Well-Posedness of a Navier-Stokes/Mean Curvature Flow system

TL;DR

This paper proves the well-posedness of a coupled Navier-Stokes and mean curvature flow system modeling two-phase incompressible fluids with a sharp interface, establishing existence of strong solutions under certain conditions.

Contribution

It demonstrates the existence of strong solutions for a coupled Navier-Stokes and mean curvature flow system, linking sharp interface models to diffuse interface limits.

Findings

Existence of strong solutions for small times

Coupling of Navier-Stokes with mean curvature flow

Connection to diffuse interface models

Abstract

We consider a two-phase flow of two incompressible, viscous and immiscible fluids which are separated by a sharp interface in the case of a simple phase transition. In this model the interface is no longer material and its evolution is governed by a convective mean curvature flow equation, which is coupled to a two-phase Navier-Stokes equation with Young-Laplace law. The problem arises as a sharp interface limit of a diffuse interface model, which consists of a Navier-Stokes system coupled with an Allen-Cahn equation. We prove existence of strong solutions for sufficiently small times and regular initial data.

Equations

$$\text{textlbrackdbl}f\text{textrbrackdbl}(x):={\text{lim}}_{r\to 0}f(x+r{\nu }_{\Gamma (t)}(x))-f(x-r{\nu }_{\Gamma (t)}(x))$$

$${\partial }_{t}v+v\cdot \nabla v-{\mu }^{\pm }\Delta v+\nabla p$$

$$\text{div}v$$

$$-\text{textlbrackdbl}T(v,p)\text{textrbrackdbl}{\nu }_{\Gamma (t)}$$

$$\text{textlbrackdbl}v\text{textrbrackdbl}$$

$$v{|}_{\partial \Omega }$$

$$v{|}_{t=0}$$

$$V_{\Gamma (t)}-{\nu }_{\Gamma (t)}\cdot v{|}_{\Gamma (t)}$$

$$\Gamma (0)$$

$$\frac{d}{dt}|{\Omega }^{+}(t)|={\int }_{\Gamma (t)}{V}_{\Gamma (t)}d\sigma ={\int }_{\Gamma (t)}v\cdot {\nu }_{\Gamma (t)}d\sigma ={\int }_{{\Omega }^{+}(t)}\text{div}vdx=0.$$

$$V_{\Gamma (t)}=H_{\Gamma (t)}\text{ for }t>0.$$

$$W_{p,0}^k(\Omega ):={\overline{C_0^{\infty }(\Omega )}}^{W_p^k(\Omega )}\text{ and }{W}_p^{-k}(\Omega ):=W_{p^{\prime },0}^k(\Omega {)}^{\prime },$$

$$W_{p,(0)}^k(\Omega ):=W_p^k(\Omega )\cap L_{(0)}^p(\Omega )\text{ and }{W}_{p,(0)}^{-k}(\Omega ):=W_{p^{\prime },(0)}^k(\Omega {)}^{\prime }.$$

$$X:\Sigma \times (-a,a)\to {\mathbb{R}}^n:(s,r)\mapsto s+r{\nu }_{\Sigma }(s)$$

$${\Theta }_h(x):=x+\chi (\frac{d_{\Sigma }(x)}{a})h(\Pi (x)){\nu }_{\Sigma }(\Pi (x))\text{ for all }x\in {\mathbb{R}}^n.$$

$$\det D{\Theta }_h\ge c>0\text{ and }\Vert D{\Theta }_h{\Vert }_{\infty }+\Vert D{\Theta }_h^{-1}{\Vert }_{\infty }\le C(1+\Vert h{\Vert }_{C^1(\Sigma )})$$

$${\varphi }_h:{\Sigma }_a\to \mathbb{R}:x\mapsto d_{\Sigma }(x)-h(\Pi (x)),$$

$$D{\Theta }_h^{-\top }{|}_x{\nu }_{\Sigma }{|}_{\Pi (x)}\cdot {\nu }_{\Sigma }{|}_{\Pi (x)}=1\text{ for all }x\in \overline{{\Sigma }_{a/3}}.$$

$$X_0:=W_p^{1-\frac{1}{p}}(\Sigma )\text{ and }{X}_1:=W_p^{3-\frac{1}{p}}(\Sigma ).$$

$${\Theta }_h(x,t):={\Theta }_{h(.,t)}(x)\text{ for }(x,t)\in {\mathbb{R}}^n\times [0,T]$$

$$V_T:=\{h\in {\mathbb{E}}_1(T)\cap C^0([0,T];U):\Vert h{\Vert }_{{\mathbb{E}}_1(T)}+\Vert h(0){\Vert }_{X_{\gamma }}<R_0\}.$$

$$E:X_0=W_p^{1-\frac{1}{p}}(\Sigma )\to W_p^1({\Sigma }_{\overline{a}})\text{ with }E\in \mathcal{L}(W_p^{k-\frac{1}{p}}(\Sigma ),W_p^k({\Sigma }_{\overline{a}})),k=1,2,3$$

$${\tilde{X}}_{\gamma }=W_p^{3-\frac{2}{p}}({\Sigma }_{\overline{a}})\hookrightarrow C^2(\overline{{\Sigma }_{\overline{a}}}).$$

$${\tilde{\Theta }}_h(x):=x+\chi (\frac{d_{\Sigma }(x)}{a})Eh(x){\nu }_{\Sigma }(\Pi (x))\text{ for all }x\in {\mathbb{R}}^n$$

$$\Vert h\circ \Pi -Eh{\Vert }_{C^1(\overline{{\Sigma }_{\overline{a}}})}\le \epsilon \Vert h{\Vert }_{X_{\gamma }}.$$

$$\det D{\tilde{\Theta }}_h\ge \tilde{c}>0\text{ and }D{\tilde{\Theta }}_h^{-\top }{|}_x{\nu }_{\Sigma }{|}_{\Pi (x)}\cdot {\nu }_{\Sigma }{|}_{\Pi (x)}\ge \frac{1}{2}\text{ for all }x\in \overline{{\Sigma }_{a/3}}.$$

$$h(t)\in U_0:=\{h_0\in U:\Vert h_0{\Vert }_{X_{\gamma }}<\overline{R_0}\}$$

$$\Vert D{\Theta }_h-D{\tilde{\Theta }}_h{\Vert }_{\infty }\le \Vert \chi (\frac{d_{\Sigma }(.)}{a}){\nu }_{\Sigma }(\Pi (.))(h\circ \Pi -Eh){\Vert }_{C^1(\overline{{\Sigma }_{\overline{a}}})}\le C\epsilon R.$$

$$|D{\tilde{\Theta }}_h^{-\top }(x)-D{\Theta }_h^{-\top }(x)|\le \tilde{C}(R)\epsilon$$

$$|D{\tilde{\Theta }}_h^{-\top }{|}_x{\nu }_{\Sigma }{|}_{\Pi (x)}\cdot {\nu }_{\Sigma }{|}_{\Pi (x)}-1|=|(D{\tilde{\Theta }}_h^{-\top }-D{\Theta }_h^{-\top }){|}_x{\nu }_{\Sigma }{|}_{\Pi (x)}\cdot {\nu }_{\Sigma }{|}_{\Pi (x)}|\le \tilde{C}(R)\epsilon \le \frac{1}{2}$$

$$A(h):=D{\tilde{\Theta }}_h^{-\top },{\nabla }_h:=A(h)\nabla ,{\nabla }_{h}u:=({\nabla }_h{u}_k^{\top }{)}_{k=1}^n\text{ and }{\text{div}}_{h}u:=\text{Tr}{\nabla }_{h}u.$$

$$K(h):\Sigma \times [0,T]\to \mathbb{R}:(x,t)\mapsto H_{{\Gamma }_h(t)}{|}_{{\Theta }_h(x,t)}.$$

$${\nu }_h{|}_{(x,t)}:={\nu }_{{\Gamma }_h(t)}{|}_{{\tilde{\Theta }}_h(x,t)}=\frac{A(h){|}_{(x,t)}{\nu }_{\Sigma }{|}_x}{|A(h){|}_{(x,t)}{\nu }_{\Sigma }{|}_x|}\text{ for all }(x,t)\in \Sigma \times (0,T).$$

$${\partial }_{t}u+u\cdot {\nabla }_{h}u-{\mu }^{\pm }{\text{div}}_h{\nabla }_{h}u+{\nabla }_h\tilde{q}$$

Full text

\setkomafont

disposition \setkomafontsection

ection]section

**Well-Posedness of a Navier-Stokes/Mean Curvature Flow system

** Helmut Abels and Maximilian Moser

*Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 31, D-93053 Regensburg, Germany

E-mail: Helmut.Abels@mathematik.uni-regensburg.de, Maximilian1.Moser@mathematik.uni-regensburg.de

March 10, 2024

Abstract. We consider a two-phase flow of two incompressible, viscous and immiscible fluids which are separated by a sharp interface in the case of a simple phase transition. In this model the interface is no longer material and its evolution is governed by a convective mean curvature flow equation, which is coupled to a two-phase Navier-Stokes equation with Young-Laplace law. The problem arises as a sharp interface limit of a diffuse interface model, which consists of a Navier-Stokes system coupled with an Allen-Cahn equation. We prove existence of strong solutions for sufficiently small times and regular initial data.

**Mathematics Subject Classification (2000):

Primary: 35R35; Secondary 35Q30, 76D27, 76D45, 76T99.

**

**Key words: Two-phase flow, Navier-Stokes system, free boundary problems, mean curvature flow.

**

1 Introduction

We consider the flow of two immiscible, incompressible Newtonian fluids with phase transition in a bounded, smooth domain $\Omega\subseteq\mathbb{R}^{n},n=2,3$. At time $t\in(0,T)$ the fluids fill domains $\Omega^{\pm}(t)$ separated by the interface $\Gamma(t):=\partial\Omega^{+}(t)$. For simplicity we set the densities to one. Moreover, we assume that the fluids have constant viscosities $\mu^{\pm}>0$. Then the stress tensor has the form $T(v,p)=2\mu^{\pm}\textup{sym}(\nabla v)-pI$, where $\textup{sym}(\nabla v):=\frac{1}{2}(\nabla v+\nabla v^{\top})$, $v:\Omega\times(0,T)\rightarrow\mathbb{R}^{n}$ is the velocity and $p:\Omega\times(0,T)\rightarrow\mathbb{R}$ is the pressure in Eulerian coordinates.

To formulate the model, we need some notation. We denote by $\nu_{\Gamma(t)}$ the outer unit normal to $\Gamma(t)=\partial\Omega^{+}(t)$ and by $V_{\Gamma(t)},H_{\Gamma(t)}$ the normal velocity and mean curvature (for convenience the sum of the principal curvatures) with respect to $\nu_{\Gamma(t)}$. Furthermore the jump of a quantity $f$ defined on $\Omega^{\pm}(t)$ across $\Gamma(t)$ (with respect to $\nu_{\Gamma(t)}$) is defined as

[TABLE]

for all $x\in\Gamma(t)$.

As model we consider the following Navier-Stokes/mean curvature flow system

[TABLE]

where $\sigma>0$ is a surface tension constant and $\Gamma_{0}$ is the initial interface. Here the velocity $v$, the pressure $p$ and the family of interfaces $\{\Gamma(t)\}_{t\in(0,T)}$ are to be determined. Equations (1.1)-(1.2) describe the (local) mass and momentum conservation of the fluids and (1.3) is the balance of forces at the interface, see Landau and Lifshitz [15], §61. Moreover we require continuity of $v$ on $\Gamma(t)$ in (1.4) and the no-slip condition for $v$ on $\partial\Omega$ in (1.5). (1.6) and (1.8) are the initial conditions for $v$ and the family of hypersurfaces $\{\Gamma(t)\}_{t\in(0,T)}$, respectively. Equation (1.7) without the curvature term $H_{\Gamma(t)}$ means that the interface is just transported by the fluids. Then (1.1)-(1.8) reduces to the classical two-phase Navier-Stokes system with surface tension, which was studied e.g. by Köhne, Prüss and Wilke [14]. For this system the total mass of each fluid is preserved since the densities are constant and the enclosed volume of $\Gamma(t)$ stays the same. Formally this follows from (references in) Escher and Simonett [7], p. 623, the divergence theorem and (1.2):

[TABLE]

The additional curvature term $H_{\Gamma(t)}$ on the right hand side in (1.7) allows for a change of total masses and couples the Navier-Stokes system to the mean curvature flow

[TABLE]

Therefore we call (1.1)-(1.8) the Navier-Stokes/mean curvature flow system. The system is obtained as a sharp interface limit of a Navier-Stokes-Allen-Cahn system. Another motivation is the regularizing effect of the curvature term in (1.7). But we can also view it as a simple model for a two-phase flow with phase transition. In the case that the Navier-Stokes equations are modified for shear thickening non-Newtonian fluid of power-law type Liu, Sato, and Tonegawa in [16] have shown existence of weak solutions. In the present contribution we prove local existence of strong solutions to (1.1)-(1.8) in an $L^{p}$-setting. The result is part of the second author’s Master thesis, which was supervised by the first author.

We apply a similar strategy as A. and Wilke [2], who study local well-posedness and qualitative behaviour of solutions for a two-phase Navier-Stokes-Mullins-Sekerka system. The equations are transformed with a variant $\tilde{\Theta}_{h}$ of the Hanzawa transform to a reference surface $\Sigma$ and disjoint domains $\Omega^{\pm}$, where $\Omega=\Omega^{+}\cup\Omega^{-}\cup\Sigma$ is similar as above. Here $h:\Sigma\times(0,T)\rightarrow\mathbb{R}$ is the “height function” of $\{\Gamma(t)\}_{t\in(0,T)}$ with respect to $\Sigma$. The coupling with the Navier-Stokes part in (1.7) is of lower order. Therefore we solve the Navier-Stokes-part (1.1)-(1.6) for a given appropriate family of hypersurfaces $\{\Gamma(t)\}_{t\in(0,T)}$ or equivalently for a given suitable time-dependent height function $h$. Then we insert the obtained velocity field $u(h)$ into equation (1.7) and (1.7)-(1.8) turns into an abstract evolution equation for $h$. The latter is solved by using the theory of maximal $L^{p}$-regularity.

The paper is organized as follows. In Section 2 we describe the transformation to a fixed reference hypersurface (and related domains) and analyse the occurring terms, especially the transformed mean curvature. In Section 3 we consider the Navier-Stokes part for a given family of hypersurfaces $\{\Gamma_{h}(t)\}_{t\in(0,T)}$ and in Section 4 we show the local well-posedness for the transformed system. Definitions and some properties of the used Banach-space valued function spaces are summarized in Appendix A. In Appendix B we prove a maximal regularity result for a two-phase Stokes system which is needed for Section 3.

We always assume $\Omega\subseteq\mathbb{R}^{n},n=2,3$, to be a bounded, connected and smooth domain and throughout let $\Sigma\subseteq\Omega$ be a compact, connected and smooth hypersurface, that separates $\Omega$ in two disjoint, connected domains $\Omega^{\pm}$ with $\Sigma=\partial\Omega^{+}$ and outer unit normal $\nu_{\Sigma}$.

Acknowledgements: The second author is grateful to Mathias Wilke for many helpful discussions and his lecture series on “maximal regularity”.

2 Preliminaries

2.1 Notation and Scalar-Valued Function Spaces

Let $\mathbb{R}_{+}:=[0,\infty)$ and $\mathbb{R}^{n}_{+}:=\mathbb{R}^{n-1}\times(0,\infty)$ for $n\in\mathbb{N},n\geq 2$. The Euclidean norm in $\mathbb{R}^{n}$ is denoted by $|.|$. For matrices $A,B\in\mathbb{R}^{n\times n}$ let $A:B$ be the matrix-product and $|A|:=\sqrt{A:A}$ be the induced norm. The set of invertible matrices in $\mathbb{R}^{n\times n}$ is $GL(n,\mathbb{R})$. For $A\in GL(n,\mathbb{R})$ we set $A^{-\top}:=(A^{-1})^{\top}$.

For metric spaces $X$ denote by $B_{X}(x,r)$ the ball with radius $r>0$ around $x\in X$. For normed spaces $X,Y$ over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ the set of bounded, linear operators $T:X\rightarrow Y$ is $\mathcal{L}(X,Y)$ and $\mathcal{L}(X):=\mathcal{L}(X,X)$. If $X,Y$ are additionally complete and $U\subseteq X$ is open, for $k\in\mathbb{N}$ the set of all $k$-times continuously differentiable functions $f:U\rightarrow Y$ is denoted by $C^{k}(U;Y)$. The set of all bounded $f\in C^{k}(U;Y)$ with bounded derivatives is $BC^{k}(U;Y)$ with usual norm.

If $(X_{0},X_{1})$ is admissible and $\theta\in(0,1),1\leq p\leq\infty$, we denote by $(X_{0},X_{1})_{\theta,p}$ the real interpolation spaces and with $\|.\|_{(X_{0},X_{1})_{\theta,p}}\equiv\|.\|_{\theta,p}$ the norm, see Lunardi [17], Bergh and Löfström [5] and Triebel [22].

Let $\Omega\subseteq\mathbb{R}^{n},n\in\mathbb{N}$, be open and nonempty. Spaces of continuous and continuously differentiable functions $f:\overline{\Omega}\rightarrow\mathbb{R}$ are defined as usual. Now let $1\leq p\leq\infty$ and $k\in\mathbb{N}$. Then the Lebesgue and Sobolev spaces are denoted by $L^{p}(\Omega)=:W_{p}^{0}(\Omega)$ and $W^{k}_{p}(\Omega)$, respectively. Moreover, we set

[TABLE]

where $p^{\prime}$ is the conjugate exponent to $p$, as well as $L^{p}_{(0)}(\Omega):=\{f\in L^{p}(\Omega):\int_{\Omega}f\,dx=0\}$,

[TABLE]

The closure of the divergence-free $\phi\in C_{0}^{\infty}(\Omega)^{n}$ in $L^{p}(\Omega)^{n}$ is denoted by $L^{p}_{\sigma}(\Omega)$. Besides, we need the Besov spaces $B^{s}_{p,q}(\mathbb{R}^{n})$ for $0<s<\infty$ and $1\leq p,q\leq\infty$. We set $W^{s}_{p}(\mathbb{R}^{n}):=B^{s}_{p,p}(\mathbb{R}^{n})$ for all $s\in(0,\infty)\textbackslash\mathbb{N},1\leq p\leq\infty$. Respective spaces can also be defined on bounded, smooth domains $\Omega\subseteq\mathbb{R}^{n}$ and, if $n\geq 2$, on their boundaries $\partial\Omega$. For embeddings, interpolation results and trace theorems, compare e.g. Triebel [22], [23].

2.2 Hanzawa Transformation $\Theta_{h}$

For the construction of $\Theta_{h}$ we use that there is an $a>0$, such that

[TABLE]

is a $C^{\infty}$-diffeomorphism onto $\Sigma_{a}:=B_{a}(\Sigma)$. We set $(\Pi,d_{\Sigma}):=X^{-1}$. Then it holds $\nabla d_{\Sigma}=\nu_{\Sigma}\circ\Pi$. Here $\Pi$ is called projection onto $\Sigma$ and $d_{\Sigma}$ signed distance function. For the proof cf. Hildebrandt [11], Chapter 4.6, Theorems 1-3.

Let $a<\textup{dist}(\Sigma,\partial\Omega)$ and $a_{0}<\frac{a}{4}$ fixed, $\chi\in C^{\infty}(\mathbb{R})$ be a cutoff function such that $|\chi^{\prime}|\leq 4$ and $\chi(s)=1$ for $|s|\leq\frac{1}{3}$, as well as $\chi(s)=0$ for $|s|\geq\frac{2}{3}$. For $h\in C^{2}(\Sigma)$ with $\|h\|_{\infty}<a_{0}$ we define the Hanzawa transformation $\Theta_{h}$ by

[TABLE]

Essential properties of $\Theta_{h}$ are listed in

Lemma 2.1

The Hanzawa transformation $\Theta_{h}:\Omega\rightarrow\Omega$ is a $C^{2}$-diffeomorphism and $\Theta_{h}$ is the identity on $\mathbb{R}^{n}\textbackslash\Sigma_{2a/3}$, in particular nearby $\partial\Omega$. Moreover,

[TABLE]

with $c,C>0$ independent of $h$.

Proof. The first part can be shown as in Kneisel [12], Lemma 2.2.1 and one also obtains a local representation for $D\Theta_{h}$. The latter yields the estimates by compactness of $\Sigma$. $\square$

Remark 2.2

Let $h\in C^{2}(\Sigma)$ such that $\|h\|_{\infty}<a_{0}$. Then $\Gamma_{h}:=\Theta_{h}(\Sigma)$ is a connected and compact $C^{2}$-hypersurface as zero level set of

[TABLE]

where $|\nabla\phi_{h}|\geq 1$ as one can show $\nabla\phi_{h}|_{x}\cdot\nu_{\Sigma}|_{\Pi(x)}=1$ for all $x\in\Sigma_{a}$ similar to Kneisel [12], p. 13. Moreover, $\nu_{\Gamma_{h}}(x):=\nabla\phi_{h}(x)/|\nabla\phi_{h}(x)|$ is the outer unit normal to $\Gamma_{h}=\partial(\Theta_{h}(\Omega^{+}))$ at $x\in\Gamma_{h}$. Furthermore, because of $\phi_{h}(\Theta_{h}(x))=d_{\Sigma}(x)$ on $\overline{\Sigma_{a/3}}$, we get the identity $\nabla\phi_{h}|_{\Theta_{h}(x)}=D\Theta_{h}^{-\top}|_{x}\,\nu_{\Sigma}|_{\Pi(x)}$ for $x\in\overline{\Sigma_{a/3}}$ and this implies

[TABLE]

2.3 Function Spaces for the Height Function and Modification $\tilde{\Theta}_{h}$ of $\Theta_{h}$

In the following we specify, in which function spaces we consider height functions $h$. Let $p>n+2$ be fixed. We set

[TABLE]

Since $3-\frac{3}{p}-\frac{n-1}{p}>2$ we have $X_{\gamma}:=(X_{0},X_{1})_{1-\frac{1}{p},p}=W^{3-\frac{3}{p}}_{p}(\Sigma)\hookrightarrow C^{2}(\Sigma).$ Moreover, we define $\mathbb{E}_{1}(T):=L^{p}(0,T;X_{1})\cap W^{1}_{p}(0,T;X_{0})$. Theorem A.6 yields $\mathbb{E}_{1}(T)\hookrightarrow C^{0}([0,T];X_{\gamma})$. Hence

[TABLE]

is well-defined for $h\in\mathbb{E}_{1}(T)\cap C^{0}([0,T];U)$ where $U:=\{h\in X_{\gamma}:\|h\|_{\infty}<a_{0}\}$. Let $R_{0}>0$ arbitrary and fixed from now on. It suffices to consider height functions in the space

[TABLE]

The restriction due to the estimate is negligible as we are interested in local well-posedness, but it is needed to show suitable properties of a modification of the Hanzawa transformation $\Theta_{h}$ which we need for technical reasons: if we transform differential operators in space, the term $D\Theta_{h}^{-\top}$ always appears. In order to show appropriate regularity properties in space of $D\Theta_{h}^{-\top}$, it is complicated (but possible) to work with the Hanzawa transformation directly. However, it seems difficult to obtain suitable regularity in time for $D\Theta_{h}^{-\top}$ since derivatives of $h$ in space are present. It is much simpler to replace $h\circ\Pi$ in $\Theta_{h}$ by an extension $Eh$ of $h$, where for $\overline{a}:=\frac{3a}{4}$

[TABLE]

is a suitable extension operator. Then we have $E\in\mathcal{L}(X_{\gamma},\tilde{X}_{\gamma})$, where we have set $\tilde{X}_{\gamma}:=(W^{1}_{p}(\Sigma_{\overline{a}}),W^{3}_{p}(\Sigma_{\overline{a}}))_{1-\frac{1}{p},p}$. Here because of $p>n+2$ it holds

[TABLE]

So we define the modification $\tilde{\Theta}_{h}$ of the Hanzawa transformation $\Theta_{h}$ as follows:

Definition 2.3

Let $p>n+2,R_{0}>0$ as above and $T>0$. For $h_{0}\in U$ we set

[TABLE]

and for $h\in V_{T}$ let $\tilde{\Theta}_{h}(x,t):=\tilde{\Theta}_{h(.,t)}(x)$ for all $(x,t)\in\mathbb{R}^{n}\times[0,T]$.

The following lemma shows that similar properties as for $\Theta_{h}$ also hold for the modification $\tilde{\Theta}_{h}$ if the extension operator $E$ is chosen properly.

Lemma 2.4

Let $\overline{a}=\frac{3a}{4}$ and $\varepsilon>0$. Then $E$ can be chosen such that for all $h\in X_{\gamma}$

[TABLE]

Moreover, for $R>0$ there is an $\varepsilon=\varepsilon(R)>0$ such that with the extension operator $E$ with respect to $\varepsilon$ the following hold: $\tilde{\Theta}_{h}:\Omega\rightarrow\Omega$ is for all $h\in U$ with $\|h\|_{X_{\gamma}}\leq R$ a $C^{2}$-diffeomorphism, the identity on $\mathbb{R}^{n}\textbackslash\Sigma_{2a/3}$ and we have

[TABLE]

Remark 2.5

Theorem A.6 implies $\mathbb{E}_{1}(T)\hookrightarrow C^{0}([0,T];X_{\gamma})$ and the embedding constant is bounded independent of $T>0$ if we add the term $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm. Now we can do the following: we choose $R=\overline{R_{0}}>R_{0}$ in Lemma 2.4 (and so $E$) such that

[TABLE]

holds for all $h\in V_{T}$ and $0\leq t\leq T$. In particular for $h\in V_{T}$ the last part in Lemma 2.4 is valid for $\tilde{\Theta}_{h}(.,t)$ instead of $\tilde{\Theta}_{h}$. Here $0\leq t\leq T$ and $0<T<\infty$ are arbitrary.

Proof of Lemma 2.4. Similar to A. and Wilke [2] one can construct a new operator $E$ that fulfils (2.4) by scaling $E$ in normal direction of $\Sigma$. Now let $R>0$ arbitrary. First of all, $\tilde{\Theta}_{h}$ is well-defined for all $h\in X_{\gamma}$, a $C^{2}$-mapping and the identity on $\mathbb{R}^{n}\textbackslash\Sigma_{2a/3}$, as the second part is cut off properly. Since $\tilde{\Theta}_{h}=\Theta_{h}$ on $\Sigma$, we can use the properties of $\Theta_{h}$. For all $h\in U$ with $\|h\|_{X_{\gamma}}\leq R$ it holds $\|h\circ\Pi-Eh\|_{C^{1}(\overline{\Sigma_{\overline{a}}})}\leq\varepsilon R$ due to (2.4) and therefore

[TABLE]

Since $\{D\Theta_{h}(x):x\in\mathbb{R}^{n},h\in U,\|h\|_{X_{\gamma}}\leq R\}$ is contained in a compact set $K\subseteq GL(n,\mathbb{R})$, we obtain for $\varepsilon(R)>0$ small that $\det D\tilde{\Theta}_{h}\geq\frac{c}{2}>0$. In particular $D\tilde{\Theta}_{h}$ is invertible and the inverse mapping theorem implies that $\tilde{\Theta}_{h}:\Omega\rightarrow\Omega$ is a $C^{2}$-diffeomorphism if we show bijectivity of $\tilde{\Theta}_{h}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. This can be done for a suitable choice of $\varepsilon(R)$ as in A. and Wilke [2], p. 49 using Lemma 2.1 and (2.5).

It remains to show the second estimate. Lemma 2.1 and (2.5) yield because of $D\tilde{\Theta}_{h}=D\Theta_{h}(I-D\Theta_{h}^{-1}(D\Theta_{h}-D\tilde{\Theta}_{h}))$ together with a Neumann series argument

[TABLE]

for all $x\in\mathbb{R}^{n}$ and $h\in U$ with $\|h\|_{X_{\gamma}}\leq R$ if $\varepsilon=\varepsilon(R)>0$ is sufficiently small. Equation (2.1) in Remark 2.2 implies

[TABLE]

for all $x\in\overline{\Sigma_{a/3}}$ if $\varepsilon(R)>0$ is chosen suitably. $\square$

2.4 The Transformed Equations

With $\tilde{\Theta}_{h}$ we transform the Navier-Stokes/mean curvature flow system (1.1)-(1.8) to $\Sigma$ and $\Omega^{\pm}$, respectively. Therefore we fix the height function $h\in V_{T}$ and let $v:\Omega\times(0,T)\rightarrow\mathbb{R}^{n}$ be a vector-valued and $\tilde{p}:\Omega\times(0,T)\rightarrow\mathbb{R}$ be a scalar-valued function. Then we define $u(x,t):=v(\tilde{\Theta}_{h}(x,t),t)$ and $\tilde{q}(x,t):=\tilde{p}(\tilde{\Theta}_{h}(x,t),t)$ for $(x,t)\in\Omega\times(0,T)$. Moreover, let

[TABLE]

Let $t\in[0,T]$. We set $\Gamma_{h}(t):=\Gamma_{h(.,t)}$ and denote the outer unit normal by $\nu_{\Gamma_{h}(t)}$ and by $H_{\Gamma_{h}(t)}$ the corresponding mean curvature. Furthermore, we define

[TABLE]

Applying the chain rule to $\phi_{h(.,t)}\circ\tilde{\Theta}_{h}(.,t)$ with $\phi_{h(.,t)}$ as in Remark 2.2 implies

[TABLE]

Moreover, we have $V_{\Gamma_{h}(t)}|_{\tilde{\Theta}_{h}(x,t)}=\partial_{t}(\Theta_{h}(x,t))\cdot\nu_{\Gamma_{h}(t)}|_{\Theta_{h}(x,t)}=\partial_{t}h(x,t)\,\nu_{\Sigma}|_{x}\cdot\nu_{h}|_{(x,t)}$ for the normal velocity. Since $\tilde{\Theta}_{h}$ is the identity in a neighbourhood of $\partial\Omega$, the no-slip condition on $v$ is retained. Therefore (1.1)-(1.8) becomes

[TABLE]

where we set $T_{h}(u,\tilde{q}):=2\mu^{\pm}\text{sym}(\nabla_{h}u)-\tilde{q}I$ as well as

[TABLE]

Moreover $u\cdot\nabla_{h}u\equiv(u\cdot\nabla_{h})u$ and $\textup{div}_{h}$ is applied row-wise to $\nabla_{h}u$ in (2.6). For this system we show local well-posedness. As a preparation, in the next two Subsections 2.5 and 2.6 we show properties of $\tilde{\Theta}_{h},A(h)=D\tilde{\Theta}_{h}^{-\top}$ and of the transformed mean curvature $K(h)$, respectively.

2.5 Properties of $\tilde{\Theta}_{h}$ and $D\tilde{\Theta}_{h}^{-\top}$

For later use we must know the regularity of $\tilde{\Theta}_{h}$ and $A(h)=D\tilde{\Theta}_{h}^{-\top}$. This is provided by

Lemma 2.6

Let $U_{0}$ as in Remark 2.5 and $0<T\leq T_{0}$. Then $\tilde{\Theta}_{.}\in BC^{1}(U_{0};C^{2}(\overline{\Omega})^{n})$, $A\in BC^{1}(U_{0};C^{1}(\overline{\Omega})^{n\times n})$ and $\tilde{\Theta}_{.}\in BC^{1}(V_{T};\tilde{X}_{T}),A\in BC^{1}(V_{T};X_{T})$, where

[TABLE]

for some $\tau>0$. In the time-dependent case the mappings and their derivatives are bounded by a $C>0$ independent of $0<T\leq T_{0}$ if we add $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm.

Proof. Let $h\in U_{0}$ or $h\in V_{T}$. First, we show the properties of $\tilde{\Theta}_{h}$. It holds

[TABLE]

Here $E$ is the extension operator from Remark 2.5 that induces $E\in\mathcal{L}(X_{\gamma},\tilde{X}_{\gamma})$ with $\tilde{X}_{\gamma}\hookrightarrow C^{2}(\overline{\Sigma_{\overline{a}}})$ by (2.3) and $E\in\mathcal{L}(\mathbb{E}_{1}(T),\tilde{\mathbb{E}}_{1}(T))$ with

[TABLE]

where the operator norm is bounded by a $C>0$ independent of $T>0$. We need suitable embeddings for $\tilde{\mathbb{E}}_{1}(T)$: Theorem A.7 yields for $0<\theta<1-\frac{1}{p}$

[TABLE]

and the embedding constant is bounded independent of $T>0$ if we add the term $\|\tilde{h}(0)\|_{\tilde{X}_{\gamma}}$ in the $\tilde{\mathbb{E}}_{1}(T)$-norm. For $\theta=\frac{n}{2p}+\frac{k-1}{2}+\delta,k=1,2$ and $\delta>0$ small we obtain because of $p>n+2$

[TABLE]

Embedding theorems and (2.15) imply $\tilde{\mathbb{E}}_{1}(T)\hookrightarrow C^{\tau}([0,T];C^{2}(\overline{\Sigma_{\overline{a}}}))\cap C^{\frac{1}{2}+\tau}([0,T];C^{1}(\overline{\Sigma_{\overline{a}}}))$ for some $\tau>0$. Since $\tilde{\Theta}_{h}$ is affine linear in $h$, we get the properties of $\tilde{\Theta}_{h}$.

Now we deduce the assertions for $A(h)=D\tilde{\Theta}_{h}^{-\top}$. The first part yields analoguous properties for $D\tilde{\Theta}_{h}$. Moreover, the space $X_{T}$ is an algebra with pointwise multiplication and a product estimate holds because this is valid for the Hölder spaces and we also have Lemma A.9. Additionally, Lemma 2.4 implies $\det D\tilde{\Theta}_{h}\geq c>0$ for all $h\in U_{0}$ and $h\in V_{T}$, respectively. The inverse formula, Lemma A.3 and Lemma A.9 give

[TABLE]

Furthermore, the mappings are bounded and in the time-dependent case bounded by a $C>0$ independent of $0<T\leq T_{0}$ if we add $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm. The implicit function theorem applied to

[TABLE]

at $(h,A(h))$ for $(Z,X)=(U_{0},C^{1}(\overline{\Omega})^{n\times n})$ or $(V_{T},X_{T})$ implies $A\in C^{1}(U_{0};C^{1}(\overline{\Omega})^{n\times n})$ and $A\in C^{1}(V_{T};X_{T})$, respectively, with

[TABLE]

From the properties of $D\tilde{\Theta}_{h}$ and (2.16) the claim follows. $\square$

As a consequence we obtain

Corollary 2.7

Let $U_{0}$ be as in Remark 2.5, $0<T\leq T_{0}$ and $\tau>0$ as in Lemma 2.6. Then $\nu_{h},a_{1}(h)$ and $a_{2}(h)$ as in Subsection 2.4 are well-defined for $h\in U_{0}$ and $h\in V_{T}$, respectively, and have the regularity

[TABLE]

In the time-dependent case the mappings and their derivatives are bounded by a $C>0$ independent of $0<T\leq T_{0}$ if we add the term $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm.

Proof. Lemma 2.4 and Remark 2.5 yield $|A(h)\nu_{\Sigma}|_{\Pi}|\geq\frac{1}{2}$ on $\overline{\Sigma_{a/3}}$. Thus we can extend $\nu_{\Sigma}$ by $\nu_{\Sigma}|_{\Pi}$ smoothly to $\overline{\Sigma_{a/3}}$ and obtain the assertion by Lemma 2.6. $\square$

2.6 Mean Curvature

For $h\in U$ we denote by $H_{\Gamma_{h}}$ the mean curvature of $\Gamma_{h}=\Theta_{h}(\Sigma)$ with respect to $\nu_{\Gamma_{h}}$. This subsection is devoted to the transformed mean curvature $K(h):=H_{\Gamma_{h}}\circ\Theta_{h}:\Sigma\rightarrow\mathbb{R}$. In particular we show that $K(h)$ has a quasilinear structure in terms of $h$ and the principal part for $h=0$ is given by the Laplace-Beltrami operator $\Delta_{\Sigma}$.

Lemma 2.8

Let $p>n+2$. Then there are $P\in C^{1}(U;\mathcal{L}(X_{1},X_{0}))$ and $Q\in C^{1}(U;X_{0})$ with $P(0)=\Delta_{\Sigma}$ and $K(h)=P(h)h+Q(h)$ for all $h\in U\cap X_{1}$.

Proof. Let $\varphi_{l}:V_{l}\rightarrow U_{l}\subseteq\Sigma$ for $l=1,...,N$ be suitable parametrizations of $\Sigma$ with $\Sigma=\bigcup_{l=1}^{N}U_{l}$ and $V_{l}\subseteq\mathbb{R}^{n-1}$. Then $\tilde{\varphi}_{l}:=(\varphi_{l},\textup{id}):V_{l}\times(-a,a)\rightarrow U_{l}\times(-a,a)$ are appropriate parametrizations of $\Sigma\times(-a,a)$. Now fix $l$. The Euclidean metric $g^{\textup{eucl}}$ on $\Sigma_{a}$ induces via the diffeomorphism $X$ from Subsection 2.2 a Riemannian metric $g^{X}$ on $\Sigma\times(-a,a)$. We denote by $w_{ij},\Gamma^{k}_{ij}$ for $i,j,k=1,...,n$ the local representation of $g^{X}$ and the Christoffel symbols with respect to $\tilde{\varphi}_{l}$, respectively, and $w^{ij}$ is as usual. Moreover, we set $w_{ij}(h)|_{s}:=w_{ij}|_{(s,h(s))}$ for $s\in U_{l}$. $\Gamma_{ij}^{k}(h)$ and $w^{ij}(h)$ are defined analogously. As in Escher and Simonett [8], proof of Lemma 3.1 on p. 274 ff. and Remark 3.2 on p. 277 it follows that

[TABLE]

where $P_{l}$ and $Q_{l}$ are altered by a factor $-(n-1)$ here. Additionally let $p_{jk},p_{j},q,l_{.}$ be as in Escher and Simonett [8] modified by the same factor. Then $P_{l}(0)$ is a local representation of $\Delta_{\Sigma}$. One can show that the $w_{ij}|_{(s,r)}$ and $\partial_{i}\partial_{j}X\cdot\partial_{m}X|_{(s,r)}$ for $i,j,m=1,...,n$ are at most quadratic in $r\in(-a,a)$ with smooth coefficients in $s\in\overline{U_{l}}$. Because of $\|h\|_{\infty}<a_{0}<\frac{a}{4}$ and $w_{ij}(h)=w_{ij}(.,h(.))$ for $i,j=1,...,n$, a compactness argument yields

[TABLE]

for a $c>0$ independent of $h$.

Using $X_{\gamma}\hookrightarrow C^{2}(\Sigma)$, the product rule for the Fréchet-derivative, the inverse formula as well as Lemma A.3 for the local representations in Escher and Simonett [8] implies

[TABLE]

Now the claim follows utilizing a suitable partition of unity. $\square$

Furthermore, we need properties of $K$, when we insert a time-dependent height function $h\in V_{T}$. This gives us

Lemma 2.9

Let $p>n+2,1<q<p$ and $0<T\leq T_{0}$. Then it holds

[TABLE]

Moreover, $K$ and its derivative are bounded by a constant $C(p,q,T_{0})>0$ when we add the term $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm.

Proof. We use the same notation as in the proof of Lemma 2.8. Then

[TABLE]

where the $a_{\alpha}$ correspond to $p_{ij},p_{j}$ and $q$, respectively, and $\partial_{s}^{\alpha}$ is the $\alpha$-th derivative with respect to the canonical coordinates.

First we estimate the $a_{\alpha}$ in suitable Hölder spaces. As in the proof of Lemma 2.6 one can show $\mathbb{E}_{1}(T)\hookrightarrow C^{\tau}([0,T];C^{2}(\Sigma))\cap C^{\frac{1}{2}+\tau}([0,T];C^{1}(\Sigma))$ for $\tau>0$ small because of $p>n+2$ and the embedding constant is bounded independent of $T>0$ if we add the term $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm. Since the $w_{ij}$ for $i,j=1,...,n$ are at most quadratic in $h$ with smooth coefficients in $\overline{U_{l}}$ product estimates in Hölder spaces yield

[TABLE]

Now (2.17) implies111 In fact, the same regularity as for the $w_{ij}(.)\circ\varphi_{l}$ should be obtained. But this is not needed. using the inverse formula, product estimates and Lemma A.3 that

[TABLE]

Hence this also holds for $\Gamma_{ij}^{k}(.)\circ\varphi_{l}$ and $(l_{.})^{r}$, where $i,j,k=1,...,n$ and $r\in\mathbb{R}$. Using the mapping properties in the local representations, we obtain for $\alpha\in\mathbb{N}_{0}^{n-1},|\alpha|\leq 2$

[TABLE]

Furthermore, all above terms and their derivatives are bounded by a $C>0$ independent of $T>0$ if we add $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm. So this also holds for the $a_{\alpha}$. The exponent $q$ does not appear here.

To show the desired properties of $K$ we use (2.18). Therefore we need another embedding for $\mathbb{E}_{1}(T)$ to estimate $\partial_{s}^{\alpha}h$ suitably. Theorem A.5 and a well-known interpolation inequality for real interpolation spaces (see Corollary 1.7 in Lunardi [17]) yield

[TABLE]

for $\theta\in(0,1)$ and the embedding constant is bounded independent of $T>0$. We set $\theta=\frac{1}{2}(1+\frac{1}{p})+\varepsilon$ for $\varepsilon>0$ small. Then for $\varepsilon=\varepsilon(p,q)>0$ small it holds that

[TABLE]

We infer $\mathbb{E}_{1}(T)\hookrightarrow B_{p,\infty}^{\frac{1}{2}(1-\frac{1}{p})-\varepsilon}(0,T;W^{2}_{p}(\Sigma))$ and, again, the embedding constant is bounded independent of $T>0$. This implies

[TABLE]

for $i,j=1,...,n-1$ and the mappings and their derivatives are bounded independent of $T>0$. Since pointwise multiplication is a product on $C^{1}(\overline{V_{l}})\times W^{1-\frac{1}{p}}_{p}(V_{l})\rightarrow W^{1-\frac{1}{p}}_{p}(V_{l})$ in sense of Definition A.1, using Lemma A.4 and (2.18)-(2.19) yields

[TABLE]

where $K(.)\circ\varphi_{l}$ and the derivative are bounded by a constant $C>0$ independent of $T>0$. Now we use the embeddings $L^{p}(V_{l})\hookrightarrow L^{q}(V_{l})$ and $W_{p}^{1-\frac{1}{p}}(V_{l})\hookrightarrow W_{q}^{1-\frac{1}{q}}(V_{l})$ as well as

[TABLE]

where the latter can be shown by using Hölder’s inequality and the embedding constant is bounded by $C(p,q,T_{0})>0$. We infer from Simon [21], Corollary 15 that

[TABLE]

and the embedding constant is also bounded by a $C(p,q,T_{0})>0$. The claim follows by using a suitable partition of unity. $\square$

3 Two-Phase Navier-Stokes System for Given Time-Dependent Interface

We consider the Navier-Stokes part (2.6)-(2.11) for a fixed time-dependent height function $h\in V_{T}$ (or equivalently for a given family of hypersurfaces $\{\Gamma_{h}(t)\}_{t\in(0,T)}$) and show unique solvability and some properties of the solution operator. Let as before $p>n+2,R_{0}>0$ be fixed as well as $X_{0},X_{1},X_{\gamma},U,\mathbb{E}_{1}(T)$ and $V_{T}$ for $T>0$ as in Subsection 2.3. Moreover, let $2<q<3$ and $h\in V_{T}$ with $h_{0}:=h(0)$. Then we set

[TABLE]

Now let $v_{0}\in Y_{\gamma,h_{0}}\cap L^{q}_{\sigma}(\Omega)$. In order to write the Navier-Stokes part (2.6)-(2.11) as an abstract fixed-point equation, we rearrange the equations such that the left hand side is the same as for the linear Stokes system in Appendix B. This yields

[TABLE]

where $A(h)=D\tilde{\Theta}_{h}^{-\top},\nabla_{h}=A(h)\nabla$ and $K(h),\nu_{h}$ are as in Subsection 2.4 as well as

[TABLE]

We introduce the space $Y_{T,h_{0},v_{0}}:=\{(u,\tilde{q})\in Y_{T}:u|_{t=0}=v_{0}\circ\tilde{\Theta}_{h_{0}}\}$ for the transformed velocity and pressure $(u,\tilde{q})$, where $Y_{T}:=Y_{T}^{1}\times Y_{T}^{2}$ and with $\Omega_{0}:=\Omega^{+}\cup\Omega^{-}$ we define

[TABLE]

Then the existence result for the Navier-Stokes part reads as follows:

Theorem 3.1

Let $p>n+2$ and $2<q<3$ with $1+\frac{n+2}{p}>\frac{n+2}{q}$. Then for $R>0$ there are $0<\varepsilon=\varepsilon(R)<R_{0}$ and $T_{0}=T_{0}(R)>0$ such that for all $0<T\leq T_{0}$ and

[TABLE]

the Navier-Stokes part (3.1)-(3.6) has a unique solution $F_{T}(h,v_{0}):=(u,\tilde{q})(h)\in Y_{T,h(0),v_{0}}$ and it holds that $\|F_{T}(h,v_{0})\|_{Y_{T}}\leq C_{R}$ as well as $F_{T}(h,v_{0})|_{[0,\tilde{T}]}=F_{\tilde{T}}(h|_{[0,\tilde{T}]},v_{0})$ for all $0<\tilde{T}\leq T$.

Additionally, for $h_{j}\in V_{T,\varepsilon}$ with $h_{0}^{j}:=h_{j}(0)$ as well as $v_{0}^{j}\in Y_{\gamma,h_{j}(0)}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}^{j}\|_{Y_{\gamma,h_{j}(0)}}\leq R$ and $\tilde{v}_{0}^{j}:=v_{0}^{j}\circ\tilde{\Theta}_{h_{0}^{j}}$ for $j=1,2$ we have the estimate

[TABLE]

For the proof we reformulate (3.1)-(3.6) as an abstract fixed-point equation

[TABLE]

where for $w=(u,\tilde{q})\in Y_{T}$ we define

[TABLE]

Lemma B.1 and Theorem B.3 imply that $L:Y_{T}\rightarrow Z_{T}$ is an isomorphism, where

[TABLE]

with $\tilde{Z}_{T}:=\tilde{Z}_{T}^{1}\times\tilde{Z}_{T}^{2}\times(\tilde{Z}_{T}^{3})^{n}\times Y_{\gamma}$ and the spaces

[TABLE]

Here $Y_{T},Z_{T}$ and $\tilde{Z}_{T}$ are Banach spaces with canonical norms, see Remark B.2.

At this point we see why we have to apply222 A. and Wilke [2] transform with $\tilde{\Theta}_{h}\circ\tilde{\Theta}_{h_{0}}^{-1}$. Then for fixed $h_{0}$ one could omit the dependence on $v_{0}$. But here we have chosen the other strategy for the following reasons: On one hand it is difficult to obtain suitable properties of $\tilde{\Theta}_{h_{0}}^{-1}$ and on the other hand $\Gamma_{0}:=\tilde{\Theta}_{h_{0}}(\Sigma)$ would have the role of $\Sigma$ in Theorem B.3 but one needs a priori $C^{2,1}$-regularity for the interface there, see Shimizu [20]. the fixed-point argument in the space $Y_{T,h_{0},v_{0}}$ which depends on $h_{0}$ and $v_{0}$. Without the additional condition for $u|_{t=0}$ in $Y_{T,h(0),v_{0}}$ the compatibility condition for $G(w;h,v_{0})$ in $Z_{T}$ would not be fulfilled in general. But it is essential that $G(.;h,v_{0})$ maps to $Z_{T}$ so that we can apply $L^{-1}$. For this reason we also subtracted the mean value in the second component of $G(w;h,v_{0})$. In fact, (3.1)-(3.6) and the formulation (3.8) are equivalent because of

Remark 3.2

Let $2<q<3,u\in W^{1}_{q,0}(\Omega)^{n}$ and $h\in U_{0}$ with $U_{0}$ as in Remark 2.5. Then

[TABLE]

Proof. The second equation is equivalent to $\textup{div}_{h}u=0$. For $u\in W^{1}_{q,0}(\Omega)^{n}$ holds $\int_{\Omega}\textup{div}\,u\,dx=0$ and this shows the reverse direction. The other part follows as in A. and Wilke [2], p. 51. $\square$

The crucial step for showing Theorem 3.1 is to prove the following lemma which states the properties of the right hand side $G$.

Lemma 3.3

Let $p>n+2$ and $2<q<3$ with $1+\frac{n+2}{p}>\frac{n+2}{q}$ as well as $0<T\leq 1$. Then for all $w\in Y_{T},\,h\in V_{T}$ and $v_{0}\in Y_{\gamma,h(0)}\cap L^{q}_{\sigma}(\Omega)$ we have that $G(w;h,v_{0})$ is contained in $\tilde{Z}_{T}$ and for $w\in Y_{T,h(0),v_{0}}$ it holds $G(w;h,v_{0})\in Z_{T}$. If $\|v_{0}\|_{Y_{\gamma,h(0)}}$ is bounded by a constant $R>0$, then $\|G(0;h,v_{0})\|_{\tilde{Z}_{T}}\leq C_{R}$.

Moreover, let $R_{1},R_{2}>0$. Then for $0<\varepsilon<R_{0}$ and $h_{j}\in V_{T,\varepsilon}$ with $h_{0}^{j}:=h_{j}(0)$, $v_{0}^{j}\in Y_{\gamma,h_{j}(0)}\cap L^{q}_{\sigma}(\Omega)$ and $\tilde{v}_{0}^{j}:=v_{0}^{j}\circ\tilde{\Theta}_{h_{j}(0)}$ as well as $w_{j}=(u_{j},\tilde{q}_{j})\in Y_{T}$ with $u_{0}^{j}:=u_{j}(0)$, $\|u_{0}^{j}\|_{Y_{\gamma}}\leq R_{1}$ and $\|w_{j}\|_{Y_{T}}\leq R_{2}$ for $j=1,2$ the following estimate holds:

[TABLE]

where $\delta:=\min\{\frac{\tau}{2},\frac{1}{q}-\frac{1}{p}\}>0$ and $\tau>0$ is as in Lemma 2.6.

Also as preparation for the proof of Lemma 3.3, in the following we discuss where the conditions for the exponent $q$ come from:

Remark 3.4

First of all, we have the restriction $2\leq q<3$ because of Theorem B.3 below. Theorem 3.1 gives us the velocity field $u(h)\in Y_{T}^{1}$ in dependence of the height function $h$. Later we will insert it in equation (2.12) which should hold in $L^{p}(0,T;X_{0})$. In order to show local well-posedness afterwards, we need enough regularity for $\textup{tr}_{\Sigma}\,u$. This means

[TABLE]

Since $Y_{\gamma}=\{v\in L^{p}(\Omega):v|_{\Omega^{\pm}}\in W^{2-\frac{2}{q}}_{q}(\Omega^{\pm})\}^{n}\cap W^{1}_{q,0}(\Omega)^{n}$, we get from Lemma A.8 below that

[TABLE]

and the embedding constant is bounded independent of $T>0$ if we add the term $\|u(0)\|_{Y_{\gamma}}$ in the $Y_{T}^{1}$-norm. Now, for $\theta\in(0,1)$ we use the reiteration theorem in Lunardi [17], Corollary 1.24 to the result

[TABLE]

Therefore Lemma A.5 yields $Y_{T}^{1}|_{\Omega^{\pm}}\hookrightarrow L^{s}(0,T;B_{q,q}^{2-2(1-\theta)\frac{1}{q}}(\Omega^{\pm})^{n})$ with $s=\frac{q}{\theta}$. To achieve (3.9) we want to choose $\theta\in(0,1)$ such that $\frac{q}{\theta}=s>p$ and that the Besov-space embeds into $W^{1}_{p}(\Omega^{\pm})^{n}$. Because of $q\leq p$ the latter is fulfilled for

[TABLE]

Hence (3.9) holds by the trace theorem if

[TABLE]

Since $p>n+2$ necessary conditions on $q$ are $q>2$, if $n=2$, and $q>\frac{5}{2}$, if $n=3$. So we directly wrote $q>2$ instead of $q\geq 2$. On the other hand the conditions are fulfilled for many $p,q$.

Another point is that we have to estimate $u\cdot\nabla_{h}u$ in the proof of Lemma 3.3 and therefore want to use the embedding $Y_{T}^{1}\hookrightarrow L^{r}(0,T;C^{0}(\overline{\Omega}))^{n}$ for some $r>q$. Embeddings over $\Omega^{\pm}$ yield $Y_{\gamma}\hookrightarrow C^{0}(\overline{\Omega})^{n}$ for $q>2$, if $n=2$ and $q>\frac{5}{2}$, if $n=3$, since then $2-\frac{2+n}{q}>0$ holds. Hence from (3.10) we even obtain $Y_{T}^{1}\hookrightarrow C^{0}([0,T];C^{0}(\overline{\Omega})^{n})$ and the embedding constant is bounded independent of $T>0$ if we add $\|u(0)\|_{Y_{\gamma}}$ in the $Y_{T}^{1}$-norm.

Altogether, for simplicity we restrict ourselves in Theorem 3.1 and Lemma 3.3 to exponents $p>n+2$ and $2<q<3$ with $1+\frac{n+2}{p}>\frac{n+2}{q}$.

Proof of Lemma 3.3. For simplicity we consider $0<T\leq 1$. Properties of $\tilde{\Theta}_{h}$ and $A(h)$ were shown in Lemma 2.6. The mean value theorem implies for all $h\in V_{T,\varepsilon}$ and $t\in[0,T]$

[TABLE]

Now let $w,h,v_{0}$ and $(u_{j},\tilde{q}_{j})=w_{j},u_{0}^{j}$ as well as $h_{j},h_{0}^{j}$ and $v_{0}^{j},\tilde{v}_{0}^{j}$ for $j=1,2$ as in the lemma. We begin with the first component of $G$ which we have to estimate in $\tilde{Z}_{T}^{1}=L^{q}(0,T;L^{q}(\Omega)^{n})$. Lemma 2.6 implies $(\nabla-\nabla_{h})\tilde{q}\in\tilde{Z}_{T}^{1}$ and (3.11) yields

[TABLE]

From Lemma 2.6 we obtain $\mu^{\pm}(\textup{div}_{h}\nabla_{h}u-\Delta u)\in\tilde{Z}_{T}^{1}$ and null additions imply

[TABLE]

Together with this estimate we infer from (3.11)

[TABLE]

Furthermore, Remark 3.4 yields $Y_{T}^{1}\hookrightarrow C^{0}([0,T];Y_{\gamma})\hookrightarrow C^{0}([0,T];C^{0}(\overline{\Omega}))^{n}$ and the embedding constant is bounded independent of $T>0$ if we add $\|u(0)\|_{Y_{\gamma}}$ in the $Y_{T}^{1}$-norm. Hence we get $u\cdot\nabla_{h}v\in\tilde{Z}_{T}^{1}$ for $u,v\in Y_{T}^{1}$ because of Lemma 2.6 and null additions imply

[TABLE]

Additionally we obtain $\nabla_{h}u\,\partial_{t}\tilde{\Theta}_{h}\in\tilde{Z}_{T}^{1}$ from Lemma 2.6, the embedding for $Y_{T}^{1}$ mentioned above and $W^{1}_{p}(\Omega)\hookrightarrow C^{0}(\overline{\Omega})$ as $p>n+2$. Moreover, null additions yield

[TABLE]

Now we consider the second component of $G$. First of all, Lemma 2.6 implies that $g(h)u=\textup{Tr}(\nabla u-\nabla_{h}u)=(I-A(h)):\nabla u\in L^{q}(0,T;W^{1}_{q}(\Omega_{0}))$ and (3.11) yields with a null addition

[TABLE]

Therefore this also follows for $G(w;h,v_{0})_{2}$ in $L^{q}(0,T;W^{1}_{q,(0)}(\Omega_{0}))$. Apart from that we have $G(w;h,v_{0})_{2}=g(h)u$ in $L^{q}(0,T;W^{-1}_{q,(0)}(\Omega))$ since constants drop out when integrating with $\phi\in W^{1}_{q^{\prime},(0)}(\Omega)$. In this space we need to show the existence of a weak derivative. To this end let $\eta\in C_{0}^{\infty}(0,T)$. We approximate $u$ and $A(h)$ suitably. Using convolution one can show that there are $(u^{m})_{m\in\mathbb{N}}$ in $C^{\infty}([0,T];W^{2}_{q}(\Omega_{0})\cap W^{1}_{q,0}(\Omega))^{n}$ and $(A^{m})_{m\in\mathbb{N}}$ in $C^{\infty}([0,T];C^{1}(\overline{\Omega})^{n\times n})$ with $u^{m}\rightarrow u$ in $Y_{T}^{1}$ and $A^{m}\rightarrow A(h)$ in the space $C^{0}([0,T];C^{1}(\overline{\Omega})^{n\times n})\cap W^{1}_{p}(0,T;L^{p}(\Omega)^{n\times n})$ for $m\rightarrow\infty$. Now the product rule yields

[TABLE]

Choose $\phi\in W^{1}_{q^{\prime},(0)}(\Omega)$ arbitrary. Then for $m\in\mathbb{N}$ we obtain

[TABLE]

Here $\partial_{t}g^{m}=-\partial_{t}A^{m}:\nabla u^{m}+(I-A^{m}):\nabla\partial_{t}u^{m}$ and therefore

[TABLE]

where we used integration by parts for the second term. In this equation we want to pass to the limit. Lemma A.8 implies $Y_{T}^{1}\hookrightarrow C^{0}([0,T];Y_{\gamma})$ and the embedding constant is bounded independent of $T>0$ if we add $\|u(0)\|_{Y_{\gamma}}$ in the $Y_{T}^{1}$-norm. This yields

[TABLE]

Furthermore we have $\partial_{t}u^{m}\cdot\textup{div}A^{m}\overset{m\rightarrow\infty}{\longrightarrow}\partial_{t}u\cdot\textup{div}\,A$ and $(I-A^{m})\partial_{t}u^{m}\overset{m\rightarrow\infty}{\longrightarrow}(I-A(h))\partial_{t}u$ in $L^{q}(0,T;L^{q}(\Omega))$ (vector-valued for the second one). Now it holds $W^{1}_{q^{\prime}}(\Omega)\hookrightarrow L^{r_{2}}(\Omega)$ for $n=2,3$ with $\frac{1}{r_{2}}:=\frac{1}{q^{\prime}}-\frac{1}{n}\in(0,1)$ and because of

[TABLE]

the mapping $L^{r_{1}}(\Omega)\rightarrow W^{-1}_{q,(0)}(\Omega):g\mapsto\langle g,.\rangle$ is bounded and linear. This is also true for

[TABLE]

Since $\langle.,\phi\rangle:W^{-1}_{q,(0)}(\Omega)\rightarrow\mathbb{R}$ is bounded and linear for $\phi\in W^{1}_{q^{\prime},(0)}(\Omega)$ we can pass to the limit in (3.12)-(3.13) for fixed $\phi$ and then we put outside the duality product. Hence $g(h)u\in W^{1}_{q}(0,T;W^{-1}_{q,(0)}(\Omega))$ and the derivative is given by

[TABLE]

For the difference of the derivatives the above observations together with null additions and $\textup{div}\,I=0$ yield

[TABLE]

Now we show $\textup{tr}_{\Sigma}(G(w;h,v_{0})_{2}|_{\Omega^{+}})\in\tilde{Z}_{T}^{3}$ and the related difference estimate. To this end let $\textup{ext}_{T}\in\mathcal{L}(W^{2,1}_{q}(\Omega^{+}_{T}),W^{2,1}_{q}(\Omega^{+}\times\mathbb{R}_{+}))$ for $T>0$ be extension operators in time such that

[TABLE]

holds for some $M>0$. To construct such operators one can e.g. first extend suitably from $\Omega$ to $\mathbb{R}^{n}$ and then proceed as in Amann [4], Lemma 7.2 utilizing that the Laplace-operator has maximal $L^{p}$-regularity, cf. Prüss and Simonett [18], Theorem 6.1.8.

We can extend $\textup{ext}_{T}u$ by reflection to a $\tilde{u}$ defined on $\Omega^{+}\times\mathbb{R}$. From Grubb [10], (A.8)-(A.12) we know that $\nabla\tilde{u}\in H^{1,\frac{1}{2}}_{q}(\Omega^{+}\times\mathbb{R})^{n\times n}\hookrightarrow W^{1,\frac{1}{2}}_{q}(\Omega^{+}\times\mathbb{R})^{n\times n}$, where the latter embedding holds because of $q\in(2,3)$. Moreover, we have the estimate

[TABLE]

We decompose $A(h)=\tilde{A}(h)+A(h(0))$ with $\tilde{A}(h):=A(h)-A(h(0))$ and extend $\tilde{A}(h)$ by reflection at $T$ to $[0,2T]$ and then by [math] to all of $\mathbb{R}_{+}$. Then $\tilde{A}(h)$ is contained in $C^{\frac{1}{2}+\tau}(\mathbb{R}_{+};C^{0}(\overline{\Omega})^{n\times n})\cap C^{\tau}(\mathbb{R}_{+};C^{1}(\overline{\Omega})^{n\times n})$ and the norm is estimated by the respective one of $A(h)-A(h(0))$. Since $\tilde{A}(h)=0$ on $(2T,\infty)$, we can apply Lemma A.4 to $I=(0,2T+1)$ and $\tilde{A}(h)|_{\Omega^{+}}$. Afterwards we extend by [math] and obtain

[TABLE]

This then also holds for the mean value and e.g. Denk, Hieber and Prüss [6], Lemma 3.5 implies $\textup{tr}_{\Sigma}(G(w;h,v_{0})_{2}|_{\Omega^{+}})\in\tilde{Z}_{T}^{3}$. Furthermore, a null addition yields

[TABLE]

where we used Lemma 2.6, (3.11) and $T\leq 1$.

Next we study the third component of $G$. Properties of $\nu_{.}$ were shown in Corollary 2.7. As in (3.11) we obtain $\|\nu_{\Sigma}-\nu_{h}\|_{C^{0}([0,T];C^{1}(\Sigma)^{n})}\leq C(T^{\tau}+\varepsilon)$. Similar as before we get $\textup{\textlbrackdbl}2\mu^{\pm}\textup{sym}(\nabla u)\textup{\textrbrackdbl},\,\textup{\textlbrackdbl}2\mu^{\pm}\text{sym}(\nabla u-\nabla_{h}u)\textup{\textrbrackdbl}\in(\tilde{Z}_{T}^{3})^{n\times n}$ and the following estimates hold:

[TABLE]

Since by definition $\textup{\textlbrackdbl}\tilde{q}\textup{\textrbrackdbl}\in\tilde{Z}_{T}^{3}$ and a product on $C^{1}(\Sigma)\times W_{q}^{1-\frac{1}{q}}(\Sigma)\rightarrow W_{q}^{1-\frac{1}{q}}(\Sigma)$ is given by pointwise multiplication, Lemma A.4 implies

[TABLE]

Corollary 2.7 yields $[\nu_{h}]_{C^{\frac{1+\tau}{2}}([0,T];C^{0}(\Sigma))^{n}}\leq CT^{\tau/2}$ and with null additions we infer

[TABLE]

Invoking Lemma 2.9 and Corollary 2.7 together with Lemma A.4 and the product rule, we obtain $\sigma K(.)\nu_{.}\in BC^{1}(V_{T};(\tilde{Z}_{T}^{3})^{n})$ where $\sigma K(.)\nu_{.}$ and the derivative is bounded independent of $0<T\leq 1$ if we add $\|h(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm. In particular we get

[TABLE]

For the fourth component we use that $\tilde{\Theta}_{h(0)}:\Omega\rightarrow\Omega$ is a $C^{2}$-diffeomorphism and the identity in a neighbourhood of $\partial\Omega$ by Lemma 2.4 and Remark 2.5. Moreover, because of Lemma 2.6 we have that $|\det\,A(h(0))|$ and the $C^{2}(\overline{\Omega})$-norm of $\tilde{\Theta}_{h(0)}$ are bounded by a constant independent of $h\in V_{T}$. Therefore $W^{k}_{q}(\tilde{\Theta}_{h(0)}(\Omega^{\pm}))^{n}\rightarrow W^{k}_{q}(\Omega^{\pm})^{n}:v\mapsto v\circ\tilde{\Theta}_{h(0)}$ for $k=0,1,2$ is bounded, linear and the operator norm is bounded independent of $h\in V_{T}$. The latter follows from Adams and Fournier [3], Theorem 3.41 and its proof. By interpolation this also holds for $Y_{\gamma,h(0)}\rightarrow Y_{\gamma}:v\mapsto v\circ\tilde{\Theta}_{h(0)}$. We obtain $G(w;h,v_{0})_{4}=v_{0}\circ\tilde{\Theta}_{h(0)}\in Y_{\gamma}$ and, if $\|v_{0}\|_{Y_{\gamma,h(0)}}\leq R$ for some $R>0$, we have $\|v_{0}\circ\tilde{\Theta}_{h(0)}\|_{Y_{\gamma}}\leq C_{R}$. The estimate for the difference of $\tilde{v}_{0}^{j}$ is trivial.

Altogether we have proven $G(w;h,v_{0})\in\tilde{Z}_{T}$ and the difference estimate in the lemma. Additionally $G(0;h,v_{0})=(0,\,0,\,-\sigma K(h)\nu_{h},\,v_{0}\circ\tilde{\Theta}_{h(0)})^{\top}$ and the above implies $\|G(0;h,v_{0})\|_{\tilde{Z}_{T}}\leq C_{R}$. It remains to show for $w\in Y_{T,h(0),v_{0}}$ the compatibility condition for $G(w;h,v_{0})$ in $Z_{T}$ which is equivalent to $\textup{div}\,(u(0))=(g(h)u)|_{t=0}$ in $W^{-1}_{q,(0)}(\Omega)$ since constants drop out when integrating with $\phi\in W^{1}_{q^{\prime},(0)}(\Omega)$. Lemma A.8 implies that $Y_{T}^{1}\hookrightarrow C^{0}([0,T];Y_{\gamma})$ and therefore

[TABLE]

Since $L^{q}(\Omega)\rightarrow W^{-1}_{q,(0)}(\Omega):g\mapsto\langle g,.\rangle$ is bounded and linear, it suffices to show $\textup{div}\,u(0)=g(h(0))u(0)$ or, equivalently, $\textup{div}_{h(0)}u(0)=0$ in $L^{q}(\Omega)$. But by chain rule we know $(\textup{div}\,v_{0})\circ\tilde{\Theta}_{h(0)}=\textup{div}_{h(0)}(u(0))$. Since $v_{0}$ is divergence-free, the claim follows. $\square$

After this preparation we can show the existence result for the Navier-Stokes part.

Proof of Theorem 3.1. Let $p,q$ be as in the theorem and $R>0$ be arbitrary. Then for $h\in V_{T}$ and $v_{0}\in Y_{\gamma,h(0)}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}\|_{Y_{\gamma,h(0)}}\leq R$ Lemma 3.3 yields

[TABLE]

We set $C_{L}:=\sup_{0<T\leq 1}\|L^{-1}\|_{\mathcal{L}(Z_{T},Y_{T})}$ and choose $R_{1}:=C_{R}$ and $R_{2}:=2C_{R}C_{L}$ in Lemma 3.3. Here $C_{L}$ is well-defined because of Theorem B.3. Then (3.16) implies for $0<\varepsilon<R_{0}$ and $0<T\leq 1$ as well as $h\in V_{T,\varepsilon}$ and $w\in Y_{T,h(0),v_{0}}$ with $\|w\|_{Y_{T}}\leq R_{2}$ the estimate

[TABLE]

where $\delta>0$. For $T_{0}=T_{0}(R),\varepsilon=\varepsilon(R)>0$ small we obtain from (3.16) for all $0<T\leq T_{0}$

[TABLE]

Lemma 3.3 yields $\|L^{-1}G(w_{1};h,v_{0})-L^{-1}G(w_{2};h,v_{0})\|_{Y_{T}}\leq C_{L}C_{R_{1},R_{2}}(T_{0}^{\delta}+\varepsilon)\|w_{1}-w_{2}\|_{Y_{T}}$ for all $w_{j}\in Y_{T,h(0),v_{0}}$ with $\|w_{j}\|_{Y_{T}}\leq R_{2}$ for $j=1,2$. We choose $\varepsilon=\varepsilon(R)>0$ and $T_{0}=T_{0}(R)>0$ small such that

[TABLE]

holds. Then $L^{-1}G(.;h,v_{0}):\overline{B_{Y_{T,h(0),v_{0}}}(0,R_{2})}\rightarrow\overline{B_{Y_{T,h(0),v_{0}}}(0,R_{2})}$ is well-defined for all $0<T\leq T_{0}$ and a strict contraction. Here $Y_{T,h(0),v_{0}}$ is a complete metric space as closed subset of a Banach space. Banach’s fixed-point theorem yields a unique fixed-point $F_{T}(h,v_{0})$ in $\overline{B_{Y_{T,h(0),v_{0}}}(0,R_{2})}$ and it holds

[TABLE]

since $h|_{[0,\tilde{T}]}\in V_{\tilde{T},\varepsilon}$ as well as $F_{T}(h,v_{0})|_{[0,\tilde{T}]}\in\overline{B_{Y_{\tilde{T},h(0),v_{0}}}(0,R_{2})}$ and $L^{-1}G(.;h,v_{0})$ is compatible with restrictions in time on $[0,\tilde{T}]$ for $0<\tilde{T}\leq T$. The uniqueness in $Y_{T,h(0),v_{0}}$ can be shown with the compatibility for restrictions in time and (3.17).

It remains to prove the Lipschitz dependence. To this end let $h_{j},h_{0}^{j},v_{0}^{j}$ and $\tilde{v}_{0}^{j}$ for $j=1,2$ be as in the theorem. Then Lemma 3.3 implies

[TABLE]

Now from (3.18) the claim follows. $\square$

As preparation for the proof of the local well-posedness for (2.6)-(2.13) we show that the term coming from the Navier-Stokes part in (2.12) is sufficiently regular:

Corollary 3.5

Let $p,q$ and $\varepsilon,T_{0}$ for $R>0$ as in Theorem 3.1. For $0<T\leq T_{0},h\in V_{T,\varepsilon}$ and $v_{0}\in Y_{\gamma,h(0)}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}\|_{Y_{\gamma,h(0)}}\leq R$ for $j=1,2$ let

[TABLE]

where $a_{1}(h)$ is defined in Subsection 2.4 and $F_{T}(h,v_{0})$ stems from Theorem 3.1. Then for a $s>p$ it holds $G_{T}(h,v_{0})\in L^{s}(0,T;X_{0})$ and $\|G_{T}(h,v_{0})\|_{L^{s}(0,T;X_{0})}\leq C(R)$ as well as

[TABLE]

Moreover, for $h_{j},h_{0}^{j}:=h_{j}(0),v_{0}^{j}$ and $\tilde{v}_{0}^{j}:=v_{0}^{j}\circ\tilde{\Theta}_{h_{0}^{j}},j=1,2$ as in Theorem 3.1 it holds

[TABLE]

Proof. From Remark 3.4 we know that $Y_{T}^{1}\rightarrow L^{s}(0,T;X_{0})^{n}:u\mapsto\textup{tr}_{\Sigma}u$ is bounded, linear for some $s>p$ and the operator norm is bounded independent of $T>0$ if we add $\|u(0)\|_{Y_{\gamma}}$ in the $Y_{T}^{1}$-norm. Hence $\textup{tr}_{\Sigma}(F_{T}(h,v_{0})_{1})\in L^{s}(0,T;X_{0})^{n}$ and from Theorem 3.1 and (3.16) we deduce $\|\textup{tr}_{\Sigma}(F_{T}(h,v_{0})_{1})\|_{L^{s}(0,T;X_{0})^{n}}\leq C(R)$ as well as

[TABLE]

Since pointwise multiplication is a product on $C^{1}(\Sigma)\times X_{0}\rightarrow X_{0}$, we obtain $G_{T}(h,v_{0})\in L^{s}(0,T;X_{0})$ with $\|G_{T}(h,v_{0})\|_{L^{s}(0,T;X_{0})}\leq\tilde{C}(R)$ by Corollary 2.7 and (3.20) together with a null addition yields the estimate in the lemma. Equation (3.19) follows from the respective one for $F_{T}$ in Theorem 3.1. $\square$

4 Local Well-Posedness

In this section we show the local well-posedness for the transformed Navier-Stokes/mean curvature flow system (2.6)-(2.13). Let $p>n+2$ and $2<q<3$ with $1+\frac{n+2}{p}>\frac{n+2}{q}$. For $\varepsilon,T_{0}>0$ to $R>0$ as in Theorem 3.1 and $0<T\leq T_{0}$ as well as $h\in V_{T},h_{0}\in X_{\gamma}$ with $\|h_{0}\|_{X_{\gamma}}\leq\varepsilon$ and $v_{0}\in Y_{\gamma,h(0)}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}\|_{Y_{\gamma,h(0)}}\leq R$ the system is equivalent to the abstract evolution equation

[TABLE]

where $G_{T}$ is defined in Corollary 3.5. Here “equivalent” means that $(u,\tilde{q},h)\in Y_{T}\times V_{T}$ solves (2.6)-(2.13) if and only if $h$ fulfils (4.1)-(4.2) and it holds $(u,\tilde{q})=F_{T}(h,v_{0})$.

In order to show local well-posedness for (4.1)-(4.2), we use that by Lemma 2.8 the transformed mean curvature $K(h)$ has a quasilinear structure with respect to the height function $h$ and for $h=0$ the principal part is given by the Laplace-Beltrami operator $\Delta_{\Sigma}$. Here $\Delta_{\Sigma}:X_{1}\rightarrow X_{0}$ has maximal $L^{p}$-regularity on finite intervals because of Theorem 6.4.3 in Prüss and Simonett [18]. To give a simple proof for the local well-posedness, we proceed as in the case of quasilinear parabolic equations, compare e.g. Köhne, Prüss and Wilke [13], Theorem 2.1 (for $\mu=1$ there). We especially cannot apply this result directly since $G_{T}$ is a non-local operator. That means $G_{T}(h,v_{0})$ at point $x_{0}\in\Sigma$ and time $t_{0}\in[0,T]$ in general depends on values $h(x,t)$ with $(x,t)\in\Sigma\times[0,T]$ outside a neighbourhood of $(x_{0},t_{0})$. Nevertheless we can essentially adapt the proof of [13] (for $\mu=1$) since by Corollary 3.5 we have that $G_{T}$ is sufficiently regular and compatible with restrictions in time on $[0,\tilde{T}]$ for $0<\tilde{T}\leq T$. Intuitively the velocity and pressure at time $T$ depend on the evolution of the entire interface from the beginning up to time $T$.

The result for the local well-posedness of the transformed Navier-Stokes/mean curvature flow system (2.6)-(2.13) reads as follows:

Theorem 4.1

Let $p>n+2$ and $2<q<3$ with $1+\frac{n+2}{p}>\frac{n+2}{q}$. For $R>0$ there exist $T_{0}=T_{0}(R),\varepsilon_{0}=\varepsilon_{0}(R)>0$ such that for all $0<T\leq T_{0}$ and $h_{0}\in X_{\gamma}$ with $\|h_{0}\|_{X_{\gamma}}\leq\varepsilon_{0}$ and $v_{0}\in Y_{\gamma,h_{0}}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}\|_{Y_{\gamma,h_{0}}}\leq R$ there is a unique solution $(u,\tilde{q},h)(h_{0},v_{0})\in Y_{T}\times V_{T}$ of (2.6)-(2.13) and

[TABLE]

holds for $h_{0}^{j}\in X_{\gamma}$ with $\|h_{0}^{j}\|_{X_{\gamma}}\leq\varepsilon_{0}$ and $v_{0}^{j}\in Y_{\gamma,h_{j}(0)}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}^{j}\|_{Y_{\gamma,h_{j}(0)}}\leq R$ as well as $\tilde{v}_{0}^{j}:=v_{0}^{j}\circ\tilde{\Theta}_{h_{0}^{j}}$ for $j=1,2$.

Proof. With $P,Q$ as in Lemma 2.8 and $U_{0}$ as in Remark 2.5 we denote $\tilde{B}(h):=a_{2}(h)\,P(h)$ and $\tilde{F}(h):=a_{2}(h)\,Q(h)$ for all $h\in U_{0}$. Because of Theorem 6.4.3 in Prüss and Simonett [18] there is a $w>0$ such that $\Delta_{\Sigma}-wI:X_{1}\rightarrow X_{0}$ has maximal $L^{p}$-regularity on $\mathbb{R}_{+}$. Therefore we consider $B:=\tilde{B}-wI$ and $F:=\tilde{F}+wI$. Since $a_{2}(0)=1$ we have $B(0)=\Delta_{\Sigma}-wI$. Moreover, Corollary 2.7 yields $B\in C^{1}(U_{0};\mathcal{L}(X_{1},X_{0}))$ and $F\in C^{1}(U_{0};X_{0})$. The embedding $X_{\gamma}\hookrightarrow L^{\infty}(\Sigma)$ implies $\overline{B_{X_{\gamma}}(0,\delta_{0})}\subseteq U_{0}$ for $0<\delta_{0}<R_{0}$ small. So if $\delta_{0}$ is sufficiently small, by the mean value theorem, there is a $L>0$ such that

[TABLE]

holds for all $h_{1},h_{2}\in\overline{B_{X_{\gamma}}(0,\delta_{0})}$ and $v\in X_{1}$.

Now fix $R>0$ and let $T_{0}=T_{0}(R),\varepsilon=\varepsilon(R)>0$ be as in Theorem 3.1. Then for $0<T\leq T_{0}$ and $h\in V_{T}$ as well as $h_{0}\in X_{\gamma}$ with $\|h_{0}\|_{X_{\gamma}}\leq\varepsilon$ and $v_{0}\in Y_{\gamma,h_{0}}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}\|_{Y_{\gamma,h_{0}}}\leq R$ the system (2.6)-(2.13) or (4.1)-(4.2), respectively, are equivalent to

[TABLE]

Here $B(h)$ and $F(h)$ are defined in the sense of $V_{T}\subseteq C^{0}([0,T];U_{0})$ by Remark 2.5. In particular, it holds $B(h)\in C^{0}([0,T];\mathcal{L}(X_{1},X_{0}))$.

The idea for solving (4.4)-(4.5) is as follows: we fix $B(h)$ at $h=0$ and apply Banach’s fixed-point theorem to $H(.;h_{0},v_{0})$ where $H(h;h_{0},v_{0}):=f$ is the unique solution of

[TABLE]

The unique solvability of (4.6)-(4.7) will follow from Prüss and Simonett [18], Theorem 3.5.5 if we extend the right hand side by zero to $\mathbb{R}_{+}$. To this end we need suitable sets for the height function. For $0<\delta<\min\{\varepsilon,\delta_{0}\},0<r<R_{0}-\delta_{0}$ and $0<T\leq T_{0}$ we set

[TABLE]

For $r,\delta>0$ small we show $\mathbb{B}_{r,T,h_{0}}\subseteq V_{T}\cap C^{0}([0,T];\overline{B_{X_{\gamma}}(0,\delta_{0})})$. Therefore let $h\in\mathbb{B}_{r,T,h_{0}}$.

Because of $\|h\|_{\mathbb{E}_{1}(T)}+\|h_{0}\|_{X_{\gamma}}\leq r+\delta<R_{0}$ and our choice of $\delta_{0}$ it suffices to show $\|h(t)\|_{X_{\gamma}}\leq\delta_{0}$ for all $t\in[0,T]$ if $r,\delta>0$ are small independent of $h$. By Prüss and Simonett [18], Theorem 3.5.5 there is a unique solution $h_{0}^{\ast}\in\mathbb{E}_{1}(\mathbb{R}_{+})$ of

[TABLE]

Since $h(0)=h_{0}^{\ast}(0)$, we get from Theorem A.6 with a null addition

[TABLE]

For $r,\delta>0$ small we obtain $\|h(t)\|_{X_{\gamma}}\leq\delta_{0}$ for all $t\in[0,T]$ and thus the above inclusion.

Now we can define $H(.;h_{0},v_{0})\equiv H_{r,T}(.;h_{0},v_{0}):\mathbb{B}_{r,T,h_{0}}\rightarrow\mathbb{E}_{1}(T):h\mapsto f$ where $f$ is the unique solution to (4.6)-(4.7). Then $h\in\mathbb{B}_{r,T,h_{0}}$ is a solution of (4.4)-(4.5) if and only if $h$ is a fixed-point of $H(.;h_{0},v_{0})$. To apply Banach’s theorem we need the following:

For $r,\delta,T>0$ small, all $g_{j}\in\overline{B_{X_{\gamma}}(0,\delta)}$ and $h_{j}\in\mathbb{B}_{r,T,g_{j}}$ as well as $v_{0}^{j}\in Y_{\gamma,g_{j}}\cap L^{q}_{\sigma}(\Omega)$ with $\|v_{0}^{j}\|_{Y_{\gamma,g_{j}}}\leq R$ and $\tilde{v}_{0}^{j}:=v_{0}^{j}\circ\tilde{\Theta}_{g_{j}}$ for $j=1,2$ we will show the following inequality:

[TABLE]

Moreover, for all $r>0$ there are $\delta(r),T(r)>0$ such that for all $0<\delta\leq\delta(r)$ and $0<T\leq T(r)$ the estimate $\|H(0;h_{0},v_{0})\|_{\mathbb{E}_{1}(T)}\leq\frac{r}{2}$ is valid, where $h_{0}\in\overline{B_{X_{\gamma}}(0,\delta)}$.

Now let us prove this. From Prüss and Simonett [18], Theorem 3.5.5 we know that there exists a $C>0$ independent of $T>0$ such that

[TABLE]

Here because of (4.3) it holds $\|F(h_{1})-F(h_{2})\|_{\mathbb{E}_{0}(T)}\leq LT^{\frac{1}{p}}\|h_{1}-h_{2}\|_{C^{0}([0,T];X_{\gamma})}$ and with $g_{j}^{\ast}$ as in (4.8) for $g_{j},j=1,2$ instead of $h_{0}$ we conclude from Theorem A.6

[TABLE]

Furthermore (4.3) together with (4.9) and (4.10) implies

[TABLE]

Additionally, for some $s>p$ Corollary 3.5 and Hölder’s inequality yield

[TABLE]

For $r,\delta,T>0$ small we obtain the claimed difference estimate. It remains to treat $\|H(0;h_{0},v_{0})\|_{\mathbb{E}_{1}(T)}$. Theorem 3.5.5 in Prüss and Simonett [18] and Corollary 3.5 imply

[TABLE]

So if $\delta(r),T(r)>0$ are appropriate, we get $\|H(0;h_{0},v_{0})\|_{\mathbb{E}_{1}(T)}\leq\frac{r}{2}$ for $0<\delta\leq\delta(r)$ and $0<T\leq T(r)$. Altogether we have proven everything claimed above.

After this preparation we can choose $r,\delta,T_{0}>0$ small such that everything before holds for $r,\delta,T_{0}$ and for $2r,\delta,T_{0}$. Then this also follows for $\tilde{r},\delta,T$ where $\tilde{r}\in\{r,2r\}$ and $0<T\leq T_{0}$. Now we choose $h_{0}=g_{1}=g_{2}\in\overline{B_{X_{\gamma}}(0,\delta)}$ and $v_{0}=v_{0}^{1}=v_{0}^{2}$ above and obtain that $H_{\tilde{r},T}(.;h_{0},v_{0})$ is a strict contraction on $\mathbb{B}_{\tilde{r},T,h_{0}}$ where the latter is a complete metric space as closed subset of a Banach space. Banach’s theorem now yields a unique fixed-point $h^{\ast}_{\tilde{r},T}(h_{0},v_{0})\in\mathbb{B}_{\tilde{r},T,h_{0}}$ of $H_{\tilde{r},T}(.;h_{0},v_{0})$ in $\mathbb{B}_{\tilde{r},T,h_{0}}$ which is a solution of (4.4)-(4.5) and

[TABLE]

solves the transformed Navier-Stokes/mean curvature flow system (2.6)-(2.13). The Lipschitz-dependence of $(u^{\ast},\tilde{q}^{\ast},h^{\ast})$ on $(h_{0},\tilde{v}_{0})$ where $\tilde{v}_{0}:=v_{0}\circ\tilde{\Theta}_{h_{0}}$ is obtained as follows: For $h^{\ast}$ we use $H_{r,T}(h^{\ast}(h_{0},v_{0});h_{0},v_{0})=h^{\ast}(h_{0},v_{0})$ in the difference estimate proven before. Then for $(u^{\ast},\tilde{q}^{\ast})$ we utilize Theorem 3.1.

It remains to show uniqueness in $Y_{T}\times V_{T}$ for fixed $h_{0}\in\overline{B_{X_{\gamma}}(0,\delta)}$ and $v_{0}$ as in the theorem. In the following we write $h^{\ast}_{\tilde{r},T}$ and $(u^{\ast},\tilde{q}^{\ast},h^{\ast})$ instead of $h^{\ast}_{\tilde{r},T}(h_{0},v_{0})$ and $(u^{\ast},\tilde{q}^{\ast},h^{\ast})(h_{0},v_{0})$, respectively. In particular it holds $h^{\ast}=h^{\ast}_{r,T}$. We need that the fixed-point $h^{\ast}_{\tilde{r},T}$ is the same for $\tilde{r}\in\{r,2r\}$ and remains the same up to restriction in time if we shrink $T$. Therefore let $0<\tilde{T}\leq T$ and $\tilde{r}\in\{r,2r\}$. For $h\in\mathbb{B}_{\tilde{r},T,h_{0}}$ we have $h|_{[0,\tilde{T}]}\in\mathbb{B}_{\tilde{r},\tilde{T},h_{0}}$ and because of Corollary 3.5 it holds $G_{T}(h,v_{0})|_{[0,\tilde{T}]}=G_{\tilde{T}}(h|_{[0,\tilde{T}]},v_{0})$. Hence we obtain

[TABLE]

Since $H_{r,\tilde{T}}(h;h_{0},v_{0})=H_{2r,\tilde{T}}(h;h_{0},v_{0})$ for all $h\in\mathbb{B}_{r,\tilde{T},h_{0}}\subseteq\mathbb{B}_{2r,\tilde{T},h_{0}}$ the uniqueness of the respective fixed-point yields

[TABLE]

Now the uniqueness in $Y_{T}\times V_{T}$ can be shown by a typical contradiction argument. $\square$

Appendix A Banach-Space-Valued Functions

A.1 Spaces of (Hölder-)Continuous Functions

Let $X$ be a Banach space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For a closed interval $I\subseteq\mathbb{R}$ we denote by $C^{0}(I;X),C_{b}^{0}(I;X),BUC(I;X)$ and $C^{0,\alpha}(I;X)$ for $\alpha\in(0,1]$ the set of continuous; continuous and bounded; bounded and uniformly continuous; bounded and Hölder-continuous functions $f:I\rightarrow X$, respectively. In the latter case we write $[f]_{C^{0,\alpha}(I;X)}$ for the semi-norm and we set $C^{\alpha}(I;X):=C^{0,\alpha}(I;X)$ if $\alpha\in(0,1)$. Moreover, let $C^{0,0}(I;X):=C^{0}(I;X)$. If $I\subseteq\mathbb{R}$ is an open interval, then $C^{\infty}(\overline{I};X)$ is the set of all smooth $f:I\rightarrow X$ such that $f$ and all derivatives can be extended continuously to $\overline{I}$ and $C_{0}^{\infty}(I;X)$ denotes the set of all $f\in C^{\infty}(\overline{I};X)$ with compact support in $I$. For suitable product estimates we need:

Definition A.1

Let $X,Y,Z$ be Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Then $B:X\times Y\rightarrow Z$ is called product if $B$ is bilinear and $\|B(f,g)\|_{Z}\leq C_{0}\|f\|_{X}\|g\|_{Y}$ holds for all $(f,g)\in X\times Y$ and some $C_{0}>0$.

With a null addition one can show that for any closed, bounded interval $I\subseteq\mathbb{R}$ and $\alpha\in[0,1]$ the product $B$ induces a product on the corresponding Hölder-spaces to the exponent $\alpha$ and in the estimate one can choose the same $C_{0}$. For estimates of nonlinear terms we use

Lemma A.2

Let $I\subseteq\mathbb{R}$ a closed, bounded interval, $n,m\in\mathbb{N}$ and $\Omega\subseteq\mathbb{R}^{n}$ open and bounded, $U\subseteq\mathbb{R}^{m}$ open, $F:U\rightarrow\mathbb{R}$ be $C^{1}$ and $K\subseteq U$ compact and convex. For $\alpha\in[0,1]$ let

[TABLE]

and $\tilde{F}(u)(t)(x):=F(u(t)(x))$ for $u\in M$ and $t\in I,x\in\overline{\Omega}$. Then for $u\in M$ with $\|u\|_{C^{0,\alpha}(I;C^{0}(\overline{\Omega})^{m})}\leq R$ it holds $\tilde{F}(u)\in C^{0,\alpha}(I;C^{0}(\overline{\Omega}))$ with $\|\tilde{F}(u)\|_{C^{0,\alpha}(I;C^{0}(\overline{\Omega}))}\leq C(R)$ where $C(R)>0$ is independent of $I$ and $\alpha$.

Moreover, if $F$ is $C^{2}$, then for all $u,v\in M$ with norm estimated by $R$ there is a $C(R)>0$ independent of $I$ and $\alpha$ such that $\|\tilde{F}(u)-\tilde{F}(v)\|_{C^{0,\alpha}(I;C^{0}(\overline{\Omega}))}\leq C(R)\|u-v\|_{C^{0,\alpha}(I;C^{0}(\overline{\Omega})^{m})}$.

Proof. For $u,v\in M$ and $t\in I,x\in\overline{\Omega}$ the mean value theorem implies

[TABLE]

Now the first part directly follows from the compactness of $K$. If additionally $F$ is $C^{2}$ we apply the first part to $G:U\times U\rightarrow\mathbb{R}:(\tilde{u},\tilde{v})\mapsto\int_{0}^{1}DF(s\tilde{u}+(1-s)\tilde{v})\,ds$ and $K\times K$ instead of $K$ using a product estimate. $\square$

In the following lemma let all functions have values in $\mathbb{R}$.

Lemma A.3

Let $I\subseteq\mathbb{R}$ be a closed, bounded interval, $\Omega\subseteq\mathbb{R}^{n}$ be a bounded, connected domain, $\alpha\in[0,1]$ and $k=0,1$. For $\beta\in\mathbb{R}$ and $c_{0}>0$ we consider $F(h):=h^{\beta}$ acting on $U_{k}:=\{h\in C^{k}(\overline{\Omega}):h(x)>c_{0}\text{ for all }x\in\overline{\Omega}\}$ and on

[TABLE]

respectively. Then $F\in C^{1}(U_{k};C^{k}(\overline{\Omega}))$ and $F\in C^{1}(V_{\alpha,k};C^{0,\alpha}(I;C^{k}(\overline{\Omega})))$ and for $R>0$ arbitrary and all $h\in U_{k}$ and $h\in V_{\alpha,k}$ with norm less or equal $R$, respectively, it holds that $F(h)$ and $F^{\prime}(h)$ is bounded by a constant $C_{R,\beta,c_{0}}>0$ independent of $I$.

Proof. For $\beta\in\mathbb{N}_{0}$ the claim follows from the product rule. Therefore it is enough to consider the case $-\infty<\beta<1$. By scaling we can also assume $c_{0}=1$. Now one uses that for $\beta<1$ and $x,y\geq 1$ it holds that $|x^{\beta}-y^{\beta}|\leq c_{\beta}|x-y|$ and $x^{\beta}\leq C_{\beta}(1+|x|)$. Moreover, for $k=1$ we have $\partial_{x_{i}}(F(h))=\beta\,h^{\beta-1}\partial_{x_{i}}h$ for $i=1,...,n$. From this the continuity of $F$ on respective spaces and the estimate for $F(h)$ follows. In both cases our candidate for the derivative is $G(h)(\rho):=\beta h^{\beta-1}\rho$ on corresponding spaces. Replacing $\beta-1$ by $\beta$ above one can show that $G$ is well-defined, continuous and satisfies the desired estimate for $F^{\prime}$ in the lemma. To verify the definition of the Fréchet-derivative one uses that for small $\rho$

[TABLE]

If $\rho$ is bounded by a small $\varepsilon>0$ in the respective norm, Lemma A.2 yields for $k=0$ that the first part can be estimated by $c(h,\varepsilon)$ in the corresponding norm. Then product estimates yield the case $k=0$ and the case $k=1$ follows from this by replacing $\beta-1$ by $\beta$ and using the identity for $\partial_{x_{i}}(F(h))$. $\square$

A.2 Lebesgue-, Sobolev- and Besov-Spaces

Let $I\subseteq\mathbb{R}$ be measurable, $X$ be a Banach space and $1\leq p\leq\infty$. Then $L^{p}(I;X)$ are the usual Bochner spaces and $W^{k}_{p}(I;X)$ for $I$ open and $k\in\mathbb{N}$ are the $X$-valued Sobolev spaces.

Moreover, we need vector-valued variants of some fractional order Besov- and Sobolev-spaces: Therefore let $s\in(0,1),1\leq p\leq\infty$ and $I$ be an interval in $\mathbb{R}$. Then we define $B^{s}_{p,\infty}(I;X):=\{f\in L^{p}(I;X):\|f\|_{B^{s}_{p,\infty}(I;X)}<\infty\}$ where the norm is given by $\|f\|_{B^{s}_{p,\infty}(I;X)}:=\|f\|_{L^{p}(I;X)}+[f]_{B^{s}_{p,\infty}(I;X)}$ with

[TABLE]

For $s\in(0,1)$ and $1\leq p<\infty$ we set $W^{s}_{p}(I;X):=\{f\in L^{p}(I;X):\|f\|_{W^{s}_{p}(I;X)}<\infty\}$ where the norm is given by $\|f\|^{p}_{W^{s}_{p}(I;X)}:=\|f\|_{L^{p}(I;X)}^{p}+[f]^{p}_{W^{s}_{p}(I;X)}$ and

[TABLE]

All the above spaces are Banach spaces since $L^{p}(I;X)$ and $L^{1}(I\times I)$ are complete. The above definitions equal those in Simon [21] up to equivalent norms. Moreover, for $s>0$ and $1<p<\infty$ the Bessel-potential spaces are denoted by $H^{s}_{p}(I;X)$.

Additionally, we need spaces with mixed regularity. Let $\Omega\subseteq\mathbb{R}^{n}$ be open, $r,s\in\mathbb{N}$, $1<p<\infty$ and $T>0$. Then we define $W_{p}^{r,s}(\Omega_{T}):=L^{p}(0,T;W_{p}^{r}(\Omega))\cap W^{s}_{p}(0,T;L^{p}(\Omega))$. If $T=\infty$ we write $\Omega\times\mathbb{R}_{+}$ instead of $\Omega_{T}$. Further anisotropic spaces are defined analogously. For properties of such spaces cf. the appendix in Grubb [10] and Denk, Hieber and Prüss [6].

We need product estimates for Hölder spaces with suitable Sobolev- and Besov-spaces:

Lemma A.4

Let $X,Y,Z$ be Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $B:X\times Y\rightarrow Z$ be a product in sense of Definition A.1. Moreover, let $I\subseteq\mathbb{R}$ be a finite interval, $s\in(0,1)$ and $1\leq p\leq\infty$. Then $C^{s}(\overline{I};X)\times B^{s}_{p,\infty}(I;Y)\rightarrow B^{s}_{p,\infty}(I;Z):(f,g)\mapsto B(f,g)$ is a product and in the product estimate one can choose the same constant as for $B$.

If $s\in(0,1),1\leq p<\infty$ and $\varepsilon>0$ is such that $s+\varepsilon<1$ holds, then $B$ induces a product on $C^{s+\varepsilon}(\overline{I};X)\times W^{s}_{p}(I;Y)\rightarrow W^{s}_{p}(I;Z)$ and the constant in the estimate can be chosen independent of $|I|$ if $|I|\leq T$ for a fixed $T>0$.

Proof. First one can show that $B$ induces a product on $C^{0}(\overline{I};X)\times L^{p}(I;X)\rightarrow L^{p}(I;X)$ and in the estimate the same constant can be chosen. Then the first part follows using a null addition. Now let $f\in C^{s+\varepsilon}(\overline{I};X)$ and $g\in W^{s}_{p}(I;Y)$. Then a null addition yields

[TABLE]

By Tonelli and $I-x\subseteq[-|I|,|I|]$ the double integral can be estimated by $\|g\|_{L^{p}(I;Y)}^{p}\frac{2|I|^{\varepsilon p}}{\varepsilon p}$. $\square$

The following lemma yields embeddings for suitable intersection spaces:

Theorem A.5

Let $X_{0},X_{1},X$ be Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ with $X_{1}\hookrightarrow X\hookrightarrow X_{0}$ and $I\subseteq\mathbb{R}$ be an open interval. Assume that for a $\theta\in(0,1)$ the following interpolation inequality holds: $\|x\|_{X}\leq C_{0}\|x\|_{X_{0}}^{1-\theta}\|x\|_{X_{1}}^{\theta}$ for all $x\in X_{0}\cap X_{1}$. Then

If $1\leq p,p_{0},p_{1}\leq\infty$ with $\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$, then $L^{p_{0}}(I;X_{0})\cap L^{p_{1}}(I;X_{1})\hookrightarrow L^{p}(I;X)$. 2. 2.

If $1\leq p<\infty$, then $L^{p}(I;X_{1})\cap W^{1}_{p}(I;X_{0})\hookrightarrow B^{1-\theta}_{p,\infty}(I;X)$.

In both cases analogous interpolation inequalities hold with $C_{0}$ and $3C_{0}$, respectively.

Proof. The first part follows from Hölder’s inequality. Now let $f\in L^{p}(I;X_{1})\cap W^{1}_{p}(I;X_{0})$ and $0<h\leq 1$ be arbitrary. It remains to estimate $[f]_{B^{1-\theta}_{p,\infty}(I;X)}$. The first part yields

[TABLE]

Using $f(x+h)-f(x)=h\int_{0}^{1}f^{\prime}(x+th)\,dt$, Hölder’s inequality and Tonelli’s theorem we get $\|f(.+h)-f\|^{p}_{L^{p}(I_{h};X_{0})}\leq|h|^{p}\|f^{\prime}\|_{L^{p}(I;X_{0})}^{p}$. Thus the claim follows. $\square$

The following theorem characterizes the trace of functions in $\mathbb{E}_{1}(T)$ at $t=0$.

Theorem A.6

Let $X_{0},X_{1}$ Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ with $X_{1}\hookrightarrow X_{0}$ and let $1<p<\infty$. Then for $\mathbb{E}_{1}(\mathbb{R}_{+}):=L^{p}(\mathbb{R}_{+};X_{1})\cap W^{1}_{p}(0,\infty;X_{0})$ it holds that

[TABLE]

and an equivalent norm on $X_{\gamma}$ is given by the induced quotient norm. Moreover we have $\mathbb{E}_{1}(\mathbb{R}_{+})\hookrightarrow C_{b}^{0}(\mathbb{R}_{+};X_{\gamma})$ and $\mathbb{E}_{1}(T)\hookrightarrow C^{0}([0,T];X_{\gamma})$ for $0<T<\infty$ where the embedding constant is bounded independent of $T>0$ if we add $\|u(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm. In particular this also holds for $u(0)=0$.

Proof. Up to $\mathbb{E}_{1}(\mathbb{R}_{+})\hookrightarrow C_{b}^{0}(\mathbb{R}_{+};X_{\gamma})$ this directly follows from the trace method, cf. Lunardi [17], Section 1.2. The embedding $\mathbb{E}_{1}(T)\hookrightarrow C^{0}([0,T];X_{\gamma})$ can be shown by applying the one on $\mathbb{R}_{+}$ to suitable extensions. E.g. one can first use reflection at $T$ and then multiply by a cutoff function. It remains to show the property of the embedding constant. In case $u(0)=0$ we extend $u$ as before and for $u(0)\neq 0$ we subtract an $\tilde{u}\in\mathbb{E}_{1}(\mathbb{R}_{+})$ with $\tilde{u}(0)=u(0)$ such that $\|\tilde{u}\|_{\mathbb{E}_{1}(\mathbb{R}_{+})}\leq 2\|u(0)\|_{1-\frac{1}{p},p}^{\textup{tr}}$ holds and apply the first part to $u-\tilde{u}$. $\square$

By interpolation we get a whole scale of embeddings from Theorem A.6:

Theorem A.7

Let $X_{0},X_{1}$ Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ with $X_{1}\hookrightarrow X_{0}$. Moreover, let $1<p<\infty$ and $T>0$. Then for $0<\theta<1-\frac{1}{p}$ and $1\leq q\leq\infty$ it holds

[TABLE]

and the embedding constant is bounded independent of $T>0$ if we add $\|u(0)\|_{X_{\gamma}}$ in the $\mathbb{E}_{1}(T)$-norm, where $\mathbb{E}_{1}(T)$ and $X_{\gamma}$ are as in Theorem A.6.

Proof. It is well-known that $W^{1}_{p}(0,T;X_{0})\hookrightarrow C^{0,1-\frac{1}{p}}([0,T];X_{0})$ and the semi-norm is estimated by $\|u^{\prime}\|_{L^{p}(0,T;X_{0})}$ for all $u\in W^{1}_{p}(0,T;X_{0})$. Now one interpolates with the embedding from Theorem A.6 and uses the reiteration theorem in Lunardi [17], Corollary 1.24 and the interpolation inequality in [17], Corollary 1.7. $\square$

As another consequence of Theorem A.6 we obtain

Lemma A.8

Let $\Omega\subseteq\mathbb{R}^{n},n\geq 2$ be a bounded $C^{\infty}$-domain and $\Sigma\subseteq\Omega$ be a connected, compact and smooth hypersurface that separates $\Omega$ in two disjoint, connected domains $\Omega^{\pm}$ with $\Sigma=\partial\Omega^{+}$. Moreover, let $0<T<\infty,2\leq p<\infty$ and $\Omega_{0}:=\Omega^{+}\cup\Omega^{-}$. Then

[TABLE]

with $X_{1}:=W^{2}_{p}(\Omega_{0})\cap W^{1}_{p,0}(\Omega)$ and $Y_{\gamma}:=\{v\in L^{p}(\Omega):v|_{\Omega^{\pm}}\in W^{2-\frac{2}{p}}_{p}(\Omega^{\pm})\}\cap W^{1}_{p,0}(\Omega)$. The embedding constant is bounded independent of $T>0$ if we add $\|u(0)\|_{Y_{\gamma}}$ in the norm.

Proof. We define $X_{\gamma}:=(L^{p}(\Omega),X_{1})_{1-\frac{1}{p},p}$ and $\tilde{X}_{\gamma}:=(L^{p}(\Omega),W^{2}_{p}(\Omega_{0}))_{1-\frac{1}{p},p}$. Then Theorem A.6 implies $\mathbb{E}_{1}(T)\hookrightarrow C^{0}([0,T];X_{\gamma})$ and $\tilde{\mathbb{E}}_{1}(T):=L^{p}(0,T;W^{2}_{p}(\Omega_{0}))\cap W^{1}_{p}(0,T;L^{p}(\Omega))$ embeds to $C^{0}([0,T];\tilde{X}_{\gamma})$. The embedding constants are bounded in a similar way as in the theorem. E.g. from the $K$-method, cf. Lunardi [17], Section 1.1 we obtain

[TABLE]

with equivalent norms. Hence it remains to show that for the continuous representative of $u\in\mathbb{E}_{1}(T)$ additionally $u(t)\in W^{1}_{p,0}(\Omega)$ for all $t\in[0,T]$ holds. Therefore we use the characterization in Theorem A.6 to conclude $X_{\gamma}\hookrightarrow\tilde{X}_{\gamma}$. Here $X_{1}$ is dense in $X_{\gamma}$ by Lunardi [17], Proposition 1.17. For $v\in X_{1}$ it holds $\textup{\textlbrackdbl}v\textup{\textrbrackdbl}=\textup{tr}_{\Sigma}(v|_{\Omega^{-}})-\textup{tr}_{\Sigma}(v|_{\Omega^{+}})=0$ and $\textup{tr}_{\partial\Omega}\,v=0$. By density this carries over to all $v\in X_{\gamma}$ because of $X_{\gamma}\hookrightarrow\tilde{X}_{\gamma}\hookrightarrow W^{1}_{p}(\Omega_{0})$ for $2\leq p<\infty$ and the continuity of the respective boundary trace operators. Using the definition of weak derivative and partial integration as well as the trace characterization of $W^{1}_{p,0}(\Omega)$ we get $X_{\gamma}\subseteq\tilde{X}_{\gamma}\cap W^{1}_{p,0}(\Omega)=Y_{\gamma}$. $\square$

Finally, we need some properties of certain types of intersection spaces:

Lemma A.9

Let $\Omega\subseteq\mathbb{R}^{n}$ a bounded, connected domain, $0<T\leq T_{0}$ and $1\leq p<\infty$. Then $W_{T}:=C^{0}([0,T];C^{0}(\overline{\Omega}))\cap W^{1}_{p}(0,T;L^{p}(\Omega))$ is an algebra with pointwise multiplication and it holds $\|fg\|_{W_{T}}\leq C\|f\|_{W_{T}}\|g\|_{W_{T}}$ with $C>0$ independent of $T,T_{0}$ and $p$.

Moreover, for $f\in W_{T}$ with $f(t)(x)\geq c_{0}>0$ for all $(x,t)\in\overline{\Omega}\times[0,T]$ also $1/f\in W_{T}$ and we have $\partial_{t}(1/f)=-\partial_{t}f/f^{2}$. If additionally $\|f\|_{W_{T}}\leq R$, then $\|\frac{1}{f}\|_{W_{T}}\leq c_{R}$ with $c_{R}>0$ independent of $0<T\leq T_{0}$.

Proof. Using convolution one can show that $C^{\infty}([0,T];C^{0}(\overline{\Omega}))$ is dense in $W_{T}$. With this one can directly prove the first part and that a product rule holds. The second part follows by density utilizing Lemma A.3 and that the evaluation at any point in $\overline{\Omega}$ gives a bounded, linear functional on $C^{0}(\overline{\Omega})$. $\square$

Appendix B Maximal regularity for a two-phase Stokes system

Let $2\leq q<3$, $\Omega\subseteq\mathbb{R}^{n},n\geq 2$ be a bounded $C^{\infty}$-domain and $\Sigma\subseteq\Omega$ a compact, smooth and connected hypersurface that divides $\Omega$ into two disjoint and connected domains $\Omega^{\pm}$ with $\Sigma=\partial\Omega^{+}$ and outer unit normal $\nu_{\Sigma}$. We define $\Omega_{0}:=\Omega^{+}\cup\Omega^{-}$ and consider the following two-phase Stokes system

[TABLE]

where $\mu^{\pm}>0$ and $T(u,\tilde{q}):=2\mu^{\pm}\textup{sym}(\nabla u)-\tilde{q}I$ in $\Omega^{\pm}$. With the aid of Shimizu [20] we show an optimal existence result in an $L^{q}$-setting: For $(u,\tilde{q})$ we introduce the space $Y_{T}:=Y_{T}^{1}\times Y_{T}^{2}$ where

[TABLE]

Here we require mean value [math] for $\tilde{q}$ in order to get uniqueness later. This implies necessary conditions for the data $(f,g,a,u_{0})$ if (B.1)-(B.6) hold:

Lemma B.1

Let $2\leq q<3,T>0$ and $(u,\tilde{q})\in Y_{T}$. Then $(f,g,a,u_{0})$ defined by (B.1)-(B.6) is contained in $\tilde{Z}_{T}:=\tilde{Z}_{T}^{1}\times\tilde{Z}_{T}^{2}\times(\tilde{Z}_{T}^{3})^{n}\times Y_{\gamma}$ with $\tilde{Z}_{T}^{1}:=L^{q}(0,T;L^{q}(\Omega))^{n}$ and

[TABLE]

Additionally, the compatibility condition $\textup{div}\,u_{0}=g|_{t=0}$ in $W^{-1}_{q,(0)}(\Omega)$ is valid and we have $\|(f,g,a,u_{0})\|_{\tilde{Z}_{T}}\leq C(\|(u,\tilde{q})\|_{Y_{T}}+\|u_{0}\|_{Y_{\gamma}})$ with $C>0$ independent of $(u,\tilde{q})$ and $T>0$.

Remark B.2

The spaces are equipped with the natural norms. In particular we include the term $\|\textup{\textlbrackdbl}\tilde{q}\textup{\textrbrackdbl}\|_{\tilde{Z}_{T}^{3}}$ in the $Y_{T}^{2}$-norm and $\|\textup{tr}_{\Sigma}(g|_{\Omega^{+}})\|_{\tilde{Z}_{T}^{3}}$ in the $\tilde{Z}_{T}^{2}$-norm. One can directly verify that all spaces are Banach spaces.

Proof of Lemma B.1. The assertion for $f$ and $L^{q}$-in-time properties of $g$ can be directly shown. Lemma A.8 implies $u_{0}\in Y_{\gamma}$. Weak differentiability of $g$ follows from

[TABLE]

for all $\phi\in W^{1}_{q^{\prime}}(\Omega)$ and almost all $t\in(0,T)$. The latter identity also implies the compatibility condition. Results on mixed order Sobolev spaces yield together with $\textup{\textlbrackdbl}\tilde{q}\textup{\textrbrackdbl}\in\tilde{Z}^{3}_{T}$ that $a,\textup{tr}_{\Sigma}(g|_{\Omega^{+}})\in\tilde{Z}_{T}^{3}$ and the uniform estimate in the lemma can be shown by extending functions in time with suitable operators, cf. proof of Lemma 3.3, below (3.14). $\square$

Our maximal regularity result for (B.1)-(B.6) is

Theorem B.3

Let $2\leq q<3$ and $0<T\leq T_{0}$. Let $(f,g,a,u_{0})$ be contained in the space $Z_{T}:=\{(f,g,a,u_{0})\in\tilde{Z}_{T}:\textup{div}\,u_{0}=g|_{t=0}\text{ in }W^{-1}_{q,(0)}(\Omega)\}$. Then the Stokes system (B.1)-(B.6) has exactly one solution $(u,\tilde{q})$ in $Y_{T}$. Moreover, there is a $C>0$ independent of $(f,g,a,u_{0})$ and $0<T\leq T_{0}$ such that $\|(u,\tilde{q})\|_{Y_{T}}\leq C\|(f,g,a,u_{0})\|_{\tilde{Z}_{T}}$.

Remark B.4

Further necessary conditions are $\textup{tr}_{\partial\Omega}\,g\in W^{1-\frac{1}{q},\frac{1}{2}(1-\frac{1}{q})}_{q}(\partial\Omega_{T})$ and $\textup{tr}_{\Sigma}(g|_{\Omega^{-}})\in\tilde{Z}_{T}^{3}$. But these follow indirectly from the proof of Theorem B.3. 2. 2.

For simplicity we restricted to $2\leq q<3$. In principle also for other scales of $q$ statements are possible. But then e.g. other necessary conditions can arise. For instance, if $q>3$ another compatibility condition appears involving the jump of the stress tensor, cf. Shimizu [20].

Proof of Theorem B.3. The case $(a,g,u_{0})=0$ follows from the announced result by Shimizu [20], Theorem 2. The main step for the proof is the investigation of model problems. For $\mathbb{R}^{n}$ with $\mathbb{R}^{n-1}$ as interface cf. Shibata and Shimizu [19]. For simplicity we do not give a proof of Theorem 2 in [20]. But we reduce the general case to the above one similar to A. and Wilke [2], proof of Theorem A.1.

First of all, we reduce to the case $(g,u_{0})|_{\Omega^{+}}=0$. Therefore we consider

[TABLE]

and apply Theorem 1.1 in A. [1] (for $\Gamma_{1}=\emptyset,\Gamma_{2}=\Sigma$ and $f,a=0$ there). To this end we have to show the properties of $g|_{\Omega^{+}}$ and $u_{0}|_{\Omega^{+}}$. First it holds

[TABLE]

by definition of $\tilde{Z}_{T}^{2}$. Moreover, for $w\in L^{q}_{(0)}(\Omega)$ and $\phi\in W^{1}_{q^{\prime},0}(\Omega^{+})$

[TABLE]

where $e_{0}\phi\in W^{1}_{q^{\prime},0}(\Omega)$ is the extension by zero of $\phi$ and $\tilde{e}_{0}\phi:=e_{0}\phi-\frac{1}{|\Omega|}\int_{\Omega}(e_{0}\phi)(y)\,dy$. In particular this is also valid for $g(t)$ instead of $w$ for almost all $t\in(0,T)$. Since

[TABLE]

is bounded and linear, we obtain $g|_{\Omega^{+}}\in W^{1}_{q}(0,T;W^{-1}_{q}(\Omega^{+}))$, the norm is estimated by $C\|g\|_{\tilde{Z}_{T}^{2}}$ and for all $\phi\in W^{1}_{q^{\prime},0}(\Omega^{+})$ we have

[TABLE]

This yields $\textup{div}(u_{0}|_{\Omega^{+}})=(g|_{\Omega^{+}})|_{t=0}$ in $W^{-1}_{q}(\Omega^{+})$ because of $u_{0}\in Y_{\gamma}$ and the compatibility condition in $Z_{T}$. Now Theorem 1.1 in A. [1] yields a solution $v^{+}\in W^{2,1}_{q}(\Omega^{+}_{T})^{n}$ of (B.7)-(B.8) with

[TABLE]

where $C_{T_{0}}>0$ is independent of $0<T\leq T_{0}$. We extend $v^{+}$ to $\tilde{v}^{+}\in Y^{1}_{T}$ using a suitable extension operator. Because of Lemma B.1 we have reduced to the case $(g,u_{0})|_{\Omega^{+}}=0$.

Now we want to trace the latter one back to the case $(g,u_{0})=0$. Therefore we look at

[TABLE]

and again apply Theorem 1.1 in A. [1] (for $\Gamma_{1}=\partial\Omega^{-},\Gamma_{2}=\emptyset$ and $f,a=0$ there). Since $g|_{\Omega^{+}}=0$ it holds $g|_{\Omega^{-}}\in L^{q}(0,T;W^{1}_{q,(0)}(\Omega^{-}))$. Furthermore, for $w\in L^{q}_{(0)}(\Omega)$ with $w|_{\Omega^{+}}=0$ and all $\phi\in W^{1}_{q^{\prime}}(\Omega^{-})$ we have

[TABLE]

where $E:W^{1}_{q^{\prime}}(\Omega^{-})\rightarrow W^{1}_{q^{\prime}}(\Omega)$ is an extension operator and $\tilde{E}\phi:=E\phi-\frac{1}{|\Omega|}\int_{\Omega}(E\phi)(y)\,dy$. Since

[TABLE]

is bounded and linear, we obtain $g|_{\Omega^{-}}\in W^{1}_{q}(0,T;W^{1}_{q^{\prime}}(\Omega^{-})^{\prime})$, the norm is estimated by $C\|g\|_{\tilde{Z}_{T}^{2}}$ and for all $\phi\in W^{1}_{q^{\prime}}(\Omega^{-})$ it holds that

[TABLE]

Since $u_{0}|_{\Omega^{+}}=0$ and $u_{0}\in Y_{\gamma}$ it follows that $\textup{div}(u_{0}|_{\Omega^{-}})=(g|_{\Omega^{-}})|_{t=0}$ in $W^{1}_{q^{\prime}}(\Omega^{-})^{\prime}$ with the compatibility condition in $Z_{T}$. By Theorem 1.1 in A. [1] there exists a solution $v^{-}\in W^{2,1}_{q}(\Omega^{-}_{T})^{n}$ of (B.9)-(B.11) such that

[TABLE]

where $C_{T_{0}}>0$ is independent of $0<T\leq T_{0}$. Because of $v^{-}|_{\partial\Omega^{-}}=0$, we can extend $v^{-}$ by zero to $e_{0}(v^{-})\in Y^{1}_{T}$. Lemma B.1 implies that we have reduced to the case $(g,u_{0})=0$.

To trace this one back to the case $(g,a,u_{0})=0$ we define for any $a:\Sigma\rightarrow\mathbb{R}^{n}$ the normal and tangential component $a_{\nu}:=a\cdot\nu_{\Sigma}\,\nu_{\Sigma}$ and $a_{\tau}:=a-a_{\nu}$, respectively. First we reduce to the case $(g,a_{\tau},u_{0})=0$. As in A. [1], proof of Lemma 2.5, 2. there is an $A\in W^{2,1}_{q}(\Omega^{+}_{T})^{n}$ such that $\|A\|_{W^{2,1}_{q}(\Omega^{+}_{T})^{n}}\leq C\|a\|_{(\tilde{Z}_{T}^{3})^{n}}$ as well as

[TABLE]

But we also want $A$ to be divergence free. To this end one shows as in the proof of Lemma B.1 using $A|_{\Sigma}=0$ that $\textup{div}\,A\in L^{q}(0,T;W^{1}_{q,0}(\Omega^{+}))\cap W^{1}_{q}(0,T;W^{1}_{q^{\prime}}(\Omega^{+})^{\prime})$ and the norm is estimated by $C\|A\|_{W^{2,1}_{q}(\Omega^{+})^{n}}$. Now we apply the Bogovskiĭ-operator $B$ to $\textup{div}\,A$, cf. Geissert, Heck and Hieber [9], Theorem 2.5, and obtain

[TABLE]

since $\textup{div}\,A$ has mean value [math]. Additionally, the norm of $B(\textup{div}A)$ is estimated by $C\|A\|_{W^{2,1}_{q}(\Omega^{+})^{n}}$. Therefore the extension by zero to $\Omega$ of $\tilde{A}:=A-B(\textup{div}\,A)$ has the desired properties and we reduced to the case $(g,a_{\tau},u_{0})=0$.

Finally, to reduce the latter one to the case $(g,a,u_{0})=0$, we subtract a suitable extension of $a\cdot\nu_{\Sigma}\in\tilde{Z}_{T}^{3}$ from the pressure: By the trace theorem there is a

[TABLE]

We extend $p^{+}$ by [math] to $\Omega$ and subtract the mean value which does not alter the jump. Hence we obtain a $\tilde{p}^{+}\in Y_{T}^{2}$ with $\|\tilde{p}^{+}\|_{Y_{T}^{2}}\leq C\|a\|_{(\tilde{Z}_{T}^{3})^{n}}$ and $-\textup{\textlbrackdbl}\tilde{p}^{+}\textup{\textrbrackdbl}\,\nu_{\Sigma}=a_{\nu}$.

Altogether we reduced the general case to the case $(g,a,u_{0})=0$. $\square$