Analysis of shape optimization problems for unsteady fluid-structure interaction

Shape optimization via the method of mappings is investigated for unsteady fluid-structure interaction (FSI) problems that couple the Navier–Stokes equations and the Lamé system. Building on recent existence and regularity theory we prove Fréchet differentiability results for the state with respect to domain variations. These results form an analytical foundation for optimization und inverse problems governed by FSI systems. Our analysis develops a general framework for deriving local-in-time continuity and differentiability results for parameter dependent nonlinear systems of partial differential equations. The main part of the paper is devoted to conducting this analysis for the FSI problem, transformed to a shape reference domain. The underlying shape transformation—actually we work with the corresponding shape displacement instead—represents the shape and the main result proves the Fréchet differentiability of the solution of the FSI system with respect to the shape transformation.


Introduction
Shape optimization for fluid-structure interaction (FSI) has many important applications in engineering and other fields. So far, most of the research devoted to this challenging class of optimization problems mainly targeted at numerical approaches, e.g. for biomedical applications [38], naval architecture [37] or wind engineering [27,45], using direct differentiation [39] or adjoint based gradient computation [25], while a rigorous supporting theory is scarce. In this paper, we build on recent work by Raymond and Vanninathan [43] on the existence and regularity of solutions to an unsteady FSI problem. We extend these results and prove continuity and Fréchet differentiability of the solution of an unsteady Navier-Stokes-Lamé-system with respect to domain variations. Existence and regularity theory for FSI is challenging due to the hyperbolic nature of the elasticity equation, which leads to a lack of regularity that needs to be compensated by hidden regularity results. To the authors' knowledge, analytical results for unsteady FSI models that consider elastic structures in fluids are so far restricted to cases with stationary interfaces [2,14], a priori known time-dependency of the domain [8], very smooth data [11,12] or geometrical constraints on the interface [28,33,43], whereas differentiability results are only available for steady FSI models [41,51].
The fluid-structure interaction model that is considered in this paper couples the transient Navier-Stokes equations with the Lamé system and is formulated in a fully Lagrangian framework: denote by Ω f (t) and Ω s (t), respectively, the domains occupied by the fluid and the solid, respectively, at time t. Further, let Ω (t) denote the interior of Ω f (t) ∪Ω s (t) and Γ i (t) := ∂Ω f (t) ∩ ∂Ω s (t) the fluid-solid interface. In the considered setting, Ω (t) =Ω is timeindependent, while Ω f (t), Ω s (t), and Γ i (t) change with time. The Lagrangian framework uses the displacement field induced by the velocity field to obtain a transformation between the reference domain Ω f and the physical domain Ω f (t). For the solid, the Lagrangian formulation provides the standard framework and the displacement field induces a transformation between the reference domain Ω s and Ω s (t). Let T f > 0, Γ i := ∂Ω f ∩ ∂Ω s , Γ f ⊂ ∂Ω f \Γ i , Γ s ⊂ ∂Ω s \Γ i and the space-time cylinders be denoted by Q T :=Ω × (0, T), Q T f , Q T s , Σ T i :=Γ i × (0, T), Σ T f and Σ T s for all 0 < T T f . A coupled Navier-Stokes-Lamé system in Lagrangian coordinates can be written in the form i , σ f ,y (v,p)n f = σ s,y (ŵ)n f +Ĥ(v,p) onΣ T i , ∂ ttŵ − div y (σ s,y (ŵ)) = 0 inQ T s , w = 0 onΣ T s , w(·, 0) = 0, ∂ tŵ (·, 0) =ŵ 1 inΩ s . denote the fluid velocity and pressure, ŵ the solid displacement, and v 0 as well as ŵ 1 appropriate initial conditions. We define the underlying transformation bŷ for any t ∈ (0, T) and its inverse Υ (·, t) := (χ(·, t)) −1 as well as F Υ := F −1 χ , which exist if T > 0 is sufficiently small and the initial data are smooth enough, see [43]. Then the right hand side terms read where cof denotes the cofactor matrix. We define ĝ(v) := (I − det( F χ ) F Υ )v, such that div y (ĝ(v)) =Ĝ(v) due to Piola's identity.
Shape optimization problems can be analyzed with different, yet closely related, techniques. On the one hand, shape calculus can be used to investigate functionals Ĵ (Ω) that depend on the domain Ω . The Eulerian derivative dĴ(Ω,V) can be represented by the Hadamard-Zolésio shape gradient, which is the representation of the shape gradient as a distribution that is supported on the design boundary and only acts on the normal boundary variation V ·n f [13,42,47]. If a state equation is involved, then the Eulerian derivative depends on the shape derivative of the state and can also be expressed using an adjoint state. An alternative approach is the method of mappings [3,17,20,32,40,46], also called perturbation of identity, which parametrizes the shape by a bi-Lipschitz homeomorphism τ : R d → R d via Ω =τ (Ω), where Ω ⊂ R d is a nominal domain (or shape reference domain). Optimization can be performed based on the function J :τ →Ĵ(τ (Ω)). An underlying state equation is then transformed to Ω and derivatives of J can be obtained via sensitivities or adjoints. The Hadamard-Zolésio shape gradient representation can be derived from this approach essentially by an integration by parts. The method of mappings directly yields an optimal control setting in Banach spaces. Further, it fits well in the theoretical setting of the FSI model that was introduced above since it also employs the idea of domain transformations. In this paper we use the method of mappings to transform the fully Lagrangian FSI system to a shape reference domain. The major part of the paper is devoted to the study of existence, uniqueness and especially continuity and Fréchet differentiability of solutions to the transformed FSI system with respect to transformations of the domain.
The investigations in this paper have several important connections to inverse problems. Shape identification and other inverse problems for FSI systems have many interesting applications in engineering, e.g. wind turbines [27] and naval structures [50], in hemodynamics [6,7], i.e. blood flows, and in other fields. Since shape variations belong to the most challenging types of parametric dependencies that can arise in PDEs, our differentiability results for the state with respect to shape variations can be transferred to many other parametric dependencies in FSI systems and often the analysis then would become less complex. The theory and methods of nonlinear inverse problems often make use of Fréchet derivatives of where m, σ are chosen such that s = m + σ, m ∈ N 0 and for 0 < σ < 1 the semi-norm | · | σ,(0,T),X is defined by The choice of the norm on the spaces H s ((0,T),X) is crucial for the theoretical analysis which requires the knowledge of the T-dependency of appearing constants. More precisely, for

Method of successive approximations
The method of successive approximations is a well known approach for establishing existence and uniqueness results for nonlinear partial differential equations. On an abstract level, the FSI system can be considered as a nonlinear partial differential equation of the form where y ∈ Y and Y is a Banach sace. As in [43] we write this in the form By = F(y), where F(y) := By − A(y) and B is a linear operator that represents the principal part of the FSI system, i.e. the PDE operator in a linear FSI system. For our setting we will show that the system By = f has a unique solution y = Sf, where S ∈ L(W, Y), with W being a Banach space. Existence and uniqueness of solutions is now studied via the fixed point equation Unique solvability of (9) on a closed subset Ỹ ⊂ Y can be shown if y → SF(y) maps Ỹ into itself and is a contraction on Ỹ . This, e.g. is the case if S L(W,Y) L S and if F :Ỹ → W is Lipschitz continuous with a constant L F < 1 LS . Uniqueness on Ỹ then also follows.

Framework for continuity and differentiability results
One can extend the considerations of the previous section to an equation with parameter or control z in a Banach space Z. As before, we consider solutions of the fixed point equation where F(y, z) := By − A(y, z), B is as in section 2.2 and S ∈ L(W, Y) is the solution operator of By = f .
Then, for all z ∈Z , the system (11) has a unique solution y(z) and z → y(z) is Lipschitz continuous on Z : In addition, let y(z) lie in the relative interior of Ỹ and denote by Ỹ L the linear subspace parallel to the affine hull aff(Ỹ). Assume that F is Fréchet differentiable at (y(z), z), where (y, z)-variations are taken in Ỹ L × Z .
Proof. For any fixed z ∈Z , (12) implies the Lipschitz continuity of the mapping F(·, z) :Ỹ → W. Using (12), (13), and the properties of F , L F and L S shows that the map y ∈Ỹ → SF(y, z) ∈Ỹ is a well-defined contraction. The existence of a unique solution y(z) ∈Ỹ is thus ensured by the method of successive approximations. Now (14) follows from (12). For showing differentiability, we fix z ∈Z and assume that F is differentiable at (y(z), z) in the way stated in the theorem. Let h ∈ Z be arbitrarily fixed. Since y(z) is a relative interior point of Ỹ , we obtain from (12) that, for all d 1 , d 2 ∈Ỹ L , there holds Thus, since L F < 1 LS , the method of successive approximations applied to the fixed point equation δ h y(z) = SδF(y(z), z)(δ h y(z), h) posed in Ỹ L , see (15), yields a unique solution δ h y(z) ∈Ỹ L ⊂ Y which by linearity of (15) depends linearly on h. Let h Z be sufficiently small. Then z + h ∈Z and, as h → 0, Therefore, The rest of the paper is dedicated to the application of this argumentation to shape optimization for the FSI problem via the method of mappings. The parameter z then corresponds to a domain transformation that represents a variation of a reference shape domain.

Existence and uniqueness results for Navier-Stokes-Lamé system
In order to have the theoretical tools at hand that will be used for showing differentiability of the state with respect to domain variations the main results of [43] are recalled. Since, in contrast, the analysis will be carried out on a nominal domain Ω instead of Ω , the statements are presented for a general domain Ω ∈ {Ω,Ω}. We will work under the following assumption 1 on the unique solvability of the Stokes equations and the elastic wave equation. We will see in lemma 2 that according to [43] assumption 1 is satisfied for particular boundary conditions and geometrical settings. , where, for a Banach space X, D X denotes the closure of D w.r.t. X, (6) and (7) are valid.

Remark 1.
The main difficulty in finding a setting that fulfills assumption 1 is the improved regularity for the normal stress of the Lamé system on the boundary Σ T i . [43] provides a setting that fulfills the assumption, see lemma 2. Another restriction is the validity of the requirements for the fluid system. Existing results require, e.g. that Γ i ∩ Γ f = ∅, see [19, definition 7.2 and theorem 7.5].
Proof. See [43, sections 3 and 4]. □ Remark 2. As we will see, any geometrical configuration satisfying assumption 1, for example the one in lemma 2, can be used as shape reference domain Ω for shape optimization. The shape reference domain is mapped by a C 1 -diffeomorphism to the ALE reference domain Ω for the FSI system. We will show in theorem 3 that there exists a suitable open neighborhood in H 2+ (Ω) d of C 1 -diffeomorphisms containing the identity, see (30) and lemma 4, such that the solution of the corresponding FSI-system pulled-back to the reference shape domain depends continuously differentiable on the transformation.
For ∈ ( 1 2 , 1), the function spaces the norms with analogous definitions on the spaces S T and S T , and, for v 0 ∈ V 0 , the metric spaces (20) are defined. Due to trace theorems and interpolation theorems the modified norms on E T and S T , S T , S T are equivalent to the standard norms on these function spaces. However, the appearing equivalence constant might depend on T without further knowledge about this dependency. Since the dependency of the appearing constants on T is a key point in the theoretical analysis it is therefore necessary to work with the modified norms defined above.

Remark 3.
The following adaptions have been made compared to [43]: , which is not needed for estimating the right hand side terms and is not compatible to our choice of the norm, but other norms of interpolation and trace spaces. • In the theoretical setting considered here, it is not guaranteed that which is required in [43, theorem 5.1] to use a trace theorem and give a meaning to g| Σ T f = 0. However, the proof of [43, lemma 4 The following continuity result that is also part of the proof of theorem 2 corresponds to [43, theorem 5

.1] with the modification that only
which is possible by Remark 3. It will be needed for showing the Fréchet-differentiability of the state with respect to domain variations.

Lemma 3. Let assumption 1 be fulfilled. Assume that
Furthermore, let

Then, the system
admits a unique solution (v, p, w) ∈ E T × P T × W T and the states depend continuously on the initial data and the right hand sides, more precisely,

The constant C S depends on T f but is independent of T.
For obtaining time independent continuity estimates for the Stokes equations and the Lamé system with respect to the right hand sides, the partial differential equations are split into several systems that have either zero initial conditions or the right hand sides are obtained by lifting initial values to the interval (0, ∞) [43, sections 3 and 4]. The systems are extended to the time-interval (T − T f ,T) or (0,T f ) of length T f . In the first case, the temporal fractional order of the right hand side terms is smaller than 1 2 or with additional zero initial conditions. Property P3 of the norm yields continuity of the extension-by-zero- 3 2 }, X is a Hilbert space and C is independent of T. Now, the solution theory for the equations can be applied on these extended systems yielding constants C T f that might depend on T f but do not depend on T. Due to property P2 of the norm the equivalence constants of · H s ((T−T f ,T),X) to an equivalent norm on H s ((T − T f , T), X) might depend on T f but not on T (using (6) with r = 2 + , s = 1 + 2 and (7) explains why we can add norms in the definition of the norm on E T ). Now, by property P4 of the norm we obtain the estimates on the time interval (0, T) with constants independent of T. In the second case, the right hand sides can be bounded above by a constant times the norm of the initial values [35, p 22, remark 3.3], where the appearing constant does not depend on T and we use property P5 of the norm. In order to obtain an existence and regularity result for the coupled system, a fixed point argument is used [43, theorem 5.1], which requires property P6 of the norm. The extension of the local-in-time result to arbitrary time intervals requires P8.
The main result of [43] is given by the following theorem (with the same adaptions as in lemma 3), which shows existence and uniqueness of solutions to the FSI problem if some additional requirements are met.

Theorem 2.
Let assumption 1 be fulfilled. Denote by T f > 0 a fixed terminal time, let C S be the constant from lemma 3, and K 0 : Assume that one can find some 0 < T * < T f such that for all 0 < T T * the following estimates hold for and for some α > 0, a positive constant C that does not depend on T but only on T f and a polynomial χ. Furthermore, let Then, there exists T > 0 and M 0 < ∞ such that the system admits a unique solution Proof. This theorem corresponds to a large extent to [43, theorem 2.1], where the requirements (22) and (23) replace [43, proposition 6.1]. As a first step the system (24) is reformulated as a fixed point system that can be started with homogeneous F , G and H. To this end, To prove the existence of solutions to the system (24) or the equivalent system (26), the method of successive approximations is used. Therefore, we show that there exists some M 0 > K 0 such that the mapping is well-defined and a contraction with respect to the norm In order to show the contraction property we consider arbitrary (23) we know that

Due to lemma 3 and the inequalities
Thus, M is a well-defined contraction and we can apply the fixed point theorem of Banach in order to show existence and uniqueness of the solution to the fixed point equation

Shape optimization via the method of mappings approach
We consider shape optimization problems governed by the FSI model (1). This results in an optimization problem Here, Ô ad denotes the set of admissible domains, Ĵ :Ê T ×P T ×Ŵ T ×Ô ad → R is an objective function, and Ê (v,p,ŵ,Ω) = 0 if and only if v, p, ŵ fulfill (1), wherê and Ẑ T is a suitable Banach space. There exist different approaches to shape optimization that are closely related to each other. In particular, one can use shape derivatives in the Hadamard-Zolésio sense or one can apply the method of mappings (also called perturbation of identity method) [40]. In this paper, we use the method of mappings for a couple of reasons. It is based on domain transformations of a nominal domain Ω to represent shapes and thus fits very well to the arbitrary Lagrangian-Eulerian (ALE) approach of which the fully Lagrangian formulation (1) is a special case. Moreover, the method of mappings transforms the shape optimization problem to a nonlinear optimal control problem in a Banach space setting, which is attractive from a theoretical as well as a numerical perspective. The set of admissible domains Ô ad := {Ω ⊂ R d :Ω =τ (Ω),τ ∈T ad } comprises all domains that can be obtained by trans- and to optimize over ũ τ ∈Ũ ad instead of τ ∈T ad . Thus, we obtain the optimization problem which yields an optimal control setting with the control ũ τ .
Since the analysis will be carried out on the nominal domain Ω , the geometric assumptions are needed on Ω instead of Ω . We require that two transformations with the same normal displacement of the design boundary part result in the same ALE domains and that the support of the transformation is disjoint from the support of the initial velocity v 0 .

Navier-Stokes-Lamé system on the nominal domain
We apply the method of mappings approach to the FSI problem. In order to maintain the structure required by theorem 2 we have to ensure that the right hand side of the transformed elasticity equation remains 0. For this purpose, the set of admissible transformations is chosen such that τ |Ω s = id z for all τ ∈T ad , i.e. ũ τ |Ω s = 0 for all ũ τ ∈Ũ ad . The transformation of the Navier-Stokes-Lamé system (1) from the reference domain Ω to the shape reference domain Ω via τ yields the system where σ f ,z (ṽ,p) := 2ν z (ṽ) −pI, σ s,z (w) := λtr( z (w))I + 2µ z (w), z (w) := 1 2 (D zw + (D zw ) ), v 0 =v 0 •τ , w 1 =ŵ 1 •τ and the nonlinear terms F , G and H are defined bỹ and thus F χ (z, t) : Let T be defined by (18) and , which is a closed linear subspace of H 2+ (Ω) d , be endowed with the norm Furthermore, let α 1 > I H 1+ (Ω f ) d×d . We consider solutions of the FSI problem for transformations id z +ũ τ induced by displacements ũ τ ∈Ṽ, wherẽ V := {ũ τ ∈Ũ : id z +ũ τ can be extended to an orientation-preserving C 1 -diffeomorphism which by lemma 4 is an open subset of Ũ . In particular, if Ũ ad ⊂Ṽ, then our results will hold at any admissible design displacement. Alternatively, the current design of the ALE domain could be viewed as the reference shape domain, making it correspond to ũ τ = 0, and our results then can be applied to study continuity and differentiability w.r.t. variations of this domain.

We use the extension operator to obtain
By a Sobolev embedding we obtain also that τ v,R d is C 1 .
Since W 1,∞ (R d ) and C 0,1 (R d ), are equal with equivalent norms, see [24, theorem 4.1, remark 4.2], there there exists c > 0 such that any f ∈ W 1,∞ (R d ) d has a Lipschitz continuous representative with modulus c f W 1,∞ (R d ) d .
We now show that τ v,R d : R d → R d is bijective. In fact for any fixed z ∈ R d , the equation τ v,R d (z) = z can be written as For sufficiently small ρ, the map A(z ; ·) is a contraction since, for any z 1 , z 2 ∈ R d , with (31) Hence, by the Banach fixed point theorem, if ρ is sufficiently small, then for any z ∈ R d there exists a unique z ∈ R d with τ v,R d (z) = z .
We show next that τ −1 v,R d is C 1 . From (31) and Hence, for ρ > 0 small enough we obtain det(D zτ v,R d (z)) > ω/2 for all z ∈ R d and thus [44, pp 336 and 297] also 1 Finally, with a constant C > 0 we obtain Then Ṽ ρ is by lemma 4 an open subset of Ũ and we will study the differentiability of the solution of (28) on Ṽ ρ at ũ τ = 0. The choice of the space of admissible transformations restricts the shape optimization to the optimal design of the fluid domain, but keeps the interface in the Lagrangian frame fixed. The boundedness properties of Ṽ allow us to establish estimates of the right hand sides in (28). The following lemma is a helpful tool that takes the special structure of the right hand side terms into account.

Let M j > 0, Ỹ ⊂ Y and Z ⊂ Z be such that T j (y, z) Sj M j for all (y, z) ∈Ỹ ×Z, 1 j k.
Then, there exists a constant C > 0 that is independent of T such that T (y, z) S CΠ j M j for all (y, z) ∈Ỹ ×Z.

Let in addition to
with a constant C > 0 that is independent of T. 3. Let (y 1 , z 1 ) be an element of the relative interior of Ỹ ×Z and T j :Ỹ ×Z → S j be Fréchet differentiable in (y 1 , z 1 ) for all 1 j k. Then, T :Ỹ ×Z → S is Fréchet differentiable in (y 1 , z 1 ).

Let in addition to
Proof. We recursively apply lemma A.1 in order to get continuity of m :

It holds
where Ŝ j := H 1 ((0, T), X j,1 ) is endowed with the norm · Ŝ j := ( · 2 H 1 ((0,T),Xj,1) + · (0) 2 Xj,1 ) In order to show the boundedness in the norm · S the initial values have to be bounded appropriately. However, this is ensured by the continuity properties of the multilinear form m. Moreover, property P1 of the norm is used. The assertions now follow directly as in lemma 5. □ Furthermore, the following results will be needed for establishing the required right hand side estimates.
for a constant C that is independent of T. Proof.

The bilinear form
Here, we recall that the norm on S T is defined by (19). 2. By [43, lemma A.7] we know that for a constant C independent of T. The proof of this lemma also shows that Let C now denote a generic constant independent of T. In order to bound ∂ tf −1 (·, 0) L 2 (Ω f ) , we consider G ∈ C ∞ (R) such that G(0) = 0 and G(x) = x −1 for all x ω. Then, These estimates imply for a constant independent of T and it remains to estimate ∂ tf . We obtain with lemma A.1, 2.
Combining the estimates implies the assertion. □ Lemmas 5, 6 and 7 allow to estimate products of rational functions in terms of the norms of the factors. We start by estimating the appearing factors.

Consider
(Ω) d×d , as well as, with P7, for almost every t ∈ (0, T) due to the definition of · Ẽ T . Boundedness follows with property P1 of the norm. Fréchet differentiability and continuity now follow by linearity of T 1 and due to Hence, boundedness, continuity and differentiability follow from the definition of Ṽ , (32) and (35). 2. Each component of the cofactor matrix cof( F χ ) can be written as a finite sum of terms a · x 1 · x 2 , where x 1 , x 2 denote components of the matrix F χ and a ∈ {−1, 1}. Therefore, cof( F χ ) is a sum of bilinear forms with factors T 1 (ṽ,ũ τ ) := a( F χ ) i,j and T 2 (ṽ,ũ τ ) := ( F χ ) k,l for i, j, k, l ∈ {1, 2, 3}. Due to the estimates in 1.(a) we know that 2}. Therefore, the continuity estimate and Fréchet differentiability follow from lemma 6. 3. Since det( F χ ) is a polynomial of degree 3 in the components of the matrix F χ , the assertions can be proved similar to 2. 4. (a) Since det( F χ ) is a cubic polynomial in the components of F χ and det( F χ )(·, t) α > 0 for all t ∈ [0, T α ], the assertion follows from lemma 7.2 which implies

and lemma 7, since with
which yields with 1. the Fréchet differentiability.
With the above lemmas the required right hand side estimates can be established.

Lemma 9.
Let assumption 1 be satisfied. Let T f > 0 and ρ = ρ(0) be given by lemma 4. There exist 0 < T * T f , α 1 > 0, as well as, for each of the following terms, a constant C > 0 independent of T but dependent on T f and a polynomial χ such that for all 0 < T < T * , ṽ,ṽ 1 ,ṽ 2 ∈Ẽ T,M0,ṽ0 , p,p 1 ,p 2 ∈P T,M0,ṽ0 and ũ τ ,ũ 1 τ ,ũ 2 τ ∈Ṽ ρ we havẽ as well as, Proof. The compatibility conditions (40) are fulfilled, due to the choice of Ṽ , which ensures that suppũ τ ∩ suppṽ 0 = ∅ and therefore H (ṽ,p,ũ τ )(·, 0) = 0 and g(ṽ,ũ τ )(·, 0) = 0. The boundary condition on Σ T f ensures that g(ṽ,ũ τ )| Σ T f = 0. The right hand side terms F , H , G and g are sums of multilinear forms as introduced in lemmas 5 and 6. In lemma 8 boundedness, continuity and Fréchet differentiability of the corresponding factors are shown. Thus, it suffices to establish an appropriate boundedness estimate such that the product of the appearing M j in lemma 5 have the structure C (T α + ũ τ H 2+ (Ω f ) d ) for a suitable α 0 and C which is independent of T. The explicit time dependency is obtained by using the extension and re-striction properties P3, P4 and P5 of the norm and by using P6. The time dependency for the corresponding constants M j ,1 and M j ,2 follows with similar arguments. The desired continuity estimates, as well as, Fréchet differentiability can be deduced from lemma 5 if (33) and thus are kept in mind, which by lemma A.4 and the definition of Ṽ implies Moreover, since for arbitrary matrices A, B ∈ R d×d the cofactor-matrix is a polynomial of degree d − 1 in every entry, we have that where χ i,j is a polynomial of degree d − 2 in the entries of A and B for 1 i, j 3. Thus, We show boundedness of F , H , G and g. In order to obtain the estimates we have to split the terms such that the initial values of selected factors vanish at t = 0. To this end, we decomposẽ F(ṽ,p,ũ τ ) =F 1 (ṽ,ũ τ ) +F 2 (ṽ,ũ τ ) +F 3 (ṽ,ũ τ ) +F 4 (ṽ,ũ τ ) +F 5 (ṽ,p,ũ τ ) +F 6 (ṽ,p,ũ τ ), where Since the ideas for the estimates for the different summands of F , H , G and g are similar we just present the proofs for F 2 , F 6 , H 1 , G 1 and g 1 . Let C denote a generic constant independent of T. In the following argumentation we frequently use lemma A.4 in order to ensure that X 1 , . . . , X k are chosen such that multilinear forms m(x 1 , . . . , x k ) := x 1 · . . . · x k fulfill the requirements of lemma 5. The notation S i , M i , M i,1 , M i,2 , s i for i ∈ {1, . . . , k} is defined by lemma 5.
As seen in the previous estimates, due to lemma 5, the derivation of the continuity estimates and Fréchet differentiability is straightforward if one knows how to show boundedness of the multilinear forms. We thus only address boundedness in the following.