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Global weak solutions for an Newtonian Fluid interacting with a Koiter Type Shell under natural boundary conditions

We consider an viscous, incompressible Newtonian fluid flowing through a thin elastic structure. The motion of the structure is described by the equations of a linearised Koiter shell, whose motion is restricted to transverse displacements. The fluid and the structure are coupled by the continuity of velocities and an equilibrium of surface forces on the interface between fluid and structure. On a fixed in- and outflow region we prescribe natural boundary conditions. We show that weak solutions exist as long as the shell does not self-intersect.


INTRODUCTION
Fluid-structure interaction (FSI) problems are common in nature, most prominently seen in the blood flow through vessels. Blood generally exhibits a non-Newtonian behaviour (see [11]), while the vascular wall consists of many layers all having different biomechanical properties (see [33]). If medium or big blood vessels are considered one can use a simplified model. Firstly, we treat blood as a Newtonian fluid. Since only a small part of the circulatory system is considered we introduce artificial in-and outflow boundary conditions. Secondly, we will model the vascular wall as a linearly elastic Koiter shell whose motion is restricted to transverse displacements.
Due to the apparent regularity incompatibilities between the parabolic fluid phase and the hyperbolic dispersive solid phase, the interaction between an elastic structure and a fluid is exceedingly difficult, see e. g. [3,12,8,18,25] and the references therein. For the existence proof of weak solutions, two different strategies have been proven successful. The first one is working directly on the fluid domain and therefore preserving the structure of the equations. Since by the non-cylindrical space-time domain the usual Bochner-space theory is not applicable, the key element in this approach is to find an appropriate compactness argument. This was accomplished in [22], [24] and [23], by generalising methods from [8], [18]. There, the existence of a global-in-time weak solution to the interaction of an incompressible generalised Newtonian fluid completely surrounded with an linearised transversal Koiter shell was shown. The second approach consists of transforming the fluid equation by an ALE mapping to a reference domain and using a semi-discrete, operator splitting Lie scheme. This method was used in [29], [30] and [31] to show the existence of a global-in-time weak solution to the interaction between a Newtonian fluid and a (semi-) linear transversal Koiter shell, where the flow is driven by a dynamical pressure condition and no other external forces apply. Furthermore, in [32] the Navier slip boundary condition was used for the coupling between the fluid and an elastic structure in a two-dimensional setting.
The present paper is based on the first authors Ph.D. thesis [16] and extends the result of [24] to the case of an in-and outflow region. We will use the same method as in [24] and thus the structure and the arguments of the present paper are similar to [24]. Since the in-and outflow region admits a flow through the domain, various new difficulties have to 1 be solved. Moreover, since the reference domain has no C 4 -boundary, but only a Lipschitz one, special care has to be taken to transfer the compactness argument to our situation. In comparison to [30] and [31], we consider a different in-and outflow condition and a more general geometry. Furthermore, we allow external forces acting on the fluid and the shell. This paper is organised as follows: In the next Subsections, we introduce our setting and formally derive an a priori estimate for the resulting system. In Section 2 we develop the mathematical framework for our non-Lipschitz in-and outflow domains. In particular, in Subsection 2.1 we define a generalised trace operator and show some density results for functions vanishing on a part of the boundary, and in Subsection 2.4 we construct a suitable extension operator of test functions for the shell equation. In Section 3, we can finally formulate the main result of this paper, the existence of a global-in-time weak solution. Crucial for the proof of this result is that one can generalise the compactness result from [24] to our setting, which is possible due to our preparatory work (see Subsection 3.1). The proof of the main Theorem is then carried out by looking first at a decoupled, regularised and linearised problem, then a fixed point argument to restore the coupling, and finally a limiting process to eliminate the regularisation.
1.1. Koiter's energy and statement of the problem. By Ω ⊂ R 3 we denote a reference domain with ∂ Ω = Γ ∪ M, where Γ is assumed to be the fixed in-and outflow region and ∂ Γ = ∂ M = / 0. Moreover, let M represent the middle surface of the thin elastic shell of thickness 2 ε s > 0 in its rest state. The deformation of the shell is then given by the displacement η relatively to M. By ν we denote the unit normal on ∂ Ω, by g and h the first and second fundamental form of M induced by the ambient Euclidean space, and by dA the surface measure of M or ∂ Ω, respectively. As in [24,30] we restrict the deformations to transverse displacements, i. e. η = η ν. Following [24], we assume further that the elastic shell, clamped on ∂ M, consists of a homogeneous, isotropic material whose linear elastic behaviour is characterised by the Lamé constants λ and µ, and the elastic energy is given by Koiter's energy for linearly elastic shells and transverse displacements K(η) := K(η, η) with K(η, ζ ) := 1 2 M ε s C, σ (η) ⊗ σ (ζ ) + ε 3 s 3 C, ξ (η) ⊗ ξ (ζ ) dA.
Here C denotes the elasticity tensor of the shell, C αβ γδ = 4µλ λ + 2µ g αβ g γδ + 2µ g αγ g β δ + g αδ g β γ and σ (η) = −h η, ξ (η) = ∇ 2 η − k η are the linearised strain tensors, where k αβ := h σ α h αβ , ∇ denotes the Levi-Civita connection of M, and ∆ is the corresponding Laplacian. We refer the interested reader to [9,10] for additional details to the Koiter model. As shown in [10,, K(η, ζ ) is a symmetric bilinear form (and therefore K(η) a quadratic form) which is coercive on H 2 0 (M). Moreover, as has been shown by partial integration in [22,Chapter 3], the L 2gradient of this Koiter energy satisfies 2 K(η, ζ ) = M grad L 2 K(η) ζ dA, η, ζ ∈ H 2 0 (M). (1.1) Throughout the paper, we use standard notations for (vector valued) function spaces. We denote the fluid domain, which depends on a time t ∈ I and is a priori not known, by Ω η(t) ⊂ R 3 . Then the motion of a homogeneous, incompressible, Newtonian fluid on the space-time domain Ω I η := t∈I {t} × Ω η(t) is governed by the system ρ ∂ t u + ρ (u · ∇)u = div 2σ Du − π Id + f in Ω I η , where u is the velocity field, Du the symmetric part of the gradient of u, π the pressure field, ρ the density, σ the dynamic viscosity, f an external body force and Id the 3 × 3 unit matrix. We will assume no-slip boundary conditions on the deformed boundary and natural boundary conditions on Γ, i. e. u(·, · + η ν) = ∂ t η ν on I × M, (1.4) 2σ Du − π Id ν = ρ 2 (u · ν) u on I × Γ. (1.5) Using the map Φ η(t) : M → ∂ Ω η(t) \ Γ, Φ η(t) (q) := η(t, q) ν(q) to parametrize the deformed boundary, the force exerted by the fluid on this boundary is given by where ν η(t) is the outer normal of Ω η(t) . The external forces acting on the shell along the outer normal are thereby composed of F · ν and some given external force g. Using Hamilton's principle, the displacement η of the shell is a stationary point of the integrated difference between the kinetic and potential energy of the shell and therefore satisfies the corresponding Euler-Lagrange equation Since the shell is clamped, we have η = 0, ∇η = 0 on I × ∂ M. (1.7) Finally we prescribe the initial conditions u(·, 0) = u 0 in Ω η 0 and η(·, 0) = η 0 , ∂ t η(·, 0) = η 1 in M. (1.8) In the following, we will analyse the system (1.2)-(1.8).
1.2. Formal a priori estimates. We take (sufficiently smooth) solutions u and η of the fluid-and shell equation. Multiplying the fluid equation (1.2) with u(t) and integrating over Ω η(t) leads to ρ Ω η(t) Using Reynolds transport theorem and our boundary conditions, i. e. the domain velocity equals the fluid velocity on the moving boundary ∂ Ω η(t) \ Γ and vanishes on Γ, the first term of the equation can be written as where ν η(t) denotes the outer normal and dA η(t) the surface measure on ∂ Ω η(t) . By partial integration and taking the divergence constraint into account, the convective term vanishes except a boundary term ρ Ω η(t) u(t) · ∇ u(t) · u(t) dx = ρ 2 ∂ Ω η(t) |u(t)| 2 u(t) · ν η(t) dA η(t) . (1.11) Likewise we get by partial integration, the divergence constraint, the symmetry of Du and the the boundary condition (1.5) as well as ν η(t) = ν on the fixed boundary Γ Hence, using (1.10), (1.11) and (1.12), the equation (1.9) can be written as 2σ Du(t) ν η(t) − π(t) ν η(t) · u(t) dA η(t) .
Multiplying equation (1.6) with ∂ t η(t), integrating over M and using the bilinearity of the Koiter-Energy as well as (grad (1.14) Using the definition of F, the boundary condition (1.4) and a change of variables, adding (1.13) and (1.14) leads to the energy equality d dt We denote the expression in the parentheses on the left-hand side of (1.15) by E(t), and set E 0 := E(0), which depends by our initial conditions (1.8) only on η 0 , η 1 and u 0 . The coercivity of the Koiter Energy K and Grönwall's inequality (see [6,Appendix]) imply the estimate esssup t∈(0,T ) Again by the coercivity of the Koiter Energy the quantity is bounded by a constant depending only on the data. This (spatial) regularity of the displacement η(t, ·) ∈ H 2 (M) is not enough to ensure Lipschitz continuity, but only Hölder continuity C 0,λ (M) for any λ < 1. Hence, the boundary of the moving domain Ω η(t) is not necessarily Lipschitz and the classical partial integration theorem, trace operators, etc. cannot be used. In the next Section we will develop the necessary framework for the moving domain, using a special reference domain. For the sake of better readability, we set the constants ε s , ρ s , ρ and σ equal to 1 throughout the paper, but emphasize that all the computations hold with the original constants.

MOVING DOMAINS
We assume that Ω ⊂ R 3 is a bounded domain with ∂ Ω ∈ C 0,1 and ∂ Ω = Γ ∪ M, where M, Γ = / 0 are compact, oriented, embedded two-dimensional C 4 -manifolds with smooth non-empty boundaries and ∂ M = ∂ Γ. Furthermore, we assume M to be connected and Γ to be the finite union of connected components Γ i , where Γ i is subset of a hyperplane perpendicular to M, i. e. the continuous extension of the outer normal ν on int M to ∂ M ∩ ∂ Γ i is perpendicular to the outer normal of the hyperplane. For α > 0 the open α-tube S α and the half-closed tube S α around int M are given by We assume 1 that there exists a κ > 0 such that the map Λ : M × (−κ, κ) → S κ , Λ(q, s) := q+s ν(q) is a C 3 -diffeomorphism, and write Λ −1 (x) = (q(x), s(x)) for the inverse mapping. For 0 < α ≤ κ we divide the boundary of S α into the parts where the outer normal is again defined through int M. Furthermore, we can assume that the sets M α + , M α − and Γ α s are disjoint and the representations through x = q + β ν(q) are unique. Hence, the orthogonality assumption implies ∂ S α ∈ C 0,1 . We will require that the domain's deformation is taking place inside Ω ∪ S κ . To ensure that the deformed moving boundary does not interfere with the fixed in-and outflow boundary Γ, we assume that for some ε > 0, where ν is the extension of the outer normal on int Γ. Finally, we assume that for all 0 < α < κ it holds that int Γ α,c We call a domain Ω fulfilling these requirements an admissible in-and outflow domain. Its simplest example is the straight tubular cylinder. For the rest of paper, we will always assume that Ω is an admissible in-and outflow domain. Definition 2.1. Let η : M → (−κ, κ) be continuous. We set the moving domain Ω η as Figure 1. Furthermore, for 0 < α < κ we set B α := Ω ∪ S α .
By our formal a priori estimate, we cannot expect the deformation and therefore the Hanzawa transform to be Lipschitz. Hence, a change of variables by the Hanzawa transform does not hold in the classical sense. Since the Hanzawa transformation inherits the regularity of the displacement η and preserves the convergence of η n in a suitable way by the preceding Lemma, an approximation argument shows that a change of variables is still possible. However, we have a small loss of regularity since the Jacobian determinant det dΨ η is not L ∞ . A careful inspection of this approximation argument, which can be found in [16], also gives a bound for the continuity constant.
is continuous from L p (Ω η ) to L r (Ω) and from W 1,p (Ω η ) to W 1,r (Ω). The continuity constant only depends on Ω, p, r, η H 2 (M) and the choice of β in the definition of the Hanzawa transform. An analogous statement holds for Ψ −1 η instead of Ψ η .
Combining the last two results we also get the following Lemma. Lemma 2.6. Let 1 < p ≤ ∞, η n η weakly in H 2 (M) with η L ∞ (M) < κ and let Ψ η n , Ψ η be defined by the same cut-off function β .
where the functions are extended by 0. Similar results hold for the convergence in the appropriate L p -spaces.
To preserve divergence constraint, we introduce the Piola transform T η ϕ of ϕ as pushforward of (det is an isomorphism between the Lebesgueand Sobolev spaces on Ω, respectively Ω η , as long as the order of differentiability is not larger than one. Furthermore, T η preserves the divergence-free property, as can be seen in [28,Theorem 7.20]. Using Lemma 2.5, we can obtain the usual Sobolev embeddings by transforming to the reference domain, but we have to take into account a loss of regularity. Also, the following trace operator, which compares the fluid velocity and the boundary velocity (which is given in Lagrange coordinates) is continuous. Definition 2.7. Let 1 < p ≤ ∞ and η ∈ H 2 (M) with η L ∞ (M) < κ. For 1 < r < p we define the linear, continuous trace operator tr η by Since the regularity of ∂ Ω η does not guarantee the existence of an outer normal, we will derive an suitable analogue, which implies a formula of Green type. We approximate η ∈ H 2 (M) by (η n ) n∈N ⊂ C 2 (M) in H 2 (M) and choose the same cut-off function β for the definitions of Φ η n . Since int M is a two-dimensional C 4 -manifold, there exist (locally) orthonormal, tangential C 3 vector fields e 1 , e 2 with e 1 × e 2 = ν. We set the (local) vector fields v n 1 := dΦ η n e 1 , v n 2 := dΦ η n e 2 and consider dΦ η n as a linear map from the parallelogram spanned by e 1 , e 2 into the parallelogram spanned by v n 1 , v n 2 . Then we get |v n 1 × v n 2 | = | det dΦ η n | |e 1 × e 2 | = | det dΦ η n | |ν| = | det dΦ η n |. By Φ η n ∈ C 2 (M), the outer normal ν η n of Ω η n admits the representation ν η n • Φ η n = v n 1 × v n 2 / |v n 1 × v n 2 |. We set v n := v n 1 × v n 2 and get v n = (ν η n • Φ η n ) | det dΦ η n |. Thus v n is independent of the choice of e 1 , e 2 and consequently defined globally on int M. For q ∈ int M and a curve c on int M with c(0) = q and d dt t=0 (2.8) In here, h j i denotes the components of the Weingarten map with respect to the orthonormal basis e 1 , e 2 . Hence, we have where H is the mean curvature and G the Gauss curvature of int M. Remark 2.11. We have already shown the following properties of v η and γ(η).
Using the scaled pseudonormal, we can now show a Green's type formula. Proof.
At the construction of the scaled pseudonormal we already saw that v η n → v η in L r (M) for any 1 ≤ r < ∞. Using Lemma 2.4, Lemma 2.6 and the definition of tr η , taking the limit n → ∞ yields Now passing to the limit k → ∞ and → ∞ finally proofs (2.13).
Although domains with Hölder-continuous boundary generally do not admit a Korn's inequality (cf. [1]), we can -allowing the typical loss of regularity -show a similar statement.

By a change of variables and
where we used the embedding To close this Section, we define the following Banach spaces.
with the norm ϕ L p σ (U) = ϕ E p (U) = ϕ L p (U) + div ϕ L p (U) admit a generalised trace operator tr n for the normal component as long as ∂U is Lipschitz. In particular, this trace operator is defined by an approximation argument through Green's formula from Proposition 2.12 as an element tr n u ∈ (H 1−1/p ,p (∂U)) * , and admits the representation tr n ϕ, g = ∂U ϕ · ν g dA for ϕ ∈ C ∞ (U), g ∈ H 1−1/p ,p (∂U).
It should be noted, that by mollification C ∞ 0 (U) is dense in E p (U) as long as ∂U ∈ C 0 (cf. [24,Prop. A.1]). To extend the trace operator to our deformed domain, we will use Green's formulae from Proposition 2.12.
Proof. We will define tr n η ϕ for ϕ ∈ C 1 (Ω η ). Then, by density, the claim follows. For b ∈ H 1−1/r,r (∂ Ω) we set i. e. tr n η ϕ ∈ (H 1−1/r,r (∂ Ω)) * . Since Ω is a bounded Lipschitz domain, there exists a linear, continuous extension operator F : H 1−1/r,r (∂ Ω) → W 1,r (Ω) (see [14,Satz 6.41]). By Lemma 2.5 and p < r, the map F • Ψ −1 η : is linear and continuous with an appropriately bounded continuity constant. Using the definition of tr η and the extension property of F, we have Thus, by definition of tr η n and Proposition 2.12 we have By this inequality, we can extend tr n η continuously to the space E p (Ω η ), which shows the first part of our claim. For the second part, we observe that r <r and therefore tr η ψ ∈ H 1−1/r,r (∂ Ω) for ψ ∈ W 1,r (Ω η ). Thus, an approximation of ϕ, the definition of tr η n for smooth functions and (2.13) shows the desired identity.
To obtain a trace operator defined only on a part of the boundary, we restrict tr n η to the space of test functions vanishing on the rest of the boundary. For a measurable subset γ ⊂ ∂ Ω and 1 < r < ∞ we therefore set W 1,r ∂ Ω\γ (Ω) := {u ∈ W 1,r (Ω) | u| ∂ Ω\γ = 0} and H as the restriction of tr n η to H 1−1/r,r 00 (γ).

Moreover, for an open subset
be seen by taking ϕ ∈ C ∞ 0 (V ) and observing the identity This space admits the following density result.
. At first, we will construct a suitable extension of u which vanishes on the whole boundary. Again by Remark 2.23, the extension u by zero gives u ∈ L 2 σ (B α ). Since B α is an bounded Lipschitz domain, B α ⊂⊂ B for some smooth bounded domain B, see Figure 2. Moreover, by [20] (with λ j = 0 since B α is connected), there exists an extension u ∈ L 2 σ (B) of u, whose support is compactly contained in B. In particular, this function vanishes on the boundary, i. e. we have [14,Satz 6.10] we have the extension E(ψ) ∈ W 1,r (B). Hence, (2.26) and the extension properties imply Again this identity can be interpreted as vanishing boundary values for v. We set Then F is linear and continuous, i. e. H(U) := {u ∈ L 2 σ (U) | Fu = 0} is a closed subspace of L 2 σ (U) and therefore -equipped with the L 2 scalar product -a Hilbert space. We will now show the density of . This implies our claim by restriction of the approximating functions to Ω η . We argue similarly as in [38,Theorem 1.6]. Obviously it holds V (U) ⊂ H(U). We will show that every functional G ∈ H(U) * vanishing on V (U) is the zero functional. Hahn-Banach's theorem then implies the density.
for all u ∈ H(U).
In particular, U g · ϕ dx = 0 for all ϕ ∈ V (U). By the theorem of De Rham (see [ The following result ensures that we can approximate functions of H M (Ω η ), while keeping the support uniformly away from the moving boundary. This Lemma adapts [24, Lemma A.13] to our situation and is crucial for the compactness theorem. Lemma 2.28. Let 0 < α < κ and C, ε > 0 be given. Then there exists ρ > 0 such that for all realised through the extension by 0 to R 3 and the L 2 Riesz representation.
Proof. We suppose the claim is false. Hence, we find sequences (ρ n ) n∈N In particular, we have (M), we have by the weak convergence of ϕ n and Green's formula from Proposition 2.21 , by density it suffices to show tr n η ϕ, b| ∂ Ω = 0 for all b ∈ C ∞ (Ω) vanishing in a neighbourhood of Γ, see [4]. For such a function b we get, again by Green's formula from Proposition 2.12, We would like to pass to the sequence ϕ n on the right hand side, but first we have to extend b • Ψ −1 η appropriately. Using the operator from [14, Satz 6.10], which is locally defined as a reflection, we can extend Furthermore, by a change of variables through the inverse Hanzawa transform (which can be extended smoothly using the same construction on the set Using the uniform convergence, it follows that supp Ψ ⊂ Ω η n −ρ n for n big enough. By our assumption we have (2.29) Choosing a ball B big enough, the compact embedding H 2 for n big enough, a contradiction to (2.29).

2.2.
Time-dependent function spaces. We will use an obvious substitute for the classical Bochner spaces in our moving domain. For I := (0, T ) with 0 < T < ∞ and a continuous η : It should be noted, that ∇, div and D are acting only with respect to the space variable and the derivatives are (partial) weak derivatives on Ω I η . These spaces, equipped with the canonical norms, are Banach spaces. Moreover, motivated by the formal a priori estimate, we define the following spaces It means that we renounce the vanishing boundary values for Y I and the divergence constraint for X I η . For η ∈ C 0 (I × M) and an appropriate cut-off function β , applying the (stationary) Hanzawa transformation Ψ η(t) at every time t ∈ I also defines a map between I × Ω and Ω I η . Clearly, we have the following result.
As shown in [24], Y I embeds into a space of Hölder-continuous functions.
As in the stationary case, the Hanzawa transform preserves convergence.
Lemma 2.32. Let (η n ) n∈N ⊂ Y I with η n L ∞ (I×M) ≤ α < κ be a bounded sequence and η n → η uniformly in I × M. Moreover, let the cut-off function β be chosen uniformly for the corresponding Hanzawa transforms. Then . Proof. First we remark that using weak and weak-* convergent subsequences, we can deduce η ∈ Y I with η L ∞ (I×M) ≤ α. The first two claims follow from the uniform convergence of η n and the definition of the Hanzawa transform (one should note that for the inverse functions, a case differentiation has to be made, see [16,Lemma 2.13]). For s ≥ 2, the embeddings H 2 (M) → → W 1,s (M) → → L 2 (M) are continuous and compact. By the definition of Y I and Aubin-Lions Lemma, η n converges strongly towards η in L s (I;W 1,s (M)). Taking into account the characterisations of the (spatial) Jacobians (2.3) of the transforms, the claimed convergences follow.
Remark 2.33. We can transfer the results of the preceding Sections to the time-dependent case by either repeating the proofs using Proposition 2.30 and Lemma 2.32 or applying them at every time t ∈ I. In particular, for η ∈ C 2 (I ×M) the corresponding time-dependent Piola transform T η is an isomorphism between the Lebesgue and Sobolev spaces on I × Ω respectively Ω I η , as long as the order of differentiability is not larger than one.
The next Lemma shows that our spaces are closed under the compatibility condition tr η u = ∂ t η ν on I × M in a suitable sense.
for some δ ∈ Y I , η ∈ Y I , u ∈ X I δ , where u n , u are spatially extended by zero. Then tr δ u = ∂ t η ν on I × M.
Proof. Again we choose the cut-off function β for the Hanzawa transforms uniformly. Let ϕ ∈ L 3 (I × Ω). By Hölder's inequality and a change of variables, we get . Together with the weak convergence of u n , (2.35) implies the weak convergence of u n • Ψ δ n towards u • Ψ δ in L 3/2 (I × Ω). Furthermore, by our Korn-type inequality and the time-dependent version of Lemma 2.5, u n • Ψ δ n is bounded uniformly in L 2 (I;W 1,5/3 (Ω)), i. e. u n • Ψ δ n converges weakly towards u • Ψ δ in L 2 (I,W 1,5/3 (Ω)). By linearity and continuity of the trace operator and the definition of tr η the sequence tr δ n u n converges weakly towards tr δ u in L 2 (I × M). Hence, the compatibility condition tr δ n u n = ∂ t η n ν on I × M and the weak convergence of ∂ t η n imply tr δ u = ∂ t η ν on I × M.

2.3.
Regularization of the displacement. To avoid the usual loss of regularity by a transformation to and from the reference domain, we construct a regularisation of the displacements. Since the initial data has to be adapted to the regularised moving domain, special care has to be taken. That means, we use a special mollification kernel and approximate from "above". Hence, our regularisation cannot be linear and does not preserve zero boundary conditions. Let (ϕ k ,U k ) N k=1 be a finite atlas of M with subordinate partition of unity (ψ k ) N k=1 . We extend a given δ ∈ C 0 (I ×M) constantly by δ (0, ·) and δ (T, ·) to (−∞, 0)×M and (T, ∞)×M, respectively. By the generalised reflection In here, the summand with index k is extended by 0 to M. By basic properties of the mollification (see e. g. [5,Proposition II.2.25]) and the reflection we get: Remark 2.38. Let (δ n ) n∈N ⊂ Y I be a bounded sequence. Then R ε δ n is also bounded in Y I independently of 0 < ε < ε 0 and n ∈ N (see [ Hence, R ε δ (0, ·) is independent of δ apart from δ (0, ·) = η 0 . By the continuous embedding H 2 0 (M) → C 0, 3 4 (M) and the properties of the generalised reflection we have Together with the identity above, this implies for z ∈ ϕ k (U k ) In the definition of R ε δ , only δ and w ε depend (non trivially) on time. Thus, the classical properties of mollification and reflection imply the following convergences.

Divergence-free extension operator.
To extend a test function to the shell equation defined on M to a test function of the fluid equation, we have to construct a divergence-free extension. As in our moving domains, we first look at the situation in the stationary case.
Then there exists a linear, continuous extension operator For a fixed N ∈ N, the continuity constant can be chosen uniformly with respect to η ∈ H 2 (M) satisfying η H 2 (M) ≤ N and and get with the same arguments as in [24,Proposition 2.19], i. e. using properties of the tubular neighbourhood, that F η is a linear and continuous operator from W 1,p 0 (M) to W 1,p (S α ) satisfying div F η b = 0. In particular, the continuity constant is bounded as claimed in the Lemma and we have the identity Since Ω is an admissible in-and outflow domain, the boundary of S α is given by the disjoint sets M α − , M α + and Γ α s , where the common boundary of S α and Ω \ S α is given by M α − , see where ν α is the outer normal on S α and ν α,c the outer normal on the Lipschitz domain Ω \ S α . Using the trace operator 2 , we have ξ ∈ H 1−1/p,p (∂ (Ω \ S α )) with  Using the same construction as in the preceding Lemma but with the unique very weak solution from [35, Theorem 0.3]), we can also extend a given L p (M) function. Since this very weak solution coincides with the unique weak solution of the Stokes system if regular data is given, a simple approximation argument and Remark 2.23 show that the extended function is divergence-free and fulfils an appropriate identity for the normal trace. Proposition 2.45. Let η ∈ H 2 (M) with η L ∞ (M) < α < κ and 2 ≤ p < ∞. Then there exists an linear, continuous extension operator By the construction of our extension operators, convergence of the displacements η n imply also the "uniform" convergence of the extended functions.
Proof. By the uniform convergence of η n , uniformly in S α . Therefore, F η n b converges uniformly to F η b in S α , where the convergence is also uniform with respect to b ∈ L p (M) satisfying b L p (M) ≤ 1 (or b ∈ H 2 0 (M) satisfying b H 2 0 (M) ≤ 1). Moreover, by the definition of the boundary data ξ and the linearity and continuity of the solution operator to the Stokes system, this convergence carries over to the whole extension operator.
Next, we look at the induced time-dependent extension operator. (2.48) Taking into account L p (I; H 2 (M)) → L 2 (I; L ∞ (M)), we deduce that both ∂ t F η b| I×S α ∈ L 2 (I; L 2 (S α )) and the trace ∂ t F η b| I×M α − ∈ L 2 (I; L 2 (M α − )) are appropriately bounded. Using the characterisation of the generalised time derivative by the difference quotient (see [6,Proposition A.6] and [6,Proposition A.7]) and the properties of the solution operator of the Stokes system, ∂ t F η b ∈ L 2 (I, L 2 (B α )) follows with a continuity constant bounded as claimed.
Analogously we can argue for the induced operator from Proposition 2.45.
The application of the extension operator from Proposition 2.45 at almost all times defines an linear, continuous extension operator For higher spatial regularity, the operator coincides with the one from Lemma 2.47. Moreover, b ∈ C(I; L q (M)) implies F η b ∈ C(I; L q σ (B α )). For a fixed N ∈ N, the continuity constant can be chosen uniformly with respect to η ∈ Y I satisfying η L ∞ (I×M) < α and η Y I < N.
The next Lemma states some convergence properties of the time-dependent extension operators.
Lemma 2.50. Let 0 < α < κ, N ∈ N and η, η n ∈ Y I with η Y I , η n Y I < N and η L ∞ (I×M) , η n L ∞ (I×M) < α for all n ∈ N. a) Let 1 ≤ p ≤ ∞, 2 ≤ q < ∞ and b ∈ L p (I; L q (M)). If η n converges uniformly towards η in I × M, then F η n b → F η b in L p (I; L q (B α )). Moreover, for b ∈ C(I; L q (M)), one has F η n b → F η b in C(I; L q (B α )). b) If b n converges weakly towards b in L 2 (I × M), F η n b n converges weakly towards Proof. a) Let b ∈ C(I; L q (M)). By the definition of F η b on I × S α and the extension to I × B α through the solution of the Stokes system, F η n b converges towards F η b in C(I; L q (B α )). By density and the uniform bound on the continuity constants from Corollary 2.49, the claim follows.
b) By the definition of F η b on I × S α , F η n b n converges weakly towards F η b in L 2 (I × S α ) (note that the integral term converges uniformly). Also the boundary values for the Stokes equation is also compact, i. e. η n converges strongly towards η in L 2p (I;W 1,2p (M)). Hence, by the characterisation of the spatial derivative (2.42), ∂ i F η n b converges towards ∂ i F η b in L p (I; L p (S α )). To treat the time derivative, we use the interpolation inequality to show the strong convergence of ∂ t η n towards ∂ t η in L 6 (I; L 2 (M)). Arguing again by interpolation and Sobolev's embedding, the embeddings are continuous, i. e. b ∈ L 3 (I; L ∞ (M)). By the characterisation of the time derivative (2.48), ∂ t F η n b converges towards ∂ t F η b in L 2 (I; L 2 (S α )). Using the properties of the Stokes operator, these convergences carry over to the extension operator defined on I × B α , which finishes the proof.

MAIN RESULT
Since our regularity is not sufficient to treat the force-coupling boundary term F, we add the weak formulations of the shell and the fluid and couple the test functions in a way that the force-coupling term vanishes. This implies that the test functions depend on the solution of the shell equation. For η ∈ Y I with η L ∞ (I×M) < κ, we therefore define the canonically normed space T I η as the set of all couples (b, ϕ), where b ∈ H 1 (I; L 2 (M)) ∩ L 3 (I; H 2 0 (M)), ϕ ∈ W η := H 1 (I; L 2 (Ω η(t) )) ∩ L 3 (I;W 1,3 (Ω η(t) )) ∩ L ∞ (I; L 4 (Ω η(t) )), with b(T, ·) = 0, ϕ(T, ·) = 0, div ϕ = 0 (see Remark 2.36) and for which ϕ − F η b can be approximated by functions ϕ n ∈ W η , ϕ n (T, ·) = 0, div ϕ n = 0 vanishing in a neighbourhood of the moving boundary. This implies tr η ϕ = b ν on I × M. We call (f, g, u 0 , η 0 , Hence, our notion of a weak solution to (1.2)-(1.8) is the following.
If the admissible data is sufficiently small, we have T * = ∞. If T * < ∞, we find a T < T * and a weak solution (η, u) such that the maximal displacement η L ∞ (I×M) is arbitrary close to κ.
Remark 3.5. For T * < ∞ the displacement of our solution is arbitrary close to κ, at which point different parts of the shell could touch each other, i. e. reach a situation which is not covered by our mathematical model.

3.1.
Compactness. By the non-linearity of the convective term, the weak convergences implied by the formal a priori estimate for some approximate solutions are insufficient to pass to the limit but a compactness argument is required. Because of our noncylindrical domain, classical arguments like Aubin-Lions [26] are not applicable. In particular, the representation of the dual spaces and an appropriate notion of a generalised time derivative are not clear. The proof of Aubin-Lions Lemma uses basically of the fundamental theorem of calculus, an application of Ehrling's Lemma and an Arzela-Ascoli argument. Lengeler used in [22] (see also [24]) the weak formulation of his problem instead of the fundamental theorem of calculus (and some modified Ehrling Lemma) to prove compactness. A careful analysis of his proof shows that replacing his extension operator F η n M η n by our extension operator F η and using the framework developed above, especially the density result from Lemma 2.28, the result also holds in our situation. More precisely, we have the following generalisation of [24, Proposition 3.8]: Lemma 3.6. Let 0 < α < κ and (f, g, u 0 , η 0 , η 1 ) be admissible data, (δ n ) n∈N ⊂ Y I a bounded sequence with δ n L ∞ (I×M) < α and assume that δ n → δ uniformly in I × M for some δ ∈ Y I with δ L ∞ (I×M) < α. Moreover, let (η n , u n , v n , u n 0 , η n 1 ) n∈N be a bounded sequence in Y I × X I δ n × X I δ n × L 2 (Ω δ n (0) ) × L 2 (M) satisfying Du n : Dϕ dx dt for all (b, ϕ) ∈ T I δ n . Furthermore, let tr δ n u n = ∂ t η n ν on I × M and ∂ t η n ∂ t η weakly in L 2 (I, L 2 (M)), for some η ∈ Y I , u ∈ X I δ , where u n , u are spatially extended by zero. Then (∂ t η n , u n ) converges strongly towards Note that the structure of the proof of Lemma 3.6 and of the compactness result in [22], [24] is the same, but lengthy and technically demanding. The proof of Lemma 3.6 can be found in [16,Lemma 4.4]. Since already the proof in [24] is densely written we give, for the convenience of the reader, full details in the Appendix. Note that by the assumptions of the preceding Lemma, tr δ u = ∂ t η ν is implied by Lemma 2.34. This fact is missing in [22], [24].

Construction of basis functions.
Since we do not transform our system to the reference domain, we have to construct an appropriate set of basis functions, at least in case of a given "smooth" deformation δ ∈ C 4 (I × M) with δ L ∞ (I×M) < κ . Therefore we chose a basis ( Y k ) k∈N of H 2 0 (M) and a basis ( X k ) k∈N of the canonically normed space X(Ω) := {ϕ ∈ W 1,3 (Ω) | div ϕ = 0, ϕ| M = 0}.
We extend Y k to F 0 Y k ∈ X(Ω) by Lemma 2.41 and set ( W 2k , W 2k ) := (0, X k ), is dense in the space T (I, Ω).

3.3.
The decoupled, regularised and linearised problem. To obtain a weak solution of (1.2)-(1.8), we first decouple the dependency of the moving domain from the solution of the shell equation, i. e. we prescribe some displacement δ with δ (0, ·) = η 0 . We will later restore this coupling by a fixed point argument, hence we have to choose δ in a space which Y I embeds compactly into. Therefore, we prescribe δ ∈ C(I × M) with δ = 0 on I × ∂ M and δ L ∞ (I×M) ≤ α < κ. We further regularise the displacement, therefore we also have to adapt the initial fluid velocity u 0 ∈ L 2 σ (Ω η 0 ) and by the compatibility condition also the initial velocity of the shell equation η 1 ∈ L 2 (M). To avoid the usual loss of regularity by transformation, we use the regularisation from Proposition 2.37 which approximates δ at t = 0 from "above". By Proposition 2.37 and Proposition 2.39, there exists 0 < ε 0 = ε 0 (α, η 0 ) such that holds for all 0 < ε < ε 0 and all δ ∈ C(I × M) satisfying δ (0, ·) = η 0 , δ L ∞ (I×M) ≤ α, i. e.
In order to derive a uniform energy estimate, we multiply (3.19) with α j n (t), take the sum over j = 1, . . . , n and get f n · u n dx.

(3.25)
Thus, Reynolds transport theorem yields where ν R ε δ (t) is the outer normal of Ω R ε δ (t) and v R ε δ (t) the boundary velocity. By our assumption for the moving domain, the deformations happen only along the outer normal on M. In particular, the outer normal on ∂ M is perpendicular to the normal on Γ R ε δ (t) by the orthogonality assumption for the reference domain. Hence, transforming the boundary integral to the boundary of the reference domain and taking into account the compatibility condition (3.24) and Remark 2.11, the identity follows. Arguing as in our formal a priori estimate, we deduce the energy inequality esssup t∈(0,T 1 ) E(R ε δ , η n , u n ,t) ≤ E 0 (R ε δ (0, ·), η 0 , ∂ t η n (0, ·), u n (0, ·)) (3.26) for all 0 < T 1 ≤ T . Hence, by the coercivity of the Koiter energy and the convergences (3.21), (3.22) and (3.14) the couples (η n , u n ) are uniformly bounded in Y I × X I R ε δ as claimed in the Lemma. Using the compact embedding Y I → → C 0 (I × M), we get for a subsequence η n * η weakly-* in L ∞ (I, H 2 0 (M)) and uniformly in I × M, thus η ∈ Y I . By our Korn-type inequality, the spatial extensions of u n , ∇u n and Du n by zero are uniformly bounded in L ∞ (I; L 2 (R 3 )), L 2 (I; L 13/7 (R 3 )) and L 2 (I; L 2 (R 3 )), respectively. For the convenience we use the notation ∇u and Du also outside Ω I R ε δ but we emphasize that the usual meaning of the symbols only hold on the inside. Hence, u n * u weakly-* in L ∞ (I; L 2 (R 3 )), ∇u n ∇u weakly in L 2 (I; L 13/7 (R 3 )), Du n Du weakly in L 2 (I; L 2 (R 3 )), Since div u n = 0 in Ω I R ε δ the convergences imply div u = 0 in Ω I R ε δ , i. e. u ∈ X I R ε δ . Moreover, (3.26) and the lower semi-continuity of Koiter's energy and the norms imply (3.18) and the uniform bound on (η, u) as claimed by the Lemma. To show (3.16), we first take ϕ ∈ C 1 0 ([0, T )). Using again Reynolds transport theorem, the orthogonality assumption for the reference domain, the compatibility condition (3.24) and Remark 2.11, we have Therefore, by multiplying (3.19) with ϕ, integration over I and integration by parts with respect to time, we get Du n : D(ϕ W j ) dx dt By the linearisation of the convective term and (3.21), (3.22), (3.27) and (3.28) we can pass to the limit n → ∞. Thus, (η, u) satisfy (3.16) for all test functions (b, ϕ) ∈ span (ϕW k , ϕW k ) ϕ ∈ C 1 0 ([0, T )), k ∈ N . By Proposition 3.11, this set is dense in T I R ε δ , hence (η, u) fulfils (3.16) for all (b, ϕ) ∈ T I R ε δ . Since (3.20) implies η n (0, ·) = η 0 , we deduce that η(0, ·) = η 0 , using (3.27) as well. Moreover, by (3.24) and Lemma 2.34, it follows that tr R ε δ u = ∂ t η ν and the proof is complete.
3.4. The regularised problem. To restore the coupling between the displacement and the domain as well as the convective term, we use a set-valued fixed-point theorem, i. e. the Bohnenblust-Karlin theorem [2,Theorem 17.57]. By choosing a sufficiently small time interval together with our compactness result, we satisfy the assumptions of the fixed-point theorem and get a weak solution of the regularised problem.
Proof. We define I 0 := (0, T 0 ) and chose a δ The associated weak solution (η, u) of the regularised, decoupled and linearised problem from Lemma 3.17 satisfies the energy inequality (3.32) and, by the coercivity of the Koiter energy, the estimate In here, the constant can be chosen independent of the admissible data, δ , v and ε, if there exits an upper bound on the norms f L 2 (I 0 ×B κ ) , g L 2 (I 0 ×M) , η 0 H 2 0 (M) , η 1 L 2 (M) and u 0 L 2 (Ω η 0 ) and as long as δ satisfies δ (0, ·) = η 0 and δ L ∞ (I 0 ×M) ≤ α. Extending u by zero in the spatial direction, we also get the estimate u L 2 (I 0 ×R 3 ) ≤ √ T 0 c 0 . Hence, we set c 1 := max{c 0 , √ T 0 c 0 }. By Lemma 2.31, the embedding Y I 0 → C 0,1−θ (I 0 ;C 0 (M)) is linear and continuous for some 1 2 < θ < 1 with an operator norm denoted by c 2 . Since η 0 L ∞ (M) < κ, we can choose 0 < T ≤ T 0 which satisfies and depends only on c 1 , c 2 , and a positive lower bound on κ − η 0 L ∞ (M) . We set I := (0, T ) and notice that this particular choice of T , together with the initial condition η(0, ·) = η 0 and the θ -Hölder-continuity of η, implies η L ∞ (I×M) < α. We define the space and the non-empty, closed, convex subset We define F : D ⊂ Z → 2 Z as the set-valued map which assigns to (δ , v) ∈ D the set of all weak solutions (η, u) of the decoupled, regularised and linearised problem on the interval I to the admissible data (f, g, u 0 , η 0 , η 1 ) and the functions δ and v, which satisfy for all 0 < T 1 ≤ T the inequality (3.18) and the estimates η L ∞ (I×M) ≤ α, where u is again extended by 0 in the spatial direction. Taking into account the boundary and initial conditions of the weak solution, F maps D into its power set, F(D) ⊂ 2 D . To show that F has a fixed point, we use the theorem of Bohnenblust-Karlin, see [2,Theorem 17.57]. Therefore, we have to check prerequisites, i. e. we have to show that for all (δ , v) ∈ D the set F(δ , v) is non-empty and convex, the graph of F is closed, and the image of F is relatively compact in Z. By our choice of T , Lemma 3.17 implies that for all (δ , v) ∈ D the set F(δ , v) is non-empty. By the linearity of the decoupled, regularised and linearised problem and the coercivity and bilinearity of the Koiter energy, some straightforward computations show that F(δ , v) is convex. To show the relative compactness of F(D) in Z, we take a sequence (η n , u n ) n∈N ⊂ F(D). Thus, there exists a sequence (δ n , v n ) n∈N ⊂ D with (η n , u n ) ∈ F(δ n , v n ). Since ε is fixed, by Proposition 2.37 (R ε δ n ) n∈N is bounded in C 4 (I × M). Using the Arzela-Ascoli theorem, we get, for a subsequence,  for some η ∈ Y I , and u n * u weakly-* in L ∞ (I; L 2 (R 3 )), ∇u n ∇u weakly in L 2 (I; L 13/7 (R 3 )), Du n Du weakly in L 2 (I; L 2 (R 3 )), (3.38) for some u ∈ X I ξ , whereby the functions are extended spatially by zero. Hence, we emphasize again that the symbols ∇u and Du have their usual meaning only on the set Ω I ξ . Since the weak solutions satisfy the identity (3.16), we can use the compactness result (Lemma 3.6) to obtain the strong convergences ∂ t η n → ∂ t η in L 2 (I × M), u n → u in L 2 (I × R 3 ).
(3.39) By (3.37) and (3.39), a subsequence of (η n , u n ) n∈N converges towards (η, u) in Z, therefore F(D) is relatively compact in Z. It remains to show that F has a closed graph. To this end, we consider some sequences for a (δ , v) ∈ Z and a (η, u) ∈ Z. We will prove that (δ , v), (η, u) ∈ D and (η, u) ∈ F(δ , v).
Since D is closed, (δ , v), (η, u) ∈ D follows. With the same arguments as above, we find a subsequence such that (3.35), (3.37), (3.38) and (3.39) hold. By the properties of the regularisation operators, (3.35) and (3.40) imply ξ = R ε δ and Furthermore, from (η n , u n ) ∈ F(δ n , v n ) and our convergences through the lower semicontinuity of the norms and the continuity of the Koiter energy we can deduce that for all 0 < T 1 ≤ T the couple (η, u) satisfies the energy inequality (3.18) and the estimates Furthermore, (η n , u n ) ∈ F(δ n , v n ) implies η n (0, ·) = η 0 on M and tr R ε δ n u n = ∂ t η n ν on I × M. By the uniform convergence of η n and Lemma 2.34, we also have η(0, ·) = η 0 and tr R ε δ u = ∂ t η ν. To finally show the identity (3.16) for (η, u) and all test functions (b, ϕ) ∈ T I R ε δ , we will again use the fact that (η n , u n ) ∈ F(δ n , v n ). Therefore, Du n : Dϕ n dx dt holds for all (b n , ϕ n ) ∈ T I R ε δ n . Since we cannot use (b, ϕ) ∈ T I R ε δ directly as a test function in (3.42), we have to use the special structure of this space. Hence, we take b ∈ H 1 (I; L 2 (M)) ∩ L 3 (I; H 2 0 (M)) with b(T, ·) = 0. By Lemma 2.47, (b, F R ε δ n b) ∈ T I R ε δ n . Taking into account the convergences from Lemma 2.50 and Lemma 2.46 as well as (3.35), (3.37), (3.38), (3.39) and (3.41), we can pass to the limit n → ∞ in the identity (3.42) tested with (b, F R ε δ n b) and obtain that (η, u) satisfies (3.16) for (b, F R ε δ n b). Due to our definition of T I R ε δ , it remains to show that (η, u) satisfies the identity (3.16) for all test functions (0, ϕ) with ϕ ∈ W R ε δ , ϕ = 0 in a neighbourhood of the moving boundary, div ϕ = 0 and ϕ(T, ·) = 0. For such a test function (0, ϕ), the uniform convergence of R ε δ n towards R ε δ implies supp ϕ ⊂ Ω I R ε δ n for n big enough. Therefore, silently extending this function by zero in the spatial direction, we have that (0, ϕ) ∈ T I R ε δ n is an admissible test function for the identities (3.42) for n big enough. Using again the convergences from above, we can pass to the limit n → ∞ to show that (η, u) is a weak solution of the decoupled, regularised and linearised problem, i. e. (η, u) ∈ F(δ , v). Hence, F has a closed graph and we can use theorem of Bohnenblust-Karlin, [2,Theorem 17.57]), to obtain a fixed point and therefore a weak solution of the regularised problem. The energy inequality (3.32) then follows from the definition of the map F.

3.5.
Limiting Process. Taking weak solutions of the regularised problem to the parameter ε, we can now pass to the limit ε → 0 to obtain a weak solution for our problem (1.2)-(1.8).
Proof. (of Theorem 3.3) We choose some T 0 ∈ (0, ∞) and set I 0 := (0, T 0 ). Further we set α := 1 2 ( η 0 L ∞ (M) + κ), ε n := 1 n and take n ∈ N big enough so that ε n < ε 0 (with ε 0 = ε 0 (α, η 0 ) as in Section 3.3) holds. By Lemma 3.31, we get the existence of an time interval I = (0, T ) with 0 < T ≤ T 0 independently of n and weak solutions (η n , u n ) of the regularised problem to the regularization parameter ε n fulfilling η n L ∞ (I×M) ≤ α and (3.32) for all 0 < T 1 ≤ T . Using the coercivity of K and (3.14) we deduce the estimate η n Y I + u n X I for a subsequence and for an η ∈ Y I . Since by Remark 2.38 R ε n η n is uniformly bounded in Y I , we similarly get a subsequence with R ε n η n * ζ weakly-* in L ∞ (I, H 2 (M)) and uniformly in I × M, ∂ t R ε n η n * ∂ t ζ weakly-* in L ∞ (I, L 2 (M)). (3.45) Taking Proposition 2.37 into account, we have This implies ζ = η, by the convergence properties of the regularization operator from Lemma 2.40. As usual, to treat the convective term in the fluid part, we need further information from the parabolic embeddings. By our choice of ε 0 we have R ε n η n L ∞ (I×M) ≤ α+κ 2 < κ. Hence, using our Korn-type inequality and the Sobolev embeddings, the sequence (u n ) n∈N is uniformly bounded in L 2 (I; L 11/2 (Ω R εn η n (t) )) and in L 2 (I;W 1,r (Ω R εn η n (t) )) for all 1 ≤ r < 2. The parabolic embedding (see [27,Korollar 2.10]) and the Hölder interpolation (see [13,Proposition 3.1]) yield L ∞ (I, L 2 (Ω δ n (t) )) ∩ L 2 (I, L 11/2 (Ω R εn η n (t) )) → L q (I, L q (Ω R εn η n (t) )) for q = 16/5. Extending u n , ∇u n and Du n by zero to I × R 3 , we get, for a further subsequence, the convergences u n * u weakly-* in L ∞ (I; L 2 (R 3 )), u n u weakly in L q (I; L q (R 3 )) and weakly in L 2 (I; L 11/2 (R 3 )), ∇u n ∇u weakly in L 2 (I; L 20/11 (R 3 )), Du n Du weakly in L 2 (I; L 2 (R 3 )), for an u ∈ X I η which is also extended by zero in an appropriate fashion. Using the compactness result (Lemma 3.6), we deduce that Since (η n , u n ) are weak solutions to the regularised problem, we have η n (0, ·) = η 0 and tr R εn η n u n = ∂ t η n ν. By the uniform convergence from (3.44) and Lemma 2.34, η(0, ·) = η 0 and tr η u = ∂ t η ν follows. Also, due to the lower semi-continuity of the norms and the Koiter energy, the energy estimates (3.14) of the regularised solutions imply the energy estimate (3.4) for all 0 < T 1 ≤ T . To show the integral identity (3.2) for all (b, ϕ) ∈ T I η , we have to use the special structure of the space of test functions. Let b ∈ H 1 (I; L 2 (M)) ∩ L 3 (I; H 2 0 (M)) with b(T, ·) = 0. Then, using the properties of our extension operator F R εn η n , i. e. Lemma 2.47, we have (b, F R εn η n b) ∈ T I R εn η n . Hence for all n ∈ N the tuple (b, F R εn η n b) is an admissible test function for the corresponding integral identity of the regularised weak solution (3.30). By Lemma 2.50, F R εn η n b converges to F η b in L ∞ (I, L 4 (B α )). Thanks to the convergence of R ε n u n → u in L 2 (I; L 5 (R 3 )) by (3.48) and the weak convergence of ∇u n ∇u in L 2 (I; L 20/11 (R 3 )) by (3.46), we get for the first part of the convective term I Ω Rε n ηn(t) (R ε n u n · ∇)u n · F R εn η n b dx dt → In here, we also used the spatial extension by zero. Arguing similarly for the rest of the terms in (3.30) tested with (b, F R εn η n b), i. e. taking into account the convergences (3.44)-(3.48) as well as the convergences from Lemma 2.50 and Lemma 2.46, the integral identity (3.2) holds for all (b, F R ε δ n b) with b ∈ H 1 (I; L 2 (M)) ∩ L 3 (I; H 2 0 (M)) and b(T, ·) = 0. By the definition of our space of test functions, it only remains to show the identity (3.2) for all (0, ϕ) with ϕ ∈ W η , ϕ = 0 in a neighbourhood of the moving boundary, div ϕ = 0 and ϕ(T, ·) = 0. For such a test function (0, ϕ), the uniform convergence of R ε n η n towards η implies supp ϕ ⊂ Ω I R εn η n for n big enough. Hence, silently extending this function by zero in the spatial direction, we have that (0, ϕ) ∈ T I R εn η n is an admissible test function for the regularised identities (3.30) for n big enough. Using again the convergences (3.44)-(3.48), we can pass to the limit n → ∞ to show that (η, u) is a weak solution to our problem on the interval I = (0, T ).
To extend this local existence to an global result, we observe that by the coercivity of the Koiter energy, the energy inequality (3.4) implies the estimate where the constant c only depends on u 0 L 2 (Ω η 0 ) , η 1 L 2 (M) , f L 2 ((0,∞)×B κ ) and g L 2 ((0,∞)×M) . By the compatibility condition tr η u = ∂ t η ν and η L ∞ (I×M) ≤ α < κ, the estimate (3.49) implies that the quintuple (f(· − s), g(· − s), u(s), η(s), ∂ t η(s)) is an admissible data for almost all s ∈ I. Repeating the fist part of the proof, we get for almost all s ∈ I a local weak solution to this data, whereby the length of the time interval (0, T ) can be chosen independent of s by Lemma 3.31. Choosing an s 0 ∈ I outside of an appropriate null set close enough to T and shifting the local weak solution by s 0 , we get the existence of a weak solution (η 0 , u 0 ) on the time interval (s 0 , T + T /2). Considering the initial conditions and an the alternative integral identity (4.6), we see that the combined function (again denoted by (η, u)) is a weak solution to (1.2)-(1.8) on the time interval (0, T + T /2). Since (3.4) holds for (η 0 , u 0 ) as well as for (η, u) (for clarity we denote the initial energy for (η 0 , u 0 ) by E 0 (s 0 ) and notice that E 0 (s 0 ) ≤ esssup t∈(0,T ) E(t)), we have esssup t∈(s 0 ,T + T /2) Hence, the combined solution also fulfils the energy inequality (3.4) and therefore the estimate (3.49) with the same constant. Iterating this procedure, we obtain a maximal time 0 < T * ≤ ∞ such that for all 0 < T < T * a weak solution (η, u) of (1.2)-(1.8) exists on the interval I := (0, T ), fulfilling the energy inequality (3.4). Since by (3.49) the norms of the initial data for the extension stay uniformly bounded, Lemma 3.31 implies that T * < ∞ is only possible if the norm of the displacement converges to κ. Since small enough data implies a uniform bound for the displacement (see (3.33)), we have T * = ∞ for this case.

APPENDIX
For the convenience of the reader, we include the proof of the compactness result. As already mentioned, we will need a modified Ehrling's Lemma.
Analogously, we assume that the second claim is wrong. Then there exist ε > 0 and bounded sequences (η n ) n∈N , (η n ) n∈N ⊂ H 2 (M), (v n ) n∈N ⊂ V 2 (Ω η n ) and (ṽ n ) n∈N ⊂ V 2 (Ωη n ), which satisfy η n L ∞ (M) ≤ α, η n L ∞ (M) ≤ α and, after some scaling, v n V 2 (Ω ηn ) + ṽ n V 2 (Ωη n ) = 1 as well as As in the first part of the proof, we find a subsequence with v n → v,ṽ n →ṽ in L 2 (R 3 ). By the continuity of the Piola-transform (notice δ ∈ C 4 (M)), for all ϕ ∈ H M (Ω) with ϕ H M (Ω) ≤ 1 it follows Hence, the left-hand side of (4.3) converges and is therefore bounded. This implies Since by Lemma 2.25 the space X(Ω) is dense in H M (Ω), the left-hand side of (4.3) converges to zero as well, a contradiction to ε > 0.
Considering (4.8), the sequence c b,n ∈ C 0, 1 12 (I) is uniformly bounded with respect to n ∈ N and b ∈ B 1 (0; H 2 0 (M)). In particular, by Arzela-Ascoli's theorem, for any fixed b ∈ H 2 0 (M) a subsequence of (c b,n ) n∈N converges uniformly on I. Since Lemma 2.50 and our assumptions imply that (c b,n ) n∈N converges to c b weakly in L 2 (I), the sequence c b,n converges to c b uniformly on I. We will show that converges uniformly to 0 on I. To this end, we recall that the sequence (∂ t η n , u n ) n∈N is bounded in L ∞ (I, L 2 (M)) × L ∞ (I; L 2 (Ω δ n (t) )), and chose a countable dense subset I 0 of I such that the functions u n (s, ·) ∈ L 2 (B α ) and ∂ t η n (s, ·) ∈ L 2 (M) are bounded in their respective norms for all n ∈ N and s ∈ I 0 . Using a diagonal sequence argument, we get a (not further denoted) subsequence such that for all s ∈ I 0 we have ∂ t η n (s, ·) η * s weakly in L 2 (M) and u n (s, ·) u * s weakly in L 2 (B α ). Since the embedding H 1 (M) → L 2 (M) is compact, by Schauder's theorem the embedding through the dual operator (L 2 (M)) * → (H 1 (M)) * is also compact. Therefore, ∂ t η n (s, ·) → η * s in (H 1 (M)) * and, analogously, (u n (s, ·)) n∈N → u * s in (H 1 (B α )) * . By the estimate Ω δn(s) u n (s, ·) · F δ n b(s, ·) dx − Ω δ (s) u * s · F δ b(s, ·) dx + 2 M ∂ t η n (s, ·) − η * s b(s, ·) dA ≤ Ω δn(s) u n (s, ·) − u * s · F δ n b(s, ·) dx + Ω δ (s) u * s · F δ n b(s, ·) − F δ b(s, ·) dx + 2 ∂ t η n (s, ·) − η * s (H 1 (M)) * b(s, ·) H 1 (M) ≤ u n (s, ·) − u * s (H 1 (B α )) * F δ n b(s, ·) H 1 (B α ) + u * s L 2 (B α ) F δ n b(s, ·) − F δ b(s, ·) L 2 (B α ) + 2 ∂ t η n (s, ·) − η * s (H 1 (M)) * b(s, ·) H 1 (M) , the estimate (4.7) and Lemma 2.46 imply the convergence of (c b,n ) n∈N on s ∈ I 0 , uniformly with respect to b ∈ B 1 (0; H 2 0 (M)). Since we already identified the weak limit of (c b,n ) n∈N , we also have, for the original sequence, c b,n → c b on I 0 uniformly with respect to b ∈ B 1 (0; H 2 0 (M)). By the uniform bound on c b,n ∈ C 0, 1 12 (I), for s, s ∈ I we get where the constant c is independent of n, m ∈ N, s, s ∈ I and b ∈ B 1 (0; H 2 0 (M)). Let ε > 0. Since I is compact and I 0 dense in I, we get a finite subset I ε 0 of I 0 , such that for all s ∈ I there exists s ∈ I ε 0 with c |s − s | 1 12 < ε 2 . Since I ε 0 is finite, the convergence from above implies c b,n (s ) − c b,m (s ) < ε 2 for all s ∈ I ε 0 and n, m ≥ N uniformly with respect to b ∈ B 1 (0; H 2 0 (M)). Therefore we have the uniform convergence of c b,n (s) to c b,m (s) independently of b ∈ B 1 (0; H 2 0 (M)) and therefore the uniform convergence of h n to 0 on I.
By the uniform convergence of h n to 0, this implies | I c ∂ t η n ,n − c ∂ t η n dt| → 0. We now consider the inequality I Ω δn(t) u n · F δ n ∂ t η n dx dt − I Ω δ (t) u · F δ ∂ t η dx dt u · F δ ∂ t η n dx + 2 M ∂ t η ∂ t η n dA dt u · F δ ∂ t η n − F δ ∂ t η dx dt + 2 I M ∂ t η ∂ t η n − ∂ t η) dA dt ≤ I c ∂ t η n ,n − c ∂ t η n dt + I Ω δ (t) u · F δ ∂ t η n − F δ ∂ t η dx dt The weak convergence of ∂ t η n , Corollary 2.49 and the convergence from above therefore imply (4.5).
Hence, for a (in the following fixed) λ > 0 small enough and all n ∈ N big enough, we have Using the uniform convergence of h λ n and the bound on u n in X I δ n , we therefore deduce