1 Introduction

1.1 The fluid–structure interaction problem

The interactions of fluids with elastic structures are important for many applications ranging from hydro- and aero-elasticity over bio-mechanics to hydrodynamics. We are interested in the case, where a viscous incompressible fluid interacts with a flexible shell which is located at one part of the boundary (or even describes the complete boundary) of the underlying domain \(\Omega \subset \mathbb {R}^2\) denoted by \(\omega .\) The shell, described by a function \(\eta :(0,T)\times \omega \rightarrow \mathbb {R},\) reacts to the surface forces induced by the fluid and deforms the domain \(\Omega \) to \(\Omega _{\eta (t)},\) where the function \({\varvec{\varphi }}_{\eta (t)}\) describes the coordinate transform (see Figs.  and below) and \(\textbf{n}_\eta \) is the normal at the deformed boundary.

The motion of the fluid is governed by the Navier–Stokes equations

$$\begin{aligned} \varrho _f\big (\partial _t \textbf{u}+(\textbf{u}\cdot \nabla )\textbf{u}\big )=\mu \Delta \textbf{u}-\nabla \pi +\textbf{f},\quad {{\,\textrm{div}\,}}\textbf{u}=0, \end{aligned}$$

in the moving domain \(\Omega _\eta ,\) where \(\textbf{u}:(0,T)\times \Omega _\eta \rightarrow \mathbb {R}^2\) is the velocity field and \(\pi :(0,T)\times \Omega _\eta \rightarrow \mathbb {R}\) the pressure function. The function \(\textbf{f}:(0,T)\times \mathbb {R}^2\rightarrow \mathbb {R}^2\) is a given volume force. The equations are supplemented with initial conditions and the boundary condition \(\textbf{u}\circ {\varvec{\varphi }}_\eta =\partial _t\eta \textbf{n}\) at the flexible part of the boundary with normal \(\textbf{n}\) (see Sect. 2.6 for the precise definition of a solution). There exist various models in literature to model the behaviour of the shell and a typical example is given by

$$\begin{aligned} \varrho _s \partial _t^2\eta -\gamma \partial _t\partial _y^2\eta +\alpha \partial _y^4\eta =g-\textbf{n}{\varvec{\tau }}\circ {\varvec{\varphi }}_\eta \textbf{n}_\eta \,{\textrm{d}}y_{\textbf{n}_\eta } \end{aligned}$$

on \(\omega \) supplemented with initial and boundary conditions. Here \(\varrho _s,\gamma \) and \(\alpha \) are positive constants and the function \(g:(0,T)\times \omega \rightarrow \) is a given forcing term. The quantity \({\varvec{\tau }}\) denotes the Cauchy stress of the fluid given by Newton’s rheological law, that is \({\varvec{\tau }}=\mu \big (\nabla \textbf{u}+\nabla \textbf{u}^\top \big )-\pi {\mathbb {I}}_{2\times 2}.\) The model (1.1)–(1.2) has been suggested, in particular, for blood vessels (where the 2D geometry is often sufficient), see [8, 15].

There exists already results concerning the existence of local-in-time strong solutions to the coupled system (1.1)–(1.2), see [5, 6, 11, 13, 14].Footnote 1 Rather recently, even the existence of a global-in-time strong solution has been shown in [10]. All these papers are concerned with a simplified geometrical set-up, where the domain \(\Omega \) is given by a rectangle and the flexible part of the boundary is flat, see Fig. 1. In this case the transformation between the reference domain and the moving domain is particularly easy, which simplifies the mathematical analysis significantly. While it is natural to start the investigation which such an idealised model, this model is not suitable for most real-world applications such as blood vessels. In the case of a more realistic non-flat geometry as in Fig. 2 only the existence of weak solutions to (1.1)–(1.2) is known, see [12, 17, 18]\(^1.\) For a weak solution, the kinetic energy \(\Vert \textbf{u}\Vert _{L^2(\Omega _\eta )}\) is bounded, the velocity gradient belongs to \(L^2\) and we have

$$\begin{aligned} \eta \in W^{1,\infty } \big (I; L^2(\omega ) \big )\cap W^{1,2} \big (I; W^{1,2}(\omega ) \big )\cap L^\infty \big (I; W^{2,2}(\omega ) \big ). \end{aligned}$$

This can be seen formerly by testing the momentum equation by \(\textbf{u}\) and the shell equation by \(\partial _t\eta \) (note that the boundary terms cancel due to the condition \(\textbf{u}\circ {\varvec{\varphi }}_\eta =\partial _t\eta \textbf{n}\)). With a weak solution at our disposal we are confronted with the question whether it enjoys additional regularity properties (in this case we speak about strong solutions) and is, in fact, unique. These properties are not only of theoretical interest but also crucial for robust numerical simulations. The analysis of regularity properties of solutions to (1.1)–(1.2) in the framework of Fig. 2 is the purpose of the present paper. Our main result shows that under natural assumptions on the data there is a unique global-in-time strong solution to (1.1)–(1.2), see Theorem 2.5 for the precise statement. Here strong means that both equations hold in the strong sense, that is, all quantities exist as measurable functions. In particular, all terms in the momentum equation (1.1) belong, in fact, to \(L^2\) and we have

$$\begin{aligned} \eta \in W^{1,\infty } \big (I; W^{1,2}(\omega ) \big )&\cap W^{1,2} \big (I; W^{2,2}(\omega ) \big )\cap W^{2,2}\big (I;L^2(\omega )\big )\nonumber \\&\cap L^\infty \big (I; W^{3,2}(\omega ) \big )\cap L^2\big (I;W^{4,2}(\omega )\big ) . \end{aligned}$$
Fig. 1
figure 1

Domain transformation in the simplified set-up

Fig. 2
figure 2

Domain transformation in the general set-up

1.2 Stokes systems in irregular domains

As in [10] the crucial tool in our analysis of (1.1)–(1.2) is an elliptic estimate for the Stokes system

$$\begin{aligned} \mu \Delta \textbf{u}-\nabla \pi =-\textbf{g},\quad {{\,\textrm{div}\,}}\textbf{u}=0, \end{aligned}$$

in a bounded domain \({\mathcal {O}}\subset \mathbb {R}^n,\) \(n=2,3,\) (supplemented with homogeneous boundary conditions). To be more precise, we require an inequality of the form

$$\begin{aligned} \Vert \textbf{u}\Vert _{W^{2,p}({\mathcal {O}})}+\Vert \pi \Vert _{W^{1,p}({\mathcal {O}})}\lesssim \Vert \textbf{g}\Vert _{L^p({\mathcal {O}})} \end{aligned}$$

for \(p\in (1,\infty )\) (in fact, \(p=2\) is sufficient for the application to (1.1)–(1.2)). Such an estimate is well-known if the boundary of the underlying domains belongs to the class \(C^2.\) We will apply (1.6) to (1.1) with \(\textbf{g}=\varrho _f\big (\partial _t \textbf{u}+(\textbf{u}\cdot \nabla )\textbf{u}\big )\) and \({\mathcal {O}}=\Omega _{\eta (t)}\) for a fixed t. Hence the regularity of \({\mathcal {O}}\) is determined by \(\eta \) which only belongs to \(W^{2,2},\) see (1.3). A version of (1.6) for the simplified framework from Fig. 1 is proved in [10]. It is, however, based on some cancellations which are not available in the general case. On the other hand, the question about minimal assumptions on the regularity of \(\partial {\mathcal {O}}\) for (1.6) is of independent interest and seems to be missing in literature. The only comparable result can be found in [3] (which is, in turn, based on results from [7]), where an estimate for Lipschitz domains (that is, \(\partial {\mathcal {O}}\in W^{1,\infty }\)) is shown, which controls fractional derivatives (of order 3/2 for \(\textbf{u}\) and 1/2 for \(\pi \)). The method from [3] is, unfortunately, designed specifically for Lipschitz domains and does not seem to apply in a more general framework.

In Theorem 3.2 we offer an exhaustive picture concerning the maximal regularity theory for the Stokes system (1.5) in irregular domains in the framework of fractional Sobolev spaces. This is based on the theory of Sobolev multipliers from [16] which has not been used in fluid mechanics before. Our assumptions on the boundary coincide with those made in [16, Chapter 14] for the Laplace equation which are known to be optimal. As a special case we obtain estimate (1.6) for \(p=2\) provided the Lipschitz constant of \(\partial {\mathcal {O}}\) is small and \(\partial {\mathcal {O}}\) belongs—roughly speaking—to the class \(W^{3/2,2}\) (we will make these concepts precise in Sect. 3.1). The relation between both spaces is that \(W^{3/2,2}\) is the trace-space of \(W^{2,2}\) (the space for the velocity field in (1.6)) in the sense that the linear mapping

$$\begin{aligned} W^{2,2}({\mathbb {R}}^n)\ni \varphi \mapsto \varphi (\cdot ,0)\in W^{3/2,2}({\mathbb {R}}^{n-1}) \end{aligned}$$

is continuous.

1.3 The acceleration estimate

With estimate (1.6) at our disposal we return to the fluid–structure interaction problem (1.1)–(1.2). We aim at testing the structure equation (1.2) with \(\partial _t^2\eta \) and seek for an appropriate test-function for the momentum equation. Due to the condition \(\textbf{u}\circ {\varvec{\varphi }}_\eta =\partial _t\eta \textbf{n}\) at the boundary we have

$$\begin{aligned} \frac{\textrm{D}}{\textrm{D}t}\textbf{u}&=\partial _t\textbf{u}(t,x+\eta \textbf{n})+ (\textbf{u}(t,x+\eta \textbf{n})\cdot \nabla )\textbf{u}(t,x+\eta \textbf{n})\\&=\partial _t\textbf{u}(t,x+\eta \textbf{n})+\partial _t\eta \textbf{n}\cdot \nabla \textbf{u}(t,x+\eta \textbf{n})\quad \text {on}\ \partial \Omega _\eta . \end{aligned}$$

Hence the material derivative of the velocity field is the corresponding test-function for the momentum equation. To be precise we use

$$\begin{aligned} \partial _t\textbf{u}+({\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\cdot \nabla )\textbf{u}, \end{aligned}$$

where \({\mathscr {F}}_\eta \) is an appropriate extension operator, see Sect. 2.5. The drawback with the function in (1.8) is that it is not solenoidal. This problem is overcome in [10] by using instead

$$\begin{aligned} \partial _t\textbf{u}+({\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\cdot \nabla ) \textbf{u}-(\textbf{u}\cdot \nabla ){\mathscr {F}}_\eta (\partial _t\eta \textbf{n}). \end{aligned}$$

In fact, in the simplified geometric set-up used in [10] it is possible to construct an extension which is at the same time solenoidal and satisfies

$$\begin{aligned} \nabla {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\cdot \textbf{n}=0\quad \text {on}\ \partial \Omega _\eta . \end{aligned}$$

In conclusion, the function in (1.9) is solenoidal and equals to \(\partial _t^2\eta \) at the boundary. In the general case, some elementary calculations based on Fourier expansion reveal that both conditions cannot hold simultaneously. Therefor, the existing extension operators from [12, 18] are solenoidal but do not satisfy (1.10). In Sect. 5.1 we propose an alternative approach which is based on (1.8) and an elementary extension operator introduced in Sect. 2.5. It is not solenoidal but, different to those from [12, 18], has the usual regularisation property (which is inverse to the trace embedding from (1.7)). Accordingly, me must introduce the pressure function and estimate it. This can be done with the help of (1.6), see the proof of Proposition 5.1 for details.

In order to implement these ideas rigorously we first prove the existence of a local-in-time strong solution in Sect. 4. As in previous papers, where the flat geometry is considered, we follow a standard approach based on a transformation of (1.1) to the reference geometry, linearisation and a fixed point argument. Our situation is, however, technically more complicated due to the non-trivial transformation map between the reference and moving geometry.

2 Preliminaries

2.1 Conventions

For notational simplicity we set all physical constants in (1.1)–(1.2) to 1. The analysis is not effected as long as they are strictly positive. We write \(f\lesssim g\) for two non-negative quantities f and g if there is a \(c>0\) such that \(f\le \,c g.\) Here c is a generic constant which does not depend on the crucial quantities. If necessary we specify particular dependencies. We write \(f\approx g\) if \(f\lesssim g\) and \(g\lesssim f.\) We do not distinguish in the notation for the function spaces between scalar- and vector-valued functions. However, vector-valued functions will usually be denoted in bold case. For simplicity we supplement (1.2) with periodic boundary conditions and identify \(\omega \) (which represents the complete boundary of \(\Omega \)) with the interval (0, 1). We consider periodic function spaces for zero-average functions. It is only a technical matter to consider instead (1.2) on a nontrivial subset of \(\partial \Omega \) together with zero boundary conditions for \(\eta \) and \(\partial _y\eta ,\) see, e.g., [12] or [2] for the corresponding geometrical set-up. We shorten the time interval (0, T) by I.

2.2 Classical function spaces

Let \({\mathcal {O}}\subset \mathbb {R}^m,\) \(m\ge 1,\) be open. Function spaces of continuous or \(\alpha \)-Hölder-continuous functions, \(\alpha \in (0,1),\) are denoted by \(C(\overline{{\mathcal {O}}})\) or \(C^{0,\alpha }(\overline{{\mathcal {O}}})\) respectively. Similarly, we write \(C^1(\overline{{\mathcal {O}}})\) and \(C^{1,\alpha }(\overline{{\mathcal {O}}})\) for spaces of functions which are (\(\alpha \)-Hölder) continuously differentiable. We denote as usual by \(L^p({\mathcal {O}})\) and \(W^{k,p}({\mathcal {O}})\) for \(p\in [1,\infty ]\) and \(k\in {\mathbb {N}}\) Lebesgue and Sobolev spaces over \({\mathcal {O}}.\) For a bounded domain \({\mathcal {O}}\) the space \(L^p_\perp ({\mathcal {O}})\) denotes the subspace of \(L^p({\mathcal {O}})\) of functions with zero mean, that is \(.\) We denote by \(W^{s,p}_0({\mathcal {O}})\) the closure of the smooth and compactly supported functions in \(W^{s,p}({\mathcal {O}}).\) If \(\partial {\mathcal {O}}\) is regular enough, this coincides with the functions vanishing \({\mathcal {H}}^{m-1}\)-a.e. on \(\partial {\mathcal {O}}.\) We also denote by \(W^{-k,p}({\mathcal {O}})\) the dual of \(W^{k,p}_0({\mathcal {O}}).\) Finally, we consider subspaces \(W^{1,p}_{{{\,\textrm{div}\,}}}({\mathcal {O}})\) and \(W^{1,p}_{0,{{\,\textrm{div}\,}}}({\mathcal {O}})\) of divergence-free vector fields which are defined accordingly. The space \(L^p_{{{\,\textrm{div}\,}}}({\mathcal {O}})\) is defined as the closure of the smooth and compactly supported solenoidal functions in \(L^p({\mathcal {O}}).\) We will use the shorthand notations \(L^p_x\) and \(W^{k,p}_x\) in the case of n-dimensional domains (typically spaces defined over \(\Omega \subset \mathbb {R}^n\) or \(\Omega _\eta \subset \mathbb {R}^n\)) and \(L^p_y\) and \(W^{k,p}_y\) for \((n-1)\) dimensional sets (typically spaces of periodic functions defined over \(\omega \subset \mathbb {R}\)).

For a separable Banach space \((X,\Vert \cdot \Vert _X)\) we denote by \(L^p(0,T;X)\) the set of (Bochner-) measurable functions \(u:(0,T)\rightarrow X\) such that the mapping \(t\mapsto \Vert u(t)\Vert _{X}\in L^p(0,T).\) The set C([0, T]; X) denotes the space of functions \(u:[0,T]\rightarrow X\) which are continuous with respect to the norm topology on \((X,\Vert \cdot \Vert _X).\) For \(\alpha \in (0,1]\) we write \(C^{0,\alpha }([0,T];X)\) for the space of Hölder-continuous functions with values in X. The space \(W^{1,p}(0,T;X)\) consists of those functions from \(L^p(0,T;X)\) for which the distributional time derivative belongs to \(L^p(0,T;X)\) as well. The space \(W^{k,p}(0,T;X)\) is defined accordingly. We use the shorthand \(L^p_tX\) for \(L^p(0,T;X).\) For instance, we write \(L^p_tW^{1,p}_x\) for \(L^p(0,T;W^{1,p}({\mathcal {O}})).\) Similarly, \(W^{k,p}_tX\) stands for \(W^{k,p}(0,T;X).\)

2.3 Fractional differentiability and Sobolev multipliers

For \(p\in [1,\infty )\) the fractional Sobolev space (Sobolev–Slobodeckij space) with differentiability \(s>0\) with \(s\notin {\mathbb {N}}\) will be denoted by \(W^{s,p}({\mathcal {O}}).\) For \(s>0\) we write \(s=\lfloor s\rfloor +\lbrace s\rbrace \) with \(\lfloor s\rfloor \in \mathbb {N}_0\) and \(\lbrace s\rbrace \in (0,1).\) We denote by \(W^{s,p}_0({\mathcal {O}})\) the closure of the smooth and compactly supported functions in \(W^{1,p}({\mathcal {O}}).\) For \(s>\frac{1}{p}\) this coincides with the functions vanishing \({\mathcal {H}}^{m-1}\)-a.e. on \(\partial {\mathcal {O}}\) provided \(\partial {\mathcal {O}}\) is regular enough. We also denote by \(W^{-s,p}({\mathcal {O}})\) for \(s>0\) the dual \(W^{s,p}(\mathcal O)\). Similar to the case of unbroken differentiabilities above we use the shorthand notations \(W^{s,p}_x\) and \(W^{s,p}_y.\) We will denote by \(B^s_{p,q}(\mathbb {R}^m)\) the standard Besov spaces on \(\mathbb {R}^m\) with differentiability \(s>0,\) integrability \(p\in [1,\infty ]\) and fine index \(q\in [1,\infty ].\) They can be defined (for instance) via Littlewood-Paley decomposition leading to the norm \(\Vert \cdot \Vert _{B^s_{p,q}(\mathbb {R}^m)}.\) We refer to [19,20,21] for an extensive picture. The Besov spaces \(B^s_{p,q}({\mathcal {O}})\) for a bounded domain \({\mathcal {O}}\subset \mathbb {R}^m\) are defined as the restriction of functions from \(B^s_{p,q}(\mathbb {R}^m),\) that is

$$\begin{aligned} B^s_{p,q}({\mathcal {O}})&:=\{f|_{{\mathcal {O}}}:\,f\in B^s_{p,q}(\mathbb {R}^m)\},\\ \Vert g\Vert _{B^s_{p,q}({\mathcal {O}})}&:=\inf \{ \Vert f\Vert _{B^s_{p,q}(\mathbb {R}^m)}:\,f|_{{\mathcal {O}}}=g\}. \end{aligned}$$

If \(s\notin {\mathbb {N}}\) and \(p\in (1,\infty )\) we have \(B^s_{p,p}({\mathcal {O}})=W^{s,p}({\mathcal {O}}).\)

In accordance with [16, Chapter 14] the Sobolev multiplier norm is given by

$$\begin{aligned} \Vert \varphi \Vert _{{\mathcal {M}}(W^{s,p}({\mathcal {O}}))}:= \sup _{\textbf{v}:\,\Vert \textbf{v}\Vert _{W^{s-1,p}({\mathcal {O}})}=1} \Vert \nabla \varphi \cdot \textbf{v}\Vert _{W^{s-1,p}({\mathcal {O}})}, \end{aligned}$$

where \(p\in [1,\infty ]\) and \(s\ge 1.\) The space \({\mathcal {M}}(W^{s,p}({\mathcal {O}}))\) of Sobolev multipliers is defined as those objects for which the \({\mathcal {M}}(W^{s,p}({\mathcal {O}}))\)-norm is finite. By mathematical induction with respect to s one can prove for Lipschitz-continuous functions \(\varphi \) that membership to \({\mathcal {M}}(W^{s,p}({\mathcal {O}}))\) in the sense of (2.1) implies that

$$\begin{aligned} \sup _{w:\,\Vert w\Vert _{W^{s,p}({\mathcal {O}})}=1}\Vert \varphi \,w\Vert _{W^{s,p}({\mathcal {O}})}<\infty . \end{aligned}$$

The quantity (2.2) also serves as customary definition of the Sobolev multiplier norm in the literature but (2.1) is more suitable for our purposes. Note that in our applications we always assume that the functions in question are Lipschitz continuous such that the implication above is given.

Let us finally collect some useful properties of Sobolev multipliers. By [16, Corollary 14.6.2] we have

$$\begin{aligned} \Vert \phi \Vert _{{\mathcal {M}}(W^{s,p}(\mathbb {R}^{m}))}\lesssim \Vert \nabla \phi \Vert _{L^{\infty }(\mathbb {R}^m)}, \end{aligned}$$

provided that one of the following conditions holds:

  • \(p(s-1)<m\) and \(\phi \in B^{s}_{\varrho ,p}(\mathbb {R}^{m})\) with \(\varrho \in [\frac{m}{s-1}, \infty ]\);

  • \(p(s-1)=m\) and \(\phi \in B^{s}_{\varrho ,p}(\mathbb {R}^m)\) with \(\varrho \in (p,\infty ].\)

Note that the hidden constant in (2.3) depends on the \(B^{s}_{\varrho ,p}(\mathbb {R}^{m})\)-norm of \(\phi .\) By [16, Corollary 4.3.8] it holds

$$\begin{aligned} \Vert \phi \Vert _{{\mathcal {M}}(W^{s,p}(\mathbb {R}^{m}))}\approx \Vert \nabla \phi \Vert _{W^{s-1,p}(\mathbb {R}^{m})} \end{aligned}$$

for \(p(s-1)>m.\) Finally, we note the following rule about the composition with Sobolev multipliers which is a consequence of [16, Lemma 9.4.1]. For open sets \({\mathcal {O}}_1,{\mathcal {O}}_2\subset \mathbb {R}^m,\) \(u\in W^{s,p}({\mathcal {O}}_2)\) and a Lipschitz continuous function \({\varvec{\phi }}:{\mathcal {O}}_1\rightarrow {\mathcal {O}}_2\) with Lipschitz continuous inverse and \({\varvec{\phi }}\in {\mathcal {M}}(W^{s,p}({\mathcal {O}}_1))\) we have

$$\begin{aligned} \Vert u\circ {\varvec{\phi }}\Vert _{W^{s,p}({\mathcal {O}}_1)}\lesssim \Vert u\Vert _{W^{s,p}({\mathcal {O}}_2)} \end{aligned}$$

with constant depending on \({\varvec{\phi }}.\) Using Lipschitz continuity of \({\varvec{\phi }}\) and \({\varvec{\phi }}^{-1},\) estimate (2.5) is obvious for \(s\in (0,1].\) The general case can be proved by mathematical induction with respect to s.

2.4 Function spaces on variable domains

The spatial domain \(\Omega \) is assumed to be an open bounded subset of \({\mathbb {R}}^n,\) \(n=2,3,\) with smooth boundary and an outer unit normal \({\textbf{n}}.\) Wee assume that \(\partial \Omega \) can be parametrised by an injective mapping \({{\varvec{\varphi }}}\in C^k(\omega ;\mathbb {R}^n)\) for some sufficiently large \(k\in \mathbb {N}.\) If \(n=3\) we suppose for all points \(y=(y_1,y_2)\in \omega \) that the pair of vectors \(\partial _i {{\varvec{\varphi }}}(y),\) \(i=1,2,\) are linearly independent. If \(n=2\) the corresponding assumption simply asks for \(\partial _y{\varvec{\varphi }}\) not to vanish. For a point x in the neighborhood or \(\partial \Omega \) we can define the functions y and s by

$$\begin{aligned} y(x)=\arg \min _{y\in \omega }|x-{\varvec{\varphi }}(y)|,\quad s(x)=(x-y(x))\cdot \textbf{n}(y(x)). \end{aligned}$$

Moreover, we define the projection \(\textbf{p}(x)={\varvec{\varphi }}(y(x)).\) We define \(L>0\) to be the largest number such that sy and \(\textbf{p}\) are well-defined on \(S_L,\) where

$$\begin{aligned} S_L=\{x\in \mathbb {R}^n:\,{\textrm{dist}}(x,\partial \Omega )<L\}. \end{aligned}$$

Due to the smoothness of \(\partial \Omega \) for L small enough we have \({|{s(x)}|}=\min _{y\in \omega }|x-{\varvec{\varphi }}(y)|\) for all \(x\in S_L.\) This implies that \(S_L=\{s\textbf{n}(y)+y:(s,y)\in (-L,L)\times \omega \}.\) For a given function \(\eta : I \times \omega \rightarrow \mathbb {R}\) we parametrise the deformed boundary by

$$\begin{aligned} {{\varvec{\varphi }}}_\eta (t,y)={{\varvec{\varphi }}}(y) + \eta (t,y){\textbf{n}}(y), \quad y \in \omega ,\,t\in I. \end{aligned}$$

By possibly decreasing L,  one easily deduces from this formula that \(\Omega _{\eta }\) does not degenerate, that is

$$\begin{aligned} \partial _y{\varvec{\varphi }}_\eta (t,y)\ne 0\quad \text {if} \ n&=2,\quad \partial _1{\varvec{\varphi }}_\eta \times \partial _2{\varvec{\varphi }}_\eta (t,y)\ne 0\quad \text {if} \ n=3,\nonumber \\ \textbf{n}(y)\cdot \textbf{n}_{\eta (t)}(y)&>0,\quad \,y \in \omega ,\,t\in I, \end{aligned}$$

provided \(\Vert \eta \Vert _{L^\infty _{t,x}}<L.\) Here \(\textbf{n}_{\eta (t)}\) is the normal of the domain \(\Omega _{\eta (t)}\) defined through

$$\begin{aligned} \partial \Omega _{\eta (t)}={\{{{{\varvec{\varphi }}}(y) + \eta (t,y){\textbf{n}}(y):y\in \omega }\}}. \end{aligned}$$

With some abuse of notation we define deformed space-time cylinder

$$\begin{aligned} I\times \Omega _\eta =\bigcup _{t\in I}{\{{t}\}}\times \Omega _{\eta (t)}\subset \mathbb {R}^{1+n}. \end{aligned}$$

The corresponding function spaces for variable domains are defined as follows.

Definition 2.1

(Function spaces) For \(I=(0,T),\) \(T>0,\) and \(\eta \in C({\overline{I}}\times \omega )\) with \(\Vert \eta \Vert _{L^\infty (I\times \omega )}< L\) we define for \(1\le p,r\le \infty \)

$$\begin{aligned} L^p(I;L^r(\Omega _\eta ))&:=\Big \{v\in L^1(I\times \Omega _\eta ): \begin{array}{c} v(t,\cdot )\in L^r(\Omega _{\eta (t)})\,\,\text {for a.e. }t,\\ \Vert v(t,\cdot )\Vert _{L^r(\Omega _{\eta (t)})}\in L^p(I) \end{array}\Big \},\\ L^p(I;W^{1,r}(\Omega _\eta ))&:=\big \{v\in L^p(I;L^r(\Omega _\eta )):\,\,\nabla v\in L^p(I;L^r(\Omega _\eta ))\big \}. \end{aligned}$$

For various purposes it is useful to relate the time dependent domain and the fixed domain. This can be done by the means of the Hanzawa transform. Its construction can be found in [12, pages 210, 211]. Note that variable domains in [12] (see also [1]) are defined via functions \(\zeta :\partial \Omega \rightarrow \mathbb {R}\) rather than functions \(\eta :\omega \rightarrow \mathbb {R}\) (clearly, one can link them by setting \(\zeta =\eta \circ {\varvec{\varphi }}^{-1}\)). For any \(\eta :\omega \rightarrow (-L,L)\) we define the Hanzawa transform \({\varvec{\Psi }}_\eta : \Omega \rightarrow \Omega _\eta \) by

$$\begin{aligned} {\varvec{\Psi }}_\eta (x){=} {\left\{ \begin{array}{ll}\textbf{p}(x)+\Big (s(x)+\eta (y(x))\phi (s(x))\Big ) \textbf{n}(y(x)),&{}\quad \text {if } \text {dist}(x,\partial \Omega )<L,\\ \quad x, &{}\quad \text {elsewhere}.\\ \end{array}\right. }\nonumber \\ \end{aligned}$$

Here \(\phi \in C^\infty ({\mathbb {R}})\) is such that \(\phi \equiv 0\) in neighborhood of \(-L\) and \(\phi \equiv 1\) in a neighborhood of 0. Due to the size of L,  we find that \({\varvec{\Psi }}_\eta \) is a homomorphism such that \({\varvec{\Psi }}_\eta |_{\Omega \setminus S_L}\) is the identity. We clearly have for \(k\in \mathbb {N}\) and \(p\in [1,\infty ]\)

$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta \Vert _{W^{k,p}_x}\lesssim 1+\Vert \eta \Vert _{W^{k,p}_y},\quad \eta \in W^{k,p}(\omega ), \end{aligned}$$

as well as

$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta -{\varvec{\Psi }}_\zeta \Vert _{W^{k,p}_x}\lesssim \Vert \eta -\zeta \Vert _{W^{k,p}_y},\quad \eta ,\zeta \in W^{k,p}(\omega ), \end{aligned}$$

where the hidden constant only depends on the reference geometry.

If \(\Vert \eta \Vert _{L^\infty _y}<\alpha <L\) and \(\Vert \nabla \eta \Vert _{L^\infty _y}< R\) for some \(\alpha ,R>0\) the inverseFootnote 2\({\varvec{\Psi }}_\eta ^{-1}:\Omega _\eta \rightarrow \Omega \) satisfies for \(k\in \mathbb {N}\) and \(p\in [1,\infty ]\)

$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta ^{-1}\Vert _{W^{k,p}_x}\lesssim 1+\Vert \eta \Vert _{W^{k,p}_y},\quad \eta \in W^{k,p}(\omega ), \end{aligned}$$

as well as

$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta ^{-1}-{\varvec{\Psi }}^{-1}_\zeta \Vert _{W^{k,p}_x}\lesssim \Vert \eta -\zeta \Vert _{W^{k,p}_y},\quad \eta ,\zeta \in W^{k,p}(\omega ), \end{aligned}$$

if \(\Vert \zeta \Vert _{L^\infty _y}<\alpha \) and \(\Vert \nabla \zeta \Vert _{L^\infty _y}< R.\) In (2.12) and (2.13) the hidden constant depends on the reference geometry (which is assumed to be sufficiently smooth), on \(L-\alpha \) and R. Similarly, we obtain fractional estimates, that is

$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta \Vert _{W^{s,p}_x}&\lesssim 1+\Vert \eta \Vert _{W^{s,p}_y},\quad \eta \in W^{s,p}(\omega ), \end{aligned}$$
$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta ^{-1}\Vert _{W^{s,p}_x}&\lesssim 1+\Vert \eta \Vert _{W^{s,p}_y},\quad \eta \in W^{s,p}(\omega ), \end{aligned}$$

for \(s>0 \) with \(s\notin \mathbb {N}\) and

$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta -{\varvec{\Psi }}_\zeta \Vert _{W^{s,p}_x}&\lesssim \Vert \eta -\zeta \Vert _{W^{k,p}_y},\quad \eta ,\zeta \in W^{s,p}(\omega ), \end{aligned}$$
$$\begin{aligned} \Vert {\varvec{\Psi }}_\eta ^{-1}-{\varvec{\Psi }}^{-1}_\zeta \Vert _{W^{s,p}_x}&\lesssim \Vert \eta -\zeta \Vert _{W^{s,p}_y},\quad \eta ,\zeta \in W^{s,p}(\omega ). \end{aligned}$$

Finally, it holds

$$\begin{aligned} \Vert \partial _t{\varvec{\Psi }}_\eta \Vert _{W^{s,p}_x}\lesssim 1+\Vert \partial _t\eta \Vert _{W^{s,p}_y},\quad \eta \in W^{1,1}(I;W^{s,p}(\omega )), \end{aligned}$$

uniformly in time.

2.5 Extensions on variable domains

In this subsection we construct an extension operator which extends functions from \(\omega \) to the moving domain \(\Omega _\eta \) for a given function \(\eta \) defined on \(\omega ,\) where \(n=2,3.\) We follow [2, Section 2.3]. Since \(\Omega \) is assumed to be sufficiently smooth, it is well-known that there is an extension operator \({\mathscr {F}}_\Omega \) which extends functions from \(\partial \Omega \) to \(\mathbb {R}^n\) and satisfies

$$\begin{aligned} {\mathscr {F}}_\Omega :W^{\sigma ,p}(\partial \Omega )\rightarrow W^{\sigma +1/p,p}(\mathbb {R}^n) \end{aligned}$$

for all \(p\in [1,\infty ]\) and \(\sigma >0\), as well as \({\mathscr {F}}_\Omega v|_{\partial \Omega }=v.\) Now we define \({\mathscr {F}}_\eta \) by

$$\begin{aligned} {\mathscr {F}}_\eta b={\mathscr {F}}_\Omega ((b\textbf{n})\circ {\varvec{\varphi }}^{-1})\circ {{\varvec{\Psi }}}_\eta ^{-1},\quad b\in W^{\sigma ,p}(\omega ), \end{aligned}$$

where \({\varvec{\varphi }}\) is the function in the parametrisation of \(\Omega .\) If \(\eta \) is regular enough, \({\mathscr {F}}_\eta \) behaves as a classical extension. To be more precise, we can use the formula

$$\begin{aligned} \nabla {\mathscr {F}}_\eta b=\nabla {\mathscr {F}}_\Omega ((b\textbf{n})\circ {\varvec{\varphi }}^{-1})\circ {{\varvec{\Psi }}}_\eta ^{-1}\nabla {{\varvec{\Psi }}}_\eta ^{-1}, \end{aligned}$$

estimate (2.12) and (2.19) to obtain the following.

Lemma 2.2

Let \(\eta \in C^{0,1}(\omega )\) with \(\Vert \eta \Vert _{L^\infty _y}<\alpha <L.\) The operator \({\mathscr {F}}_\eta \) defined in (2.20) satisfies for all \(p\in (1,\infty ],\)Footnote 3\(\sigma \in (0,1-\tfrac{1}{p}]\) and \(s\in (0,\tfrac{1}{p}),\)

$$\begin{aligned} {\mathscr {F}}_\eta : W^{\sigma ,p}(\omega )\rightarrow W^{\sigma +1/p,p}(\Omega \cup S_\alpha ),\quad {\mathscr {F}}_\eta : L^{p}(\omega )\rightarrow W^{s,p}(\Omega \cup S_\alpha ) \end{aligned}$$

and \(({\mathscr {F}}_\eta b)\circ {\varvec{\varphi }}_\eta =b\textbf{n}\) on \(\omega \) for all \(b\in L^{p}(\omega ).\) In particular,  we have

$$\begin{aligned} \Vert {\mathscr {F}}_\eta b\Vert _{W^{\sigma +1/p,p}(\Omega \cup S_\alpha )}\lesssim \Vert b\Vert _{W^{\sigma ,p}(\omega )},\quad \Vert {\mathscr {F}}_\eta b\Vert _{W^{s,p}(\Omega \cup S_\alpha )}\lesssim \Vert b\Vert _{L^{p}(\omega )}, \end{aligned}$$

where the hidden constant depends only on \(\Omega ,p,\sigma ,\) \(\Vert \nabla \eta \Vert _{L^\infty _y}\) and \(L-\alpha .\)

2.6 The concept of solutions and the main results

In this section we introduce the framework for the system (1.1)–(1.2) and present our main results concerning the regularity of solutions. We start with the definition of a weak solution. Note that different to the previous subsection we assume again that \(n=2.\)

Definition 2.3

(Weak solution) Let \((\textbf{f}, g, \eta _0, \textbf{u}_0, \eta _1)\) be a dataset such that

$$\begin{aligned}{} & {} \textbf{f}\in L^2\big (I; L^2_{\textrm{loc}}({\mathbb {R}}^2)\big ),\quad g \in L^2\big (I; L^2(\omega )\big ), \quad \eta _0 \in W^{2,2}(\omega ) \quad \text {with } \Vert \eta _0 \Vert _{L^\infty ( \omega )} < L,\nonumber \\{} & {} \textbf{u}_0\in L^2_{\mathrm {{{\,\textrm{div}\,}}}}(\Omega _{\eta _0}) \text { is such that }\textbf{u}_0\circ {\varvec{\varphi }}_{\eta _0} =\eta _1 \textbf{n}\text { on }\omega , \quad \eta _1 \in L^2(\omega ). \end{aligned}$$

We call the tuple \((\eta ,\textbf{u})\) a weak solution to the system (1.1)–(1.2) with data \((\textbf{f}, g, \eta _0, \eta _1,\textbf{u}_0)\) provided that the following holds:

  1. (a)

    The structure displacement \(\eta \) satisfies

    $$\begin{aligned} \eta \in W^{1,\infty }&\big (I; L^2(\omega ) \big )\cap W^{1,2} \big (I; W^{1,2}(\omega ) \big )\cap L^\infty \big (I; W^{2,2}(\omega ) \big )\\&\text {with} \Vert \eta \Vert _{L^\infty (I \times \omega )} <L, \end{aligned}$$

    as well as \(\eta (0)=\eta _0\) and \(\partial _t\eta (0)=\eta _1.\)

  2. (b)

    The velocity field \(\textbf{u}\) satisfies

    $$\begin{aligned} \textbf{u}\in L^\infty \big (I; L^2(\Omega _{\eta }) \big )\cap L^2 \big (I; W^{1,2}_{{{\,\textrm{div}\,}}}(\Omega _{\eta }) \big ) \quad \text {with} \quad \textbf{u}\circ {\varvec{\varphi }}_\eta =\partial _t \eta {\textbf{n}}\quad \text {on}\quad I\times \omega , \end{aligned}$$

    as well as \(\textbf{u}(0)=\textbf{u}_0.\)

  3. (c)

    For all \((\phi , {{\varvec{\phi }}}) \in C^\infty ({\overline{I}}\times \omega ) \times C^\infty _{{{\,\textrm{div}\,}}}({\overline{I}}\times \mathbb {R}^2; \mathbb {R}^2)\) with \(\phi (T,\cdot )=0,\) \({{\varvec{\phi }}}(T,\cdot )=0\) and \({\varvec{\phi }}\circ {\varvec{\varphi }}_\eta =\phi {\textbf{n}}\) on \(I\times \omega ,\) we have

    $$\begin{aligned}&\int _I \frac{{\textrm{d}}}{\,{\textrm{d}} t}\bigg (\int _{\Omega _{\eta }}\textbf{u}\cdot {{\varvec{\phi }}}\,{\textrm{d}}x+\int _\omega \partial _t \eta \, \phi \,{\textrm{d}}y\bigg )\,{\textrm{d}} t\\&\quad =\int _I \int _{\Omega _{\eta }}\big ( \textbf{u}\cdot \partial _t {{\varvec{\phi }}} + \textbf{u}\otimes \textbf{u}: \nabla {{\varvec{\phi }}} \big ) \,{\textrm{d}}x\,{\textrm{d}} t\\&\qquad \ -\int _I \int _{\Omega _{\eta }}\big ( \nabla \textbf{u}:\nabla {{\varvec{\phi }}} -\textbf{f}\cdot {{\varvec{\phi }}} \big ) \,{\textrm{d}}x\,{\textrm{d}} t\\&\qquad \ + \int _I \int _\omega \big (\partial _t \eta \, \partial _t\phi -\partial _t\partial _y\eta \,\partial _y \phi - g\, \phi \big )\,{\textrm{d}}y\,{\textrm{d}} t\\&\qquad \ -\int _I\int _\omega \partial _y^2\eta \,\partial _y^2 \phi \,{\textrm{d}}y\,{\textrm{d}} t. \end{aligned}$$

The existence of a weak solution can be shown as in [12]. The term \(\partial _t\partial _y^2\eta \) is not included there, but it does not alter the arguments. Note that we use a pressure-free formulation (that is, with test-function satisfying additionally \({{\,\textrm{div}\,}}{\varvec{\phi }}=0\)) here. If the solution possess more regularity, the pressure can be recovered by setting

$$\begin{aligned} {\tilde{\pi }}_0:=\Delta _{\eta }^{-1}{{\,\textrm{div}\,}}((\nabla \textbf{u})\textbf{u}),\quad \pi _0:={\tilde{\pi }}_0-({\tilde{\pi }}_0)_{\Omega _{\eta }}. \end{aligned}$$

For \({\mathcal {O}}\subset \mathbb {R}^n\) open and bounded with normal \(\textbf{n}_{{\mathcal {O}}}\) we denote by \(\Delta ^{-1}_{{\mathcal {O}}}{{\,\textrm{div}\,}}\) the solution operator to the equation

$$\begin{aligned} \Delta h={{\,\textrm{div}\,}}\textbf{g}\quad \text {in}\ {\mathcal {O}},\qquad \textbf{n}_{{\mathcal {O}}}\cdot (\nabla h-\textbf{g})=0\quad \text {on}\ \partial {\mathcal {O}}. \end{aligned}$$

We must complement \(\pi _0\) by a function depending on time only being uniquely determined by the structure equation. Setting \(\pi (t)=\pi _0(t)+c_\pi (t)\) and testing the structure equation with 1 we obtain

$$\begin{aligned} c_\pi (t)\int _{\omega }\textbf{n}\cdot \textbf{n}_\eta |\partial _y{\varvec{\varphi }}_\eta |\,{\textrm{d}}y&=\int _{\omega }\textbf{n}\big (\nabla \textbf{u}+\nabla \textbf{u}^\top -\pi _0{\mathbb {I}}_{2\times 2}\big )\circ {\varvec{\varphi }}_\eta \textbf{n}_\eta |\partial _y{\varvec{\varphi }}_\eta |\,{\textrm{d}}y\nonumber \\&\quad +\int _\omega \partial _t^2\eta \,{\textrm{d}}y-\int _\omega g\,{\textrm{d}}y. \end{aligned}$$

Since \(\Omega _\eta \) is Lipschitz uniformly in time the operator \(\Delta _{\Omega \eta }^{-1}{{\,\textrm{div}\,}}\) has the usual properties. In particular, it is continuous \(L^2\rightarrow W^{1,2}\) such that

$$\begin{aligned} \int _I\int _{\Omega _\eta }|\nabla \pi |^2\,{\textrm{d}}x\,{\textrm{d}}t&\lesssim \int _I\int _{\Omega _\eta }|(\nabla \textbf{u})\textbf{u})|^2\,{\textrm{d}}x\,{\textrm{d}}t\\&\lesssim \bigg ( \int _I\Vert \textbf{u}\Vert ^4_{L^4(\Omega _\eta )}\bigg )^{\frac{1}{2}}\bigg (\int _I\Vert \nabla \textbf{u}\Vert ^4_{L^{4}(\Omega _\eta )}\,{\textrm{d}} t\bigg )^{\frac{1}{2}} \end{aligned}$$

by Ladyshenskaya’s inequality (using again that \(\Omega _\eta \) is Lipschitz uniformly in time). Hence we have \(\pi \in L^2(I\;W^{1,2}(\Omega _\eta ))\) provided the right-hand side is finite (which is the case if \(\textbf{u}\) and \(\nabla \textbf{u}\) belong to \(L^4\) in space-time). This is the case for a strong solution which is defined as follows.

Definition 2.4

(Strong solution) We call the triple \((\eta ,\textbf{u},\pi )\) a strong solution to (1.1)–(1.2) provided \((\eta ,\textbf{u})\) is a weak solution to (1.1)–(1.2), it satisfies

$$\begin{aligned}&\eta \in W^{1,\infty } \big (I; W^{1,2}(\omega ) \big )\cap W^{1,2} \big (I; W^{2,2}(\omega ) \big )\cap L^\infty \big (I; W^{3,2}(\omega ) \big ) \cap W^{2,2}(I;L^2(\omega )),\\&\textbf{u}\in W^{1,2} \big (I; L^{2}(\Omega _{\eta }) \big )\cap L^2 \big (I; W^{2,2}(\Omega _{\eta }) \big ),\quad \pi \in L^2 \big (I; W^{1,2}(\Omega _{\eta }) \big ), \end{aligned}$$

and we have \(\nabla \pi =\nabla \Delta _{\Omega _\eta }^{-1}{{\,\textrm{div}\,}}((\nabla \textbf{u})\textbf{u}).\)

For a strong solution \((\eta ,\textbf{u},\pi )\) the momentum equation holds in the strong sense, that is we have

$$\begin{aligned} \partial _t \textbf{u}+(\nabla \textbf{u})\textbf{u}=\Delta \textbf{u}-\nabla \pi +\textbf{f} \end{aligned}$$

a.a. in \(I\times \Omega _\eta .\) The beam equation together with the regularity properties above yield \(\eta \in L^2(I;W^{4,2}(\omega )).\) Hence the beam equation holds in the strong sense as well, that is we have

$$\begin{aligned} \partial _t^2\eta -\partial _t\partial _y^2\eta +\partial _y^4\eta =g-\textbf{n}{\varvec{\tau }}\circ {\varvec{\varphi }}_\eta \partial _y{\varvec{\varphi }}_\eta ^\perp \end{aligned}$$

a.a. in \(I\times \omega .\) Note that for a strong solution the Cauchy stress \({\varvec{\tau }}=\nabla \textbf{u}+\nabla \textbf{u}^\top -\pi {\mathbb {I}}_{2\times 2}\) possesses enough regularity to be evaluated at the moving boundary (this is due to the trace theorem and the uniform Lipschitz continuity of \(\Omega _\eta \)).

We are finally ready to state our main result concerning the existence of a unique strong solution to (1.1)–(1.2).

Theorem 2.5

Let \(n=2.\) Suppose that the dataset \((\textbf{f}, g, \eta _0, \textbf{u}_0, \eta _1)\) satisfies in addition to (2.21) that

$$\begin{aligned} g \in L^2\big (I; W^{1,2}(\omega )\big ), \quad \eta _0 \in W^{3,2}(\omega ) , \quad \eta _1 \in W^{1,2}(\omega ),\quad \textbf{u}_0\in W^{1,2}_{\mathrm {{{\,\textrm{div}\,}}}}(\Omega _{\eta _0}). \end{aligned}$$

Then there is a unique strong solution to (1.1)–(1.2) in the sense of Definition 2.4. The interval of existence is of the form \(I = (0, t),\) where \(t < T\) only in case \(\Omega _{\eta (s)}\) approaches a self-intersection when \(s\rightarrow t\) or it degeneratesFootnote 4 (namely,  if \(\lim _{s\rightarrow t}\partial _y{\varvec{\varphi }}_\eta (s,\omega )=0\) or \(\lim _{s\rightarrow t}\textbf{n}(y)\cdot \textbf{n}_{\eta (s)}(y)=0\) for some \(y\in \omega ).\)

The proof of Theorem 2.5 can be found in Sect. 5.

Remark 2.6

The result of Theorem 2.5 heavily hinges on the dissipative term \(-\partial _t\partial _y^2\eta \) in the shell equation. It is a difficult open problem to understand if a similar result can be achieved in the purely elastic case (that is, \(\gamma =0\) in (1.2)) even for the flat geometry studied in [10].

Remark 2.7

A drawback of Theorem 2.5 compared to the corresponding statement for the flat geometry from [10] is that we can currently not exclude a self-intersection of the moving domain for arbitrary times. It would be a of great interest to prove a distance estimate as in [10, Section 4.2] in the present set-up.

3 The Stokes equations in non-smooth domains

This section is devoted to the study of the Stokes equations in a domain \({\mathcal {O}}\subset \mathbb {R}^n,\) \(n=2,3,\) with minimal regularity. We start by introducing the necessary framework to parametrise the boundary of the underlying domain by local maps of a certain regularity. This yields, in particular, a rigorous definition of a \(B^s_{\rho ,q}\)-boundary. In Sect. 3.2 we consider the steady Stokes system. This will be crucial for the acceleration estimate for the fluid–structure problem in Sect. 5.1 (we explain in Remark 3.4 how to parametrise the sets \(\Omega _\eta \) introduced in Sect. 2.4 by local maps).

3.1 Parametrisation of domains

Let \({\mathcal {O}}\subset \mathbb {R}^n\) be a bounded open set. We assume that \(\partial {{\mathcal {O}}}\) can be covered by a finite number of open sets \({\mathcal {U}}^1,\dots ,{\mathcal {U}}^\ell \) for some \(\ell \in {\mathbb {N}},\) such that the following holds. For each \(j\in \{1,\dots ,\ell \}\) there is a reference point \(y^j\in \mathbb {R}^n\) and a local coordinate system \(\{e^j_1,\dots ,e_n^j\}\) (which we assume to be orthonormal and set \({\mathcal {Q}}_j=(e_1^j|\dots |e_n^j)\in {\mathbb {R}}^{n\times n}\)), a function \(\varphi _j:{\mathbb {R}}^{n-1}\rightarrow {\mathbb {R}}\) and \(r_j>0\) with the following properties:

  • (A1) There is \(h_j>0\) such that

    $$\begin{aligned} {\mathcal {U}}^j=\{x={\mathcal {Q}}_jz+y^j\in {\mathbb {R}}^n:\,z=(z',z_n)\in \mathbb {R}^n,\,|z'|<r_j,\, |z_n-\varphi _j(z')|<h_j\}. \end{aligned}$$
  • (A2) For \(x\in {\mathcal {U}}^j\) we have with \(z={\mathcal {Q}}_j^\top (x-y^j)\)

    • \(x\in \partial {{\mathcal {O}}}\) if and only if \(z_n=\varphi _j(z');\)

    • \(x\in {{\mathcal {O}}}\) if and only if \(0<z_n-\varphi _j(z')<h_j;\)

    • \(x\notin {{\mathcal {O}}}\) if and only if \(0>z_n-\varphi _j(z')>-h_j.\)

  • (A3) We have that

    $$\begin{aligned} \partial {{\mathcal {O}}}\subset \bigcup _{j=1}^\ell {\mathcal {U}}^j. \end{aligned}$$

In other words, for any \(x_0\in \partial {{\mathcal {O}}}\) there is a neighborhood U of \(x_0\) and a function \(\varphi :{\mathbb {R}}^{n-1}\rightarrow {\mathbb {R}}\) such that after translation and rotationFootnote 5

$$\begin{aligned} U \cap {{\mathcal {O}}} = U \cap G,\quad G = {\{{(x',x_n)\in \mathbb {R}^n \,:\, x' \in \mathbb {R}^{n-1}, x_n > \varphi (x')}\}}. \end{aligned}$$

The regularity of \(\partial {{\mathcal {O}}}\) will be described by means of local coordinates as just described.

Definition 3.1

Let \({{\mathcal {O}}}\subset \mathbb {R}^n\) be a bounded domain, \(s>0\) and \(1\le \rho ,q\le \infty .\) We say that \(\partial {{\mathcal {O}}}\) belongs to the class \(B^s_{\rho ,q}\) if there is \(\ell \in {\mathbb {N}}\) and functions \(\varphi _1,\dots ,\varphi _\ell \in B^s_{\rho ,q}({\mathbb {R}}^{n-1})\) satisfying (A1)(A3).

Clearly, a similar definition applies for a Lipschitz boundary (or a \(C^{1,\alpha }\)-boundary with \(\alpha \in (0,1)\)) by requiring that \(\varphi _1,\dots ,\varphi _\ell \in W^{1,\infty }({\mathbb {R}}^{n-1})\) (or \(\varphi _1,\dots ,\varphi _\ell \in C^{1,\alpha }({\mathbb {R}}^{n-1})\)). We say that the local Lipschitz constant of \(\partial {{\mathcal {O}}},\) denoted by \(\textrm{Lip}(\partial {{\mathcal {O}}}),\) is (smaller or) equal to some number \(L>0\) provided the Lipschitz constants of \(\varphi _1,\dots ,\varphi _\ell \) are not exceeding L. Our main result depends on the assumption of a sufficiently small local Lipschitz constant. While this seems rather restrictive at first glance, it appears quite natural when looking closer. Indeed, it holds, for instance, if the regularity of \(\partial {{\mathcal {O}}}\) is better than Lipschitz (such as \(C^{1,\alpha }\) for some \(\alpha >0\)). By means of the transformations \({\mathcal {Q}}_j\) introduced above, we can assume that the reference point \(y^j\) in question is the origin and that \(\nabla \varphi _j(0)=0.\) Choosing \(r_j\) in (A1) small enough (which can be achieved simply by allowing more sets in the cover \({\mathcal {U}}^1,\dots ,{\mathcal {U}}^l\)) we have

$$\begin{aligned} |\nabla \varphi _j(z')|=|\nabla \varphi _j(z')-\nabla \varphi _j(0)|\le \,r_j^\alpha [\nabla \varphi _j]_{C^\alpha }\ll 1 \end{aligned}$$

for all \(z'\) with \(|z'|\le r_j.\)

In order to describe the behaviour of functions defined in \({{\mathcal {O}}}\) close to the boundary we need to extend the functions \(\varphi _1,\dots ,\varphi _\ell \) from (A1)(A3) to the half space \({\mathbb {H}} := {\{{z = (z',z_n)\,:\, z_n > 0}\}}.\) Hence we are confronted with the task of extending a function \(\phi \,:\, \mathbb {R}^{n-1}\rightarrow \mathbb {R}\) to a mapping \(\Phi \,:\, {\mathbb {H}} \rightarrow \mathbb {R}^n\) that maps the 0-neighborhood in \({\mathbb {H}}\) to the \(x_0\)-neighborhood in \({{\mathcal {O}}}.\) The mapping \((z',0) \mapsto (z',\phi (z'))\) locally maps the boundary of \({\mathbb {H}}\) to the one of \(\partial {{\mathcal {O}}}.\) We extend this mapping using the extension operator of Maz’ya and Shaposhnikova [16, Section 9.4.3]. Let \(\zeta \in C^\infty _c(B_1(0'))\) with \(\zeta \ge 0\) and \(\int _{\mathbb {R}^{n-1}} \zeta (x')\,{\textrm{d}}x'=1.\) Let \(\zeta _t(x') := t^{-(n-1)} \zeta (x'/t)\) denote the induced family of mollifiers. We define the extension operator

$$\begin{aligned} ({\mathcal {T}}\phi )(z',z_n)=\int _{\mathbb {R}^{n-1}} \zeta _{z_n}(z'-y')\phi (y')\,{\textrm{d}}y',\quad (z',z_n) \in {\mathbb {H}}, \end{aligned}$$

where \(\phi :\mathbb {R}^n\rightarrow \mathbb {R}\) is a Lipschitz function with Lipschitz constant K. Then the estimate

$$\begin{aligned} {\Vert {\nabla ({\mathcal {T}} \phi )}\Vert }_{B_{\rho ,q}^{s}(\mathbb {R}^{n})}\le c{\Vert {\nabla \phi }\Vert }_{B_{\rho ,q}^{s-\frac{1}{p}}(\mathbb {R}^{n-1})} \end{aligned}$$

follows from [16, Theorem 8.7.2]. Moreover, [16, Theorem 8.7.2] yields

$$\begin{aligned} \Vert {\mathcal {T}}\phi \Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\lesssim \Vert \phi \Vert _{{\mathcal {M}}(W^{s-1/p,p}(\mathbb {R}^{n-1}))}. \end{aligned}$$

It is shown in [16, Lemma 9.4.5] that (for sufficiently large N,  i.e., \(N \ge c(\zeta ) K+1\)) the mapping

$$\begin{aligned} \alpha _{z'}(z_n) \mapsto N\,z_n+({\mathcal {T}} \phi )(z',z_n) \end{aligned}$$

is for every \(z' \in \mathbb {R}^{n-1}\) one to one and the inverse is Lipschitz with its gradient bounded by \((N-K)^{-1}.\) Now, we define the mapping \({\varvec{\Phi }}\,:\, {\mathbb {H}} \rightarrow \mathbb {R}^n\) as a rescaled version of the latter one by setting

$$\begin{aligned} {\varvec{\Phi }}(z',z_n)&:= \big (z', \alpha _{z'}(z_n)\big ) = \big (z', \,z_n + ({\mathcal {T}} \phi )(z',z_n/K)\big ). \end{aligned}$$

Thus, \({\varvec{\Phi }}\) is one-to-one (for sufficiently large \(N=N(K)\)) and we can define its inverse \({\varvec{\Psi }}:= {\varvec{\Phi }}^{-1}.\) The mapping \({\varvec{\Phi }}\) has the Jacobi matrix of the form

$$\begin{aligned} J = \nabla {\varvec{\Phi }}= \begin{pmatrix} {\mathbb {I}}_{(n-1)\times (n-1)}&{}0 \\ \partial _{z'} ({\mathcal {T}} \phi )&{} 1+ 1/N\partial _{z_n}{\mathcal {T}} \phi \end{pmatrix}. \end{aligned}$$

Since \({|{\partial _{z_n}{\mathcal {T}} \phi }|} \le K,\) we have

$$\begin{aligned} \frac{1}{2} < 1-K/N \le {|{\det (J)}|} \le 1+K/N\le 2\end{aligned}$$

using that N is large compared to K. Finally, we note the implication

$$\begin{aligned} {\varvec{\Phi }}\in {\mathcal {M}}(W^{s,p}({\mathbb {H}}))\,\,\Rightarrow \,\,{\varvec{\Psi }}\in {\mathcal {M}}(W^{s,p}({\mathbb {H}})), \end{aligned}$$

which holds, for instance, if \({\varvec{\Phi }}\) is Lipschitz continuous, cf. [16, Lemma 9.4.2]. In fact, one can prove (3.7) with the help of (2.5) and (3.6).

3.2 The steady Stokes problem

In this section we consider the steady Stokes system

$$\begin{aligned} \Delta \textbf{u}-\nabla \pi =-\textbf{f},\quad {{\,\textrm{div}\,}}\textbf{u}=0,\quad \textbf{u}|_{\partial {{\mathcal {O}}}}=\textbf{u}_{\partial }, \end{aligned}$$

in a domain \({{\mathcal {O}}}\subset \mathbb {R}^n\) with unit normal \(\textbf{n}.\) The result given in the following theorem is a maximal regularity estimate for the solution in terms of the right-hand side and the boundary datum under minimal assumption on the regularity of \(\partial {\mathcal {O}}.\)

Theorem 3.2

Let \(n=2,3.\) Let \(p\in (1,\infty ),\) \(s\ge 1+\frac{1}{p}\) and

$$\begin{aligned} \varrho \ge p\quad \text {if}\ p(s-1)\ge n,\qquad \varrho \ge \tfrac{p(n-1)}{p(s-1)-1}\quad \text {if}\ p(s-1)< n, \end{aligned}$$

such that \(n\big (\frac{1}{p}-\frac{1}{2}\big )+1\le s.\) Suppose that \({{\mathcal {O}}}\) is a \(B^{\theta }_{\varrho ,p}\)-domain for some \(\theta >s-1/p\) with locally small Lipschitz constant,  \(\textbf{f}\in W^{s-2,p}({{\mathcal {O}}})\) and \(\textbf{u}_{\partial }\in W^{s-1/p,p}(\partial {{\mathcal {O}}})\) with \(\int _{\partial {{\mathcal {O}}}}\textbf{u}_\partial \cdot \textbf{n}\,{\textrm{d}}{\mathcal {H}}^{n-1}=0.\) Then there is a unique solution to (3.8) and we have

$$\begin{aligned} \Vert \textbf{u}\Vert _{W^{s,p}({{\mathcal {O}}})}+\Vert \pi \Vert _{W^{s-1,p} ({{\mathcal {O}}})}\lesssim \Vert \textbf{f}\Vert _{W^{s-2,p}({{\mathcal {O}}})}+ \Vert \textbf{u}_{\partial }\Vert _{W^{s-1/p,p}(\partial {{\mathcal {O}}})}. \end{aligned}$$

The constant in (3.10) depends on the local Lipschitz constant of \(\partial {\mathcal {O}}\) as well as the \(B^{\theta }_{\varrho ,p}\)-norms of the local charts in the parametrisation of \(\partial {\mathcal {O}}.\)

Remark 3.3

The theorem holds under the slightly weaker assumption that \(\partial {{\mathcal {O}}}\in {\mathcal {M}}(W^{s-1/p,p})(\delta )\) for \(\delta \) sufficiently small. This means that the functions \(\varphi _1,\dots ,\varphi _\ell \) from the parametrisation of \(\partial {{\mathcal {O}}}\) belong to the multiplier space \({\mathcal {M}}(W^{s,p}(\mathbb {R}^{n-1}))\) with norm bounded by \(\delta .\) This is a sharp assumption for the corresponding theory for the Laplace equation, cf. [16, Chapter 14]. The relationship between \({\mathcal {M}}(W^{s-1/p,p})(\delta )\) and Besov spaces can be seen from (2.3) and (2.4).


By use of a standard extension operator we can assume that \(\textbf{u}_\partial =0.\) Otherwise we can solve the homogeneous problem with solution \(\tilde{\textbf{u}}\) and set

$$\begin{aligned} \textbf{u}:={\tilde{\textbf{u}}}+{\mathcal {E}}_{{{\mathcal {O}}}}\textbf{u}_\partial -{{\,\textrm{Bog}\,}}_{{\mathcal {O}}}({{\,\textrm{div}\,}}{\mathcal {E}}_{{{\mathcal {O}}}}\textbf{u}_\partial ) \end{aligned}$$


$$\begin{aligned} {\mathcal {E}}_{{{\mathcal {O}}}}:W^{s-1/p,p}(\partial {{\mathcal {O}}})\rightarrow W^{s,p}({{\mathcal {O}}}) \end{aligned}$$

is a continuous linear extension operator and \({{\,\textrm{Bog}\,}}_{{\mathcal {O}}}\) the Bogovskii-operator. The latter solves the divergence equation (with respect to homogeneous boundary conditions on \(\partial {{\mathcal {O}}}\)) and satisfies

$$\begin{aligned} {{\,\textrm{Bog}\,}}_{{\mathcal {O}}}{{\,\textrm{div}\,}}:W^{s,p}\cap \bigg \{\textbf{w}:\,\int _{\partial {{\mathcal {O}}}}\textbf{w}\cdot \textbf{n}\,{\textrm{d}}{\mathcal {H}}^{n-1}=0\bigg \}\rightarrow W^{s,p}_0({{\mathcal {O}}}) \end{aligned}$$

for all \(s\ge 1\) and \(p\in (1,\infty ).\) See [9, Section III.3] for the case \(s\in \mathbb {N}_0,\) the case of fractional s follows by interpolation.

Our assumption \(n\big (\frac{1}{p}-\frac{1}{2}\big )+1\le s\)  implies

$$\begin{aligned} W^{s-2,p}({{\mathcal {O}}})\hookrightarrow W^{-1,2}({{\mathcal {O}}}),\quad W^{s-1/p,p}(\partial {{\mathcal {O}}})\hookrightarrow W^{1/2,2}(\partial {{\mathcal {O}}}), \end{aligned}$$

such that a unique weak solution \((\textbf{u},\pi )\in W^{1,2}_{0,{{\,\textrm{div}\,}}}({{\mathcal {O}}})\times L^2_{\perp }({{\mathcal {O}}})\) to (3.8) exists. Furthermore, let us suppose that \(\textbf{u}\) and \(\pi \) are sufficiently smooth. We will remove this restriction at the end of the proof. By assumption there is \(\ell \in {\mathbb {N}}\) and functions \(\varphi _1,\dots ,\varphi _\ell \in B^{\theta }_{\rho ,p}({\mathbb {R}}^{n-1})\) satisfying (A1)(A3). We clearly find an open set \({\mathcal {U}}^0\Subset {{\mathcal {O}}}\) such that \({{\mathcal {O}}}\subset \cup _{j=0}^\ell {\mathcal {U}}^j.\) Finally, we consider a decomposition of unity \((\xi _j)_{j=0}^\ell \) with respect to the covering \({\mathcal {U}}^0,\dots ,{\mathcal {U}}^\ell \) of \({{\mathcal {O}}}.\) For \(j\in \{1,\dots ,\ell \}\) we consider the extension \({\varvec{\Phi }}_j\) of \(\varphi _j\) given by (3.4) with inverse \({\varvec{\Psi }}_j.\)

Let us fix \(j\in \{1,\dots ,\ell \}\) and assume, without loss of generality, that the reference point \(y_j=0\) and that the outer normal at 0 is pointing in the negative \(x_n\)-direction (this saves us some notation regarding the translation and rotation of the coordinate system). We multiply \(\textbf{u}\) by \(\xi _j\) and obtain for \(\textbf{u}_j:=\xi _j\textbf{u},\) \(\Pi _j:=\xi _j\pi \) and \(\textbf{f}_j:=\xi _j\textbf{f}\) the equation

$$\begin{aligned}&S\Delta \textbf{u}_j-\nabla \Pi _j=[\Delta ,\xi _j]\textbf{u}-[\nabla ,\xi _j]\Pi -\textbf{f}_j,\nonumber \\&{{\,\textrm{div}\,}}\textbf{u}_j=\nabla \xi _j\cdot \textbf{u},\quad \textbf{u}_j|_{\partial {{\mathcal {O}}}}=0, \end{aligned}$$

with the commutators \([\Delta ,\xi _j]=\Delta \xi _j+2\nabla \xi _j\cdot \nabla \) and \([\nabla ,\xi _j]=\nabla \xi _j.\) Finally, we set \(\textbf{v}_j:=\textbf{u}_j\circ {\varvec{\Phi }}_j,\) \(\theta _j:=\Pi _j\circ {\varvec{\Phi }}_j,\) \(\textbf{g}_j:=\textrm{det}(\nabla {\varvec{\Phi }}_j)([\Delta ,\xi _j]\textbf{u}-[\nabla ,\xi _j]\Pi -\textbf{f}_j)\circ {\varvec{\Phi }}_j,\) \(h_j=\textrm{det}(\nabla {\varvec{\Phi }}_j)(\nabla \xi _j\cdot \textbf{u})\circ {\varvec{\Phi }}_j\) and obtain the equations

$$\begin{aligned} \begin{aligned}&{{\,\textrm{div}\,}}\big (\textbf{A}_j\nabla \textbf{v}_j)-{{\,\textrm{div}\,}}({\textbf{B}}_j\theta _j)=\textbf{g}_j, \quad {\textbf{B}}_j^\top :\nabla \textbf{v}_j=h_j,\quad \textbf{v}_j|_{\partial {\mathbb {H}}}=0, \end{aligned} \end{aligned}$$

where \(\textbf{A}_j:=\textrm{det}(\nabla {\varvec{\Phi }}_j)\nabla {\varvec{\Psi }}_j^\top \circ {\varvec{\Phi }}_j\nabla {\varvec{\Psi }}_j\circ {\varvec{\Phi }}_j\) and \({\textbf{B}}_j:=\textrm{det}(\nabla {\varvec{\Phi }}_j)\nabla {\varvec{\Psi }}_j\circ {\varvec{\Phi }}_j\) (note that we have \({{\,\textrm{div}\,}}{\textbf{B}}_j=0\) due to the Piola identity). This can be rewritten as

$$\begin{aligned}&\Delta \textbf{v}_j-\nabla \theta _j={{\,\textrm{div}\,}}\big (({\mathbb {I}}_{n\times n}-\textbf{A}_j)\nabla \textbf{v}_j)+{{\,\textrm{div}\,}}(({\textbf{B}}_j-{\mathbb {I}}_{n\times n})\theta _j)+\textbf{g}_j,\nonumber \\&\qquad \qquad {{\,\textrm{div}\,}}\textbf{v}_j=({\mathbb {I}}_{n\times n}-{\textbf{B}}_j)^\top :\nabla \textbf{v}_j+h_j,\quad \textbf{v}_j|_{\partial {\mathbb {H}}}=0. \end{aligned}$$


$$\begin{aligned} {\mathcal {S}} (\textbf{v},\theta )&={\mathcal {S}} _1(\textbf{v})+{\mathcal {S}}_2 (\theta ),\\ {\mathcal {S}}_1 (\textbf{v})&={{\,\textrm{div}\,}}\big (({\mathbb {I}}_{n\times n}-\textbf{A}_j)\nabla \textbf{v}),\\ {\mathcal {S}}_2 (\theta )&={{\,\textrm{div}\,}}(({\textbf{B}}_j-{\mathbb {I}}_{n\times n})\theta ),\\ {\mathfrak {s}}(\textbf{v})&=({\mathbb {I}}_{n\times n}-{\textbf{B}}_j)^\top :\nabla \textbf{v}, \end{aligned}$$

we can finally write (3.14) as

$$\begin{aligned} \begin{aligned} \Delta \textbf{v}_j-\nabla \theta _j={\mathcal {S}} (\textbf{v}_j,\theta _j)+\textbf{g}_j, \quad {{\,\textrm{div}\,}}\textbf{v}_j={\mathfrak {s}}(\textbf{v}_j)+h_j,\quad \textbf{v}_j|_{\partial {\mathbb {H}}}=0, \end{aligned} \end{aligned}$$

in \({\mathbb {H}}.\) Estimates for the Stokes system on the half space are well-known: We apply [9, Thm. IV 2.1] to (3.15) which yields

$$\begin{aligned} \Vert \nabla \textbf{v}_j\Vert _{L^p_x}+\Vert \theta _j\Vert _{L^p_x}\lesssim \Vert {\mathcal {S}} (\textbf{v}_j,\theta _j)+\textbf{g}_j\Vert _{W^{-1,p}_x}+\Vert {\mathfrak {s}}(\textbf{v}_j)+h_j\Vert _{L^p_x}. \end{aligned}$$

Similarly, we obtain from [9, Thm. IV 3.3] for \(k\ge 2\)

$$\begin{aligned} \Vert \nabla ^k\textbf{v}_j\Vert _{L^p_x}+\Vert \nabla ^{k-1}\theta _j\Vert _{L^p_x}&\lesssim \Vert \nabla ^{k-2}({\mathcal {S}} (\textbf{v}_j,\theta _j)+\textbf{g}_j)\Vert _{L^{p}_x}\nonumber \\&+\Vert \nabla ^{k-1}({\mathfrak {s}}(\textbf{v}_j)+h_j)\Vert _{L^p_x}.\qquad \end{aligned}$$

Since \(\textbf{v}_j\) is compactly supported (with support included in \(\xi _j\circ {\varvec{\Phi }}_j\)) we conclude by Poincaré’s inequality

$$\begin{aligned} \Vert \textbf{v}_j\Vert _{W^{k,p}_x}+\Vert \theta _j\Vert _{W^{k-1,p}_x}\lesssim \Vert {\mathcal {S}} (\textbf{v}_j,\theta _j)+\textbf{g}_j\Vert _{W^{k-2}_x}+\Vert {\mathfrak {s}}(\textbf{v}_j)+h_j\Vert _{W^{k-1,p}_x} \end{aligned}$$

for all \(k\ge 1.\) Interpolation implies

$$\begin{aligned} \Vert \textbf{v}_j\Vert _{W^{s,p}_x}+\Vert \theta _j\Vert _{W^{s-1,p}_x}\lesssim \Vert {\mathcal {S}} (\textbf{v}_j,\theta _j)+\textbf{g}_j\Vert _{W^{s-2}_x}+\Vert {\mathfrak {s}}(\textbf{v}_j)+h_j\Vert _{W^{s-1,p}_x} \end{aligned}$$

for all \(s\ge 1.\) Our remaining task consists in estimating the right-hand side. In order to estimate \(\Vert {\mathcal {S}}(\textbf{v},\theta )\Vert _{W^{s-2,p}_x}\) and \(\Vert {\mathfrak {s}}(\textbf{v})\Vert _{W^{s-1,p}_x}\) we use the Sobolev multiplier norm introduced in (2.1). By our assumptions on \(\varphi _j\) we infer from (2.3) and (2.4) that \(\phi _j\in {\mathcal {M}}(W^{s-1/p,p}({\mathbb {H}})).\) Thus \({\varvec{\Phi }}_j\in {\mathcal {M}}(W^{s,p}({\mathbb {H}}))\) by (3.4), (3.3), and (3.9) and \({\varvec{\Psi }}_j\in {\mathcal {M}}(W^{s,p}({\mathbb {H}}))\) by (3.7). Hence we obtain by (3.5), (2.5) and the definitions of \(\textbf{A}_j\) and \({\varvec{\Phi }}_j\)

$$\begin{aligned}&\Vert {\mathcal {S}}_1(\textbf{v})\Vert _{W^{s-2,p}({\mathbb {H}})}\\ {}&\quad \lesssim \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert ({\mathbb {I}}_{n\times n}-\textbf{A}_j)\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \nabla \textbf{v}\Vert _{W^{s-1,p}_x}\\&\quad \lesssim \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\le 1}\Vert (1-\textrm{det}(\nabla {\varvec{\Phi }}_j))\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \textbf{v}\Vert _{W^{s,p}_x}\\&\qquad + \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert \textrm{det}(\nabla {\varvec{\Phi }}_j)({\mathbb {I}}_{n\times n}-\nabla {\varvec{\Psi }}_j^\top \circ {\varvec{\Phi }}_j)\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \textbf{v}\Vert _{W^{s,p}_x}\\&\qquad + \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert \textrm{det}(\nabla {\varvec{\Phi }}_j)\nabla {\varvec{\Psi }}_j^\top \circ {\varvec{\Phi }}_j({\mathbb {I}}_{n\times n}-\nabla {\varvec{\Psi }}_j\circ {\varvec{\Phi }}_j)\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \textbf{v}\Vert _{W^{s,p}_x}\\&\quad \lesssim \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\Vert \textbf{v}\Vert _{W^{s,p}_x}\\ {}&\qquad + \Vert {\varvec{\Phi }}_j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}^n\sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert ({\mathbb {I}}_{n\times n}-\nabla {\varvec{\Psi }}_j\circ {\varvec{\Phi }}_j)\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \textbf{v}\Vert _{W^{s,p}_x}\\&\qquad + \Vert {\varvec{\Phi }}_j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}^n\Vert {\varvec{\Psi }}_j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert ({\mathbb {I}}_{n\times n}\\&\qquad -\nabla {\varvec{\Psi }}_j\circ {\varvec{\Phi }}_j)\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \textbf{v}\Vert _{W^{s,p}_x}\\&\quad \lesssim \Bigg (\Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}+ \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert ({\mathbb {I}}_{n\times n}-\nabla {\varvec{\Psi }}_j\circ {\varvec{\Phi }}_j)\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Bigg )\Vert \textbf{v}\Vert _{W^{s,p}_x}, \end{aligned}$$


$$\begin{aligned}&\sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert ({\mathbb {I}}_{n\times n}-\nabla {\varvec{\Psi }}_j\circ {\varvec{\Phi }}_j)\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\\&\quad =\sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert ({\mathbb {I}}_{n\times n}-\textrm{det}(\nabla {\varvec{\Psi }}_j)\textrm{cof}(\nabla {\varvec{\Phi }}_j\circ {\varvec{\Phi }}_j))\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})} \\&\quad \le \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert (1-\textrm{det}(\nabla {\varvec{\Psi }}_j))\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\\&\qquad +\sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}_x}\le 1}\Vert \textrm{det}(\nabla {\varvec{\Psi }}_j)({\mathbb {I}}_{n\times n}-\textrm{cof}(\nabla {\varvec{\Phi }}_j\circ {\varvec{\Phi }}_j))\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\\&\quad \lesssim \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\Vert {\varvec{\Psi }}_j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}^n\lesssim \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}. \end{aligned}$$

So we finally have

$$\begin{aligned} \Vert {\mathcal {S}}_1(\textbf{v})\Vert _{W^{s-2,p}({\mathbb {H}})}&\lesssim \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\Vert \textbf{v}\Vert _{W^{s,p}({\mathbb {H}})} \end{aligned}$$

and, similarly,

$$\begin{aligned} \Vert {\mathcal {S}}_2(\theta )\Vert _{W^{s-2,p}({\mathbb {H}})}&\lesssim \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\le 1} \Vert ({\textbf{B}}_j-{\mathbb {I}}_{n\times n})\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \theta \Vert _{W^{s-1,p}({\mathbb {H}})}\\&\lesssim \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\Vert \theta \Vert _{W^{s-1,p}({\mathbb {H}})}, \end{aligned}$$

as well as

$$\begin{aligned} \Vert {\mathfrak {s}}(\textbf{v})\Vert _{W^{s-1,p}({\mathbb {H}})}&\lesssim \sup _{\Vert \textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\le 1}\Vert ({\textbf{B}}_j-{\mathbb {I}}_{n\times n})\textbf{w}\Vert _{W^{s-1,p}({\mathbb {H}})}\Vert \nabla \textbf{v}\Vert _{W^{s-1,p}({\mathbb {H}})}\\&\lesssim \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\Vert \textbf{v}\Vert _{W^{s,p}({\mathbb {H}})}, \end{aligned}$$

By (3.3) we have

$$\begin{aligned} \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\lesssim \Vert \varphi _j\Vert _{{\mathcal {M}}(W^{s-1/p,p}({\mathbb {H}}))}. \end{aligned}$$

Finally, in the case \(p\le n\) the right-hand side can be bounded by the Lipschitz constant due to by (2.3) and the embedding \(B_{p,p}^{\theta }\hookrightarrow B^{s-1/p}_{\rho ,p}\) for \(\theta >s-1/p\) and \(\varrho \) satisfying (3.9). Hence it is conveniently small by our assumption. If \(p>n\) we have by (2.4)

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {T}}\phi _j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}&\lesssim \Vert \nabla ({\mathcal {T}}\phi _j)\Vert _{W^{s-1,p}({\mathbb {H}})}\lesssim \Vert \varphi _j\Vert _{W^{s-1/p,p}(\mathbb {R}^{n-1})}\\ {}&\lesssim \Vert \varphi _j\Vert _{W^{\theta ,p}(\mathbb {R}^{n-1})}^\alpha \Vert \varphi _j\Vert _{W^{1,p}(\mathbb {R}^{n-1})}^{1-\alpha }\\&\lesssim \Vert \varphi _j\Vert _{W^{\theta ,p}(\mathbb {R}^{n-1})}^\alpha \Vert \varphi _j\Vert _{W^{1,\infty }(\mathbb {R}^{n-1})}^{1-\alpha } \end{aligned} \end{aligned}$$

for an appropriate choice of \(\alpha \in (0,1).\) This is again suitably small. We conclude that

$$\begin{aligned} \Vert {\mathcal {S}}(\textbf{v}_j,\theta _j)\Vert _{W^{s-2,p}_x}+\Vert {\mathfrak {s}}(\textbf{v})\Vert _{W^{s-1,p}_x}\le \delta \big (\Vert \textbf{v}_j\Vert _{W^{s,p}({\mathbb {H}})}+\Vert \theta _j\Vert _{W^{s-1,p}({\mathbb {H}})}\big ) \end{aligned}$$

for some small \(\delta >0.\) On the other hand, we have

$$\begin{aligned} \Vert \textbf{g}_j\Vert _{W^{s-2,p}_x}&\lesssim \Vert \textbf{u}\circ {\varvec{\Phi }}_j\Vert _{W^{s-1,p}_x}+ \Vert \pi \circ {\varvec{\Phi }}_j\Vert _{W^{s-2,p}_x}+ \Vert \textbf{f}\circ {\varvec{\Phi }}_j\Vert _{W^{s-2}_x}\\&\lesssim \Vert \textbf{u}\Vert _{W^{s-1,p}_x}+ \Vert \pi \Vert _{W^{s-2,p}_x}+\Vert \textbf{f}\Vert _{W^{s-2}_x}, \end{aligned}$$

where the hidden constant depends on \(\textrm{det}(\nabla {\varvec{\Phi }}_j)\) and \(\Vert {\varvec{\Phi }}_j\Vert _{{\mathcal {M}}(W^{s,p}({\mathbb {H}}))}\) being controlled by (3.6), (3.20) and (3.21) (see (2.5) for the composition with Sobolev multipliers). Similarly, we obtain

$$\begin{aligned} \Vert h_j\Vert _{W^{s-1,p}_x}&\lesssim \Vert \textbf{u}\Vert _{W^{s-1,p}_x}. \end{aligned}$$

Plugging this and (3.22) into (3.19) shows for all \(j\in \{1,\dots ,\ell \}\)

$$\begin{aligned} \Vert \textbf{v}_j\Vert _{W^{s,p}_x}+\Vert \theta _j\Vert _{W^{s-1,p}_x}\lesssim \Vert \textbf{u}\Vert _{W^{s-1}_x}+ \Vert \pi \Vert _{W^{s-2}_x}+\Vert \textbf{f}\Vert _{W^{s-2}_x} \end{aligned}$$

provided \(\delta \) is sufficiently small. Clearly, the same estimate (even without the first two terms on the right-hand side) holds for \(j=0\) by local regularity theory for the Stokes system. Choosing \(s_0\in \mathbb {R}\) such that \(W^{1,2}({{\mathcal {O}}})\hookrightarrow W^{s_0,p}({{\mathcal {O}}}),\) there is \(\alpha \in (0,1)\) such that

$$\begin{aligned} \Vert \textbf{u}\Vert _{W^{s-1,p}_x}&\le \Vert \textbf{u}\Vert _{W^{s,p}_x}^{\alpha }\Vert \textbf{u}\Vert _{W^{s_0,p}_x}^{1-\alpha }\lesssim \Vert \textbf{u}\Vert _{W^{s,p}_x}^{\alpha }\Vert \textbf{u}\Vert _{W^{1,2}_x}^{1-\alpha } \\ {}&\lesssim \Vert \textbf{u}\Vert _{W^{s,p}_x}^{\alpha }\Vert \textbf{f}\Vert _{W^{-1,2}_x}^{1-\alpha }\lesssim \Vert \textbf{u}\Vert _{W^{s,p}_x}^{\alpha }\Vert \textbf{f}\Vert _{W^{s-2,p}_x}^{1-\alpha } \end{aligned}$$

by the assumption \(n\big (\frac{1}{p}-\frac{1}{2}\big )+1\le s\)  and the standard energy estimate for the Stokes system. Hence we obtain

$$\begin{aligned} \Vert \textbf{u}\Vert _{W^{s-1,p}_x}&\le \kappa \Vert \textbf{u}\Vert _{W^{s,p}_x}+c(\kappa )\Vert \textbf{f}\Vert _{W^{s-2,p}_x} \end{aligned}$$

for any \(\kappa >0.\) Similarly,

$$\begin{aligned} \Vert \pi \Vert _{W^{s-2,p}_x}&\le \kappa \Vert \pi \Vert _{W^{s-1,p}_x}+c(\kappa )\Vert \textbf{f}\Vert _{W^{s-2,p}_x} \end{aligned}$$

using that \(\Vert \pi \Vert _{L^2_x}\lesssim \Vert \textbf{f}\Vert _{W^{-1,2}_x}\) as well. Plugging (3.24) and (3.25) into (3.23) summing over \(j=0,1,\dots ,\ell \) and choosing \(\kappa \) small enough proves the claim provided \(\textbf{u}\) and \(\pi \) are sufficiently smooth. Let us finally remove this assumption which is not a priori given. Applying a standard regularisation procedure (by convolution with mollifying kernel) to the functions \(\varphi _1,\dots ,\varphi _\ell \) from (A1)(A3) in the parametrisation of \(\partial {{\mathcal {O}}}\) we obtain a smooth boundary. Classically, the solution to the corresponding Stokes system is smooth. Such a procedure is standard and has been applied, for instance, in [4, Section 4]. It is possible to do this in a way that the original domain is included in the regularised domain to which we extend the function \(\textbf{f}\) by means of an extension operator. The regularisation applied to the \(\varphi _j's\) converges on all Besov spaces with \(p<\infty .\) It does not converge on \(W^{1,\infty }(\mathbb {R}^{n-1}),\) but the regularisation does not expand the \(W^{1,\infty }(\mathbb {R}^{n-1})\)-norm, which is sufficient. Following the arguments above we obtain (3.10) for the regularised problem with a uniform constant. The limit passage is straightforward since (3.8) is linear. \(\square \)

Remark 3.4

In Sect. 5.1 we have to apply Theorem 3.2 in the case \(n=2\) to the domain \({\mathcal {O}}={\Omega }_{\eta (t)}\) for a fixed t. We exclude self-intersection and degeneracy by assumption. In the framework of Theorem 2.5 we have \(\eta \in L^\infty (I;W^{2,2}(\omega ))\) and \({\Omega }_{\eta (t)}\) is defined in accordance with (2.8). We must argue that \(\partial {\Omega }_\eta \in B^{s}_{2,2}({{\mathcal {O}}})\) (in the sense of Definition 3.1) for some \(s>\frac{3}{2}\) and has a small local Lipschitz constant (both uniformly in time). While the Besov regularity is initially clear, we have to introduce local coordinates to control the Lipschitz constant appropriately. Eventually, we must check the Besov regularity again. Given \(x_0\in \partial {\Omega }_{\eta (t)}\) for some \(t\in I\) fixed we can rotate the coordinate system such that \(\textbf{n}_{\eta (t)}(y(x_0))=(0,1)^\top \) (recall that \(\textbf{n}_{\eta (t)}\) is well-defined since \(\partial _y{\varvec{\varphi }}_\eta \ne 0\) by assumption). Accordingly, it holds

$$\begin{aligned}\partial _y{\varvec{\varphi }}_{\eta (t)}(y(x_0))=\begin{pmatrix}\partial _y\varphi ^1_{\eta (t)}(y(x_0))\\ \partial _y\varphi ^2_{\eta (t)}(y(x_0))\end{pmatrix}=\begin{pmatrix}1\\ 0\end{pmatrix}. \end{aligned}$$

Hence the function \(\varphi ^1_{\eta (t)}\) is invertible in a neighborhood \({\mathcal {U}}\) of \(y(x_0).\) We define in \(\varphi ^1_{\eta (t)}({\mathcal {U}})\) the function

$$\begin{aligned} {\widetilde{{\varvec{\varphi }}}}_{x_0}(z)=\begin{pmatrix}z\\ {\widetilde{\varphi }}_{x_0}(z))\end{pmatrix}=\begin{pmatrix}z\\ \varphi ^2_{\eta (t)}((\varphi ^1_{\eta (t)})^{-1}(z))\end{pmatrix}. \end{aligned}$$

It describes the boundary \(\partial {\Omega }_{\eta (t)}\) close to \(x_0.\) One easily checks with \(z_0=\varphi ^1_{\eta (t)}(y(x_0))\) that \(\partial _z{\widetilde{\varphi }}_{x_0}(z_0)=0\) such that \(\partial _z{\widetilde{\varphi }}\) is small close to \(z_0.\) Also, we obtain from the chain rule and the one-dimensional Sobolev embedding that \({\widetilde{\varphi }}_{x_0}\in W^{2,2}\) and hence \({\widetilde{\varphi }}_{x_0}\in W^{s,2}=B^s_{2,2}\) for all \(s\in (1,2)\) in a neighborhood of \(z_0.\)

Remark 3.5

A result in the spirit of Theorem 3.2 is proved in [5, Lemma 3.1]. However, it only applies in spaces of high regularity with \(s\ge 3\) (and only the case \(p=2\) is considered) which is too restrictive for our application in Sect. 5. Moreover, it is assumed that the global parametrisation is a small perturbation of a smooth reference domain (such as the half space). First of all, the assumption of a global parametrisation restricts the result to applications in fluid–structure interaction as generally only local charts are available on bounded domains. Second, global smallness is a small data assumption, while local smallness (as the small Lipschitz constant) can be achieved by local re-parametrisation as in Remark 3.4.

4 Local strong solutions

In this subsection we prove the existence of a unique strong solution to (1.1)–(1.2) which exists locally in time. We consider again only the case \(n=2.\)

Proposition 4.1

Let \(n=2.\) Suppose that the assumptions of Theorem 2.5 hold. There is \(T^*>0\) such that there is a unique strong solution to (1.1)–(1.2) in \(I^*=(0,T^*)\) in the sense of Definition 2.4.

The strategy to prove Proposition 4.1 is rather standard and similar to previous papers [10, 11, 13]:

  • We transform the system to the reference domain, cf. Lemma 4.2.

  • We linearise the system from Lemma 4.2 and obtain estimates for the linearised system, cf. Lemma 4.3.

  • We construct a contraction map for the linearised problem in Lemma 4.4 (by choosing the end-time small enough) which gives the local solution to (1.1)–(1.2).

4.1 The transformed problem

For a solution \((\eta ,\textbf{u},\pi )\) to (1.1)–(1.2) we define \({\overline{\pi }}=\pi \circ {\varvec{\Psi }}_\eta \) and \(\overline{\textbf{u}}=\textbf{u}\circ {\varvec{\Psi }}_\eta ,\) where \({\varvec{\Psi }}_\eta \) is defined in (2.9). We also introduce

$$\begin{aligned} \textbf{h}_\eta ({\overline{\textbf{u}}})&=-(J_{\eta }-J_{\eta _0})\partial _t{\overline{\textbf{u}}}- J_\eta (\nabla {\varvec{\Psi }}_\eta ^{-1}\circ {\varvec{\Psi }}_\eta \nabla {\overline{\textbf{u}}})\big (\partial _t{\varvec{\Psi }}_\eta ^{-1}\circ {\varvec{\Psi }}_\eta +{\overline{\textbf{u}}}\big )+J_\eta \textbf{f}\circ {\varvec{\Psi }}_\eta ,\\ \textbf{A}_\eta&=J_\eta \big (\nabla {\varvec{\Psi }}_\eta ^{-1}\circ {\varvec{\Psi }}_\eta \big )^{\top }\nabla {\varvec{\Psi }}_\eta ^{-1}\circ {\varvec{\Psi }}_\eta ,\quad {\textbf{B}}_\eta =J_\eta \nabla {\varvec{\Psi }}_\eta ^{-1}\circ {\varvec{\Psi }}_\eta ,\\ \textbf{H}_\eta ({\overline{\textbf{u}}},{\overline{\pi }})&=(\textbf{A}_{\eta _0}-\textbf{A}_\eta )\nabla {\overline{\textbf{u}}}-({\textbf{B}}_{\eta _0}-{\textbf{B}}_{\eta }){\overline{\pi }},\quad h_\eta ({\overline{\textbf{u}}})=({\textbf{B}}_{\eta _0}-{\textbf{B}}_{\eta }):\nabla {\overline{\textbf{u}}}, \end{aligned}$$

where \(J_\eta =|\textrm{det}\nabla {\varvec{\Psi }}_\eta |.\) We see that \((\eta ,{\overline{\textbf{u}}},{\overline{\pi }})\) is a strong solution to the coupled system

$$\begin{aligned} J_{\eta _0}\partial _t{\overline{\textbf{u}}}+{{\,\textrm{div}\,}}\big ({\textbf{B}}_{\eta _0}{\overline{\pi }}\big )&-{{\,\textrm{div}\,}}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )=\textbf{h}_\eta ({\overline{\textbf{u}}})-{{\,\textrm{div}\,}}\textbf{H}_\eta ({\overline{\textbf{u}}},{\overline{\pi }}), \end{aligned}$$
$$\begin{aligned} {\textbf{B}}_{\eta _0}:\nabla {\overline{\textbf{u}}}&=h_\eta ({\overline{\textbf{u}}}), \end{aligned}$$
$$\begin{aligned} \partial _t^2\eta -\partial _t\partial _y^2\eta +\partial _y^4\eta&=-\textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\nonumber \\&+\textbf{n}\textbf{H}_\eta ({\overline{\textbf{u}}},{\overline{\pi }})\circ {\varvec{\varphi }}\,\textbf{n}, \end{aligned}$$

in \(I\times \Omega \) and we have

$$\begin{aligned} {\overline{\textbf{u}}}\circ {{\varvec{\varphi }}} =\partial _t \eta \textbf{n}\quad \text {on}\ I\times \omega . \end{aligned}$$

Here Eqs. (4.1) and (4.3) are understood in the strong sense (satisfied a.a. in \(I\times \Omega \) and \(I\times \omega \) respectively).

We call \((\eta ,{\overline{\textbf{u}}},{\overline{\pi }})\) a strong solution to (4.1)–(4.4) provided

$$\begin{aligned} \begin{aligned} \eta \in W^{1,\infty } \big (I; W^{1,2}(\omega ) \big )\cap W^{1,2} \big (I; W^{2,2}(\omega ) \big )\cap L^\infty \big (I; W^{3,2}(\omega ) \big ) \cap W^{2,2}(I;L^2(\omega )),\\ {\overline{\textbf{u}}} \in L^\infty \big (I; W^{1,2}(\Omega ) \big )\cap L^2 \big (I; W^{2,2}(\Omega ) \big )\cap W^{1,2} \big (I; L^{2}(\Omega ) \big ),\quad {\overline{\pi }}\in L^2 \big (I; W^{1,2}(\Omega ) \big ). \end{aligned} \end{aligned}$$

Note that we construct a weak solution to Eq. (4.3) meaning we have

$$\begin{aligned}&\int _I \int _\omega \big (\partial _t \eta \, \partial _t\phi -\partial _t\partial _y\eta \,\partial _y \phi - g\, \phi \big )\,{\textrm{d}}y\,{\textrm{d}} t-\int _I\int _\omega \partial _y^2\eta \,\partial _y^2 \phi \,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\quad =-\int _\omega \textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\phi \,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\qquad \ +\int _\omega \textbf{n}\textbf{H}_\eta ({\overline{\textbf{u}}},{\overline{\pi }})\circ {\varvec{\varphi }}\,\textbf{n}\phi \,{\textrm{d}}y\,{\textrm{d}} t\end{aligned}$$

for all \(\phi \in C^\infty ({{\overline{I}}}\times \omega ).\) However, one can use the regularity properties (4.5) to infer that \(\eta \in L^2(I;W^{4,2}(\omega ))\) such that all quantities in (4.6) are, in fact, \(L^2\)-functions and we have indeed a strong solution.

We obtain the following characterisation regarding (4.1)–(4.18).

Lemma 4.2

Suppose that the dataset \((\textbf{f}, g, \eta _0, \textbf{u}_0, \eta _1)\) satisfies (2.21) and (2.25). Then \((\eta ,\textbf{u},\pi )\) is a strong solution to (1.1)–(1.2) (in the sense of Definition 2.4) if and only if \((\eta ,{\overline{\textbf{u}}},{\overline{\pi }})\) is a strong solution to (4.1)–(4.4).


Transforming the momentum equation to the reference domain we obtain

$$\begin{aligned}&J_{\eta }\partial _t{\overline{\textbf{u}}}+{{\,\textrm{div}\,}}\big ({\textbf{B}}_{\eta }{\overline{\pi }}\big )-{{\,\textrm{div}\,}}\big (\textbf{A}_{\eta }\nabla {\overline{\textbf{u}}}\big )\\&\quad =- J_\eta (\nabla {\varvec{\Psi }}_\eta ^{-1}\circ {\varvec{\Psi }}_\eta \nabla {\overline{\textbf{u}}})\big (\partial _t{\varvec{\Psi }}_\eta ^{-1}\circ {\varvec{\Psi }}_\eta +{\overline{\textbf{u}}}\big )+J_\eta \textbf{f}\circ {\varvec{\Psi }}_\eta , \end{aligned}$$

while the incompressibility constraint gives \({\textbf{B}}_{\eta }^\top :\nabla {\overline{\textbf{u}}}=0.\) Reordering terms and recalling the definitions of \(B_{\eta },\) \(\textbf{A}_{\eta _0},\) \(\textbf{f}_\eta \) and \(\textbf{H}_\eta \) yields

$$\begin{aligned} J_{\eta _0}\partial _t{\overline{\textbf{u}}}+{{\,\textrm{div}\,}}\big ({\textbf{B}}_{\eta _0}{\overline{\pi }}\big )-{{\,\textrm{div}\,}}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )&=\textbf{h}_\eta ({\overline{\textbf{u}}})-{{\,\textrm{div}\,}}\textbf{H}_\eta ({\overline{\textbf{u}}},{\overline{\pi }})\end{aligned}$$


$$\begin{aligned} B_{\eta _0}:\nabla {\overline{\textbf{u}}}&=h_\eta ({\overline{\textbf{u}}}). \end{aligned}$$

Allowing now a couple of test-functions \((\phi , {{\varvec{\phi }}}) \in C^\infty ({\overline{I}}\times \omega ) \times C^\infty ({\overline{I}}\times \mathbb {R}^3)\) and \({\varvec{\phi }}\circ {\varvec{\varphi }}_\eta = \phi {\textbf{n}},\) see Definition 2.3 (c), rewriting the terms for the fluid equation as above (that is, setting \(\overline{{\varvec{\phi }}}={\varvec{\phi }}\circ {\varvec{\Psi }}_\eta \)) and integrating by parts yields

$$\begin{aligned}&\int _I \int _\omega \big (\partial _t \eta \, \partial _t\phi -\partial _t\partial _y\eta \,\partial _y \phi - g\, \phi \big )\,{\textrm{d}}y\,{\textrm{d}} t-\int _I\int _\omega \partial _y^2\eta \,\partial _y^2 \phi \,{\textrm{d}}y\,{\textrm{d}} t\\&\quad =-\int _\omega \textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\phi \,{\textrm{d}}y\,{\textrm{d}} t\\&\qquad \ +\int _\omega \textbf{n}\textbf{H}_\eta ({\overline{\textbf{u}}},{\overline{\pi }})\circ {\varvec{\varphi }}\,\textbf{n}\phi \,{\textrm{d}}y\,{\textrm{d}} t\end{aligned}$$

due to (4.7). Note that the terms on the right-hand side are the boundary terms which arise due to the integration by parts. This finishes the proof as all the manipulations can be reversed for strong solutions given the regularity of \({\overline{\textbf{u}}}\) and \({\overline{\pi }}\) assumed in (4.5).

\(\square \)

4.2 The linearised problem

We will now consider solutions to the linearised problem for a given right-hand side, that is we analyse for given functions \(\textbf{h},\textbf{H}\) and h the system

$$\begin{aligned} J_{\eta _0}\partial _t{\overline{\textbf{u}}}&+{{\,\textrm{div}\,}}\big ({\textbf{B}}_{\eta _0}{\overline{\pi }}\big )-{{\,\textrm{div}\,}}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )=\textbf{h}-{{\,\textrm{div}\,}}\textbf{H}, \end{aligned}$$
$$\begin{aligned} {\textbf{B}}_{\eta _0}:\nabla {\overline{\textbf{u}}}&=h, \end{aligned}$$
$$\begin{aligned} \nonumber \partial _t^2\eta -\partial _t\partial _y^2\eta +\partial _y^4\eta&=g-\textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\\&\quad +\textbf{n}\textbf{H}\circ {\varvec{\varphi }}\,\textbf{n}, \end{aligned}$$
$$\begin{aligned}&{\overline{\textbf{u}}}\circ {\varvec{\varphi }}=\partial _t \eta {\textbf{n}}\quad \text {on}\ I\times \omega , \end{aligned}$$
$$\begin{aligned}&{\overline{\textbf{u}}}(0)={\overline{\textbf{u}}}_0,\quad \eta (0)=\eta _0,\quad \partial _t\eta (0)=\eta _1. \end{aligned}$$

Note that (4.8)–(4.12) is linear in \((\eta ,{\overline{\textbf{u}}},{\overline{\pi }})\) such that we expect strong solutions globally in time belonging to the regularity class specified in (4.5).

Lemma 4.3

Suppose that the dataset \((\textbf{h},\textbf{H},h, g, \eta _0, \overline{\textbf{u}}_0, \eta _1)\) satisfies

$$\begin{aligned}&\textbf{h}\in L^2\big (I; L^2(\Omega )\big ),\quad \textbf{H}\in L^2\big (I; W^{1,2}(\Omega )\big ),\nonumber \\&h\in L^2\big (I; W^{1,2}(\Omega )\big )\cap W^{1,2}\big (I; W^{-1,2}(\Omega )\big ) \cap \{h(0,\cdot )=0\},\nonumber \\&g \in L^2\big (I; W^{1,2}(\omega )\big ), \quad \eta _0 \in W^{3,2}(\omega ) \text { with } \Vert \eta _0 \Vert _{L^\infty ( \omega )} < L, \quad \eta _1 \in W^{1,2}(\omega ),\nonumber \\&{\overline{\textbf{u}}}_0\in W^{1,2}(\Omega ) \text { is such that }{\overline{\textbf{u}}}_0\circ {\varvec{\varphi }}=\eta _1 \textbf{n}\text { and }{\textbf{B}}_{\eta _0}:\nabla {\overline{\textbf{u}}}_0=0. \end{aligned}$$

Then there is a strong solution to (4.8)–(4.12) satisfying the estimate

$$\begin{aligned}&\sup _I\int _\Omega |\nabla {\overline{\textbf{u}}}|^2\,{\textrm{d}}x+ \int _I\int _\Omega \big (|\nabla ^2{\overline{\textbf{u}}}|^2+ |\partial _t{\overline{\textbf{u}}}|^2+|{\overline{\pi }}|^2+|\nabla {\overline{\pi }}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\nonumber \\&\qquad +\sup _I\int _{\omega }\big ( |\partial _t\partial _y\eta |^2+ |\partial _y^3\eta |^2\big )\,{\textrm{d}}y+\int _I\int _{\omega } \big (|\partial _t\partial _y^2\eta |^2+|\partial _t^2\eta |^2\big ) \,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\quad \lesssim \int _\Omega |\nabla {\overline{\textbf{u}}}_0|^2\,{\textrm{d}}x+\int _I\int _\Omega \big (|\textbf{h}|^2+|\nabla \textbf{H}|^2+|\nabla h|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t+\int _I\Vert \partial _t h \Vert _{W^{-1,2}(\Omega )}^2\,{\textrm{d}} t\nonumber \\&\qquad +\int _\omega \big (|\partial _y^3\eta _0|^2+|\partial _y\eta _1|^2\big ) \,{\textrm{d}}y+\int _I\int _\omega |\partial _y g|^2\,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\qquad +\overline{{\mathcal {E}}}(0)+\int _I\int _\Omega | \textbf{H}|^2\,{\textrm{d}}x\,{\textrm{d}}t+\int _I\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t, \end{aligned}$$

where the energy \(\overline{{\mathcal {E}}}\) is given by

$$\begin{aligned} \overline{{\mathcal {E}}}(t)=\frac{1}{2}\int _{\Omega } |{\overline{\textbf{u}}}(t)|^2 \,{\textrm{d}}x+ \frac{1}{2}\int _{\omega } |\partial _t\eta |^2 \,{\textrm{d}}y+ \frac{1}{2}\int _{\omega } |\partial _y^2\eta |^2 \,{\textrm{d}}y. \end{aligned}$$


Let us initially suppose that \(h=0,\) that is we have \(B_{\eta _0}:\nabla {\overline{\textbf{u}}}=0.\) We proceed formally; a rigorous proof can be obtained by working with a Galerkin approximation. Testing with \(({\overline{\textbf{u}}},\partial _t\eta ),\) using \(B_{\eta _0}:\nabla {\overline{\textbf{u}}}=0,\) ellipticity of \(\textbf{A}_{\eta _0}\) (which follows from \(\Vert \eta _0\Vert _{L^\infty _y}<L\)) as well as the cancellation of the boundary terms due to (4.11) yields

$$\begin{aligned}&\overline{{\mathcal {E}}}(t) +\int _0^t \int _{\Omega }|\nabla {\overline{\textbf{u}}} |^2 \,{\textrm{d}}x\,{\textrm{d}}\sigma +\int _0^t\int _{\omega }|\partial _t\partial _y\eta |^2 \,{\textrm{d}}y\,{\textrm{d}}\sigma \\&\quad \le \overline{{\mathcal {E}}}(0) + \int _0^t\int _{\Omega }{\overline{\textbf{u}}}\cdot \textbf{h}\,{\textrm{d}}x\,{\textrm{d}}\sigma + \int _0^t\int _{\Omega }\nabla {\overline{\textbf{u}}}:\textbf{H}\,{\textrm{d}}x\,{\textrm{d}}\sigma + \int _0^t\int _\omega g\,\partial _t\eta \,{\textrm{d}}y\,{\textrm{d}}\sigma . \end{aligned}$$

This implies

$$\begin{aligned}&\sup _I\int _\Omega |{\overline{\textbf{u}}}|^2\,{\textrm{d}}x+\int _I\int _\Omega |\nabla {\overline{\textbf{u}}}|^2\,{\textrm{d}}x\,{\textrm{d}} t\nonumber \\&\qquad +\sup _I\int _\omega |\partial _t\eta |^2\,{\textrm{d}}y+\sup _I\int _\omega |\partial _y^2\eta |^2\,{\textrm{d}}y+\int _I\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\quad \lesssim \overline{{\mathcal {E}}}(0)+\int _I\int _\Omega \big (|\textbf{h}|^2+|\textbf{H}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t+\int _I\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t. \end{aligned}$$

Similarly, we can test by \((\partial _t{\overline{\textbf{u}}},\partial _t^2\eta )\) noticing that the coefficients in (4.8) and (4.9) are independent of time and that \(\partial _t{\overline{\textbf{u}}}\) and \(\partial _t^2\eta \) match again at the boundary due to (4.11). We obtain

$$\begin{aligned}&\int _0^t\int _\Omega |\partial _t{\overline{\textbf{u}}}|^2\,{\textrm{d}}x\,{\textrm{d}}\sigma + \frac{1}{2}\int _{\Omega }|\nabla {\overline{\textbf{u}}}|^2 \,{\textrm{d}}x+\frac{1}{2}\int _{\omega }|\partial _t\partial _y\eta |^2 \,{\textrm{d}}y+\int _0^t\int _{\omega }|\partial _t^2\eta |^2 \,{\textrm{d}}y\,{\textrm{d}}\sigma \\&\quad = \frac{1}{2}\int _{\Omega }|\nabla {\overline{\textbf{u}}}_0|^2 \,{\textrm{d}}x+ \int _0^t\int _{\Omega }\partial _t{\overline{\textbf{u}}}\cdot \textbf{h}\,{\textrm{d}}x\,{\textrm{d}}\sigma -\int _0^t\int _{\Omega }\partial _t{\overline{\textbf{u}}}:{{\,\textrm{div}\,}}\textbf{H}\,{\textrm{d}}x\,{\textrm{d}}\sigma \\&\qquad +\int _\omega \textbf{n}\textbf{H}\circ {\varvec{\varphi }}\,\textbf{n}\partial _t^2\eta \,{\textrm{d}}y\,{\textrm{d}} t+\frac{1}{2}\int _{\omega }|\partial _y\eta _1|^2 \,{\textrm{d}}y+ \int _0^t\int _\omega g\,\partial _t^2\eta \,{\textrm{d}}y\,{\textrm{d}}\sigma \\&\qquad -\int _0^t\int _\omega \partial _y^4\eta \,\partial _t^2\eta \,{\textrm{d}}y\,{\textrm{d}}\sigma , \end{aligned}$$

such that, using

$$\begin{aligned}&-\int _{0}^t\int _{\omega }\partial _y^4\eta \,\partial _t^2\eta \,{\textrm{d}}y\,{\textrm{d}}\sigma =\int _{0}^t\int _{\omega }\partial _t(\partial _y^3\eta \,\partial _t\partial _y\eta )\,{\textrm{d}}y\,{\textrm{d}}\sigma +\int _{0}^t\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}}\sigma \\&\quad \le \tfrac{1}{2}\sup _{I}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y+2\sup _{I}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y+\int _{I}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}}\sigma , \end{aligned}$$

we have

$$\begin{aligned}&\int _I\int _\Omega |\partial _t{\overline{\textbf{u}}}|^2\,{\textrm{d}}x\,{\textrm{d}}t+\sup _I \int _{\Omega }|\nabla {\overline{\textbf{u}}}|^2 \,{\textrm{d}}x+\sup _I\int _{\omega }|\partial _t\partial _y\eta |^2 \,{\textrm{d}}y+\int _I\int _{\omega }|\partial _t^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\quad \lesssim \int _{\Omega }|\nabla {\overline{\textbf{u}}}_0|^2 \,{\textrm{d}}x+\int _{\omega }|\partial _y\eta _1|^2 \,{\textrm{d}}y+\int _I\int _\Omega \big (|\textbf{h}|^2+|\nabla \textbf{H}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\nonumber \\&\qquad +\int _I\int _{\partial \Omega }|\textbf{H}|^2\,{\textrm{d}}{\mathcal {H}}^1\,{\textrm{d}} t+\int _I\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t+\sup _{I}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y\nonumber \\&\qquad +\int _{I}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y. \end{aligned}$$

Now we differentiate the structure equation in space by testing with \(\partial _t\partial _y^2\eta \) which leads to

$$\begin{aligned}&\frac{1}{2}\int _{\omega } |\partial _t\partial _y\eta |^2 \,{\textrm{d}}y+ \frac{1}{2}\int _{\omega } |\partial _y^3\eta |^2 \,{\textrm{d}}y+\int _0^t\int _{\omega }|\partial _t\partial _y^2\eta |^2 \,{\textrm{d}}y\,{\textrm{d}}\sigma \\&\quad = \frac{1}{2}\int _{\omega } |\partial _y\eta _1 |^2 \,{\textrm{d}}y+ \frac{1}{2}\int _{\omega } |\partial _y^3\eta _0 |^2 \,{\textrm{d}}y+\int _0^t\int _\omega g\,\partial _y^2\partial _t\eta \,{\textrm{d}}y\,{\textrm{d}}\sigma \\&\qquad +\int _I\int _\omega \textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\,\partial _t\partial _y^2\eta \,{\textrm{d}}y\,{\textrm{d}} t\\&\qquad -\int _I\int _\omega \textbf{n}\textbf{H}\circ {\varvec{\varphi }}\,\textbf{n}\,\partial _t\partial _y^2\eta \,{\textrm{d}}y\,{\textrm{d}} t. \end{aligned}$$

Let us explain how to control the last two integrals in the above. By the trace theorem, smoothness of \(\textbf{A}_{\eta _0},\) \(B_{\eta _0}\) and \({\varvec{\varphi }}\) as well as interpolation we have

$$\begin{aligned}&\int _I\int _\omega \textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\,\partial _t\partial _y^2\eta \,{\textrm{d}}y\,{\textrm{d}} t\\&\quad \le \int _{I}\Vert \textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\Vert _{W^{1/2,2}(\omega )}\Vert \partial _t\partial _y^2\eta \Vert _{W^{-1/2,2}(\omega )}\,{\textrm{d}} t\\&\quad \le \,c\int _{I}\big (\Vert \nabla {\overline{\textbf{u}}}\Vert _{W^{1/2,2}(\partial \Omega )}+\Vert {\overline{\pi }}\Vert _{W^{1/2,2}(\partial \Omega )}\big )\Vert \partial _t\eta \Vert _{W^{3/2,2}(\omega )}\,{\textrm{d}} t\\&\quad \le \,c\int _{I}\big (\Vert \nabla {\overline{\textbf{u}}}\Vert _{W^{1,2}(\Omega _\eta )}+\Vert {\overline{\pi }}\Vert _{W^{1,2}(\Omega )}\big )\Vert \partial _t\eta \Vert _{W^{1,2}(\omega )}^{1/2}\Vert \partial _t\eta \Vert _{W^{2,2}(\omega )}^{1/2}\,{\textrm{d}} t\\&\quad \le \,\kappa \int _{I}\big (\Vert \nabla {\overline{\textbf{u}}}\Vert _{W^{1,2}_x}^2+\Vert {\overline{\pi }}\Vert _{W^{1,2}_x}^2\big )\,{\textrm{d}} t+\kappa \int _I\Vert \partial _t\eta \Vert _{W^{2,2}_y}^2\,{\textrm{d}} t\\&\qquad +c(\kappa )\int _I\Vert \partial _t\eta \Vert _{W^{1,2}_y}^2\,{\textrm{d}} t. \end{aligned}$$

In order to control the pressure we write similarly to (2.22)

$$\begin{aligned} {\overline{\pi }}={\overline{\pi }}_0+c_{{\overline{\pi }}}, \end{aligned}$$

where \(({\overline{\pi }}_0)_\Omega =0\) and \(c_{{\overline{\pi }}}\) is a function of time only. The latter satisfies

$$\begin{aligned} c_{{\overline{\pi }}}(t)\int _{\omega }\textbf{n}{\textbf{B}}_{\eta _0}\circ {\varvec{\varphi }}\textbf{n}\,{\textrm{d}}y&=\int _{\omega }\textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}-{\textbf{B}}_{\eta _0}{\overline{\pi }}_0\big )\circ {\varvec{\varphi }}\textbf{n}\,{\textrm{d}}y\\ {}&\quad -\int _{\omega }\textbf{n}\textbf{H}\circ {\varvec{\varphi }}\textbf{n}\,{\textrm{d}}y+\int _\omega \partial _t^2\eta \,{\textrm{d}}y-\int _\omega g\,{\textrm{d}}y\end{aligned}$$

due to Eq. (4.10). Noticing that \(B_{\eta _0}\) is uniformly elliptic we infer from Poincaré’s inequality

$$\begin{aligned} \int _{I}\Vert {\overline{\pi }}\Vert ^2_{W^{1,2}_x}\,{\textrm{d}} t&\lesssim \int _{I}\Vert \nabla {\overline{\pi }}\Vert ^2_{L^{2}_x}\,{\textrm{d}} t+\int _{I}c_{{\overline{\pi }}}^2\,{\textrm{d}} t\\&\lesssim \int _{I}\Vert \nabla {\overline{\pi }}\Vert ^2_{L^{2}_x}\,{\textrm{d}} t+\int _{I}\int _\omega |\partial _t^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t+\int _{I}\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t\\&\quad +\int _{I}\Vert \textbf{H}\Vert _{L^{2}(\partial \Omega )}^2\,{\textrm{d}} t+\int _{I}\Vert {\overline{\pi }}_0\Vert _{L^{2}(\partial \Omega )}^2\,{\textrm{d}} t+\int _{I}\Vert \nabla {\overline{\textbf{u}}}\Vert _{L^{2}(\partial \Omega )}^2\,{\textrm{d}} t. \end{aligned}$$

By the trace theorem and \(({\overline{\pi }}_0)_\Omega =0\) we obtain

$$\begin{aligned} \int _{I}\Vert \nabla {\overline{\textbf{u}}}\Vert _{W^{1/2,2}(\Omega )}^2\,{\textrm{d}} t&\lesssim \int _{I^*}\Vert \nabla {\overline{\textbf{u}}}\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t,\quad \int _{I}\Vert \textbf{H}\Vert _{W^{1/2,2}(\Omega )}^2\,{\textrm{d}} t\lesssim \int _{I}\Vert \textbf{H}\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t,\\ \int _{I}\Vert {\overline{\pi }}_0\Vert _{L^{2}(\partial \Omega )}^2\,{\textrm{d}} t&\lesssim \int _{I}\Vert {\overline{\pi }}_0\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t\lesssim \int _{I}\Vert \nabla {\overline{\pi }}_0\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t=\int _{I}\Vert \nabla {\overline{\pi }}\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t. \end{aligned}$$

Similarly, it holds

$$\begin{aligned} -\int _I\int _\omega \textbf{n}\textbf{H}\circ {\varvec{\varphi }}\,\textbf{n}\,\partial _t\partial _y^2\eta \,{\textrm{d}}y\,{\textrm{d}} t&\le \int _{I}\Vert \textbf{H}\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t+\kappa \int _I\Vert \partial _t\eta \Vert _{W^{2,2}_y}^2\,{\textrm{d}} t\\ {}&\quad +c(\kappa )\int _I\Vert \partial _t\eta \Vert _{W^{1,2}_y}^2\,{\textrm{d}} t\end{aligned}$$

We conclude

$$\begin{aligned}&\sup _I\int _{\omega } |\partial _t\partial _y\eta |^2 \,{\textrm{d}}y+ \sup _I\int _{\omega } |\partial _y^3\eta |^2 \,{\textrm{d}}y+\int _I\int _{\omega }|\partial _t\partial _y^2\eta |^2 \,{\textrm{d}}y\,{\textrm{d}} t\\&\quad \le \,c(\kappa )\bigg (\int _{\omega } |\partial _y\eta _1 |^2 \,{\textrm{d}}y+ \int _{\omega } |\partial _y^3\eta _0 |^2 \,{\textrm{d}}y+\int _I\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t+\overline{{\mathcal {E}}}(0)\bigg ) \\&\qquad +c\int _{I}\Vert \textbf{H}\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t+\kappa \bigg (\int _{I}\Vert \nabla {\overline{\textbf{u}}}\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t+\int _{I}\Vert \nabla {\overline{\pi }}\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t\bigg ), \end{aligned}$$

where \(\kappa >0\) is arbitrary. Now we consider the fluid equation (4.8)–(4.9) and transform it by means of \({\varvec{\Psi }}_{\eta _0}^{-1},\) that is, we set \({\underline{\pi }}={\overline{\pi }}\circ {\varvec{\Psi }}_{\eta _0}^{-1}\) and \(\underline{\textbf{u}}={\overline{\textbf{u}}}\circ {\varvec{\Psi }}_{\eta _0}^{-1}.\) Arguing as in the beginning of the proof of Lemma 4.2 and noticing that this transformation is independent of time we get

$$\begin{aligned} \partial _t{\underline{\textbf{u}}}+\nabla {\underline{\pi }}-\Delta {\underline{\textbf{u}}}&=J_{\eta _0}^{-1}\big (\textbf{h}\circ {\varvec{\Psi }}^{-1}_{\eta _0}-({{\,\textrm{div}\,}}\textbf{H})\circ {\varvec{\Psi }}_{\eta _0}^{-1}\big ),\quad {{\,\textrm{div}\,}}{\underline{\textbf{u}}}=0, \end{aligned}$$

in \(I\times \Omega _{\eta _0}\) together with

$$\begin{aligned} {\underline{\textbf{u}}}\circ {{\varvec{\varphi }}_{\eta _0}} =\partial _t \eta {\textbf{n}}\quad \text {on}\ I\times \omega . \end{aligned}$$

Maximal regularity theory for the classical Stokes problem (and smoothness of \({\varvec{\varphi }}_{\eta _0}\)) yields

$$\begin{aligned}&\int _I\int _{\Omega _{\eta _0}}\big (|\nabla ^2{\underline{\textbf{u}}}|^2+|\partial _t{\underline{\textbf{u}}}|^2+|\nabla {\underline{\pi }}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\\&\quad \lesssim \int _I\int _{\Omega _{\eta _0}}\big (|\textbf{h}\circ {\varvec{\Psi }}_{\eta _0}^{-1}|^2+|({{\,\textrm{div}\,}}\textbf{H})\circ {\varvec{\Psi }}_{\eta _0}^{-1}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\\&\qquad +\int _I\Vert \partial _t\eta \Vert ^2_{W^{3/2,2}(\omega )}\,{\textrm{d}} t\end{aligned}$$

such that, for \(\kappa >0\) arbitrary,

$$\begin{aligned}&\int _I\int _{\Omega }\big (|\nabla ^2{\overline{\textbf{u}}}|^2+|\partial _t{\overline{\textbf{u}}}|^2+|\nabla {\overline{\pi }}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\\&\quad \le \,c\int _I\int _{\Omega }\big (|\textbf{h}|^2+|\nabla \textbf{H}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t+c(\kappa )\int _I\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\\&\qquad +\kappa \int _I\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\end{aligned}$$

using again interpolation and transforming back to \(\Omega .\)

Collecting all the estimate and choosing \(\kappa \) small enough proves the claim for \(h=0.\) Let us now explain how to remove this restriction. We consider the steady Stokes-type system

$$\begin{aligned} {{\,\textrm{div}\,}}\big ({\textbf{B}}_{\eta _0}{{\overline{p}}}\big )-{{\,\textrm{div}\,}}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{v}}}\big )=0,\quad B_{\eta _0}:\nabla {\overline{\textbf{v}}}=h,\quad {\overline{\textbf{v}}}|_{\partial \Omega }=0, \end{aligned}$$

in \(\Omega \) for a given function \(h:\Omega \rightarrow \mathbb {R}.\) We denote the solution operator, which maps h to \({\overline{\textbf{v}}},\) by \({\mathcal {A}}_{\eta _0}^{-1}\) and We claim that the estimates

$$\begin{aligned} \int _{\Omega }|\nabla {\mathcal {A}}^{-1}_{\eta _0}h|^2\,{\textrm{d}}x\lesssim \int _{\Omega }|h|^2\,{\textrm{d}}x,\quad \int _{\Omega }|\nabla ^2{\mathcal {A}}^{-1}_{\eta _0}h|^2\,{\textrm{d}}x\lesssim \int _{\Omega }|\nabla h|^2\,{\textrm{d}}x, \end{aligned}$$

hold. Indeed, transforming (4.19) by means of \({\varvec{\Psi }}_{\eta _0}^{-1}\) (that is, setting \({\underline{p}}={{\overline{p}}}\circ {\varvec{\Psi }}_{\eta _0}^{-1}\) and \(\underline{\textbf{v}}={\overline{\textbf{v}}}\circ {\varvec{\Psi }}_{\eta _0}^{-1}\)) we obtain the system

$$\begin{aligned} \nabla {\underline{p}}-\Delta {\underline{\textbf{v}}}=0,\quad {{\,\textrm{div}\,}}{\underline{\textbf{v}}}=h\circ {\varvec{\Psi }}_{\eta _0}^{-1},\quad {\overline{\textbf{v}}}|_{\partial \Omega _{\eta _0}}=0, \end{aligned}$$

in \(\Omega _{\eta _0}.\) The estimates

$$\begin{aligned} \int _{\Omega }|\nabla {\underline{\textbf{v}}}|^2\,{\textrm{d}}x\lesssim \int _{\Omega }|h\circ {\varvec{\Psi }}_{\eta _0}^{-1}|^2\,{\textrm{d}}x,\quad \int _{\Omega }|\nabla ^2{\underline{\textbf{v}}}|^2\,{\textrm{d}}x\lesssim \int _{\Omega }|\nabla (h\circ {\varvec{\Psi }}_{\eta _0}^{-1})|^2\,{\textrm{d}}x, \end{aligned}$$

are classical and yield (4.20) by transformation (and smoothness of \({\varvec{\Psi }}_{\eta _0}\)). If \({\overline{\textbf{u}}}\) satisfies (4.8)–(4.9) for a given function h,  then \({\overline{\textbf{u}}}-{\mathcal {A}}_{\eta _0}^{-1}h\) satisfies the problem with homogenous constraint (note that (4.10) and (4.11) do not change as \({\mathcal {A}}_{\eta _0}^{-1}h\) vanishes at the boundary) with the additional term \(J_{\eta _0}\partial _t{\mathcal {A}}_{\eta _0}^{-1}h\) on the right-hand side of the momentum equation. Applying the previously proved estimate for the problem with homogeneous constraint we obtain the additional term

$$\begin{aligned} \int _I\int _{\Omega }\big (|\partial _t{\mathcal {A}}_{\eta _0}^{-1}h|^2+|\nabla ^2{\mathcal {A}}_{\eta _0}^{-1}h|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t&\lesssim \int _I\int _\Omega |\nabla h|^2\,{\textrm{d}}x\,{\textrm{d}}t+\int _I\Vert \partial _t h\Vert _{W^{-1,2}(\Omega )}^2\,{\textrm{d}} t\\ \sup _I\int _{\Omega }|\nabla {\mathcal {A}}_{\eta _0}^{-1}h|^2&\lesssim \int _I\int _{\Omega }\big (|\partial _t{\mathcal {A}}_{\eta _0}^{-1}h|^2+|\nabla ^2{\mathcal {A}}_{\eta _0}^{-1}h|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\\&\lesssim \int _I\int _\Omega |\nabla h|^2\,{\textrm{d}}x\,{\textrm{d}}t+\int _I\Vert \partial _t h\Vert _{W^{-1,2}(\Omega )}^2\,{\textrm{d}} t\end{aligned}$$

using also (4.20) and \(h(\cdot ,0)=0.\) The proof is now completed. \(\square \)

4.3 The fixed point argument

We consider now for \((\zeta ,{\overline{\textbf{w}}},{{\overline{q}}})\) given the problem

$$\begin{aligned} J_{\eta _0}\partial _t{\overline{\textbf{u}}}+{{\,\textrm{div}\,}}\big ({\textbf{B}}_{\eta _0}{\overline{\pi }}\big )-{{\,\textrm{div}\,}}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}\big )&=\textbf{h}_{\zeta }({\overline{\textbf{w}}})-{{\,\textrm{div}\,}}\textbf{H}_{\zeta }({\overline{\textbf{w}}},{{\overline{q}}}), \end{aligned}$$
$$\begin{aligned} B_{\eta _0}:\nabla {\overline{\textbf{u}}}&=h_{\zeta }({\overline{\textbf{w}}}), \end{aligned}$$
$$\begin{aligned} \partial _t^2\eta -\partial _t\partial _y^2\eta +\partial _y^4\eta&=-\textbf{n}\big (\textbf{A}_{\eta _0}\nabla {\overline{\textbf{u}}}-{\textbf{B}}_{\eta _0}{\overline{\pi }}\big )\circ {\varvec{\varphi }}\,\textbf{n}\end{aligned}$$
$$\begin{aligned}&\quad +\textbf{n}\textbf{H}_{\zeta }({\overline{\textbf{w}}},{{\overline{q}}})\circ {\varvec{\varphi }}\,\textbf{n}, \nonumber \\ {\overline{\textbf{u}}}\circ {\varvec{\varphi }}&=\partial _t \eta {\textbf{n}}\quad \text {on}\quad I\times \omega . \end{aligned}$$

We consider the solution map \({\mathscr {T}}_{\eta _0}\) which maps \((\zeta ,{\overline{\textbf{w}}},{{\overline{q}}})\) to the solution \((\eta ,{\overline{\textbf{u}}},{\overline{\pi }})\) of (4.23)–(4.26) (existence of which follows from Lemma 4.3). Setting \(I^*=(0,T^*)\) for some small \(T^*>0\) we must prove that it is a contraction on the space

$$\begin{aligned} {\mathscr {Y}}^*:=&\,\,W^{1,\infty } \big (I^*; W^{1,2}(\omega ) \big )\cap W^{1,2} \big (I^*; W^{2,2}(\omega ) \big )\cap L^\infty \big (I^*; W^{3,2}(\omega ) \big ) \cap W^{2,2}(I^*;L^2(\omega ))\\&\times L^\infty \big (I^*; W^{1,2}(\Omega ) \big )\cap W^{1,2} \big (I^*; L^{2}(\Omega ) \big )\cap L^2 \big (I^*; W^{2,2}(\Omega ) \big )\times L^2 \big (I^*; W^{1,2}(\Omega )\big ) \end{aligned}$$

complemented with the norm

$$\begin{aligned} \Vert (\eta ,{\overline{\textbf{u}}},{\overline{\pi }})\Vert _{\mathscr {Y}^*}^2:=&\sup _{I^*}\int _\Omega |{\overline{\textbf{u}}}|^2\,{\textrm{d}}x+\int _{I^*}\int _\Omega |\nabla {\overline{\textbf{u}}}|^2\,{\textrm{d}}x\,{\textrm{d}} t+\sup _{I^*}\int _\Omega |\nabla {\overline{\textbf{u}}}|^2\,{\textrm{d}}x\\&+\int _{I^*}\int _\Omega \big (|\nabla ^2{\overline{\textbf{u}}}|^2+|\partial _t{\overline{\textbf{u}}}|^2+|{\overline{\pi }}|^2+|\nabla {\overline{\pi }}|^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\\&+\sup _{I^*}\int _{\omega }\big ( |\partial _t\eta |^2+ |\partial _y^2\eta |^2\big )\,{\textrm{d}}y+\int _{I^*}\int _{\omega } |\partial _t\partial _y\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\\&+\sup _{I^*}\int _{\omega }\big ( |\partial _t\partial _y\eta |^2+ |\partial _y^3\eta |^2\big )\,{\textrm{d}}y+\int _{I^*}\int _{\omega } \big (|\partial _t\partial _y^2\eta |^2+|\partial _t^2\eta |^2\big ) \,{\textrm{d}}y\,{\textrm{d}} t\end{aligned}$$

given by the energy estimate from Lemma 4.3. This is the content of the following lemma in which we denote by \(B_{R}^{{\mathscr {Y}}^*}(0)\) the ball in \({\mathscr {Y}}^*\) with radius R around the origin.

Lemma 4.4

Suppose that \((\textbf{f},g,\eta _0,\textbf{u}_0,\eta _1)\) satisfies (2.21) and (2.25). There are \(R\gg 1\) and \(T^{*}\ll 1\) such that

$$\begin{aligned} {\mathscr {T}}_{\eta _0}:B_{R}^{{\mathscr {Y}}^*}(0)\cap \{\eta (0)=\eta _0\}\rightarrow B_{R}^{{\mathscr {Y}}^*}(0)\cap \{\eta (0)=\eta _0\} \end{aligned}$$

is a contraction.


First of all, we choose R sufficiently large compared to the dataset \((\textbf{f},g,\eta _0,\textbf{u}_0,\eta _1).\) We intend to control the Lipschitz constants of the mappings

$$\begin{aligned} {\mathscr {Y}}^*\ni (\zeta ,{\overline{\textbf{w}}},{{\overline{q}}})&\mapsto \textbf{h}_\zeta ({\overline{\textbf{w}}})\in L^2(I^*;L^2(\Omega )),\nonumber \\ {\mathscr {Y}}^*\ni (\zeta ,{\overline{\textbf{w}}},{{\overline{q}}})&\mapsto \textbf{H}_\zeta ({\overline{\textbf{w}}},{{\overline{q}}})\in L^2(I^*;W^{1,2}(\Omega )),\nonumber \\ {\mathscr {Y}}^*\ni (\zeta ,{\overline{\textbf{w}}},{{\overline{q}}})&\mapsto h_\zeta ({\overline{\textbf{w}}})\in L^2(I^*;W^{1,2}(\Omega ))\cap W^{1,2}(I^*,W^{-1,2}(\Omega )). \end{aligned}$$

Since all of them map \((\eta _0,0,0)\) to the origin, this will also imply that

$$\begin{aligned} {\mathscr {T}}_{\eta _0}:B_{R}^{{\mathscr {Y}}^*}(0)\cap \{\eta (0)=\eta _0\}\rightarrow B_{R}^{{\mathscr {Y}}^*}(0). \end{aligned}$$

As far as \(\textbf{H}\) is concerned, we haveFootnote 6

$$\begin{aligned}&\int _{I^*}\Vert \textbf{H}_{\zeta _1}({\overline{\textbf{w}}}_1,{{\overline{q}}}_1)-\textbf{H}_{\zeta _2}({\overline{\textbf{w}}}_2,{{\overline{q}}}_2)\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t\\ {}&\quad \lesssim \int _{I^*}\Vert (\textbf{A}_{\eta _0}-\textbf{A}_{ \zeta _1})(\nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2)\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t+\int _{I^*}\Vert (\textbf{A}_{ \zeta _1}-\textbf{A}_{ \zeta _2})\nabla {\overline{\textbf{w}}}_2\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t\\&\quad +\int _{I^*}\Vert (B_{ \zeta _1}-B_{\eta _0})({{\overline{q}}}_1-{{\overline{q}}}_2)\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t+\int _{I^*}\Vert ({\textbf{B}}_{ \zeta _1}-{\textbf{B}}_{ \zeta _2}){{\overline{q}}}_2\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t, \end{aligned}$$

where, by the embeddings \(W^{1,2}(I^*;L^\infty (\omega ))\hookrightarrow L^\infty (I^*;L^\infty (\omega ))\) and \(W^{1,2}(\omega )\hookrightarrow L^{\infty }(\omega )\) as well as (2.11)–(2.13),

$$\begin{aligned}&\int _{I^*}\Vert (\textbf{A}_{\eta _0}-\textbf{A}_{ \zeta _1})(\nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2)\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t+\int _{I^*}\Vert (\textbf{A}_{ \zeta _1}-\textbf{A}_{ \zeta _2})\nabla {\overline{\textbf{w}}}_2\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim T^{*}\sup _{I^*}\Vert \partial _y^2 \zeta _1-\partial _y^2\eta _0\Vert _{L^\infty _y}^2\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}\\ {}&\qquad +\sup _{I^*}\Vert \partial _y\zeta _1- \partial _y\eta _0\Vert _{L^\infty _y}^2\int _{ I^{*}}\Vert \nabla ^2{\overline{\textbf{w}}}_1-\nabla ^2{\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\qquad + T^{*}\sup _{I^*}\Vert \partial _y^2 \zeta _1-\partial _y^2 \zeta _2\Vert _{L^\infty _y}^2\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}\\ {}&\qquad +\sup _{I^*}\Vert \partial _y\zeta _1- \partial _y\zeta _2\Vert _{L^\infty _y}^2\int _{ I^{*}}\Vert \nabla ^2{\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\quad \lesssim T^{*}\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}+T^*\int _{ I^{*}}\Vert \nabla ^2{\overline{\textbf{w}}}_1-\nabla ^2{\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\qquad + T^{*}\sup _{I^*}\Vert \partial _y^3 \zeta _1-\partial _y^3 \zeta _2\Vert _{L^2_y}^2+T^*\int _{I^*}\Vert \partial _t \partial _y^2\zeta _1-\partial _t \partial _y^2\zeta _2\Vert _{L^{2}_y}^2\,{\textrm{d}} t\\&\quad \lesssim T^*\Vert ( \zeta _1,{\overline{\textbf{w}}}_1,{\overline{q}}_1)-( \zeta _2,{\overline{\textbf{w}}}_2,{\overline{q}}_2)\Vert _{{\mathscr {Y}}^*}^2 \end{aligned}$$

as well as

$$\begin{aligned}&\int _{I^*}\Vert ({\textbf{B}}_{ \zeta _1}-{\textbf{B}}_{\eta _0})({{\overline{q}}}_1-{{\overline{q}}}_2)\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t+\int _{I^*}\Vert ({\textbf{B}}_{ \zeta _1}-{\textbf{B}}_{ \zeta _2}){{\overline{q}}}_2\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim \sup _{I^*}\Vert \partial _y^2\zeta _1-\partial _y^2\eta _0\Vert _{L^4_y}^2\int _{I^*}\Vert {{\overline{q}}}_1-{{\overline{q}}}_2\Vert ^2_{L^4_x}\,{\textrm{d}} t\\ {}&\qquad +\sup _{I^*}\Vert \partial _y\zeta _1- \partial _y\eta _0\Vert _{L^\infty _y}^2\int _{ I^{*}}\Vert \nabla {{\overline{q}}}_1-\nabla {{\overline{q}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\qquad + \sup _{I^*}\Vert \partial _y^2\zeta _1-\partial _y^2\zeta _2\Vert _{L^4_y}^2\int _{I^*}\Vert {{\overline{q}}}_2\Vert ^2_{L^4_x}\,{\textrm{d}} t\\ {}&\qquad +\sup _{I^*}\Vert \partial _y\zeta _1- \partial _y\zeta _2\Vert _{L^\infty _y}^2\int _{ I^{*}}\Vert \nabla {{\overline{q}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\quad \lesssim \sup _{I^*}\Vert \zeta _1-\eta _0\Vert _{W^{5/2,2}_y}^2\Vert {{\overline{q}}}_1-{{\overline{q}}}_2\Vert ^2_{W^{1,2}_x}\,{\textrm{d}} t\\ {}&\qquad +\sup _{I^*}\Vert \partial _y^2\zeta _1- \partial _y^2\eta _0\Vert _{L^2_y}^2\int _{ I^{*}}\Vert \nabla {{\overline{q}}}_1-\nabla {{\overline{q}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\qquad + \sup _{I^*}\Vert \zeta _1-\zeta _2\Vert _{W^{5/2,2}_y}^2\int _{I^*}\Vert {{\overline{q}}}_2\Vert ^2_{W^{1,2}_x}\,{\textrm{d}} t\\ {}&\qquad +\sup _{I^*}\Vert \partial _y^2\zeta _1- \partial _y^2\zeta _2\Vert _{L^2_y}^2\int _{ I^{*}}\Vert \nabla {{\overline{q}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\quad \lesssim \sqrt{T^*}\int _{I^*}\Vert {{\overline{q}}}_1-{{\overline{q}}}_2\Vert ^2_{W^{1,2}_x}\,{\textrm{d}} t+T^*\int _{ I^{*}}\Vert \nabla {{\overline{q}}}_1-\nabla {{\overline{q}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\qquad + \sqrt{T^*}\bigg (\sup _{I^*}\Vert \partial _y^3\zeta _1-\partial _y^3\zeta _2\Vert ^2_{L^{2}_y}+\int _{I^*}\Vert \partial _t\partial _y^2\zeta _1-\partial _t\partial _y^2\zeta _2\Vert ^2_{L^2_y}\,{\textrm{d}} t\bigg )\\ {}&\qquad +T^*\int _{I^*}\Vert \partial _t\partial _y^2\zeta _1- \partial _t\partial _y^2\zeta _2\Vert _{L^2_y}^2\,{\textrm{d}} t\\&\quad \lesssim \sqrt{T^*}\Vert ( \zeta _1,{\overline{\textbf{w}}}_1,{\overline{q}}_1)-( \zeta _2,{\overline{\textbf{w}}}_2,{\overline{q}}_2)\Vert _{{\mathscr {Y}}^*}^2. \end{aligned}$$

Note that we also used the embedding

$$\begin{aligned} W^{1,2}(I^*;W^{2,2})\cap L^\infty (I^*;W^{3,2}(\Omega ))&\hookrightarrow C^{1/4}(I^*;W^{5/2,2}(\omega ))\\ {}&\hookrightarrow C^{1/4}(I^*;W^{2,4}(\omega )). \end{aligned}$$

Similarly, it holds

$$\begin{aligned}&\int _{I^*}\Vert h_{\zeta _1}({\overline{\textbf{w}}}_1)-h_{\zeta _2}({\overline{\textbf{w}}}_2)\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim \int _{I^*}\Vert ({\textbf{B}}_{\zeta _0}-{\textbf{B}}_{ \zeta _1}):(\nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2)\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t\\ {}&\qquad +\int _{I^*}\Vert ({\textbf{B}}_{ \zeta _1}-{\textbf{B}}_{ \zeta _2}):\nabla {\overline{\textbf{w}}}_2\Vert _{W^{1,2}_x}^2\,{\textrm{d}} t\\ {}&\lesssim T^{*}\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}+T^*\int _{ I^{*}}\Vert \nabla ^2{\overline{\textbf{w}}}_1-\nabla ^2{\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}\,{\textrm{d}} t\\&\qquad + T^{*}\sup _{I^*}\Vert \partial _y^3 \zeta _1-\partial _y^3 \zeta _2\Vert _{L^2_y}^2+T^*\int _{I^*}\Vert \partial _t \partial _y^2\zeta _1-\partial _t \partial _y^2\zeta _2\Vert _{L^{2}_y}^2\,{\textrm{d}} t\\&\quad \lesssim T^*\Vert ( \zeta _1,{\overline{\textbf{w}}}_1,{\overline{q}}_1)-( \zeta _2,{\overline{\textbf{w}}}_2,{\overline{q}}_2)\Vert _{{\mathscr {Y}}^*}^2. \end{aligned}$$

As far as the \(W^{1,2}_tW^{-1,2}_x\)-norm is concerned, we use the embeddings

$$\begin{aligned}&L^2(I^*;W^{3,2}(\omega ))\cap W^{2,2}(I^*;L^2(\omega ))\hookrightarrow W^{1,2}(I^*;W^{1,4}(\omega )),\\&W^{1,2}(I^*;W^{2,2}(\omega ))\cap L^\infty (I^*;W^{3,2}(\omega ))\hookrightarrow C^{1/8}(I^*;W^{11/4,2}(\omega )), \end{aligned}$$

and obtain

$$\begin{aligned}&\int _{I^*}\Vert \partial _t(h_{\zeta _1}({\overline{\textbf{w}}}_1)-h_{\zeta _2}({\overline{\textbf{w}}}_2))\Vert _{W^{-1,2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim \int _{I^*}\Vert \partial _t{\textbf{B}}_{ \zeta _1}:(\nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2)\Vert _{W^{-1,2}_x}^2\,{\textrm{d}} t\\ {}&\qquad +\int _{I^*}\Vert \partial _t({\textbf{B}}_{ \zeta _1}-{\textbf{B}}_{ \zeta _2}):\nabla {\overline{\textbf{w}}}_2\Vert _{W^{-1,2}_x}^2\,{\textrm{d}} t\\&\qquad +\int _{I^*}\Vert ({\textbf{B}}_{\eta _0}-{\textbf{B}}_{ \zeta _1}):\partial _t(\nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2)\Vert _{W^{-1,2}_x}^2\,{\textrm{d}} t\\ {}&\qquad +\int _{I^*}\Vert ({\textbf{B}}_{ \zeta _1}-{\textbf{B}}_{ \zeta _2}):\nabla \partial _t{\overline{\textbf{w}}}_2\Vert _{W^{-1,2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim \sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert _{L^{2}_x}^2\int _{I^*}(1+\Vert \partial _t\partial _y\zeta _1\Vert _{L^4_y}^2)\,{\textrm{d}} t\\ {}&\qquad +\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_2\Vert _{L^{2}_x}^2\int _{I^*}\Vert \partial _t\partial _y(\zeta _1-\zeta _2)\Vert _{L^4_y}^2\,{\textrm{d}} t\\&\qquad +\sup _{I^*}\Vert \partial _y^2(\zeta _0-\zeta _1)\Vert _{L^\infty _y}\int _{I^*}\Vert \partial _t({\overline{\textbf{w}}}_1-{\overline{\textbf{w}}}_2)\Vert _{L^{2}_x}^2\,{\textrm{d}} t\\ {}&\qquad +\sup _{I^*}\Vert \partial _y^2(\zeta _1-\zeta _2)\Vert _{L^\infty _y}\int _{I^*}\Vert \partial _t{\overline{\textbf{w}}}_2\Vert _{L^{2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim \sqrt{T^*}\sup _{I^*}\Vert \partial _y^3(\zeta _1-\zeta _2)\Vert _{L^2_y}\bigg (\int _{I^*}\Vert \partial _t\partial _y^2(\zeta _1-\zeta _2)\Vert ^2_{L^2_y}\,{\textrm{d}} t\bigg )^{\frac{1}{2}}\\&\qquad +\sqrt{T^{*}}\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert ^2_{L^2_x}+ \root 4 \of {T^{*}}\sup _{I^*}\Vert \partial _t({\overline{\textbf{w}}}_1-{\overline{\textbf{w}}}_2)\Vert _{L^2}^2\,{\textrm{d}} t\\&\qquad +\root 4 \of {T^*}\bigg (\sup _{I^*}\Vert \partial _y^3\zeta _1-\partial _y^3\zeta _2\Vert ^2_{L^{2}_y}+\int _{I^*}\Vert \partial _t\partial _y^2\zeta _1-\partial _t\partial _y^2\zeta _2\Vert ^2_{L^2_y}\,{\textrm{d}} t\bigg )\\&\quad \lesssim \root 4 \of {T^*}\Vert ( \zeta _1,{\overline{\textbf{w}}}_1,{\overline{q}}_1)-( \zeta _2,{\overline{\textbf{w}}}_2,{\overline{q}}_2)\Vert _{{\mathscr {Y}}^*}^2. \end{aligned}$$

Moreover, we gain using again (2.12) and (2.13)

$$\begin{aligned}&\int _{I^*}\Vert \textbf{h}_{\zeta _1}({\overline{\textbf{w}}}_1,{{\overline{q}}}_1)-\textbf{h}_{\zeta _2}({\overline{\textbf{w}}}_2,{{\overline{q}}}_2)\Vert _{L^{2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim \int _{I^*}\Vert (J_{ \zeta _1}-J_{\eta _0})(\partial _t{\overline{\textbf{w}}}_1-\partial _t{\overline{\textbf{w}}}_2)\Vert _{L^2_x}^2\,{\textrm{d}} t+\int _{I^*}\Vert (J_{ \zeta _2}-J_{ \zeta _1})\partial _t{\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\qquad +\int _{I^*}\Vert J_{ \zeta _1}\nabla {\varvec{\Psi }}_{ \zeta _1}^{-1}\circ {\varvec{\Psi }}_{ \zeta _1}(\nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2)\partial _t{\varvec{\Psi }}_{ \zeta _1}^{-1}\circ {\varvec{\Psi }}_{ \zeta _1}\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\qquad +\int _{I^*}\Vert \big (J_{ \zeta _1}\nabla {\varvec{\Psi }}_{ \zeta _1}^{-1}\circ {\varvec{\Psi }}_{ \zeta _1}-J_{ \zeta _2}\nabla {\varvec{\Psi }}_{ \zeta _2}^{-1}\circ {\varvec{\Psi }}_{ \zeta _2}\big )\nabla {\overline{\textbf{w}}}_2\partial _t{\varvec{\Psi }}_{ \zeta _1}^{-1}\circ {\varvec{\Psi }}_{ \zeta _1}\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\qquad +\int _{I^*}\Vert J_{ \zeta _2}\nabla {\varvec{\Psi }}_{ \zeta _2}^{-1}\circ {\varvec{\Psi }}_{ \zeta _2}\nabla {\overline{\textbf{w}}}_2\big (\partial _t{\varvec{\Psi }}_{ \zeta _1}^{-1}\circ {\varvec{\Psi }}_{ \zeta _1}-\partial _t{\varvec{\Psi }}_{ \zeta _2}^{-1}\circ {\varvec{\Psi }}_{ \zeta _2}\big )\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\qquad +\int _{I^*}\Vert J_{ \zeta _1}\nabla {\varvec{\Psi }}_{ \zeta _1}^{-1}\circ {\varvec{\Psi }}_{ \zeta _1}(\nabla {\overline{\textbf{w}}}_1{\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2{\overline{\textbf{w}}}_2)\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\qquad +\int _{I^*}\Vert (J_{ \zeta _1}\nabla {\varvec{\Psi }}_{ \zeta _1}^{-1}\circ {\varvec{\Psi }}_{ \zeta _1}-J_{ \zeta _2}\nabla {\varvec{\Psi }}_{ \zeta _2}^{-1}\circ {\varvec{\Psi }}_{ \zeta _2})\nabla {\overline{\textbf{w}}}_2{\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\qquad +\int _{I^*}\Vert J_{\zeta _1}(\textbf{f}\circ {\varvec{\Psi }}_{ \zeta _1}-\textbf{f}\circ {\varvec{\Psi }}_{ \zeta _2})\Vert _{L^2_x}^2\,{\textrm{d}} t+\int _{I^*}\Vert (J_{\zeta _1}-J_{\zeta _2})\textbf{f}\circ {\varvec{\Psi }}_{ \zeta _2}\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\quad \lesssim \sup _{I^*}\Vert \partial _y\zeta _1- \partial _y\eta _0\Vert _{L^\infty _y}^2\int _{I^*}\Vert \partial _t{\overline{\textbf{w}}}_1-\partial _t{\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\,{\textrm{d}} t\\ {}&\qquad +\sup _{I^*}\Vert \partial _y\zeta _2- \partial _y\zeta _1\Vert ^2_{L^\infty _y}\int _{I^*}\Vert \partial _t{\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\,{\textrm{d}} t\\&\qquad +T^*\sup _{I^*}\big (1+\Vert \partial _y\zeta _1\Vert ^2_{L^\infty _y}\big )\sup _{I^*}\big (1+\Vert \partial _t \zeta _1\Vert ^2_{L^\infty _y}\big )\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\\&\qquad +T^*\sup _{I^*}\Vert \partial _y\zeta _1- \partial _y\zeta _2\Vert _{L^\infty _y}^2\sup _{I^*}\big (1+\Vert \partial _t \zeta _1\Vert _{L^\infty _y}^2\big )\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\\&\qquad +T^*\sup _{I^*}\big (1+\Vert \partial _y\zeta _2\Vert ^2_{L^\infty _y}\big )\sup _{I^*}\Vert \partial _t \zeta _1-\partial _t \zeta _2\Vert _{L^\infty _y}^2\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\\&\qquad +\sup _{I^*}\big (1+\Vert \partial _y\zeta _1\Vert _{L^\infty _y}^2\big )\sup _{I^*}\Vert {\overline{\textbf{w}}}_1-{\overline{\textbf{w}}}_2\Vert ^2_{L^4_x}\int _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1\Vert _{L^4_x}^2\,{\textrm{d}} t\\&\qquad +\sup _{I^*}\big (1+\Vert \partial _y\zeta _1\Vert _{L^\infty _y}^2\big )\sup _{I^*}\Vert {\overline{\textbf{w}}}_1\Vert ^2_{L^4_x}\int _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert _{L^4_x}^2\,{\textrm{d}} t\\&\qquad +\sup _{I^*}\Vert \partial _y\zeta _1- \partial _y\zeta _2\Vert _{L^\infty _y}^2\sup _{I^*}\Vert {\overline{\textbf{w}}}_2\Vert ^2_{L^4_x}\int _{I^*}\Vert \nabla {\overline{\textbf{w}}}_2\Vert _{L^4_x}^2\,{\textrm{d}} t\\&\qquad +T^*\sup _{I^*}\Vert \nabla \textbf{f}\Vert _{L^2_x}^2\sup _{I^*}\Vert \zeta _1- \zeta _2\Vert ^2_{L^\infty _y}. \end{aligned}$$

Finally, using the parabolic embeddings

$$\begin{aligned}&L^\infty (I^*,L^2(\Omega ))\cap L^2(I^*;W^{1,2}(\Omega ))\hookrightarrow L^4(I^*;L^4(\Omega )),\\&W^{1,2}(I^*;L^\infty (\omega ))\hookrightarrow L^\infty (I^*;L^\infty (\omega )), \end{aligned}$$

we obtain

$$\begin{aligned}&\int _{I^*}\Vert \textbf{h}_{\zeta _1}({\overline{\textbf{w}}}_1,{{\overline{q}}}_1)-\textbf{h}_{\zeta _2}({\overline{\textbf{w}}}_2,{{\overline{q}}}_2)\Vert _{L^{2}_x}^2\,{\textrm{d}} t\\&\quad \lesssim T^*\Vert \partial _t{\overline{\textbf{w}}}_1-\partial _t{\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\,{\textrm{d}} t+T^*\int _{I^*}\Vert \partial _t( \partial _y^2\zeta _2- \partial _y^2\zeta _1)\Vert ^2_{L^2_y}\,{\textrm{d}} t\\&\qquad +T^*\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert _{L^2_x}^2 +T^*\sup _{I^*}\Vert \partial _y^2\zeta _1- \partial _y^2\zeta _2\Vert _{L^\infty _y}^2\\ {}&\qquad +T^*\sup _{I^*}\Vert \partial _t \zeta _1-\partial _t \zeta _2\Vert _{L^\infty _y}^2+\sqrt{T^*}\sup _{I^*}\Vert {\overline{\textbf{w}}}_1-{\overline{\textbf{w}}}_2\Vert ^2_{W^{1,2}_x}\\ {}&\qquad +\sqrt{T^*}\bigg (\int _{I^*}\Vert \nabla ^2{\overline{\textbf{w}}}_1-\nabla ^2{\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\,{\textrm{d}} t+\sup _{I^*}\Vert \nabla {\overline{\textbf{w}}}_1-\nabla {\overline{\textbf{w}}}_2\Vert _{L^2_x}^2\bigg )\\&\qquad +T^*\sup _{I^*}\Vert \zeta _1- \zeta _2\Vert ^2_{L^\infty _y}\\&\quad \lesssim \sqrt{T^*}\Vert ( \zeta _1,{\overline{\textbf{w}}}_1,{\overline{q}}_1)-( \zeta _1,{\overline{\textbf{w}}}_2,{\overline{q}}_2)\Vert _{{\mathscr {Y}}^*}^2. \end{aligned}$$

In conclusion, the Lipschitz constants of the mappings in (4.27) can be made arbitrarily small if we choose \(T^*\) appropriately. Combining this observation with the estimate from Lemma 4.3 gives the claim. \(\square \)

Proof of Proposition 4.1

Combining Lemmas 4.24.4 yields the claim by a standard fixed point argument. \(\square \)

5 Regularity estimates

This section is devoted to the proof of Theorem 2.5 for the fluid–structure-interaction problem (1.1)–(1.2). With the local strong solution from Proposition 4.1 at hand we have a sufficiently smooth object such that the following computations are well-defined. The heart of our analysis is an acceleration estimate in Proposition 5.1. It implies that there is no blow-up in finite time such that the global solution can be constructed by gluing local solutions together.

Let us start with the standard energy estimate (which is even satisfied by weak solutions). Let \((\eta ,\textbf{u},\pi )\) be the unique local strong solution from Proposition 4.1. We choose \((\textbf{u},\partial _t\eta )\) as a test-function in the weak formulation, cf. Definition 2.3(c). This yields

$$\begin{aligned}&{\mathcal {E}}(t) +\int _0^t \int _{\Omega _{\eta }}|\nabla \textbf{u}|^2 \,{\textrm{d}}x\,{\textrm{d}}\sigma +\int _0^t\int _{\omega }|\partial _t\partial _y\eta |^2 \,{\textrm{d}}y\,{\textrm{d}}\sigma \nonumber \\&\quad \le {\mathcal {E}}(0) + \int _0^t\int _{\Omega _\eta }\textbf{u}\cdot \textbf{f}\,{\textrm{d}}x\,{\textrm{d}}\sigma + \int _0^{t}\int _\omega g\,\partial _t\eta \,{\textrm{d}}y\,{\textrm{d}}\sigma ,\nonumber \\&{\mathcal {E}}(t)=\frac{1}{2}\int _{\Omega _{\eta (t)}} |\textbf{u}(t)|^2 \,{\textrm{d}}x+ \frac{1}{2}\int _{\omega } |\partial _t\eta (t) |^2 \,{\textrm{d}}y+ \frac{1}{2}\int _{\omega } |\partial _y^2\eta (t) |^2 \,{\textrm{d}}y\end{aligned}$$

and hence

$$\begin{aligned}&\sup _{I^*}{\mathcal {E}}(t)+ \int _{I^*}\int _{\Omega _{\eta }}|\nabla \textbf{u}|^2 \,{\textrm{d}}x\,{\textrm{d}} t+\int _{I^*}\int _{\omega }|\partial _t\partial _y\eta |^2 \,{\textrm{d}}y\,{\textrm{d}} t\\ {}&\quad \lesssim {\mathcal {E}}(0)+\int _{I^*}\int _{\Omega _\eta }|\textbf{f}|^2\,{\textrm{d}}x\,{\textrm{d}}t+\int _{I^*}\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t=:C_0 \end{aligned}$$

for all \(t\in I^*=(0,T^*).\) This implies

$$\begin{aligned}&\sup _{I^*}\Vert \textbf{u}\Vert _{L^2_x}^2+\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^2_x}^2\,{\textrm{d}} t\lesssim \,C_0, \end{aligned}$$
$$\begin{aligned}&\sup _{I^*}\Vert \partial _t\eta \Vert _{L^2_y}^2+\sup _{I^*}\Vert \partial _y^2\eta \Vert _{L^2_y}^2+\int _{I^*}\Vert \partial _t\partial _y\eta \Vert _{L^2_y}^2\,{\textrm{d}} t\lesssim \,C_0. \end{aligned}$$

5.1 The acceleration estimate

The acceleration estimate is the heart of our analysis. It heavily relies on the elliptic estimate for the Stokes system in irregular domains given in Theorem 3.2. A further difference to [10] is that we cannot work with a solenoidal extension operator (as explained in Sect. 1.3). Hence we must estimate the pressure function.

Proposition 5.1

Suppose that the assumptions of Theorem 2.5 hold and let \((\eta ,\textbf{u},\pi )\) be the unique local strong solution from Proposition 4.1. Then we have the estimate

$$\begin{aligned}&\sup _{I^*}\int _{\Omega _\eta }|\nabla \textbf{u}|^2\,{\textrm{d}}x+ \int _{I^*}\int _{\Omega _\eta }\big (|\nabla ^2\textbf{u}|^2+| \partial _t\textbf{u}|^2+|\nabla \pi |^2\big )\,{\textrm{d}}x\,{\textrm{d}}t\nonumber \\&\qquad +\sup _{I^*}\int _{\omega }\big ( |\partial _t\partial _y\eta |^2+ |\partial _y^3\eta |^2\big )\,{\textrm{d}}y+\int _{I^*}\int _{\omega } \big (|\partial _t\partial _y^2\eta |^2+|\partial _t^2\eta |^2\big ) \,{\textrm{d}}y\,{\textrm{d}} t\\&\quad \lesssim \int _{\Omega _{\eta _0}}|\nabla \textbf{u}_0|^2\,{\textrm{d}}x+\int _\omega \big (|\partial _y^3\eta _0|^2+|\partial _y\eta _1|^2\big )\,{\textrm{d}}y+\int _{I^*}\int _\omega |\partial _y g|^2\,{\textrm{d}}y\,{\textrm{d}} t+1, \nonumber \end{aligned}$$

where the hidden constant depends on \(C_0.\)


We aim at testing the structure equation with \(\partial _t^2\eta \) and seek for an appropriate test-function for the momentum equation. Due to the condition \(\textbf{u}(t,x+\eta \textbf{n})=\partial _t\eta \textbf{n}\) at the boundary we have

$$\begin{aligned} \frac{\textrm{D}}{\textrm{D}t}\textbf{u}&=\partial _t\textbf{u}(t,x+\eta \textbf{n})+\textbf{u}(t,x+\eta \textbf{n})\cdot \nabla \textbf{u}(t,x+\eta \textbf{n})\\&=\partial _t\textbf{u}(t,x+\eta \textbf{n})+\partial _t\eta \textbf{n}\cdot \nabla \textbf{u}(t,x+\eta \textbf{n}) \end{aligned}$$

such that the material derivative of the velocity field is the corresponding test-function for the momentum equation. We use

$$\begin{aligned} {\varvec{\phi }}&=\partial _t\textbf{u}+{\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\cdot \nabla \textbf{u}\end{aligned}$$

and \(\phi =\partial _t^2\eta \) as test-function. Here \({\mathscr {F}}_\eta \) is the extension operator introduced in Sect. 2.5. Note that we have

$$\begin{aligned} \Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{L^2_x}&\lesssim \Vert \partial _t\eta \Vert _{L^2_y},\quad \Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{W^{1,2}_x}\lesssim \Vert \partial _t\eta \Vert _{W^{1,2}_y}, \end{aligned}$$

as a consequence of Lemma 2.2. From the momentum equation in the strong form (2.23) we obtain

$$\begin{aligned}&\int _{I^*}\int _{\Omega _\eta }\big (\partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\big )\cdot \big (\partial _t\textbf{u}+\mathscr {F}_\eta (\partial _t\eta \textbf{n})\cdot \nabla \textbf{u}\big )\,{\textrm{d}}x\,{\textrm{d}}t\\&\quad =\int _{I^*}\int _{\Omega _\eta }{{\,\textrm{div}\,}}{\varvec{\tau }}\cdot \big (\partial _t\textbf{u}+\mathscr {F}_\eta (\partial _t\eta \textbf{n})\cdot \nabla \textbf{u}\big )\,{\textrm{d}}x\,{\textrm{d}}t\\&\qquad +\int _{I^*}\int _{\Omega _\eta }\textbf{f}\cdot \big (\partial _t\textbf{u}+\mathscr {F}_\eta (\partial _t\eta \textbf{n})\cdot \nabla \textbf{u})\big )\,{\textrm{d}}x\,{\textrm{d}}t, \end{aligned}$$

where \({\varvec{\tau }}=\nabla \textbf{u}+\nabla \textbf{u}^\top -\pi {\mathbb {I}}_{2\times 2}\) is the Cauchy stress. On the other hand, from the structure equation (2.24) multiplied by \(\partial _t^2\eta ,\) and the formal computation

$$\begin{aligned} \int _{0}^t\int _{\omega }\partial _y^2\eta \,\partial _t^2\partial _y^2\eta \,{\textrm{d}}y\,{\textrm{d}}\sigma&=-\int _{0}^t\int _{\omega }\partial _t(\partial _y^3\eta \,\partial _t\partial _y\eta )\,{\textrm{d}}y\,{\textrm{d}}\sigma +\int _{0}^t\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}}\sigma \\&\le \tfrac{1}{2}\sup _{I^*}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y+2\sup _{I^*}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y\\ {}&\quad +\int _{I^*}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}}\sigma \end{aligned}$$

we obtain

$$\begin{aligned} \int _{I^*}&\int _\omega |\partial _t^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t+\sup _{I^*}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\\ {}&\lesssim \int _\omega |\partial _y\eta _1|^2\,{\textrm{d}}y+\sup _{I^*}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y\\&+\int _{I^*}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y+\int _{I^*}\int _\omega (g+\textbf{F})\,\partial _t^2\eta \,{\textrm{d}}y\,{\textrm{d}} t\end{aligned}$$

with \(\textbf{F}=-\textbf{n}{\varvec{\tau }}\circ {\varvec{\varphi }}_\eta \partial _y{\varvec{\varphi }}^\perp _\eta .\) Combining both, using Reynold’s transport theorem (applied to \(\int _{\Omega _{\eta (t)}}|\nabla \textbf{u}(t)|^2\,{\textrm{d}}x\)) and Young’s inequality and writing

$$\begin{aligned} {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\cdot \nabla \textbf{u}=\textbf{u}\cdot \nabla \textbf{u}+{\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\cdot \nabla \textbf{u}-\textbf{u}\cdot \nabla \textbf{u}\end{aligned}$$


$$\begin{aligned}&\sup _{I^*}\int _{\Omega _\eta }|\nabla \textbf{u}|^2\,{\textrm{d}}x+\int _{I^*}\int _{\Omega _\eta }|\partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}|^2\,{\textrm{d}}x\,{\textrm{d}}t\nonumber \\&\qquad +\int _{I^*}\int _\omega |\partial _t^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t+\sup _{I^*}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\quad \lesssim \int _{I^*}\int _{\Omega _\eta }|\textbf{u}\cdot \nabla \textbf{u}|^2\,{\textrm{d}}x\,{\textrm{d}}t+\int _{I^*}\int _{\partial \Omega _\eta }(\partial _t\eta \textbf{n})\circ {\varvec{\varphi }}_\eta ^{-1}\cdot \textbf{n}_\eta \circ {\varvec{\varphi }}_\eta ^{-1}|\nabla \textbf{u}|^2{\textrm{d}}{\mathcal {H}}^1\,{\textrm{d}} t\nonumber \\&\qquad -\int _{I^*}\int _{\Omega _\eta }\big (\partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\big )\cdot \big ({\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\cdot \nabla \textbf{u}\big )\,{\textrm{d}}x\nonumber \\&\qquad -\int _{I^*}\int _{\Omega _\eta }\nabla \textbf{u}:\big ({\mathscr {F}}_\eta (\partial _t\eta \textbf{n})^\top \nabla ^2\textbf{u}+\nabla {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\nabla \textbf{u}^\top \big )\,{\textrm{d}}x\,{\textrm{d}}t\nonumber \\&\qquad +\int _{I^*}\int _{\Omega _\eta }\pi \,{{\,\textrm{div}\,}}\big ({\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\nabla \textbf{u}\big )\,{\textrm{d}}x\,{\textrm{d}}t+\int _{I^*}\int _{\Omega _\eta }|\textbf{f}|^2\,{\textrm{d}}x\,{\textrm{d}}t+\int _{\Omega _{\eta _0}}|\nabla \textbf{u}_0|^2\,{\textrm{d}}x\nonumber \\&\qquad +\int _\omega |\partial _y\eta _1|^2\,{\textrm{d}}y+\sup _{I^*}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y+\int _{I^*}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t+\int _{I^*}\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t\nonumber \\&\quad =:{\textrm{I}}+\cdots +{\textrm{XI}}. \end{aligned}$$

In order to control the first term we make use of Theorem 3.2. Due to (5.3) its application can be justified by Remark 3.4. We estimate for \(\kappa >0\) arbitrary by Ladyshenskaya’s inequality (recalling that \(\partial \Omega _\eta \) is Lipschitz uniformly in time by (5.3)) and (5.2)

$$\begin{aligned} \begin{aligned} \textrm{I}&\le \int _{I^*}\Vert \textbf{u}\Vert ^2_{L^4_x}\Vert \nabla \textbf{u}\Vert _{L^4_x}^2\,{\textrm{d}} t\le \,c\int _{I^*}\Vert \textbf{u}\Vert _{L^2_x}\Vert \textbf{u}\Vert ^2_{W^{1,2}_x}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}\,{\textrm{d}} t\\&\le \,c\int _{I^*}\Vert \textbf{u}\Vert ^2_{W^{1,2}_x}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}+\Vert \textbf{f}\Vert _{L^2_x}+\Vert \partial _t\eta \Vert _{W^{3/2,2}_y}\big )\,{\textrm{d}} t\\&\le \,\kappa \int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}^2+\Vert \textbf{f}\Vert ^2_{L^2_x}+\Vert \partial _t\eta \Vert ^2_{W^{3/2,2}_y}\big )\,{\textrm{d}} t\\ {}&\quad +c(\kappa )\bigg (\int _{I^*}\Vert \nabla \textbf{u}\Vert ^4_{L^2_x}\,{\textrm{d}} t+1\bigg ), \end{aligned} \end{aligned}$$

where the first part of the \(\kappa \)-term can be absorbed in the left-hand side of (5.6). Note that we also used the estimate

$$\begin{aligned} \Vert \partial _t\eta \textbf{n}\circ {\varvec{\varphi }}_\eta ^{-1}\Vert _{W^{3/2,2}_y}\lesssim \Vert \partial _t\eta \Vert _{W^{3/2,2}_y}, \end{aligned}$$

which is a consequence of (5.3) and the definition \({{\varvec{\varphi }}}_\eta ={{\varvec{\varphi }}}+ \eta {\textbf{n}}.\) In fact, \({\varvec{\varphi }}_\eta ^{-1}\) is uniformly bounded in time in the space of Sobolev multipliers on \(W^{3/2,2}(\omega )\) by (2.3), (2.4) and (3.7) (together with the assumption \(\partial _y{\varvec{\varphi }}_\eta \ne 0\)) such that the transformation rule (2.5) applies. Similarly to (5.7) we obtain

$$\begin{aligned} \textrm{III}&\le \,\kappa \int _{I^*}\Vert \partial _t\textbf{u}+\nabla \textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}^2\,{\textrm{d}} t+c(\kappa )\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^4_x}^2\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert ^2_{L^4_x}\,{\textrm{d}} t\\&\le \,\kappa \int _{I^*}\Vert \partial _t\textbf{u}+\nabla \textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}^2\,{\textrm{d}} t\\ {}&\quad +c(\kappa )\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^2_x}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{L^2_x}\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{W^{1,2}_x}\,{\textrm{d}} t. \end{aligned}$$

Due to (5.3), (5.5) and Theorem 3.2 (which applies by (5.3), cf. Remark 3.4) the second term can be estimated by

$$\begin{aligned}&\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^2_x}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{W^{1,2}_x}\,{\textrm{d}} t\\&\quad \le \,c(\kappa ')\bigg (\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^2_x}^4\,{\textrm{d}} t+\int _{I^*}\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{W^{1,2}_x}^4\,{\textrm{d}} t\bigg )+\kappa '\int _{I^*}\Vert \nabla \textbf{u}\Vert ^2_{W^{1,2}_x}\,{\textrm{d}} t\\&\quad \le \,c(\kappa ')\bigg (\int _{I^*}\Vert \nabla \textbf{u}\Vert ^4_{L^2_x}\,{\textrm{d}} t+\int _{I^*}\Vert \partial _t\eta \Vert _{W^{1,2}_y}^4\,{\textrm{d}} t\bigg )\\&\qquad +\kappa '\int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}^2+\Vert \textbf{f}\Vert _{L^2_x}^2+\Vert \partial _t\eta \Vert ^2_{W^{3/2,2}_y}\big )\,{\textrm{d}} t, \end{aligned}$$

where \(\kappa '>0\) is arbitrary (recall also (5.8)). Furthermore, we have by the trace theorem (recall that the boundary of \(\Omega _\eta \) is Lipschitz continuous uniformly in time by (5.3)), Sobolev’s embedding and interpolationFootnote 7

$$\begin{aligned} {\textrm{II}}&\le \int _{I^*}\Vert \partial _t\eta \circ {\varvec{\varphi }}_\eta ^{-1}\Vert _{L^\infty (\partial \Omega _\eta )}\Vert \nabla \textbf{u}\Vert _{L^{2}(\partial \Omega _\eta )}^2\,{\textrm{d}}y\\ {}&\le \,c\int _{I^*}\Vert \partial _t\eta \Vert _{L^{2}(\omega )}^{1/2}\Vert \partial _t\eta \Vert _{W^{1,2}(\omega )}^{1/2}\Vert \nabla \textbf{u}\Vert _{W^{3/4,2}(\Omega _\eta )}^2\,{\textrm{d}}y\\&\le \,c\int _{I^*}\Vert \partial _t\eta \Vert _{W^{1,2}_y}^{1/2}\Vert \textbf{u}\Vert _{W^{1,2}_x}^{1/2}\Vert \textbf{u}\Vert _{W^{2,2}_x}^{3/2}\,{\textrm{d}}y\\&\le \,c\int _{I^*}\Vert \partial _t\eta \Vert _{W^{1,2}_y}^{1/2}\Vert \textbf{u}\Vert _{W^{1,2}_x}^{1/2}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}+\Vert \textbf{f}\Vert _{L^2_x}+\Vert \partial _t\eta \Vert _{W^{3/2,2}_y}\big )^{3/2}\,{\textrm{d}}y\\&\le \,\kappa \int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}^2+\Vert \textbf{f}\Vert _{L^2_x}^2+\Vert \partial _t\eta \Vert ^2_{W^{3/2,2}_y}\big )\,{\textrm{d}} t\\&\quad +c(\kappa )\bigg (\int _{I^*}\Vert \nabla \textbf{u}\Vert ^4_{L^2_x}\,{\textrm{d}} t+\int _{I^*}\Vert \partial _t\eta \Vert _{W^{1,2}_y}^4\,{\textrm{d}} t\bigg ). \end{aligned}$$

Note that we used again Theorem 3.2 and (5.8). Similarly, it holds

$$\begin{aligned} {\textrm{IV}}&\le \,\kappa \int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^2_x}^2+\Vert \textbf{f}\Vert _{L^2_x}^2+\Vert \partial _t\eta \Vert ^2_{W^{3/2,2}_y}\big )\,{\textrm{d}} t\\&\quad +c(\kappa )\bigg (\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^2_x}^{4}\,{\textrm{d}} t+\int _{I^*}\Vert \partial _t\eta \Vert _{W^{1,2}_y}^{4}\,{\textrm{d}} t+1\bigg ). \end{aligned}$$

For \(\textrm{V}\) we write

$$\begin{aligned} {\textrm{V}}&=-\int _{I^*}\int _{\Omega _\eta }\nabla \pi \cdot {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\nabla \textbf{u}\,{\textrm{d}}x\,{\textrm{d}}t\\&\quad +\int _{I^*}\int _{\partial \Omega _\eta }\pi \,{\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\nabla \textbf{u}\,\textbf{n}_\eta \circ {\varvec{\varphi }}_\eta ^{-1}\,{\textrm{d}}{\mathcal {H}}^1\,{\textrm{d}} t, \end{aligned}$$


$$\begin{aligned}&\int _{I^*}\int _{\Omega _\eta }\nabla \pi \cdot {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\nabla \textbf{u}\,{\textrm{d}}x\,{\textrm{d}}t\\&\quad \le \int _{I^*}\Vert \nabla \pi \Vert _{L^2_x}\,\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{L^4_x}\Vert \nabla \textbf{u}\Vert _{L^4_x}\,{\textrm{d}} t\\&\quad \le \,c \int _{I^*}\Vert \nabla \pi \Vert _{L^2_x}\,\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{W^{1/2,2}_x}\Vert \nabla \textbf{u}\Vert _{L^2_x}^{\frac{1}{2}}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}^{\frac{1}{2}}\,{\textrm{d}} t\\&\quad \le \,c \int _{I^*}\Vert \nabla \pi \Vert _{L^2_x}\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert ^{1/2}_{L^{2}_x}\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{W^{1,2}_x}^{1/2}\Vert \nabla \textbf{u}\Vert _{L^2_x}^{\frac{1}{2}}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}^{\frac{1}{2}}\,{\textrm{d}} t\\&\quad \le \kappa \int _{I^*}\Vert \nabla \pi \Vert _{L^2_x}^{2}\,{\textrm{d}} t+\kappa \int _{I^*}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t\\&\qquad +c(\kappa )\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^2_x}^{4}\,{\textrm{d}} t+c(\kappa )\int _{I^*}\Vert \partial _t\eta \Vert _{W^{1,2}_y}^{4}\,{\textrm{d}} t\end{aligned}$$

by (5.5) and (5.3). Note that we used again interpolation, the trace theorem as well as Sobolev’s embedding \(W^{1/2,2}(\Omega _\eta )\hookrightarrow L^4(\Omega _\eta )\) (recall again that the boundary \(\partial \Omega _\eta \) is uniformly Lipschitz by (5.3)). As far as the second term in \(\textrm{IV}\) is concerned we estimate

$$\begin{aligned}&\int _{I^*}\int _{\partial \Omega _\eta }\pi \,{\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\nabla \textbf{u}\,\textbf{n}_\eta \circ {\varvec{\varphi }}_\eta ^{-1}\,{\textrm{d}}{\mathcal {H}}^1\,{\textrm{d}} t\\ {}&\quad \le \int _{I^*}\Vert \pi \Vert _{L^4(\partial \Omega _\eta )}\Vert {\mathscr {F}}_\eta (\partial _t\eta \textbf{n})\Vert _{L^2(\partial \Omega _\eta )}\Vert \nabla \textbf{u}\Vert _{L^4(\partial \Omega _\eta )}\,{\textrm{d}} t\\&\quad \le \,c\int _{I^*}\Vert \pi \Vert _{W^{1/4,2}(\partial \Omega _\eta )}\Vert \nabla \textbf{u}\Vert _{W^{1/4,2}(\partial \Omega _\eta )}\,{\textrm{d}} t\\&\quad \le \,c\int _{I^*}\Vert \pi \Vert _{W^{3/4,2}(\Omega _\eta )}\Vert \nabla \textbf{u}\Vert _{W^{3/4,2}(\Omega _\eta )}\,{\textrm{d}} t\\&\quad \le \,c\int _{I^*}\Vert \pi \Vert _{W^{1,2}(\Omega _\eta )}\Vert \nabla \textbf{u}\Vert _{W^{1,2}(\Omega _\eta )}^{3/4}\Vert \nabla \textbf{u}\Vert _{L^{2}(\Omega _\eta )}^{1/4}\,{\textrm{d}} t\\&\quad \le \kappa \int _{I^*}\Vert \pi \Vert ^2_{W^{1,2}_x}\,{\textrm{d}} t+\kappa \int _{I^*}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t+c(\kappa )\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t\end{aligned}$$

using Sobolev’s embedding \(W^{1/4,2}(\partial \Omega _\eta )\hookrightarrow L^4(\partial \Omega _\eta ),\) (5.5) and (5.3) and the trace embedding \(W^{3/4,2}(\Omega _\eta )\hookrightarrow W^{1/4,2}(\partial \Omega _\eta ).\) Different to first term in \(\textrm{V}\) above we must estimate here also the \(L^2\)-norm of the pressure, for which we use (2.22) (noticing that \(\int _{\omega }\textbf{n}\cdot \textbf{n}_\eta |\partial _y{\varvec{\varphi }}_\eta |\,{\textrm{d}}y\) is strictly positive by our assumption of non-degeneracy). We have

$$\begin{aligned} \int _{I^*}\Vert \pi \Vert ^2_{W^{1,2}_x}\,{\textrm{d}} t&\lesssim \int _{I^*}\Vert \nabla \pi \Vert ^2_{L^{2}_x}\,{\textrm{d}} t+\int _{I^*}c_\pi ^2\,{\textrm{d}} t\\&\lesssim \int _{I^*}\Vert \nabla \pi \Vert ^2_{L^{2}_x}\,{\textrm{d}} t+\int _{I^*}\int _\omega |\partial _t^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t+\int _{I^*}\int _\omega |g|^2\,{\textrm{d}}y\,{\textrm{d}} t\\ {}&\quad +\int _{I^*}\Vert \pi _0\Vert _{L^{2}(\partial \Omega _\eta )}^2\,{\textrm{d}} t+\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^{2}(\partial \Omega _\eta )}^2\,{\textrm{d}} t, \end{aligned}$$

where the last term can be estimated by

$$\begin{aligned} \int _{I^*}\Vert \nabla \textbf{u}\Vert _{W^{1/2,2}(\Omega _\eta )}^2\,{\textrm{d}} t&\lesssim \int _{I^*}\Vert \nabla \textbf{u}\Vert _{W^{1,2}(\Omega _\eta )}\Vert \nabla \textbf{u}\Vert _{L^{2}(\Omega _\eta )}\,{\textrm{d}} t\\&\lesssim \int _{I^*}\Vert \nabla \textbf{u}\Vert _{W^{1,2}_x}^{2}\,{\textrm{d}} t+\int _{I^*}\Vert \nabla \textbf{u}\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t. \end{aligned}$$

Similarly, we infer from Poincaré’s inequality

$$\begin{aligned} \int _{I^*}\Vert \pi _0\Vert _{L^{2}(\partial \Omega _\eta )}^2\,{\textrm{d}} t&\lesssim \int _{I^*}\Vert \nabla \pi _0\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t+\int _{I^*}\Vert \pi _0\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t\\ {}&\lesssim \int _{I^*}\Vert \nabla \pi _0\Vert _{L^{2}_x}^{2}\,{\textrm{d}} t=\int _{I^*}\Vert \nabla \pi \Vert _{L^{2}_x}^{2}\,{\textrm{d}} t\end{aligned}$$

using that \((\pi _0)_{\Omega _{\eta }}=0\) by definition. As for the estimates for \(\textrm{I}-\textrm{III}\) the \(\kappa \) terms in the above can now be controlled by means of Theorem 3.2. Combining everything, choosing \(\kappa \) small enough and using (5.3) once more we conclude that

$$\begin{aligned}&\sup _{I^*}\int _{\Omega _\eta }|\nabla \textbf{u}|^2\,{\textrm{d}}x+\int _{I^*}\int _{\Omega _\eta }|\partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}|^2 +\int _{I^*}\int _\omega |\partial _t^2\eta |^2\,{\textrm{d}}x\,{\textrm{d}} t\nonumber \\&\quad \lesssim \int _{I^*}\Vert \nabla \textbf{u}\Vert ^4_{L^2_x}\,{\textrm{d}} t+\int _{I^*}\Vert \partial _t\partial _y\eta \Vert _{L^{2}_y}^4\,{\textrm{d}} t+\int _{I^*}\Vert \textbf{f}\Vert _{L^2_x}^2\,{\textrm{d}} t+\Vert \nabla \textbf{u}_0\Vert _{L^2_x}^2\nonumber \\&\qquad +\sup _{I^*}\Vert \partial _y^3\eta \Vert _{L^2_y}^2\,{\textrm{d}} t+\int _{I^*}\Vert \partial _t\partial _y^2\eta \Vert _{L^2_y}^2\,{\textrm{d}} t+\Vert \partial _y\eta _1\Vert ^2_{L^2_y}+1. \end{aligned}$$

Testing the structure equation by \(\partial _t\partial _y^2\eta \) yields

$$\begin{aligned}&\frac{1}{2}\sup _{I^*}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t+\int _{I^*}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y+\frac{1}{2}\sup _{I^*}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y\\&\quad =\frac{1}{2}\int _\omega |\partial _y\eta _1|^2\,{\textrm{d}}y+\frac{1}{2}\int _\omega |\partial _y^3\eta _0|^2\,{\textrm{d}}y+\int _{I^*}\int _\omega (g+\textbf{F})\,\partial _t\partial _y^2\eta \,{\textrm{d}}y\,{\textrm{d}} t, \end{aligned}$$

where (arguing as in (5.8) to control \(\textbf{F}\) by \({\varvec{\tau }}\) and arguing as for \(\textrm{I}-\textrm{III}\) and \(\textrm{V}\) to estimate \({\varvec{\tau }}\))

$$\begin{aligned}&\int _{I^*}\int _\omega \textbf{F}\cdot \partial _t\partial _y^2\eta \,{\textrm{d}}y\,{\textrm{d}} t\le \int _{I^*}\Vert \textbf{F}\Vert _{W_y^{1/2,2}(\omega )}\Vert \partial _t\partial _y^2\eta \Vert _{W^{-1/2,2}(\omega )}\,{\textrm{d}} t\\ {}&\quad \le \,c\int _{I^*}\Vert {\varvec{\tau }}\Vert _{W^{1/2,2}(\partial \Omega _\eta )}\Vert \partial _t\eta \Vert _{W^{3/2,2}(\omega )}\,{\textrm{d}} t\\&\quad \le \,c\int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^{2}_x}+\Vert \textbf{f}\Vert _{L^2_x}+\Vert \partial _t\eta \Vert _{W^{3/2,2}_y}+\Vert \partial _t^2\eta \Vert _{L^2_y}+\Vert g\Vert _{L^2_y}\big )\Vert \partial _t\eta \Vert _{W^{3/2,2}_y}\,{\textrm{d}} t\\&\quad \le \kappa \int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^{2}_x}^2+\Vert \textbf{f}\Vert _{L^2_x}^2+\Vert \partial _t^2\eta \Vert _{L^2_y}^2+\Vert g\Vert _{L^2_y}^2\big )\,{\textrm{d}} t+c(\kappa )\int _{I^*}\Vert \partial _t\eta \Vert _{W^{3/2,2}_y}^2\,{\textrm{d}} t\\&\quad \le \kappa \int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^{2}_x}^2+\Vert \partial _t^2\eta \Vert _{L^2_y}^2+\Vert \partial _t\eta \Vert _{W^{2,2}_y}^2+\Vert \textbf{f}\Vert _{L^2_x}^2\big )\,{\textrm{d}} t\\ {}&\qquad +c(\kappa )\int _{I^*}\Vert \partial _t\eta \Vert _{W^{1,2}_y}^2\,{\textrm{d}} t+c(\kappa )\int _{I^*}\Vert g\Vert _{L^{2}_y}^2\,{\textrm{d}} t\end{aligned}$$

using interpolation in the last step. Hence we have

$$\begin{aligned}&\sup _{I^*}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}x\,{\textrm{d}} t+\int _{I^*}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y+\sup _{I^*}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y\nonumber \\&\quad \le \kappa \int _{I^*}\big (\Vert \partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}\Vert _{L^{2}}^2+\Vert \partial _t\partial _y^2\eta \Vert _{L^{2}_y}^2+\Vert \partial _t^2\eta \Vert _{L^2_y}^2\big )\,{\textrm{d}} t+ c(\kappa ){\tilde{C}}_0 \end{aligned}$$

by (5.3). Here we denoted

$$\begin{aligned} {\tilde{C}}_0=C_0+\int _\omega |\partial _y\eta _1|^2\,{\textrm{d}}y+\int _\omega |\partial _y^3\eta _0|^2\,{\textrm{d}}y. \end{aligned}$$

Combining (5.9) and (5.10) implies

$$\begin{aligned}&\sup _{I^*}\int _{\Omega _\eta }|\nabla \textbf{u}|^2\,{\textrm{d}}x+\int _{I^*}\int _{\Omega _\eta }|\partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}|^2 +\int _{I^*}\int _\omega |\partial _t^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\\&\qquad +\sup _{I^*}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}x\,{\textrm{d}} t+\int _{I^*}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}x+\sup _{I^*}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y\\&\quad \le \,c\bigg (\int _{I^*}\big (\Vert \nabla \textbf{u}\Vert ^2_{L^2_x}+\Vert \partial _t\partial _y\eta \Vert _{L^2_y}^2\big )^2\,{\textrm{d}} t+1\bigg ). \end{aligned}$$

By Gronwall’s lemma, using that \(\int _{I^*}\big (\Vert \nabla \textbf{u}\Vert ^2_{L^2_x}+\Vert \partial _t\partial _y\eta \Vert _{L^2_y}^2\big )\,{\textrm{d}} t\le \,c\) by (5.2) and (5.3) we obtain

$$\begin{aligned}&\sup _{I^*}\int _{\Omega _\eta }|\nabla \textbf{u}|^2\,{\textrm{d}}x+\int _{I^*}\int _{\Omega _\eta }|\partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}|^2\,{\textrm{d}}x+\int _{I^*}\int _\omega |\partial _t^2\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t\le \,c,\nonumber \\&\sup _{I^*}\int _\omega |\partial _t\partial _y\eta |^2\,{\textrm{d}}y\,{\textrm{d}} t+\int _{I^*}\int _\omega |\partial _t\partial _y^2\eta |^2\,{\textrm{d}}y+\sup _{I^*}\int _\omega |\partial _y^3\eta |^2\,{\textrm{d}}y\le c. \end{aligned}$$

We can now use the momentum equation and Theorem 3.2 again to obtain (recall (5.8))

$$\begin{aligned}&\int _{I^*}\int _{\Omega _\eta }|\nabla ^2\textbf{u}|^2\,{\textrm{d}}x\,{\textrm{d}} t+\int _{I^*}\int _{\Omega _\eta }|\nabla \pi |^2\,{\textrm{d}}x\,{\textrm{d}} t\nonumber \\&\quad \le \,c\int _{I^*}\int _{\Omega _\eta }|\partial _t\textbf{u}+\textbf{u}\cdot \nabla \textbf{u}|^2\,{\textrm{d}}x\,{\textrm{d}}t+\int _{I^*}\Vert \textbf{f}\Vert _{L^2_x}^2\,{\textrm{d}} t\nonumber \\&\quad +\int _{I^*}\Vert \partial _t\eta \Vert _{W^{3/2,2}_y}^2\,{\textrm{d}} t\le \,c, \end{aligned}$$

which completes the proof. \(\square \)

Proof of Theorem 2.5

By Propositions 5.1 and 4.1 we can obtain a strong solution in the interval \((T^*,2T^*)\) with initial data \(\textbf{u}(T^*),\eta (T^*),\partial _t\eta (T^*)\) with a corresponding regularity estimate. This procedure can now be repeated until the moving boundary approaches a self-intersection or degenerates (that is \(\partial _y{\varvec{\varphi }}_\eta (T,y)=0\) for some \(y\in \omega \)). In the latter case Theorem 3.2 is not applicable anymore, cf. Remark 3.4. \(\square \)