## 1 Introduction

We consider the incompressible sharp interface two-phase Navier-Stokes equations. To this end, let the hypersurface (interface) $$\Gamma (t)$$ divide $$\mathbb {R}^{n+1}$$ into two open domains $$\Omega _1(t)$$ and $$\Omega _2(t)=\mathbb {R}^{n+1}\setminus \overline{\Omega _1}(t)$$, $$i=1,2$$, occupied by two viscous incompressible immiscible capillary Newtonian fluids with constant densities $$\rho _i>0$$ and constant viscosities $$\mu _i>0$$, $$i=1,2$$. We set

\begin{aligned} \Omega (t):=\Omega _1(t)\cup \Omega _2(t) \end{aligned}

and with the indicator functions $$1_{\Omega _i}$$

\begin{aligned} \rho =\rho _1 1_{\Omega _1}+\rho _2 1_{\Omega _2},\quad \mu =\mu _1 1_{\Omega _1}+\mu _2 1_{\Omega _2}. \end{aligned}

Moreover, we denote by $$\nu (t,\cdot )$$ the normal field on $$\Gamma (t)$$ pointing form $$\Omega _1(t)$$ to $$\Omega _2(t)$$, by $$V(t,\cdot )$$ the normal velocity of the interface $$\Gamma (t)$$ and by $$\kappa (t,\cdot )$$ the mean curvature of $$\Gamma (t)$$ with respect to $$\nu (t,\cdot )$$. Then $$\kappa (t,x)$$ is negative when $$\Omega _1(t)$$ is convex close to $$x\in \Gamma (t)$$ and is for sufficiently smooth $$\Gamma (t)$$ given by

\begin{aligned} \kappa (t,\cdot )=-{\text{ div }}_\Gamma \nu (t,\cdot ) \end{aligned}

(note that this coincides with $$-{\text{ div }}\nu (t,\cdot )$$ if $$\nu (t,\cdot )$$ admits a differentiable extension to a neighborhood of $$\Gamma (t)$$). Finally, if v is defined and admits boundary traces on both domains $$\Omega _i(t)$$ then

\begin{aligned}{}[v]=(v|_{\Omega _2(t)}-v|_{\Omega _1(t)})|_{\Gamma (t)} \end{aligned}

denotes the jump of v accross $$\Gamma (t)$$. The two-phase Navier-Stokes equations with surface tension then read

\begin{aligned} \begin{aligned} \rho (\partial _t u+u\cdot \nabla u)-\mu \Delta u+\nabla q&=c&\text{ in } \Omega (t),\\ {\text{ div }}u&=0&\text{ in } \Omega (t),\\ -[S(u,q;\mu ) \nu ]&=\sigma \kappa \nu&\text{ on } \Gamma (t),\\ [u]&=0&\text{ on } \Gamma (t),\\ V&=u^\top \nu&\text{ on } \Gamma (t),\\ u(0)&=u_0&\text{ on } \Omega (0),\\ \Gamma (0)&=\Gamma _0, \end{aligned} \end{aligned}
(1)

with the velocity u, the pressure q, the stress tensor $$S(u,q;\mu )=-qI+\mu (\nabla u+\nabla u^\top )$$ and the surface tension coefficient $$\sigma >0$$. Here, c denotes some control.

The conditions on the interface ensure the balance between surface tension and the jump of the normal stress on the interface, the continuity of the velocity across the interface and the transport of the interface by the fluid velocity.

We note that the first four equations can be written in weak form on the whole domain by

\begin{aligned}&\int _{\mathbb {R}^{n+1}} \left( \partial _t (\rho u)+{\text{div}}(\rho u\otimes u)-c)^\top \varphi +S(u,q;\mu ):\nabla \varphi \right) \,dx =\int _{\Gamma (t)} \sigma \kappa \nu ^\top \varphi \,dS(x)\\\nonumber&\quad \forall \,\varphi \in C_c^1(\mathbb {R}^{n+1};\mathbb {R}^{n+1}), \end{aligned}
(2)
\begin{aligned}&\int _{\mathbb {R}^{n+1}} {\text{div}}(u) \,\psi \,dx=0\quad \forall \,\psi \in C_c^1(\mathbb {R}^{n+1}). \end{aligned}
(3)

Our aim is to study the differentiability properties of local solutions with respect to $$u_0$$ and c. To this end, we will work in an $$L_p$$-maximal regularity setting proposed in [22], see also [20, 23].

There exist several papers on the existence and uniqueness of local solutions for (1). In [8, 9, 24, 25] Lagrangian coordinates are used to obtain local well-posedness. Since this approach makes it difficult to establish smoothing of the unknown interface, [20, 22, 23] use a transformation to a fixed domain and are then able to show local well-posedness in an $$L_p$$ maximum regularity setting for the case $$c=0$$ [20, 22] or for the case of gravitation [23]. Moreover, they prove that the interface as well as the solution become instantaneously real analytic. Since we are considering a distributed control c of limited regularity, the instant analyticity is in general lost.

While optimal control problems for the Navier-Stokes equations have been studied by many researchers, see for example [12, 15, 19, 26], there are only a few contributions in the context of two-phase Navier-Stokes equations, mainly for phase-field formulations with semidiscretization in time. In [18] optimal boundary control of a time-discrete Cahn-Hilliard-Navier-Stokes system with matched densities is studied. By using regularization techniques, existence of optimal solutions and optimality conditions are derived. Analogous results for distributed optimal control with unmatched densities for the diffuse interface model of [1] have been obtained in [17]. Using the same model, [14] derive based on the stable time discretization proposed in [13] necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. Moreover, the differentiability of the control-to-state mapping for the semidiscrete problem is shown. Optimal control of a binary fluid described by its density distribution, but without surface tension, is studied in [4]. Different numerical approaches for the optimal control of two-phase flows are discussed in [5].

In this paper we derive differentiability results of the solution of the two-phase Navier-Stokes equations (1) with respect to controls. The results can be used to state optimality conditions and to justify the application of derivative based optimization methods. To the best of our knowledge, this is the first work providing differentiability properties of control-to-state mappings for sharp interface models of two-phase Navier-Stokes flow. The analysis is involved, since the moving interface renders a variational analysis difficult. Therefore it is beneficial, to first consider a transformed problem with fixed interface. However, since most numerical approaches are working in physical coordinates, we derive also differentiability results for the original problem. Since the normal derivative of the velocity is in general discontinuous at the interface, the sensitivities of the velocity are discontinuous across the interface. Moreover, the pressure is in general discontinuous at the interface and thus differentiability properties with respect to controls in strong spaces hold only away from the interface while at the interface differentiability properties can only be expected in the weak topology of measures. The same applies to phase indicators which are often used in Volume-of-Fluid (VoF)-type approaches. In order to obtain a PDE-formulation for the sensitivity equations, we work with a Volume-of-Fluid (VoF)-type formulation based on a discontinuous phase indicator and derive carefully a corresponding sensitivity equation.

We build on the quite recent existence and uniqueness results obtained for sufficiently small data by [22], see also [20, 23]. We consider first a formulation, where the interface is transformed to a hyperplane. By using $$L_p$$-maximal regularity of a linear system and applying a refined version of a fixed point theorem, we show differentiability of the transformed state with respect to controls in the maximum regularity spaces. A similar technique was recently used in [16] to show differentiability properties for shape optimization of fluid-structure interaction, but the analysis of the fixed point iteration is very different from two-phase flows considered here. In fact, the main difficulties in fluid-structure interaction arise from the coupling of a hyperbolic equation for the solid with the Navier-Stokes equations for the fluid while in two phase flows the moving interface and the surface tension are the main challenge. In a second step we deduce differentiability results for the control-to-state map in the physical coordinates. Finally, we derive an equivalent Volume-of-Fluid (VoF)-type formulation based on a discontinuous phase indicator that is governed by a multidimensional transport equation. By using the obtained differentiability results, we are able to justify a sensitivity system for the VoF-type formulation, which invokes measure-valued solutions of the linearized transport equation. This can be used as an analytical foundation to study the convergence of discrete sensitivity schemes for VoF-type methods. Moreover, we obtain the differentiability of objective functionals invoking the velocity field or the discontinuous pressure or phase indicator field and state formulas for the derivative.

The paper is organized as follows. In Sect. 2, the transformed problem is formulated. In Sect. 3, existence, uniqueness and differentiability of the control-to-state mapping is shown. The analysis starts in 3.1 for the transformed problem with flat interface. In 3.2 differentiability results for the original problem in physical coordinates are derived. In 3.3 the VoF-type formulation and its sensitivity equation are justified. In Sect. 4 we derive some analytical settings for the application of optimization methods. In 4.1 we consider objective functionals involving the velocity field and state differentiability results. Subsequently, we discuss in 4.2 objective functionals involving the pressure field or the phase indicator, obtain their differentiability with respect to controls as well as a formula for the derivative.

## 2 Transformation to a flat interface

In this paper, we consider as in Prüss and Simonett [22] the problem in $$n+1$$ dimensions, where $$\Gamma _0$$ is the graph of a sufficiently smooth function $$h_0:\mathbb {R}^n\rightarrow \mathbb {R}$$, i.e.,

\begin{aligned} \Gamma _0&=\left\{ (x,y)\in \mathbb {R}^n\times \mathbb {R}: y=h_0(x)\right\} ,\\ \Omega _1(0)&=\left\{ (x,y)\in \mathbb {R}^n\times \mathbb {R}: y<h_0(x)\right\} ,\\ \Omega _2(0)&=\left\{ (x,y)\in \mathbb {R}^n\times \mathbb {R}: y>h_0(x)\right\} . \end{aligned}

The interface has then the form

\begin{aligned} \Gamma (t)=\left\{ (x,h(t,x)):\, x\in \mathbb {R}^n \right\} , \end{aligned}

where $$h: [0,t_0]\times \mathbb {R}^n\rightarrow \mathbb {R}$$ with $$h(0,\cdot )=h_0$$ and $$t_0>0$$ is some final time. We note that the case of bounded fluid domains is considered in [20]. The analysis of this paper should also extend to this setting, but the presentation would be more technical.

If $$h(t,\cdot )$$ has second derivatives then normal and curvature of the interface $$\Gamma (t)$$ are given by

\begin{aligned} \begin{aligned} {\hat{\nu }}(t,x)=\nu (t,x,h(t,x))&=\frac{1}{\sqrt{1+|\nabla h(t,x)|^2}}\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) ,\\ {\hat{\kappa }}(t,x)=\kappa (t,x,h(t,x))&={\text{div}}_x \left( \frac{\nabla h(t,x)}{\sqrt{1+|\nabla h(t,x)|^2}}\right) =\Delta h-G_\kappa (h), \end{aligned} \end{aligned}
(4)

where $$\nabla h$$ and $$\Delta h$$ denote the gradient and Laplacian of h with respect to x and

\begin{aligned} G_\kappa (h)=\frac{|\nabla h|^2 \Delta h}{\left( 1+\sqrt{1+|\nabla h|^2}\right) \sqrt{1+|\nabla h|^2}}+ \frac{\nabla h^\top \nabla ^2 h \nabla h}{\left( 1+|\nabla h|^2 \right) ^{3/2}}. \end{aligned}
(5)

Following [22], we now transform the problem to $${\dot{\mathbb {R}}}^{n+1}=\{(x,y)\in \mathbb {R}^{n+1}: y\ne 0\}$$ with a flat interface at $$y=0$$ by using the transformation

\begin{aligned} {\hat{u}}(t,x,y)=\left( {\begin{array}{c}v(t,x,y)\\ w(t,x,y)\end{array}}\right) :=u \left( t,x,h(t,x)+y \right) , \pi (t,x,y):=q \left( t,x,h(t,x)+y \right) . \end{aligned}
(6)

Analogously, let with $$\mathbb {R}_\pm ^{n+1}=\{(x,y)\in \mathbb {R}^n\times \mathbb {R}:\, \pm y>0\}$$

\begin{aligned} {\hat{\rho }}(t,x,y)&=\rho (t,x,h(t,x)+y)=\chi _{\mathbb {R}_-^{n+1}}(x,y) \rho _1+\chi _{\mathbb {R}_+^{n+1}}(x,y)\rho _2,\\ {\hat{\mu }}(t,x,y)&=\mu (t,x,h(t,x)+y)=\chi _{\mathbb {R}_-^{n+1}}(x,y) \mu _1+\chi _{\mathbb {R}_+^{n+1}}(x,y) \mu _2. \end{aligned}

As in [22], we work with the following function spaces. Let $$\Omega \subset \mathbb {R}^m$$ be open and X be a Banach space. $$L_p(\Omega ;X)$$, $$H_p^s(\Omega ;X)$$, $$1\le p\le \infty$$, $$s\in \mathbb {R}$$, denote the X-valued Lebesgue and Bessel potential spaces of order s, respectively. We note that $$H_p^k(\Omega ;X)=W_p^k(\Omega ;X)$$ for $$k\in \mathbb {N}_0$$, $$1<p<\infty$$ with the Sobolev-Slobodetskiǐ spaces $$W_p^k$$. Moreover, we will use the fractional Sobolev-Slobodetskiǐ spaces $$W_p^s(\Omega ;X)$$, $$1\le p<\infty$$, $$s\in (0,\infty )\setminus \mathbb {N}$$, with norm

\begin{aligned} \Vert g\Vert _{W_p^s(\Omega ;X)}=\Vert g\Vert _{W_p^{[s]}(\Omega ;X)} +\sum _{|\alpha |=[s]}\left( \int _\Omega \int _\Omega \frac{\Vert \partial ^\alpha g(x)-\partial ^\alpha g(y)\Vert _X^p}{|x-y|^{m+(s-[s])p}}\,dx\,dy\right) ^{1/p} \end{aligned}

We recall that $$W_p^s(\Omega ;X)=B_{pp}^s(\Omega ;X)$$ for $$s\in (0,\infty )\setminus \mathbb {N}$$ with the Besov space $$B_{pp}^s$$. Finally, the homogeneous Sobolev space $$\dot{H}_p^1(\Omega )$$ is defined by

\begin{aligned} \dot{H}_p^1(\Omega ):=\left( \left\{ g\in L_{1,loc}(\Omega ): \Vert \nabla g\Vert _{L_p(\Omega )}<\infty \right\} ,\Vert \cdot \Vert _{\dot{H}_p^1(\Omega )}\right) ,\quad \Vert g\Vert _{\dot{H}_p^1(\Omega )}:=\Vert \nabla g\Vert _{L_p(\Omega ;\mathbb {R}^m)}. \end{aligned}

Then $$\dot{H}_p^1(\Omega )$$ is for connected $$\Omega$$ a Banach space if we factor out the constant functions and equip the resulting space with the corresponding quotient norm.

Finally, for $$\Omega \subset \mathbb {R}^m$$ open or closed we denote by $$BUC(\Omega ;X)$$ and $$BC(\Omega ;X)$$ the space of bounded uniformly continuous and the space of bounded continuous functions equipped with the supremum norm, respectively. Analogously, $$BUC^k(\Omega ;X)$$ and $$BC^k(\Omega ;X)$$, $$k\in \mathbb {N}_0$$, are defined for k-times continuously differentiable functions with bounded uniformly continuous or bounded continuous derivatives up to order k. If $$\Omega$$ is compact, we may briefly write $$C^k(\Omega ;X)$$, since boundedness und uniform continuity are automatically satisfied.

To state the transformed problem, we follow [22] and we use a fixed point formulation consisting of a linearized Stokes problem with nonlinear right hand side. In fact, define with $$J=(0,t_0)$$ and $$p>n+3$$ the space $$\mathbb {E}(t_0)$$ by

\begin{aligned} \nonumber \mathbb {E}_1(t_0)&= \left\{ {\hat{u}}\in H_p^1 \left( J;L_p \left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) \cap L_p \left( J;H_p^2 \left( {\dot{\mathbb {R}}}^{n+1},\mathbb {R}^{n+1}\right) \right) : [{\hat{u}}]=0 \right\} ,\\ \nonumber \mathbb {E}_2(t_0)&=L_p \left( J;\dot{H}_p^1 \left( {\dot{\mathbb {R}}}^{n+1}\right) \right) ,\\ \mathbb {E}_3(t_0)&=W_p^{1/2-1/(2p)} \left( J;L_p \left( \mathbb {R}^{n}\right) \right) \cap L_p \left( J;W_p^{1-1/p}\left( \mathbb {R}^{n}\right) \right) ,\\ \nonumber \mathbb {E}_4(t_0)&=W_p^{2-1/(2p)}\left( J;L_p \left( \mathbb {R}^{n}\right) \right) \cap H_p^1 \left( J;W_p^{2-1/p}\left( \mathbb {R}^{n}\right) \right) \\ \nonumber&\quad \cap W_p^{1/2-1/(2p)}\left( J;H_p^2 \left( \mathbb {R}^{n}\right) \right) \cap L_p \left( J;W_p^{3-1/p}\left( \mathbb {R}^{n}\right) \right) ,\\ \nonumber \mathbb {E}(t_0)&=\left\{ ({\hat{u}},\pi ,r,h)\in \mathbb {E}_1(t_0)\times \mathbb {E}_2(t_0)\times \mathbb {E}_3(t_0)\times \mathbb {E}_4(t_0):[\pi ]=r \right\}, \end{aligned}
(7)

and denote by

\begin{aligned} L ({\hat{u}},\pi ,r,h)=(f,f_d,g_v,g_w,g_h),~ ({\hat{u}}(0),h(0))=({\hat{u}}_0,h_0),~ ({\hat{u}},\pi ,r,h)\in \mathbb {E}(t_0) \end{aligned}
(8)

(i.e., $$r=[\pi ]$$ by the definition of $$\mathbb {E}(t_0)$$) the Stokes problem with free boundary

\begin{aligned} \begin{aligned} {\hat{\rho }} \partial _t {\hat{u}}-{\hat{\mu }} \Delta {\hat{u}}+\nabla \pi&=f&\text{ in } {\dot{\mathbb {R}}}^{n+1},\\ {\text{ div }}{\hat{u}}&= f_d&\text{ in } {\dot{\mathbb {R}}}^{n+1},\\ - [{\hat{\mu }} \partial _y v]-[{\hat{\mu}} \nabla _x w]&=g_v&\text{ on } \mathbb {R}^{n},\\ - 2[{\hat{\mu }} \partial _y w]+[\pi ]-\sigma \Delta h&=g_w&\text{ on } \mathbb {R}^{n},\\ [{\hat{u}}]&= 0&\text{ on } \mathbb {R}^{n},\\ \partial _t h-\gamma w&=g_h&\text{ on } \mathbb {R}^{n},\\ {\hat{u}}(0)={\hat{u}}_0,\quad h(0)&=h_0 \end{aligned} \end{aligned}
(9)

for $$t>0$$. Here, $$[{\hat{u}}]$$ denotes the jump across the transformed interface $$y=0$$ and $$\gamma w(x)=w(x,0)$$ denotes the trace of a function $$w: {\dot{\mathbb {R}}}^{n+1}\rightarrow \mathbb {R}$$ at $$y=0$$ satisfying $$[w]=0$$. Then it is shown in [22] that the transformation (6) leads to the following problem for $${\hat{u}}=(v,w), \pi , h$$

\begin{aligned} \begin{aligned} L ({\hat{u}},\pi ,[\pi ],h)&=\left( {\hat{c}}+F({\hat{u}},\pi ,h),F_d({\hat{u}},h),G_v({\hat{u}},[\pi ],h),G_w({\hat{u}},h),H({\hat{u}},h)\right) ,\\ ({\hat{u}}(0),h(0))&=({\hat{u}}_0,h_0), \end{aligned} \end{aligned}
(10)

where the right hand sides are given by $$F=(F_v,F_w)$$ and

\begin{aligned} \begin{aligned} F_v(v,w,\pi ,h)=&\,{\hat{\mu }} \left( -2(\nabla h\cdot \nabla _x)\partial _y v+ |\nabla h|^2 \partial _y^2 v-\Delta h \partial _y v\right) + \partial _y \pi \nabla h\\&+{\hat{\rho }} \left( -(v\cdot \nabla _x)v+(\nabla h^\top v)\partial _y v- w \partial _y v\right) +{\hat{\rho }} \partial _t h \partial _y v,\\ F_w(v,w,h)=&\,{\hat{\mu }} \left( -2(\nabla h\cdot \nabla _x)\partial _y w+ |\nabla h|^2 \partial _y^2 w-\Delta h \partial _y w\right) \\&+{\hat{\rho }} \left( -(v\cdot \nabla _x)w+(\nabla h^\top v)\partial _y w- w \partial _y w\right) +{\hat{\rho }} \partial _t h \partial _y w,\\ F_d(v,h) =&\, \nabla h^\top \partial _y v,\\ G_v(v,w,[\pi ],h)=&-\left[ {\hat{\mu }} (\nabla _x v+(\nabla _x v)^\top )\right] \nabla h+ |\nabla h|^2 [{\hat{\mu }} \partial _y v]+\left( \nabla h^\top [{\hat{\mu}} \partial _y v]\right) \nabla h\\&-[{\hat{\mu}} \partial _y w]\nabla h+ \left( [\pi ]-\sigma (\Delta h-G_\kappa (h))\right) \nabla h,\\ G_w(v,w,h)=&-\nabla h^\top [{\hat{\mu}} \partial _y v]- \nabla h^\top [{\hat{\mu}} \nabla _x w]+|\nabla h|^2 [{\hat{\mu }} \partial _y w]-\sigma G_\kappa (h),\\ H(v,w,h) =&-(\gamma v)^\top \nabla h. \end{aligned} \end{aligned}
(11)

Note that all terms except $$G_\kappa (h)$$ are polynomials in $$(v,w,\pi ,[\pi ],h)$$ and derivatives of $$(v,w,\pi ,h)$$. Moreover, all terms are linear with respect to second derivatives and $$G_\kappa (h)$$ is the pointwise superposition of a smooth function with $$\nabla h$$ and $$\nabla ^2 h$$.

### Remark 1

The transformed version of the deformation tensor $$D(u)=\nabla u+\nabla u^\top$$ is given by $${{\mathcal {D}}}({\hat{u}},h)={{\mathcal {D}}}(v,w,h)$$, where

\begin{aligned} {{\mathcal {D}}}({\hat{u}},h)=\nabla {\hat{u}}+\nabla {\hat{u}}^\top - \left( {\begin{array}{c}\nabla h \partial _y {\hat{u}}^\top \\ 0\end{array}}\right) -\left( {\begin{array}{c}\nabla h \partial _y {\hat{u}}^\top \\ 0\end{array}}\right) ^\top . \end{aligned}

Then the compatibility condition (14) can with $${\hat{\nu }}(0,x)=\frac{1}{\sqrt{1+|\nabla h_0(x)|^2}}\left( {\begin{array}{c}-\nabla h_0(x)\\ 1\end{array}}\right)$$ equivalently be written as

\begin{aligned} \begin{aligned}&[{\hat{\mu }} {{\mathcal {D}}}({\hat{u}}_0,h_0){\hat{\nu }}(0)-{\hat{\mu }}({\hat{\nu }}(0)^\top {{\mathcal {D}}}({\hat{u}}_0,h_0){\hat{\nu }}(0)){\hat{\nu }}(0)]=0,\\&{\text{ div }}{\hat{u}}_0=F_d({\hat{u}}_0,h_0),\quad [{\hat{u}}_0]=0. \end{aligned} \end{aligned}
(12)

## 3 Well-posedness and differentiability with respect to controls

### 3.1 Results for the transformed problem

By applying a fixed point theorem to (10), the following result is shown in [22] for $${\hat{c}}=0$$.

### Theorem 2

Let $$p>n+3$$ and consider the case $$c=0$$, i.e. $${\hat{c}}=0$$. Let

\begin{aligned} \mathbb {U}_{{\hat{u}}}:=W_p^{2-2/p}({\dot{\mathbb {R}}}^{n+1},\mathbb {R}^{n+1}),\quad \mathbb {U}_h:=W_p^{3-2/p}(\mathbb {R}^n) \end{aligned}
(13)

and let with $$J=(0,t_0)$$ the space $$\mathbb {E}(t_0)$$ be defined by (7).

Then for any $$t_0>0$$ there exists $${\hat{\varepsilon }}_0={\hat{\varepsilon }}_0(t_0)>0$$ such that for all initial values

\begin{aligned} ({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h \end{aligned}

satisfying, with $$u_0(x,h_0(x)+y)={\hat{u}}_0(x,y)$$, the compatibility conditions

\begin{aligned}{}[\mu D(u_0)\nu (0)-\mu (\nu (0)^\top D(u_0) \nu (0))\nu (0)]=0,\quad {\text{ div }}u_0=0,\quad [u_0]=0, \end{aligned}
(14)

as well as the smallness condition

\begin{aligned} \Vert {\hat{u}}_0\Vert _{\mathbb {U}_{{\hat{u}}}}+\Vert h_0\Vert _{\mathbb {U}_h}\le {\hat{\varepsilon }}_0 \end{aligned}

there exists a unique solution of the transformed problem (10) with

\begin{aligned} ({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0). \end{aligned}

Moreover, $$({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0)$$ depends continuously on $$({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h$$ satisfying (14).

### Proof

See [22, Thm. 6.3]. $$\square$$

Our first aim is to study the differentiability properties of the control-to-state map $$({\hat{u}}_0,{\hat{c}})\mapsto ({\hat{u}},\pi ,[\pi ],h)$$. Note that we consider also the case $${\hat{c}}\ne 0$$. The proof is carried out by an appropriate extension of the fixed point argument for (10) based on Theorem 7.

To apply the fixed point argument, the following $$L_p$$-maximum regularity result of [22] for the linearized problem (9) will be essential.

### Theorem 3

Let $$1<p<\infty$$ be fixed, $$p\ne 3/2,3$$ and assume that $$\rho _i,\mu _i$$ are positive constants. For arbitrary $$t_0>0$$ let $$J=(0,t_0)$$ and let $$\mathbb {E}_1(t_0),\ldots ,\mathbb {E}_4(t_0), \mathbb {U}_{{\hat{u}}},\mathbb {U}_h$$ be defined by (7), (26). Set

\begin{aligned} \begin{aligned} \mathbb {F}_1(t_0)&=L_p(J;L_p(\mathbb {R}^{n+1},\mathbb {R}^{n+1})),\\ \mathbb {F}_2(t_0)&=H_p^1(J;\dot{H}_p^{-1}(\mathbb {R}^{n+1}))\cap L_p(J;H_p^1({\dot{\mathbb {R}}}^{n+1})),\\ \mathbb {F}_3(t_0)&=W_p^{1/2-1/(2p)}(J;L_p(\mathbb {R}^{n},\mathbb {R}^{n+1}))\cap L_p(J;W_p^{1-1/p}(\mathbb {R}^{n},\mathbb {R}^{n+1})),\\ \mathbb {F}_4(t_0)&=W_p^{1-1/(2p)}(J;L_p(\mathbb {R}^{n}))\cap L_p(J;W_p^{2-1/p}(\mathbb {R}^{n})),\\ \mathbb {F}(t_0)&=\mathbb {F}_1(t_0)\times \mathbb {F}_2(t_0)\times \mathbb {F}_3(t_0)\times \mathbb {F}_4(t_0). \end{aligned} \end{aligned}
(15)

Then for all initial values $$({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h$$ and $$(f,f_d,g,g_h)\in \mathbb {F}(t_0)$$ satisfying the compatibility conditions

\begin{aligned} {\text{ div }}{\hat{u}}_0=f_d(0) ~~ \text{ on } {\dot{\mathbb {R}}}^{n+1},\quad [{\hat{u}}_0]=0 ~~&\text{ on } \mathbb {R}^n ~~\text{ if } p>3/2, \end{aligned}
(16)
\begin{aligned} \text{ and } \text{ in } \text{ addition }\quad [-{\hat{\mu }} \partial _y v_0]-[{\hat{\mu }} \nabla _x w_0]=g_v(0) ~~&\text{ on } \mathbb {R}^n ~~\text{ if } p>3, \end{aligned}
(17)

there exists a unique solution $$({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0)$$ of (9) and the solution map

\begin{aligned} (f,f_d,g,g_h,{\hat{u}}_0,h_0)\in \mathbb {F}(t_0)\times \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\mapsto ({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0) \end{aligned}

is continuous.

### Proof

This follows from [22, Thm. 5.1] and [22, Lem. 6.1, (e)]. $$\square$$

For homogeneous initial data we obtain immediately

### Corollary 4

Let $$p>3$$ and define in addition to $$\mathbb {E}(t_0)$$ and $$\mathbb {F}(t_0)$$ the spaces

\begin{aligned} {}_0\mathbb {E}(t_0)&:=\{({\hat{u}},\pi ,r,h)\in \mathbb {E}(t_0):~ {\hat{u}}(0)=0,~~ r(0)=0,~~ h(0)=0\},\\ {}_0\mathbb {F}(t_0)&:=\{(f,f_d,g,g_h)\in \mathbb {F}(t_0):~ f_d(0)=0,~~ g(0)=0,~~ g_h(0)=0\} \end{aligned}

with initial value 0 for all components that admit a trace at $$t=0$$. Then (9) has a unique and continuous solution map

\begin{aligned} (f,f_d,g,g_h,0,0)\in {}_0\mathbb {F}(t_0)\times \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\mapsto ({\hat{u}},\pi ,[\pi ],h)\in {}_0\mathbb {E}(t_0). \end{aligned}

The fixed point argument relies on the following properties of the right hand sides (11) of (10).

### Lemma 5

Let $$p>n+3$$ and set for $$({\hat{u}},\pi ,r,h)\in \mathbb {E}(t_0)$$

\begin{aligned} N({\hat{u}},\pi ,r,h):=\left( F({\hat{u}},\pi ,h),F_d({\hat{u}},h),G({\hat{u}},r,h),H({\hat{u}},h)\right) , \end{aligned}
(18)

with $$F=(F_v,F_w)$$, $$G=(G_v,G_w)$$, $$F_d$$ and H defined in (11). Then the mapping $$N: \mathbb {E}(t_0)\rightarrow \mathbb {F}(t_0)$$ is well defined and real analytic, more precisely,

\begin{aligned} N\in C^\omega \left( \mathbb {E}(t_0),\mathbb {F}(t_0)\right) ,\quad N(0)=0,\quad DN(0)=0. \end{aligned}

Moreover,

\begin{aligned} DN({\hat{u}},\pi ,r,h)\in {{\mathcal {L}}}\left( {}_{0} \mathbb {E}(t_0),{}_{0} \mathbb {F}(t_0)\right) \quad \forall \, ({\hat{u}},\pi ,r,h)\in \mathbb {E}(t_0). \end{aligned}

### Proof

See [22, Prop. 6.2] $$\square$$

Moreover, we will need the following analogue for the spaces of the initial values.

### Lemma 6

Let $$p>n+3$$, $$\mathbb {U}_{{\hat{u}}}, \mathbb {U}_h$$ be as in (13) and set

\begin{aligned} \mathbb {U}_{{\hat{u}},c}:=\{{\hat{u}}_0=(v_0,w_0) \in \mathbb {U}_{{\hat{u}}}: [{\hat{u}}_0]=0\}. \end{aligned}

Then with $$G=(G_v,G_w)$$ and H defined in (11) the mappings

\begin{aligned} ({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h&\mapsto v_0^\top \nabla h_0\in W^{2-2/p}_p({\dot{\mathbb {R}}}^{n+1}), \end{aligned}
(19)
\begin{aligned} ({\hat{u}}_0,h_0) \in \mathbb {U}_{{\hat{u}},c}\times \mathbb {U}_h&\mapsto H(v_0,h_0)\in W^{2-3/p}_p(\mathbb {R}^n), \end{aligned}
(20)
\begin{aligned} ({\hat{u}}_0,r_0,h_0)\in \mathbb {U}_{{\hat{u}}} \times W^{1-2/p}_p(\mathbb {R}^n) \times \mathbb {U}_h&\mapsto G({\hat{u}}_0,r_0,h_0)\in W^{1-2/p}_p(\mathbb {R}^n) \end{aligned}
(21)

are real analytic and the first derivatives vanish in $$({\hat{u}}_0,r_0,h_0)=0$$.

### Proof

Since $$p>n+3$$ we have $$W^{1-2/p}_p({\dot{\mathbb {R}}}^{n+1}) \hookrightarrow BUC({\dot{\mathbb {R}}}^{n+1})$$ and thus $$W^{s}_p({\dot{\mathbb {R}}}^{n+1})$$ is a multiplication algebra, i.e. a Banach algebra under the operation of multiplication, for all $$s\ge 1-2/p$$, see e.g. [6, Lem. 4.1, Rem. 6.4]. As a consequence, (19) is a continuous bilinear form and thus in $$C^\omega (\mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h,W^{2-2/p}_p({\dot{\mathbb {R}}}^{n+1}))$$.

Similarly, $$W^{s}_p(\mathbb {R}^{n})$$ is a multiplication algebra for all $$s\ge 1-2/p$$. Since the trace operator $${\hat{u}}_0 \in \mathbb {U}_{{\hat{u}},c} \mapsto \gamma v_0\in W^{2-3/p}_p(\mathbb {R}^n)$$ is continuous, (20) is a continuous bilinear form and thus real analytic.

Finally $$G({\hat{u}}_0,r_0,h_0)$$ in (21) is a polynomial in $$W^{1-2/p}_p(\mathbb {R}^n)$$-functions and in functions of the form $$\nabla h_0 / (a+(1+\nabla h_0^\top \nabla h_0)^{k/2})$$ with $$a\ge 0$$ and $$k\in \{1,2\}$$. The function $$\Psi : v\in \mathbb {R}^n \mapsto v/(a+(1+v^\top v)^{k/2})$$ is smooth with bounded derivatives and $$\Psi (0)=0$$. Since $$2-2/p>n/p$$ implies $$h_0\in \mathbb {U}_h \mapsto \nabla h_0\in W^{2-2/p}_p(\mathbb {R}^n)\hookrightarrow W^1_{(2-2/p)p}(\mathbb {R}^n)\cap BUC(\mathbb {R}^n)$$, it is well known that

\begin{aligned} h_0\in \mathbb {U}_h \mapsto \Psi (\nabla h_0)\in W^{2-2/p}_p(\mathbb {R}^n) \end{aligned}
(22)

is well defined and continuous, see [6, Thm. 1.1]. It is also differentiable. In fact, for $$d\in \mathbb {U}_h$$

\begin{aligned} \Psi (\nabla h_0+\nabla d)-\Psi (\nabla h_0)-\Psi '(\nabla h_0) \nabla d =\int _0^1 \left( \Psi '(\nabla h_0+\tau \nabla d)-\Psi '(\nabla h_0)\right) \nabla d\,d\tau, \end{aligned}

where the integrand is in $$BUC([0,1]\times \mathbb {R}^n)$$. Moreover, since $$v\mapsto \Psi '(\nabla v)-\Psi '(0)$$ is smooth with bounded derivatives and vanishes at 0, the integrand is continuous from $$[0,1]\rightarrow W^{2-2/p}_p(\mathbb {R}^n)$$ again by [6, Thm. 1.1]. Hence the integral is also a Bochner integral and thus by using the multiplication algebra property there is $$C>0$$ with

\begin{aligned}&\Vert \Psi (\nabla h_0+\nabla d)-\Psi (\nabla h_0)-\Psi '(\nabla h_0) \nabla d\Vert _{W^{2-2/p}_p(\mathbb {R}^n)}\\&\le C \int _0^1 \Vert \Psi '(\nabla h_0+\tau \nabla d)-\Psi '(\nabla h_0)\Vert _{W^{2-2/p}_p(\mathbb {R}^n)}\,d\tau \Vert \nabla d\Vert _{W^{2-2/p}_p(\mathbb {R}^n)} =o(\Vert d\Vert _{\mathbb {U}_h}), \end{aligned}

since $$d\in \mathbb {U}_h \mapsto \Psi '(\nabla h_0+\nabla d)-\Psi '(0)\in W^{2-2/p}_p(\mathbb {R}^n;\mathbb {R}^n)$$ is continuous at $$d=0$$ by [6, Thm. 1.1]. Now we can iteratively show that (22) is real analytic. In fact, we can write $$d\in \mathbb {U}_h \mapsto \Psi '(\nabla h_0) \nabla d=(\Psi '(\nabla h_0)-\Psi '(0)) \nabla d+\Psi '(0) \nabla d\in W^{2-2/p}_p(\mathbb {R}^n)$$. The second term is a constant mapping in $${{\mathcal {L}}}(\mathbb {U}_h,W^{2-2/p}_p(\mathbb {R}^n))$$. Moreover, as before $$h_0\in \mathbb {U}_h \mapsto \Psi '(\nabla h_0)-\Psi '(0)\in W^{2-2/p}_p(\mathbb {R}^n;\mathbb {R}^n)$$ is well defined and continuous [6, Thm. 1.1] and by the same arguments as above also differentiable. Iterating the argument shows that (22) is real analytic.

We conlude that (21) is a polynomial in $$W^{1-2/p}_p(\mathbb {R}^n)$$-functions and in real analytic functions of the form (22). Since $$W^{1-2/p}_p(\mathbb {R}^n)$$ is a multiplication algebra, we conclude that (21) is real analytic.

By the product structure of (19)–(21) the first derivatives vanish in 0. $$\square$$

We will work with the following extension of Banach’s fixed point theorem.

### Theorem 7

1. a)

Let UWZ be real Banach spaces, let $$A\in {{\mathcal {L}}}(Z,W)$$ be an isomorphism and set $$M:=\Vert A^{-1}\Vert _{{{\mathcal {L}}}(W,Z)}$$. Let $$B_Z\subset Z$$ be a nonempty closed convex set and $$B_U\subset U$$ be a nonempty set. Moreover, let $$K: B_Z\times B_U \rightarrow W$$ be Lipschitz continuous with

\begin{aligned} \Vert K(z,u)-K({\tilde{z}},{\tilde{u}})\Vert _{W}\le L_z \Vert z-{\tilde{z}}\Vert _Z+L_u \Vert u-{\tilde{u}}\Vert _U \quad \forall \, (z,u), ({\tilde{z}},{\tilde{u}})\in B_Z\times B_U \end{aligned}

and assume that

\begin{aligned} A^{-1} K(z,u)\in B_Z\quad \forall \, (z,u)\in B_Z\times B_U\quad \text{ and }\quad M L_z<1. \end{aligned}
(23)

Then for all $$u\in B_U$$ the equation

\begin{aligned} Az=K(z,u) \end{aligned}

has a unique solution $$z=z(u)\in B_Z$$ and

\begin{aligned} \Vert z(u)-z({\tilde{u}})\Vert _Z \le \frac{L_u M}{1-M L_z} \Vert u-{\tilde{u}}\Vert _U\quad \forall \, u,{\tilde{u}}\in B_U. \end{aligned}
(24)
2. b)

Assume in addition that $$B_U$$ is a relatively open convex subset of $$u^*+U_L\subset U$$, where $$U_L$$ is a closed linear subspace of U ($$U_L=U$$ is admitted, then $$B_U\subset U$$ is convex and open), and that $$K: B_Z\times B_U \rightarrow W$$ is Fréchet differentiable. Then $$u\in B_U\mapsto z(u)\in Z$$ is Fréchet differentiable, where $$\delta z_d:= Dz(u) d$$ is for any $$d\in U_L$$ the unique solution of the problem

\begin{aligned} A \delta z_d=D_z K(z(u),u) \delta z_d+D_u K(z(u),u) d. \end{aligned}
(25)

If $$DK: B_Z\times B_U \rightarrow {{\mathcal {L}}}(Z\times U_L,W)$$ is Lipschitz continuous then also $$Dz: B_U\rightarrow {{\mathcal {L}}}(U_L,Z)$$ is Lipschitz continuous. If $$K: B_Z\times B_U \rightarrow W$$ is k-times Fréchet differentiable then $$u\in B_U\mapsto z(u)\in Z$$ is k-times Fréchet differentiable and if $$D^k K$$ is Lipschitz continuous on $$B_Z\times B_U$$ then $$D^k z$$ is Lipschitz continuous on $$B_U$$.

### Proof

a): By assumption the mapping $$T: (z,u)\in B_Z\times B_U \mapsto A^{-1} K(z,u)\in B_Z$$ is well defined and Lipschitz continuous with Lipschitz constants $$M L_z<1$$ with respect to z and $$M L_u$$ with respect to u. Hence, for all $$u\in B_U$$ there exists a unique fixed point $$z=z(u)$$ with $$z=T(z,u)$$ by Banach’s fixed point theorem.

For $$u,{\tilde{u}}\in B_U$$ we obtain

\begin{aligned} \Vert z(u)-z({\tilde{u}})\Vert _Z&=\Vert T(z(u),u)-T(z({\tilde{u}}),{\tilde{u}})\Vert _W\\&\le M L_z \Vert z(u)-z({\tilde{u}})\Vert _Z+ M L_u \Vert u-{\tilde{u}}\Vert _U \end{aligned}

and thus (24).

b): Now let in addition $$B_U$$ be relatively open in the closed affine subspace $$u^*+U_L$$. Moreover, let $$K: B_Z\times B_U \rightarrow W$$ be Fréchet differentiable and let $$u\in B_U$$ be arbitrary. Then $$\Vert D_z K(z(u),u)\Vert _{{{\mathcal {L}}}(Z,W)}\le L_z$$ and $$\Vert D_u K(z(u),u)\Vert _{{{\mathcal {L}}}(U_L,W)}\le L_u$$ and thus for any $$d\in U_L$$ the linear problem (25) has by Banach’s fixed point theorem a unique solution $$\delta z_d\in Z$$.

Since $$B_U$$ is relatively open in $$u^*+U_L$$, we find $$\delta >0$$ such that $$u+d\in B_U$$ for all $$d\in U_L$$ with $$\Vert d\Vert _U<\delta$$. Then

\begin{aligned}&A (z(u+d)-z(u)-\delta z_d) \\&=K(z(u+d),u+d)-K(z(u),u)-D_z K(z(u),u) \delta z_d-D_u K(z(u),u) d\\&=D_z K(z(u),u) (z(u+d)-z(u)-\delta z_d)+o_W(\Vert z(u+d)-z(u)\Vert _Z+\Vert d\Vert _U). \end{aligned}

Here, the Landau symbol $$o_W$$ indicates, that the term is considered in the space W. By using (24) we conclude that for $$d\in U_L$$, $$\Vert d\Vert _U\rightarrow 0$$

\begin{aligned} \Vert z(u+d)-z(u)-\delta z_d\Vert _Z \le \frac{M L_u}{1-M L_z} o_Z(\Vert d\Vert _U)=o_Z(\Vert d\Vert _U). \end{aligned}

If $$DK: B_Z\times B_U \rightarrow {{\mathcal {L}}}(Z\times U_L,W)$$ is Lipschitz continuous then (25) can be written as

\begin{aligned} A \delta z_d=K^{(1)}(\delta z_d,u;d), \end{aligned}

where $$K^{(1)}(\cdot ,\cdot ;d): Z\times B_U\rightarrow W$$ has for all $$d\in U_L$$, $$\Vert d\Vert _U\le 1$$ the Lipschitz constant $$L_z$$ with respect to $$\delta z_d$$ and a uniform Lipschitz constant with respect to u. Applying the first part of the theorem again yields that $$Dz: B_U \rightarrow {{\mathcal {L}}}(U_L,Z)$$ is Lipschitz continuous.

Repeating the argument for higher derivatives concludes the proof. $$\square$$

By applying this result to (8)–(10), we obtain the following extension of Theorem 2.

### Theorem 8

Let $$p>n+3$$ and consider any $$t_0>0$$. Let $$\mathbb {E}(t_0), \mathbb {F}(t_0)$$ be defined as in (7) and (15) and set with $$J=(0,t_0)$$

\begin{aligned} \begin{aligned}&\mathbb {U}_{{\hat{u}}}:=W_p^{2-2/p}({\dot{\mathbb {R}}}^{n+1},\mathbb {R}^{n+1}),\quad \mathbb {U}_h:=W_p^{3-2/p}(\mathbb {R}^n),\\&\mathbb {U}_{{\hat{c}}}(t_0):=\mathbb {F}_1(t_0)=L_p(J;L_p(\mathbb {R}^{n+1},\mathbb {R}^{n+1})). \end{aligned} \end{aligned}
(26)

Then for any $$t_0>0$$ there exists $${\hat{\varepsilon }}_0={\hat{\varepsilon }}_0(t_0)>0$$ such that for all data

\begin{aligned} ({\hat{u}}_0,h_0,{\hat{c}})\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h \times \mathbb {U}_{{\hat{c}}}(t_0) \end{aligned}

satisfying the compatibility condition (12) (or equivalently (14) with $$u_0(x,h_0(x)+y)={\hat{u}}_0(x,y)$$) as well as the smallness condition

\begin{aligned} \Vert {\hat{u}}_0\Vert _{\mathbb {U}_{{\hat{u}}}}+\Vert h_0\Vert _{\mathbb {U}_h}+\Vert {\hat{c}}\Vert _{\mathbb {U}_{{\hat{c}}}(t_0)}< {\hat{\varepsilon }}_0 \end{aligned}
(27)

there exists a unique solution of the transformed problem (10) with

\begin{aligned} ({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0). \end{aligned}

Moreover, the mapping

\begin{aligned} \{ ({\hat{u}}_0,h_0,{\hat{c}})\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\times \mathbb {U}_{{\hat{c}}}(t_0): ({\hat{u}}_0,h_0,{\hat{c}}) \text{ satisfy } \text{(12), } \text{(27) }\} \mapsto ({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0) \end{aligned}

is continuous and infinitely many times differentiable with respect to $$({\hat{u}}_0,{\hat{c}})$$.

### Proof

We extend the arguments in [22] and apply Theorem 7 to the transformed formulation (10).

Let $$z=({\hat{u}},\pi ,r,h) \in \mathbb {E}(t_0)$$ and write (10)

\begin{aligned} Lz=N(z)+({\hat{c}},0),\quad ({\hat{u}}(0),h(0))=({\hat{u}}_0,h_0), \end{aligned}
(28)

where N is defined in (18).

Let $$({\hat{u}}_0,h_0)$$ satisfy (12) and (27), where $${\hat{\varepsilon }}_0$$ will be adjusted later.

Following [22], we first construct $$z^*=z^*({\hat{u}}_0,h_0)\in \mathbb {E}(t_0)$$ that satisfies the equation

\begin{aligned} Lz^*=\left( 0,f_d^*,g^*,g_h^* \right) ,\quad \left( {\hat{u}}^*(0),h^*(0)\right) =({\hat{u}}_0,h_0), \end{aligned}
(29)

where $$(0,f_d^*,g^*,g_h^*)\in \mathbb {F}(t_0)$$ resolves the compatibility conditions (16), (17). Then we can write (28) equivalently as

\begin{aligned} L {\tilde{z}}=N \left( {\tilde{z}}+z^*({\hat{u}}_0,h_0)\right) +({\hat{c}},0)-L z^*({\hat{u}}_0,h_0)=:K({\tilde{z}};{\hat{u}}_0,h_0,{\hat{c}}),~~ {\tilde{z}}\in {}_0\mathbb {E}(t_0). \end{aligned}
(30)

The construction of $$z^*$$ can be accomplished as in [22]. Set

\begin{aligned} r_0({\hat{u}}_0,h_0)=[\pi _0]:=\left[ {\hat{\mu }} {\hat{\nu }}(0)^\top {{\mathcal {D}}}({\hat{u}}_0,h_0) {\hat{\nu }}(0)\right] +\sigma (\Delta h_0-G_\kappa (h_0)). \end{aligned}

The right hand side consists of several terms of $$G({\hat{u}}_0,0,h_0)$$ in (21) and thus Lemma 6 yields that the above mapping $$({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\mapsto [\pi _0]=r_0({\hat{u}}_0,h_0)\in W^{1-2/p}_p(\mathbb {R}^n)$$ is real analytic. Moreover, it is easy to check that the compatibility conditions hold

\begin{aligned} \begin{aligned} - [{\hat{\mu }} \partial _y v_0]-[{\hat{\mu}} \nabla _x w_0]&=G_v({\hat{u}}_0,[\pi _0],h_0)&\text{ on } \mathbb {R}^{n},\\ - 2[{\hat{\mu }} \partial _y w_0]+[\pi _0]-\sigma \Delta h_0&=G_w({\hat{u}}_0,h_0)&\text{ on } \mathbb {R}^{n}.\\ \end{aligned} \end{aligned}
(31)

Now let $$D_{n}=-\Delta$$ be the Laplacian in $$L_p(\mathbb {R}^{n})$$ with domain $$H_p^2(\mathbb {R}^{n})$$ and set

\begin{aligned} g^*(t):= e^{-t D_n} G({\hat{u}}_0,r_0({\hat{u}}_0,h_0),h_0),\quad g_h^*(t):=e^{-t D_n} H({\hat{u}}_0,h_0). \end{aligned}

By the real analyticity of $$r_0({\hat{u}}_0,h_0)$$ and Lemma 6 the mappings

\begin{aligned} ({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h&\mapsto G({\hat{u}}_0,r_0({\hat{u}}_0,h_0),h_0)\in W^{1-2/p}_p(\mathbb {R}^n),\\ ({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h&\mapsto H({\hat{u}}_0,h_0)\in W^{2-3/p}_p(\mathbb {R}^n) \end{aligned}

are real analytic. Now maximal $$L_p$$-regularity for $$D_n$$ yields, see e.g. [11, Lem. 8.2]

\begin{aligned}&g^*\in H_p^1 \left( J;W^{-1-1/p}_p \left( \mathbb {R}^{n}\right) \right) \cap L_p \left( J;W^{1-1/p}_p \left( \mathbb {R}^{n}\right) \right) \hookrightarrow \mathbb {F}_3(t_0),\\&g_h^*\in H_p^1 \left( J;W^{-1/p}_p \left( \mathbb {R}^{n}\right) \right) \cap L_p \left( J;W^{2-1/p}_p \left( \mathbb {R}^{n}\right) \right) \hookrightarrow \mathbb {F}_4(t_0), \end{aligned}

where the imbeddings follow by real interpolation and $$g^*, g_h^*$$ are real analytic in $$({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h$$. (31) ensures that (17) holds for $$g^*$$. Next, let

\begin{aligned} c_d^*(t)={\left\{ \begin{array}{ll} {{\mathcal {R}}}_+ e^{-t D_{n+1}} {{\mathcal {E}}}_+ v_0^\top \nabla h_0 &{} \text{ in } \mathbb {R}_+^{n+1},\\ {{\mathcal {R}}}_- e^{-t D_{n+1}} {{\mathcal {E}}}_- v_0^\top \nabla h_0 &{} \text{ in } \mathbb {R}_-^{n+1}, \end{array}\right. } \end{aligned}

where $${{\mathcal {E}}}_\pm \in {{\mathcal {L}}}(W_p^{2-2/p}(\mathbb {R}_\pm ^{n+1}),W_p^{2-2/p}(\mathbb {R}^{n+1})))$$ are extension operators and $${{\mathcal {R}}}_\pm$$ are the restrictions to $$\mathbb {R}_\pm ^{n+1}$$. Now $$({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h \mapsto v_0^\top \nabla h_0\in W_p^{2-2/p}({\dot{\mathbb {R}}}^{n+1})$$ is by Lemma 6 real analytic. By $$L_p$$-regularity for $$D_{n+1}$$ $$c_d^*\in H_p^1(J;L_p(\mathbb {R}^{n+1}))\cap L_p(J;H_p^2({\dot{\mathbb {R}}}^{n+1}))$$ and thus

\begin{aligned} f_d^*:=\partial _y c_d^*\in \mathbb {F}_2(t_0)\quad \text{ with }\quad f_d^*(0)=F_d(v_0,h_0) \end{aligned}

is real analytic with respect to $$({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h$$. Hence, also (16) holds for $$f_d^*$$ and we conclude that $$R^*:=(0,f_d^*,g^*,g_h^*)\in \mathbb {F}(t_0)$$ satisfies the compatibility conditions (16), (17) and by construction $$({\hat{u}}_0,h_0)\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\mapsto R^*\in \mathbb {F}(t_0)$$ is real analytic. Hence, by Theorem 3 the linear problem (29) has a unique solution $$z^*=z^*({\hat{u}}_0,h_0)$$ that is real analytic and by Lemma 6 the first derivative vanishes in 0, i.e., $$Dz^*(0,0)=0$$.

Now consider (30). By construction of $$z^*$$ the right hand side of (30) is in $${}_0\mathbb {F}(t_0)$$. Denote by $$L_0 \in {{\mathcal {L}}}({}_0\mathbb {E}(t_0),{}_0\mathbb {F}(t_0))$$ the restriction of L which is an isomorphism by Corollary 4. Hence, (30) can be written as

\begin{aligned} L_0 {\tilde{z}}=N({\tilde{z}}+z^*({\hat{u}}_0,h_0)) +({\hat{c}},0)-L z^*({\hat{u}}_0,h_0)=:K({\tilde{z}};{\hat{u}}_0,h_0,{\hat{c}}),~~ {\tilde{z}}\in {}_0\mathbb {E}(t_0). \end{aligned}
(32)

To apply Theorem 7 we set now with suitable $${\hat{\varepsilon }}_0>0$$ and $$\delta >0$$

\begin{aligned} B_U({\hat{\varepsilon }}_0)&:= \left\{ ({\hat{u}}_0,h_0,{\hat{c}})\in \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\times \mathbb {U}_{{\hat{c}}}(t_0):\, ({\hat{u}}_0,h_0,{\hat{c}}) \text{ satisfy } \text{(12), } \text{(27) }\right\} ,\\ B_Z(\delta )&:= \left\{ {\tilde{z}}\in {}_0\mathbb {E}(t_0):\, \Vert \tilde{z}\Vert _{{}_0\mathbb {E}(t_0)}\le \delta \right\} , \end{aligned}

where $${\hat{\varepsilon }}_0,\delta >0$$ will be adjusted later.

Let $$M=\Vert L_0^{-1}\Vert _{{{\mathcal {L}}}({}_0\mathbb {F}(t_0),{}_0\mathbb {E}(t_0))}$$. We know by Lemma 5 and the properties of $$z^*$$ that the right hand side

\begin{aligned} ({\tilde{z}},{\hat{u}}_0,h_0,{\hat{c}})\in {}_0\mathbb {E}(t_0)\times \mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\times \mathbb {U}_{{\hat{c}}}(t_0) \mapsto K({\tilde{z}};{\hat{u}}_0,h_0,{\hat{c}})\in \mathbb {F}(t_0) \end{aligned}
(33)

is real analytic with

\begin{aligned} K(0)=0,\quad D_{({\tilde{z}},{\hat{u}}_0,h_0)} K(0)=0. \end{aligned}

Hence, the Lipschitz constant $$L_z$$ of K with respect to $${\tilde{z}}$$ is arbitrary small close to 0 and the Lipschitz constant of K with respect to $$({\hat{u}}_0,h_0,{\hat{c}})$$ is $$L_u=2$$ close enough to 0 (note that the Lipschitz constant with respect to $${\hat{c}}$$ is 1). Hence, if we set $$\delta =4 M {\hat{\varepsilon }}_0$$ then for $${\hat{\varepsilon }}_0$$ small enough K has the Lipschitz constants $$L_z=1/(2M)$$ and $$L_u=2$$ on $$B_Z(\delta )\times B_U({\hat{\varepsilon }}_0)$$. Hence, for all $$({\tilde{z}},{\hat{u}}_0,h_0,{\hat{c}})\in B_Z(\delta )\times B_U({\hat{\varepsilon }}_0)$$

\begin{aligned} \begin{aligned}&\Vert L_0^{-1} K({\tilde{z}};{\hat{u}}_0,h_0,{\hat{c}})\Vert _{{}_0\mathbb {E}(t_0)} \\ {}&\le M L_z \Vert {\tilde{z}}\Vert _{{}_0\mathbb {E}(t_0)} + M L_u (\Vert {\hat{u}}_0\Vert _{\mathbb {U}_{{\hat{u}}}}+\Vert h_0\Vert _{\mathbb {U}_h}+\Vert {\hat{c}}\Vert _{\mathbb {U}_{{\hat{c}}}(t_0)}) < \frac{1}{2} \delta +2M {\hat{\varepsilon }}_0 = \delta . \end{aligned} \end{aligned}
(34)

Thus, (23) is satisfied and (32) has by Theorem 7 for all $$({\hat{u}}_0,h_0,{\hat{c}})\in B_U({\hat{\varepsilon }}_0)$$ a unique solution $${\tilde{z}}={\tilde{z}}({\hat{u}}_0,h_0,{\hat{c}})\in B_Z(\delta )$$ satisfying the Lipschitz stability (24). Since also the real analytic operator $$z^*({\hat{u}}_0,h_0)\in \mathbb {E}(t_0)$$ is Lipschitz continuous on $$B_U({\hat{\varepsilon }}_0)$$, the solution $$z({\hat{u}}_0,h_0,{\hat{c}})={\tilde{z}}+z^*\in \mathbb {E}(t_0)$$ is unique and Lipschitz continuous on $$B_U({\hat{\varepsilon }}_0)$$.

Now let $$({\hat{u}}_0^*,h_0^*,{\hat{c}}^*)\in B_U({\hat{\varepsilon }}_0)$$ be arbitrary. Then $$\{({\hat{u}}_0,h_0^*,{\hat{c}})\in B_U({\hat{\varepsilon }}_0)\}$$ is a relatively open subset of an affine subspace of $$\mathbb {U}_{{\hat{u}}}\times \mathbb {U}_h\times \mathbb {U}_{{\hat{c}}}(t_0)$$. Since (33) is real analytic, it follows from Theorem 7, b) that $${\tilde{z}}({\hat{u}}_0,h_0^*,{\hat{c}})\in {}_0\mathbb {E}(t_0)$$ is infinitely many times differentiable with respect to $$({\hat{u}}_0,{\hat{c}})$$ and the same holds for $$z({\hat{u}}_0,h_0^*,{\hat{c}})={\tilde{z}}+z^*\in \mathbb {E}(t_0)$$. $$\square$$

### 3.2 Results for the original problem

We transfer now the results of Theorem 8 for the transformed problem (10) to the original problem (1). To this end, we define for $$h_0\in \mathbb {U}_h$$ the spaces

\begin{aligned} \begin{aligned}&\mathbb {U}_{u}(h_0):=W_p^{2-2/p}\left( \mathbb {R}^{n+1}\setminus \Gamma (0),\mathbb {R}^{n+1}\right) ,\quad \mathbb {U}_{c}(t_0):=L_p \left( J;H^1_p \left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) . \end{aligned} \end{aligned}
(35)

Since the pressure $$\pi (t,\cdot )$$ is only determined up to a constant, we select from now on without restriction the unique representative satisfying (note that $$[\pi ]$$ is uniquely determined in (10))

\begin{aligned} \pi \in \mathbb {E}_1(t_0),\quad \int _{[-1,1]^n} \pi (t,x,0-)\,dx=0\quad \text{ for } \text{ a.a. } t\in (0,t_0). \end{aligned}
(36)

Then with the convention (36) and by the trace theorem, we find a Poincaré constant $$C_P>0$$ with

\begin{aligned} \Vert \pi (t,\cdot )\Vert _{H_p^1({\dot{\mathbb {R}}}^{n+1})}\le C_P \left( \left\| \pi (t,\cdot )\right\| _{\dot{H}_p^1({\dot{\mathbb {R}}}^{n+1})}+\left\| [\pi ]\right\| _{W_p^{1-1/p}(\mathbb {R}^n)}\right). \end{aligned}
(37)

The following imbeddings will be useful.

### Lemma 9

Let $$p>n+3$$. Then the following imbeddings hold with $$J=(0,t_0)$$, $$t_0>0$$.

\begin{aligned}&\mathbb {E}_1(t_0) \hookrightarrow C \left( {\bar{J}};BUC^1 \left( {\dot{\mathbb {R}}}^{n+1},\mathbb {R}^{n+1}\right) \right) \cap C \left( {\bar{J}};BUC \left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) , \end{aligned}
(38)
\begin{aligned}&\mathbb {E}_1(t_0) \hookrightarrow H^1_p \left( J\times \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \cap C \left( {\bar{J}};H^1_p \left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) , \end{aligned}
(39)
\begin{aligned}&\mathbb {E}_1(t_0) \hookrightarrow H^1_p \left( J\times \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \cap C \left( {\bar{J}};H^1_{{\tilde{p}}}\left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) \quad \forall \,{\tilde{p}}\in [p,\infty ), \end{aligned}
(40)
\begin{aligned}&\mathbb {E}_4(t_0) \hookrightarrow C^1 \left( {\bar{J}};BC^1 \left( \mathbb {R}^n \right) \right) \cap C \left( {\bar{J}};BC^2 \left( \mathbb {R}^{n}\right) \right) . \end{aligned}
(41)

### Proof

For the imbeddings (38), (41) see [22, Lem. 6.1]. Moreover, it is obvious that

\begin{aligned} \mathbb {E}_1(t_0) \hookrightarrow H^1_p(J\times {\dot{\mathbb {R}}}^{n+1},\mathbb {R}^{n+1}) \end{aligned}

and also $$\mathbb {E}_1(t_0) \hookrightarrow C({\bar{J}};W_p^{2-2/p}({\dot{\mathbb {R}}}^{n+1},\mathbb {R}^{n+1}))$$ holds, see [2, Theorem III.4.10.2]. Since the functions $${\hat{u}}\in \mathbb {E}_1(t_0)$$ are continuous by (38) and thus $$[{\hat{u}}]=0$$, this implies the imbedding (39). Now (40) follows from interpolation between (38) and (39). $$\square$$

### Theorem 10

Let $$({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0)$$, $$h_0\in \mathbb {U}_h$$, $$u_0\in \mathbb {U}_u(h_0)$$, and consider, see (6),

\begin{aligned}&u(t,x,y)={\hat{u}}\left( t,x,y-h(t,x)\right) ,~ q(t,x,y)=\pi \left( t,x,y-h(t,x)\right) ,\\&u_0(x,y)={\hat{u}}_0 \left( x,y-h_0(x)\right) .\nonumber \end{aligned}
(42)

Then there exist constants $$C(\Vert h\Vert _{\mathbb {E}_4(t_0)})>0$$ and $$C(\Vert h_0\Vert _{\mathbb {U}_h})$$ such that

\begin{aligned} \begin{aligned}&\Vert u\Vert _{W_p^1(J\times \mathbb {R}^{n+1},\mathbb {R}^{n+1})}+ \left( \int _J \Vert u(t)\Vert _{H_p^2(\mathbb {R}^{n+1}\setminus \Gamma (t),\mathbb {R}^{n+1})}^p\,dt\right) ^{1/p} \le C (\Vert h\Vert _{\mathbb {E}_4(t_0)}) \Vert {\hat{u}}\Vert _{\mathbb {E}_1(t_0)},\\\nonumber&\left( \int _J \Vert q(t)\Vert _{\dot{H}_p^1(\mathbb {R}^{n+1}\setminus \Gamma (t),\mathbb {R}^{n+1})}^p\,dt\right) ^{1/p} \le C(\Vert h\Vert _{\mathbb {E}_4(t_0)}) \Vert \pi \Vert _{\mathbb {E}_2(t_0)},\\&\left( \int _J \Vert [q(t)]\Vert _{W_p^{1-1/p}(\Gamma (t))}^p\,dt\right) ^{1/p} \le C (\Vert h\Vert _{\mathbb {E}_4(t_0)}) \Vert [\pi ]\Vert _{L_p(J;W_p^{1-1/p}(\mathbb {R}^n))},\\&\Vert {\hat{u}}_0\Vert _{\mathbb {U}_{{\hat{u}}}}\le C (\Vert h_0\Vert _{\mathbb {U}_h}) \Vert u_0\Vert _{\mathbb {U}_{u}(h_0)}. \end{aligned} \end{aligned}
(43)

### Proof

Let $$({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0)$$ and consider, see (6),

\begin{aligned} u(t,x,y)={\hat{u}}\left( t,x,y-h(t,x)\right) . \end{aligned}

By (41) the mapping $$T_{h(t)}: (x,y)\mapsto (x,y-h(t,x))$$ is for all $$t\in [0,t_0]$$ a $$C^2$$-diffeomorphism with $$T_{h(t)}^{-1}(x,y)=(x,y+h(t,x))$$ and $$\text{ det }(DT_{h(t)}(x,y))=1$$. By (39) the chain rule for Sobolev functions can be applied and yields $$u\in H_p^1(J\times \mathbb {R}^{n+1},\mathbb {R}^{n+1})$$ with

\begin{aligned} \partial _t u(t,x,y)&=\partial _t {\hat{u}}\left( t,T_{h(t)}(x,y)\right) -\partial _y {\hat{u}}\left( t,T_{h(t)}(x,y)\right) \partial _t h(t,x),\\ \partial _{(x,y)} u(t,x,y)&=\partial _{(x,y)} {\hat{u}}\left( t,T_{h(t)}(x,y)\right) DT_{h(t)}(x,y). \end{aligned}

Moreover, again by (41) and $$\nabla {\hat{u}}\in L_p(J;H^1_p({\dot{\mathbb {R}}}^{n+1},\mathbb {R}^{n+1,n+1}))$$ we have

\begin{aligned} \Vert u\Vert _{H_p^1(J\times \mathbb {R}^{n+1},\mathbb {R}^{n+1})}+\left( \int _J \Vert u(t)\Vert _{H_p^2(\mathbb {R}^{n+1}\setminus \Gamma (t),\mathbb {R}^{n+1})}^p\,dt\right) ^{1/p}\le C \left( \Vert h\Vert _{\mathbb {E}_4(t_0)}\right) \Vert {\hat{u}}\Vert _{\mathbb {E}_1(t_0)}. \end{aligned}

Completely analogous one obtains

\begin{aligned} \left( \int _J \Vert q(t)\Vert _{\dot{H}_p^1(\mathbb {R}^{n+1}\setminus \Gamma (t),\mathbb {R}^{n+1})}^p\,dt\right) ^{1/p} \le C \left( \Vert h\Vert _{\mathbb {E}_4(t_0)}\right) \Vert \pi \Vert _{\mathbb {E}_2(t_0)}. \end{aligned}

Now consider $${\hat{r}}=[\pi ]$$ and $$r=[q]$$ then $$r(t,x,h(t,x))={\hat{r}}(t,x)$$ and

\begin{aligned} \Vert r(t,\cdot )\Vert _{W_p^{1-1/p}(\Gamma (t))}^p&=\int _{\mathbb {R}^n}\int _{\mathbb {R}^n} \frac{|r(t,x,h(t,x))-r(t,{\tilde{x}},h(t,{\tilde{x}}))|^p}{\left( \sqrt{|x-{\tilde{x}}|^2+|h(t,x)-h(t,\tilde{x})|^2}\right) ^{n+p-1}}\\&\quad \qquad \qquad \cdot \sqrt{1+|\nabla h(t,x)|^2} \sqrt{1+|\nabla h(t,{\tilde{x}})|^2}\, dx\,d{\tilde{x}}\\&\le \int _{\mathbb {R}^n}\int _{\mathbb {R}^n} \frac{|{\hat{r}}(t,x)-{\hat{r}}(t,{\tilde{x}})|^p}{|x-{\tilde{x}}|^{n+p-1}}\, dx\,d{\tilde{x}} (1+|h(t,\cdot )|_{BC^1(\mathbb {R}^n)})^2\\&\le \Vert {\hat{r}}(t,\cdot )\Vert _{W_p^{1-1/p}(\mathbb {R}^n)}^p C \left( \Vert h\Vert _{\mathbb {E}_4(t_0)}\right) ^p. \end{aligned}

Similarly, one obtains also the estimate for $$\Vert {\hat{u}}_0\Vert _{\mathbb {U}_{{\hat{u}}}}$$, see [22, Proof Thm 1.1]. $$\square$$

### Lemma 11

Consider the transformation (42), where we choose for $$\pi \in \mathbb {E}_2(t_0), [\pi ]\in \mathbb {E}_3(t_0)$$ the unique representative $$\pi$$ satisfying (36).

Then for all $${\tilde{p}}\in [p,\infty )$$ the mapping

\begin{aligned} \left( {\hat{u}},\pi ,[\pi ],h \right) \in \mathbb {E}(t_0)\mapsto u\in C \left( {\bar{J}};L_{{\tilde{p}}}\left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) \end{aligned}
(44)

is continuously differentiable with derivative

\begin{aligned}&\left( \delta {\hat{u}},\delta \pi ,[\delta \pi ],\delta h \right) \in \mathbb {E}(t_0)\mapsto \delta u(t,x,y)\\&\quad = \delta {\hat{u}}\left( t,x,y-h(t,x)\right) -\partial _y {\hat{u}}\left( t,x,y-h(t,x)\right) \delta h(t,x). \end{aligned}

Let $${{\mathcal {E}}}_{\pm }\in {{\mathcal {L}}}(H^l_p(\mathbb {R}^{n+1}_\pm ),H^l_p(\mathbb {R}^{n+1}))$$ be extension operators for $$l=1,2$$ and set

\begin{aligned} \begin{aligned} {\hat{u}}_\pm (t,\cdot )&={{\mathcal {E}}}_{\pm } {\hat{u}}(t,\cdot ),\quad u_\pm (t,x,y)={\hat{u}}_\pm (t,T_{h(t)}(x,y)),\\ \pi _\pm (t,\cdot )&={{\mathcal {E}}}_{\pm } \pi (t,\cdot ),\quad q_\pm (t,x,y)=\pi _\pm (t,T_{h(t)}(x,y)). \end{aligned} \end{aligned}
(45)

Then the mappings

\begin{aligned} \left( {\hat{u}},\pi ,[\pi ],h \right) \in \mathbb {E}(t_0)&\mapsto u_\pm \in L_p \left( J;H_p^1 \left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) , \end{aligned}
(46)
\begin{aligned} \left( {\hat{u}},\pi ,[\pi ],h \right) \in \mathbb {E}(t_0)&\mapsto q_\pm \in L_p \left( J;L_p \left( \mathbb {R}^{n+1}\right) \right) \end{aligned}
(47)

are continuously differentiable with derivative

\begin{aligned}&(\delta {\hat{u}},\delta \pi ,[\delta \pi ],\delta h)\in \mathbb {E}(t_0)\mapsto \left( {\begin{array}{c}\delta u_\pm \\ \delta q_\pm \end{array}}\right) (t,x,y)\\&\quad =\left( {\begin{array}{c}\delta {\hat{u}}_\pm \\ \delta \pi _\pm \end{array}}\right) (t,x,y-h(t,x)) -\partial _y \left( {\begin{array}{c}{\hat{u}}_\pm \\ \pi _\pm \end{array}}\right) (t,x,y-h(t,x)) \delta h(t,x). \end{aligned}

### Proof

Define as in the previous proof the $$C^2$$-diffeomorphisms $$T_{h(t)}: (x,y)\mapsto (x,y-h(t,x))$$. Then $$u(t,x,y)={\hat{u}}(t,T_{h(t)}(x,y))$$. Let $$({\hat{u}},\pi ,[\pi ],h),(\delta {\hat{u}},\delta \pi ,[\delta \pi ],\delta h)\in \mathbb {E}(t_0)$$ be arbitrary. We recall the well known fact that for any $$v\in C({\bar{J}};L_{{\tilde{p}}}(\mathbb {R}^{n+1}))$$, $$p\le {\tilde{p}}<\infty$$, it holds

\begin{aligned} \sup _{t\in J}\left\| v \left( t,T_{(h+\delta h)(t)}(\cdot )\right) -v \left( t,T_{h(t)}(\cdot )\right) \right\| _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})} \rightarrow 0\quad \text{ as } \Vert \delta h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}\rightarrow 0, \end{aligned}
(48)

which can be shown by an approximation of v through a sequence of continuous functions with compact support. Similarly, for $$v\in L_p(J;L_p(\mathbb {R}^{n+1}))$$ one has

\begin{aligned} \int _J \left\| v\left( t,T_{(h+\delta h)(t)}(\cdot )\right) -v\left( t,T_{h(t)}(\cdot )\right) \right\| _{L_p(\mathbb {R}^{n+1})}^p\,dt \rightarrow 0\quad \text{ as } \Vert \delta h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}\rightarrow 0. \end{aligned}
(49)

Consider the remainder term

\begin{aligned} \begin{aligned} R_u(t,x,y):=&\left( {\hat{u}}+\delta {\hat{u}}\right) \left( t,T_{(h+\delta h)(t)}(x,y)\right) -{\hat{u}}\left( t,T_{h(t)}(x,y)\right) \\&-\delta {\hat{u}}\left( t,T_{h(t)}(x,y)\right) +\partial _y {\hat{u}}\left( t,T_{h(t)}(x,y)\right) \delta h(t,x). \end{aligned} \end{aligned}
(50)

Let $$p\le {\tilde{p}}<\infty$$ be arbitrary. We obtain

\begin{aligned}&\Vert R_u\Vert _{C({\bar{J}};L_{{\tilde{p}}}(\mathbb {R}^{n+1}))} \le \sup _{t\in J}\left\| \int _0^1 (\partial _y \delta {\hat{u}}(t,T_{(h+\tau \delta h)(t)}(\cdot ))\,d\tau \, \delta h(t,\cdot )\right\| _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})}\\&+\sup _{t\in J}\left\| \int _0^1 (\partial _y {\hat{u}}(t,T_{(h+\tau \delta h)(t)}(\cdot )) -\partial _y {\hat{u}}(t,T_{h(t)}(\cdot )))\,d\tau \, \delta h(t,\cdot )\right\| _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})} \\ \le&\, \Vert \delta {\hat{u}}\Vert _{C({\bar{J}};H^1_{{\tilde{p}}}(\mathbb {R}^{n+1})} \Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}\\&+\sup _{t\in J,\tau \in [0,1]} \left\| \partial _y {\hat{u}}(t,T_{(h+\tau \delta h)(t)}(\cdot )) -\partial _y {\hat{u}}(t,T_{h(t)}(\cdot ))\right\| _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})}\Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}\\ =&\, \Vert \delta {\hat{u}}\Vert _{C({\bar{J}};H^1_{{\tilde{p}}}(\mathbb {R}^{n+1})} \Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}+ o\left( \Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}\right) \\ =&\, o\left( \Vert \delta {\hat{u}}\Vert _{\mathbb {E}_1(t_0)}+\Vert \delta h\Vert _{\mathbb {E}_4(t_0)}\right) . \end{aligned}

Here, we have used (47) and the imbeddings (40), (41). This shows that (43) is Fréchet differentiable. The continuity of the derivative follows from the fact that for $$({\hat{u}}_1,\pi _1,[\pi _1],h_1)\rightarrow ({\hat{u}},\pi ,[\pi ],h)$$ in $$\mathbb {E}(t_0)$$ we have

\begin{aligned}&\sup _{t\in J}\left\| \delta {\hat{u}}\left( t,T_{h_1(t)}(\cdot )\right) -\delta {\hat{u}}\left( t,T_{h(t)}(\cdot )\right) \right\| _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})}\\&\quad =\sup _{t\in J}\left\| \int _0^1 \partial _y \delta {\hat{u}}\left( t,T_{(h+\tau (h_1-h))(t)}(\cdot )\right) \,d\tau \, (h_1-h)(t,\cdot )\right\| _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})}\\&\quad \le \Vert \delta {\hat{u}}\Vert _{C({\bar{J}};H^1_{{\tilde{p}}}(\mathbb {R}^{n+1}))} \Vert h_1-h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}\\&\quad \le C \Vert \delta {\hat{u}}\Vert _{\mathbb {E}_1(t_0)} \Vert h_1-h\Vert _{\mathbb {E}_4(t_0)} \end{aligned}

as well as $$\Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}\le C \Vert \delta h\Vert _{\mathbb {E}_4(t_0)}$$ and

\begin{aligned}&\sup _{t\in J} \Vert \partial _y {\hat{u}}_1(t,T_{h_1(t)}(\cdot )) -\partial _y {\hat{u}}(t,T_{h(t)}(\cdot ))\Vert _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})}\\&\quad \le \Vert {\hat{u}}_1-{\hat{u}}\Vert _{C({\bar{J}};H_{{\tilde{p}}}^1(\mathbb {R}^{n+1}))}+ \sup _{t\in J} \Vert \partial _y {\hat{u}}_1(t,T_{h_1(t)}(\cdot )) -\partial _y {\hat{u}}(t,T_{h(t)}(\cdot ))\Vert _{L_{{\tilde{p}}}(\mathbb {R}^{n+1})}\rightarrow 0, \end{aligned}

where we have used (47).

The continuous differentiability of (46) follows very similarly by using (48) instead of (47) and by applying (36), (37) and Theorem 10.

Finally, consider (45), (44). Then $${\hat{u}}_\pm , \delta {\hat{u}}_\pm \in L_p(J;H^2_p(\mathbb {R}^{n+1},\mathbb {R}^{n+1}))$$. Define the remainder terms $$R_{u_\pm }$$ as in (49) with $${\hat{u}},{\hat{\delta }} u$$ replaced by $${\hat{u}}_\pm ,\delta {\hat{u}}_\pm$$. After differentiation a calculation as above yields

\begin{aligned}&\Vert \nabla R_{u_\pm }\Vert _{L_p(J;L_p(\mathbb {R}^{n+1}))}\\ \le&\left\| \int _0^1 (\partial _y \nabla \delta {\hat{u}}_\pm \left( t,T_{(h+\tau \delta h)(t)}(\cdot )\right) ^\top DT_{(h+\tau \delta h)(t)}\,d\tau \, \delta h(t,\cdot )\right\| _{L_p(J;L_p(\mathbb {R}^{n+1}))}\\&+\left\| \int _0^1 \partial _y \delta {\hat{u}}_\pm \left( t,T_{(h+\tau \delta h)(t)}(\cdot )\right) \,d\tau \, \nabla \delta h(t,\cdot )^\top \right\| _{L_p(J;L_p(\mathbb {R}^{n+1}))}\\&+\left\| \int _0^1 \nabla \left( \partial _y {\hat{u}}_\pm \left( t,T_{(h+\tau \delta h)(t)}(\cdot )\right) -\partial _y {\hat{u}}_\pm \left( t,T_{h(t)}(\cdot )\right) \right) ^\top DT_{h(t)}\,d\tau \, \delta h(t,\cdot )\right\| _{L_p(J;L_p(\mathbb {R}^{n+1}))}\\&+\left\| \int _0^1 \partial _y \nabla {\hat{u}}_\pm \left( t,T_{(h+\tau \delta h)(t)}(\cdot )\right) ^\top \left( DT_{(h+\tau \delta h)(t)}-DT_{h(t)}\right) \,d\tau \, \delta h(t,\cdot )\right\| _{L_p(J;L_p(\mathbb {R}^{n+1}))}\\&+\left\| \int _0^1 \left( \partial _y {\hat{u}}_\pm \left( t,T_{(h+\tau \delta h)(t)}(\cdot )\right) -\partial _y {\hat{u}}_\pm \left( t,T_{h(t)}(\cdot )\right) \right) \,d\tau \, \nabla \delta h(t,\cdot )^\top \right\| _{L_p(J;L_p(\mathbb {R}^{n+1}))}\\ \le&\, \Vert \delta {\hat{u}}_\pm \Vert _{L_p(J;H^2_p(\mathbb {R}^{n+1}))} \left( 1+\Vert h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}+\Vert \delta h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}\right) \Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}\\&+\Vert \delta {\hat{u}}_\pm \Vert _{L_p(J;H^1_p(\mathbb {R}^{n+1}))}\Vert \delta h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}\\&+\Vert \partial _y \nabla {\hat{u}}_\pm \left( t,T_{(h+\tau \delta h)(t)}(\cdot )\right) -\partial _y \nabla {\hat{u}}_\pm \left( t,T_{h(t)}(\cdot )\right) \Vert _{L_p(J;L_p(\mathbb {R}^{n+1}))}\\&\cdot \left( 1+\Vert h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}\right) \Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))} +\Vert {\hat{u}}_\pm \Vert _{L_p(J;H_p^2(\mathbb {R}^{n+1}))}\Vert \delta h\Vert _{C({\bar{J}};BC(\mathbb {R}^n))}^2\\&+\Vert \partial _y {\hat{u}}_\pm \left( t,T_{(h+\tau \delta h)(t)}(\cdot )\right) -\partial _y {\hat{u}}_\pm \left( t,T_{h(t)}(\cdot )\right) \Vert _{L_p(J;L_p(\mathbb {R}^{n+1}))} \Vert \delta h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}\\ =&\,o \left( \Vert \delta {\hat{u}}\Vert _{L_p(J;H^2_p(\mathbb {R}^{n+1}))}+\Vert \delta h\Vert _{C({\bar{J}};BC^1(\mathbb {R}^n))}\right) =o \left( \Vert \delta {\hat{u}}\Vert _{\mathbb {E}_1(t_0)}+\Vert \delta h\Vert _{\mathbb {E}_4(t_0)}\right) . \end{aligned}

Here we have used (48) and the imbedding (41). The continuity of the derivative follows with very similar estimates. $$\square$$

Similarly, we have

### Lemma 12

Let $$\mathbb {U}_c(t_0)=L_p(J;H^1_p(\mathbb {R}^{n+1}))$$. Then the mapping

\begin{aligned} (c,h)\in \mathbb {U}_c(t_0)\times \mathbb {E}_4(t_0)\mapsto {\hat{c}}(c,h) \in \mathbb {U}_{{\hat{c}}}(t_0) \end{aligned}
(51)

with $${\hat{c}}(c,h)(t,x,y)=c(t,x,y+h(t,x))$$ is continuously differentiable with derivative

\begin{aligned} (\delta c,\delta h)\in \mathbb {U}_c(t_0)\times \mathbb {E}_4(t_0)\mapsto \delta c(t,x,y+h(t,x))+\partial _y c(t,x,y+h(t,x)) \delta h(t,x). \end{aligned}

### Proof

The proof is the same as for (46). $$\square$$

For the original data $$(u_0,h_0,c)$$ we obtain the following existence and differentiability result.

### Theorem 13

Let $$p>n+3$$ and $$\mathbb {U}_{u}(h_0), \mathbb {U}_{c}(t_0)$$ be defined by (35). Then for any $$t_0>0$$ there exists $$\varepsilon _0=\varepsilon _0(t_0)>0$$ such that for all data

\begin{aligned} (h_0,c)\in \mathbb {U}_h \times \mathbb {U}_c(t_0),\quad u_0\in \mathbb {U}_{u}(h_0) \end{aligned}

satisfying the compatibility condition (14) as well as the smallness condition

\begin{aligned} \Vert u_0\Vert _{\mathbb {U}_{u}(h_0)}+\Vert h_0\Vert _{\mathbb {U}_h}+\Vert c\Vert _{\mathbb {U}_c(t_0)}< \varepsilon _0 \end{aligned}
(52)

there exists a unique solution of the transformed problem (10) with

\begin{aligned} \left( {\hat{u}},\pi ,[\pi ],h \right) \in \mathbb {E}(t_0). \end{aligned}

Moreover, for any $$h_0$$ with $$\Vert h_0\Vert _{\mathbb {U}_h}<\varepsilon _0$$ the mapping

\begin{aligned} \{ (u_0,c)\in \mathbb {U}_u(h_0)\times \mathbb {U}_{c}(t_0): (u_0,h_0,c) \text{ satisfy } \text{(14), } \text{(51) }\} \mapsto ({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0) \end{aligned}

is continuously differentiable.

By the chain rule in Lemma 11, also the original state (uq) depends continuously differentiable on $$(u_0,c)$$ with the spaces given in (43), (45), (46).

### Proof

We adapt the fixed point argument in the proof of Theorem 8. Let

\begin{aligned} {\hat{c}}(c,h)(t,x,y)=c(t,x,y+h(t,x)). \end{aligned}
(53)

The only difference compared to the situation in Theorem 8 results from the fact that $${\hat{c}}(c,h)$$ depends now on h. Hence, the fixed point equation (32) changes to

\begin{aligned} L_0 {\tilde{z}}=K \left( {\tilde{z}};{\hat{u}}_0,h_0,{\hat{c}}\left( c,{\tilde{z}}+z^*({\hat{u}}_0,h_0)\right) \right) ,~~ {\tilde{z}}\in {}_0\mathbb {E}(t_0). \end{aligned}
(54)

Let $${\hat{\varepsilon }}_0>0$$ be as in Theorem 8. We have

\begin{aligned} \Vert {\hat{c}}(c,h)\Vert _{\mathbb {U}_{{\hat{u}}}}=\Vert c\Vert _{\mathbb {U}_{{\hat{u}}}} \end{aligned}
(55)

and the last estimate in (43) shows that for $$\varepsilon _0>0$$ small enough (51) implies (27).

Hence, for all $$(u_0,h_0,c)$$ satisfying (51) we have $$({\hat{u}}_0,h_0,{\hat{c}}(c,h))\in B_U({\hat{\varepsilon }}_0)$$ (note that (54) holds independently of h) and thus by (34)

\begin{aligned}&\left\| L_0^{-1} K({\tilde{z}};{\hat{u}}_0,h_0,{\hat{c}})\right\| _{{}_0\mathbb {E}(t_0)}<\delta . \end{aligned}

Finally, the Lipschitz constant of $$K({\tilde{z}};{\hat{u}}_0,h_0,{\hat{c}})$$ with respect to $${\hat{c}}$$ is 1 and the mapping (50), (52) is by Lemma 12 continuously differentiable and the Lipschitz constant with respect to h is bounded by $$\Vert c\Vert _{\mathbb {U}_c(t_0)}<\varepsilon _0$$. Hence, for $$\varepsilon _0>0$$ small enough, (53) is a contraction and the existence, uniqueness and continuous differentiability follow as in the proof of Theorem 8.

Lemma 11 and the chain rule yield now the continuous differentiability of the original state (uq) with respect to $$(u_0,c)$$ for the spaces given in (43), (45), (46). $$\square$$

### 3.3 Volume-of-fluid type formulation

Our aim is finally to derive a Volume-of-Fluid (VoF) type formulation with corresponding sensitivity equation that is satisfied by the solution (uq) of the problem (1) and its sensitivities $$(\delta u,\delta q)$$. This provides an analytical foundation to derive and analyze appropriate numerical VoF schemes for sensitivity calculations.

Let $$\alpha :\mathbb {R}^{n+1}\rightarrow [0,1]$$ be a phase indicator satisfying the transport equation

\begin{aligned} \partial _t \alpha +u\cdot \nabla \alpha =0&\quad \text{ in } J\times \mathbb {R}^{n+1},\quad \alpha (0)=1_{\Omega _1(0)}\quad \text{ on } \mathbb {R}^{n+1}. \end{aligned}
(56)

We note that for $$u\in L_1(J;W^1_\infty (\mathbb {R}^{n+1};\mathbb {R}^{n+1}))$$ with $${\text{ div }}u=0$$ a.e. any distributional solution $$\alpha \in L_1(J;L_{1,loc}(\mathbb {R}^{n+1}))$$ is also a distributional solution of

\begin{aligned} \partial _t \alpha +{\text{div}}(u \alpha )=0\quad \text{ in } J\times \mathbb {R}^{n+1}, \quad \alpha (0)=1_{\Omega _1(0)}\quad \text{ on } \mathbb {R}^{n+1}. \end{aligned}
(57)

We define now

\begin{aligned} \rho (\alpha )=\alpha \rho _1+(1-\alpha )\rho _2,\quad \mu (\alpha )=\alpha \mu _1+(1-\alpha )\mu _2. \end{aligned}

We will show that the unique solution (uq) of (1) according to Theorem 13 satisfies the VoF-type formulation

\begin{aligned} \nonumber&\int _{\mathbb {R}^{n+1}} \left( \partial _t \left( \rho (\alpha ) u \right) +{\text{div}}\left( \rho (\alpha ) u\otimes u \right) \right)(t,x,y)^\top \varphi (x,y) \\&\qquad \qquad \left. +S(u,q;\mu (\alpha ))(t,x,y):\nabla \varphi (x,y)\right) \,d(x,y) \qquad \forall \,\varphi \in C_c^1 \left( \mathbb {R}^{n+1};\mathbb {R}^{n+1}\right) \end{aligned}
(58)
\begin{aligned} \nonumber&=-\lim _{\varepsilon \searrow 0}\int _{\mathbb {R}^{n+1}} \sigma \frac{\nu _{\varepsilon }(t,x,y)^\top }{|\nu _{\varepsilon }(t,x,y)|} \left( D\varphi -{\text{div}}(\varphi )I \right) (x,y) \nabla \alpha (t,x,y)\,d(x,y),\\&\int _{\mathbb {R}^{n+1}} {\text{div}}(u)\, \psi \,dx=0\quad \forall \,\psi \in C_c^1 \left( \mathbb {R}^{n+1}\right) , \end{aligned}
(59)
\begin{aligned}&\alpha \text{ satisfies } \text{(56) }, \end{aligned}
(60)

where $$\nu _{\varepsilon }$$ is a suitable smoothed normal computed from $$\nabla \alpha$$, see (70) below.

In order to deal with the sensitivity equation, it will be beneficial to consider measure-valued solutions of the general equation

\begin{aligned} \partial _t \delta \alpha +{\text{div}}(u \,\delta \alpha )=b\quad \text{ in } J\times \mathbb {R}^{n+1},\quad \delta \alpha (0)=\delta \alpha _0\quad \text{ on } \mathbb {R}^{n+1}. \end{aligned}
(61)

For $$u\in L_1(J;W^1_\infty (\mathbb {R}^{n+1};\mathbb {R}^{n+1}))$$ we can define uniquely the continuous mapping $$(x,y)\mapsto X(t;x,y)$$, where X(txy) satisfies the characteristic equation

\begin{aligned} \partial _t X(t;x,y)= u \left( t,X(t;x,y)\right) ,~~ t\in J,\quad X(0;x,y)=(x,y). \end{aligned}
(62)

In the following, we denote by $${{\mathcal {M}}}_{loc}(\mathbb {R}^{n+1})$$ the space of locally bounded Radon measures.

### Proposition 14

Let $$u\in L_1(J;W^1_\infty (\mathbb {R}^{n+1};\mathbb {R}^{n+1}))$$. Then for any $$\delta \alpha _0\in {{\mathcal {M}}}_{loc}(\mathbb {R}^{n+1})$$ there exists a unique distributional solution of (60) in $$C({\bar{J}};{{\mathcal {M}}}_{loc}(\mathbb {R}^{n+1})-\text{ weak}^*)$$, given by

\begin{aligned} \delta \alpha (t)=X(t)(\delta \alpha _0)+\int _0^t X(t-s)(b(s)). \end{aligned}
(63)

Here, X is the forward flow defined by (61) and $$\delta \alpha _t=X(t)(\delta \alpha _0)$$ is the measure satisfying

\begin{aligned} \int _{\mathbb {R}^{n+1}} \phi (x,y) \,d\delta \alpha _t(x,y)= \int _{\mathbb {R}^{n+1}} \phi \left( X(t;x,y)\right) \,d\delta \alpha _0(x,y)\quad \forall \,\phi \in C_c \left( \mathbb {R}^{n+1}\right) . \end{aligned}

### Proof

For $$u\in L_1(J;C^1(\mathbb {R}^{n+1};\mathbb {R}^{n+1}))$$, see [21, Thm. 3.1 and 3.3]. Since the characteristics are unique and stable also for $$u\in L_1(J;W^1_\infty (\mathbb {R}^{n+1};\mathbb {R}^{n+1}))$$, the proofs directly extend to this case, see also [3]. $$\square$$

### Proposition 15

If $${\hat{u}}\in \mathbb {E}_1(t_0)$$, $$[{\hat{u}}]=0$$ and u is given by (42) then (55) as well as (56) have a unique solution given by

\begin{aligned} \alpha \left( t,X(t;x,y)\right) =1_{\Omega _1(0)}(x,y) \end{aligned}
(64)

and thus $$\alpha (t,\cdot )=1_{\Omega _1(t)}$$.

Moreover, for $$\varepsilon _0$$ from Theorem 13 and any $$h_0$$ with $$\Vert h_0\Vert _{\mathbb {U}_h}<\varepsilon _0$$ the mapping

\begin{aligned} \left\{ (u_0,c)\in \mathbb {U}_u(h_0)\times \mathbb {U}_{c}(t_0): (u_0,h_0,c) \text{ satisfy } \text{(14), } \text{(51) }\right\} \mapsto \alpha \in C \left( {\bar{J}};{{\mathcal {M}}}_{loc}\left( \mathbb {R}^{n+1}\right) -\text{ weak}^{*}\right) \end{aligned}

is continuously differentiable. The derivative

\begin{aligned} (\delta u_0,\delta c)\in \mathbb {U}_u(h_0)\times \mathbb {U}_{c}(t_0) \mapsto \delta \alpha \in C \left( {\bar{J}};{{\mathcal {M}}}_{loc}\left( \mathbb {R}^{n+1}\right) -\text{ weak}^{*}\right) \end{aligned}

is given by the unique measure-valued solution of

\begin{aligned} \partial _t \delta \alpha +{\text{div}}(u\, \delta \alpha )=-{\text{div}}(\delta u\, \alpha )\quad \text{ in } J\times \mathbb {R}^{n+1}, \quad \delta \alpha (0)=0\quad \text{ on } \mathbb {R}^{n+1}. \end{aligned}
(65)

Finally, $$\delta \alpha$$ satisfies

\begin{aligned} \int _{\mathbb {R}^{n+1}} \phi (x,y)\,d\delta \alpha (t)(x,y)=\int _{\mathbb {R}^n} \phi \left( x,h(t,x)\right) \delta h(t,x)\,dx. \end{aligned}
(66)

### Proof

If $${\hat{u}}\in \mathbb {E}_1(t_0)$$, $$[{\hat{u}}]=0$$ and u is given by (42) then $$u\in C({\bar{J}};W^1_\infty (\mathbb {R}^{n+1};\mathbb {R}^{n+1}))$$ by (38), (41). Now it is well known that (63) provides the unique weak solution of (55) in $$L_{1,loc}(J\times \mathbb {R}^{n+1})$$, see [3, Prop. 2.2] and [10, Cor. II.1]. Since $${\text{div}}(u)=0$$ a.e., it is also a distributional solution of (56), which is unique by Proposition 14.

Let now $$(u_0,h_0,c), (\delta u_0,0,\delta c)\in \mathbb {U}_u(h_0)\times \mathbb {U}_h\times \mathbb {U}_c(t_0)$$ be such that $$(u_0,h_0,c)$$ and $$(u_0,h_0,c)+(\delta u_0,0,\delta c)$$ satisfy the conditions of Theorem 13. Denote by $$({\hat{u}},\pi ,[\pi ],h)$$ the unique solution of (10) for data $$(u_0,h_0,c)$$ and by $$({\hat{u}}^s,\pi ^s,[\pi ^s],h^s)$$ the one for data $$(u_0,h_0,c)+s\,(\delta u_0,0,\delta c)$$. Let (uq) and $$(u^s,q^s)$$ be the corresponding states in physical coordinates according to (6) and let $$\alpha =1_{\Omega _1(t)}, \alpha ^s=1_{\Omega _1^s(t)}$$ be the corresponding solutions of (56). Finally, let $$(\delta u,\delta h,\delta q)$$ be the directional derivatives (sensitivities) in direction $$(\delta u_0,0,\delta c)$$ which exist by Theorem 13. We show that

\begin{aligned} \frac{\alpha ^s-\alpha }{s} \rightarrow \delta \alpha \quad \text{ in } C \left( {\bar{J}};{{\mathcal {M}}}_{loc}\left( \mathbb {R}^{n+1}\right) -\text{ weak}^{*}\right) \text{ as } s\rightarrow 0, \end{aligned}
(67)

where $$\delta \alpha$$ solves (64). Let $$\phi \in C_c(\mathbb {R}^{n+1})$$ be arbitrary. Then

\begin{aligned} \int _{\mathbb {R}^{n+1}} \frac{\alpha ^s-\alpha }{s}(t,x,y)\phi (x,y)\,d(x,y)&= \int _{\mathbb {R}^n} \int _{h(t,x)}^{h^s(t,x)} \frac{1}{s} \phi (x,y)\,d(x,y)\\&\rightarrow \int _{\mathbb {R}^n} \phi \left( x,h(t,x)\right) \delta h(t,x)\,dx \end{aligned}

as $$s\rightarrow 0$$ uniformly in $$t\in {\bar{J}}$$, where we have used the differentiability result of Theorem 13. Moreover, it is obvious that the middle term is continuous with respect to t. Hence, (66) is proven and we have only to show that $$\delta \alpha$$ solves (64).

To this end, let $$\varphi \in C_c^1(J\times \mathbb {R}^{n+1})$$ be arbitrary. Since $$\alpha ,\alpha ^s$$ are distributional solutions of (56), we have

\begin{aligned} 0&=\int _J \int _{\mathbb {R}^{n+1}} -\left( \left(\partial _t \varphi + (u\cdot \nabla )\varphi \right) \tfrac{\alpha ^s-\alpha }{s}+\alpha ^s \left( \tfrac{u^s-u}{s}\cdot \nabla \right) \varphi \right)(t,x,y)\,d(x,y)\,dt\\&\rightarrow \int _J \int _{\mathbb {R}^{n+1}} -\left( \left(\partial _t \varphi +(u\cdot \nabla ) \varphi \right) \delta \alpha +\alpha (\delta u\cdot \nabla )\varphi \right)(t,x,y)\,d(x,y)\,dt \end{aligned}

as $$s\rightarrow 0$$. For the limit transition, we have used $$u\in C(\bar{J};W^1_\infty (\mathbb {R}^{n+1};\mathbb {R}^{n+1})$$, (66) and that by Theorem 13$$~~\alpha ^s=1_{\Omega ^s(t)}\rightarrow \alpha =1_{\Omega_1 (t)}$$ in $$L_{2,loc}(J\times \mathbb {R}^{n+1})$$ and $$\frac{u^s-u}{s}\rightarrow \delta u$$ in $$C({\bar{J}};L_p(\mathbb {R}^{n+1}))$$. Hence, $$\delta \alpha$$ is a distributional solution of (64), which is unique by Proposition 14. $$\square$$

The next step is to express the surface tension term by using the phase indicator $$\alpha$$ such that its sensitivities can be expressed by using the measure $$\delta \alpha$$.

We first rewrite the surface tension term in the weak formulation (2).

### Lemma 16

Let $$\varphi \in C_c^1(\mathbb {R}^{n+1};\mathbb {R}^{n+1})$$. Then with the curvature $$\kappa (t)$$ of $$\Gamma (t)$$ according to (4) one has the identity

\begin{aligned} \nonumber&\int _{\Gamma (t)} (\sigma \kappa \nu )(t,x,y)^\top \varphi (x,y)\,dS(x,y)\\\nonumber&\quad =\int _{\mathbb {R}^n} \sigma \,{\text{div}}_x\left( \frac{\nabla h(t,x)}{\sqrt{1+|\nabla h(t,x)|^2}}\right) \left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) ^\top \varphi (x,h(t,x))\,dx\\&\quad =\int _{\mathbb {R}^n}\sigma \frac{(\nabla h(t,x)^\top ,-1)}{\sqrt{1+|\nabla h(t,x)|^2}} \left( D\varphi (x,h(t,x))-{\text{ div }}(\varphi )(x,h(t,x))I \right) \left( {\begin{array}{c}\nabla h(t,x)\\ -1\end{array}}\right) \,dx. \end{aligned}
(68)

### Proof

The first identity follows directly from (4). The second one follows from integration by parts and reflects the well known identity from differential geometry, see for example [7, Lem. 2.1]

\begin{aligned} \int _{\Gamma (t)} (\kappa \nu )(t,x,y)^\top \varphi (x,y)\,dS(x,y)= -\int _{\Gamma (t)} \nabla _T \mathop {\text {id}}\nolimits _{\Gamma (t)}(x,y) : \nabla _T \varphi (x,y)\,dS(x,y), \end{aligned}

where $$\nabla _T \varphi _i=\nabla \varphi _i-\nu ^\top \nabla \varphi _i \nu$$ is the tangential derivative. $$\square$$

To compute the interface normal from $$\nabla \alpha$$, we use the following simple fact.

### Lemma 17

Let $$\psi \in C_c^1(\mathbb {R}^{n+1};\mathbb {R}^{n+1})$$. Then

\begin{aligned} -\int _{\mathbb {R}^{n+1}} \psi (x,y)^\top \nabla \alpha (t,x,y)\,d(x,y)&= \int _{\Gamma (t)} \psi (x,y)^\top \nu (t,x,y)\,dS(x,y)\\&= \int _{\mathbb {R}^n} \psi \left( x,h(t,x)\right) ^\top \left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) \,dx. \end{aligned}

### Proof

By the definition of distributional derivatives one has

\begin{aligned}&-\int _{\mathbb {R}^{n+1}} \psi (x,y)^\top \nabla \alpha (t,x,y)\,d(x,y)= \int _{\mathbb {R}^{n+1}} {\text{div}}(\psi )(x,y)\alpha (t,x,y)\,d(x,y)\\&\quad = \int _{\Omega _1(t)} {\text{div}}(\psi )(x,y)\,d(x,y)= \int _{\Gamma (t)} \psi (x,y)^\top \nu (t,x,y)\,dS(x,y)\\&\quad =\int _{\mathbb {R}^n} \psi (x,h(t,x))^\top \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }\,dx. \end{aligned}

$$\square$$

Let now $$\delta \in (0,1/2)$$ and

\begin{aligned} \psi _\delta \in C_c^1((-1,1)),\quad \psi _\delta |_{[-1+\delta ,1-\delta ]}\equiv 1,\quad \psi _\delta (-s)=\psi _\delta (s)\quad \forall \,s\in \mathbb {R}, \int _\mathbb {R}\psi _\delta (s)\,ds=1 \end{aligned}

and set

\begin{aligned} \phi _{\varepsilon }(x,y)=\frac{1}{\varepsilon ^{n}} \psi _\delta (y/\varepsilon )\prod _{i=1}^n\psi _\delta (x_i/\varepsilon ). \end{aligned}

To recover a mollified normal (not necessarily of unit length) we use

\begin{aligned} \nu _{\varepsilon }(t,x,y)&:= -\int _{\mathbb {R}^{n+1}} \phi _{\varepsilon } \left( ({\tilde{x}},{\tilde{y}})-(x,y)\right) \nabla \alpha (t,{\tilde{x}},{\tilde{y}})\,d({\tilde{x}},{\tilde{y}}). \end{aligned}
(69)

Then by Lemma 17

\begin{aligned} \nu _{\varepsilon }(t,x,y)&=\int _{\Gamma (t)} \phi _{\varepsilon }\left( ({\tilde{x}},{\tilde{y}})-(x,y)\right) \nu (t,{\tilde{x}},{\tilde{y}})\,dS({\tilde{x}},{\tilde{y}})\\&=\int _{\mathbb {R}^n} \phi _{\varepsilon }\left( ({\tilde{x}},h(t,\tilde{x}))-(x,y)\right) \textstyle {\left( {\begin{array}{c}-\nabla h(t,{\tilde{x}})\\ 1\end{array}}\right) }\,d{\tilde{x}}. \end{aligned}

Now assume that

\begin{aligned} |\nabla h|\le 1-\delta \quad \text{ on } x+[-\varepsilon ,\varepsilon ]^n. \end{aligned}
(70)

Then we have by the definition of $$\phi _{\varepsilon }$$

\begin{aligned} \nu _{\varepsilon }(t,x,h(t,x))&= \frac{1}{\varepsilon ^n}\int _{\mathbb {R}^n} \prod _{i=1}^n\psi _\delta (({\tilde{x}}_i-x_i)/\varepsilon ) \textstyle {\left( {\begin{array}{c}-\nabla h(t,{\tilde{x}})\\ 1\end{array}}\right) }\,d{\tilde{x}}. \end{aligned}
(71)

### Lemma 18

Let (69) hold. If $$h\in C({\bar{J}};BC^2(\mathbb {R}^n))$$ then there is $$C>0$$ such that

\begin{aligned} |\nu _{\varepsilon }(t,x,h(t,x))-(-\nabla h(t,x),1)^\top | \le C \varepsilon \quad \forall \, (t,x)\in J\times \mathbb {R}^n. \end{aligned}

On compact subsets the error is $$o(\varepsilon )$$.

### Proof

Since $$\nabla h$$ has a uniform Lipschitz constant with respect to x the first assertion follows immediately from (70). Moreover, since $$\nabla h(t,{\tilde{x}})=\nabla h(t,x)+\nabla ^2 h(t,x) ({\tilde{x}}-x) +o(\Vert {\tilde{x}}-x\Vert )$$, the $$o(\varepsilon )$$ is obtained by the symmetry of $$\psi _\delta$$. $$\square$$

The variation of $$\nu _{\varepsilon }$$ is

\begin{aligned} \begin{aligned} \delta \nu _{\varepsilon }(t,x,y)&:= -\int _{\mathbb {R}^{n+1}} \phi _{\varepsilon }(({\tilde{x}},{\tilde{y}})-(x,y)) \nabla d\delta \alpha (t)({\tilde{x}},{\tilde{y}})\\&=\int _{\mathbb {R}^{n+1}} \nabla \phi _{\varepsilon }(({\tilde{x}},{\tilde{y}})-(x,y)) d\delta \alpha (t)({\tilde{x}},{\tilde{y}}) \end{aligned} \end{aligned}
(72)

with the measure-valued solution of (64).

### Lemma 19

Let (69) hold. If $$\delta h\in C({\bar{J}};BC^2(\mathbb {R}^n))$$ then there is $$C>0$$ such that

\begin{aligned} |\delta \nu _{\varepsilon }(t,x,h(t,x))-(-\nabla \delta h(t,x),0)^\top | \le C \varepsilon \quad \forall \, (t,x)\in J\times \mathbb {R}^n. \end{aligned}

On compact subsets the error is $$o(\varepsilon )$$.

### Proof

Then by (65) and (71)

\begin{aligned} \delta \nu _{\varepsilon }(t,x,y)&= \int _{\mathbb {R}^n} \nabla \phi _{\varepsilon }(({\tilde{x}},h(t,{\tilde{x}}))-(x,y)) \delta h(t,{\tilde{x}})\,d{\tilde{x}}. \end{aligned}

Setting $$y=h(t,x)$$ and using (69) we obtain

\begin{aligned} \delta \nu _{\varepsilon }(t,x,h(t,x))&= \frac{1}{\varepsilon ^n}\int _{\mathbb {R}^n} \left( {\begin{array}{c}\nabla _{{\tilde{x}}} \prod _{i=1}^n\psi _\delta (({\tilde{x}}_i-x_i)/\varepsilon )\\ 0\end{array}}\right) \delta h(t,{\tilde{x}})\,d{\tilde{x}}\\ {}&= \frac{1}{\varepsilon ^n}\int _{\mathbb {R}^n} \prod _{i=1}^n\psi _\delta (({\tilde{x}}_i-x_i)/\varepsilon ) \textstyle {\left( {\begin{array}{c}-\nabla \delta h(t,{\tilde{x}})\\ 0\end{array}}\right) }\,d{\tilde{x}}. \end{aligned}

The remaining proof is identical to the one of Lemma 18. $$\square$$

We are now in the position to show the following result.

### Theorem 20

If (69) holds for the solution (uq) of (1) according to Theorem 13 (which is satisfied for $$\varepsilon _0>0$$ small enough) then it satisfies the VoF-type formulation (57)–(59).

Let vice versa $$(u,q,\alpha )$$ be a solution of the VoF-type formulation (57)–(59), where $$\alpha (t)$$ is the indicator function of a domain $$\Omega _1(t)=\{(x,y)\in \mathbb {R}^n\times \mathbb {R}: y=h(t,x)\}$$. If (uqh) has the regularity as in Theorem 13, then (uqh) coincides with the solution of (1) according to Theorem 13.

### Proof

Let (uq) be the solution of (1) according to Theorem 13. Then it solves clearly also the weak formulation (2)–(3). Since the solution of (56) is $$\alpha =1_{\Omega _1}(t)$$ by Proposition 15, the formulations (57)–(59) and (2)–(3) are equivalent if the right hand side of (57) coincides with the surface tension force term (67). To show this we note that Lemma 17 yields for any $$\varepsilon >0$$

\begin{aligned}&-\int _{\mathbb {R}^{n+1}} \sigma \frac{\nu _{\varepsilon }(t,x,y)^\top }{|\nu _{\varepsilon }(t,x,y)|} (D\varphi -{\text{div}}(\varphi )I)(x,y) \nabla \alpha (t,x,y)\,d(x,y)\\ =&\int _{\mathbb {R}^n} \sigma \frac{\nu _{\varepsilon }(t,x,h(t,x))^\top }{|\nu _{\varepsilon }(t,x,h(t,x))|} (D\varphi -{\text{div}}(\varphi )I)(x,h(t,x))\textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }\,dx. \end{aligned}

Now the uniform convergence of $$\nu _{\varepsilon }(t,x,h(t,x))$$ to $$\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right)$$ for $$\varepsilon \searrow 0$$ by Lemma 18 yields the convergence of the above term to (67).

Let vice versa $$(u,q,\alpha )$$ be a solution of the VoF-type formulation (57)–(59) such that (uqh) satisfies the regularity assumptions of Theorem 13. Then again $$\alpha =1_{\Omega _1}(t)$$, where the normal velocity of $$\Gamma (t)$$ is $$u^\top \nu$$. Moreover, $$[u]=0$$ on $$\Gamma (t)$$ by the regularity of u and clearly the first two PDEs in (1) follow. Finally, the jump condition in the third line of (1) follow from (57)–(59) (or (2)–(3)) by choosing test functions of the form

\begin{aligned} \varphi _\tau (t,x,y)&=\phi (t,x) \psi _\tau (y-h(t,x)) \end{aligned}
(73)

with $$\phi \in C_c^\infty (J\times \mathbb {R}^n;\mathbb {R}^{n+1})$$, $$\psi _\tau (s)=\psi (s/\tau )$$, $$\psi \in C_c^1((-1,1))$$, $$\psi \ge 0$$, $$\psi (0)=1$$, $$\psi (-s)=\psi (s)$$ and letting $$\tau \searrow 0$$. $$\square$$

Finally, we can justify the following VoF-type formulation for computing the sensitivities $$(\delta u,\delta q)$$. Due to the limited spatial regularity of $$\partial _t u$$, we have to state time derivatives on the interface in weak form.

\begin{aligned}&\int _{J\times \mathbb {R}^{n+1}} \bigl (\partial _t (\rho (\alpha ) \delta u) +{\text{ div }}(\rho (\alpha ) (\delta u\otimes u+u\otimes \delta u))+\delta c\bigr)^\top \varphi \,d(t,x,y)\nonumber \\&+\int _{J\times \mathbb {R}^{n+1}} S(\delta u,\delta q;\mu (\alpha )):\nabla \varphi \,d(t,x,y)\nonumber \\&+\int _{J}\int _{\mathbb {R}^{n+1}} (\rho _2-\rho _1) u^\top \bigl (\partial _t\varphi +u\cdot \nabla \varphi \bigr ) d\delta \alpha (t)(x,y)\,dt\nonumber \\&-\int _{J}\int _{\mathbb {R}^{n+1}} \left[ S(u,q;\mu (\alpha ))\right] :\nabla \varphi \,d\delta \alpha (t)(x,y) dt= \\&=\lim _{\varepsilon \searrow 0}-\int _{J\times \mathbb {R}^{n+1}} \sigma \left( \frac{\delta \nu _{\varepsilon }^\top }{|\nu _{\varepsilon }|}- \frac{\delta \nu _{\varepsilon }^\top \nu _{\varepsilon }\nu _{\varepsilon }^\top }{|\nu _{\varepsilon }|^3} \right) \left( D\varphi -{\text{ div }}(\varphi )I \right) \nabla \alpha \,d(t,x,y) \nonumber \\&\qquad -\int _{J\times \mathbb {R}^{n+1}} \sigma \frac{\nu _{\varepsilon }^\top }{|\nu _{\varepsilon }|} \left( D\varphi -{\text{ div }}(\varphi )I \right) \nabla d\delta \alpha (t)(x,y)\qquad \forall \,\varphi \in C_c^2(J\times \mathbb {R}^{n+1};\mathbb {R}^{n+1}),\end{aligned}
(74)
$$\int _{J\times \mathbb {R}^{n+1}} {\text{div}}(\delta u)\, \psi \,d(t,x,y)=0\quad \forall \,\psi \in C_c^1 \left( J\times \mathbb {R}^{n+1}\right) ,$$
(75)
\begin{aligned}&\delta \alpha \text{ satisfies } \text{(64) }, \end{aligned}
(76)
\begin{aligned}&\delta u(0)=\delta u_0, \end{aligned}
(77)

where $$\nu _{\varepsilon }$$ and $$\delta \nu _{\varepsilon }$$ are given by (70) and (71).

We need the following Lemma

### Lemma 21

Let $$\psi \in C_c^1(\mathbb {R}^{n+1};\mathbb {R}^{n+1})$$. Then

\begin{aligned}&-\int _{\mathbb {R}^{n+1}} \psi (x,y)^\top \nabla d\delta \alpha (t)(x,y)\\&= \int _{\mathbb {R}^n} \partial _y \psi (x,h(t,x))^\top \left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) \delta h(t,x)+\psi (x,h(t,x))^\top \left( {\begin{array}{c}-\nabla \delta h(t,x)\\ 0\end{array}}\right) \,dx\\&=\int _{\mathbb {R}^n} {\text{div}}(\psi )(x,h(t,x)) \delta h(t,x)\,dx. \end{aligned}

### Proof

By the definition of distributional derivatives one has with (65)

\begin{aligned}&-\int _{\mathbb {R}^{n+1}} \psi (x,y)^\top \nabla d\delta \alpha (t)(x,y)= \int _{\mathbb {R}^{n+1}} {\text{div}}(\psi )(x,y) d\delta \alpha (t)(x,y)\\&=\int _{\mathbb {R}^n} {\text{div}}\left( \psi )(x,h(t,x)\right) \delta h(t,x)\,dx. \end{aligned}

On the other hand, integration by parts yields

\begin{aligned}&\int _{\mathbb {R}^n} \delta h(t,x)\partial _y \psi (x,h(t,x))^\top \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }+\psi \left( x,h(t,x)\right) ^\top \textstyle {\left( {\begin{array}{c}-\nabla \delta h(t,x)\\ 0\end{array}}\right) }\,dx\\&= \int _{\mathbb {R}^n} \delta h(t,x)\partial _y \psi \left( x,h(t,x)\right) ^\top \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }\\&\quad +\left( \sum _{i=1}^n \partial _{x_i}\psi _i \left( x,h(t,x)\right) + \partial _y \psi \left( x,h(t,x)\right) ^\top \textstyle {\left( {\begin{array}{c}\nabla h(t,x)\\ 0\end{array}}\right) }\right) \delta h(t,x)\,dx\\&=\int _{\mathbb {R}^n} {\text{div}}(\psi )\left( x,h(t,x)\right) \delta h(t,x)\,dx. \end{aligned}

$$\square$$

### Theorem 22

Let (uq) be the solution of (1) according to Theorem 13 and let (69) hold (which is satisfied for $$\varepsilon _0>0$$ small enough). Moreover, let $$(\delta u,\delta q)$$ be the sensitivities of (uq) in Theorem 13 corresponding to $$(\delta u_0,\delta c)$$. Then $$(\delta u,\delta q)$$ solve the linearized VoF-type system (73)–(76).

Let vice versa $$(u,q,\alpha )$$ be a solution of the VoF-type formulation (57)–(59), where $$\alpha (t)$$ is the indicator function of a domain $$\Omega _1(t)=\{(x,y)\in \mathbb {R}^n\times \mathbb {R}: y=h(t,x)\}$$. If (uqh) has the regularity as in Theorem 13 and $$(\delta u,\delta q,\delta \alpha )$$ is a solution of (73)–(76) such that $$(\delta u,\delta q)$$ has the regularity as in Theorem 13, then $$(\delta u,\delta q)$$ coincide with the sensitivities according to Theorem 13.

### Proof

Let $$(u_0,h_0,c), (\delta u_0,0,\delta c)\in \mathbb {U}_u(h_0)\times \mathbb {U}_h\times \mathbb {U}_c(t_0)$$ be such that $$(u_0,h_0,c)$$ and $$(u_0,h_0,c)+(\delta u_0,0,\delta c)$$ satisfy the conditions of Theorem 13. Denote now by $$({\hat{u}},\pi ,[\pi ],h)$$ the unique solution of (10) for data $$(u_0,h_0,c)$$ and by $$({\hat{u}}^s,\pi ^s,[\pi ^s],h^s)$$ the one for data $$(u_0,h_0,c)+s\,(\delta u_0,0,\delta c)$$. Let (uq) and $$(u^s,q^s)$$ be the corresponding states in physical coordinates according to (6) and let $$\alpha =1_{\Omega _1(t)}, \alpha ^s=1_{\Omega _1^s(t)}$$ be the corresponding solutions of (56). Finally, let $$(\delta u,\delta h,\delta q)$$ be the directional derivatives (sensitivities) in direction $$(\delta u_0,0,\delta c)$$ which exist by Theorem 13. By the differentiability result of Theorem 13 we know that with the extensions $$u_\pm ,q_\pm$$ in (44), see (43), (45), (46)

\begin{aligned} \frac{u^s-u}{s}&\rightarrow \delta u&\text{ in } C \left( {\bar{J}};L_p \left( \mathbb {R}^{n+1};\mathbb {R}^{n+1}\right) \right) , \end{aligned}
(78)
\begin{aligned} \frac{u_\pm ^s-u_\pm }{s}&\rightarrow \delta u_\pm&\text{ in } L_p \left( J;H_p^1 \left( \mathbb {R}^{n+1};\mathbb {R}^{n+1}\right) \right) ,\end{aligned}
(79)
\begin{aligned} \frac{q_\pm ^s-q_\pm }{s}&\rightarrow \delta q_\pm&\text{ in } L_p \left( J;L_p \left( \mathbb {R}^{n+1};\mathbb {R}^{n+1}\right) \right) . \end{aligned}
(80)

We derive now the different terms in (73). Let

\begin{aligned} \Omega _s=\left\{ (t,x,y):\alpha ^s=\alpha \right\} ,\quad \Omega _s^c=\left\{ (t,x,y):\alpha ^s(t)\ne \alpha \right\}. \end{aligned}

We have for arbitrary $$\varphi \in C_c^2(J\times \mathbb {R}^{n+1};\mathbb {R}^{n+1})$$

\begin{aligned}&\int _{J\times \mathbb {R}^{n+1}}\!\!\!\frac{-1}{s} \left( \left( \rho (\alpha ^s) u^s-\rho (\alpha ) u \right) ^\top \partial _t\varphi +\rho (\alpha ^s) (u^s)^\top (u^s\cdot \nabla \varphi ) -\rho (\alpha ) u^\top (u\cdot \nabla \varphi )\right) \,d(t,x,y)\\ =&\int _{\Omega _s}\frac{-1}{s} \left( \rho (\alpha ) (u^s-u)^\top \partial _t\varphi +\rho (\alpha ) \left( (u^s)^\top (u^s\cdot \nabla \varphi ) -u^\top (u\cdot \nabla \varphi )\right) \right) \,d(t,x,y)\\&+\int _{\Omega _s^c}\frac{-1}{s} \left( \left( \rho (\alpha ^s) u^s-\rho (\alpha ) u \right) ^\top \partial _t\varphi +\rho (\alpha ^s)(u^s)^\top (u^s\cdot \nabla \varphi ) -\rho (\alpha ) u^\top (u\cdot \nabla \varphi )\right) \,d(t,x,y). \end{aligned}

By (77), (78) one obtains

\begin{aligned}&\int _{\Omega _s}\frac{-1}{s} \left( \rho (\alpha ) (u^s-u)^\top \partial _t\varphi +\rho (\alpha ) \left( (u^s)^\top (u^s\cdot \nabla \varphi ) -u^\top (u\cdot \nabla \varphi )\right) \right) \,d(t,x,y)\\&\rightarrow \int _{J\times \mathbb {R}^{n+1}}-\left( \rho (\alpha ) \delta u^\top \partial _t\varphi +\rho (\alpha )(\delta u^\top (u\cdot \nabla \varphi )+u^\top (\delta u\cdot \nabla \varphi ) )\right) \,d(t,x,y)\\&=\int _{J\times \mathbb {R}^{n+1}} \bigl (\partial _t (\rho (\alpha ) \delta u) +{\text{div}}(\rho (\alpha ) (\delta u\otimes u+u\otimes \delta u)))^\top \varphi \,d(t,x,y). \end{aligned}

For the second summand we have by Theorem 13

\begin{aligned}&\int _{\Omega _s^c}\frac{-1}{s} \left( \left( \rho (\alpha ^s) u^s-\rho (\alpha ) u \right) ^\top \partial _t\varphi +\rho (\alpha ^s)(u^s)^\top (u^s\cdot \nabla \varphi ) -\rho (\alpha ) u^\top (u\cdot \nabla \varphi )\right) \,d(t,x,y)\\&=\int _{J\times \mathbb {R}^n}\!\!\!\frac{-1}{s}\int _{h(t,x)}^{\max (h(t,x),h^s(t,x))} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left( \left( \rho _1 u^s-\rho _2 u \right) ^\top \partial _t\varphi +\rho _1(u^s)^\top (u^s\cdot \nabla \varphi ) -\rho _2 u^\top (u\cdot \nabla \varphi )\right) \,d(t,x,y)\\&+\int _{J\times \mathbb {R}^n}\!\!\!\frac{-1}{s}\int _{h^s(t,x)}^{\max (h(t,x),h^s(t,x))} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left( \left( \rho _2 u^s-\rho _1 u \right) ^\top \partial _t\varphi +\rho _2(u^s)^\top (u^s\cdot \nabla \varphi ) -\rho _1 u^\top (u\cdot \nabla \varphi )\right) \,d(t,x,y)\\ \rightarrow&\int _{J\times \mathbb {R}^n} (\rho _2-\rho _1) u^\top \left( \partial _t\varphi +u\cdot \nabla \varphi \right) \left( t,x,h(t,x)\right) \delta h(t,x)\,d(t,x) \\ =&\int _{J}\int _{\mathbb {R}^{n+1}} (\rho _2-\rho _1) u^\top \left( \partial _t\varphi +u\cdot \nabla \varphi \right) d\delta \alpha (t)(x,y)\,dt, \end{aligned}

where we have used (65), (38) and (41) in the last step.

For the next term in (73) we note that

\begin{aligned} \begin{aligned}&\int _{J\times \mathbb {R}^{n+1}} \frac{1}{s}\left( S \left( u^s,q^s;\mu (\alpha ^s)\right) -S \left( u,q;\mu (\alpha )\right) \right) :\nabla \varphi \,d(t,x,y)\\&\quad =\int _{\Omega _s} \frac{1}{s}\left( S(u^s-u,q^s-q;\mu (\alpha )\right) :\nabla \varphi \,d(t,x,y)\\&\qquad +\int _{\Omega _s^c} \frac{1}{s}\left( S(u^s,q^s;\mu (\alpha ^s)\right) - S \left( u,q;\mu (\alpha )\right) :\nabla \varphi \,d(t,x,y). \end{aligned} \end{aligned}
(81)

Now (78), (79) yield

\begin{aligned}&\int _{\Omega _s} \frac{1}{s}\left( S(u^s-u,q^s-q;\mu (\alpha )\right) :\nabla \varphi \,d(t,x,y) \rightarrow \int _{J\times \mathbb {R}^{n+1}} S \left( \delta u,\delta q;\mu (\alpha )\right) :\nabla \varphi \,d(t,x,y). \end{aligned}

Moreover, by using (38) and Theorem 8 we have

\begin{aligned}&\int _{\Omega _s^c} \frac{1}{s}\left( S\left(u^s,q^s;\mu (\alpha ^s)\right) - S \left( u,q;\mu (\alpha )\right)\right) :\nabla \varphi \,d(t,x,y)\\ =&\int _{J\times \mathbb {R}^n}\frac{1}{s}\int _{h(t,x)}^{\max (h(t,x),h^s(t,x))} \left( S(u^s_-,q^s_-;\mu _1)-S(u_+,q_+;\mu _2)\right) :\nabla \varphi \,d(t,x,y)\\&+\int _{J\times \mathbb {R}^n}\frac{1}{s}\int _{h^s(t,x)}^{\max (h(t,x),h^s(t,x))} \left( S(u^s_+,q^s_+;\mu _2)-S(u_-,q_-;\mu _1)\right) :\nabla \varphi \,d(t,x,y)\\ \rightarrow&-\int _{J\times \mathbb {R}^n} \left[ S(u,q;\mu (\alpha ))\right] (t,x,h(t,x))\delta h(t,x):\nabla \varphi (t,x,h(t,x))\,d(t,x)\\ =&-\int _{J}\int _{\mathbb {R}^{n+1}} \left[ S(u,q;\mu (\alpha ))\right] :\nabla \varphi \,d\delta \alpha (t)(x,y) dt. \end{aligned}

Here, we have used (65) and (38) in the last step.

Finally, the surface tension term (67) has with the abbreviations

\begin{aligned} {\tilde{\nu }}(t,x)=\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) ,\quad \delta {\tilde{\nu }}(t,x)=\left( {\begin{array}{c}-\nabla \delta h(t,x)\\ 0\end{array}}\right) \end{aligned}

by Theorem 13 and (41) the directional derivative

\begin{aligned} \begin{aligned}&\int _{J\times \mathbb {R}^{n}} \sigma \left( \frac{\delta {\tilde{\nu }}^\top }{|{\tilde{\nu }}|}- \frac{\delta {\tilde{\nu }}^\top {\tilde{\nu }}{\tilde{\nu }}^\top }{|{\tilde{\nu }}|^3} \right) (t,x)\left( D\varphi -{\text{div}}(\varphi )I \right) \left( t,x,h(t,x)\right) {\tilde{\nu }}(t,x)\,d(t,x)\\&+\int _{J\times \mathbb {R}^{n}} \sigma \frac{{\tilde{\nu }}^\top }{|{\tilde{\nu }}|} \Bigl (\partial _y \left( D\varphi -{\text{div}}(\varphi )I \right) \left( t,x,h(t,x)\right) \delta h(t,x) {\tilde{\nu }}(t,x)\\&\qquad \qquad +\left( D\varphi -{\text{div}}(\varphi )I \right) \left( t,x,h(t,x)\right) \delta {\tilde{\nu }}(t,x)\Bigr )\,d(t,x). \end{aligned} \end{aligned}
(82)

Now the first integral on the right hand side of (73) converges to the first intergal in (81) by first applying Lemma 17 and then Lemmas 18 and 19 . By using first Lemma 21 (note that $$\nu _\varepsilon (t,x,y)$$ depends close to $$\Gamma (t)$$ only on x by (69), see (70)), and then Lemma 18 and the fact that $$\nabla \delta h$$ is continuous by (41), the second integral on the right hand side of (73) converges to the second intergal in (81).

(74), (76) are obvious and (75) follows by Proposition 15.

Let vice versa $$(u,q,\alpha )$$ be a solution of the VoF-type formulation (57)–(59) and $$(\delta u,\delta q,\delta \alpha )$$ a solution of (73)–(76) with the regularities as in Theorem 13. By Theorem 20 (uqh) coincides with the solution of (1) in Theorem 13 and (64) implies by Proposition 15 that $$\delta \alpha$$ and $$\delta h$$ correspond to each other via (65). Hence, (73)–(76) ensure that (uq) satisfy the linearization of (1) on $$\Omega (t)$$ and that $$\delta \alpha$$ provides for given $$\delta u$$ the correct $$\delta h$$.

It remains to show that (73)–(76) implies the correct linearized jump condition. Denote the tested surface tension term from (67) for $$\phi \in C_c^\infty (J\times \mathbb {R}^n;\mathbb {R}^{n+1})$$ by

\begin{aligned} K(h;\phi ):=\int _{J\times \mathbb {R}^n} \sigma\, \textstyle {{\text{div}}_x\left( \frac{\nabla h(t,x)}{\sqrt{1+|\nabla h(t,x)|^2}}\right) \left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) ^\top } \phi (t,x)\,d(t,x). \end{aligned}

The jump condition in strong form is equivalent to

\begin{aligned} \begin{aligned}&-\int _{J\times \mathbb {R}^n} \phi (t,x)^\top \left[ S(u,q;\mu )(t,x,h(t,x))\right] \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }\,d(t,x)= K(h;\phi ) \end{aligned} \end{aligned}
(83)

for all $$\phi \in C_c^\infty (J\times \mathbb {R}^n;\mathbb {R}^{n+1})$$ with $$S(u,q;\mu )=-qI+\mu (\nabla u+\nabla u^\top )$$. In the transformed variables the jump condition reads

\begin{aligned}&-\int _{J\times \mathbb {R}^n} \phi (t,x)^\top \left[ {\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})(t,x,0)\right] \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }\,d(t,x)=K(h;\phi ) \end{aligned}

for all $$\phi \in C_c^\infty (J\times \mathbb {R}^n;\mathbb {R}^{n+1})$$, where with the notation of Remark 1$$~~{\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})=-\pi I+{\hat{\mu }} {{\mathcal {D}}}({\hat{u}},h)$$. Thus, the sensitivities $$\delta {\hat{u}},\delta \pi ,\delta h$$ satisfy the linearized jump condition

\begin{aligned} \begin{aligned}&-\int _{J\times \mathbb {R}^n} \phi (t,x)^\top \left( \left[ \partial _{({\hat{u}},\pi ,h)}{\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})\cdot (\delta {\hat{u}},\delta {\hat{\pi }},\delta h)(t,x,0)\right] \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) } \right. \\&\qquad \qquad \left. +\left[ {\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})(t,x,0)\right] \textstyle {\left( {\begin{array}{c}-\nabla \delta h(t,x)\\ 0\end{array}}\right) }\right) \,d(t,x) =\partial _h K(h;\phi )\cdot \delta h \end{aligned} \end{aligned}
(84)

for all $$\phi \in C_c^\infty (J\times \mathbb {R}^n;\mathbb {R}^{n+1})$$. We show now, that under the regularity ensured by Theorem 13, (83) is implied by the weak formulation (73) by using test functions of the form

\begin{aligned} \varphi _\tau (t,x,y)&=\phi (t,x) \psi _\tau (y-h(t,x)) \end{aligned}
(85)

with $$\phi \in C_c^\infty (J\times \mathbb {R}^n;\mathbb {R}^{n+1})$$, $$\psi _\tau (s)=\psi (s/\tau )$$, $$\psi \in C_c^1((-1,1))$$, $$\psi \ge 0$$, $$\psi (0)=1$$, $$\psi (-s)=\psi (s)$$ for $$\tau \searrow 0$$. Then

\begin{aligned} \nabla \varphi _\tau (t,x,y)&= \psi _\tau '(y-h(t,x)) \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) } \phi (t,x)^\top + \psi _\tau (y-h(t,x)) \textstyle {\left( {\begin{array}{c}\nabla \phi (t,x)\\ 0\end{array}}\right) },\nonumber \\ \partial _t \varphi _\tau (t,x,y)&=- \phi (t,x)\psi _\tau '(y-h(t,x))\partial _t h(t,x)+ \partial _t \phi (t,x) \psi _\tau (y-h(t,x)).\nonumber \end{aligned}

We test the weak form (57) also in time and rewrite it in the transformed variables. This results in

\begin{aligned} \nonumber&\int _{J\times \mathbb {R}^{n+1}} \left( \left( \partial _t (\rho (\alpha ) u)+{\text{div}}(\rho (\alpha ) u\otimes u)\right) (t,x,y)^\top \varphi \,+S(u,q;\mu ):\nabla \varphi \right) d(t,x,y)=\\&\int _{J\times \mathbb {R}^{n+1}} \Bigl (-{\hat{\rho }}(x,y) {\hat{u}}^\top \bigl (\partial _t \varphi (t,x,y+h(t,x))+{\hat{u}}\cdot \nabla \varphi (t,x,y+h(t,x))\bigr )\\\nonumber&\quad + {\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})(t,x,y):\nabla \varphi (t,x,y+h(t,x))\Bigr )\,d(t,x,y) =K(h;\varphi (\cdot ,\cdot ,h(\cdot ,\cdot ))). \end{aligned}
(86)

For the right hand side we have used that as in the proof of Theorem 20 the right hand side of (57) coincides with (67).

For the test function (84) we obtain

\begin{aligned}&\partial _t \varphi _\tau (t,x,y+h(t,x))+{\hat{u}}\cdot \nabla \varphi _\tau (t,x,y+h(t,x))\\&= -\phi (t,x) \psi _\tau '(y) (\partial _th(t,x)+v^\top \nabla h(t,x)-w) +\psi _\tau (y) (\partial _t \phi (t,x)+v\cdot \nabla \phi (t,x)). \end{aligned}

Moreover, for any $$({\bar{x}},{\bar{y}})\in \Gamma ({\bar{t}})$$ and (x(t), y(t)) with $$(x'(t),y'(t))=u(x(t),y(t))$$, $$(x({\bar{t}}),y({\bar{t}}))=({\bar{x}},{\bar{y}})$$ one has $$y(t)-h(t,x(t))=0$$ and thus

\begin{aligned} 0&=y'(t)-\partial _t h(t,x(t))-\partial _x h(t,x(t)) x'(t)\\&= w(t,x(t),y(t))-\partial _t h(t,x(t))-v(t,x(t),y(t))^\top \nabla h(t,x(t)). \end{aligned}

Hence,

\begin{aligned} \partial _t \varphi _\tau (t,x,y+h(t,x))+{\hat{u}}\cdot \nabla \varphi _\tau (t,x,y+h(t,x)) =\psi _\tau (y) (\partial _t \phi (t,x)+v\cdot \nabla \phi (t,x)), \end{aligned}

and inserting $$\varphi _\tau$$ in (85) yields

\begin{aligned}&\int _{J\times \mathbb {R}^{n+1}} \Bigl (-{\hat{\rho }}(x,y) {\hat{u}}^\top \psi _\tau (y) (\partial _t \phi (t,x)+v\cdot \nabla \phi (t,x))\\&\quad + {\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }}): \left( \phi (t,x) \psi _\tau '(y) \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }+ \psi _\tau (y) \textstyle {\left( {\begin{array}{c}\nabla \phi (t,x)\\ 0\end{array}}\right) }\right) \Bigr )\,d(t,x,y)=K(h;\phi ). \end{aligned}

We have already observed that the right hand side of (57) coincides with (67) and the right hand side of (73) with the derivative (81) of (67). Since $$K(h;\phi )$$ corresponds to using $$\varphi _\tau$$ in (67), $$\partial _h K(h;\phi )\cdot \delta h$$ can be expressed as the sum of the right hand side of (73) with $$\varphi =\varphi _\tau$$ and of the right hand side of (57) with $$\varphi =-\partial _y \varphi _\tau \delta h$$ (note again that $$\varphi _\tau$$ depends on h).

Similarly taking the derivative of the left hand side of (85) corresponds to the sum of the left hand side of (73) with $$\varphi =\varphi _\tau$$ and of the left hand side of (85) with $$\varphi =-\partial _y \varphi _\tau \delta h$$ (note that $$\varphi _\tau$$ depends on h) and is given by (we differentiate equivalently the middle term of (85) in transformed variables)

\begin{aligned}&\int _{J\times \mathbb {R}^{n+1}} \Bigl (-{\hat{\rho }}(x,y) \psi _\tau (y) \bigl (\delta {\hat{u}}^\top (\partial _t \phi (t,x)+v\cdot \nabla \phi (t,x)) + {\hat{u}}^\top (\partial _t \phi (t,x)+\delta v\cdot \nabla \phi (t,x))\bigr )\\&\quad + \partial _{({\hat{u}},\pi ,h)}{\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})(x,0)\cdot (\delta {\hat{u}},\delta {\hat{\pi }},\delta h): \left( \phi (t,x) \psi _\tau '(y) \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }+ \psi _\tau (y) \textstyle {\left( {\begin{array}{c}\nabla \phi (t,x)\\ 0\end{array}}\right) }\right) \\&\quad + {\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})(x,0): \phi (t,x) \psi _\tau '(y) \textstyle {\left( {\begin{array}{c}-\nabla \delta h(t,x)\\ 0\end{array}}\right) }\Bigr ) \,d(t,x,y)=\partial _h K(h;\phi )\cdot \delta h. \end{aligned}

By the assumed regularity for $$\tau \searrow 0$$ all terms containing the factor $$\psi _\tau (y)$$ tend to zero and the remaining terms converge to

\begin{aligned}&\int _{J\times \mathbb {R}^{n}} \left( -\left[ \partial _{({\hat{u}},\pi ,h)}{\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})(x,0)\cdot (\delta {\hat{u}},\delta {\hat{\pi }},\delta h)\right] : \textstyle {\left( {\begin{array}{c}-\nabla h(t,x)\\ 1\end{array}}\right) }\right. \\&\qquad \left. - \left[ {\hat{S}}({\hat{u}},\pi ,h;{\hat{\mu }})(x,0)\right] : \textstyle {\left( {\begin{array}{c}-\nabla \delta h(t,x)\\ 0\end{array}}\right) }\right) \phi (t,x) \,d(t,x)=\partial _h K(h;\phi )\cdot \delta h. \end{aligned}

This is exactly the linearized jump condition (83). $$\square$$

## 4 Analytical settings for the application of optimization methods

The results of this paper justify the application of derivative based optimization methods. We discuss now some possible settings. While some are canonical, the treatment of optimization problems involving the state across the interface, in particular the pressure or the position of the interface, requires care, since the pressure and phase indicator field are discontinuous across the interface.

Let $$p>n+3$$ and $$\mathbb {U}_{u}(h_0), \mathbb {U}_{c}(t_0)$$ be defined by (35). Let $$h_0\in \mathbb {U}_h$$ with $$\Vert h_0\Vert _{\mathbb {U}_h}$$ small enough

\begin{aligned} U_{ad}:=\left\{ (u_0,c)\in \mathbb {U}_u(h_0)\times \mathbb {U}_{c}(t_0): (u_0,h_0,c) \text{ satisfy } \text{(14), } \text{(51) }\right\} \end{aligned}

and consider the control-to-state mapping

\begin{aligned} (u_0,c)\in U_{ad} \subset \mathbb {U}_u(h_0)\times \mathbb {U}_{c}(t_0) \mapsto ({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0) \end{aligned}

given by (10), which is differentiable at least for sufficiently small controls by Theorem 13, and the corresponding original state

\begin{aligned} (u,q)(t,x,y)=({\hat{u}},\pi )(1,x,y-h(t,x)) \end{aligned}

solving (1). Denote the solution operator by

\begin{aligned} (u,q)=({{\mathcal {S}}}^u(u_0,c),{{\mathcal {S}}}^q(u_0,c))={{\mathcal {S}}}(u_0,c). \end{aligned}

Now we are interested in optimization problems of the form

\begin{aligned} \min _{(u_0,c)} {{\mathcal {J}}}({{\mathcal {S}}}(u_0,c),u_0,c) \text{ s.t. } (u_0,c) \in U_{ad}, \end{aligned}

where additional constraints would be possible. We discuss now analytical settings, for which the continuous differentiability of the reduced objective function $$(u_0,c) \in U_{ad}\mapsto {{\mathcal {J}}}({{\mathcal {S}}}(u_0,c),u_0,c)$$ is ensured.

### 4.1 Objective functions involving the velocity field

We consider first the simpler case, where the state-dependence of the objective function involves only the velocity, i.e.,

\begin{aligned} (u_0,c) \in U_{ad}\mapsto {{\mathcal {J}}}({{\mathcal {S}}}(u_0,c),u_0,c)= {{\mathcal {J}}}^u({{\mathcal {S}}}^u(u_0,c),u_0,c). \end{aligned}
(87)

Using the differentiability results of Theorem 13 with the space given in (43), the reduced objective functional (86) is continuously differentiable, if the mapping

\begin{aligned} (u,u_0,c)\in C \left({\bar{J}}; L_{{\tilde{p}}}\left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right)\right) \times \mathbb {U}_u(h_0)\times \mathbb {U}_{c}(t_0) \mapsto {{\mathcal {J}}}^u(u,u_0,c) \end{aligned}

is continuously differentiable for some $${\tilde{p}}\in [p,\infty )$$. This applies for example in the case of least squares functionals or many other types of tracking functionals. The derivative is easily obtained by the chain rule and by using the sensitivities $$\delta u$$, where $$\delta u$$ can be obtained by the VoF-type formulation (73) or by using the linearization of the transformed problem (10) together with Lemma 11 and Lemma 12.

If the objective function (86) evaluates the velocity field u only in an open observation domain $$J\times \Omega _o$$ with positive distance from the interface $$\bigcup _{t\in J} (\{t\}\times \Gamma (t))$$, then it is by Theorem 13 sufficient if the mapping

\begin{aligned}&(u,u_0,c)\in \left( C \left( {\bar{J}}; L_{{\tilde{p}}}\left( \mathbb {R}^{n+1},\mathbb {R}^{n+1}\right) \right) \cap L_p \left( J;H_p^1 \left( \Omega _o,\mathbb {R}^{n+1}\right) \right) \right) \times \mathbb {U}_u(h_0)\times \mathbb {U}_{c}(t_0) \\&\quad \mapsto {{\mathcal {J}}}^u(u,u_0,c) \end{aligned}

is continuously differentiable for some $${\tilde{p}}\in [p,\infty )$$.

### 4.2 Objective functions involving the pressure or phase indicator

Since the pressure field is discontinuous across the interface, the differentiability results of Theorem 13 apply only for extensions $$q_\pm$$ of the pressure across the interface. We show now that certain types of objective functionals are nevertheless differentiable. As before we work with the unique representative of the pressure satisfying (36).

We consider objective functions of the form

\begin{aligned} \begin{aligned} (u_0,c) \in U_{ad}&\mapsto {{\mathcal {J}}}({{\mathcal {S}}}(u_0,c),u_0,c) ={{\mathcal {J}}}^q({{\mathcal {S}}}^q(u_0,c),u_0,c)\\&=\int _J \int _{\mathbb {R}^{n+1}} \ell ({{\mathcal {S}}}^q(u_0,c)(t,x,y),t,x,y)\,d(x,y)\,dt. \end{aligned} \end{aligned}
(88)

Here, $$\ell : \mathbb {R}\times J\times \mathbb {R}^n\times \mathbb {R}\rightarrow \mathbb {R}$$ is a continuous function, twice differentiable with respect to q and y such that there are $$R>0, C_\ell >0$$ with (other settings are possible)

\begin{aligned} \begin{aligned}&{\text {supp}}(\ell (q,\cdot ))\subset \left\{ (t,x,y)\in J\times \mathbb {R}^n\times \mathbb {R}:\, |(x,y)|\le R \right\} \quad \forall \,q\in \mathbb {R},\\&|\partial ^2_{(q,y)} \ell (q,t,x,y)|\le C_\ell \left( 1+|q|^{p-2}\right) \quad \forall \,(t,x,y)\in J\times \mathbb {R}^n\times \mathbb {R}. \end{aligned} \end{aligned}
(89)

This implies the estimate

\begin{aligned} |\partial _q\ell (q,t,x,y)|&\le |\partial _q\ell (0,t,x,y)|+\left| \int _0^q \partial ^2_{qq} \ell (s,t,x,y)\,ds\right| \le C_\ell ' (1+|q|^{p-1}) \end{aligned}

with the constant $$C_\ell '=\Vert \partial _q\ell (0,\cdot )\Vert _{L_\infty (\mathbb {R}^{n+1})}+2 C_\ell$$ as well as

\begin{aligned} |\ell (q,x,y)|\le |\ell (0,x,y)|+\left| \int _0^q \partial _q \ell (s,x,y)\,ds\right| \le C_\ell '' (1+|q|^{p}) \end{aligned}
(90)

with $$C_\ell ''=\Vert \ell (0,\cdot )\Vert _{L_\infty (\mathbb {R}^{n+1})}+2 C_\ell '$$.

Now, using the transformation $$({\hat{x}},{\hat{y}})=T_{h(t)}(x,y)=(x,y-h(t,x))$$ we can rewrite (87) as

\begin{aligned} \begin{aligned} {{\mathcal {J}}}^q(q,u_0,c)&= \int _J \int _{\mathbb {R}^{n+1}} \ell \left( q(t,x,y),t,x,y \right) \,d(x,y)\,dt\\&= \int _J \int _{\mathbb {R}^{n+1}} \ell \left( \pi (t,x,y-h(t,x)),t,x,y \right) \,d(x,y)\,dt\\&=\int _J \int _{\mathbb {R}^{n+1}} \ell \left( \pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}})\right) \,d({\hat{x}},{\hat{y}})\,dt\\&=\int _J \int _{\dot{R}^{n+1}} \ell \left( \pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}})\right) \,d({\hat{x}},{\hat{y}})\,dt. \end{aligned} \end{aligned}
(91)

We will now show the following result.

### Theorem 23

Let (88) hold. Then with the convention (36) the objective function (87) is continuously differentiable with derivative

\begin{aligned} ({{\mathcal {J}}}^q)' \left( {{\mathcal {S}}}^q(u_0,c),u_0,c \right) \cdot \left( {\begin{array}{c}\delta u_0\\ \delta c\end{array}}\right)&=\int _J \int _{{\dot{\mathbb {R}}}^{n+1}} \partial _q\ell \left( q(t,x,y),t,x,y \right) \delta q(t,x,y)\, d(x,y)\,dt\\&\quad-\int _J \int _{\mathbb {R}^{n}} \left[ \ell \left( q(t,x,h(t,x)),t,x,h(t,x)\right) \right] \delta h(t,x)\,dx\,dt\\&=\int _J \int _{{\dot{\mathbb {R}}}^{n+1}} \partial _q\ell \left( q(t,x,y),t,x,y \right) \delta q(t,x,y)\, d(x,y)\,dt\\&\quad-\int _J \int _{\mathbb {R}^{n}} \left[ \ell \left( q(t,x,h(t,x)),t,x,h(t,x)\right) \right] d\delta \alpha (t)(x,y)\,dt, \end{aligned}

where $$[\ell (q(t,x,h(t,x)),t,x,h(t,x))]$$ is the jump across $$\Gamma (t)$$ at (xh(tx)) and the sensitivities $$\delta q$$, $$\delta \alpha$$ can be obtained by the VoF-type formulation (73) or $$\delta \pi$$, $$\delta h$$ can be computed by using the linearization of the transformed problem (10) together with Lemma 12 and $$\delta q$$ by using Lemma 11.

### Remark 24

If we consider an objective function $${{\mathcal {J}}}^\alpha$$ of the form (87) with the pressure q replaced by the phase indicator $$\alpha$$ then an analogue of Theorem 23 holds and the derivative simplifies to

\begin{aligned}&({{\mathcal {J}}}^\alpha )'({{\mathcal {S}}}^\alpha (u_0,c),u_0,c)\cdot \left( {\begin{array}{c}\delta u_0\\ \delta c\end{array}}\right) \\&\quad =-\int _J \int _{\mathbb {R}^{n}} \left[ \ell \left( \alpha (t,x,h(t,x)),t,x,h(t,x)\right) \right] d\delta \alpha (t)(x,y)\,dt, \end{aligned}

since $$\delta \alpha$$ has its support on the complement of $${\dot{\mathbb {R}}}^{n+1}$$.

### Remark 25

A practically relevant example satisfying (88) is a tracking type functional for the pressure q or the phase indicator $$\alpha$$ of the form

\begin{aligned} {{\mathcal {J}}}^q \left( {{\mathcal {S}}}^q(u_0,c),u_0,c)\right) = \int _J \int _{\mathbb {R}^{n+1}} \psi (x,y) \left( {{\mathcal {S}}}^q(u_0,c)-q_d \right) ^2(t,x,y)\,d(x,y)\,dt,\quad \end{aligned}

with desired pressure field $$q_d\in BC^2(J\times \mathbb {R}^{n+1})$$ and a weighting function $$\psi \in C_c^2(\mathbb {R}^{n+1})$$ (or similarly with $${{\mathcal {S}}}^\alpha$$ and desired phase indicator $$\alpha _d\in BC^2(J\times \mathbb {R}^{n+1};[0,1])$$). Note that discontinuous $$q_d$$ can lead to nonsmoothness if the jump set of q and $$q_d$$ coincide on a set of positiv surface measure.

In order to cover discontinuous $$q_d$$ (or $$\alpha _d$$) a variant with smoothed observation of the form

\begin{aligned} {{\mathcal {J}}}^q \left( {{\mathcal {S}}}^q(u_0,c),u_0,c)\right) = \int _J \int _{\mathbb {R}^{n+1}} \psi (x,y) \left( \phi * \left( {{\mathcal {S}}}^q(u_0,c)-q_d \right) \right) ^2(t,x,y)\,d(x,y)\,dt \end{aligned}

with a mollifier $$\phi \in C_c^2(\mathbb {R}^{n+1})$$ (the convolution is only in space) can be shown to be continuously differentiable similarly as for (87) under assumption (88).

We use the following auxiliary result.

### Lemma 26

Let (88) hold. Then the mapping

\begin{aligned}&(\pi ,h)\in L_p(J;L_\infty (\mathbb {R}^{n+1}))\times L_p(J;L_\infty (\mathbb {R}^{n}))\\&\mapsto \widehat{{\mathcal {J}}}^q(\pi ,h):= \int _J \int _{\dot{R}^{n+1}} \ell (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}}))\,d({\hat{x}},{\hat{y}})\,dt \end{aligned}

is continuously differentiable with derivative

\begin{aligned}&(\widehat{{\mathcal {J}}}^q)'(\pi ,h)\cdot \left( {\begin{array}{c}\delta \pi \\ \delta h\end{array}}\right) \\&\quad =\int _J \int _{\dot{R}^{n+1}} \left( {\begin{array}{c}\partial _q\ell \\ \partial _y\ell \end{array}}\right) (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}}, {\hat{y}}+h(t,{\hat{x}}))^\top \left( {\begin{array}{c}\delta \pi (t,{\hat{x}},{\hat{y}})\\ \delta h(t,{\hat{x}})\end{array}}\right) \,d({\hat{x}},{\hat{y}})\,dt. \end{aligned}

### Proof

We have by using (88) and its consequence (89)

\begin{aligned} \widehat{{\mathcal {J}}}^q(\pi ,h)&\le \int _J \int _{\dot{R}^{n+1}} |\ell (\pi (t,x,y-h(t,x)),t,x,y)|\,d(x,y)\,dt\\&\le \int _J \int _{|(x,y)|\le R} C_\ell '' (1+|\pi (t,x,y-h(t,x))|^p)\,d(x,y)\,dt\\&\le C_\ell '' (2R)^{n+1} \int _J (1+\Vert \pi (t,\cdot )\Vert _{L_\infty (\mathbb {R}^{n+1})}^p) \le C_\ell '' (2R)^{n+1} (t_0+\Vert \pi \Vert _{L_p(J;L_\infty (\mathbb {R}^{n+1}))}^p). \end{aligned}

To show the differentiability, we note that by Taylor expansion and (89)

\begin{aligned}&\Biggl |\ell ((\pi +\delta \pi )(t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+(h+\delta h)(t,{\hat{x}})) -\ell (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}}))\\&-\left( {\begin{array}{c}\partial _q\ell \\ \partial _y\ell \end{array}}\right) (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}}, {\hat{y}}+h(t,{\hat{x}}))^\top \left( {\begin{array}{c}\delta \pi (t,{\hat{x}},{\hat{y}})\\ \delta h(t,{\hat{x}})\end{array}}\right) \Biggr |\\ \le&\, C_\ell (1+(|\pi (t,{\hat{x}},{\hat{y}})|+|\delta \pi (t,{\hat{x}},{\hat{y}})|)^{p-2}) (|\delta \pi (t,{\hat{x}},{\hat{y}})|^2+|\delta h(t,{\hat{x}})|^2). \end{aligned}

Hence, we obtain with the asserted derivative by using (88)

\begin{aligned}&\left| \widehat{{\mathcal {J}}}^q(\pi +\delta \pi ,h+\delta h)-\widehat{{\mathcal {J}}}^q(\pi ,h) -(\widehat{{\mathcal {J}}}^q)'(\pi ,h)\cdot \left( {\begin{array}{c}\delta \pi \\ \delta h\end{array}}\right) \right| \\&\le \int _J \int _{\genfrac{}{}{0.0pt}2{{\dot{\mathbb {R}}}^{n+1}\cap ~~}{[-R,R]^{n+1}}} \!\!\!\!C_\ell \left( 1+ \left( |\pi (t,{\hat{x}},{\hat{y}})|+|\delta \pi (t,{\hat{x}},{\hat{y}})|\right) ^{p-2}\right) \left( |\delta \pi (t,{\hat{x}},{\hat{y}})|^2+|\delta h(t,{\hat{x}})|^2 \right) \,d({\hat{x}},{\hat{y}})\,dt\\&\le (2R)^{n+1} C_\ell \int _J \left( 1+\Vert |\pi (t,\cdot )|+|\delta \pi (t,\cdot )|\Vert _{L_\infty }^{p-2}\right) \cdot \left( \Vert \delta \pi (t,\cdot )\Vert _{L_\infty }^2+ \Vert \delta h(t,\cdot )\Vert _{L_\infty }^2 \right) \,dt\\&\le (2R)^{n+1} C_\ell \left( t_0^{(p-2)/p}+\Vert |\pi |+|\delta \pi |\Vert _{L_p(J;L_\infty )}^{p-2}\right) \left( \Vert \delta \pi \Vert _{L_p(J;L_\infty )}^2+ \Vert \delta h\Vert _{L_p(J;L_\infty )}^2 \right) \\&=O \left( \Vert \delta \pi \Vert _{L_p(J;L_\infty )}^2+\Vert \delta h\Vert _{L_p(J;L_\infty )}^2 \right) , \end{aligned}

where we have applied Hölder’s inequality in the last step. The continuity of the derivative follows with (89) by very similar calculations. $$\square$$

### Proof of Theorem 23

From Theorem 8, Theorem 10 and Lemma 12 we know that

\begin{aligned} (u_0,c) \in U_{ad}\mapsto ({\hat{u}},\pi ,[\pi ],h) \in \mathbb {E}(t_0) \end{aligned}

is continuously differentiable and with the convention (36) the mapping $$({\hat{u}},\pi ,[\pi ],h) \in \mathbb {E}(t_0) \mapsto \pi \in L_p(J;H_p^1({\dot{\mathbb {R}}}^{n+1}))$$ is linear and continuous by (37). Moreover, by $$p>n+3$$ we have clearly $$L_p(J;H_p^1({\dot{\mathbb {R}}}^{n+1}))\hookrightarrow L_p(J;L_\infty (\mathbb {R}^{n+1}))$$ and $$\mathbb {E}_4(t_0) \hookrightarrow L_p(J;L_\infty (\mathbb {R}^{n}))$$. Hence, the mapping

\begin{aligned} (u_0,c) \in U_{ad}\mapsto (\pi ,h) \in L_p(J;L_\infty (\mathbb {R}^{n+1}))\times L_p(J;L_\infty (\mathbb {R}^{n})) \end{aligned}

is continuously differentiable and thus (87) is continuously differentiable by Lemma 26 and the last representation of (87) in (90). The derivative is given by Lemma 26. Using that

\begin{aligned} \frac{d}{d{\hat{y}}} \ell (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}})) =&\,\partial _y\ell (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}}))\\&+\partial _q\ell (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}})) \partial _{{\hat{y}}}\pi (t,{\hat{x}},{\hat{y}}), \end{aligned}

integration by parts yields

\begin{aligned} (\widehat{{\mathcal {J}}}^q)'(\pi ,h)\cdot \left( {\begin{array}{c}\delta \pi \\ \delta h\end{array}}\right) =&\int _J \int _{\dot{R}^{n+1}} \left( {\begin{array}{c}\partial _q\ell \\ \partial _y\ell \end{array}}\right) (\pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}}, {\hat{y}}+h(t,{\hat{x}}))^\top \left( {\begin{array}{c}\delta \pi (t,{\hat{x}},{\hat{y}})\\ \delta h(t,{\hat{x}})\end{array}}\right) \,d({\hat{x}},{\hat{y}})\,dt\\ =&\,\int _J \int _{\dot{R}^{n+1}} \partial _q\ell \left( \pi (t,{\hat{x}},{\hat{y}}),t,{\hat{x}},{\hat{y}}+h(t,{\hat{x}})\right) \left( -\partial _{{\hat{y}}}\pi (t,{\hat{x}},{\hat{y}}) \delta h(t,{\hat{x}})+ \delta \pi (t,{\hat{x}},{\hat{y}})\right) \, d({\hat{x}},{\hat{y}})\,dt\\&-\int _J \int _{\mathbb {R}^{n}} \left[ \ell (\pi (t,{\hat{x}},0),t,{\hat{x}},h(t,{\hat{x}}))\right] \delta h(t,{\hat{x}})\,d{\hat{x}}\,dt\\ =&\int _J \int _{\mathbb {R}^{n+1}} \partial _q\ell \left( q(t,x,y),t,x,y \right) \delta q(t,x,y)\, d({\hat{x}},{\hat{y}})\,dt\\&-\int _J \int _{\mathbb {R}^{n}} \left[ \ell \left( q \left( t,x,h(t,x)\right) ,t,x,h(t,x)\right) \right] \delta h(t,x)\,dx\,dt. \end{aligned}

Here, we have used Lemma 11 and (65) in the last step. $$\square$$