Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

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 form 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.

The conditions on the interface ensure that the surface tension balances the jump of the normal stress on the interface the balance of 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 ∂t(ρu) + div(ρu ⊗ u) − c) ⊤ ϕ + S(u, q; µ) : ∇ϕ dx = Γ(t) σκν ⊤ ϕ dS(x) (2) Our aim is to study the differentiability properties of local solutions with respect to u0 and c.To this end, we will work in an Lp-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 Lp 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 Lp-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 Volumeof-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 section 2, the transformed problem is formulated.In section 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 section 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.

Transformation to a flat interface
In this paper, we consider as in Prüss and Simonett [22] the problem in n + 1 dimensions, where Γ0 is the graph of a sufficiently smooth function h0 : R n → R, i.e., The interface has then the form where h : [0, t0] × R n → R with h(0, •) = h0 and t0 > 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, •) has second derivatives then normal and curvature of the interface Γ(t) are given by where ∇h and ∆h denote the gradient and Laplacian of h with respect to x and Following [22], we now transform the problem to Ṙn+1 = {(x, y) ∈ R n+1 : y = 0} with a flat interface at y = 0 by using the transformation û(t, x, y) = v(t, x, y) w(t, x, y) := u(t, x, h(t, x) + y), π(t, x, y) := q(t, x, h(t, x) + y).(6) Analogously, let with R n+1 As in [22], we work with the following function spaces.Let Ω ⊂ R m be open and X be a Banach space.Lp(Ω; X), H s p (Ω; X), 1 ≤ p ≤ ∞, s ∈ R, denote the X-valued Lebesgue and Bessel potential spaces of order s, respectively.We note that H k p (Ω; X) = W k p (Ω; X) for k ∈ N0, 1 < p < ∞ with the Sobolev-Slobodetski ǐ spaces W k p .Moreover, we will use the fractional Sobolev-Slobodetski ǐ spaces We recall that W s p (Ω; X) = B s pp (Ω; X) for s ∈ (0, ∞) \ N with the Besov space B s pp .Finally, the homogeneous Sobolev space Ḣ1 p (Ω) is defined by Then Ḣ1 p (Ω) is for connected Ω a Banach space if we factor out the constant functions and equip the resulting space with the corresponding quotient norm.
Finally, for Ω ⊂ R m open or closed we denote by BU C(Ω; X) and BC(Ω; X) the space of bounded uniformly continuous and the space of bounded continuous functions equipped with the supremum norm, respectively.Analogously, BU C k (Ω; X) and BC k (Ω; X), k ∈ N0, are defined for k-times continuously differentiable functions with bounded uniformly continuous or bounded continuous derivatives up to order k.If Ω is compact, we may briefly write C k (Ω; 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, denote by (i.e., r = [π] by the definition of E(t0)) the Stokes problem with free boundary for t > 0. Here, [û] denotes the jump across the transformed interface y = 0 and γw(x) = w(x, 0) denotes the trace of a function w : Ṙn+1 → R at y = 0 satisfying [w] = 0.
Then it is shown in [22] that the transformation (6) leads to the following problem for û = (v, w), π, h where the right hand sides are given by Note that all terms except Gκ(h) are polynomials in (v, w, π, [π], h) and derivatives of (v, w, π, h).Moreover, all terms are linear with respect to second derivatives and Gκ(h) is the pointwise superposition of a smooth function with ∇h and ∇ 2 h.Remark 1.The transformed version of the deformation tensor D(u) = ∇u + ∇u ⊤ is given by D(û, h) = D(v, w, h), where Then the compatibility condition (13) can with ν(0, x) equivalently be written as 3 Well-posedness and differentiability with respect to controls

Results for the transformed problem
By applying a fixed point theorem to (9), the following result is shown in [22] for ĉ = 0.
Our first aim is to study the differentiability properties of the control-to-state map (û0, ĉ) → (û, π, [π], h).Note that we consider also the case ĉ = 0.The proof is carried out by an appropriate extension of the fixed point argument for (9) based on Theorem 7.
For homogeneous initial data we obtain immediately.
Corollary 4. Let p > 3 and define in addition to E(t0) and F(t0) the spaces with initial value 0 for all components that admit a trace at t = 0. Then (8) has a unique and continuous solution map The fixed point argument relies on the following properties of the right hand sides (10) of ( 9).
Lemma 5. Let p > n + 3 and set for (û, π, r, h) ∈ E(t0) with F = (Fv, Fw), G = (Gv, Gw), F d and H defined in (10).Then the mapping N : E(t0) → F(t0) is a well defined and real analytic, more precisely, Moreover, we will need the following analogue for the spaces of the initial values.
Lemma 6.Let p > n + 3, U û, U h be as in (12) and set Then with G = (Gv, Gw) and H defined in (10) the mappings are real analytic and the first derivatives vanish in (û0, r0, h0) = 0.
Proof.Since p > n + 3 we have W  20) is a continuous bilinear form and thus real analytic.
where the integrand is in BU C([0, 1]×R n ).Moreover, since v → Ψ ′ (∇v)−Ψ ′ (0) is smooth with bounded derivatives and vanishes at 0, the integrand is continuous from [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 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 (R n )-functions and in real analytic functions of the form (22). Since W 1−2/p p (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.
We will work with the following extension of Banach's fixed point theorem.
a) Let U, W.Z be real Banach spaces, let A ∈ L(Z, W ) be an isomorphism and set M := A −1 L(W,Z) .Let BZ ⊂ Z be a nonempty closed convex set and BU ⊂ U be a nonempty set.Moreover, let K : BZ × BU → W be Lipschitz continuous with Then for all u ∈ BU the equation has a unique solution z = z(u) ∈ BZ and Proof.a): By assumption the mapping T : (z, u) ∈ BZ × BU → A −1 K(z, u) ∈ BZ is well defined and Lipschitz continuous with Lipschitz constants M Lz < 1 with respect to z and M Lu with respect to u.Hence, for all u ∈ BU there exists a unique fixed point z = z(u) with z = T (z, u) by Banach's fixed point theorem.For u, ũ ∈ BU we obtain and thus (24).Since BU is relatively open in u * + UL, we find δ > 0 such that u By using (24) we conclude that for d ∈ UL, d U → 0 If DK : BZ × BU → L(Z × UL, W ) is Lipschitz continuous then (25) can be written as where K (1)  By applying this result to ( 7)-( 9), we obtain the following extension of Theorem 2.

Results for the original problem
We transfer now the results of Theorem 8 for the transformed problem (9) to the original problem (1).To this end, we define for f0 ∈ U h the spaces Since the pressure π(t, •) is only determined up to a constant, we select from now on without restriciton the unique representative satisfying (note that [π] is uniquely determined in ( 9)) Then with the convention (36) and by the trace theorem, we find a Poincaré constant CP > 0 with The following imbeddings will be useful.
Then for all p ∈ [p, ∞) the mapping is continuously differentiable with derivative )) 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) → (x, y − h(t, x)).
For the original data (u0, h0, c) we obtain the following existence and differentiability result.
Theorem 13.Let p > n + 3 and Uu(h0), Uc(t0) be defined by (35).Then for any t0 > 0 there exists ε0 = ε0(t0) > 0 such that for all data satisfying the compatibility condition (13) as well as the smallness condition there exists a unique solution of the transformed problem (9) with Moreover, for any h0 with h0 U h < ε0 the mapping is continuously differentiable.By the chain rule, also the original state (u, q) depends continuously differentiable on (u0, 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 ĉ(c, h) depends now on h.Hence, the fixed point equation (32) changes to Let ε0 > 0 be as in Theorem 8. We have and the last estimate in (43) shows that for ε0 > 0 small enough (51) implies (27).
Hence, for all (u0, h0, c) satisfying (51) we have (û0, h0, ĉ(c, h)) ∈ BU (ε0) (note that (54) holds independently of h) and thus by (34) Finally, the Lipschitz constant of K(z; û0, h0, ĉ) with respect to ĉ is 1 and the mapping (50), ( 52) is by Lemma 12 continuously differentiable and the Lipschitz constant with respect to h is bounded by c Uc (t 0 ) < ε0.Hence, for ε0 > 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 (u0, c) for the spaces given in (43), (45), (46).

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 (δu, δq).This provides an analytical foundation to derive and analyze appropriate numerical VoF schemes for sensitivity calculations.
Let α : R n+1 → [0, 1] be a phase indicator satisfying the transport equation We note that for u ∈ L1( We define now We will show that the unique solution (u, q) of (1) according to Theorem 13 satisfies the VoF-type formulation where νε is a suitable smoothed normal computed from ∇α, see (70) below.
In order to deal with the sensitivity equation, it will be beneficial to consider measure-valued solutions of the general equation we can define uniquely the continuous mapping (x, y) → X(t; x, y), where X(t; x, y) satisfies the characteristic equation ∂tX(t; x, y) = u(t, X(t; x, y)), t ∈ J, X(0; x, y) = (x, y). (61) In the following, we denote by M loc (R n+1 ) the space of locally bounded Radon measures.
The next step is to express the surface tension term by using the phase indicator α such that its sensitivities can be expressed by using the measure δα.
We first rewrite the surface tension term in the weak formulation (2).Lemma 16.Let ϕ ∈ C 1 c (R n+1 ).Then with the curvature κ(t) of Γ(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.
To compute the interface normal from ∇α, we use the following simple fact.
Proof.By the definition of distributional derivatives one has Let now δ ∈ (0, 1/2) and and set To recover a mollified normal (not necessarily of unit length Then by Lemma 17 Then we have by the definition of φε On compact subsets the error is o(ε).

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., (u0, c) ∈ U ad → J (S(u0, c), u0, c) = J u (S u (u0, c), u0, c). (80) Using the differentiability results of Theorem 13 with the space given in (43), the reduced objective functional (80) is continuously differentiable, if the mapping is continuously differentiable for some p ∈ [p, ∞).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 δu, where δu can be obtained by the VoF-type formulation (72) or by using the linearization of the transformed problem (9) together with Lemma 11 and Lemma 12.
If the objective function (80) evaluates the velocity field u only in an open observation domain J × Ωo with positive distance from the interface t∈J ({t} × Γ(t)), then it is by Theorem 13 sufficient if the mapping is continuously differentiable for some p ∈ [p, ∞).

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± of the pressure across the interface.We show now that certain types of objective functionals are nevertheless differentiable.We consider objective functions of the form (u0, c) ∈ U ad → J (S(u0, c), u0, c) = J q (S q (u0, c), u0, c) = J R n+1 ℓ(S q (u0, c)(t, x, y), x, y) d(x, y) dt.
Remark 24.If we consider an objective function J α of the form (81) with the pressure q replaced by the phase indicator α then an analogue of Theorem 23 holds and the derivative simplifies to [ℓ(α(t, x, h(t, x)), x, h(t, x))]dδα(t)(x, y) dt, since δα has its support on the complement of Ṙn+1 .
We use the following auxiliary result.
Proof.We have by using (82) and its consequence (83) Hence, we obtain with the asserted derivative J q (π + δπ, h + δh) − J q (π, h) − ( J q ) ′ (π, h) b): Now let in addition BU is relatively open in the closed affine subspace u * + UL.Moreover, let K : BZ × BU → W be Fréchet differentiable and let u ∈ BU be arbitrary.Then DzK(z(u), u) L(Z,W ) ≤ Lz and DuK(z(u), u) L(U L ,W ) ≤ Lu and thus for any d ∈ UL the linear problem (25) has by Banach's fixed point theorem a unique solution δz d ∈ Z.