{u}^{q-2}}_{\LL_{q'}(\Sigma_\tau)} \leq C \norm{u}_{\WW_q^{(1,2)}(\Omega_\tau)}^{q-1}$ and $u \abs{u}^{q-2} \in \LL_{q'}(\Omega_\tau)$ with $\norm{u \abs{u}^{q-2}}_{\LL_{q'}(\Omega_\tau)} \leq C \norm{u}_{\LL_q(\Omega_\tau)}^{q-1}$.
Further, $\nabla u \cdot \nabla (u \abs{u}^{q-2}) = (q-1) \abs{\nabla u}^2 \abs{u}^{q-2} \geq 0$.
Approximating $u \in \WW_q^{(1,2)}(\Omega_\tau)$ by functions in $\CC^2(\overline{\Omega}_\tau)$ and employing Hölder's inequality for the left-hand side in (<ref>) and the boundary term on the right hand side, we may thus infer that the identity (<ref>) is valid for all $u \in \WW_q^{(1,2)}(\Omega_\tau)$, the integral $\int_{\Omega_\tau} \nabla u \cdot \nabla (u \abs{u}^{q-2}) \dd(t,\vec x)$ exists for all $u \in \WW_q^{(1,2)}(\Omega_\tau)$ and can be bounded by $C \norm{u}_{\WW_q^{(1,2)}(\Omega_\tau)}^q$.
It follows that $\abs{v}^{q/2} \in \LL_2((0,T);\WW^1_2(\Omega))$ with $\norm{ \abs{v}^{q/2} }_{\LL_2(\Omega_\tau)} = \norm{v}_{\LL_q(\Omega_\tau)}^{q/2}$ and $\nabla \abs{v}^{q/2} = \frac{q}{2} v \abs{v}^{q/2-2} \nabla v$ with $\LL_2$-norm $\norm{ \nabla \abs{v}^{q/2} }_{\LL_2(\Omega_\tau)} = \frac{q}{2} \norm{ \abs{v}^{q/2-1} \nabla v }_{\LL_2(\Omega_\tau)}$.
Moreover, the $\Sigma$-trace of $\abs{v}^{q/2} \in \LL_2((0,T);\WW_2^1(\Omega))$ is $\big( \abs{v}^{q/2} \big)|_\Sigma = \abs{v|_\Sigma}^{q/2} \in \LL_2(\Sigma_\tau)$, so that

\begin{align*}
\norm{v|_\Sigma}_{\LL_q(\Sigma_\tau)}^q
&= \norm{\abs{v}^{\nicefrac{q}{2}}|_\Sigma}_{\LL_2(\Sigma_\tau)}^2
\leq C \norm{\abs{v}^{\nicefrac{q}{2}}|_\Sigma}_{\LL_2((0,\tau);\HH^{s-1/2}(\Sigma))}^2
\\
&\leq C \norm{\abs{v}^{\nicefrac{q}{2}}}_{\LL_2((0,\tau);\HH^s(\Omega))}^2
\leq C \norm{\abs{v}^{\nicefrac{q}{2}}}_{\LL_2((0,\tau);\HH^1(\Omega))}^{2s} \norm{\abs{v}^{\nicefrac{q}{2}}}_{\LL_2((0,\tau);\LL_2(\Omega))}^{2(1-s)}
\\
&= C \norm{\abs{v}^{\nicefrac{q}{2}}}_{\LL_2((0,\tau);\HH^1(\Omega))}^{2s} \norm{v}_{\LL_q(\Omega_\tau)}^{(1-s)q}.
\end{align*}

Adjusting the norm for which we want to estimate the surface trace $v|_\Sigma$ in the previous result, we obtain the following variant using the duality method, cf. [22].

Let $p, q \in (1, \infty]$, $T > 0$ and $\alpha, \beta > 0$ be fixed, and let $\tau \in (0,T]$.
Assume that $\vec v \in \fs U_{\tilde p}^\Omega(T)$ with $\tilde p > d + 2$.
Let $f \in \LL_p(\Omega_\tau)$, $g^\Sigma \in \WW_p^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2p})}(\Sigma_\tau)$ and $v_0 \in \WW_p^{2-2/p}(\Omega)$ be given and assume that $- \beta \partial_{\vec \nu} v_0|_\Sigma = g^\Sigma|_{t=0}$ if $p > 3$.
Then there is a constant $C = C(q,T) > 0$ such that for every strong solution $v \in \WW_p^{(1,2)}(\Omega_\tau)$ of the initial-boundary value problem

α∂_t v + v⃗ ·∇v - (d ∇v)
= f

on  Ω_τ,

- d ∂_νv
= g^Σ
on  Σ_τ,

= v_0

in  Ω,

the following estimate is valid:

\begin{align*}
&\norm{v}_{\LL_q(\Omega_\tau)} + \norm{v|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\ &\qquad
\leq C \left( \norm{f}_{\WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{g^\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{v^0}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} \right),
\end{align*}

where we write

\begin{align*}
\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)
&:= \begin{cases}
\{ g \in \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau): \, g|_{t = \tau} = 0 \},
&q < \frac{3}{2},
\vspace{2pt}
\\
\WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau),
&q \geq \frac{3}{2},
\end{cases}
\\
\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)
&:= \begin{cases}
\{ h \in \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau): \, h|_{t = \tau} = 0 \},
&q < 3,
\vspace{2pt}
\\
\WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau),
&q \geq 3.
\end{cases}
\end{align*}

In the spirit of the duality technique, let arbitrary $\Theta \in \LL_{q'}(\Omega_\tau)$ and $\eta \in \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)$ such that (if $q < \frac{3}{2}$) $\eta|_{t = \tau} = 0$ be given and let $\theta \in \WW_{q'}^{(1,2)}(\Omega_\tau)$ be the unique solution to the (time-reversed) dual problem

\[
\begin{cases}
- \alpha \partial_t \theta - \vec v \cdot \nabla \theta - \dv(d \nabla \theta)
= \Theta \quad
&\text{in } \Omega_\tau,
\\
- d \partial_{\vec \nu} \theta
= \eta \quad
&\text{on } \Sigma_\tau,
\\
\theta(\tau,\cdot)
= 0 \quad
&\text{in } \Omega.
\end{cases}
\]

Thanks to $\LL_{q'}$-maximal regularity, this solution $\theta \in \WW_{q'}^{(1,2)}(\Omega_\tau)$ exists and is uniquely determined.
Moreover, thanks to the condition $\theta(\tau,\cdot) = 0$ there is a constant $C = C(q,T)$, which is independent of $\tau \in (0,T]$ such that

\[
\norm{\theta}_{\WW_{q'}^{(1,2)}(\Omega_\tau)}
\leq C \big( \norm{\Theta}_{\LL_{q'}(\Omega_\tau)} + \norm{\eta}_{\WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)} \big).
\]

Using integration by parts and Green's formula, we may conclude that (recall that $\theta(\tau,\cdot) = 0$ in $\Omega$, $\dv \vec v = 0$ in $\Omega$ and $\partial_{\vec \nu} v \equiv 0$ on $\Sigma_\tau$)

\begin{align*}
&\int_{\Omega_\tau} v \Theta \dd (t,\vec x)
+ \int_{\Sigma_\tau} v \eta \dd (t, \sigma(\vec x))
\\
&= - \int_{\Omega_\tau} v (\alpha \partial_t \theta + \vec v \cdot \nabla \theta + \dv(d \nabla \theta)) \, \dd (t,\vec x)
- \int_{\Sigma_\tau} v d \partial_{\vec \nu} \theta \, \dd (t, \sigma(\vec x))
\\
&= \int_{\Omega_\tau} (\alpha \partial_t v + \vec v \cdot \nabla v - \dv(d \nabla v)) \theta \dd (t, \vec x)
- \int_{\Sigma_\tau} \theta d \partial_{\vec \nu} v \dd (t, \sigma(\vec x))
- \int_{\Omega} v^0 \theta(0,\cdot) \dd \vec x
\\
&= \int_{\Omega_\tau} f \theta \dd (t, \vec x)
+ \int_{\Sigma_\tau} g^\Sigma \theta \dd (t, \sigma(\vec x))
- \int_{\Omega} v^0 \theta(0,\cdot) \dd \vec x.
\end{align*}

The right-hand side can be estimated by

\begin{align*}
&\norm{f}_{\WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} \norm{\theta}_{\WW_{q'}^{(1,2)}(\Omega_\tau)}
+ \norm{g^\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} \norm{\theta}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)}
\\ &\qquad
+ \norm{v^0}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} \norm{\theta(0,\cdot)}_{\WW_{q'}^{2-2/q'}(\Omega)}
\\
C \big( \norm{f}_{\WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{g^\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{v^0}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} \big)
\\ &\qquad
\cdot \big( \norm{\Theta}_{\LL_{q'}(\Omega_\tau)} + \norm{\eta}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)} \big).
\end{align*}

Taking the supremum over all such $\Theta$ and $\eta$ with norm less or equal $1$, and using that $\LL_q(\Omega_\tau)$ is isometrically isomorphic to $\LL_{q'}(\Omega_\tau)^\ast$, we find that

\begin{align*}
&\norm{v}_{\LL_q(\Omega_\tau)} + \norm{v|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq C \big( \norm{f}_{\WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{g^\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{v^0}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} \big).
\end{align*}

Similar estimates are also valid for reaction-diffusion-advection systems on the surface $\Sigma = \partial \Omega$:

Let $\Sigma$ be the boundary of a bounded $\CC^{3-}$-domain $\Omega \subseteq \R^d$, $p, q \in (1, \infty]$, $T > 0$ and $\alpha, \beta > 0$ be fixed, and let $\tau \in (0,T]$.
Let $f^\Sigma \in \LL_p(\Sigma_\tau)$ and $v^{\Sigma,0} \in \WW_p^{2-2/p}(\Sigma)$.
Then there is a constant $C = C(q,T) > 0$ such that for every strong solution $v^\Sigma \in \WW_p^{(1,2)}(\Sigma_\tau)$ of the initial value problem

∂_t v^Σ- _Σ(d^Σ∇_Σv^Σ)
= f^Σ
on  Σ_τ,

= v^Σ,0

on  Σ
the following estimate is valid:

\[
\norm{v^\Sigma}_{\LL_q(\Sigma_\tau)}
\leq C \left( \norm{f^\Sigma}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} + \norm{v^{\Sigma,0}}_{\WW_{q'}^{2-2/q'}(\Sigma)^\ast} \right).
\]

For arbitrary $\Theta^\Sigma \in \LL_{q'}(\Sigma_\tau)$, let us consider the dual, time-reversed problem

\[
\begin{cases}
- \partial_t \Psi^\Sigma - \dv_\Sigma(d^\Sigma \nabla_\Sigma \Psi^\Sigma)
= \Theta^\Sigma \quad
&\text{on } \Sigma_\tau,
\\
\Psi^\Sigma(\tau,\cdot)
= 0 \quad
&\text{on } \Sigma,
\end{cases}
\]

which admits $\LL_{q'}$-maximal regularity, hence, has a unique solution $\Psi^\Sigma \in \WW_{q'}^{(1,2)}(\Sigma_\tau)$ such that

\[
\norm{\Psi^\Sigma}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)}
\leq C \norm{\Theta^\Sigma}_{\LL_{q'}(\Sigma_\tau)}
\]

for some constant $C > 0$, which is not only independent of $\Theta^\Sigma$, but of $\tau \in (0, T]$ as well since we only consider zero final data $\Psi^\Sigma(\tau,\cdot) = 0$.
We then calculate that

\begin{align*}
\int_{\Sigma_\tau} v^\Sigma \cdot \Theta^\Sigma \dd (t, \sigma(\vec x))
&= - \int_{\Sigma_\tau} v^\Sigma (\partial_t \Psi^\Sigma + \dv_\Sigma (d^\Sigma \nabla_\Sigma \Psi^\Sigma)) \dd(t, \sigma(\vec x))
\\
&= \int_{\Sigma_\tau} (\partial_t v^\Sigma - \dv_\Sigma(d^\Sigma \nabla_\Sigma v^\Sigma)) \Psi^\Sigma \dd (t, \sigma(\vec x))
+ \int_\Sigma v^{\Sigma,0} \Psi^{\Sigma}(0,\cdot) \dd \sigma(\vec x)
\\
&= \int_{\Sigma_\tau} f^\Sigma \Psi^\Sigma \dd (t, \sigma(\vec x))
+ \int_\Sigma v^{\Sigma,0} \Psi^{\Sigma}(0,\cdot) \dd \sigma(\vec x),
\end{align*}

which we can estimate by

\begin{align*}
&\norm{f^\Sigma}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} \norm{\Psi^\Sigma}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)}
+ \norm{v^{\Sigma,0}}_{\WW_{q'}^{2-2/q'}(\Sigma)^\ast} \norm{\Psi^\Sigma}_{\WW_{q'}^{2-2/q'}(\Sigma)}
\\
&\leq \big(\norm{f^\Sigma}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} + \norm{v^{\Sigma,0}}_{\WW_{q'}^{2-2/q'}(\Sigma)^\ast} \big) \norm{\Psi^\Sigma}_{\WW_{q'}^{2-2/q'}(\Sigma)^\ast},
\end{align*}

and the assertion follows by taking the supremum over all $\Theta^\Sigma \in \LL_{q'}(\Sigma_\tau)$ with $\norm{\Theta^\Sigma}_{\LL_{q'}(\Sigma_\tau)} \leq 1$.

We will also need a version of the duality estimate related to the positive part.
Such positive versions are well-established for the pure bulk phase case, where we once again refer to M. Pierre's survey [52].

Let $T > 0$, $q \in (1, \infty]$ and $\Omega \subseteq \R^d$ be a bounded domain with $\CC^{3-}$-boundary $\Sigma = \partial \Omega$.
Let $\vec v \in \fs U_{\tilde p}^\Omega(T)$ for some $\tilde p > d+2$.
Given $f, f_i \in \LL_q(\Omega_\tau)$, $g^\Sigma, g_i^\Sigma \in \WW_q^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2p})}(\Sigma_\tau)$ and $v^0, v^0_i \in \WW_q^{2-2/q}(\Omega) $ (with compatibility condition $\partial_{\vec \nu} v^0 = g^\Sigma|_{t=0}$ and $\partial_{\vec \nu} v^0_i = g_i^\Sigma|_{t=0}$ if $q > 3$), let $v, v_i \in \WW_q^{(1,2)}(\Omega_\tau)$ be the (unique) strong $\WW_q^{(1,2)}$-solutions to the parabolic initial-boundary value problems

∂_t v + v⃗ ·∇v - ( d ∇v)
= f

in  (0, τ) ×Ω,

d ∂_ν⃗ v
= g^Σ
on  (0, τ) ×Σ,

= v^0

in  Ω

α_i ∂_t v_i + β_i v⃗ ·∇v_i - ( d_i ∇v_i)
= f_i

in  (0, τ) ×Ω,

d_i ∂_ν⃗ v_i
= g_i^Σ
on  (0, τ) ×Σ,

= v^0_i

in  Ω,

where $\tau \in (0, T)$, and positive constants $d, \alpha_i, d_i > 0$ and $\beta_i \in \R$ are given.
Then there is a constant $C = C(q,\Omega,T) > 0$ such that

\begin{align*}
&\norm{v^+}_{\LL_q(\Omega_\tau)} + \norm{v^+|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq C \big(\sum_{j=1}^N \norm{v_j}_{\LL_q(\Omega_\tau)} + \norm{v_j^+|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{(v^0 - \sum_{j=1}^N \alpha_j v^0_j)^+}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast}
\\ &\qquad
+ \norm{(f - \sum_{j=1}^N f_j)^+}_{\LL_q(\Omega_\tau)} + \norm{(g^\Sigma - \sum_{j=1}^N g_j^\Sigma)^+}_{\WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} \big).
\end{align*}

We proceed similar to the proof of the dual estimates, however, this time restrict ourselves to functions $\Theta \in \LL_{q'}(\Omega_\tau)$ and $\eta \in \mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)$ which are non-negative.
As in the proof of Lemma <ref>, we consider the dual problem

\[
\begin{cases}
- \partial_t \Psi - \vec v \cdot \nabla \Psi - \dv(d \nabla \Psi)
= \Theta \quad
&\text{in } \Omega_\tau,
\\
d \partial_{\vec \nu} \Psi
= \eta \quad
&\text{on } \Sigma_\tau,
\\
\Psi(\tau,\cdot)
= 0 \quad
&\text{in } \Omega.
\end{cases}
\]

By $\LL_{q'}$-maximal regularity of this problem and the comparison principle for parabolic equations with inhomogeneous Neumann data, we obtain a unique and non-negative solution $\Psi \in \WW_{q'}^{(1,2)}(\Omega_\tau)$ such that

\[
\norm{\Psi}_{\WW_{q'}^{(1,2)}(\Omega_\tau)}
\leq C \big( \norm{\Theta}_{\LL_{q'}(\Omega_\tau)} + \norm{\eta}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)} \big)
\]

for a constant $C > 0$ which is independent of the time-horizon $\tau \in (0,T]$ (thanks to the homogeneous final condition $\Psi(\tau,\cdot) = 0$) and the data $(\Theta, \eta)$.
Then, using the PDE solved by $\theta$ and $v$, and via integration by parts, we find that

\begin{align*}
&\int_{\Omega_\tau} v \Theta \dd (t, \vec x)
+ \int_{\Sigma_\tau} v \eta \dd (t, \sigma(\vec x))
\\
&= - \int_{\Omega_\tau} v (\partial_t \Psi + \vec v \cdot \nabla \Psi + \dv(d \nabla \Psi)) \dd (t, \vec x)
- \int_{\Sigma_\tau} v \partial_{\vec \nu} \Psi \dd (t, \sigma(\vec x))
\\
&= \int_{\Omega_\tau} (\partial_t v + \vec v \cdot \nabla v - \dv(d \nabla v)) \Psi \dd (t, \vec x)
- \int_{\Sigma_\tau} \Psi d \partial_{\vec \nu} v \dd (t, \sigma(\vec x))
+ \int_\Omega v^0 \Psi(0,\cdot) \dd \vec x
\\
&= \int_{\Omega_\tau} f \Psi \dd (t, \vec x)
+ \int_{\Sigma_\tau} g^\Sigma \Psi \dd (t, \sigma(\vec x))
+ \int_\Omega v^0 \Psi(0,\cdot) \dd \vec x.
\end{align*}

The latter terms can be written as

\begin{align*}
&\int_{\Omega_\tau} f \Psi \dd (t, \vec x)
+ \int_{\Sigma_\tau} g^\Sigma \Psi \dd (t, \sigma(\vec x))
+ \int_\Omega v^0 \Psi(0,\cdot) \dd \vec x
\\
&= \int_{\Omega_\tau} (f - \sum_{j=1}^m f_j) \Psi \dd (t, \vec x)
+ \int_{\Sigma_\tau} (g^\Sigma - \sum_{j=1}^m g_j^\Sigma) \Psi \dd (t, \sigma(\vec x))
+ \int_\Omega (v^0 - \sum_{j=1}^m \alpha_j v_j^0) \Psi(0,\cdot) \dd \vec x
\\
+ \sum_{j=1}^m \big(
\int_{\Omega_\tau} f_j \Psi \dd (t, \vec x)
+ \int_{\Sigma_\tau} g_j^\Sigma \Psi \dd (t, \sigma(\vec x))
+ \int_\Omega \alpha_j v_j^0 \Psi(0,\cdot) \dd \vec x
\big).
\end{align*}

Using the PDEs for the functions $v_i$, we further obtain

\begin{align*}
&\int_{\Omega_\tau} f_j \Psi \dd (t, \vec x)
+ \int_{\Sigma_\tau} g_j^\Sigma \Psi \dd (t, \sigma(\vec x))
+ \int_\Omega \alpha_j v_j^0 \Psi(0,\cdot) \dd \vec x
\\
&= \int_{\Omega_\tau} (\alpha_j \partial_t v_j + \beta_j \vec v \cdot \nabla v_j - \dv( d_j \nabla v_j) \Psi \dd (t, \vec x)
\\ &\quad
- \int_{\Sigma_\tau} \Psi d_j \partial_{\vec \nu} v_j \dd (t, \sigma(\vec x))
+ \int_\Omega \alpha_j v_j^0 \Psi(0,\cdot) \dd \vec x
\\
&= - \int_{\Omega_\tau} v_j \big( \alpha_j \partial_t \Psi + \beta_j \vec v \cdot \nabla \Psi + \dv(d_j \nabla \Psi) \big) \dd (t, \vec x)
+ \frac{d_j}{d} \int_{\Sigma_\tau} \eta v_j \dd (t, \sigma(\vec x)).
\end{align*}

Together, this gives

\begin{align*}
&\int_{\Omega_\tau} v \Theta \dd (t, \vec x)
+ \int_{\Sigma_\tau} v \eta \dd (t, \sigma(\vec x))
\\
&= \int_{\Omega_\tau} (f - \sum_{j=1}^m f_j) \Psi \dd (t, \vec x)
+ \int_{\Sigma_\tau} (g^\Sigma - \sum_{j=1}^m g_j^\Sigma) \Psi \dd (t, \sigma(\vec x))
+ \int_\Omega (v^0 - \sum_{j=1}^m \alpha_j v_j^0) \Psi(0,\cdot) \dd \vec x
\\ &\quad
+ \sum_{j=1}^m \big( \int_{\Omega_\tau} v_j \big( \alpha_j \partial_t \Psi + \beta_j \vec v \cdot \nabla \Psi + \dv(d_j \nabla \Psi) \big) \dd (t, \vec x)
+ \frac{d_j}{d} \int_{\Sigma_\tau} \eta v_j \dd (t, \sigma(\vec x)) \big)
\\
&\leq \norm{(f - \sum_{j=1}^m f_j)^+}_{\WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} \norm{\Psi}_{\WW_{q'}^{(1,2)}(\Omega_\tau)}
\\ &\qquad
+ \norm{(g^\Sigma - \sum_{j=1}^m g_j^\Sigma)^+}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} \norm{\Psi|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)}
\\ &\qquad
+ \norm{(v^0 - \sum_{j=1}^m \alpha_j v_j^0)^+}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} \norm{\Psi(0,\cdot)}_{\WW_{q'}^{2-2/q'}(\Omega)}
\\ &\qquad
+ \sum_{j=1}^m \norm{v_j}_{\LL_q(\Omega_\tau)} \norm{\alpha_j \partial_t \Psi + \beta_j \vec v \cdot \nabla \Psi + \dv(d_j \nabla \Psi)}_{\LL_{q'}(\Omega_\tau)}
\\ &\qquad
+ \frac{d_j}{d} \norm{v_j^+}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} \norm{\eta}_{\WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)},
\end{align*}

since we know that $\Psi \geq 0$ and $\eta \geq 0$.
Here, by $\LL_{q'}$-maximal regularity, we find that

\[
\norm{\alpha_j \partial_t \Psi + \beta_j \vec v \cdot \nabla \Psi + \dv(d_j \nabla \Psi)}_{\LL_{q'}(\Omega_\tau)}
\leq C \big( \norm{\Theta}_{\LL_{q'}(\Omega_\tau)} + \norm{\eta}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)} \big),
\]

so that taking the supremum over all non-negative $\Theta$ and $\eta$ with norm less or equal $1$, we may conclude that

\begin{align*}
&\norm{v^+}_{\LL_q(\Omega_\tau)} + \norm{v^+|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq C \big( \norm{(f - \sum_{j=1}^m f_j)^+}_{\WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{(g^\Sigma - \sum_{j=1}^m g_j^\Sigma)^+}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\ &\quad
+ \norm{(v^0 - \sum_{j=1}^m \alpha_j v_j^0)^+}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} + \sum_{j=1}^m \norm{v_j^+|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{v_j}_{\LL_q(\Omega_\tau)} \big).
\end{align*}

Similarly, a positive version of the dual estimate for parabolic equations on the surface $\Sigma = \partial \Omega$ can be derived:

Let $\Sigma \subseteq \R^d$ be the boundary of a bounded $\CC^{3-}$-domain, $\mu, \beta_i \in \CC^1(\Sigma;(0,\infty))$ and $\alpha_i \in \R$, $i = 1, \ldots, m$.
Moreover, let $p, q \in (1,\infty)$ and $T > 0$ be given.
Assume that $v^\Sigma$, $v_i^\Sigma \in \WW_p^{(1,2)}(\Sigma_\tau)$ are solutions to the parabolic initial-value problems

\[
\begin{cases}
\partial_t v^\Sigma - \dv_\Sigma (\mu \nabla_\Sigma v^\Sigma)
= f^\Sigma
\quad &
\text{on } \Sigma_\tau,
\\
= v^{\Sigma,0}
\quad &
\text{on } \Sigma
\end{cases}
\]

\[
\begin{cases}
\alpha_i \partial_t v_i^\Sigma - \dv_\Sigma(\beta_i \nabla_\Sigma v_i^\Sigma)
= f_i^\Sigma
\quad &
\text{on } \Sigma_\tau,
\\
= v_i^{\Sigma,0}
\quad &
\text{on } \Sigma
\end{cases}
\]

for some $\tau \in (0,T]$ and $f^\Sigma, f_i^\Sigma \in \LL_p(\Sigma_\tau)$, $v^{\Sigma,0}$, $v_i^{\Sigma,0} \in \WW_p^{2-2/p}(\Sigma)$.
Then there is a constant $C = C(q,T)$, independent of $\tau \in (0,T]$ and the data, such that

\begin{align*}
\\
&\leq C \big( \norm{(f^\Sigma - \sum_{j=1}^m f_j^\Sigma)^+}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} + \norm{(v^{\Sigma,0} - \sum_{j=1}^m \alpha_j v_j^{\Sigma,0})^+}_{\WW_{q'}^{2-2/q'}(\Sigma)^\ast} + \sum_{j=1}^m \norm{v_j^\Sigma}_{\LL_q(\Sigma_\tau)} \big).
\end{align*}

Similar to the bulk case, we consider an arbitrary non-negative function $\Theta^\Sigma \in \LL_{q'}(\Sigma_\tau)$ and the time-reversed dual problem

\[
\begin{cases}
- \partial_t \Psi^\Sigma - \dv( \mu \nabla_\Sigma \Psi^\Sigma)
= \Theta^\Sigma \quad
&\text{on } \Sigma_\tau,
\\
\Psi^\Sigma(\tau,\cdot)
= 0 \quad
&\text{on } \Sigma
\end{cases}
\]

which has the property of $\LL_{q'}$-maximal regularity with a maximal regularity constant which, thanks to the zero final conditions, can be chosen independent of the time horizon $\tau \in (0,T]$, i.e.
\[
\norm{\Psi^\Sigma}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)}
\leq C \norm{\Theta^\Sigma}_{\LL_{q'}(\Sigma_\tau)}.
\]

Moreover, as $\Theta^\Sigma \geq 0$, by the parabolic positivity principle, $\Psi^\Sigma \geq 0$, and, therefore, we may estimate

\begin{align*}
&\int_{\Sigma_\tau} v^\Sigma \Theta^\Sigma \dd (t, \sigma(\vec x))
\\
&= - \int_{\Sigma_\tau} (\partial_t \Psi^\Sigma + \dv_\Sigma( \mu \nabla_\Sigma \Psi^\Sigma)) v^\Sigma \dd (t, \sigma(\vec x))
\\
&= \int_{\Sigma_\tau} (\partial_t v^\Sigma - \dv_\Sigma (\mu \nabla_\Sigma v^\Sigma)) \Psi^\Sigma \dd (t, \sigma(\vec x))
+ \int_\Sigma v^{\Sigma,0} \Psi^{\Sigma}(0,\cdot) \dd \sigma(\vec x)
\\
&= \int_{\Sigma_\tau} f^\Sigma \Psi^\Sigma \dd (t, \sigma(\vec x))
+ \int_{\Sigma} v^{\Sigma,0} \Psi^{\Sigma}(0,\cdot) \dd \sigma(\vec x)
\\
&= \int_{\Sigma_\tau} (f^\Sigma - \sum_{j=1}^m f_j^\Sigma) \Psi^\Sigma \dd (t, \sigma(\vec x))
+ \int_{\Sigma} (v^{\Sigma,0} - \sum_{j=1}^m \alpha_j v_j^{\Sigma,0}) \Psi^\Sigma(0,\cdot) \dd \sigma(\vec x)
\\ &\quad
+ \sum_{j=1}^m \big( \int_{\Sigma_\tau} f_j^\Sigma \Psi^\Sigma \dd (t,\sigma(\vec x)) + \int_{\Sigma} \alpha_j v_j^{\Sigma,0} \Psi^\Sigma(0,\cdot) \dd \sigma(\vec x) \big).
\end{align*}

Here, we may estimate the terms in the last line by

\begin{align*}
&\int_{\Sigma_\tau} f_j^\Sigma \Psi^\Sigma \dd (t,\sigma(\vec x)) + \int_{\Sigma} \alpha_j v_j^{\Sigma,0} \Psi^\Sigma(0,\cdot) \dd \sigma(\vec x)
\\
&= \int_{\Sigma_\tau} \big( \alpha_j \partial_t v_j^\Sigma \cdot \Psi^\Sigma
- \dv_\Sigma (\beta_j \nabla_\Sigma v_j^\Sigma) \Psi^\Sigma \big) \dd (t, \sigma(\vec x))
+ \int_\Sigma \alpha_j v_j^\Sigma \Psi^\Sigma(0,\cdot) \dd \sigma(\vec x)
\\
&= - \int_{\Sigma_\tau} \big( \alpha_j v_j^\Sigma \partial_t \Psi^\Sigma + v_j^\Sigma \dv_\Sigma (\beta_j \nabla_\Sigma \Psi^\Sigma) \big) \dd (t,\sigma(\vec x))
\\
&\leq C \norm{v_j^\Sigma}_{\LL_q(\Omega_\tau)} \norm{\Theta^\Sigma}_{\LL_{q'}(\Omega_\tau)}.
\end{align*}

Taking the supremum over all non-negative $\Theta^\Sigma \in \LL_{q'}(\Omega_\tau)$ with norm less or equal one, we deduce that

\begin{align*}
\\
&\leq C \big( \norm{(f^\Sigma - \sum_{j=1}^m f_j^\Sigma)^+}_{\WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} + \norm{(v^{\Sigma,0} - \sum_{j=1}^m \alpha_j v_j^{\Sigma,0})^+}_{\WW_{q'}^{2-2/q'}(\Sigma)^\ast} + \sum_{j=1}^m \norm{v_j^\Sigma}_{\LL_q(\Sigma_\tau)} \big).
\end{align*}

§ GLOBAL-IN-TIME EXISTENCE

By comparing with blow-up results for ODEs modelling reaction networks, it is clear that for homogeneous Neumann boundary conditions global-in-time well-posedness of reaction-diffusion systems cannot be expected, in general, and blow-up may occur (but see [69] for a quite general result under mass dissipation and an entropy type dissipation condition).
Therefore, we cannot expect global-in-time existence of solutions without any further restriction on the structure of the chemical reaction networks in the bulk phase and on the surface.
Also note the blow-up result in [53] for inhomogeneous Dirichlet boundary conditions.
[Blow-up in the Henry model]
Consider $N = 1$ and the Henry model $s^\Sigma(c|_\Sigma, c^\Sigma) = c - c^\Sigma$, chemical reaction models $r^\Omega(c) = r^{\Sigma}(c) = c^2$, and initial data $c^0 \equiv 1$, $c^{\Sigma,0} \equiv 1$.
(Here, for simplicity, we set all physical parameters to be $1$.)
Then the unique solution $(c, c^\Sigma)(t,\cdot) \equiv (\frac{1}{1-t}, \frac{1}{1-t})$, $t \in [0,1)$, blows up in finite time.

Although this example is of purely academic nature, it hints that also for systems with coupling between bulk and surface reaction-diffusion-advection systems, further structural assumptions are needed to ensure global-in-time existence of solutions.
Throughout this section we, therefore, additionally demand the following of the chemical reaction models $\vec r^\Omega$ in the bulk phase and $\vec r^\Sigma$ on the active surface, respectively.
Note that for the case of Maxwell–Stefan diffusion models, where the diffusive flux is modelled by a indirect balance equation of forces under a summation constraint, $\LL_\infty$-bounds are trivial, but do, in general, not suffice to establish global-in-time existence of solutions, cf. [21].
[Triangular structure of $\vec r^\Omega$ and $\vec r^\Sigma$ / Intermediate sum condition]

There are lower triangular matrices $\bb Q^\Omega, \bb Q^{\Sigma} \in \R_+^{N \times N}$ with strictly positive diagonal entries and $C_\mathrm{tr}$, $C_\mathrm{tr}^\Sigma > 0$ such that

≤C_tr (  1 + ∑_j=1^N y_j )^μ^Ω e⃗

for all
y⃗ ∈[0, ∞)^N,

≤C_tr^Σ(  1 + ∑_j=1^N y^Σ_j )^μ^Σ e⃗

for all
y⃗^Σ∈[0, ∞)^N,

for some exponents $\mu^\Omega, \mu^\Sigma \geq 0$, and where $\vec e = (1, \ldots, 1)^\mathsf{T} \in \R^N$  and we write $\vec v \leq \vec w$ if $v_i \leq w_i$ for all components $i = 1, \ldots, N$.

To make a boot-strap argument feasible, and eventually establish $\LL_q$-growth bounds for the solution, we need $\LL_p$–$\LL_q$-estimates as in the following lemma, which heavily exploits the triangular structure of the chemical reaction networks.

Assume that, additionally to the standing Assumption <ref>, also the intermediate sum condition Assumption <ref> is satisfied.
Then, for

\[
p \in (\max \{ 1, \frac{(\mu^\Omega - 1)(d + 2)}{2 \mu^\Omega}, \frac{(\mu^\Sigma - 1)(d + 1)}{2 \mu^\Sigma} \}, \infty), \quad
q \in (1, \infty]
\quad \text{and} \quad
T_0 > 0,
\]

we set $T^\ast := \min \{ T_0, T_\mathrm{max} \} \in (0, \infty)$, and there is a constant $C^\ast = C^\ast(\vec c^0, \vec c^{\Sigma,0}, p,q,T^\ast)$ such that for all $\tau \in (0,T^\ast]$

\[
\norm{\vec c}_{\LL_q(\Omega_\tau;\R^N)}
+ \norm{\vec c|_\Sigma}_{\mathring {\fs G}_{q'}^\Sigma(\tau)^\ast}
+ \norm{\vec c^\Sigma}_{\LL_q(\Sigma_\tau)}
\leq C^\ast \left( 1
+ \norm{\vec c}_{\LL_p(\Omega_\tau;\R^N)}
%      + \norm{\vec c|_\Sigma}_{\LL_p(\Sigma_\tau;\R^N)}
+ \norm{\vec c^\Sigma}_{\LL_p(\Sigma_\tau;\R^N)} \right)^\mu,
\]

where $\mu = \mu(\mu^\Omega, \mu^\Sigma, p, q)$ will, in general, depend on $\mu^\Omega$, $\mu^\Sigma$ as well as on $p$ and $q$.

In case $\mu^\Omega = \mu^\Sigma = 1$, i.e. a linear intermediate sum condition, all $p, q \in (1, \infty)$ are admissible for the $\LL_p$–$\LL_q$-estimates in this lemma.

First, we consider the case where $q > p$ are such that $2 - \frac{d+2}{p} > - \frac{d+1}{q}$, $2 - \frac{d+2}{p} > - \frac{d+2}{\mu^\Omega q}$ and $2 - \frac{d+1}{p} > - \frac{d+1}{\mu^\Sigma q}$, so that the continuous embeddings $\WW_p^{(1,2)}(\Omega_T) \hookrightarrow \LL_q(\Sigma_T)$ (via the boundary trace map), $\WW_p^{(1,2)}(\Omega_\tau) \hookrightarrow \LL_{\mu^\Omega q}(\Omega_\tau)$ and $\WW_p^{(1,2)}(\Sigma_\tau) \hookrightarrow \LL_{\mu^\Sigma q}(\Sigma_\tau)$ are available.
Here, the assumption that $p > \max \{ \frac{(\mu^\Omega - 1)(d + 2)}{2 \mu^\Omega}, \frac{(\mu^\Sigma - 1)(d + 1)}{2 \mu^\Sigma} \}$ enables us to find $q > p$ such that the latter two conditions are met.
We employ the triangular structure of the chemical reaction network in the bulk phase and on the active surface, by which there are constants $C_\mathrm{tr}, C_\mathrm{tr}^\Sigma > 0$ such that for $i = 1, \ldots, N$ and $r_{ij} := \frac{q_{ij}}{q_{ii}}, r_{ij}^\Sigma = \frac{q_{ij}^\Sigma}{q_{ii}^\Sigma} \in \R$, the inequalities

\begin{align*}
r^\Omega_i(\vec c)
&\leq C_\mathrm{tr} (1 + \sum_{j=1}^N c_j)^{\mu^\Omega}
- \sum_{j < i} r_{ij} r^\Omega_j(\vec c)
\\
&= C_\mathrm{tr} (1 + \sum_{j=1}^N c_j)^{\mu^\Omega}
- \sum_{j < i} r_{ij} (\partial_t c_j + \vec v \cdot \nabla c_j - d_j \Delta c_j)
\\
r^\Sigma_i(\vec c^\Sigma)
&\leq C_\mathrm{tr}^\Sigma (1 + \sum_{j=1} c_j^\Sigma)^{\mu^\Sigma}
- \sum_{j < i} r_{ij}^\Sigma r^\Sigma_j(\vec c^\Sigma)
\\
&= C_\mathrm{tr}^\Sigma (1 + \sum_{j=1} c_j^\Sigma)^{\mu^\Sigma}
- \sum_{j < i} r_{ij}^\Sigma (\partial_t c^\Sigma_j - d_j^\Sigma \Delta_\Sigma c^\Sigma_j - s_j(c_j|_\Sigma,c_j^\Sigma))
\end{align*}

are valid.
We use these estimates in combination with the positive version of the duality estimates stated in Lemma <ref> and Lemma <ref>:
Fix any $T_0 > 0$, set $T^\ast := \min \{ T_0, T_\mathrm{max} \}$ and let $\tau \in (0, T^\ast)$, where $T_\mathrm{max} \in (0,\infty]$ is the maximal time of existence for the solution to the reaction-diffusion-advection-sorption system (<ref>) in the class $\WW_p^{(1,2)}$ for some $p \in (1,\infty)$ (and then, as it will turn out, away from $t = 0$ for every $p \in (1, \infty)$, see Lemma <ref> below).
For each component $i = 1, \ldots, N$ of the system, we consider the following auxiliary parabolic initial-boundary value problems:

∂_t z_i + v⃗ ·∇z_i - d_i Δz_i
= r^Ω_i(c⃗)

in  (0, τ) ×Ω,

- d_i ∂_ν⃗ z_i
= - k_i^de (1 + c_i^Σ)

on  (0, τ) ×Σ,

= c^0_i

on  Ω

in the bulk phase and the parabolic initial value problems
∂_t z_i^Σ- d_i Δ_Σz_i^Σ   = r^Σ_i(c⃗^Σ) + k_i^ad c_i|_Σ
on  (0, τ) ×Σ,

= c^Σ,0_i

on  Σ
on the boundary.
Choosing $r \in (1, \min \{p, 3 \})$ arbitrarily, the initial data do not necessarily have to satisfy the compatibility conditions, so that we obtain unique solutions $z_i \in \WW_r^{(1,2)}(\Omega_\tau)$ and $z_i^\Sigma \in \WW_r^{(1,2)}(\Sigma_\tau)$ of (<ref>) and (<ref>), respectively.
Moreover, since the lower and upper linear bounds $-k_i^\mathrm{de} (1 + c_i^\Sigma) \leq r^\mathrm{sorp}_i(c_i, c_i^\Sigma) \leq k_i^\mathrm{ad} (1 + c_i)$ are valid by assumption (<ref>), the comparison principle Lemma <ref> provides the estimates

\[
0 \leq c_i \leq z_i,
\quad
0 \leq c_i^\Sigma \leq z_i^\Sigma
\quad
\text{for } i = 1, \ldots N.
\]

We exploit that the systems (<ref>) and (<ref>) are quasi-linear and write $z_i = u_i + v_i + w_i$ and $z_i^\Sigma = u_i^\Sigma + v_i^\Sigma + w_i^\Sigma$ as a sum of solutions to the following quasi-autonomous, linear systems:

∂_t u_i + v⃗ ·∇u_i - d_i Δu_i
= C_tr (1 + ∑_j=1^N c_j)^μ^Ω

in  (0, τ) ×Ω,

- d_i ∂_ν⃗ u_i
= 0

on  (0, τ) ×Σ,

= 0

in  Ω

∂_t v_i + v⃗ ·∇v_i - d_i Δv_i
= r_i^Ω(c⃗) - C_tr (1 + ∑_j=1^N c_j)^μ^Ω

in  (0, τ) ×Ω,

- d_i ∂_ν⃗ v_i
= - k_i^de (1 + c_i^Σ)

on  (0, τ) ×Σ,

= 0

in  Ω

∂_t w_i + v⃗ ·∇w_i - d_i Δw_i
= 0

in  (0, τ) ×Ω,

- d_i ∂_ν⃗ w_i
= 0

on  (0, τ) ×Σ,

= c^0_i

in  Ω
in the bulk phase, whereas on the surface, we consider the problems

∂_t u_i^Σ- d_i^ΣΔ_Σu_i^Σ   = C_tr^Σ(1 + ∑_j=1^N c_j^Σ)^μ^Σ + k_i^ad (1 + c_i|_Σ)

on  (0, τ) ×Σ,

= 0

on  (0, τ) ×Σ,

∂_t v_i^Σ- d_i^ΣΔ_Σv_i^Σ   = r^Σ_i(c⃗^Σ) - C_tr^Σ(1 + ∑_j=1^N c_j^Σ)^μ^Σ

on  (0, τ) ×Σ,

= 0

on  (0, τ) ×Σ,

∂_t w_i^Σ- d_i^ΣΔ_Σw_i^Σ   = 0

on  (0, τ) ×Σ,

= c^Σ,0_i

on  (0, τ) ×Σ.

Since the strong solutions $c_j, c_j^\Sigma \geq 0$ are non-negative, by the comparison principle, we immediately find that the solutions $u_i, w_i, u_i^\Sigma, w_i^\Sigma$ to the initial (boundary) value problems (<ref>), (<ref>), (<ref>) and (<ref>) are non-negative, whereas for the solutions $v_i$ and $v_i^\Sigma$ to (<ref>) and (<ref>) non-negativity cannot be guaranteed. However, we always have the inequalities

\[
\leq c_i \leq z_i \leq u_i + v_i^+ + w_i,
\quad
\leq c_i^\Sigma \leq z_i^\Sigma \leq u_i^\Sigma + v_i^{\Sigma,+} + w_i
\quad
\text{for } i = 1, \ldots, N.
\]

Let us collect the information which we can extract from the duality estimates and positive duality estimates:

\[
\norm{u_i}_{\LL_q(\Omega_\tau)} + \norm{u_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\leq C_{p,q,T^\ast} (1 + \sum_{j=1}^N \norm{c_j^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast}),
\]

where the constant $C_{p,q,T^\ast} > 0$ may be chosen independent of $\tau < T^\ast$, since we consider zero initial data in (<ref>);

\begin{align*}
&\norm{v_i^+}_{\LL_q(\Omega_\tau)} + \norm{v_i^+|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
\leq C_{p,q,T} (1 + \norm{c_i^\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \sum_{j=1}^N \norm{c_j^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} ),
\\
&\norm{w_i}_{\LL_q(\Omega_\tau)} + \norm{w_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\leq C_{q,T^\ast}.
\end{align*}

Here, in the latter two estimates the constants $C_{p,q,T^\ast}$ and $C_{q,T^\ast}$ may (and will) depend on the initial data, but not on the particular choice of $\tau \in (0, T^\ast)$.
Similarly, on the surface we obtain the estimates

\begin{align*}
\norm{u_i^\Sigma}_{\LL_q(\Sigma_\tau)}
&\leq C_{p,q,T} (1 + \norm{c_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} + \sum_{j=1}^N \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast})
\\
&\leq C_{p,q,T} (1 + \norm{c_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \sum_{j=1}^N \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} )
\\
\intertext{again exploiting the zero initial data;}
\norm{v_i^{\Sigma,+}}_{\LL_q(\Sigma_\tau)}
&\leq C_{p,q,T^\ast} \sum_{j<i} \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\LL_q(\Sigma_\tau)};
\\
\norm{w_i^\Sigma}_{\LL_q(\Sigma_\tau)}
&\leq C_{p,q,T^\ast}.
\end{align*}

We now bring together these estimates, proceeding iteratively:
For the surface term, we find

\begin{align*}
\norm{c_1^\Sigma}_{\LL_q(\Sigma_\tau)}
&\leq \norm{z_1^\Sigma}_{\LL_q(\Sigma_\tau)}
\leq \norm{u_1^\Sigma}_{\LL_q(\Sigma_\tau)} + \norm{v_1^{\Sigma,+}}_{\LL_q(\Sigma_\tau)} + \norm{w_1^\Sigma}_{\LL_q(\Sigma_\tau)}
\\
&\leq C_{p,q,T^\ast} \big( 1 + \sum_{j=1}^N \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} + \norm{c_1|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} \big)
\end{align*}

and, for the bulk term,

\begin{align*}
&\norm{c_1}_{\LL_q(\Omega_\tau)} + \norm{c_1|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq \norm{z_1}_{\LL_q(\Omega_\tau)} + \norm{z_1|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq \norm{u_1}_{\LL_q(\Omega_\tau)} + \norm{u_1|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \norm{v_1^+}_{\LL_q(\Omega_\tau)}
\\ &\quad
+ \norm{v_1^+|_\Sigma}_{\WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \norm{w_1}_{\LL_q(\Omega_\tau)} + \norm{w_1|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq C_{p,q,T^\ast} \big( 1 + \sum_{j=1}^N \norm{(c_j)^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{c_1^\Sigma}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} \big).
\end{align*}

Together, this gives the estimate

\begin{align*}
+ \norm{c_1|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \norm{c_1^\Sigma}_{\LL_q(\Omega_\tau)}
\\
&\leq C_{q,T^\ast} \big( 1 + \sum_{j=1}^N \norm{c_j^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)\ast} \big).
\end{align*}

$i \geq 2$:
Now, assume that $i \geq 2$ and for $k = 1, \ldots, i-1$ we have shown the estimates

\begin{align*}
+ \norm{c_k|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \norm{c_k^\Sigma}_{\LL_q(\Omega_\tau)}
\\
&\leq C_{q,T^\ast} \big( 1 + \sum_{j=1}^N \norm{c_j^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} \big).
\end{align*}

We then find

\begin{align*}
\norm{c_i^\Sigma}_{\LL_q(\Sigma_\tau)}
&\leq \norm{z_i^\Sigma}_{\LL_q(\Sigma_\tau)}
\leq \norm{u_i^\Sigma}_{\LL_q(\Sigma_\tau)}
+ \norm{v_i^{\Sigma,+}}_{\LL_q(\Sigma_\tau)}
+ \norm{w_i^\Sigma}_{\LL_q(\Sigma_\tau)}
\\
&\leq C_{q,T^\ast} \big( 1 + \sum_{j=1}^N \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast}
+ \norm{c_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \sum_{j < i} \norm{c_j^\Sigma}_{\LL_q(\Sigma_\tau)} \big).
\end{align*}

\begin{align*}
&\norm{c_i}_{\LL_q(\Omega_\tau)} + \norm{c_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq \norm{z_i}_{\LL_q(\Omega_\tau)} + \norm{z_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq \norm{u_i}_{\LL_q(\Omega_\tau)} + \norm{u_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{v_i^+}_{\LL_q(\Omega_\tau)}
\\ &\quad
+ \norm{v_i^+|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{w_i}_{\LL_q(\Sigma_\tau)} + \norm{w_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
\\
&\leq C_{q,T^\ast} \big( 1 + \sum_{j=1}^N \norm{c_j^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast}
+ \sum_{j < i} \big( \norm{c_j}_{\LL_q(\Omega_\tau)} + \norm{c_j|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \norm{c_j^\Sigma}_{\LL_q(\Omega_\tau)} \big) \big).
\end{align*}

Together with the induction hypothesis, this gives

\begin{align*}
+ \norm{c_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \norm{c_i^\Sigma}_{\LL_q(\Omega_\tau)}
\\
&\leq C_{q,T^\ast} \big( 1 + \sum_{j=1}^N \norm{c_j^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} \big),
\end{align*}

so that by iteration we obtain the estimate

\begin{align}
&\sum_{j=1}^N \big( \norm{c_j}_{\LL_q(\Omega_\tau)}
+ \norm{c_j|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast}
+ \norm{c_j^\Sigma}_{\LL_q(\Omega_\tau)} \big)
\nonumber \\ &\quad
\leq C_{q,T^\ast} \big( 1 + \sum_{j=1}^N \big( \norm{c_j^{\mu^\Omega}}_{\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast} + \norm{(c_j^\Sigma)^{\mu^\Sigma}}_{\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast} \big) \big).
\label{eqn:general_estimate}
\end{align}

From the estimate (<ref>), we may now derive the $\LL_p$–$\LL_q$-estimates.
First, let us consider the case that $p < q \leq \infty$ (hence, $1 \leq q' < p'$ for the corresponding Hölder conjugates) such that $\frac{1}{q} \geq \frac{1}{p} - \frac{d+2}{2}$, $2 - \frac{d+2}{p} \geq - \frac{d+2}{\mu^\Sigma q}$.
Then, we may employ the continuous and dense embeddings

\[
\mathring \WW_{q'}^{(1,2)}(\Omega_\tau)
\hookrightarrow \LL_{(\mu^\Omega p)'}(\Omega_\tau)
\quad \text{and} \quad
\mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)
\hookrightarrow \LL_{(\mu^\Sigma p)'}(\Sigma_\tau),
\]

and, hence, we may conclude that

\[
\LL_p(\Omega_\tau)
\hookrightarrow \mathring \WW_{q'}^{(1,2)}(\Omega_\tau)^\ast
\quad \text{and} \quad
\LL_p(\Sigma_\tau)
\hookrightarrow \mathring \WW_{q'}^{(1,2)}(\Sigma_\tau)^\ast
\]

are continuously embedded.
Therefore, we obtain that

\begin{align*}
&\sum_{j=1}^N \big( \norm{c_j}_{\LL_q(\Omega_\tau)}
+ \norm{c_j^\Sigma}_{\LL_q(\Omega_\tau)}
+ \norm{c_j|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)} \big)
\\
&\leq C_{p,q,T^*} \big( 1 + \sum_{j=1}^N \big( \norm{c_j}_{\LL_p(\Omega_\tau)}^{\mu^\Omega} + \norm{c_j^\Sigma}_{\LL_p(\Sigma_\tau)}^{\mu^\Sigma} \big) \big)
\quad \text{for } p < q \leq \infty \text{ s.t.\ } \frac{1}{p} \geq \frac{1}{q} - \frac{d+2}{2}.
\end{align*}

Iterating this procedure, if necessary, we obtain that

\begin{align*}
&\sum_{j=1}^N \big( \norm{c_j}_{\LL_q(\Omega_\tau)}
+ \norm{c_j^\Sigma}_{\LL_q(\Omega_\tau)}
+ \norm{c_j|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)} \big)
\\
&\leq C_{p,q,T^*} \big( 1 + \sum_{j=1}^N \big( \norm{c_j}_{\LL_p(\Omega_\tau)} + \norm{c_j^\Sigma}_{\LL_p(\Sigma_\tau)} \big) \big)^\mu
\end{align*}

for all $p > \max \{ 1, \frac{(\mu^\Omega - 1)(d + 2)}{2 \mu^\Omega}, \frac{(\mu^\Sigma - 1)(d + 1)}{2 \mu^\Sigma} \}, \, q \in (1,\infty)$,
where $\mu \geq 0$ will, in general, be a multiple of $\mu^\Omega$ or $\mu^\Sigma$.

Regarding the term $c_j|_\Sigma$, this is not yet the estimate in the norm which we are actually looking for.
To improve on the norm, we consider the bulk reaction-diffusion-advection problems

(∂_t + v⃗ ·∇- d_i Δ) c_i
= f_i := r_i^Ω(c⃗)

in  Ω_τ,

= h_i := c_i|_Σ
on  Σ_τ,

= c_i^0

in  Ω.

For the initial data, we may and will w.l.o.g. assume that $c_i^0 \in \CC^2(\overline{\Omega})$.
Moreover, by the estimates which we have already established and polynomial boundedness of the chemical reaction rates, we may conclude that $r_i^\Omega(\vec c) \in \bigcap\limits_{q \in (1,\infty)} \LL_q(\Omega_\tau)$ and $h_i = c_i|_\Sigma \in \bigcap\limits_{q \in (1,\infty)} \mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast$.
To obtain an estimate of the form

\[
\norm{c_i|_\Sigma}_{\LL_q(\Sigma_\tau)}
\leq C \big( \norm{\vec c}_{\LL_{q \gamma^\Omega}(\Omega_\tau)}^{\gamma^\Omega} + \norm{c_i|_\Sigma}_{\mathring \WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} \big)
\quad
\text{for sufficiently large } q,
\]

we need to make some additional considerations:
As a first step, let us consider the known optimal regularity estimate

\[
\norm{v_i}_{\WW_q^{(1,2)}(\Omega_\tau)}
\leq C \big( \norm{h_i}_{\WW_q^{(1,2) \cdot (1 - \frac{1}{2q})}(\Sigma_\tau)} + \norm{v_i^0}_{\WW_q^{2-2/q}(\Omega)} + \norm{f_i}_{\LL_q(\Omega_\tau)}  \big)
\]

for the reaction-diffusion-advection systems with inhomogeneous Dirichlet data

(∂_t - v⃗ ·∇- d_i Δ) v_i
= f_i
in  Ω_τ,

v_i|_Σ   = h_i
on  Σ_τ,

= v_i^0
in  Ω.

Via its dual problem

- (∂_t + v⃗ ·∇+ d_i Δ) θ_i
= Θ_i
in  Ω_τ,

= g_i
on  Σ_τ,

= 0
in  Ω
we obtain the estimate

\[
\norm{v_i}_{\LL_q(\Omega_\tau)}
\leq C \big( \norm{h_i}_{\mathring \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{v_i^0}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} + \norm{f_i}_{\WW_{q'}^{(1,2)}(\Omega_\tau)^\ast}  \big).
\]

We now want to interpolate between these estimates.
If we would proceed directly with the real interpolation method, we can estimate the $( \LL_q(\Omega_\tau), \WW_q^{(1,2)}(\Omega_\tau) )_{\theta,q}$-norm (which is equivalent to $\norm{\cdot}_{\BB_q^{(1,2) \cdot \theta}(\Omega_\tau)}$) by the interpolation space norms

\begin{align*}
&\norm{h_i}_{( \WW_{q'}^{(1,2) \cdot (1 - \frac{1}{2q'})}(\Sigma_\tau)^\ast, \WW_q^{(1,2) \cdot (1 - \frac{1}{2q})})_{\theta,q}},
\quad
\norm{v_i^0}_{(\WW_{q'}^{2-2/q'}(\Omega)^\ast, \WW_q^{2-2/q}(\Omega))_{\theta,q}}
\\
&\quad \text{and} \quad
\norm{f_i}_{(\WW_q^{(1,2)}(\Omega_\tau)^\ast, \LL_q(\Omega_\tau))_{\theta,q}}
\end{align*}

of $h_i$, $v_i^0$ and $f_i$, respectively.
This direct approach, however, runs into the problem of how to characterize these interpolation spaces, e.g. by identifying them with suitable Besov or, more general, Triebel–Lizorkin spaces, if possible.
Unfortunately, the test functions $\Dcal(\Omega) = \CC_c^\infty(\Omega)$ are not dense in $\WW_{q'}^{2-2/q'}(\Omega)$ for sufficiently large $q' > 1$ (i.e. for small $q > 1$), and, therefore, the characterizations which are valid for the full space $\R^d$ do, in general, not remain true if one replaces $\R^d$ by an open, regular domain $\Omega \subseteq \R^d$.
In fact, in this case the dual spaces also include elements which are no distributions.
We refer to [7] for results in the context of corners, i.e. intersections of open or closed half-spaces, in $\R^d$.
In our situation, however, we want to estimate an element in the dual space for which will a-priori know that it is represented by an – at least locally – integrable function; in particular a distribution.
To avoid the problems sketched above, we use that thanks to the regularity of the domain there exist universal extension operators $\Ecal_{\Omega \rightarrow \R^d}: \LL_1(\Omega) \rightarrow \LL_1(\R^d)$ and $\Ecal_{\Omega_\tau \rightarrow \R^{1+d}}: \LL_1(\Omega_\tau) \rightarrow \LL_1(\R^{1+d})$, cf. [62], such that

\begin{align*}
\Ecal_{\Omega \rightarrow \R^d}|_{\WW_r^s(\Omega)}:
\quad
&\WW_r^s(\Omega) \rightarrow \WW_r^s(\R^d),
\\
\Ecal_{\Omega_\tau \rightarrow \R^d}|_{\WW_r^{(1,2) \cdot s}(\Omega_\tau)}:
&\WW_r^{(1,2) \cdot s}(\Omega_\tau) \rightarrow \WW_r^{(1,2) \cdot s}(\R^{1+d}),
\quad
r \in (1,\infty), \, s \in [0,2],
\end{align*}

i.e. for each $r \in (1,\infty)$, $s \in [0,2]$, these are bounded linear operators, and the action of $\Ecal_{\Omega \rightarrow \R^d}$ resp. $\Ecal_{\Omega_\tau \rightarrow \R^{1+d}}$ does not depend on $r$ and $s$, e.g. $\Ecal_{\Omega \rightarrow \R^d}^{(r,s)} u = \Ecal_{\Omega \rightarrow \R^d}^{(r',s')} u$, whenever $u \in \WW_r^s(\Omega) \cap \WW_{r'}^{s'}$ for some $r, r' \in (1,\infty)$ and $s, s' \in [0,2]$.
Thanks to the regularity of the domain $\Omega$, these exist.
Moreover, for the Dirichlet trace function, we consider a smooth decomposition $\{ \psi_k: \, k = 1, \ldots, m \}$ of unity in a neighbourhood subordinate to an open covering $\{ U_k: k = 1, \ldots, m \}$ of the compact set $\overline{\Omega}$, and $\CC^2$-isomorphisms $\Phi_k: U_k \rightarrow V_k \subseteq \R^d$ such that $\Phi_k(U_k \cap \Omega) = V_k \cap (\R^{d-1} \times (0,\infty))$ and $\Phi_k(U_k \cap \Sigma) = V_k \cap (\R^{d-1} \times \{0\})$, and the map $\Tcal$ defined by

\[
(\Tcal u)_k(t,\vec x')
= \begin{cases}
(\psi_k u)(t, \Phi_k^{-1}(\vec x',0)),
&\text{if } (x',0) \in V_k,
\\
\end{cases}
\]

Note that $\mathcal{T}|_{\WW_r^{(1,2) \cdot s}(\Sigma_\tau)}$ is a bounded linear operator from $\WW_r^{(1,2) \cdot s}(\Sigma_\tau)$ to $[\WW_r^{(1,2) \cdot s}(\R \times \R^{d-1})]^m$ for each $r \in (1, \infty)$ and $s \in [0,2]$.
In fact, it is a topologic linear isomorphism between $\WW_r^{(1,2) \cdot s}(\Sigma_\tau)$ and $\Tcal(\WW_r^{(1,2) \cdot s}(\Sigma_\tau)) \subseteq \WW_r^{(1,2) \cdot s}(\R \times \R^{d-1})$ for each $r \in (1,\infty)$ and $s \in [0,2]$.
We then find that

\begin{align*}
&\norm{\Ecal_{\Omega_\tau \rightarrow \R^{1+d}} v_i}_{\LL_q(\R^{1+d})}
\leq \norm{v_i}_{\LL_q(\Omega_\tau)}
\\
&\leq C \left( \norm{h_i}_{\mathring \WW_{q'}^{(1,2)} \cdot (1-\frac{1}{2q'})}(\Sigma_\tau) + \norm{v_i^0}_{\WW_{q'}^{2-2/q'}(\Omega)^\ast} + \norm{f_i}_{\WW_{q'}^{(1,2)}(\Omega_\tau)} \right)
\\
&\leq C \left( \norm{\Tcal h_i}_{\WW_{q}^{(1,2) \cdot (- \frac{1}{2q})}(\R^{1+d})} + \norm{\Ecal_{\Omega \rightarrow \R^d} v_i^0}_{\WW_q^{(1,2) \cdot (- 2/q)}(\R^d)} + \norm{\Ecal_{\Omega_\tau \rightarrow \R^{1+d}} f_i}_{\WW_q^{(1,2) \cdot (-1)}(\R^{1+d})} \right)
\end{align*}

\begin{align*}
&\norm{\Ecal_{\Omega_\tau \rightarrow \R^{1+d}} v_i}_{\WW_q^{(1,2)}(\Omega_\tau)}
\leq C \norm{v_i}_{\WW_q^{(1,2)}(\Omega_\tau)}
\\
&\leq C \left( \norm{h_i}_{\WW_q^{(1,2) \cdot (1 - \frac{1}{2q})}(\Sigma_\tau)} + \norm{v_i^0}_{\WW_q^{2-2/q}(\Omega)} + \norm{f_i}_{\LL_q(\Omega_\tau)} \right)
\\
&\leq C \left( \norm{\Tcal h_i}_{\WW_q^{(1,2) \cdot (1 - \frac{1}{2q})}(\R^{1+(d-1)};\R^N)} + \norm{\Ecal_{\Omega \rightarrow \R^d} v_i^0}_{\WW_q^{2-2/q}(\R^d)} + \norm{\Ecal_{\Omega_\tau \rightarrow \R^{1+d}} f_i}_{\LL_q(\R^{1+d})} \right).
\end{align*}

We may now interpolate between these inequalities with the real interpolation functor $(\cdot,\cdot)_{\theta,q}$ for some $\theta \in (0,1)$ and obtain that

\begin{align*}
&\norm{\Ecal_{\Omega \rightarrow \R^{1+d}} v_i}_{\BB_q^{(1,2) \cdot \theta}(\R^{1+d})}
\\
&\leq C \left( \norm{\Tcal h_i}_{\BB_q^{(1,2) \cdot (\theta - \frac{1}{2q})}(\R^{1+d};\R^N)} + \norm{\Ecal_{\Omega \rightarrow \R^d} v_i^0}_{\BB_q^{2(\theta-\frac{1}{q})}(\R^d)} + \norm{f_i}_{\BB_q^{(1,2) \cdot (\theta - 1)}(\R^{1+d})} \right).
\end{align*}

For the particular choice $\theta = \frac{1}{2q}$, we find that the spatial trace space of $\BB_q^{(1,2) \cdot \frac{1}{2q}}(\R^{1+d})$ for the restriction to $\R \times \Sigma$ continuously embeds into $\LL_q(\R \times \Sigma)$, so that we may conclude that

\begin{align*}
&\norm{(\Ecal_{\Omega_\tau \rightarrow \R^{1+d}} v_i)|_\Sigma}_{\LL_q(\R \times \Sigma)}
\leq C \norm{\Ecal_{\Omega_\tau \rightarrow \R^{1+d}} v_i}_{\BB_q^{(1,2) \cdot \theta}(\R^{1+d})}
\\
&\leq C \left( \norm{\Tcal h_i}_{\BB_q^{(1,2) \cdot (\theta - \frac{1}{2q})}(\R^{1+(d-1)}; \R^N)} + \norm{\Ecal_{\Omega \rightarrow \R^d} v_i^0}_{\BB_q^{2(\theta - \frac{1}{q})}(\R^d)} + \norm{f_i}_{\BB_q^{(1,2) \cdot (\theta-1)}(\R^{1+d})} \right)
\\
&\leq C \left( \norm{\Tcal h_i}_{\LL_q(\R^{1+(d-1)}; \R^N)} + \norm{\Ecal_{\Omega \rightarrow \R^d} v_i^0}_{\WW_q^{(1,2)}(\R^d)} + \norm{f_i}_{\LL_q(\R^{1+d})} \right)
\\
&\leq C \left( 1 + \norm{c_i|_\Sigma}_{\WW_{q'}^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q'})}(\Sigma_\tau)^\ast} + \norm{v_i^0}_{\WW_q^{(1,2)}(\Omega)} + \norm{c_i}_{\LL_{q\gamma^\Omega}(\Omega_\tau)}^{\gamma^\Omega} \right).
\end{align*}

Let $(\vec c, \vec c^\Sigma) \in \WW_p^{(1,2)}(\Omega_T)^N \times \WW_p^{(1,2)}(\Sigma_T)^N$ be the solution to the reaction-diffusion-advection-sorption system (<ref>) with Langmuir adsorption for some $p \in ( \max \{1, \frac{\mu^\Omega - 1}{\mu^\Omega} \frac{d+2}{2}, \frac{\mu^\Sigma - 1}{\mu^\Sigma} \frac{d + 1}{2} \}, \infty )$ and some $T > 0$.
Then, $(\vec c, \vec c^\Sigma) \in \WW_q^{(1,2)}((\varepsilon,T) \times \Omega)^N \times \WW_q^{(1,2)}((\varepsilon,T) \times \Sigma))^N$ for all $\varepsilon \in (0,T)$ and $q \in (1, \infty)$.

One might suspect that the smoothing property of the reaction-diffusion-advection-sorption system under consideration is valid independently of the particular triangular structure imposed on the reactive terms.
Note, however, that we also include the case of rather low values of $p \in (1, \infty)$.
In contrast to the case $p > \frac{d+2}{2}$, the solution space $\WW_p^{(1,2)}(\Omega_\tau) \times \WW_p^{(1,2)}(\Sigma_\tau)$ does, in general, not continuously embed into those spaces which would be required to employ Hölder regularity theory of reaction-diffusion equations.
In fact, if $\WW_p^{(1,2)}(\Omega_\tau) \hookrightarrow \CC^{(1,2) \cdot \alpha}(\overline{\Omega_\tau})$, $\WW_p^{(1,2)}(\Omega_\tau) \hookrightarrow \CC^{(1,2) \cdot (\alpha + \frac{1}{2})}(\overline{\Sigma_\tau})$ (via the boundary trace map), and $\WW_p^{(1,2)}(\Sigma_\tau) \hookrightarrow \CC^{(1,2) \cdot \alpha}(\overline{\Sigma_\tau})$ embed continuously for some $\alpha \in (0, \frac{1}{2})$ (which is always true if $p > \frac{d+2}{2}$), we could exploit the theory of quasi-autonomous, linear parabolic equations to deduce that actually $(\vec c, \vec c^\Sigma) \in \CC^{(1,2) \cdot (1 + \alpha)}((\varepsilon,\tau);\R^N) \times \CC^{(1,2) \cdot (1 + \alpha)}((\varepsilon, \tau) \times \Sigma; \R^N)$ for every $\varepsilon > 0$.
In [64], the smoothing property has been demonstrated for the case $p > d$.
In fact, from their results if follows that if $p > d$ and $(\vec c^0, \vec c^{\Sigma,0}) \in \WW_p^2(\Omega;\R^N) \times \WW_p^2(\Sigma;\R^N)$ (satisfying the compatibility conditions), there is a unique classical solution in the class

\begin{align*}
(\vec c, \vec c^\Sigma)
&\in \CC([0,\tau]; \LL_p(\Omega; \R^N) \times \LL_p(\Sigma;\R^N))
\cap \CC^1((0,\tau]; \CC(\overline{\Omega};\R^N) \times \CC(\Sigma; \R^N))
\\ &\qquad
\cap \CC((0,\tau]; \CC^2(\overline{\Omega}; \R^N) \times \CC^2(\Sigma; \R^N)).
\end{align*}

Note that the condition $p > d$ is sufficient to imply that $\WW_p^2(\Omega) \hookrightarrow \CC^{1+\alpha}(\overline{\Omega})$ for $\alpha := 1 - \frac{d}{p} \in (0,1)$, and then $\WW_p^2(\Sigma) \hookrightarrow \CC^{1+\alpha}(\Sigma)$ as well, so that the tools from Hölder regularity theory for parabolic equations become applicable.

By the $\LL_p$–$\LL_q$-estimates of Lemma <ref> (which crucially depend on the triangular structure!), we know that $\vec c \in \LL_q(\Omega_T;\R^N)$ and $\vec c|_\Sigma$, $\vec c^\Sigma \in \LL_q(\Sigma_T;\R^N)$ for all $q \in (1, \infty)$.
By the polynomial boundedness of the chemical reaction terms $\vec r^\Omega(\vec c)$ and $\vec r^\Sigma(\vec c^\Sigma)$ as well as the sorption model $\vec r^\mathrm{sorp}(\vec c|_\Sigma, \vec c^\Sigma)$, we may, thus, conclude that these are functions in $\LL_q(\Omega_T)$ and $\LL_q(\Sigma_T)$ for all $q \in (1, \infty)$ as well.
Therefore, the solutions $c_i^\Sigma: [0,T] \rightarrow \R$ of the problems

∂_t c_i^Σ- d_i^ΣΔ_Σc_i^Σ   = F_i^Σ+ G_i^Σ          on  Σ_T,

= c_i^Σ,0
on  Σ,

where $F_i^\Sigma = r_i^\Sigma(\vec c^\Sigma)$ and $G_i^\Sigma = s_i^\Sigma(c_i, c_i^\Sigma) \in \LL_q((\varepsilon,T) \times \Sigma)$ lie in $\WW_q^{(1,2)}((\varepsilon,T) \times \Sigma)$ for all $q \in (1, \infty)$ and $\varepsilon \in (0,T)$.
In fact, we may employ the representation

\[
= \ee^{t A_i^\Sigma} c_i^{\Sigma,0} + \int_0^t \ee^{(t-s) A_i^\Sigma} (F_i^\Sigma + G_i^\Sigma)(s,\cdot) \dd s,
\]

where the inhomogeneous term is in $\WW_q^{(1,2)}(\Sigma_T)$ since $F_i^\Sigma + G_i^\Sigma \in \LL_q(\Sigma_T)$ and by $\LL_q$-maximal regularity, for all $q \in (1,\infty)$.
For the homogeneous part $\ee^{t A_i^\Sigma} c_i^{\Sigma,0}$ we may use the smoothing effect of the analytic $C_0$-semigroup $(\ee^{t A_i^\Sigma})$ on the space $\LL_p(\Sigma)$, so that this term lies in $\WW_q^{(1,2)}((\varepsilon,T) \times \Sigma)$ for all $\varepsilon \in (0,T)$.
In particular, $c_i^\Sigma \in \CC((0,T];\WW_q^{2-2/q}(\Sigma))$ for all $q \in (1,\infty)$.
Concerning the bulk term, we see that $c_i|_{t \in [\varepsilon,T]}$ is the solution to the linear parabolic initial-boundary value problem

\[
\begin{cases}
\partial_t c_i + \vec v \cdot \nabla c_i - d_i \Delta c_i
= F_i
&\text{in } (\varepsilon,T) \times \Omega,
\\
k_i^\mathrm{ad} (1 - \frac{c_i^{\Sigma,+}}{c_i^{\Sigma,\infty}})^+ + d_i \partial_{\vec \nu} c_i
= G_i
&\text{on } (\varepsilon,T) \times \Sigma,
\\
= c_i^\varepsilon
&\text{in } \overline{\Omega}
\end{cases}
\]

for $c_i^\varepsilon := c_i(\varepsilon,0) \in \WW_p^{2-2/p}(\Omega)$, and $F_i = r_i^\Omega(\vec c) \in \LL_q((\varepsilon,T) \times \Omega)$ for all $q \in (1, \infty)$, $G_i = k_i^\mathrm{de} c_i^\Sigma \in \WW_q^{(1,2)}((\varepsilon,T) \times \Sigma)$ for all $q \in (q, \infty)$.
Now, we may write $c_i|_{t \in [\varepsilon,T]} = u_i + v_i + w_i$, where $u_i \in \WW_q^{(1,2)}((\varepsilon,T) \times \Omega$) for all $q \in (1,\infty)$ with $\frac{1}{q} \geq \frac{1}{p} - \frac{1}{d+2}$ is the solution to the parabolic initial-boundary value problem

\[
\begin{cases}
\partial_t u_i - d_i \Delta u_i
= F_i - \vec v \cdot \nabla c_i
&\text{in } (\varepsilon,T) \times \Omega,
\\
k_i^\mathrm{ad} (1 - \frac{c_i^{\Sigma,+}}{c_i^{\Sigma,\infty}})^+ u_i + d_i \partial_{\vec \nu} u_i
= G_i
&\text{on } (\varepsilon,T) \times \Sigma,
\\
= 0
&\text{on } \overline{\Omega},
\end{cases}
\]

and where we may use the fact that $c_i \in \WW_p^{(1,2)}(\Omega_T) \hookrightarrow \WW_q^{(1,2) \cdot \frac{1}{2}}(\Omega_T) \hookrightarrow \LL_q((\varepsilon,T);\LL_q(\Sigma))$ for $q \in (1,\infty)$ such that $\frac{1}{q} \geq \frac{1}{p} - \frac{1}{d+2}$.
Moreover, the functions $v_i$ and $w_i$ are constructed by taking some $w_i^\varepsilon \in \bigcap_{q \in (1,\infty)} \WW_q^{2-2/q}(\Omega)$ such that $- d_i \partial_{\vec \nu} w_i = - k_i^\mathrm{ad} (1 - \frac{c_i^{\Sigma,+}(\varepsilon,\cdot)}{c_i^{\Sigma,\infty}})^+ (c_i(\varepsilon,0) + w_i^\varepsilon)$ and then defining $v_i$ and $w_i$ as the solutions to the initial-boundary value problems

\[
\begin{cases}
\partial_t v_i - d_i \Delta v_i
= 0
&\text{in } (\varepsilon,T) \times \Omega,
\\
d_i \partial_{\vec \nu} v_i
= 0
&\text{on } (\varepsilon,T) \times \Sigma,
\\
= c_i(\varepsilon,0) - w_i^\varepsilon
&\text{on } \overline{\Omega}
\end{cases}
\]

\[
\begin{cases}
\partial_t w_i - d_i \Delta w_i
= 0
&\text{in } (\varepsilon,T) \times \Omega,
\\
k_i^\mathrm{ad} (1 - \frac{c_i^{\Sigma,+}}{c_i^{\Sigma,\infty}})^+ w_i + d_i \partial_{\vec \nu} w_i
= k_i^\mathrm{ad} (1 - \frac{c_i^{\Sigma,+}}{c_i^{\Sigma,\infty}})^+ v_i
&\text{on } (\varepsilon,T) \times \Sigma,
\\
= w_i^\varepsilon
&\text{on } \Sigma.
\end{cases}
\]

Here, by $\LL_q$-maximal regularity, the latter problem has a solution $w_i \in \WW_q^{(1,2)}((\varepsilon,T) \times \Omega)$ for all $q \in (1,\infty)$.
Moreover, by classical smoothing properties of the Neumann semigroup on $\LL_p(\Omega)$, we obtain $v_i \in \WW_q^{(1,2)}((2\varepsilon,T) \times \Omega)$ for all $q \in (1,\infty)$.
Putting things together, this means that $c_i \in \WW_q^{(1,2)}((2\varepsilon,T) \times \Omega)$ for every $\varepsilon > 0$, and, in particular, $c_i \in \CC((0,T];\WW_q^{2-2/q}(\Omega))$ for every $q \in (1,\infty)$ such that $\frac{1}{q} \geq \frac{1}{p} - \frac{1}{d+2}$.
Repeating the argument for $p' \in (1,\infty)$ such that $\frac{1}{p'} \geq \frac{1}{p} - \frac{1}{d+2}$ and possibly multiple iterations, we obtain that actually $c_i \in \WW_q^{(1,2)}((\varepsilon,T) \times \Omega)$ for all $\varepsilon > 0$, and thus $c_i \in \CC((0,T];\WW_q^{2-2/q}(\Omega))$ for all $q \in (1, \infty)$.

If the strong $\WW_p^{(1,2)}$-solution $(\vec c, \vec c^\Sigma) \in \WW_p^{(1,2)}(\Omega_T;\R^N) \times \WW_p^{(1,2)}(\Sigma_T;\R^N)$ exists for some $0 < T < \infty$ and $p \in (1, \infty)$, then its restriction to times $t \in (\varepsilon, T)$ is a $\WW_q^{(1,2)}$-solution $(\vec c, \vec c^\Sigma) \in \WW_q^{(1,2)}((\varepsilon, T) \times \Omega;\R^N) \times \WW_p^{(1,2)}((\varepsilon,T) \times \Sigma;\R^N)$ for all $q \in (1,\infty)$. In particular, the maximal time of existence $T_\mathrm{max} \in (0, \infty]$ does not depend on the choice of $p \in (1, \infty)$.
In particular, if

\[
\sup_{t \in [\varepsilon,T]} \norm{(\vec c(t,\cdot), \vec c^\Sigma(t,\cdot))}_{\WW_p^{2-2/p}}
< \infty
\]

for some $p \in (1,\infty)$, then $\sup_{t \in [\varepsilon,T]} \norm{(\vec c(t,\cdot), \vec c^\Sigma(t,\cdot))}_{\WW_q^{2-2/q}} < \infty$ for all $q \in (1, \infty)$.

Once we have the $\LL_p$–$\LL_q$-estimates at hand, we may easily infer exponential growth bounds for the $\LL_p$-norms of the solution on $\Omega_\tau$ and $\Sigma_\tau$, respectively.

For every $T_0 > 0$, $p \in (1, \infty]$, there are $M = M(p,T_0) > 0$, $\omega = \omega(p,T_0) \in \R$ (also depending on the initial data) such that

\[
\sum_{i=1}^N \big( \norm{c_i}_{\LL_p(\Omega_\tau)}
+ \norm{c_i|_\Sigma}_{\LL_p(\Sigma_\tau)}
+ \norm{c_i^\Sigma}_{\LL_p(\Sigma_\tau)} \big)
\leq M \ee^{\omega \tau},
\quad
\tau \in (0, \min \{T_0, T_\mathrm{max}\}).
\]

We start by employing the $\LL_p$–$\LL_q$-estimates of Lemma <ref> for the special case $p \in (1, \infty)$ and $q = \infty$, resulting in

\begin{align*}
&\norm{\vec c(\tau,\cdot)}_{\LL_p(\Omega)}^p
+ \norm{\vec c^\Sigma(\tau,\cdot)}_{\LL_p(\Sigma)}^p
\\
&\leq C \sup_{t \in [0, \tau]} \big(
\norm{\vec c(t,\cdot)}_{\LL_p(\Omega)}^p
+ \norm{\vec c^\Sigma(t,\cdot)}_{\LL_p(\Sigma)}^p \big)
\\
&\leq C \big(
\norm{\vec c}_{\LL_\infty(\Omega_\tau)}^p
+ \norm{\vec c^\Sigma}_{\LL_\infty(\Sigma_\tau)}^p \big)
\\
&\leq C \big( 1
+ \norm{\vec c}_{\LL_p(\Omega_\tau)}^p
+ \norm{\vec c^\Sigma}_{\LL_p(\Sigma_\tau)}^p \big)
\\
&\leq C \big[ 1
+ \int_0^\tau \big( \norm{\vec c(t,\cdot)}_{\LL_p(\Omega)}^p
+ \norm{\vec c^\Sigma(t,\cdot)}_{\LL_p(\Sigma)}^p \big) \dd t \big],
\quad
\tau \in (0, T_0] \cap (0, T_\mathrm{max}).
\end{align*}

By Gronwall's Lemma, see e.g. <cit.>), there are, thus, constants $M' = M'_{p,T_0} > 0$ and $\omega' = \omega'_{p,T_0} > 0$ (also depending on the initial data through the constant $C$ in the $\LL_p$–$\LL_q$-estimates) such that

\begin{equation}
\norm{\vec c(t,\cdot)}_{\LL_p(\Omega)}^p
+ \norm{\vec c^\Sigma(t,\cdot)}_{\LL_p(\Sigma)}^p
\leq M' \ee^{\omega' t},
\quad
t \in (0,T_0] \times (0, T_\mathrm{max}).
\label{eqn:exponential-bounds-1}
\end{equation}

Integrating over $t \in (0, \tau)$, this proves the claim for $p < \infty$.
The case $p = \infty$ follows by integrating equation (<ref>) in time for some finite $p \in (1, \infty)$ and then employing the $\LL_p$–$\LL_\infty$-estimate provided by Lemma <ref>.
Finally, we may proceed just as in the remarks after the proof of Lemma <ref> to estimate the $\LL_p$-norm of $\vec c|_\Sigma$ as well.

§.§ Statement and proof of the global existence result

After these preparations, we are ready to prove the following theorem on global-in-time existence of solutions to the bulk-surface reaction-diffusion-advection system (<ref>) for suitable initial data.

Let $J = \R_+ = [0, \infty)$, $p \in [\frac{d+2}{2}, \infty)$ and let Assumptions <ref> and <ref> be valid.
Moreover, assume that the sorption is modelled by the Langmuir sorption model.
Then, for all compatible initial data

\[
(\vec c^0, \vec c^{\Sigma,0}) \in \fs I_p^\Omega \times \fs I_p^\Sigma
\]

with $(\vec c^0, \vec c^{\Sigma}) \geq (0, 0)$, the reaction-diffusion-advection-sorption system

∂_t c_i + v⃗ ·∇c_i - (d_i ∇c_i)
= r^Ω_i(c⃗)

on  (0,∞) ×Ω,

∂_t c_i^Σ- _Σ(d_i^Σ∇_Σc_i^Σ)
= s_i^Σ(c_i, c_i^Σ) + r_i^Σ(c⃗^Σ)

on  (0, ∞) ×Σ,

- d_i ∂_νc_i
= s_i^Σ(c_i, c_i^Σ)

on  (0, ∞) ×Σ,

c_i(0, ·)
= c^0_i

in  Ω,

c_i^Σ(0, ·)
= c^Σ,0_i

on  Σ,

for $i = 1, \ldots, N$, has a unique global-in-time solution

\[
(\vec c, \vec c^\Sigma)
\in \WW^1_{p,\mathrm{loc}}([0,\infty); \LL_p(\Omega;\R^N) \times \LL_p(\Sigma;\R^N)) \cap \LL_{p,\mathrm{loc}}([0, \infty); \WW_p^2(\Omega;\R^N) \times \WW_p^2(\Sigma;\R^N)).
\]

Moreover, for every $q \in (1, \infty)$,

\[
(\vec c, \vec c^\Sigma)
\in \WW^1_{q,\mathrm{loc}}((0,\infty); \LL_q(\Omega;\R^N) \times \LL_q(\Sigma;\R^N)) \cap \LL_{q,\mathrm{loc}}((0, \infty); \WW_q^2(\Omega;\R^N) \times \WW_q^2(\Sigma;\R^N)).
\]

The result is based on the comparison principle and the dual estimates for scalar reaction-diffusion-advection equations, i.e. a technique heavily used especially by the French school around Michel Pierre, cf. the survey [52] and related papers.
Note that we need regularity of class $\WW_p^{2-2/p}$ for some $p \geq \frac{d+2}{2}$ to get a local-in time solution, which then immediately regularises, so that for any positive times $t > 0$ the solution lies in all spaces of class $\WW_q^{2-2/q}$ for all $q \in (1, \infty)$.

We employ the following strategy to establish the global-in-time existence result, where thanks to the preparations done up to now, we will be able to go through each of these steps at fast pace:

* We start with smooth initial data $(\vec c, \vec c^\Sigma) \in \bigcap_{q>1} (\fs I_q^\Omega \times \fs I_q^\Sigma)$. More general initial data can be handled later on by exploiting the smoothing effect of the parabolic system, cf. Lemma <ref>.

* We fix an arbitrary time horizon $T_0 > 0$ and consider $T^\ast = \min \{T_0, T_\mathrm{max} \}$ for the maximal time of existence $T_\mathrm{max} \in (0,\infty]$ from the local-in-time existence result.
By the unique-continuation property, it suffices to show that the solution extends to $[0,T^\ast]$ for all such $T^\ast > 0$.

* As a preliminary step, we demonstrate that the solution is in $\LL_q(\Omega_{T^\ast};\R^N) \times \LL_q(\Sigma_{T^\ast};\R^N)$ for all $q \in (1,\infty)$.

* In combination with polynomial boundedness of the chemical reaction and surface ad- and desorption models, we may then derive $\LL_p$-growth bounds for the reactive and sorption terms $\vec r^\Omega(\vec c)$, $\vec r^\Sigma(\vec c^\Sigma)$ and $\vec s^\Sigma(\vec c|_\Sigma, \vec c^\Sigma)$.

* By $\LL_q$-maximal regularity of the linearised reaction-diffusion system on the surface, we can then show that the surface part actually lies in the class $\vec c^\Sigma \in \fs E_q^\Sigma(T^\ast)$ for all $q \in (1, \infty)$.

* Only thereafter, we may exploit the $\LL_q$-maximal regularity of the reaction-diffusion-advection system in the bulk phase with inhomogeneous Neumann boundary conditions which arises from linearisation, and obtain that also $\vec c \in \fs E_q^\Omega(T^\ast)$ for all $q \in (1,\infty)$.

* We may then invoke the blow-up criterion from the local-in-time existence result to find that $T^\ast = T_0$ for every choice of $T_0 > 0$, thus $T_\mathrm{max} = \infty$, i.e. global-in-time existence.

Let us, for the moment, assume that $(\vec c, \vec c^\Sigma) \in \fs I_q^\Omega \times \fs I_q^\Sigma$ for all $q \in (1, \infty)$.
To show global existence, i.e. $T_\mathrm{max} = + \infty$, it suffices to show that there is a strong solution on each finite time interval $[0,T_0]$, where $T_0 > 0$ can be chosen arbitrary large.
Thus, let $T_0 > 0$ be arbitrary and consider $T^\ast = \min \{ T_0, T_\mathrm{max} \}$.
Then, by definition of $T_\mathrm{max}$, the strong solution exists on $[0,T^\ast)$, and for each $T \in (0,T^\ast)$, its restriction to times $t \in [0,T]$ is in the class $\fs E_p^\Omega(T) \times \fs E_p^\Sigma(T)$.
Moreover, from the local-in-time existence result for our class of reaction-diffusion-advection-sorption systems, employing the blow-up criterion and Corollary <ref>, we may conclude that

\[
= T_\mathrm{max}
\quad \Leftrightarrow \quad
\limsup_{t \rightarrow T^\ast} \norm{(\vec c, \vec c^\Sigma)(t,\cdot)}_{\WW_q^{2-2/q}(\Omega) \times \WW_q^{2-2/q}(\Sigma)}
= \infty
\quad
\text{for some } q \in (1, \infty)
\]

and, therefore, it remains to prove that $(\vec c, \vec c^\Sigma) \in \BUC([0,T^\ast); \WW_q^{2-2/q}(\Omega)^N \times \WW_q^{2-2/q}(\Sigma)^N)$ for some $q \in (1, \infty)$.
As a preliminary step, note that $(\vec c, \vec c^\Sigma) \in \BUC([0,T^\ast); \LL_q(\Omega;\R^N) \times \LL_q(\Sigma;\R^N))$ for all $q \in (1,\infty)$.
In fact, by Lemma <ref> and Lemma <ref>, for every $p \in (1,\infty]$ there is a constant $C_{p,T^\ast} > 0$ such that

\[
\norm{\vec c}_{\LL_p(\Omega_\tau)} + \norm{\vec c|_\Sigma}_{\LL_p(\Sigma_\tau)} + \norm{\vec c^\Sigma}_{\LL_p(\Sigma_\tau)}
\leq C_{p,T^\ast}
\quad
\text{for every } \tau \in (0,T^\ast).
\]

By the polynomial boundedness of the chemical reaction rates and Remark <ref>, thus

≤M^Ω(1 + c⃗__pγ^Ω(Ω_τ))^γ^Ω
≤C_pγ^Ω, T^∗,

≤M^Ω(1 + c⃗^Σ__pγ^Ω(Σ_τ))^γ^Σ

τ∈(0, T^∗).

Moreover, by the partial lower and upper linear bounds on the sorption model,

\[
\norm{s_i^\Sigma(c_i|_\Sigma, c_i^\Sigma)}_{\LL_p(\Sigma_\tau)}
\leq k_i^\mathrm{ad} \norm{c_i|_\Sigma}_{\LL_p(\Sigma_\tau)} + k_i^\mathrm{de} \norm{c_i^\Sigma}_{\LL_p(\Sigma_\tau)}
\leq C_{p,T^\ast},
\quad
\tau \in (0, T^\ast).
\]

By $\LL_p$-maximal regularity of the linearised reaction-diffusion-system on the surface

\[
\begin{cases}
\partial_t c_i^\Sigma - d_i \Delta_\Sigma c_i^\Sigma
= F_i^\Sigma + G_i^\Sigma
&\text{on } \Sigma_\tau,
\\
= c_i^{\Sigma,0}
&\text{on } \Sigma
\end{cases}
\]

for $F_i^\Sigma := r_i^\Sigma(\vec c^\Sigma)$ and $G_i^\Sigma := s_i^\Sigma(c_i|_\Sigma, c_i^\Sigma) \in \LL_q(\Sigma_\tau)$, we may conclude that

\[
\norm{c_i^\Sigma}_{\WW_{q}^{(1,2)}(\Omega_\tau)}
\leq C \big( \norm{c_i^{\Sigma,0}}_{\WW_q^{2-2/q}(\Sigma)} + \norm{F_i^\Sigma + G_i^\Sigma}_{\LL_q(\Sigma_\tau)} \big)
\leq C_{q,T^\ast}
\quad
\text{for all } \tau \in (0, T^\ast).
\]

Hence, $c_i^\Sigma \in \WW_q^{(1,2)}(\Sigma_{T^\ast})$ for all $q \in (1, \infty)$ and $i = 1, \ldots, N$, which, in particular, already implies that

\[
\sup_{t \in [0,T^\ast)} \norm{c_i^\Sigma(t,\cdot)}_{\WW_q^{2-2/q}(\Sigma)}
< \infty,
\]

as $\WW_q^{(1,2)}(\Sigma_{T^\ast}) \hookrightarrow \BUC([0,T^\ast); \WW_q^{2-2/q}(\Omega))$ embeds continuously.
With this knowledge at hand, let us take a look on the reaction-diffusion-advection-system in the bulk phase

\[
\begin{cases}
\partial_t c_i + \vec v \cdot \nabla c_i - d_i \Delta c_i
= F_i
&\text{in } \Omega_{T^\ast},
\\
k_i^\mathrm{ad} (1 - \frac{c_i^{\Sigma,+}}{c_i^{\Sigma,\infty}})^+ c_i + d_i \partial_{\vec \nu} c_i
= G_i
&\text{on } \Sigma_{T^\ast},
\\
= c_i^0
&\text{on } \overline{\Omega},
\end{cases}
\]

where $F_i := r_i^\Omega(\vec c)$ and $G_i := k_i^\mathrm{ad} c_i^\Sigma$.
Since $\beta_i := k_i^\mathrm{ad} (1 - \frac{c_i^{\Sigma,+}}{c_i^{\Sigma,\infty}})^+ \in \bigcap_{q > 1} \WW_q^1(0,T^\ast;\LL_q(\Sigma)) \cap \LL_q(0,T^\ast;\WW_\infty^1(\Sigma))$ (as we already know that $c_i^\Sigma \in \WW_{q}^{(1,2)}(\Sigma_{T^\ast})$ for all $q \in (1,\infty)$), by [30] this problem admits $\LL_q$-maximal regularity for all $q \in (1,\infty)$:
Indeed, to satisfy condition (SB) in [30], we need to ensure that there are $s_{i0}, r_{i0} \geq p$ with $\frac{2}{s_{i0}} + \frac{d-1}{r_{i0}} < 1 - \frac{1}{p} - 1 = - \frac{1}{p}$ such that $\beta_i \in \WW_{s_{i0}}^{\frac{1}{2} - \frac{1}{2p}}((0,T^\ast); \LL_{r_{i0}}(\Sigma)) \cap \LL_{s_{i0}}((0,T^\ast); \WW_{r_{i0}}^{1 - \frac{1}{p}}(\Sigma))$, which in our case is obviously true.
Therefore, $c_i \in \WW_q^{(1,2)}(\Omega_{T^\ast})$ with

\[
\norm{c_i}_{\WW_q^{(1,2)}(\Omega_{T^\ast})}
\leq C \big( \norm{F_i}_{\LL_q(\Omega_{T^\ast})} + \norm{c_i^0}_{\WW_q^{2-2/q}(\Omega)} + \norm{c_i^\Sigma}_{\WW_q^{(1,2) \cdot (\frac{1}{2} - \frac{1}{2q})}(\Sigma_{T^\ast})} \big)
\leq C_{p,T^\ast},
\quad q > 1.
\]

This, in particular, implies that

\[
(c_i, c_i^\Sigma)
\in \BUC([0,T^\ast);\WW_q^{2-2/q}(\Omega) \times \WW_q^{2-2/q}(\Sigma)).
\]

Hence, by the blow-up criterion, $T^\ast = T_0 < T_\mathrm{max}$ for each choice of $T_0 > 0$, leading to $T_\mathrm{max} = \infty$.
For more general initial data note that, by Lemma <ref>, $(\vec c(\varepsilon,\cdot), \vec c^\Sigma(\varepsilon,\cdot)) \in \bigcap_{q \in (1, \infty)} \fs I_q^\Omega \times \fs I_q^\Sigma$.
Thus we may simply consider the problem for initial data at time $t = \varepsilon > 0$.

§ FUNDING

The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 265191195, SFB 1194.

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