Abstract
We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on \(L_p\)-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting from the \(L_p\)-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.
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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
and with the indicator functions \(1_{\Omega _i}\)
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
(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
denotes the jump of v accross \(\Gamma (t)\). The two-phase Navier-Stokes equations with surface tension then read
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
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.,
The interface has then the form
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
where \(\nabla h\) and \(\Delta h\) denote the gradient and Laplacian of h with respect to x and
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
Analogously, let with \(\mathbb {R}_\pm ^{n+1}=\{(x,y)\in \mathbb {R}^n\times \mathbb {R}:\, \pm y>0\}\)
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
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
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
and denote by
(i.e., \(r=[\pi ]\) by the definition of \(\mathbb {E}(t_0)\)) the Stokes problem with free boundary
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 \({\hat{u}}=(v,w), \pi , h\)
where the right hand sides are given by \(F=(F_v,F_w)\) and
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
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
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
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
satisfying, with \(u_0(x,h_0(x)+y)={\hat{u}}_0(x,y)\), the compatibility conditions
as well as the smallness condition
there exists a unique solution of the transformed problem (10) with
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
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
there exists a unique solution \(({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0)\) of (9) and the solution map
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
with initial value 0 for all components that admit a trace at \(t=0\). Then (9) has a unique and continuous solution map
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 \(({\hat{u}},\pi ,r,h)\in \mathbb {E}(t_0)\)
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,
Moreover,
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
Then with \(G=(G_v,G_w)\) and H defined in (11) the mappings
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
is well defined and continuous, see [6, Thm. 1.1]. It is also differentiable. In fact, for \(d\in \mathbb {U}_h\)
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
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
-
a)
Let U, W, Z 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) -
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
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
Here, the Landau symbol \(o_W\) indicates, that the term is considered in the space W. By using (24) we conclude that for \(d\in U_L\), \(\Vert d\Vert _U\rightarrow 0\)
If \(DK: B_Z\times B_U \rightarrow {{\mathcal {L}}}(Z\times U_L,W)\) is Lipschitz continuous then (25) can be written as
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 \(J=(0,t_0)\)
Then for any \(t_0>0\) there exists \({\hat{\varepsilon }}_0={\hat{\varepsilon }}_0(t_0)>0\) such that for all data
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
there exists a unique solution of the transformed problem (10) with
Moreover, the mapping
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)
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
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
The construction of \(z^*\) can be accomplished as in [22]. Set
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
Now let \(D_{n}=-\Delta\) be the Laplacian in \(L_p(\mathbb {R}^{n})\) with domain \(H_p^2(\mathbb {R}^{n})\) and set
By the real analyticity of \(r_0({\hat{u}}_0,h_0)\) and Lemma 6 the mappings
are real analytic. Now maximal \(L_p\)-regularity for \(D_n\) yields, see e.g. [11, Lem. 8.2]
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
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
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
To apply Theorem 7 we set now with suitable \({\hat{\varepsilon }}_0>0\) and \(\delta >0\)
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
is real analytic with
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 \(({\tilde{z}},{\hat{u}}_0,h_0,{\hat{c}})\in B_Z(\delta )\times B_U({\hat{\varepsilon }}_0)\)
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
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))
Then with the convention (36) and by the trace theorem, we find a Poincaré constant \(C_P>0\) with
The following imbeddings will be useful.
Lemma 9
Let \(p>n+3\). Then the following imbeddings hold with \(J=(0,t_0)\), \(t_0>0\).
Proof
For the imbeddings (38), (41) see [22, Lem. 6.1]. Moreover, it is obvious that
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),
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
Proof
Let \(({\hat{u}},\pi ,[\pi ],h)\in \mathbb {E}(t_0)\) and consider, see (6),
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
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
Completely analogous one obtains
Now consider \({\hat{r}}=[\pi ]\) and \(r=[q]\) then \(r(t,x,h(t,x))={\hat{r}}(t,x)\) and
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
is continuously differentiable with derivative
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
Then the mappings
are continuously differentiable with derivative
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
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
Consider the remainder term
Let \(p\le {\tilde{p}}<\infty\) be arbitrary. We obtain
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
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
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
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
with \({\hat{c}}(c,h)(t,x,y)=c(t,x,y+h(t,x))\) is continuously differentiable with derivative
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
satisfying the compatibility condition (14) as well as the smallness condition
there exists a unique solution of the transformed problem (10) with
Moreover, for any \(h_0\) with \(\Vert h_0\Vert _{\mathbb {U}_h}<\varepsilon _0\) the mapping
is continuously differentiable.
By the chain rule in Lemma 11, also the original state (u, q) 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
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
Let \({\hat{\varepsilon }}_0>0\) be as in Theorem 8. We have
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)
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 (u, q) 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 (u, q) 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
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
We define now
We will show that the unique solution (u, q) of (1) according to Theorem 13 satisfies the VoF-type formulation
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
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(t; x, y) satisfies the characteristic equation
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
Here, X is the forward flow defined by (61) and \(\delta \alpha _t=X(t)(\delta \alpha _0)\) is the measure satisfying
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
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
is continuously differentiable. The derivative
is given by the unique measure-valued solution of
Finally, \(\delta \alpha\) satisfies
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 (u, q) 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
where \(\delta \alpha\) solves (64). Let \(\phi \in C_c(\mathbb {R}^{n+1})\) be arbitrary. Then
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
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
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]
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
Proof
By the definition of distributional derivatives one has
\(\square\)
Let now \(\delta \in (0,1/2)\) and
and set
To recover a mollified normal (not necessarily of unit length) we use
Then by Lemma 17
Now assume that
Then we have by the definition of \(\phi _{\varepsilon }\)
Lemma 18
Let (69) hold. If \(h\in C({\bar{J}};BC^2(\mathbb {R}^n))\) then there is \(C>0\) such that
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
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
On compact subsets the error is \(o(\varepsilon )\).
Proof
Setting \(y=h(t,x)\) and using (69) we obtain
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 (u, q) 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 (u, q, h) has the regularity as in Theorem 13, then (u, q, h) coincides with the solution of (1) according to Theorem 13.
Proof
Let (u, q) 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 \(\varepsilon >0\)
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 (u, q, h) 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
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.
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
Proof
By the definition of distributional derivatives one has with (65)
On the other hand, integration by parts yields
\(\square\)
Theorem 22
Let (u, q) 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 (u, q) 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 (u, q, h) 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 (u, q) 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)
We derive now the different terms in (73). Let
We have for arbitrary \(\varphi \in C_c^2(J\times \mathbb {R}^{n+1};\mathbb {R}^{n+1})\)
For the second summand we have by Theorem 13
where we have used (65), (38) and (41) in the last step.
For the next term in (73) we note that
Moreover, by using (38) and Theorem 8 we have
Here, we have used (65) and (38) in the last step.
Finally, the surface tension term (67) has with the abbreviations
by Theorem 13 and (41) the directional derivative
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 (u, q, h) 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 (u, q) 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
The jump condition in strong form is equivalent to
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
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
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
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
We test the weak form (57) also in time and rewrite it in the transformed variables. This results in
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
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
Hence,
and inserting \(\varphi _\tau\) in (85) yields
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)
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
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
and consider the control-to-state mapping
given by (10), which is differentiable at least for sufficiently small controls by Theorem 13, and the corresponding original state
solving (1). Denote the solution operator by
Now we are interested in optimization problems of the form
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.,
Using the differentiability results of Theorem 13 with the space given in (43), the reduced objective functional (86) is continuously differentiable, if the mapping
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
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
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)
This implies the estimate
with the constant \(C_\ell '=\Vert \partial _q\ell (0,\cdot )\Vert _{L_\infty (\mathbb {R}^{n+1})}+2 C_\ell\) as well as
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
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
where \([\ell (q(t,x,h(t,x)),t,x,h(t,x))]\) is the jump across \(\Gamma (t)\) at (x, h(t, x)) 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
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
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
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
is continuously differentiable with derivative
Proof
We have by using (88) and its consequence (89)
To show the differentiability, we note that by Taylor expansion and (89)
Hence, we obtain with the asserted derivative by using (88)
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
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
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
integration by parts yields
Here, we have used Lemma 11 and (65) in the last step. \(\square\)
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Change history
15 November 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10589-022-00423-6
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Acknowledgements
The authors would like to thank the referee for his/her helpful comments.
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Open Access funding enabled and organized by Projekt DEAL. The work of Johannes Haubner and Michael Ulbrich was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project Number 188264188/GRK1754—as part of the International Research Training Group IGDK 1754 Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures. The work of Elisabeth Diehl and Stefan Ulbrich was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 265191195—SFB 1194 Interaction between Transport and Wetting Processes, Project B04.
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Diehl, E., Haubner, J., Ulbrich, M. et al. Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension. Comput Optim Appl 87, 711–751 (2024). https://doi.org/10.1007/s10589-022-00415-6
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DOI: https://doi.org/10.1007/s10589-022-00415-6
Keywords
- Two-phase flow
- Surface tension
- Sharp interface
- Navier-Stokes equations
- Volume of fluid
- Differentiability
- Optimal control