---
abstract: 'We consider a fluid-structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in $L^p$-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.'
address:
- 'Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany'
- |
    Mathematisches Institut, Angewandte Analysis\
    Heinrich-Heine-Universität Düsseldorf\
    40204 Düsseldorf, Germany
author:
- Robert Denk
- Jürgen Saal
bibliography:
- 'fl\_str\_int.bib'
date: 'September 20, 2019'
title: '$L^p$-theory for a fluid-structure interaction model'
---

\[section\] \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Lemma]{}

\[theorem\][Assumption]{} \[theorem\][Definition]{}

\[theorem\][Remark]{} \[theorem\][Remarks]{} \[theorem\][Example]{} \[theorem\][Examples]{}

Ø[[O]{}]{} §[[S]{}]{} Ł[[L]{}]{}

0.5ex plus 0.5ex minus 0.5ex

Introduction and main result {#secintro}
============================

We consider the system $$\label{fsi}
    \left.
    \begin{array}{rcll}
        \rho (\partial_t u + (u \cdot \nabla) u)) -
        \mbox{div}\,T(u,\,q) & = & 0,
        & \quad t > 0,\    x \in \Omega(t), \\[0.5em]
                                                           \mbox{div}\,u & = & 0,                           & \quad t > 0,\    x \in \Omega(t), \\[0.5em]
                                    u & = & V_\Gamma, & \quad t \geq 0,\ x \in \Gamma(t), \\[0.5em]
                                   \frac1{\nu\cdot e_n}e_n^\tau T(u,\,q)\nu& = &
                       \phi_\Gamma,  & \quad t \geq 0,\ x \in \Gamma(t), \\[0.5em]
                                        \Gamma(0) = \Gamma_0,
                        \quad
                        V_\Gamma(0)=V_0,
                        \quad u(0) & = & u_0,
                        & \quad x \in \Omega(0),
    \end{array}
    \right\}$$ which represents a (one-phase) fluid-structure interaction model. The fluid with density $\rho>0$ and viscosity $\mu>0$ occupies at a time $t \geq 0$ the region $\Omega(t) \subseteq
\bR^n$ with boundary $\Gamma(t)=\partial\Omega(t)$. Furthermore, we assume the fluid to be incompressible, and we assume the stress to be given as $$T(u,\,q) = 2 \mu D(u) - q, \qquad D(u) = {\textstyle
    \frac{1}{2}} (\nabla u + (\nabla u)^{\tau}).$$ The unknowns in the model are the velocity $u$, the pressure $q$ and the interface $\Gamma$. We denote by $\nu$ the exterior unit normal field at $\Gamma$, by $V_\Gamma$ the velocity of the boundary $\Gamma$, and by $e_j$ the $j$-th standard basis vector in $\R^n$, i.e. $e_n=(0,\cdots,0,1)$.

The function $\phi_\Gamma$ describes the elastic response at $\Gamma$ which is given by a damped Kirchhoff-type plate model. Throughout the paper we assume that $\Gamma$ is given as a graph of a function $\eta:\R_+\times\R^{n-1}\to \R$, that is $$\label{gammagraph}
    \Gamma(t) = \Big\{(x',\eta(t,x'));\  x'\in
    \R^{n-1}\Big\}, \quad t \geq 0,$$ and that $\Gamma(t)$ is sufficiently flat. Thus $\Omega(t)$ is a perturbed upper half-plane.In these coordinates, the elastic response is given as $$\label{phigamma}
    \phi_\Gamma=m(\partial_t,\partial')\eta
    :=\partial_{t}^2\eta+\alpha(\Delta')^2\eta-\beta\Delta'\eta
    -\gamma\partial_t\Delta'\eta$$ for $\alpha,\gamma>0$, $\beta\in\R$, where $\Delta'$ stands for the Laplacian in $\R^{n-1}$. Finally, the initial configuration and velocity of the interface resp.the initial fluid velocity are given by $\Gamma_0$ and $V_0$ resp.$u_0= (u_0',u_0^n)$. We remark that in the formulation of the boundary conditions in lines 3 and 4 of , one has to take into account that the Kirchhoff plate model is formulated in a Lagrangian setting, whereas for the fluid an Eulerian setting is used. This is discussed in more detail in the beginning of Section 2.

The symbol of $m(\partial_t,\partial')$ is given as $$m(\lambda,\xi')=
    \lambda^2+\alpha|\xi'|^4+\beta|\xi'|^2
    +\gamma\lambda|\xi'|^2,\quad \lambda\in\C,\ \xi'\in\R^{n-1},$$ which vanishes if $$\lambda=-\frac{\gamma|\xi'|^2}{2}\pm\sqrt{\frac{\gamma^2|\xi'|^4}{4}-\alpha
    |\xi'|^4-\beta|\xi'|^2}.$$ For $\gamma>0$ , the roots of $m(\cdot,\xi')$ lie in some sector which is a subset of $\{\lambda\in\C: \Re\lambda <0\}$. This indicates that the term $-\gamma\partial_t\Delta'\eta$ in $\phi_\Gamma$ parabolizes the problem. Physically, one also speaks of structural damping of the plate.

We notice that basically the same results as proved in this note can be expected by considering layer like domains or rectangular type domains with periodic lateral boundary conditions. For simplicity, however, we restrict the approach given here to the just introduced geometry.

Model (\[fsi\]) was introduced in [@Quarteroni2000] in connection to applications to cardiovascular systems. In the 2D case, this system was investigated in [@batak2017] in the $L^2$-setting. In fact, in [@batak2017 Proposition 3.12] it is proved that the linear operator associated to (\[fsi\]) generates an analytic $C_0$-semigroup in a suitable Hilbert space setting. This exhibits the parabolic character of the problem. Therefore, it is reasonable to consider $L^p$-theory for the system (\[fsi\]) which is the main purpose of this note.

Alternative approaches to system (\[fsi\]) in the $L^2$-setting also for the hyperbolic-parabolic case, i.e. $\gamma=0$, are given, e.g., in [@cdeg2005; @grandmont2008; @Lengeler-Ruzicka14; @Lengeler; @Muha-Canic15], concerning weak solutions and, e.g., in [@bdav2004; @Coutand-Shkoller06; @lequeurre2011; @lequeurre2013] concerning (local) strong solutions. A more recent approach in an two-dimensional $L^2$-framework concerning global strong solutions is presented in [@grahil2016]. In the present paper, we develop an $L^p$-approach in general dimension for system (\[fsi\]). We show the existence of strong solutions for small data and give a precise description of the maximal regularity spaces for the unknowns. More precisely, we prove the following main result for (\[fsi\]).

\[main\] Let $n\ge 2$, $p\ge (n+2)/3$, $T>0$, and $J=(0,T)$. Assume that $$\|u_0\|_{W^{2-2/p}_p(\Omega(0))}
    +\|\eta_0\|_{W^{5-3/p}_p(\R^{n-1})}
    +\|\eta_1\|_{W^{3-3/p}_p(\R^{n-1})}
    <\kappa,$$ where $\Gamma_0=\graph(\eta_0)$ and $V_0=\graph(\eta_1)$, for some $\kappa>0$. Then, there exists a unique solution $(u,q,\Gamma)$ of system (\[fsi\]) such that $\Gamma=\graph(\eta)$ and such that $$\begin{aligned}
    u&\in H^1_p(J;L^p(\Omega(t)))\cap
    L^p(J;H^2_p(\Omega(t))),\\
    q&\in L^p(J;\dot{H}^1_p(\Omega(t))),\\
    \eta&\in \mE_\eta :=W_{p}^{9/4 - 1/(4p)}(J; L^p(\R^{n-1}))
    \cap H_{p}^2(J; W_p^{1-1/p}(\R^{n-1}))\\
  & \quad \cap L^p(J;W_p^{5-1/p}(\R^{n-1})),\end{aligned}$$ provided that $\kappa=\kappa(T)$ is small enough and that the following compatibility conditions are satisfied:

1.  $ \div u_0 = 0$,

2.  if $p>\tfrac32$, then $u_0'|_{\Gamma_0} = 0$ and $u_0^n|_{\Gamma_0} - \eta_1=0$ almost everywhere,

3.  there exists an $\eta_*\in\mE_\eta$ with $\eta_*|_{t=0} = \eta_0$, $\partial_t\eta_*|_{t=0}=\eta_1$ and $$\partial_t \eta_*\in H_{p}^1(\R_+; {}_0\dot
          H_p^{-1}(\R^n_+)),$$ where $$\partial_t \eta_*(\phi) := - \int_{\R^{n-1}} \partial_t \eta_*\phi
        dx',\quad \phi\in\dot H_{p'}^1(\R^n_+).$$

The solution depends continuously on the data.

a\) The compatibility conditions (1)–(3) are natural in the sense that they are also necessary for the existence of a strong solution. Condition (3) appears in a similar way for the two-phase Stokes problem, see, e.g., [@Pruess-Simonett16], Section 8.1.

b\) We remark that the maximal regularity space $\mE_\eta$ for $\eta$ describing the boundary is not a standard space. It is given as an intersection of three Sobolev spaces. This is due to the fact that the symbol of the complete system has an inherent inhomogeneous structure, and therefore the Newton polygon method is the correct tool to show maximal regularity. For the details, see Section 3 below.

c\) We note that in the physically relevant situations $n=2$ and $n=3$, the case $p=2$ is included. This might be of importance when considering the singular limit $\gamma\to0$ for vanishing damping of the plate.

d\) We formulated the result in the form of existence for fixed time and small data. By the same methods, one can also show short time existence for arbitrarily large data.

The proof of Theorem \[main\] is based on several ingredients: First, we transform the system to a fixed domain and consider the linearization of the transformed system. By an application of the Newton polygon approach (see, e.g., [@Denk-Kaip13] and [@Denk-Saal-Seiler08]), we obtain maximal regularity for the linearized system. To deal with the nonlinearities, we employ embedding results on anisotropic Sobolev spaces given in [@koehnesaal].

The transformed system {#sectrans}
======================

We start with a short discussion of the boundary conditions, where the Eulerian approach for the fluid has to be coupled with the Lagrangian description for the plate (see also [@Lengeler-Ruzicka14] and [@grandmont2008]). Let $\Gamma$ be given as in (\[gammagraph\]) and assume that $\eta$ is sufficiently smooth. Following the Kirchhoff plate model, in-plate motions are ignored, and the velocity of the plate at the point $(x',\eta(t,x'))^\tau$ is parallel to the vertical direction and given by $(0,\partial_t \eta(t,x'))^\tau = \partial_t   \eta(t,x') e_n$. As the fluid is assumed to adhere to the plate, we have no-slip boundary conditions for the fluid, and the equality of the velocities yields the first boundary condition $$\label{eq1-1}
  u(t,x',\eta(t,x')) = \partial_t   \eta(t,x') e_n \quad (t>0, \, x'\in \R^n).$$ The exterior normal at the point $(x',\eta(t,x'))$ of the boundary $\Gamma(t)$ is given by $$\nu=\nu(t,x')=\frac1{\sqrt{1+|\nabla'\eta(t,x')|^2}}
    \left(
    \begin{array}{c}
    \nabla'\eta(t,x')\\
    -1
    \end{array}
    \right).$$ We define the transform of variables $$\theta : J \times \R^{n}_+ \to
  \bigcup_{t \in J} \{t\} \times \Omega(t), \ (t,x',x_n)\mapsto\theta(t,x',x_n):=
  (t,x',x_n+\eta(t,x')).$$ Obviously we have $\theta^{-1}(t,x',y)=(t,x',y-\eta(t,x'))$. As it was discussed in [@Lengeler-Ruzicka14], Section 1.2, the force $F$ exerted by the fluid on the boundary is given by the evaluation of the stress tensor at the deformed boundary in the direction of the inner normal $-\nu(t,x')$. More precisely, we obtain ([@Lengeler-Ruzicka14], Eq. (1.4)) $$F = - \sqrt{1+|\nabla'\eta(t,x')}\; e_n^\tau  (T(u,q)\circ \theta(t,x)) \nu(t,x').$$ As $\sqrt{1+|\nabla' \eta|^2} = - \nu(t,x')\cdot e_n$, the equality of the forces gives the second boundary condition $$\label{eq1-2}
  \begin{aligned}
  \frac{1}{\nu(t,x')\cdot e_n} \; e_n^\tau [ T(u,q) ] (t,x',\eta(t,x')) \, \nu(t,x') & = [m(\partial_t,\partial')\eta] (t,x')\\ & \qquad (t>0,x'\in\R^{n-1}).
  \end{aligned}$$ Conditions and are the precise formulation of the boundary conditions in .

To solve the problem , we first note that by a re-scaling argument we may assume that $\rho=\mu=1$ for the density $\rho$ and viscosity $\mu$ from now on. Next, we transform the problem (\[fsi\]) to a problem on the fixed half-space $\R^n_+$, using the above transformation $\theta$. To this end, we set $J:=(0,T)$ and write $x=(x',x_n)\in\R^n_+$ with $x'\in\R^{n-1}$. With the corresponding meaning we write $v'$, $\nabla'$, etc. The pull-back is then defined as $$v:=\Theta^\ast u:= u\circ\theta, \qquad
    p:= \Theta^\ast q := q\circ\theta,$$ and correspondingly the push-forward as $$u:=\Theta_\ast v:= v\circ\theta^{-1}, \qquad
    q:= \Theta_\ast p := p\circ\theta^{-1}.$$ We also set $\Gamma_0=\Gamma(0)=\{(x',\eta_0(x'));\ x'\in\R^{n-1}\}$ and $V_0=V_\Gamma(0)=(0,\eta_1(\cdot))^\tau$.

Applying the transform of variables to (\[fsi\]) leads to the following quasilinear system for $(v,p,\eta)$: $$\label{tfsi}
    \begin{array}{rclll}
                                       \partial_t v -
                           \Delta v + \nabla
                           p& = & F_v(v,p,\eta)
                           & \mbox{in} & J
                           \times \R^n_+, \\
                                                       \div v &
                                   = &
        G(v,\eta)&  \mbox{in} & J \times \R^n_+, \\
        v'&=&0
        & \mbox{on} & J \times \R^{n-1},      \\
        \partial_t\eta - v^n & = & 0
        &  \mbox{on} & J \times \R^{n-1},      \\
          -2 \partial_n v^n + p
          -m(\partial_t,\partial')\eta& = & H_\eta(v,\eta)
          &  \mbox{on} & J \times \R^{n-1},      \\
                  v|_{t=0} &=& v_0  &\mbox{in} & \R^n_+,\\
              \eta|_{t=0}&= & \eta_0
              & \mbox{in} & \R^{n-1},\\
              \partial_t\eta|_{t=0} & = & \eta_1
              & \mbox{in} & \R^{n-1}.
    \end{array}$$ The non-linear right-hand sides are given as $$\begin{array}{rcl}
                         F_v(v,p,\eta) & = & (
                 \partial_t \eta - \Delta' \eta)
                 \partial_n v - 2  (\nabla' \eta
                 \cdot \nabla') \partial_n v +
                 |\nabla' \eta|^2 \partial^2_n v \\[0.5em]
                                         &   & \quad -
                               (v \cdot
                              \nabla) v +
                               (v' \cdot
                              \nabla'\eta) \partial_n v
                              +
                              (\nabla'\eta,0)^\tau\partial_n
                              p,               \\[0.5em]
                                 G(v,\eta) & = & \nabla'\eta
                     \cdot \partial_n v',                                                                                                                        \\[0.5em]
                             H_\eta(v,\eta) & = & -
                     \nabla'\eta \cdot \partial_n v'
                     - \nabla'\eta \cdot \nabla' v^n.                                          \end{array}$$

The linearized system
=====================

The aim of this section is to derive maximal regularity for the linearized system $$\label{linfsi}
    \begin{array}{rclll}
                                        \partial_t v -
                           \Delta v + \nabla
                           p& = & f_v
                           & \mbox{in} & \R_+
                           \times \R^n_+, \\
                                                       \div v &
                                   = &
        g&  \mbox{in} & \R_+ \times \R^n_+, \\
        v'&=&0
        & \mbox{on} & \R_+ \times \R^{n-1},      \\
        \partial_t\eta - v^n & = & 0
        &  \mbox{on} & \R_+ \times \R^{n-1},      \\
          -2 \partial_n v^n + p
          -m(\partial_t,\partial')\eta& = & f_\eta
          &  \mbox{on} & \R_+ \times \R^{n-1},      \\
                  v|_{t=0} &=& v_0  &\mbox{in} & \R^n_+,\\
              \eta|_{t=0}&= & \eta_0
              & \mbox{in} & \R^{n-1},\\
              \partial_t\eta|_{t=0} & = & \eta_1
              & \mbox{in} & \R^{n-1}.
    \end{array}$$

We will consider this system in Sobolev spaces with exponential weight with respect to the time variable. Let $\rho\in\R$ and $X$ be a Banach space. For $u\in L^p(\R_+,X)$, we define $\Psi_\rho$ as the multiplication operator with $e^{-\rho t}$, i.e. $\Psi_\rho u(t) := e^{-\rho t}u(t),\; t\in\R_+$. The spaces with exponential weights are defined by $$\begin{aligned}
  H_{p,\rho}^s(\R_+,X) & := \Psi_{-\rho}(H_p^s(\R_+,X)),\\
   W_{p,\rho}^s(\R_+,X) & := \Psi_{-\rho}(W_p^s(\R_+,X))\end{aligned}$$ with canonical norms $\|u\|_{H_{p,\rho}^s(\R_+,X) } := \| \Psi_\rho u\|_{H_p^s(\R_+,X)}$ and $\|u\|_{W_{p,\rho}^s(\R_+,X) } := \| \Psi_\rho u\|_{W_p^s(\R_+,X)}$. For $\rho\ge 0$ and $s>0$, we define ${}_0 H_{p,\rho}^s(\R_+,X)$ and ${}_0 W_{p,\rho}^s(\R_+,X)$ analogously, replacing $H_p^s$ and $W_p^s$ by ${}_0H_p^s$ and ${}_0W_p^s$, respectively. For mapping properties and interpolation results under the condition that $X$ is a UMD space, we refer, e.g., to [@Denk-Saal-Seiler08], Lemma 2.2. We also make use of homogeneous spaces, e.g., for $\Omega\subset\R^n$ we set $$\dot H_p^1(\Omega):=\{v\in L^1_{\loc}(\Omega):\ \nabla v\in
    L^p(\Omega)\}$$ and $\dot
H_{p,0}^1(\Omega):=\overline{C^\infty_c(\Omega)}^{\|\nabla\cdot\|_p}$. The corresponding dual spaces are defined as $$\dot H^{-1}_p(\Omega):=\bigl(\dot H^1_{p,0}(\Omega)\bigl)'
    \quad\text{and}\quad
    \dot H^{-1}_{p,0}(\Omega):=\bigl(\dot H^1_{p}(\Omega)\bigl)',$$ see [@Pruess-Simonett16] Section 7.2. The homogeneous Sobolev-Slobodeckii spaces over $\R^n$ are defined as usual [@Triebel15] and we have $$\dot W^{s}_p(\R^n)=\dot B^s_{pp}(\R^n)$$ for $1<p<\infty$, $n\in\N$, and $s\in\R\setminus\Z$, where the latter one denotes the homogeneous Besov space.

In the following, we denote the time trace $u\mapsto \partial_t^k u|_{t=0}$ by $\gamma_k^t$ and the trace to the boundary $u\mapsto \partial_n^k u|_{\R^{n-1}}$ by $\gamma_k$. The solution $(v,p,\eta)$ of will belong to the spaces $$\begin{aligned}
  v\in \mE_v & := H_{p,\rho}^1(\R_+;L^p(\R^n_+))\cap L^p_\rho(\R_+;H_p^2(\R^n_+)),\\
  p\in\mE_p & := L^p_\rho(\R_+; \dot H_p^1(\R^n_+)),\\
  \eta\in\mE_\eta & := W_{p,\rho}^{9/4 - 1/(4p)}(\R_+; L^p(\R^{n-1}))\cap H_{p,\rho}^2(\R_+; W_p^{1-1/p}(\R^{n-1}))\\
  & \quad \cap L^p_\rho(\R_+;W_p^{5-1/p}(\R^{n-1})).\end{aligned}$$ The function spaces for the right-hand side of are given by $$\begin{aligned}
  f_v \in \mF_v & := L^p_\rho(\R_+; L^p(\R^n_+)),\\
  g\in \mF_g & :=  H_{p,\rho}^1(\R_+;\dot H_p^{-1}(\R^n_+))\cap L^p_\rho(\R_+;H_p^1(\R^n_+)),\\
 f_\eta\in \gamma_0\mE_p & := L^p_\rho(\R_+; \dot W_p^{1-1/p}(\R^{n-1})).\end{aligned}$$ By trace results with respect to the time trace, the spaces for the initial values are given by $$\begin{aligned}
  v_0 \in \gamma_0^t\mE_v &:= W_p^{2-2/p}(\R^n_+),\\
  \eta_0  \in \gamma_0^t\mE_\eta & := W_p^{5-3/p}(\R^{n-1}),\\
  \eta_1  \in \gamma_1^t\mE_\eta & := W_p^{3-3/p}(\R^{n-1}).\end{aligned}$$ We will also need the following compatibility conditions:

1.  $ \div v_0 = g|_{t=0}$ in $\dot H_p^{-1}(\R^{n}_+)$.

2.  If $p>\tfrac32$, then $v_0'|_{\R^{n-1}} = 0$ almost everywhere in $\R^{n-1}$.

3.  If $p>\tfrac32$, then $v_0^n|_{\R^{n-1}} - \eta_1=0$ almost everywhere in $\R^{n-1}$.

4.  There exists an $\eta_*\in\mE_\eta$ with $\eta_*|_{t=0} = \eta_0$, $\partial_t\eta_*|_{t=0}=\eta_1$ and $$\label{eq2-6}
          (g,\partial_t \eta_*)\in H_{p,\rho}^1(\R_+; \dot H_{p,0}^{-1}(\R^n_+)).$$ Here we define $$(g,\partial_t \eta_*)(\phi) := \int_{\R^n_+} g\phi dx - \int_{\R^{n-1}} \partial_t \eta_*\phi dx'$$ for $\phi\in\dot H_{p'}^1(\R^n_+)$. Additionally, we have $(g|_{t=0},\eta_1)=(g|_{t=0},
           v_0^n|_{\R^{n-1}})$ in $\dot H_{p,0}^{-1}(\R^n_+)$.

We remark that only is an additional condition, as it was shown in [@Denk-Saal-Seiler08], Theorem 4.5, that for every $\eta_0\in\gamma_0^t\mE_\eta$ and $\eta_1\in\gamma_1^t\mE_\eta$ there exists an $\eta_*\in \mE_\eta$ with $\eta_*|_{t=0} = \eta_0$ and $\partial_t\eta_*|_{t=0} = \eta_1$.

The main result of this section is the following maximal regularity result.

\[2.2\] Let $p>1$, $p\neq3/2$. Then there exists a $\rho_0>0$ such that for every $\rho\ge \rho_0$, system has a unique solution $(v,p,\eta)\in \mE_v\times\mE_p\times\mE_\eta$ if and only if the data $f_v,g,f_\eta,v_0,\eta_0,\eta_1$ belong to the spaces above and satisfy the compatibility conditions . The solution depends continuously on the data.

The proof of this theorem will be done in several steps and follows from Subsections \[sec\_necessity\]–\[sec\_uniqueness\].

Necessity {#sec_necessity}
---------

Let $(v,p,\eta)\in\mE_v\times \mE_p\times \mE_\eta$ be a solution of . By standard continuity and trace results, the right-hand sides $f_v,$ and $g$ as well as the time trace $v_0$ belong to the spaces above. Noting that $\div\colon L^p(\R^n_+)\to \dot H_p^{-1}(\R^n_+)$ is continuous, we have $g = \div u \in H_p^1(\R_+;\dot H_p^{-1}(\R^n_+)) \subset C([0,\infty);\dot H_p^{-1}(\R^n_+))$, and as for all $p>1$ we also have $v_0\in W_p^{2-2/p}(\R^n_+)\subset L^p(\R^n_+)$, we obtain the compatibility condition (C1) for all $p>1$ (see also [@Pruess-Simonett16], Theorem 7.2.1).

For $f_\eta$ we have $\mE_\eta\subset H_{p,\rho}^2(\R_+;W_p^{1-1/p}(\R^{n-1})$ by the mixed derivative theorem (see, e.g., [@Denk-Saal-Seiler08], Lemma 4.3), and therefore $$\partial_t^2\eta \in L^p_\rho(\R_+; W_p^{1-1/p}(\R^{n-1})\subset \gamma_0\mE_p.$$ It is easy to see that the other terms of $m(\partial_t,\partial')\eta$ belong to the same space. By standard trace results, we also obtain $\gamma_0 u\in \gamma_0\mE_p$. Concerning the pressure, we remark that $\gamma_0\colon \dot H_p^1(\R^n_+) \to \dot W_p^{1-1/p}(\R^{n-1})$ is a retraction, see, e.g., [@Jawerth78], Theorem 2.1, and therefore $\gamma_0 p\in \gamma_0\mE_p$. This yields $f_\eta\in \gamma_0\mE_p$. For the time traces of $\eta$, we can apply [@Denk-Saal-Seiler08], Theorem 4.5 which gives $\eta_0\in\gamma_0^t\mE_\eta$ and $\eta_1\in \gamma_1^t\mE_\eta$.

If $p>\frac32$, then the boundary trace of $v_0$ exists in the space $W_p^{2-3/p}(\R^{n-1})$. This yields the compatibility conditions (C2) and (C3) as equality in the space $W_p^{2-3/p}(\R^{n-1})$, hence in particular as equality almost everywhere.

To show (C4), we can set $\eta_* := \eta$. For $\phi\in \dot H_{p'}^1(\R^n_+)$, we obtain $$(g,\partial_t\eta)(\phi) = \int_{\R^n_+}  \div u \,\phi dx  - \int_{\R^{n-1}} u^n \phi dx' = -\int_{\R^n_+} u \cdot \nabla \phi dx$$ and therefore $(g, \partial_t\eta)\in H_{p,\rho}^1(\R_+; \dot H_{p,0}^{-1}(\R^n_+))$. Setting $t=0$, we obtain $(g|_{t=0}, \eta_1) = (g|_{t=0}, v_0^n)$ as equality in $\dot H_{p,0}^{-1}(\R^n_+)$.

Reductions {#sec_reduction}
----------

We can reduce some part of the right-hand side of to zero by applying known results on the Stokes system. For this, let $(v^{(1)}, p^{(1)})\in \mE_v\times \mE_p$ be the unique solution of the Stokes problem in the half space $$\label{eq2-3}
    \begin{array}{rclll}
                                        \partial_t v^{(1)} -
                           \Delta v + \nabla
                           p^{(1)}& = & f_v
                           & \mbox{in} & \R_+
                           \times \R^n_+, \\
                                                       \div v^{(1)} &
                                   = &
        g&  \mbox{in} & \R_+ \times \R^n_+, \\
        (v^{(1)})'&=&0
        & \mbox{on} & \R_+ \times \R^{n-1},      \\
         (v^{(1)})^n & = & \partial_t\eta_*
        &  \mbox{on} & \R_+ \times \R^{n-1},      \\
                  v^{(1)}|_{t=0} &=& v_0  &\mbox{in} & \R^n_+.
    \end{array}$$ The unique solvability of follows from [@Pruess-Simonett16], Theorem 7.2.1. To show that this theorem can be applied, we remark in particular that the compatibility condition (e) in [@Pruess-Simonett16 p. 324] holds because of (C4).

Let $\tilde v := v-v^{(1)} $, $\tilde p := p - p^{(1)}$, and $\tilde\eta := \eta - \eta_*$. Then $(v,p,\eta)$ is a solution of if and only if $(\tilde v,\tilde p,\tilde \eta)$ is a solution of $$\label{eq2-4}
    \begin{array}{rclll}
                                        \partial_t \tilde v -
                           \Delta \tilde v + \nabla
                           \tilde p& = & 0
                           & \mbox{in} & \R_+
                           \times \R^n_+, \\
                                                       \div \tilde v &
                                   = &
        0&  \mbox{in} & \R_+ \times \R^n_+, \\
        \tilde v'&=&0
        & \mbox{on} & \R_+ \times \R^{n-1},      \\
         \tilde v^n -  \partial_t\tilde \eta& = &  0
        &  \mbox{on} & \R_+ \times \R^{n-1},      \\
          -2 \partial_n \tilde v^n + \tilde p
          -m(\partial_t,\partial')\tilde \eta& = & \tilde f_\eta
          &  \mbox{on} & \R_+ \times \R^{n-1},      \\
                  \tilde v|_{t=0} &=& 0  &\mbox{in} & \R^n_+,\\
              \tilde\eta|_{t=0}&= & 0
              & \mbox{in} & \R^{n-1},\\
              \partial_t\tilde\eta|_{t=0} & = & 0
              & \mbox{in} & \R^{n-1}.
    \end{array}$$ Here, $$\begin{aligned}
  \tilde f_\eta & := f_\eta + 2\partial_n (v^{(1)})^n
   - p^{(1)}  +  m(\partial_t, \partial') \eta_*.\end{aligned}$$ By the trace results in Subsection \[sec\_necessity\], we have $\tilde f_\eta \in \gamma_0\mE_p$.

Solution operators for the reduced linearized problem
-----------------------------------------------------

In the following, we show solvability for the reduced problem , omitting the tilde again. An application of the Laplace transform formally leads to the resolvent problem $$\label{resfsi}
    \begin{array}{r@{\ =\ }ll}
        \lambda v-\Delta v+\nabla p &   0&\text{in} \ \R^n_+,\\
    \mbox{div}\,v &   0&\text{in} \ \R^n_+,\\
        v'&0&\text{on} \ \partial\R^n_+,\\
        v^n- \lambda\eta&0&\text{on} \ \partial\R^n_+,\\
    -2\partial_n v^n+p-m(\lambda,\partial')\eta&f_\eta&\text{on} \
    \partial\R^n_+
    \end{array}$$ with $$m(\lambda,\partial')\eta=
    \lambda^2\eta+\alpha(\Delta')^2\eta-\beta\Delta'\eta
    -\gamma\lambda\Delta'\eta.$$ We observe that the second and the third line of imply that $$\partial_n v^n(\cdot,0)=-\nabla'\cdot v'(\cdot,0)=0.$$ Hence the fifth line reduces to $$p-m(\lambda,\partial')\eta=f_\eta\quad\text{on}\ \partial\R^n_+.$$

Applying partial Fourier transform in $x'\in\R^{n-1}$, we obtain the following system of ordinary differential equations in $x_n$ for the transformed functions $\hat v$, $\hat p$ and $\hat \eta$: $$\begin{aligned}
\omega^2 \hat v - \partial_n^2 \hat v + (i \xi',\partial_n)^\tau \hat p & =
0,
\quad x_n>0, \nonumber \\i \xi \cdot \hat v' + \partial_n \hat v^n & = 0, \quad x_n>0, \\\hv'&=0,\quad x_n=0,\\
\lambda \heta -\hat v^n & = 0, \quad x_n=0,\\
\hp-m(\lambda,|\xi'|)\heta&=\hf_\eta,\quad x_n=0,\\\end{aligned}$$ Here we have set $\omega:= \omega(\lambda,\xi'):=\sqrt{\lambda +
|\xi'|^2}$ and $$m(\lambda,\xi'):=\lambda^2+\alpha|\xi'|^4+\gamma\lambda|\xi'|^2
    +\beta|\xi'|^2.$$

Multiplying the first equation with $(i\xi',\partial_n)$ and combing it with the second one yields $(-|\xi'|^2+\partial_n^2)\hp = 0$ for $x_n>0$. The only stable solution of this equation is given by $$\label{(2.7)}
  \hp(\xi',x_n) = \hp_0(\xi') e^{-|\xi'|x_n},
  \quad \xi' \in {\mathbb R}^{n-1}, \ x_n > 0.$$ To solve the above system we employ the ansatz $$\begin{aligned}
  \hat v'(\xi',x_n) & = -\int_0^\infty k_+(\lambda,\xi',x_n,s)
  i\xi'\hp(\xi',s)  ds + \hat \phi'(\xi') e^{-\omega x_n},\label{rd_eq1}\\
  \hat v^n(\xi',x_n) & = -\int_0^\infty k_-(\lambda,\xi',x_n,s)
  \partial_n \hp(\xi',s)ds + \hat \phi^n(\xi') e^{- \omega x_n}\label{rd_eq1a}\end{aligned}$$ with $$k_{\pm} (\lambda,\xi,x_n,s) := \frac{1}{2 \omega} \left( e^{-
  \omega|x_n-s|} \pm e^{-\omega (x_n+s)} \right).$$ Here the traces $\hat p_0$ and $\hat\phi = (\hat \phi',
\hat \phi^n)^\tau$ still have to be determined. The fact that $\div v=0$ enforces $$\label{divbed}
    i\xi'\cdot\hphi'(\xi')=\omega\hphi^n(\xi').$$ The kinematic boundary condition instantly gives us $$\label{kinbc}
    \lambda\heta-\hphi^n=0.$$ Next, by utilizing (\[(2.7)\]), from the tangential boundary condition we obtain $$0=\hv'(\xi',0)=-\int_0^\infty \frac{e^{-\omega s}}{\omega}
  i\xi'\hp(\xi',s) ds + \hat \phi'(\xi'),$$ which implies $$\label{eq2-9}
    \frac{i\xi'}{\omega+|\xi'|}\hp_0=\omega\hphi'.$$ Multiplying this with $i\xi'$ and employing the relations (\[divbed\]), (\[kinbc\]) yields $$\label{tangbc}
    -\frac{|\xi'|^2}{\omega+|\xi'|}\hp_0
    =\omega^2\hphi^n=\lambda\omega^2 \heta.$$ Plugging this into the last line of the transformed system we obtain $$\label{normaltrace}
    \left(\frac{\lambda\omega^2(\omega+|\xi'|)}{|\xi'|^2}
    +m(\lambda,\xi')\right)\heta=-\hf_\eta.$$ This yields $$\label{eq2-8}
    \heta=-\frac{|\xi'|^2}{N_L(\lambda,|\xi'|)}\hf_\eta$$ with $$N_L(\lambda,|\xi'|)=|\xi'|^2m(\lambda,\xi')
    +\lambda\omega^2(\omega+|\xi'|).$$

Formula defines the solution operator for $\eta$ as a function of $f_\eta$ on the level of its Fourier-Laplace transform. The following result is based on the Newton polygon approach and shows that the solution operator is continuous on the related Sobolev spaces. In the following, we consider $(-\Delta')^{1/2}$ as an unbounded operator in $L^p_\rho(\R_+; L^p(\R^{n-1})$, and define $N_L(\partial_t,(-\Delta')^{1/2})$ by the joint $H^\infty$-calculus of $\partial_t$ and $(-\Delta')^{1/2}$ (for details, we refer to, e.g., [@Denk-Saal-Seiler08], Corollary 2.9). We will apply the Newton polygon approach on the Bessel potential scale $H_p^s$ with respect to time and on the Besov scale $B_{pp}^r$ with respect to space.

\[2.3\] a) There exists a $\rho_0>0$ such that for all $\rho\ge \rho_0$, the operator $ N_L(\partial_t, (-\Delta')^{1/2})\colon H_N \to L^p_\rho(\R_+;B_{pp}^{-1-1/p}(\R^{n-1}) ) $ is an isomorphism, where $$\begin{aligned}
   H_N &  := {}_0 H_{p,\rho}^{5/2}(\R_+;B_{pp}^{-1-1/p}(\R^{n-1}))\cap {}_0H_{p,\rho}^2(\R_+;B_{pp}^{1-1/p}(\R^{n-1})) \\
  & \quad \cap L^p_\rho(\R_+; B_{pp}^{5-1/p}(\R^{n-1})).
  \end{aligned}$$

b\) Let $\rho\ge \rho_0$. Then for every $f_\eta\in \gamma_0\mE_p$, we have $$\begin{aligned}
    \eta & := \Delta' \big[ N_L(\partial_t, (-\Delta')^{1/2})\big]^{-1}  f_\eta \in \mE_\eta,\\
    \phi^n  & := \partial_t \eta \in \gamma_0\mE_v,\\
    p_0 & := f_\eta +  m(\partial_t,\partial')\eta \in \gamma_0\mE_p.
  \end{aligned}$$

a\) We apply the Newton-polygon approach developed in [@Denk-Saal-Seiler08]. Replacing $z=|\xi'|$, the $r$-principle symbols, i.e., the leading terms of $N_L$ associated to the relation $\lambda\sim z^r$, are easily calculated as $$P_r(\lambda,z)=
    \left\{
    \begin{array}{rl}
    \alpha z^6,& 0<r<2,\\
    m_0(\lambda,z)z^2,& r=2,\\
    \lambda^2 z^2,&2<r<4,\\
    \lambda^2 z^2+\lambda^{5/2},&r=4,\\
    \lambda^{5/2},& r>4,
    \end{array}
    \right.$$ where $m_0=m$ for $\beta=0$, that is $$m_0(\lambda,z):=\lambda^2+\alpha z^4+\gamma\lambda z^2.$$ In other words, the associated Newton-polygon has the three relevant vertices $(6,0)$, $(2,2)$, and $(0,\frac52)$ and two relevant edges which again reflects the quasi-homogeneity of $N_L$.

Now, let $\vp\in(0,\pi/2)$ and $\theta\in(0,\vp/4)$. For $r\neq 2$ we then obviously have $$\label{rsymbcond}
    P_r(\lambda,z)\neq 0\quad
    \left((\lambda,z)\in\Sigma_{\pi-\vp}\times\Sigma_\theta\right).$$ For $r=2$ we deduce $$P_2(\lambda,z)=0
    \quad\Leftrightarrow \quad \lambda=\frac{z^2}{2}
    \left(-\gamma\mp \sqrt{\gamma^2-4\alpha}\right).$$ By the fact that $\gamma>0$ we see that $$\vp_0:=\pi-\arg \left(-\gamma\mp \sqrt{\gamma^2-4\alpha}\right)
    <\frac{\pi}2.$$ Thus, assuming $\vp\in(\vp_0,\pi/2)$ and $\theta\in\left(0,(\vp-\vp_0)/4\right)$ we see that (\[rsymbcond\]) is satisfied for all $r>0$. This allows for the application of [@Denk-Saal-Seiler08 Theorem 3.3] (setting $s=0$ and $r=-1-1/p$ in the notation of [@Denk-Saal-Seiler08]) which yields a).

b\) We write $$\eta =   [ N_L(\partial_t, (-\Delta')^{1/2})\big]^{-1}\Delta'   f_\eta.$$ As $\Delta'$ is an isomorphism from $\dot H_p^{2+t}(\R^{n-1})$ to $\dot H_p^{t}(\R^{n-1})$ for each $t\in\R$, by real interpolation of these spaces (see [@Jawerth78], Lemma 1.1) we see that it is also an isomorphism from $\dot B_{pp}^t(\R^{n-1})$ to $\dot B_{pp}^{t-2}(\R^{n-1})$ for each $t\in\R$. In particular, $\Delta' f_\eta \in L^p_\rho(\R_+; \dot B_{pp}^{-1-1/p}(\R^{n-1}))$. Using the fact that for $s<0$ the embedding $\dot B_{pp}^s (\R^{n-1}) \subset B_{pp}^s( \R^{n-1})$ holds (see [@Triebel14 p. 104, (3.339)], [@Triebel15 Section 3.1]), we obtain the embedding $$L^p_\rho(\R_+; \dot B_{pp}^{-1-1/p}(\R^{n-1})) \subset L^p_\rho(\R_+;  B_{pp}^{-1-1/p}(\R^{n-1})).$$ An application of a) yields $$\begin{aligned}
 \eta & \in {}_0H_{p,\rho}^{5/2}(\R_+;B_{pp}^{-1-1/p}(\R^{n-1}))  \cap {}_0H_{p,\rho}^{2}(\R_+;B_{pp}^{1-1/p}(\R^{n-1}))\\
 & \quad \cap  L^p_\rho(\R_+; B_{pp}^{5-1/p}(\R^{n-1})).
 \end{aligned}$$ Now, the mixed derivative theorem in mixed scales (see [@Denk-Kaip13], Proposition 2.7.6) implies $$\begin{aligned}
 {}_0H_{p,\rho}^{5/2}(\R_+;&B_{pp}^{-1-1/p}(\R^{n-1}))  \cap L^p_\rho(\R_+; B_{pp}^{5-1/p}(\R^{n-1}))\\
 &  \subset B_{pp,\rho}^{9/4 - 1/(4p)}(\R_+; L^p(\R^{n-1}))\end{aligned}$$ and we obtain $\eta\in \mE_\eta$.

For $u^n := \partial_t \eta$, we immediately get $$u^n  \in W_{p,\rho}^{5/4-1/(4p)}(\R_+; L^p(\R^{n-1}))\cap L^p_\rho(\R_+;W_p^{3-1/p}(\R^{n-1})\subset\gamma_0\mE_v.$$ Finally, the fact that $m(\partial_t, \partial')\eta \in \gamma_0\mE_p$ for $\eta\in \mE_\eta$ was already remarked in Subsection \[sec\_necessity\].

Due to the last result, we obtain the existence of a solution $(v,p,\eta)$ of . For $\eta$, $\phi^n$, and $p_0$ defined as in Lemma \[2.3\] b), we can define $p$ and $v$ by and –, respectively. Here, $\phi'$ is given by . As we know that $\phi^n$ and $p_0$ belong to the canonical spaces by Lemma \[2.3\] b), we get $v\in \mE_v$ and $p\in\mE_p$ by standard results on the Stokes equation (see, e.g., [@Hieber-Saal18], Section 2.6, and [@Pruess-Simonett16], Section 7.2). By construction, $(v,p,\eta)$ is a solution of .

Uniqueness of the solution {#sec_uniqueness}
--------------------------

To show that the solution of is unique, let $(v,p,\eta)$ be a solution with zero right-hand side and zero initial data. Then the Laplace transform in $t$ and partial Fourier transform in $x'$ are well-defined, and the calculations above show, in particular, that $$\heta=-\frac{|\xi'|^2}{N_L(\lambda,|\xi'|)}\hf_\eta = 0$$ for almost all $\xi'\in\R^{n-1}$. Therefore, $\eta=0$ which implies that $(v,p)$ is the solution of the Dirichlet Stokes system with zero data. Therefore, $v=0$ and $p=0$.

This finishes the proof of Theorem \[2.2\].

\[remark-finite-time\] Theorem \[2.2\] was formulated on the infinite time interval $(0,\infty)$ with exponentially weighted spaces with respect to $t$. As usual in the theory of maximal regularity, we obtain the same results on finite time intervals $t\in J = (0,T)$ with $T<\infty$ without weights, i.e., with $\rho=0$. This is due to the fact that on finite time intervals the weighted and unweighted norms are equivalent and that there exists an extension operator from $(0,T)$ to $(0,\infty)$ acting on all spaces above.

Therefore, the results of Theorem \[2.2\] hold with $\rho=0$ on the finite interval $J=(0,T)$. As we consider the nonlinear equation on a finite time interval, we will replace the function spaces above by $\mE_v := H^1(J; L^p(\R^n_+))\cap L^p(J;H_p^2(\R^n_+))$ etc., keeping the same notation.

The nonlinear system
====================

To prove mapping properties of the nonlinearities we employ sharp estimates for anisotropic function spaces provided in [@koehnesaal]. In fact, we can proceed very similar as in [@koehnesaal Section 5.2, Proposition 5.6]. For $\omega_j\in\N_0$, $j=1,\ldots,\nu$, we define a weight vector as $\omega:=(\omega_1,\ldots,\omega_\nu)$ and denote by $\dot\omega:=\mathrm{lcm}\{\omega_1,\ldots,\omega_\nu\}$ the lowest common multiple. Further, for $n=(n_1,\ldots,n_\nu)\in\N^\nu$ we write $$\R^n=\R^{n_1}\times\cdots\times\R^{n_\nu}.$$ The (generalized) Sobolev index of an $E$-valued anisotropic function space then reads as $$\frac{1}{\dot{\omega}} \left( s - \frac{\omega \cdot n}{p} \right)
    =: \left\{
        \begin{array}{rc}
                \mbox{ind}(B^{s, \omega}_{p, q}(\bR^n,\,E)), &
        s \in\R,\ 1 < p < \infty,\ 1 \leq q \leq \infty, \\[0.5em]
                \mbox{ind}(H^{s, \omega}_p(\bR^n,\,E)),      &  -\infty
< s < \infty,\ 1 < p < \infty,                        \\[0.5em]
                \mbox{ind}(W^{s, \omega}_p(\bR^n,\,E)),      &  0 \leq s
 < \infty,\ 1 \leq p < \infty,
        \end{array}
        \right.$$ where $\omega\cdot n=\sum_{j=1}^\nu \omega_jn_j$. Note that we have the corresponding definition, if $\R^n$ is replaced by a cartesian product of Intervals. For an introduction to anisotropic spaces such as $H^{s, \omega}_{p}(\R^n,E)$ we refer to [@koehnesaal] and the references cited therein. In the situation considered here we always have $\omega=(2,1)$ and the anisotropic spaces below can be represented as an intersection such as $$\begin{aligned}
    &H^{1,(2,1)}_p(J\times \R^{n-1},L^p(\R_+))\\
    &=H^{1/2}_p(J,L^p(\R^{n-1},L^p(\R_+)))
    \cap L^p(J,H^1(\R^{n-1},L^p(\R_+))),\end{aligned}$$ for instance. In this case we have $$\mbox{ind}\bigl(H^{1,(2,1)}_p(J\times \R^{n-1},L^p(\R_+))\bigr)
    =\frac12\left(1-\frac{2+n-1}{p}\right)
    =\frac12-\frac{n+1}{2p}.$$

Now, let $J=(0,T)$. By the [*mixed derivative theorem*]{}, see e.g.[@Denk-Saal-Seiler08 Lemma 4.3], we have $$H^{2}_p(J,W^{1-1/p}_p(\R^{n-1}))\cap
    L^p(J,W^{5-1/p}_p(\R^{n-1}))\hook H^1(J,W^{3-1/p}(\R^{n-1})).$$ This yields $$\label{est-ht}
\begin{split}
    \partial_t \eta &\in W^{5/4 - 1/4p}_p(J,L_p(\R^{n-1}))
    \cap L_p(J,W^{3 - 1/p}_p(\R^{n-1})) \\
    &\hook W^{1-1/2p}_p(J,L^p(\R^{n-1}))
    \cap L^p(J,W^{2-1/p}_p(\R^{n-1}))\\
    &=W^{2  - 1/p, (2,1)}_p(J\times \R^{n-1})
\end{split}$$ for $\eta \in \mE_3$. Again by the mixed derivative theorem we have $$H^{2}_p(J,W^{1-1/p}_p(\R^{n-1})) \cap L^p(J,W^{5 -
    1/p}_p(\R^{n-1})) \hookrightarrow W^{2- 1/2p}_p(J,H^1_p(\R^{n-1})),$$ which gives us $$\label{est-hx}
    \begin{split}
        \partial_j \eta &\in W^{2 - 1/2p}_p(J,L_p(\R^{n-1}))
        \cap L_p(J,W^{4 - 1/p}_p(\R^{n-1}))\\
        &= W^{4 - 1/p, (2,1)}_p(J \times
        \R^{n-1})
    \end{split}$$ for $\eta \in \mE_3$ and $j = 1,\,\dots,\,n - 1$. Analogously we obtain that $$\label{est-hxy}
    \begin{split}
    \partial_j \partial_k \eta &\in W^{3/2 - 1/2p}_p(J,L_p(\R^{n-1}))
    \cap L_p(J,W^{3 - 1/p}_p(\R^{n-1})) \\
    &= W^{3 - 1/p, (2,1)}_p(J \times \R^{n-1})\\
    &\hook\ W^{2 - 1/p, (2,1)}_p(J \times \R^{n-1})
    \end{split}$$ for $\eta \in \mE_3$ and $j,\,k = 1,\,\dots,\,n - 1$.

For the velocity we have $$\label{est-u}
                         v  \in H^{2,(2,1)}_p(J \times \R^n_+)
                 \hookrightarrow H^{2,(2,1)}_p(J \times
                 \R^{n-1},L^p(\R_+)).$$ Another application of the mixed derivative theorem yields $$\begin{aligned}
          \partial_j v & \in H^{1, (2,1)}_p(J \times \R^n_+)
          \hookrightarrow H^{1,(2,1)}_p(J \times
          \R^{n-1},L_p(\R_+)),\label{est-ux} \\
    \partial_j\partial_k v & \in L_p(J \times \R^n_+)
    = L_p(J \times \R^{n-1},L_p(\R_+)),\label{est-uxy}\end{aligned}$$ for $j,\,k = 1,\,\dots,\,n$. Taking trace this also implies $$\begin{aligned}
    v|_{\partial\R^n_+} & \in W^{1 - 1/2p}_p(J,L_p(\R^{n-1}))
    \cap L_p(J,W^{2 - 1/p}_p(\R^{n-1}))\nonumber\\
    &= W^{2 - 1/p, (2,1)}_p(J \times \R^{n-1})\label{est-ux-tr1}\\
    \partial_j v|_{\partial\R^n_+}
    & \in W^{1/2 - 1/2p}_p(J,L_p(\R^{n-1}))
    \cap L_p(J,W^{1 - 1/p}_p(\R^{n-1}))\nonumber\\
    &= W^{1 - 1/p, (2,1)}_p(J \times \R^{n-1})\label{est-ux-tr2}\end{aligned}$$ for $j = 1,\,\dots,\,n$.

Now, we denote by $L$ the linear operator on the left hand side of system (\[tfsi\]) and by $N=(F_v,G,0,0,H_\eta,0,0,0)$ its nonlinear right-hand side. Then (\[tfsi\]) is reformulated as $$L(v,p,\eta)=N(v,p,\eta)+(0,0,0,0,0,v_0,\eta_0,\eta_1).$$ We also set $$\begin{aligned}
    \widetilde\mE&:=\mE_v\times\mE_p\times\mE_\eta,\\
    \widetilde\mF&:=\mF_v\times\mF_g\times\{0\}\times\{0\}
          \times \gamma_0\mE_p
          \times\gamma_0^t\mE_v\times\gamma_0^t\mE_\eta
          \times\gamma_1^t\mE_\eta.\end{aligned}$$ The nonlinearity admits the following properties.

\[mappropnl\] Let $p\ge(n+2)/3$. Then $N\in C^\omega(\widetilde\mE,\widetilde\mF)$, $N(0)=0$, and we have $DN(0)=0$ for the Fréchet derivative of $N$.

[*Mapping properties of $F_v$.*]{} Gathering (\[est-ht\]), (\[est-hxy\]), and (\[est-ux\]) we can estimate the term $$(\partial_t \eta -  \Delta' \eta)\,\partial_n v,$$ as desired, provided the vector-valued embedding $$\begin{split}
    \label{emb-fu1}
    &\underbrace{W^{2 - 1/p, (2,1)}_p(J \times
    \R^{n-1})}_{\textrm{ind}_1 = 1 - \frac{n + 2}{2p}}
    \cdot \underbrace{H^{1, (2,1)}_p(J \times
    \R^{n-1},L^p(\R_+))}_{\textrm{ind}_2 = \frac{1}{2} - \frac{n +
    1}{2p}}\\
    &\hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times
        \R^{n-1},L^p(\R_+))}_{\textrm{ind} = - \frac{n + 1}{2p}}
\end{split}$$ does hold. Applying [@koehnesaal Theorem 1.7] this readily follows if at least one of the two indices $\mbox{ind}_1$, $\mbox{ind}_2$ is non-negative. The strictest condition to be fulfilled by [@koehnesaal Theorem 1.7], however, is $\mbox{ind}_1 + \mbox{ind}_2 \geq \mbox{ind}$ in case that both of the indices on the left-hand-side are negative which can occur for small $p$. It is easily seen that this condition is equivalent to $$\label{reqp1}
    p \geq \frac{n + 2}{3}.$$ For the terms $$2 (\nabla' \eta \cdot \nabla')\,\partial_n v,
    \quad |\nabla' \eta|^2 \partial^2_n v,
    \quad (\nabla' \eta,0)^\tau\,\partial_n p$$ we employ (\[est-hx\]), (\[est-uxy\]) and the vector-valued embeddings $$\begin{split}
    \label{emb-fu2}
    &\big[ \underbrace{W^{4 - 1/p, (2,1)}_p(J \times
    \R^{n-1})}_{\textrm{ind}_1 = 2 - \frac{n + 2}{2p}} \big]^m \cdot
    \underbrace{H^{0, (2,1)}_p(J \times
    \R^{n-1},L^p(\R_+))}_{\textrm{ind}_2 = - \frac{n +
    1}{2p}}\\
    &\hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times \R^{n-1},
    L^p(\R_+))}_{\textrm{ind} = - \frac{n + 1}{2p}}
\end{split}$$ for $m = 1,\,2$. Due to [@koehnesaal Theorem 1.9] the above embeddings are valid, provided that $\mbox{ind}_1 > 0$ or, equivalently, $$\label{reqp1weak}
    p > \frac{n + 2}{4}.$$ Next, (\[est-u\]) and (\[est-ux\]) show that we obtain the desired estimate of the term $(v \cdot \nabla)v$, if $$\underbrace{H^{2,(2,1)}_p(J \times \R^n_+)}_{\textrm{ind}_1 = 1
    - \frac{n + 2}{2p}} \cdot \underbrace{H^{1, (2,1)}_p(J \times
      \R^n_+)}_{\textrm{ind}_2 = \frac{1}{2} - \frac{n + 2}{2p}}
        \hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times
        \R^n_+)}_{\textrm{ind} = - \frac{n + 2}{2p}}.$$ This is guaranteed by [@koehnesaal Theorem 1.7] if $\max\,\{\,\mbox{ind}_1,\,\mbox{ind}_2\,\} \geq 0$. Again, for small values of $p$ both of the indices on the left-hand-side can become negative. Then [@koehnesaal Theorem 1.7] implies the embedding above if $\mbox{ind}_1 + \mbox{ind}_2 \geq \mbox{ind}$, which is equivalent to (\[reqp1\]).

Thanks to (\[est-hx\]) and (\[est-ux\]) the term $(v' \cdot \nabla' \eta)\partial_nv$ can be estimated by utilizing the embedding $$\label{3factoremb}
\begin{split}
    &\underbrace{H^{1, (2,1)}_p(J \times
    \R^{n-1},H^1_p(\R_+))}_{\textrm{ind}_1 = \frac{1}{2} - \frac{n +
    1}{2p}}\cdot
    \underbrace{W^{4 - 1/p, (2,1)}_p(J \times
    \R^{n-1})}_{\textrm{ind}_2 = 2 - \frac{n + 2}{2p}}\\
    &\cdot
    \underbrace{H^{1, (2,1)}_p(J \times
    \R^{n-1},L^p(\R_+))}_{\textrm{ind}_3 = \frac{1}{2} - \frac{n +
    1}{2p}}
    \quad\hookrightarrow\quad \underbrace{H^{0, (2,1)}_p(J \times \R^{n-1},
    L^p(\R_+))}_{\textrm{ind} = - \frac{n + 1}{2p}}.
\end{split}$$ Note that here we also employ $$H^{2, (2,1)}_p(J \times
    \R^{n}_+)
    \ \hook \ H^{1, (2,1)}_p(J \times
    \R^{n-1},H^1_p(\R_+))$$ and $H^1_p(\R_+)\cdot L^p(\R_+)\hook L^p(\R_+)$ which is valid due to the Sobolev embedding $H^1_p(\R_+)\hook L^\infty(\R_+)$ for $p>1$. Thanks to [@koehnesaal Theorem 1.7] (\[3factoremb\]) holds, if $\min\,\{\,\mbox{ind}_1,\,\mbox{ind}_2,\,\mbox{ind}_3\,\} \geq 0$. If at least one of the three indices on the left-hand side is negative, then the sum of the negative indices on the left hand side has to exceed the index on the right hand side. The most restrictive constraint hence results from $\mbox{ind}_1 + \mbox{ind}_2 + \mbox{ind}_3 \geq \mbox{ind}$, which is fullfilled if $$\label{anothconstr}
    p\ge\frac{2n+3}{6}.$$ Consequently, by our assumptions $F_v$ has the desired mapping properties, since (\[reqp1\]) also yields (\[reqp1weak\]) and (\[anothconstr\]).

[*Mapping properties of $G$.*]{} First we show $G(v,\eta) \in H^1_p(J,\,\dot{H}^{-1}_p(\bR^n_+))$. Integration by parts yields $\partial_n \in \sL(L_p(J \times
\bR^n_+),\,L_p(J,\,\dot{H}^{-1}_p(\bR^n_+)))$. Using this property and the fact that $\eta$ does not depend on $x_n$, it is sufficient to estimate the terms $$\partial_t \nabla'\eta \cdot v',
    \ \nabla' \eta \cdot \partial_t v'$$ in $L_p(J \times \bR^n_+)$. Thanks to (\[est-ht\]) and the mixed derivative theorem we know $$\partial_t \nabla'\eta \in W^{1 - 1/p, (2, 1)}_p(J \times
    \R^{n-1}).$$ The first term can thus be estimated by the vector-valued embedding $$\begin{aligned}
    &\underbrace{W^{1 - 1/p, (2,1)}_p(J \times \R^{n-1})}_{\textrm{ind}_1 =
    \frac12- \frac{n + 2}{2p}} \cdot \underbrace{H^{2, (2,1)}_p(J \times
    \R^{n-1},\,L^p(\R_+))}_{\textrm{ind}_2 = 1 - \frac{n + 1}{2p}}\\
        &\hookrightarrow \underbrace{H^{0, (2,1)}_p(J \times
        \R^{n-1},\,L^p(\R_+))}_{\textrm{ind}
        = - \frac{n +1}{2p}}.\end{aligned}$$ According to [@koehnesaal Theorem 1.7] this embedding is again valid, if we have $\max\,\{\,\mbox{ind}_1,\,\mbox{ind}_2\,\} \geq 0$ or if $\mbox{ind}_1 + \mbox{ind}_2 \geq \mbox{ind}$ in case that both indices on the left hand side are negative. The latter condition is again equivalent to (\[reqp1\]).

The second term may be estimated by employing (\[est-hx\]), the vector-valued embedding (\[emb-fu2\]) for $m = 1$, and $\partial_t v\in L^p(J\times\R^{n-1},L^p(\R_+))$ under constraint (\[reqp1weak\]).

To see that also $G(v,\eta) \in L^p(J,H^1_p(\R^n_+))$, we estimate the terms $$\partial_j \nabla' \eta \cdot \partial_n v',
    \ \nabla'\eta \cdot \partial_j \partial_n v',
    \ \nabla'\eta \cdot \partial^2_n v',
    \qquad j = 1,\,\dots,\,n - 1,$$ in $L^p(J \times \R^n_+)$. Similar as above this may be accomplished by utilizing (\[est-hx\]), (\[est-hxy\]), (\[est-ux\]), (\[est-uxy\]) in combination with the vector-valued embeddings (\[emb-fu1\]), and (\[emb-fu2\]). Once more this is feasible if (\[reqp1\]) holds.

[*Mapping properties of $H_\eta$.*]{} Note that $W^{1 - 1/p, (2,1)}_p(J\times \R^{n-1}))\,\hook\, \gamma_0\mE_p$. Hence, according to (\[est-hx\]) and (\[est-ux\]) we can estimate the terms $$-\nabla'\eta \cdot \partial_n v',\
    -\nabla'\eta \cdot \nabla'v^n$$ as desired provided that the embedding $$\begin{aligned}
    &\big[ \underbrace{W^{4 - 1/p, (2,1)}_p(J \times
    \R^{n-1})}_{\textrm{ind}_1 = 2 - \frac{n + 2}{2p}} \big] \cdot
    \underbrace{W^{1 - 1/p, (2,1)}_p(J \times
    \R^{n-1}))}_{\textrm{ind}_2 = \frac{1}{2} - \frac{n + 2}{2p}}\\
    &\hookrightarrow \underbrace{W^{1 - 1/p, (2,1)}_p(J
        \times \R^{n-1}))}_{\textrm{ind}
        = \frac{1}{2} - \frac{n + 2}{2p}}\end{aligned}$$ is at our disposal. By [@koehnesaal Theorem 1.9] this is the case if $\mbox{ind}_1 > 0$. Hence, the nonlinearity $H_\eta$ has the desired mapping properties, provided that $p>(n+2)/4$. This, in turn, is true since (\[reqp1\]) is satisfied.

Altogether we have proved the asserted embeddings, i.p. that $N(\widetilde\mE)\subset \widetilde\mF$. The claimed smoothness of $N$ as well as $N(0)=0$ and $DN(0)=0$ follow obviously by the fact that $N$ consists of polynomial nonlinearities which are of quadratic or higher order.

For a Banach space $E$ we denote by $B_E(x,r)$ the open ball in $E$ with radius $r>0$ centered in $x\in E$. Based on Theorems \[2.2\] and \[mappropnl\] we can derive well-posedness of (\[tfsi\]) for small data. For simplicity we also set $$\begin{aligned}
    \mE&:=\left\{(v,p,\eta)\in\mE_v\times\mE_p\times\mE_\eta;
    \ \partial_t\eta=v^n,\ v'=0\ \text{on } \partial\R^n_+\right\},\\
    \mF&:= \biggl\{(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1)
           \in\widetilde\mF;\
          f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1\text{ satisfy}\biggr.\\
       &\biggl.\qquad
       \text{the compatibility conditions (C1)-(C4)}\biggr\}.\end{aligned}$$

\[mainnonlinsys\] Let $p\ge (n+2)/3$ and $T>0$. Then there is a $\kappa=\kappa(T)>0$ such that for $(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1)\in B_{\widetilde\mF}(0,\kappa)$ satisfying the compatibility conditions (C2)-(C4) and $$\label{nonlincd}
    \div v_0=\nabla'\eta_0\cdot\partial_n v'_0+g|_{t=0}
    \quad \text{in } \dot{H}^{-1}_p(\R^n_+)$$ there is a unique solution $(v,p,\eta)\in\mE$ of system (\[tfsi\]). The solution depends continuously on the data.

We pick $f:=(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1)$ as assumed. System (\[tfsi\]) (including exterior forces) reads as $$\label{fullnlsys}
    L(v,p,\eta)=N(v,p,\eta)+f.$$ We first have to verify that the right hand side belongs to $\mF$. Observe that (\[nonlincd\]) gives (C1). Hence, by our assumptions the compatibility conditions (C1)-(C3) are satisfied. To see compatibility condition (C4) we have to verify that there exists an $\eta_*\in\mE_\eta$ satisfying $(\eta_*,\partial_t\eta_*)|_{t=0}=(\eta_0,\eta_1)$ and $$(g+G(v,\eta),\partial_t\eta_*)\in
    H^1_{p}(J;\dot{H}^{-1}_{p,0}(\R^n_+))$$ for every triple $(v,p,\eta)\in\mE$ such that $(v,\eta,\partial_t\eta)|_{t=0}=(v_0,\eta_0,\eta_1)$. Note that by assumption there is an extension $\eta_*\in\mE_\eta$ with the prescribed traces such that $$(g,\partial_t\eta_*)\in
    H^1_{p}(J;\dot{H}^{-1}_{p,0}(\R^n_+)).$$ Hence it suffices to prove that $$\label{nonlinginhm1}
    (\nabla'\eta\cdot\partial_n v',0)\in
    H^1_{p}(J;\dot{H}^{-1}_{p,0}(\R^n_+))$$ For $\phi\in \dot{H}^1_{p}(\R^n_+)$ we observe that thanks to $v'(x',0)=0$ we obtain $$\int_{\R_+}\phi(x)\nabla'\eta(x')\cdot\partial_n v'(x)\,dx_n
    =-\int_{\R_+}\nabla'\eta(x')\cdot v'(x)\partial_n\phi(x)\,dx_n.$$ In order to deduce (\[nonlinginhm1\]) it hence suffices to prove that $$\nabla'\eta\cdot v'
    \in H^1_{p}(J;L^p(\R^n_+)).$$ Thanks to (\[est-hx\]) and (\[est-u\]) this follows from the embedding $$\begin{aligned}
    &W^{4-1/p,(2,1)}_p(J\times \R^{n-1})
    \cdot H^{2,(2,1)}_p(J\times \R^{n-1};L^p(\R_+))\\
    &\hook\, H^{2,(2,1)}_p(J\times \R^{n-1};L^p(\R_+))
    \,\hook\, H^1_{p}(J;L^p(\R^n_+)).\end{aligned}$$ Applying once again [@koehnesaal Theorem 1.9] we see that this is fulfilled if $\mbox{ind}\bigl(W^{4-1/p,(2,1)}_p(J\times
\R^{n-1})\bigr)>0$. This, in turn, holds if $p>(n+2)/4$ which is implied by our assumption $p\ge(n+2)/3$. Thus, (\[nonlinginhm1\]) follows.

Altogether we have proved that $(f_v,g,0,0,f_\eta,v_0,\eta_0,\eta_1)\in B_{\widetilde\mF}(0,\kappa)$ satisfying the compatibility conditions (C2)-(C4) and (\[nonlincd\]) implies that $N(w)+f\in\mF$ for $w\in\overline{B_\mE(0,r)}$. Hence, (\[fullnlsys\]) is well-defined.

Now, we set $$K(w)=L^{-1}(N(w)+f), \quad w\in\overline{B_\mE(0,r)}$$ and prove that it is a contraction on $\overline{B_\mE(0,r)}$ for $r>0$ small enough. Theorem \[2.2\] yields that $L\in\sLis(\mE,\mF)$. This and the mean value theorem imply $$\begin{aligned}
    \|K(w)-K(z)\|_\mE&\le C\|N(w)-N(z)\|_\mE\\
    &\le C\sup_{v\in B_\mE(0,r)}\|DN(v)\|_{\sL(\mE,\widetilde{\mF})}
    \|w-z\|_\mE\quad (w,z\in \overline{B_\mE(0,r)}).\end{aligned}$$ Fixing $r>0$ such that $\sup_{v\in B_\mE(0,r)}\|DN(v)\|_{\sL(\mE,\widetilde{\mF})}\le 1/2C$, which is possible thanks to Theorem \[mappropnl\], we see that $K$ is contractive. The estimate above and Theorem \[mappropnl\] also imply $$\begin{aligned}
    \|K(w)\|_\mE
    &\le \|K(w)-K(0)\|_\mE+ C\|f\|_\mF\\
    &\le \frac{r}2+ C\kappa
    \quad (w\in \overline{B_\mE(0,r)}).\end{aligned}$$ Choosing $\kappa\le r/2C$ we see that $K$ is indeed a contraction on $\overline{B_\mE(0,r)}$. The contraction mapping principle gives the result.

By the equivalence of the systems (\[fsi\]) and (\[tfsi\]) given through the diffeomorphic transform introduced in Section \[sectrans\], it is clear that Theorem \[mainnonlinsys\] implies our main result Theorem \[main\].
