Here, $\mathcal{W}$ is the **Weingarten map**. It is linear in $u$ and its coefficients depends linearly on the first derivatives of the outward unit normal $\nu$, requiring then some smoothness on the boundary $\partial\Omega$. With the flat boundary $\partial\mathbb{R}_+^n$, the outward normal $\nu = -\epsilon_n$ is constant, which explains why the map $\mathcal{W}$ vanishes.

We can then exhibit the following maximal regularity result, where we identify $\Lambda^1$ with $\mathbb{C}^n$. Similar results such as Theorems 3.17 and 3.18 are also available.

**Theorem 3.22** Let $p \in (1, +\infty)$, $q \in [1, +\infty)$, $s \in (-1+1/p, 1/p+2/q)$, such that $s, s+2-2/q \notin \mathbb{N}+\frac{1}{p}$ and $(C_{s+2-2/q,p,q})$ is satisfied and let $T \in (0, +\infty]$.

For any $f \in L^q((0,T), \dot{\mathbf{B}}_{p,q,H_t}^{s, \mathcal{H}_t}(\mathbb{R}_+^n, \mathbb{C}^n))$, $u_0 \in \dot{\mathbf{B}}_{p,q,H_t}^{2+s-\frac{2}{q}, \sigma}(\mathbb{R}_+^n, \mathbb{C}^n)$, there exists a unique mild solution $(u, \nabla \mathbf{p}) \in C_b^0([0,T], \dot{\mathbf{B}}_{p,q,H_t}^{2+s-\frac{2}{q}, \sigma}(\mathbb{R}_+^n, \mathbb{C}^n)) \times L^q((0,T), \dot{\mathbf{B}}_{p,q}^s(\mathbb{R}_+^n, \mathbb{C}^n))$ of

$$ 
\left\{ 
\begin{array}{ll}
\partial_t u - \Delta u + \nabla \mathbf{p} = f, & \text{on } (0,T) \times \mathbb{R}_+^n, \\
\text{div } u = 0, & \text{on } (0,T) \times \mathbb{R}_+^n, \\
[(t\nabla u + \nabla u)\nu]_{\text{tan}|_{\partial\mathbb{R}_+^n}} = 0, & \text{on } (0,T) \times \partial\mathbb{R}_+^n, \qquad \text{(NSS)} \\
\nu \cdot u|_{\partial\mathbb{R}_+^n} = 0, & \text{on } (0,T) \times \partial\mathbb{R}_+^n, \\
u(0) = u_0, & \text{in } \dot{\mathbf{B}}_{p,q}^{2+s-\frac{2}{q}}(\mathbb{R}_+^n, \mathbb{C}^n),
\end{array}
\right. 
$$

with estimate

$$ \|u\|_{L^{\infty}((0,T), \dot{\mathbf{B}}_{p,q}^{2+s-\frac{2}{q}})} + \|(\partial_t u, \nabla^2 u, \nabla \mathbf{p})\|_{L^q((0,T), \dot{\mathbf{B}}_{p,q}^s)} \lesssim_{p,q,n} \|f\|_{L^q((0,T), \dot{\mathbf{B}}_{p,q}^s)} + \|u_0\|_{\dot{\mathbf{B}}_{p,q}^{2+s-\frac{2}{q}}} $$

In the case $q = +\infty$, if we assume in addition $u_0 \in \dot{\mathbf{B}}_p^s(A_{H,t}^2)$, we have

$$ \|(\partial_t u, \nabla^2 u, \nabla \mathbf{p})\|_{L^{\infty}((0,T), \dot{\mathbf{B}}_{p,\infty}^s)} \lesssim_{p,s,n} \|f\|_{L^{\infty}((0,T), \dot{\mathbf{B}}_{p,\infty}^s)} + \|A_{H,t} u_0\|_{\dot{\mathbf{B}}_{p,\infty}^s} $$

## A Appendix: Partial traces of differential forms

We state here a trace theorem for generalized normal and tangential traces of differential forms. The general case for vector fields in inhomogeneous function spaces is well-known, also is the differential form in the inhomogeneous case on bounded Lipschitz domains on Riemannian manifolds, see [MMS08, Section 4] and references therein.

We recall that $\nu = -\epsilon_n$ is the outer normal unit at boundary $\partial\mathbb{R}_+^n$.

**Theorem A.1** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ and let $k \in [0, n]$.

(i) For all $u \in D_p^s(\delta, \mathbb{R}_+^n, \Lambda^k)$. Then there exists a unique function $\nu \sqcup u|_{\partial\mathbb{R}_+^n} \in B_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k-1})$ called the generalized tangential trace, such that

$$ \int_{\mathbb{R}^{n-1}} \langle \nu \sqcup u|_{\partial \mathbb{R}_+^n}, \Psi|_{\partial \mathbb{R}_+^n} \rangle = \int_{\mathbb{R}_+^n} \langle u, d\Psi \rangle - \int_{\mathbb{R}_+^n} \langle \delta u, \Psi \rangle \quad (A.1) $$

for all $\Psi \in H^{1-s,p'}(\mathbb{R}_+^n, \Lambda)$, with estimates

$$ \|\nu \sqcup u|_{\partial \mathbb{R}_+^n}\|_{B_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{H^{s,p}(\mathbb{R}_+^n)} + \|\delta u\|_{H^{s,p}(\mathbb{R}_+^n)}. $$

The same result holds with corresponding estimate, for $u \in D_p^s(d, \mathbb{R}_+^n, \Lambda^k)$ we have a partial trace $\nu \wedge u|_{\partial\mathbb{R}_+^n} \in B_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k+1})$ called the generalized normal trace, satisfying the identity

$$ \int_{\mathbb{R}^{n-1}} \langle \nu \wedge u|_{\partial \mathbb{R}_+^n}, \Psi|_{\partial \mathbb{R}_+^n} \rangle = \int_{\mathbb{R}_+^n} \langle du, \Psi \rangle - \int_{\mathbb{R}_+^n} \langle u, \delta \Psi \rangle. \quad (A.2) $$
(ii) For all $u \in D_{p,q}^s(\delta, \mathbb{R}_+^n, \Lambda^k)$ we have $\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \in B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k-1})$, such that formula_3.1 holds for all $\Psi \in B_{p',q'}^{1-s}(\mathbb{R}_+^n, \Lambda)$. Moreover, we have estimates

$$ \|\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \|_{B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{B_{p,q}^s(\mathbb{R}_+^n)} + \|\delta u\|_{B_{p,q}^s(\mathbb{R}_+^n)}. $$

The same results holds with corresponding estimate, for $u \in D_{p,q}^s(d, \mathbb{R}_+^n, \Lambda^k)$ we have a partial trace $\nu \wedge u|_{\partial \mathbb{R}_+^n} \in B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k+1})$ such that (A.2) is satisfied.

(iii) For all $u \in B_{p,q}^{s+1}(\mathbb{R}_+^n, \Lambda^k)$, we have

$$ (\nu \lrcorner u \oplus \nu \wedge u)|_{\partial \mathbb{R}_+^n} \in B_{p,q}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k-1} \oplus \Lambda^{k+1}) $$
with estimate

$$ \|(\nu \lrcorner u \oplus \nu \wedge u)|_{\partial \mathbb{R}_+^n} \|_{B_{p,q}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{B_{p,q}^{s+1}(\mathbb{R}_+^n)}, $$

and everything still hold with $H^{s+1,p}$ instead of $B_{p,q}^{s+1}$, when $q = p$.

**Theorem A.2** Let $p \in (1, +\infty)$, $q \in [1, +\infty]$, $s \in (-1 + \frac{1}{p}, \frac{1}{p})$ and let $k \in [0, n]$.

(i) For all $u \in \dot{D}_p^s(\delta, \mathbb{R}_+^n, \Lambda^k)$,

* If $s \le 0$, then there exists a unique function $\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \in B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k-1})$ such that the formula_3.2 holds for all $\Psi \in H^{1-s, p'}(\mathbb{R}_+^n, \Lambda)$, with estimate

$$ \|\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \|_{B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

* If $s > 0$, for $\frac{1}{r} = \frac{1}{p} - \frac{s}{n} \in (\frac{n-1}{pn}, \frac{1}{p})$, there exists a unique function $\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \in B_{r,r'}^{-\frac{1}{r}}(\mathbb{R}^{n-1}, \Lambda^{k-1})$, such that the formula_3.3 holds for all $\Psi \in H^{1, r'}(\mathbb{R}_+^n, \Lambda^{k-1})$ with estimate

$$ \|\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \|_{B_{r,r'}^{-\frac{1}{r}}(\mathbb{R}^{n-1})} \lesssim_{r,p,s,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}. $$

The same result, up to appropriate changes, still holds for $u \in \dot{D}_p^s(d, \mathbb{R}_+^n, \Lambda^k)$ with partial trace $\nu \wedge u|_{\partial \mathbb{R}_+^n}$ satisfying the formula (A.2).

(ii) For all $u \in \dot{D}_{p,q}^s(\delta, \mathbb{R}_+^n, \Lambda^k)$,

* If $s < 0$, there exists a unique $\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \in B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k-1})$ such that the formula_3.4 holds for all $\Psi \in [S \cap B_{p',q'}^{1-s}](\mathbb{R}_+^n, \Lambda^{k-1} \oplus \Lambda^{k+1})$, with estimates

$$ \|\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \|_{B_{p,q}^{s-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. $$

* If $s > 0$, for $\frac{1}{r} = \frac{1}{p} - \frac{s}{n} \in (\frac{n-1}{pn}, \frac{1}{p})$, there exists a unique $\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \in B_{\bar{r},q}^{-\frac{1}{r}-\varepsilon}(\mathbb{R}^{n-1}, \Lambda^{k-1})$, for any sufficiently small $\varepsilon > 0$, with $\frac{1}{r} - \frac{\varepsilon}{n} = \frac{1}{\bar{r}}$, such that the formula_3.5 holds for all $\Psi \in [S \cap B_{\bar{r},q'}^{1+\varepsilon}](\mathbb{R}_+^n, \Lambda)$ with estimate

$$ \|\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \|_{B_{\bar{r},q}^{-\frac{1}{r}-\varepsilon}(\mathbb{R}^{n-1})} \lesssim_{p,s,n,\varepsilon} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}. $$

* If $s = 0$, there exists a unique $\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \in B_{r,q}^{-\frac{1}{r}-\varepsilon}(\mathbb{R}^{n-1}, \Lambda^{k-1})$, where $\frac{1}{r} = \frac{1}{p} - \frac{\varepsilon}{n}$, for any sufficiently small $\varepsilon > 0$, such that the formula_3.6 holds for all $\Psi \in [S \cap B_{r',q'}^{1+\varepsilon}](\mathbb{R}_+^n, \Lambda)$, with estimates

$$ \|\nu \lrcorner u|_{\partial \mathbb{R}_+^n} \|_{B_{r,q}^{-\frac{1}{r}-\varepsilon}(\mathbb{R}^{n-1})} \lesssim_{p,s,n,\varepsilon} \|u\|_{\dot{B}_{p,q}^0(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{B}_{p,q}^0(\mathbb{R}_+^n)}. $$
The same result, up to appropriate changes, still holds for $u \in \dot{D}_{p,q}^s(d, \mathbb{R}_+^n, \Lambda^k)$ with partial trace $\nu \wedge u|_{\partial \mathbb{R}_+^n}$ satisfying the formula (A.2).

(iii) For all $u \in [\dot{B}_{p,q}^s \cap \dot{B}_{p,q}^{s+1}](\mathbb{R}_+^n, \Lambda^k)$, if $q \neq +\infty$, we have

$$ (\nu \lrcorner u \oplus \nu \wedge u)|_{\partial \mathbb{R}_+^n} \in B_{p,q}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k-1} \oplus \Lambda^{k+1}) $$

with estimate

$$ \| (\nu \lrcorner u \oplus \nu \wedge u)|_{\partial \mathbb{R}_+^n} \|_{B_{p,q}^{s+1-\frac{1}{p}}(\mathbb{R}^{n-1})} \lesssim_{p,s,n} \|u\|_{[\dot{B}_{p,q}^s \cap \dot{B}_{p,q}^{s+1}](\mathbb{R}_+^n)} $$

and everything still hold with $(\dot{H}^{s,p}, \dot{H}^{s+1,p})$ instead of $(\dot{B}_{p,q}^s, \dot{B}_{p,q}^{s+1})$, when $q = p$.

If $q = +\infty$, we have

$$ (\nu \lrcorner u \oplus \nu \wedge u)|_{\partial \mathbb{R}_+^n} \in L^p(\mathbb{R}^{n-1}, \Lambda^{k-1} \oplus \Lambda^{k+1}) $$

with corresponding estimate.

**Remark A.3** The proof of Theorem A.1 in case of inhomogeneous function spaces follows straightforward the same proof provided for corresponding results in [MMS08, Section 4] and is somewhat sharp.

Notice that Theorem A.2 is certainly not sharp, and investigation of sharp range for partial traces could be of great interest in the treatment of inhomogeneous boundary value problems in homogeneous function spaces.

*Proof.* — Without loss of generality we only investigate the case of tangential traces $\nu \lrcorner u|_{\partial \mathbb{R}_+^n}$.

**Step 1.1:** Proof of (i), for $s \le 0$. For $u \in \dot{D}_p^s(\delta, \mathbb{R}_+^n, \Lambda^k)$, for all $\psi \in B_{p',p'}^{\frac{1-s}{p},p'}(\mathbb{R}^{n-1}, \Lambda^{k-1})$, and $\Psi \in H^{1-s,p'}(\mathbb{R}_+^n, \Lambda^{k-1})$ such that $\Psi|_{\partial \mathbb{R}_+^n} = \psi$, we can define the following functional,

$$ \kappa_u(\Psi) := \int_{\mathbb{R}_+^n} \langle u, d\Psi \rangle - \int_{\mathbb{R}_+^n} \langle \delta u, \Psi \rangle. $$

First, the map $(u, \Psi) \mapsto \kappa_u(\Psi)$ is well defined and bilinear on $\dot{D}_p^s(\delta, \mathbb{R}_+^n, \Lambda^k) \times H^{1-s,p'}(\mathbb{R}_+^n, \Lambda^{k-1})$, i.e. in particular only depends on the boundary value $\psi$ of $\Psi$. It is straightforward from duality that,

$$ \begin{aligned} |\kappa_u(\Psi)| & \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|d\Psi\|_{\dot{H}^{-s,p'}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|\Psi\|_{\dot{H}^{-s,p'}(\mathbb{R}_+^n)} \\ & \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|\Psi\|_{\dot{H}^{-1-s,p'}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|\Psi\|_{\dot{H}^{-1-s,p'}(\mathbb{R}_+^n)} \\ & \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|\Psi\|_{H^{-1-s,p'}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|\Psi\|_{H^{-1-s,p'}(\mathbb{R}_+^n)} \\ & \lesssim_{s,p,n} (\|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}) \|\Psi\|_{H^{1-s,p'}(\mathbb{R}_+^n)}, \end{aligned} $$

where above inequalities follows from $H^{-s,p'}(\mathbb{R}_+^n) \hookrightarrow \dot{H}^{-s,p'}(\mathbb{R}_+^n)$, $H^{1-s,p'}(\mathbb{R}_+^n) \hookrightarrow \dot{H}^{1-s,p'}(\mathbb{R}_+^n)$, since $-1 + \frac{1}{p} < s \le 0$, then from $H^{1-s,p'}(\mathbb{R}_+^n) \hookrightarrow H^{-s,p'}(\mathbb{R}_+^n)$.

Now, if we have $\Psi_1, \Psi_2 \in H^{1-s,p'}(\mathbb{R}_+^n, \Lambda^{k-1})$ such that $\Psi_1|_{\partial \mathbb{R}_+^n} = \Psi_2|_{\partial \mathbb{R}_+^n} = \psi$, we introduce $\Psi_0 = \Psi_1 - \Psi_2 \in H_0^{1-s,p'}(\mathbb{R}_+^n, \Lambda^{k-1})$. Therefore, let's consider $(\Phi_k)_{k \in \mathbb{N}} \subset C_c^\infty(\mathbb{R}_+^n, \Lambda^{k-1})$ such that,

$$ \Phi_k \xrightarrow{k \to +\infty} \Psi_0 \text{ in } H_0^{1-s,p'}(\mathbb{R}_+^n, \Lambda^{k-1}). $$

We can deduce,

$$ \kappa_u(\Psi_1) - \kappa_u(\Psi_2) = \kappa_u(\Psi_0) = \lim_{k \to +\infty} \left[ \int_{\mathbb{R}_+^n} \langle u, d\Phi_k \rangle - \int_{\mathbb{R}_+^n} \langle \delta u, \Phi_k \rangle \right] = 0. $$

Thus, the following equality occurs, where we also consider the extension operator $\text{Ext}_{\mathbb{R}_+^n}$ from
[Gau22, Theorem 3.1],

$$
\tilde{\kappa}_u(\psi) := \kappa_u(\mathrm{Ext}_{\mathbb{R}_+^n} \otimes \psi) = \kappa_u(\Psi),
$$

and with estimate, also obtained from [Gau22, Theorem 3.1],

$$
|\tilde{\kappa}_u(\psi)| \lesssim_{s,p,n} (\|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|du\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}) \|\psi\|_{B_{p',p'}^{\frac{1-s}{p}}(\mathbb{R}^{n-1})}.
$$

By duality, there exists a unique function depending linearly on $u$, $\nu \rfloor u|_{\partial \mathbb{R}_+^n} \in B_{p,p}^{s-\frac{1}{p}}(\mathbb{R}^{n-1}, \Lambda^{k-1})$,
such that (A.1) holds.

To guarantee that the representation formula makes sense, one may use the usual integration by parts formula with $u, \Psi \in S(\mathbb{R}_+^n, \Lambda)$.

**Step 1.2**: Proof of (i), for $s > 0$. For same assumption on $u$, as before $\Psi$, everything works similarly except the way we bounded bilinearly $(u, \Psi) \mapsto \kappa_u(\Psi)$ on $\dot{B}_p^s(\delta, \mathbb{R}_+^n, \Lambda^k) \times H^{1-s, p'}(\mathbb{R}_+^n, \Lambda^{k-1})$. For $r \in (1, +\infty)$ such that $\frac{1}{r} = \frac{1}{p} - \frac{s}{n}$, we deduce from Sobolev embeddings and duality that

$$
\begin{align*}
|\kappa_u(\Psi)| & \lesssim_{s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|d\Psi\|_{\dot{H}^{-s,p'}(\mathbb{R}_+^n)} + \|\delta u\|_{L^r(\mathbb{R}_+^n)} \|\Psi\|_{L^{r'}(\mathbb{R}_+^n)} \\
& \lesssim_{r,s,p,n} \|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|\Psi\|_{H^{1,r'}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} \|\Psi\|_{H^{1,r'}(\mathbb{R}_+^n)} \\
& \lesssim_{r,s,p,n} (\|u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{H}^{s,p}(\mathbb{R}_+^n)}) \|\Psi\|_{H^{1,r'}(\mathbb{R}_+^n)}.
\end{align*}
$$

Thus everything goes similarly.

**Step 2.1**: Proof of (ii), for $s < 0$, is very similar to the one of above **Step 1.1**.

**Step 2.2**: Proof of (ii), for $s > 0$, is somewhat similar to the one of **Step 1.2** but needs further
explanations. We use Sobolev embeddings, and generalized Hölder inequalities using Lorentz spaces,
$\dot{B}_{p,q}^s(\mathbb{R}_+^n) \hookrightarrow L^{r,q}(\mathbb{R}_+^n)$, $B_{r',q'}^\varepsilon(\mathbb{R}_+^n) \hookrightarrow L^{r',q'}(\mathbb{R}_+^n) \hookrightarrow \dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n)$, for $\varepsilon > 0$,

$$
\begin{align*}
|\kappa_u(\Psi)| & \lesssim_{s,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} \|d\Psi\|_{\dot{B}_{p',q'}^{-s}(\mathbb{R}_+^n)} + \|\delta u\|_{L^{r,q}(\mathbb{R}_+^n)} \|\Psi\|_{L^{r',q'}(\mathbb{R}_+^n)} \\
& \lesssim_{r,s,p,n} \|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} \|d\Psi\|_{L^{r',q'}(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} \|\Psi\|_{L^{r',q'}(\mathbb{R}_+^n)} \\
& \lesssim_{r,s,p,n} (\|u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)} + \|\delta u\|_{\dot{B}_{p,q}^s(\mathbb{R}_+^n)}) \|\Psi\|_{B_{r',q'}^{1+\varepsilon}(\mathbb{R}_+^n)}.
\end{align*}
$$

**Step 2.3**: Proof of (ii), for $s = 0$ is shown via similar Sobolev embeddings arguments and is left to the reader.

**Step 3**: Proof of (iii), follows from [Gau22, Proposition 3.8], with explicit representation formula for any suitable $k$-differential forms $u$:

$$
\begin{align*}
\nu \rfloor u|_{\partial \mathbb{R}_+^n} = -\epsilon_n \rfloor u|_{\partial \mathbb{R}_+^n} &= (-1)^k \sum_{1 \le \ell_1 < \dots < \ell_{k-1} < n} u_{\ell_1 \ell_2 \dots \ell_{k-1} n}(\cdot, 0) \, dx_{\ell_1} \wedge \dots \wedge dx_{\ell_{k-1}} \\
&= (-1)^k \sum_{I' \in \mathcal{I}_{n-1}^{k-1}} u_{I', n}(\cdot, 0) \, dx_{I'}.
\end{align*}
$$

A similar treatment yields the same conclusion for the boundary term $\nu \wedge u|_{\partial \mathbb{R}_+^n}$, so that one ends
the proof here.
$\blacksquare$

**Remark A.4** Let's make further comment about estimates used in above proof of Theorem A.2,
in particular the ones used in **Step 2.2**.

We recall that from Sobolev embeddings, see [Gau22, Proposition 2.31], for $0 < s_0 < s < s_1 < 1/p$, $r_0, r_1, p \in (1, +\infty)$, $1/r_j = 1/p - s_j/n$, and appropriate $q_j \in [1, +\infty]$ we have

$$
\dot{B}_{p,q_j}^{s_j}(\mathbb{R}_+^n) \hookrightarrow L^{r_j}(\mathbb{R}_+^n), \quad j \in \{0,1\}.
$$

If $(s, 1/r) = (1 - \theta)(s_0, 1/r_0) + \theta(s_1, 1/r_1)$, by real interpolation, for $q \in [1, +\infty]$ we obtain,

$$
\dot{B}_{p,q}^{s}(\mathbb{R}_{+}^{n}) = (\dot{B}_{p,q_0}^{s_0}(\mathbb{R}_{+}^{n}), \dot{B}_{p,q_1}^{s_1}(\mathbb{R}_{+}^{n}))_{\theta,q} \hookrightarrow (L^{r_0}(\mathbb{R}_{+}^{n}), L^{r_1}(\mathbb{R}_{+}^{n}))_{\theta,q} = L^{r,q}(\mathbb{R}_{+}^{n}).
$$

And one may proceed similarly for the reverse embedding,

$$
L^{r', q'}(\mathbb{R}_+^n) \hookrightarrow \dot{B}_{p', q'}^{-s}(\mathbb{R}_+^n).
$$
For more details about Lorentz spaces and their interpolation, one could consult [Lun18, Section 1, Examples 1.10, 1.11 & 1.27] and [BL76, Chapter 5, Section 5.3].

## References

[ABHN11] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander. *Vector-valued Laplace transforms and Cauchy problems*, volume 96 of *Monographs in Mathematics*. Birkhäuser/Springer Basel AG, Basel, 2nd edition, 2011.

[ACDH04] P. Auscher, T. Coulhon, X.T. Duong, and S. Hofmann. Riesz transform on manifolds and heat kernel regularity. *Ann. Scient. Éc. Norm. Sup.*, Ser. 4, 37(6):911–957, 2004.

[AM04] A. Axelsson and A. McIntosh. Hodge decompositions on weakly Lipschitz domains. In *Advances in analysis and geometry*, Trends Math., pages 3–29. Birkhäuser, Basel, 2004.

[Ama95] H. Amann. *Linear and quasilinear parabolic problems, Vol. I: Abstract linear theory*, volume 89 of *Monographs in Mathematics*. Birkhäuser Verlag Basel, 1995.

[BCD11] H. Bahouri, J.-Y. Chemin, and R. Danchin. *Fourier analysis and nonlinear partial differential equations*, volume 343 of *Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]*. Springer, Heidelberg, 2011.

[BL76] J. Bergh and J. Löfström. *Interpolation spaces. An introduction*. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.

[Bog86] M.E. Bogovskiǐ. Decomposition of $L_p(\Omega, \mathbf{R}^n)$ into the direct sum of subspaces of solenoidal and potential vector fields. *Soviet Math. Dokl.*, 33(1):161–165, 03 1986.

[Che99] J.-Y. Chemin. Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. *J. Anal. Math.*, 77(1):27–50, 1999.

[CM10] M. Costabel and A. McIntosh. On Bogovskiǐ and regularized Poincaré integral operators for the Rham complexes on Lipschitz domains. *Math. Z.*, 265:297–320, 2010.

[DC94] M. Do Carmo. *Differential Forms and Applications*. Universitext. Springer Berlin Heidelberg, Berlin, Heidelberg, 1994. Original Portuguese edition published by IMPA, 1971.

[Den22] C. Denis. Existence and uniqueness in critical spaces for the magnetohydrodynamical system in $\mathbb{R}^n$. Preprint HAL-03660738v2, June 2022.

[DHMT21] R. Danchin, M. Hieber, P.B. Mucha, and P. Tolksdorf. Free Boundary Problems via Da Prato-Grisvard Theory. *arXiv e-prints*, November 2021. arXiv:2011.07918v2.

[DHP03] R Denk, M Hieber, and J Püss. R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type. *Mem. Amer. Math. Soc.*, 166(788):xvii+114, 2003.

[DM09] R. Danchin and P. B. Mucha. A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space. *J. Funct. Anal.*, 256(3):881–927, 2009.

[DM15] R. Danchin and P. B. Mucha. Critical functional framework and maximal regularity in action on systems of incompressible flows. *Mémoires de la Société Mathématique de France*, 143, 2015.

[DPG75] G. Da Prato and P. Grisvard. Sommes d'opérateurs linéaires et équations différentielles opérationnelles. *J. Math. Pures Appl. (9)*, 54(3):305–387, 1975.

[Ege15] M. Egert. *On Kato's conjecture and mixed boundary conditions*. Sierke Verlag, Göttingen, 2015. PhD Thesis, Technischen Universität Darmstadt.

[FKS05] R. Farwig, H. Kozono, and H. Sohr. An $L^q$ approach to Stokes and Navier-Stokes in general domains. *Acta Math.*, 195:21–53, 2005.

[FKS07] R. Farwig, H. Kozono, and H. Sohr. On the Helmholtz decomposition in general unbounded domains. *Arch. Math.*, 88:239–248, 01 2007.

[FMM98] E. Fabes, O. Mendez, and M. Mitrea. Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains. *J. Funct. Anal.*, 159(2):323–368, 1998.

[FY07] H. Fujiwara and M. Yamazaki. The Helmholtz decomposition in Sobolev and Besov spaces. In *Asymptotic analysis and singularities—hyperbolic and dispersive PDEs and fluid mechanics*, volume 47 of *Adv. Stud. Pure Math.*, pages 99–116. Math. Soc. Japan, Tokyo, 2007.

[Gal11] G.P. Galdi. *An Introduction to the Mathematical Theory of the Navier-Stokes Equations*. Springer Monographs in Mathematics. Springer New York, NY, 2nd edition, 2011.

[Gau22] A. Gaudin. On homogeneous Sobolev and Besov spaces on the whole and the half space. *arXiv e-prints*, page arXiv:2211.07707, November 2022.

[Gau23] A. Gaudin. Homogeneous Sobolev global-in-time maximal regularity and related trace estimates. *arXiv e-prints*, page arXiv:2302.09862, February 2023.

[GHT13] M. Geissert, H. Heck, and C. Trunk. $H^\infty$-calculus for a system of Laplace operators with mixed order boundary conditions. *Discrete Contin. Dyn. Syst. Ser. S*, 6(5):1259–1275, 2013.
[GS91] Y. Giga and H. Sohr. Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. *J. Funct. Anal.*, 102(1):72–94, 1991.

[Gui91a] D. Guidetti. On elliptic problems in Besov spaces. *Math. Nachr.*, 152:247–275, 1991.

[Gui91b] D. Guidetti. On interpolation with boundary conditions. *Math. Z.*, 207(3):439–460, 1991.

[Haa06] M. Haase. *The functional calculus for sectorial operators*, volume 169 of *Operator Theory: Advances and Applications*. Birkhäuser Verlag, Basel, 2006.

[Hie20] M. Hieber. *Analysis of Viscous Fluid Flows: An Approach by Evolution Equations*, volume 2254 of *Lecture Notes in Mathematics*, pages 1–146. Springer, Cham, 2020.

[HM13] M. Hieber and S. Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. *Discrete Contin. Dyn. Syst.*, 33(11-12):5143–5151, 2013.

[HMM11] S. Hofmann, M. Mitrea, and S. Monniaux. Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds. *Ann. Inst. Fourier (Grenoble)*, 61(4):1323–1349 (2012), 2011.

[HNPS16] M. Hieber, M. Nesensohn, J. Prüss, and K. Schade. Dynamics of nematic liquid crystal flows: The quasilinear approach. *Ann. I. H. Poincaré - AN*, 33(2):397–408, 2016.

[Jos11] J. Jost. *Riemannian Geometry and Geometric Analysis*. Universitext. Springer Berlin Heidelberg, Berlin, Heidelberg, 6th edition, 2011.

[KW04] P. C. Kunstmann and L. Weis. *Maximal $L^p$-regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$-functional Calculus*, volume 1855 of *Lecture Notes in Mathematics*, pages 65–311. Springer Berlin Heidelberg, 2004.

[Lun18] A. Lunardi. *Interpolation theory*. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 3rd edition, 2018.

[McI86] A. McIntosh. Operators which have an $H_\infty$ functional calculus. In *Miniconference on operator theory and partial differential equations (North Ryde, 1986)*, volume 14 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 210–231. Austral. Nat. Univ., Canberra, 1986.

[MM08] M. Mitrea and S. Monniaux. The regularity of the Stokes operator and the Fujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains. *J. Funct. Anal.*, 254(6):1522–1574, 2008.

[MM09a] M. Mitrea and S. Monniaux. The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains. *Differential Integral Equations*, 22(3-4):339–356, 2009.

[MM09b] M. Mitrea and S. Monniaux. On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds. *Trans. Amer. Math. Soc.*, 361(6):3125–3157, 2009.

[MM18] A. McIntosh and S. Monniaux. Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in $L^p$ spaces on Lipschitz domains. *Rev. Mat. Iberoam.*, 34(4):1711–1753, 2018.

[MMS08] D. Mitrea, M. Mitrea, and M.-C. Shaw. Traces of Differential Forms on Lipschitz Domains, the Boundary De Rham Complex, and Hodge Decompositions. *Indiana Univ. Math. J.*, 57(5):2061–2095, 2008.

[Mon13] S. Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. *Discrete Contin. Dyn. Syst. Ser. S*, 6(5):1355–1369, 2013.

[Mon21] S. Monniaux. Existence in critical spaces for the magnetohydrodynamical system in 3D bounded Lipschitz domains. *J. Elliptic Parabol. Equ.*, 7:311–322, 2021.

[MS18] S. Monniaux and Z. Shen. Stokes problems in irregular domains with various boundary conditions. In *Handbook of mathematical analysis in mechanics of viscous fluids*, pages 207–248. Springer, Cham, 2018.

[OS16] T. Ogawa and S. Shimizu. End-point maximal $L^1$-regularity for the cauchy problem to a parabolic equation with variable coefficients. *Math. Ann.*, 365(1):661–705, 2016.

[OS21] T. Ogawa and S. Shimizu. Global well-posedness for the incompressible Navier-Stokes equations in the critical Besov space under the Lagrangian coordinates. *J. Differential Equations*, 274:613–651, 2021.

[OS22] T. Ogawa and S. Shimizu. Maximal $L^1$-regularity for parabolic initial-boundary value problems with inhomogeneous data. *J. Evol. Equ.*, 22(30):67, 2022.

[Ouh05] E. M. Ouhabaz. *Analysis of heat equations on domains*, volume 31 of *London Mathematical Society Monographs Series*. Princeton University Press, Princeton, NJ, 2005.

[Sch95] G. Schwarz. *Hodge Decomposition A Method for Solving Boundary Value Problems*. Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 1995.

[Soh01] H. Sohr. *The Navier-Stokes Equations: An Elementary Functional Analytic Approach*. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser Basel, 2001.

[SS92] C.G. Simader and H. Sohr. *A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains*, pages 1–35. WORLD SCIENTIFIC, 1992.

[Ste70] E.M. Stein. *Singular Integrals and Differentiability Properties of Functions*. Monographs in Harmonic Analysis. Princeton University Press, 1970.
[Tol17] P. Tolksdorf. *On the L<sup>p</sup>-theory of the Navier-Stokes equation on Lipschitz domains*. PhD thesis, Technische Universität Darmstadt, 2017.

[Tol18] P. Tolksdorf. On the L<sup>p</sup>-theory of the Navier-Stokes equations on three-dimensional bounded Lipschitz domains. *Math. Ann.*, 371(1-2):445-460, 2018.

[Tri83] H. Triebel. *Theory of Function Spaces I*. Modern Birkhäuser Classics. Birkhäuser Basel, 1983.

[TW20] P. Tolksdorf and K. Watanabe. The Navier-Stokes equations in exterior Lipschitz domains: L<sup>p</sup>-theory. *J. Differential Equations*, 269(7):5765-5801, 2020.

[Wei01] L. Weitz. Operator-Valued Multiplier Theorems and Maximal Lp-Regularity. *Math. Ann.*, 319(4):735-758, 2001.