Compressible Fluids Interacting with a Linear-Elastic Shell

We study the Navier–Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter’s elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies γ>127\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma > \frac{12}{7}}$$\end{document} (γ>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma >1 }$$\end{document} in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in Lengeler and Růžičkaka (Arch Ration Mech Anal 211(1):205–255, 2014) on incompressible Navier–Stokes equations.


Introduction
Fluid structure interactions have been studied intensively by engineers, physicists and also mathematicians. This is motivated by a plethora of applications anytime a fluid force is balanced by some flexible material; for instance in hydro-and aero-elasticity [7,16] or biomechanics [4]. In this work we consider the motion of an isentropic compressible fluid (in particular a gas) in a three-dimensional body. A part of the boundary is assumed to be changing in time. The displacement of the boundary is prescribed via a two dimensional surface representing a Kirchhof-Love shell. Its material properties are deduced by assuming small strains and plane stresses parallel to the middle surface. We prove the existence of a weak solution to the coupled compressible Navier-Stokes system interacting with the Kirchhof-Love shell on a part of the boundary. The time interval of existence is only restricted once a self-intersection of the moving boundary (namely the shell) is approached.

Motivation and State of Art
Over the last century mathematicians have been fascinated by the dynamics of fluid flows. The theory of (long-time) weak solutions started with the pioneering work of Leray concerning incompressible Navier-Stokes equations [34]. A compressible counterpart has been provided by Lions [37]. Lions' results have later been extended by Feireisl et al. [19,21] to physically important situations (including, in particular, monoatomic gases). Today, there exists an abundant amount of literature for both incompressible as well as compressible fluids. In the last decades fluid structure interactions have been the subject of active research. The interactions of fluids and elastic solids are of particular interest. A major mathematical difficulty is the parabolic-hyperbolic nature of the system resulting in regularity incompatibilities between the fluid-and the solid-phase. First results concerning weak solutions in the incompressible case consider regularized or damped elasticity laws, see [3,5,8,33]. The fluid interacts with an elastic shell which constitutes a moving part of the boundary of the physical domain in Lagrangian coordinates. The existence of strong solutions in short time was shown in [9]. We also refer to related studies in [13,14], where the motion of an elastic body in an incompressible fluid is considered. Long-time weak solutions in a similar setting have finally been obtained in [32] assuming a linearized elastic behavior of the shell. The authors of [32] consider a general three dimensional body in Eulerian coordinates. The elastic shell is a possibly large part of the boundary and may deform in the direction of the outer normal. Its material behavior depends on membrane and bending forces. A solution exists provided the magnitude of the displacement stays below some bound (depending only on the geometry of the reference domain) which excludes self-intersections. The results from [32] have been extended to some incompressible non-Newtonian cases in [31]; see also [24]. Results for incompressible fluids in cylindrical domains have been shown in [38,39] and [6]. The paper [38] deals with a cylindrical linear elastic/viscoelastic Koiter shell in two dimensions (the shell is prescribed by a one-dimensional curve). The papers [6,39] extend this to cylindrical three-dimensional fluid flows. Note that in [39] even nonlinear elastic behavior of the shell is allowed.
In contrast to the growing literature on incompressible fluids the knowledge about compressible fluids interacting with elastic solids is quite limited. To the best of our knowledge, the only related result is [28]. Here, a compressible fluid interacts with a structure modeled by a linear wave equation in Lagrangian coordinates. The result of [28] concerns the existence of short-time strong solutions. It is related to earlier results about the incompressible setting, see [13]. Results on long-time weak solutions from problems coupling compressible fluids with a priori unknown elastic structures seem to be missing. The aim of the present paper is to open this field by developing a compressible counterpart of the theory from [32]. More, precisely we are going to prove the existence of a weak solution to the compressible Navier-Stokes system coupled with a linear elastic Koiter-type shell. The two dimensional shell is connected to the velocity field via boundary values on the free part of the boundary. Moreover, momentum forces acting on the boundary are in equilibrium with the membrane forces and bending forces (flexural forces) of the shell.

The Model
We consider the Navier-Stokes system of an isentropic compressible viscous fluid interacting with a shell of Koiter-type of thickness ε 0 > 0. The Koiter shell model is a version of the Kirchhoff-Love shell. More precisely, it is a model reduction assuming small strains and plane stresses parallel to the middle surface of the shell. Physically this means that the shell consists of a homogeneous, isotropic material. Its mathematical formulation is as follows. Let ⊂ R 3 be the initial physical domain and let T > 0. We devide ∂ into the fixed in time part and its compact complement M, the part where the shell is located. The shell is assumed to be driven solely in the direction of the outer normal ν of , cf. [8,23]. This allows one to write the energy for elastic shells via a scalar function η : M → (a, b).
Here, the numbers a, b are fixed and depend only on the geometry of , such that self intersections of the boundary are not possible. For example, in case of a ball = B r the interval is (−r, ∞). The elastic energy of the deformation is then modeled via Koiter's energy [26, eqs.
which is the sum of two terms reflecting different material properties. The first term is the membrane part of the energy which would remain even in the case of thin films. Indeed, the term σ (ην) depends linearly on the pullback of the first fundamental form of the two dimensional surface η(M). The second term reflects the flexural part of the energy. Respectively, the argument θ depends linearly on the pullback of the second fundamental form (the change of curvature). The coefficient tensor C is a non-linear function of the first fundamental form. For more details on the derivation of this model we refer to [25,26], where Koiter's energy for nonlinear elastic shells has been introduced; see also [11,12] for a more recent exposition. Following [12, Thm. 4.2-1 and Thm. 4.2-2] one can linearize σ and θ with respect to η and obtain K (η) = m 2 η + Bη (1.1) for the L 2 -gradient K of K . Here, m > 0 depends on the shell material (to be precise on ε 0 and the Lamé constants), B is a second order differential operator and is the Laplace operator associated to the covariant derivative of the surface. In particular, it is shown in [12,, that for all η ∈ H 2 0 (M) with some c 0 > 0. Equation (1.1) is a Kirchhoff-Love shell equation for transverse displacements, cf. [10]. The technical restriction, that we only allow forces to act on the shell in (a fixed) normal direction, is the most severe restriction in our paper. Under this assumption we will, however, show long time weak solutions; they exist as long as the shell does not approach a self intersection. We also observe that long time existence results seem to be unavailable for less restrictive geometric assumptions. Even for one dimensional boundaries and for incompressible fluids no long time existence result seems to be available for less severe restrictions. Finally, observe that since η is assumed to have zero boundary values on ∂ M, there is a canonical extension by zero to ∂ , which we will use in the following without further remark.
We denote by η(t) the variable in time domain. With a slight abuse of notation we denote by I × η = t∈I {t}× η(t) the deformed time-space cylinder, defined via its boundary Recall that is a given (smooth) reference domain with outer normal ν.
Along this cylinder we observe the flow of an isentropic compressible fluid subject to the volume force f : I × η → R 3 . We seek the density : I × η → R and velocity field u : I × η → R 3 solving the following system: (1.6) Here, p( ) is the pressure which is assumed to follow the γ -law, for simplicity p( ) = a γ , where a > 0 and γ > 1. Note that in (1.3) we suppose Newton's rheological law S = S(∇u) = 2μ ∇u + ∇u T 2 − 1 3 div u I + λ + 2 3 μ div u I, with viscosity coefficients μ, λ satisfying μ > 0, λ + 2 3 μ > 0, see Remark 1.3 for the case λ + 2 3 μ = 0. The shell should respond optimally with respect to the forces, which act on the boundary. Therefore we have where S > 0 is the density of the shell. In (1.7) g : [0, T ] × M → R is a given force density and we have Here, η(t) : ∂ → ∂ η(t) is a change of coordinates and τ is the Cauchy stress.
To simplify the presentation in (1.7) we will assume that ε 0 S = 1 throughout the paper. We assume the following boundary and initial values for η: (1.9) where η 0 , η 1 : M → R are given functions such that In view of (1.4) we have to suppose the compatibility condition (1.10) By the canonical extension of η and ∂ t η by 0 to ∂ we can unify (1.4) and (1.5) to Our main result is the following existence theory for the system (1.2)-(1.9) which can be written in a natural way as a weak solution. The precise formulation can be found in Section 7, cf. (7.1) and (7.2): c(f, g, q 0 , 0 , η 0 , η 1 ). (1.12) The interval of existence is of the form I = [0, t), where t < T only in case η(s) approaches a self-intersection when s → t.
The function space for weak solutions to (1.2)-(1.9) is determined by the lefthand side of (1.12), taking into account the variable domain. For the precise assumptions on the given data, as well as the precise definition of a weak solution see Theorem 7.1 in the last section of the paper. We remark that in the three dimensional case the bound γ > 12 7 is less restrictive then the bound γ > 9 5 appearing in the pioneering work of Lions [37], but more restrictive than the bound γ > 3 2 arising in the theory by Feireisl et al. [21]. A detailed explanation can be found in the next subsection. For more information on the restrictions for the growth condition of the pressure see Remark 1.2 at the end of this section.

Mathematical Significance and Novelties
A primary task is to understand how to pass to the limit in a sequence of solutions (η n , u n , n ) to (1.2)-(1.9) which enjoys suitable regularity properties and satisfies the uniform estimate (1.12). The passage to the limit in the convective terms n u n and n u n ⊗ u n follows by local arguments combined with global integrability, see (6.13) and (6.14). Here, problems with the moving boundary can be avoided. Note that this is totally different to the incompressible system studied in [32], where huge difficulties arise due to the divergence-free constraint. As it is common for the compressible Navier-Stokes system, the major difficulty is to pass to the limit in the nonlinear pressure. A key step is to improve the (space)-integrability of the density to ensure that p( n ) actually converges to a measurable function (and not just to a measure). Locally, where the effect of the moving boundary disappears, this can be done by the standard method, see Proposition 6.3. However, our test-functions in the weak formulation are not compactly supported. This is crucial for the coupling of fluid and shell. Note, in particular, that this is different from [18], where the interaction of compressible fluids and rigid bodies is studied. In [18], at least the symmetric gradients of test-functions are supported away from the area of interaction. In our case, however, it is essential to exclude the concentration of p( n ) at the boundary. On account of the limited regularity of the moving boundary (it is not even Lipschitz continuous in three dimensions, see (1.12)) the common approach based on the Bogovskiȋ operator fails. We solve this problem inspired by a method introduced in [29] for compressible Navier-Stokes equations in irregular domains; see Propositions 6.4 and 7.4. Consequently, we can exclude the concentration of the pressure at the boundary. This, in turn, allows us to prove that the weak continuity of the effective viscous flux p( ) − (λ + 2μ) div u holds globally, see (6.35) and (7.29). In order to combine this with the renormalized continuity equation we are confronted with another problem: we do not have zero boundary conditions for the velocity at the shell. In general, it seems to be extremely difficult if not impossible to combine the properties of the effective viscous flux with the renormalized continuity equation in this case (see the remarks in [42,Section 7.12.5]). This is due to the additional boundary term which appears when extending the continuity equation to the whole space. However, due to the natural interplay between fluid flow and elastic shell, our situation can be understood as no-slip boundary conditions with respect to the moving shell. Hence, the just mentioned boundary term disappears due to the Lagrangian background of the material derivative. To make this observation accessible, a careful study of the damped continuity equation in time dependent domains is necessary. We refer to Section 3.2 and in particular Theorem 3.1, which collects the necessary regularity results for the density function on time changing domains; it implies, for instance, the respective renormalized formulation. This is the second essential tool which allows one to show strong convergence of an approximate sequence n and hence to establish the correct form of the pressure in the limit equation.
The third difficulty is to construct a sequence of solutions. In the present case, this is rather difficult to do, since the geometry and the solution are highly coupled via the partial differential equations. Hence, in order to use the ideas explained above rigorously, we need a four layer approximation of the system as follows: • Artificial pressure (δ-layer): replace p( ) = a γ by p δ ( ) = a γ +δ β where β is chosen large enough. • Artificial viscosity (ε-layer): add ε to the right-hand side of (1.2). • Regularization of the boundary (κ-layer): replace the underlying domain η by η κ where η κ is a suitable regularization of η. Accordingly, the convective terms and the pressure have to be regularized as well. • Finite-dimensional approximation (N -layer): the momentum equation has to be solved by means of a Galerkin-approximation.
The first two layers are common in the theory of compressible Navier-Stokes equations, see [21]. The third layer is needed additionally due to the low regularity of the shell described by η. By (1.12) we have η ∈ W 2,2 (M) such that Sobolev's embedding implies η ∈ W 1,q (M) for all q < ∞ but not necessarily η ∈ W 1,∞ (M). So, we do not have a Lipschitz boundary. In addition, it is necessary to regularize the convective terms in (1.2) and (1.3) (see the comments on the N -layer below for a detailed explanation).
On the last layer we are confronted with the problem that the function space depends on the solution itself. As a consequence a finite-dimensional Galerkin approximation is not possible, as the Ansatz functions depend on the solution itself. Motivated by [32] we apply a fixpoint argument in η and u for a linearized problem. (Roughly speaking we replace u ⊗ u in (1.3) by u ⊗ v and u in (1.2) by v for v given, see Section 4.3 since it is crucial for our fixed point argument that the momentum equation is linear in u. For (ζ, v) given we solve the system on the domain ζ . The domain still varies in time but is independent of the solution. Note here that is computed by solving the continuity equation with convective term independent of u. The existence of a weak solution (η, u) to the decoupled system can than be shown by the Galerkin approximation without further problems. This is due to the good a-priori information for from Theorem 3.1, see Theorem 4.4. The next difficulty is to find a fixed point of the mapping (ζ, v) → (η, u) in an appropriate function space. The compactness of the mapping situated on the shell is rather easy as we apply a proper regularization with arbitrary smoothness. The main issue is the compactness of the velocity. Inspired by ideas from [32] we can prove compactness of u n in L 2 (I × R 3 ) (where u n is extended by zero). This is based on Lemma 2.8, where we prove a variant of the Aubin-Lions compactness theorem for variable domains. It is noteworthy that we are unable to exclude a vacuum even in the situation of a damped continuity equation. To prevent the problem with the vacuum we replace on the κ-level the momentum ∂ t ( u) by ∂ t (( + κ)u) in the momentum equation, which allows us to show that u n is strongly compact in L 2 (I × R 3 ).

Outline of the Paper
In Section 2 we present basics concerning variable domains as well as the functional analytic set-up. In Section 3 we study the continuity equation (with artificial viscosity) on variable domains. The renormalized formulation is of particular importance. Section 4 is concerned with the decoupled system, its finite dimensional approximation and the fixed point argument. The main result of this section is the existence of a weak solution to the regularized system with artificial viscosity and pressure. In Section 5 we pass to the limit in the regularization (of domain and convective terms) and gain a weak solution to the system with artificial viscosity. Compactness of the density can be shown as in the fixed point argument. Hence we can pass to the limit in all nonlinearities without further difficulties. The proceeding Sections 6 and 7 deal with the vanishing artificial viscosity and vanishing artificial pressure limit respectively. Both follow a similar scheme, where the major difficulty is the strong convergence of the density. The argumentation is based on the weak continuity of the effective viscous flux, oscillation defect measures and the renormalized continuity equation. The main result of this paper (the existence of weak solutions to (1.2)-(1.9)) follows after passing to the limit with δ → 0 in Section 7. The full statement is given in Theorem 7.1. 12 7 in three space dimensions is needed to exclude concentrations of the pressure near the moving boundary. Indeed, such concentrations are excluded by constructing a test-function ϕ n whose divergence explodes at the boundary while p( n ) div ϕ n is still bounded. This requires, in particular, estimation of the integral I ηn n u n ∂ t ϕ n dx dt. Naturally, the function ϕ n depends on the distance to the boundary and as such on the shape of the moving boundary which only has low regularity. Indeed, the given a priori estimates imply that ∂ t ϕ n can only be bounded in L 2 (I ; L q ) for all q < 4 (using (1.4)). Hence, we need to know that n u n is bounded in L 2 (I ; L p ) uniformly in n for some p > 4 3 . This follows from the a priori estimates provided we have γ > 12 7 (using that n u n ∈ L 2 (I ; L 6γ /(γ +6) ) in three dimensions). In the two dimensional case we have instead ∂ t ϕ n ∈ L ∞ (I ; L q ) for all q < ∞. Consequently, no additional restrictions on γ are needed and the result holds for all γ > 1.

Remark 1.3.
Our proof requires the bulk viscosity λ + 2 3 μ to be strictly positive. In case λ + 2 3 μ = 0 it is necessary to control the full gradient by the deviatoric part of the symmetric gradient. Such a Korn-type inequality is well-known for Lipschitz domains, see [43]. In our context of domains with less regularity, a Korntype inequality for symmetric gradients is shown in [31,Prop. 2.9] following ideas of [1]. The integrability of the full gradient is, however, less than the one of the symmetric gradient. We believe that a corresponding trace-free version can be shown following similar ideas. Thus, the case λ + 2 3 μ = 0 could be included for the price that the velocity only belongs to W 1, p for all p < 2.

Preliminaries
The variable domain η can be parametrized in terms of the reference domain via a mapping η such that η : → η is invertible and η | ∂ : ∂ → ∂ η is invertible. (2.1) The explicit construction can be found below in (2.8).
Throughout the paper we will make heavily use of Reynolds transport theorem, which we will use without any further reference. This theorem says that provided all terms are well-defined. The above can easily be justified by transposition and a chain rule. The heuristic beyond is that ν is the direction in which the domain changes (which in our model is a fixed prescribed direction) and ∂ t η describes the velocity of change. Therefore, the scalar ∂ t ην • −1 η · ν η is the derivative of the change of the domain; i.e. the forces acting in direction of the outer normal of ∂ η(t) . For a couple of functions (ϕ, b) which satisfy tr η (ϕ) = b in the sense of Lemma 2.4 we have

Formal a Priori Estimates and Weak Solutions
We introduce the weak formulation of the momentum equation, which will be coupled to the material law of the shell. This is motivated by the a priori estimates. We will now derive these estimates formally. First, we multiply the momentum equation by u and integrate with respect to space (at a fixed time). We multiply the continuity equation by |u| 2 2 and integrate with respect to space (at the same time). Subtracting both and applying a chain rule yields By Reynolds' transport theorem we get d dt η To control the pressure term, we multiply the continuity equation by γ γ −1 and gain We obtain by Reynold's transport theorem and the assumed boundary values that Later we will make this step rigorous via the use of so-called renormalized formulation of the continuity equation, see Section 3.1 below. Hence, we have d dt η The boundary term represents the forces which are acting on the shell. Naturally these have to be in equilibrium with the bending and membrane potential of the shell. Formally, this is achieved by multiplying the shell equation (1.7) with ∂ t η. 1 Using once more that u • η = ∂ t ην on M we find that 1 Recall that we assume ε 0 ρ S = 1.
Thus, the right-hand sides of both equations cancel. Finally, we gain This implies, by Hölder's inequality and absorption, (1.12). In coherence with the a-priori estimates we introduce the following weak formulation of the coupled momentum equation:

Geometry
In this section we present the background for variable domains, see [32] for further details. Let ⊂ R 3 be a bounded domain with boundary ∂ of class C 4 with outer unit normal ν. In the following will be called the reference domain. We define for α > 0 the set There exists a positive number L > 0 such that the mapping is a C 3 -diffeomorphism. It is the so called Hanzawa transform. The details of this construction may be found in [30]. This is due to the fact, that for C 2domains the closest point projection is well defined in a strip around the boundary. Indeed, its inverse −1 will be written as −1 (x) = (q(x), s(x)). Here q(x) = arg min{|q − x||q ∈ ∂ } is the closest boundary point to x (which is an orthogonal projection) and s(x) = (x − q(x)) · ν(q(x)). For the sake of simpler notation we assume with no loss of generality that for a.e. t, v(t, ·) L r ( η(t) ) ∈ L p (I ) , (a) There is a homomorphism η : → η such that η | \S L is the identity; As the impact of the geometry on the PDE is quite severe we will include an explicit construction of η . Since we will use the parametrisation below locally we will assume Im(η) ⊂ [− L 2 , L 2 ], where L is a fixed size, such that given in (2.4) is well defined on the set ∂ × [−L , L]. We relate to any η : ∂ → (−L , L) the mapping η : → η , such that This can be constructed as follows.
) for all l ∈ {1, . . . , k}. We relate to any η : ∂ → (−L , L) the mapping η : → η given by (2.8) Hence, the two one-to-one relations in (2.7) are satisfied. If η ∞ < L 2 , the mapping η can be extended such that Due to the assumption η ∞ < L 2 we have that ⊂ L 2 −η ⊂ ∪ S L . The extension is obtained by setting elsewhere. (2.10) We collect a few properties of the above mapping η . Lemma 2.3. Let 1 < p ∞ and σ ∈ (0, 1]. Then: The continuity constants depend only on , p, r, σ and the respective norms of η. Proof. The first two properties can be found in [32,Lemma 2.6]. The third assertion follows by transposition rule and the fact, that ∇ η , ∇ −1 η are uniformly bounded. Indeed, let us assume that f ∈ W σ, p ( ) for some σ ∈ (0, 1). Recall that this means that In case σ = 1, the result follows directly by the transposition rule. Since the argument can be applied analogously to f • −1 η , the proof is completed.
The following lemma is a modification of [32, Cor. 2.9]: The continuity constants depend only on , p, and η W 2,2 .
Proof. The claim is a consequence of Lemma 2.3 and the continuity of the trace operator on the reference domain , which is assumed to be smooth.
The following lemma allows us to extend functions defined on the variable domain to the whole space R 3 this is not trivial for η ∈ W 2,2 (∂ ) because the boundary is not Lipschitz continuous, however, it requires the additional assumption η L ∞ (∂ ) < L 2 : Proof. If η ∈ W 1,∞ (∂ ) the result is standard. There is a continuous linear operator for any bounded Lipschtz domain A and 1 p < ∞, see, for instance [2, Thm. 5.28], where even slightly less regularity of the boundary is required.
For the general case we use the extension above to transfer from a functions space over the variable domain to a function from a functions space over the reference domain. Indeed Lemma 2.3 implies that the mapping is continuous for any 1 r < p. Since is assumed to have a uniform Lipschitz boundary it is possible to extend the function u η to the whole space. Hence, using the Extension operator on , we find In order to transform back we use the fact, that η is invertible on * η where ⊂ * η due to the assumption η L ∞ (∂ ) < L 2 , cf. (2.9). By Lemma 2.3 we find that the mapping is continuous for any 1 r <r < p. Finally, we set It is now easy to check that E η has the required properties.

Convergence on Variable Domains
Due to the variable domain the framework of Bochner spaces is not available. Hence, we cannot use the Aubin-Lions compactness theorem. In this subsection we are concerned with the question how to get compactness anyway. We start with the following definition of convergence in variable domains. Convergence in Lebesgue spaces follows from an extension by zero.
(c) Let p = ∞ and q < ∞. We say that a sequence ( Note that in case of one single η (i.e. not a sequence) the space L p (I, L q ( η )) (with 1 p < ∞ and 1 < q < ∞) is reflexive and we have the usual duality pairing provided that η is smooth enough, see [41]. Definition 2.7 can be extended in a canonical way to Sobolev spaces. We say that a sequence (g i ) ⊂ L p (I, W 1,q ( η i )) converges to g ∈ L p (I, W 1,q ( η i )) strongly with respect to (η i ), in symbols if both g i and ∇g i converges (to g and ∇g respectively) in L p (I, L q ( η i )) strongly with respect to (η i ) (in the sense of Definition 2.7 a)). We also define weak and weak * convergence in Sobolev spaces with respect to (η i ) with an obvious meaning. Note that an extension to higher order Sobolev spaces is possible but not needed for our purposes.

A Lemma of Aubin-Lions Type for Time Dependent Domains
For the next compactness lemma we require the following assumptions on the functions describing the boundary: Assume further that there are sequences ( (2.13) Remark 2.9. Assumption (A3) in Lemma 2.8 can be extended in an obvious way to the case of higher order distributional derivatives. We have chosen the version above as it is most suitable for our applications.

Proof. First we show local compactness. Consider a cube
. We can apply the classical Aubin-Lions compactness Theorem [35] for the triple and gain (2.14) Note that we do not have to take a subsequence since the original sequence already converges by (A3). Now note that (A1) implies for some α ∈ (0, 1) by interpolation. Consequently, for a given κ > 0 there is where we have set They can be chosen in such a way that we find a partition of unity (ψ k ) associated to the family Q k such that ψ k ∈ C ∞ 0 (Q k ) and In particular, by taking a diagonal sequence, we can assume that (2.14) holds with On account of (2.14) (with Q = Q k ) and (A2) the first integral on the right-hand side converges to zero. Due to (2.15) the second integral can be bounded in terms of κ. Here, we took into account the boundedness of which means we have (2.16) However, our assumptions imply that χ η i r i v i converges weakly in L q (I, L a (R 3 )) at least after taking a subsequence. As a consequence of (2.16) we can identify the limit and the claim follows.

Remark 2.10. In the case
Since weak convergence and norm convergence implies strong convergence, we find (by interpolation) that Showing such a result for the velocity field in the context of incompressible fluid mechanics is the main achievement of the paper [32]. As opposed to (A3), the time derivative is only a distribution acting on divergence-free test-functions in the incompressible case. In contrast to the compactness arguments in [32], the proof of Lemma 2.8 does not face this difficulty.

Renormalized Solutions in Time Dependent Domains
This subsection is concerned with the study of the continuity equation Observe the following interplay of the two terms of the material derivative, that shall be used many times within this work. A (strong) solution to (3.5) satisfies for In the case of our consideration we find, due to tr ζ (u) = ∂ t ην, that 2) will serve as a weak formulation of (3.1). It is worth mentioning, that by taking ψ ≡ χ [0,t] in (3.2) we find that the total mass is conserved in the sense that for all t ∈ I . Following DiPerna and Lions [15] we will introduce a renormalized formulation which will be of crucial importance for the reminder of the paper. An important observation is that the formulation in (3.2) can be extended to the whole space despite the fact that u does not have zero boundary values (this will be essential to prove strong convergence of the density, see Section 6.4). In fact, we have 3) and hence find that Now, integration by parts and Reynolds' transport theorem imply that the first line vanishes. Again, integration by parts and Reynolds' transport theorem transfer the second line in the appropriate weak formulation. Hence, we find the renormalized formulation is

The Damped Continuity Equation in Time Dependent Domains
We will need very explicit a-priori information about our approximation of the density . The necessary result is collected in Theorem 3.1 below. For the analogous results for fixed in time domains see [21, section 2.1]. We will assume that the moving boundary is prescribed by a function ζ of class C 3 (I × M). For a given function w ∈ L 2 (I ; W 1,2 ( ζ )) and ε > 0 we consider the equation On the other hand, we have This motivates the choice of the Neumann boundary values in (3.5) which implies the following neat weak formulation: . We wish to emphasize that this weak formulation is canonical with respect to the moving boundary, as it is the only formulation which preserves mass. This turns out to be the essential property to gain the necessary estimates and correlations.
) be the function describing the boundary. Assume that w ∈ L 2 (I ; W 1,2 ( ζ )) ∩ L ∞ (I × ζ ) and 0 ∈ L 2 ( ζ(0) ). Then: (a) There is a unique weak solution to (3.6) such that Then the following holds, for the canonical zero extension of ≡ χ ζ : Proof. In order to find a solution to (3.6) we discretise the system. It is standard to find a smooth orthonormal basis (ω k ) k∈N of W 1,2 ( ). Now define pointwise in t By Lemma 2.2 we still know that ω k belongs to the class C 3 ( ζ (t)). Obviously, (ω k ) k∈N forms a basis of W 1,2 ( ζ (t)). We fix the initial values as the L 2 ( ζ (0))- We are looking for a function N = β k ω k satisfying for all l = 1, with initial data N 0 . This is equivalent to and β l (0) = β l 0 . Here, the β k 's are the unknowns (as functions only on time). Now, we define the matrices A, B ∈ R N ×N by Because (ω k ) is a basis of W 1,2 ( ζ(t) ) the matrix A is positive definite. Hence (3.9) can be written as β = A −1 Bβ where β is the vector containing the β k 's. This is a linear system of ODEs which has a unique solution. In order to pass to the limit N → ∞ we need uniform a priori estimates. So, we multiply (3.9) by β l and sum over l such that We use Lemma 2.3 and the trace theorem W is given by (2.11). Interpolating W 1 2 ,2 between L 2 and W 1,2 (see see [2,Chapter 7.3]) we obtain for κ > 0 arbitrary (3.10) The same estimate holds for (I I ) N by a simple application of Young's inequality. Combining both and applying Gronwall's lemma we have shown where C depends on ξ , w ∞ and |I | only. Hence, we obtain the existence of a limit function using (2.12). Moreover, N converges weakly (weakly * ) to . The passage to the limit in (3.8) is obvious as it is a linear equation. The uniqueness is shown in the following way: assume that we have two solutions ρ 1 , ρ 2 . The differences of the two solutions 1 − 2 and ρ 2 − ρ 1 are both solutions with zero initial datum. Now we may take ϕ ≡ 1 as a test-function for both equations and find that 0 ζ (t) Hence, a) is shown.
Next we show b). We extend by zero to I × R 3 and obtain . Now, we mollify the equation in space using a standard convolution with parameter κ > 0. Then we find that the following PDE is satisfied: We observe that this equation implies in particular, that ∂ t κ is a smooth function in space. To proceed we need to use an extension operator on w. Since ζ is uniformly in C 2 there exists a continuous linear extension operator see, for instance, [2,Thm. 5.28]. Using this operator, we can reformulate (3.11) by: as well as a.e. in I . Note that a) implies that ∈ L 10/3 (I × ζ ). Now we multiply (3.12) by θ ( κ ) and obtain Due to the properties of the mollification and θ ∈ C 2 we have (at least after taking a subsequence) for all q < ∞. The same is true for θ ( κ ) and θ ( κ ). Consequently, we have and Hence, multiplying (3.14) by ψ ∈ C ∞ (I × R 3 ) and integrating over I × R 3 implies This proves b) since E ζ w ≡ w in ζ . In order to prove c) we use (3.7) for ψ = χ [0,t] and θ = θ n where θ n is a smooth approximation to θ(z) = z − = − min{z, 0}. It is possible to define θ n as a convex function such that pointwise as n → ∞ as well as uniformly in n and z. This yields On account of (3.15) and (3.16) we can pass to the limit by dominated convergence, so we have which implies θ( ) = 0 a.e. by the definition of θ and the non-negativity assumption on 0 . This implies c).

The Regularized System
The aim of this section is to prepare the existence of a weak solution to the regularized system with artificial viscosity and pressure. In order to do so we have to regularize the convective terms and the variable domain. We start by introducing a suitable regularization. Here and in the following we will use, whenever necessary, zero-extensions to the whole space for quantities which we wish to regularize via convolution without further reference.

Definition of the Regularized System
We will construct a mollification of both ζ and v. At first, for any we introduce a standard regularizer. Since we cannot extend ζ to R in time, we use convolution with half intervals. Firstly, we take τ − κ ∈ C ∞ 0 ((−κ, 0], R + ) and κ . Now, we convolute ζ with the product of τ κ and a standard mollification kernel ϕ κ on ∂ (i.e. a smooth function with ϕ κ * δ 0 and ϕ κ = 1) and define R κ ζ(t, q) = (τ κ ϕ κ * ζ )(t, q). By classical theory we have the following properties: On the other hand, for functions belonging to L 2 (I ; L 2 (R 3 )) we define ψ κ to be the standard space-time mollification kernel with parameter κ. Note that functions defined on the variable domain can be extended to the whole space by zero (i.e. a smooth function with ψ k * δ 0 and ψ κ = 1). To be precise, we will use the definition for the regularization Since we may assume that ψ κ is an even function, we find that, for u, v ∈ L 1 loc (R n+1 ), With no loss of generality, we assume that 0 , q 0 are defined in the whole space R 3 . We also set u 0 = q 0 0 and assume that u 0 ∈ L 2 (R 3 ). Finally, in accordance with (1.10), we assume that This can be achieved as done in [32][p. 234, 235] (in fact, our situation is easier as we do not have to take into account the divergence-free constraint).
The aim is therefore to get a solution to the following system: we are looking for a triple (η, , u) such that The choice of the regularization of the above system will be clear by defining the weak formulation. In fact, the weak form of the above system can be written in two equations. Every other piece of information will be imposed upon by the choice of convenient function spaces. For this reason we define the following function spaces: we set and for ζ ∈ Y I with ζ ∞ < L we define X I ζ := L 2 (I ; W 1,2 ( ζ(t) )).
A weak solution to (4.1) is a triplet (η, u, ) ∈ Y I × X I R κ η × X I R κ η that satisfies the following: (K1) The regularized weak momentum equation for all ψ ∈ C ∞ I × R 3 and we have (0) = 0 . (K3) The boundary condition tr R κ η u = ∂ t ην holds in the sense of Lemma 2.4 For more details on the interplay of the convective term and the time derivative on the boundary we refer to the next subsection.

Formal a Priori Estimates for the Regularized System
To understand the particular regularization we briefly discuss how to obtain formal a priori estimates for (4.1). By taking |u| 2 2 in the continuity equation and subtracting it from the momentum equation tested by the couple (u, ∂ t η) we find The right-hand side of the inequality is as wanted, since all dependencies on η, u can be absorbed to the left hand side. Therefore, the only term that needs an extra treatment is the pressure term. We multiply the continuity equation by γ −1 to obtain Repeating the above for θ( ) = β , we can estimate the pressure term in (4.4) accordingly and deduce the following a priori estimate: with a constant c that is independent of κ, δ, ε. The rest of this section is now dedicated to the proof of the following existence theorem: Suppose that η 0 , η 1 , 0 , q 0 , f and g are regular enough to give sense to the right-hand side of (4.5), that 0 0 a.e. and (1.10) is satisfied. Then there exists a solution (η, u, ) The solution satisfies the energy estimate (4.5).

Remark 4.3.
The restriction η ∞ < L 2 is needed for the construction of our extension operator, see Lemma 2.5. The latter one is used for the renormalized continuity equation, see Theorem 3.1 b) and, in particular, Section 6.3. This is why we keep the assumption η ∞ < L 2 during the whole construction and only relax it at the very end in Section 7.4.

Definition of the Decoupled System
The strategy for proving Theorem 4.2 is to first construct a weak solution to a decoupled system and eventually apply a fixed point theorem. Let us consider a given deformation ζ ∈ C(I × M) and a given function v ∈ L 2 (I ; R 3 ). We will decouple (4.1), by replacing there R κ η with R κ ζ and R κ u by R κ v. Firstly, we find from Theorem 3.1 that there exists a unique ∈ X R κ ζ that satisfies for all ψ ∈ C ∞ I × R 3 . Observe that exists independently of u, η.
Secondly, we repeat the interplay of the boundary deformation with the convective term for the momentum equation and find that smooth functions satisfy Observe that in the case of a fixed point u ≡ v, η ≡ ζ we find that which implies that the boundary integrals will vanish. For this reason, we will solve the decoupled momentum equation with boundary values of u, which are implicitly defined by removing the first boundary term (this is analogous to the Neumann boundary data of the decoupled continuity equation, see Section 3). For the same reason we neglect the second boundary integral as well as the very last integral. Concerning the other terms of the momentum equation, when adapting partial integration we get force terms acting on the boundary in normal direction (pressure, diffusion, exterior forces). These are then assumed to be equalized by the elastic forces of the shell. Observe here that τ is identical for the decoupled system and the coupled system. All together we require from (η, u, ) ∈ X I R κ ζ × X I R κ ζ × Y I that it satisfies the following: (N1) The regularized decoupled momentum equation holds for all test-functions (b, ϕ) ∈ C ∞ 0 (M) × C ∞ (I × R 3 ) with tr R κ ζ ϕ = bν. Moreover, we have ( u)(0) = q 0 , η(0) = η 0 and ∂ t η(0) = η 1 . (N2) The decoupled regularized continuity equation (4.6) is satisfied with initial datum (0) = 0 . (N3) The boundary condition tr R κ ζ u = ∂ t ην holds in the sense of Lemma 2.4 The a priori estimates are formally available as before for the regularized system in Section 4.2. First, one uses (u, ∂ t η) as test-function in the momentum equation and subtract the continuity equation tested with |u| 2 2 . Second, one uses the renormalized formulation (3.4) to estimate the pressure term.

Theorem 4.4. For any
for all t ∈ [0, T * ], provided that η 0 , η 1 , 0 , q 0 , f and g are regular enough to give sense to the right-hand side, that 0 0 a.e and (1.10) is satisfied. Here, the constant c is independent of all involved quantities; in particular, it is independent of v and ζ .
Proof. In order to prove Theorem 4.4 we discretise the system. It is standard to find a smooth orthonormal basis (X k ) k∈N of W 1,2 0 ( ) and a smooth orthonormal basis (Ỹ k ) k∈N of W 2,2 0 (M). We define vector fieldsỸ k by solving the homogeneous Laplace equation on with boundary datumỸ k ν (which is extended by zero to ∂ ). Note that standard results on the inverse Laplace operator guarantee thatỸ k is smooth. Now we define, pointwise in t, By Lemma 2.2 we still know that X k and Y k belong to the class C 3 ( R κ ζ (t)).
Obviously, (X k ) k∈N forms a basis of W 1,2 0 ( R κ ζ (t)). Now we choose an enumeration (ω k ) k∈N of X k ⊕ Y k . In return we associate w k := ω k • R κ ζ | ∂ R κ ζ · ν. Analogous to the arguments in [32, p. 237] we find that and in the space of test-functions Now, we can begin with the construction of the solution. First, we fix = (R κ ζ, R κ v) as the unique solution to the continuity equations subject to the initial datum 0 existence of which is guaranteed by Theorem 3.1, where ζ ≡ R κ ζ and w ≡ R κ v. Next we seek for a couple of discrete solutions (η N , u N ) ∈ Z R κ ζ of the form which solves the following discrete version of (4.8): (4.9) We can choose α k N (0) in such a way that u N (0) converges to q 0 / 0 . The system (4.9) is equivalent to a system of integro-differential equations for the vector α N = (α k N ) N k=1 ; it reads as with As ( + κ) is strictly positive (recall Theorem 3.1) and the ω k and w k from a basis the matrix A is bounded (by the integrability of ) and positive definite (due to κ > 0). Hence the inverse A −1 exists and is bounded as well. We find a continuous solution α N to (4.10) by standard arguments for ordinary integrodifferential equations. Since we wish to use it as a testfunction in the momentum equation we have to show, that ∂ t α N ∈ L 2 (I ), for some s > 1. The difficulty here is that is not weakly differentiable in time. This has to be circumvented. First observe that by the Leibnitz rule, we find that Due to (4.10) and the integrability of from Theorem 3.1 we have ∂ t (Aα N ) ∈ L ∞ (I ). Moreover, A −1 is uniformly bounded (due to κ > 0). Consequently, it suffices to prove that ∂ t A i, j ∈ L 2 (I ) to conclude the differentiability of α N . By taking the test function ω i · ω j in (4.6) we find that Since the right hand side is in L 2 (I ) (note that the ω i are smooth also in time) and the w i are smooths in time we find that ∂ t α N ∈ L s (I ) and hence ∂ t u N ∈ L s (I × R κ ζ ). The a priori estimates are now achieved by differentiating (4.9) in time, testing with (∂ t η N , u N ) and subtracting (4.6) tested by 1 2 |u N | 2 . The terms with the time derivative and the convective terms cancel and we obtain Finally, we use Theorem 3.1 b) in order to rewrite the last integral. Choosing for any convex θ ∈ C 2 (R + ; R + ) such that θ (s) = 0 for large values of s and θ(0) = 0. We approximate the function s → as γ γ −1 + δs β β−1 by a sequence of such functions and obtain By Young's inequality we can absorb the terms that depend on u N or ∂ t η N in the left hand side of (4.11) such that This implies that there is a subsequence such that for some limit function (η, u). As (4.9) is linear in (η N , u N ) we can pass to the limit and see that (η, u) solves (4.8).

A Fixed Point Argument
Now we are seeking for a fixed point of the solutions map (v, ζ ) → (u, η) on L 2 (I, L 2 (R 3 )) × C(I × ∂ ) from Theorem 4.4. As we do not know about uniqueness of the solutions constructed in Theorem 4.4 we will use the following foxed point theorem for set-valued mappings:

Proof of Theorem 4.2
We will prove Theorem 4.2 by finding a fixed point of a suitable mapping defined below. We denote I * = [0, T * ] with T * sufficiently small. We do not know about the uniqueness of solutions. Hence, we apply Theorem 4.5 to get a fixed point. We consider the sets for M = ( η 0 ∞ + L)/2 and K > 0 to be chosen later. Note that the coupling at the boundary between velocity and shell is not contained in the definition of D. This is a feature which one only gains via the fixed point and not before. Let with (v, ζ ) and satisfies the energy estimate .
Note that we extend u and η by zero to R 3 and ∂ respectively. First, we have to check that F(D) ⊂ D. We will use the a priori estimate from Theorem 4.4 to conclude independently of L, K and the size of I * . This implies that η ∈ C α (I × M), by Sobolev embedding for some α > 0, with Hölder norm independent of L and K . We obtain Therefore, we find for T * small enough (but independent of v and ζ ) such that Hence we gain F(D) ⊂ D for an appropriate choice of K ∈ R + . Next, since the problem is linear and the left-hand side of the energy inequality is convex, we find that F(ζ, v) is a convex and closed subset of Z. It remains to show that F(D) is relatively compact. Consider (η n , u n ) n ⊂ F(D). Then there exists a corresponding sequence (ζ n , v n ) n ⊂ D, such that (η n , u n ) solve (4.8), with respect to (v n , ζ n ). Due to the energy estimate we may choose subsequences such that η n * η in L ∞ (I * , W 1,2 0 (M)), (4.14) ∂ t η n * ∂ t η in L ∞ (I * , L 2 (M)), (4.15) u n * ,η u in L ∞ (I * ; L 2 ( R κ ζ )), (4.16) ∇u n η ∇u in L 2 (I * ; L 2 ( R κ ζ ))). (4.17) Note also that we can extend u n and u by zero to the whole space and gain u n * u in L ∞ (I * ; L 2 (R 3 )). (4.18) The compactness of η n in C(I * × ∂ ) follows immediately by Arcela-Ascoli's theorem, since we know that η n is uniformly Hölder continuous. The proof of the compactness of u n is much more sophisticated. We first need to show compactness of n , where n is the unique solution to (4.6) with v = v n . A direct application of Theorem 3.1 a) shows (4.19) at least after taking a subsequence. Firstly, we find for all k, l ∈ N that , k, l). Hence, there is a (not relabeled) subsequence such that (4.20) Next, we claim that for any q < 10 3 . In fact, the assumptions of Lemma 2.8 are satisfied due to (4.6). In particular, (A3) holds with H 1 n = 0, H 2 n = R κ v n + ε∇ n and h n = 0. Due to the uniform bounds on n in (4.19) and the bounds on v n encoded in the definition of D we gain strong convergence of n in L 2 by Remark 2.10 at least for a subsequence. Combining this with (4.19) proves (4.21). Now, again by Lemma 2.8, we find for the couple (κ + n )u n and u n , that for some s > 1. To be precise, we infer from (4.8) that holds locally in the sense of distributions. In particular, (A3) is satisfied with choosing p = s = 2, m arbitrary and b ∈ 6 5 , 10 3 . Here R ∈ R 3×3 is chosen appropriately. We obtain (4.22). On account of (4.21) and (4.22) we conclude (extending with 0 outside of R κ ζ ) Since strong norm convergence and weak convergence imply strong convergence the compactness is shown and the existence of a fixpoint follows by Theorem 4.5. This gives the claim of Theorem 4.2.

The Viscous Approximation
In this section we want to get rid of the regularization operator R κ in order to find a solution (η, u, ) ∈ Y I × X I η × X I η to the viscous approximation satisfying the following: (E1) The regularized momentum equation holds in the sense that with tr η ϕ = bν. Moreover, we have ( u)(0) = q 0 , η(0) = η 0 and ∂ t η(0) = η 1 . (E2) The regularized continuity equation in the sense that provided that η 0 , η 1 , 0 , q 0 , f and g are regular enough to give sense to the righthand side, that 0 0 a.e. and (1.10) is satisfied. The constant c is independent of δ, ε.
Proof (Proof of Theorem 5.1). In Theorem 4.2 we take κ := 1/n where 1/n is the regularizing parameter. We call the corresponding solution (η n , u n , n ). If n → ∞ then R 1/n → id. Now we analyze the convergence of (η n , u n , n ). The estimate from Theorem 4.2 holds uniformly with respect to n. Additionally, by testing the continuity equation with ε and using β 4 we find that n ∈ L ∞ (I ; L 2 ( R 1/n η n )), ∇ n ∈ L 2 (I ; L 2 ( R 1/n η n )) uniformly.

The Vanishing Viscosity Limit
The aim of this Section is to study the limit ε → 0 in the approximate system (5.1)-(5.2) and establish the existence of a weak solution (η, , u) to the system with artificial viscosity in the following sense. We define A weak solution is a triple (η, u, ) ∈ Y I × X I η × W I η that satisfies the following: (D1) The momentum equation in the sense that The continuity equation holds in the sense that provided that η 0 , η 1 , 0 , q 0 , f and g are regular enough to give sense to the righthand side, that 0 0 a.e. and (1.10) is satisfied. The constant c is independent of δ.

Lemma 6.2. Under the assumptions of Theorem 6.1 the continuity equation holds in the renormalized sense that is
for all ψ ∈ C ∞ (I × R 3 ) and all θ ∈ C 1 (R) with θ(0) = 0 and θ (z) = 0 for z M θ .
The proof will be split in several parts. For a given ε we gain a weak solutions (η ε , u ε , ε ) to (5.1)-(5.2) by Theorem 5.1. The estimate from Theorem 5.1 holds uniformly with respect to ε. In particular, is satisfied uniformly in ε for the time interval I . Hence, we may take a subsequence such that for some α ∈ (0, 1) we have Now, using the a-priori estimates (6.4) and the bounds that one gains (using the renormalized continuity equation from Lemma 5.2 with θ(z) = z 2 and testing with ψ ≡ 1) we find, due to β > 4, that I ηε ε|∇ ε | 2 dx dt C, (6.10) with C independent of ε. This and (6.8) imply We observe that the a-priori estimates (6.4) imply uniform bounds of ε u ε in ). Therefore, we may apply Lemma 2.8 with the choice v i ≡ u ε , r i = ε , p = s = 2, b = β and m sufficiently large to obtain ε u ε η u in L q (I, L a ( η ε )), (6.13) where a ∈ (1, 2β β+1 ) and q ∈ (1, 2). We apply Lemma 2.8 once more with the choice v i ≡ u ε , r i = ε u ε , p = s = 2, b = 2β β+1 and m sufficiently large to find that ε u ε ⊗ u ε η u ⊗ u in L 1 (I × η ε ). (6.14)

Equi-Integrability of the Pressure
First, we have to handle the problem that the pressure is merrily bounded in L 1 in space. Consequently, it might converge to a measure and not a measurable function. This is usually excluded by showing that the pressure possesses higher integrability properties. From this we deduce a weakly converging subsequence (in some Lebesgue space) and hence get a function as a limit object. In the case of a moving domain standard procedures do not apply and global higher integrability on the moving domain can not be achieved. The solution is two divide the problem in two steps: the first step is to improve the space integrability of the pressure inside the moving domain; the second step is to show that the mass of the pressure can not be concentrated on the boundary. Combining the two results will imply equiintegrability of the pressure which is equivalent to weak compactness L 1 . The next two lemmata settle that matter. The first one is happily a localized version of the standard procedure. Lemma 6.3. Let Q = J × B I × η be a parabolic cube. The following holds for any ε ε 0 (Q): with constant independent of ε.
Proof. We consider a parabolic cubeQ =J ×B with Q Q I × η . Due to (6.7) we can assume thatQ I × I η ε (by taking ε small enough). Next we wish to test with The test-function (ψ∇ −1 ε , 0) is indeed admissible in (5.1) since ψ has compact support. Moreover, regularity follows from local theory for the respective parabolic equation. In order to deal with the term involving the time derivative we use the continuity equation. We find that in the sense of distributions such that div( ε u ε + ε∇ ε ). Hence, we have ∂ t ψ ε u ε · ∇ −1 B ε dx dσ =: J 1 + · · · + J 12 .

(6.18)
Our goal is to find an estimate for the expectation of J 0 which means that we have to find suitable bounds for all the other terms. Using the continuity of the operator ∇ −1 B and Sobolev's embedding theorem, we obtain for some p > 3 that using (6.9) and β > 3. Note that, in particular, we have shown that ψ∇ −1 B ε ∈ L ∞ (I ×R 3 ) uniformly in ε. As ε ∈ L 2 (I ×Q) uniformly due to β 2 we deduce that |J 1 | C as a consequence of uniform bounds on u ε in (6.8) and the continuity of the operator ∇ 2 −1 B . Similar arguments lead to the bound for J 2 , J 3 , J 4 . The most critical is the convective term J 5 . It can be estimated using the continuity of ∇ 2 −1 B , Sobolev's embedding theorem (combined with Poincaré's inequality and the fact that u ε = 0 on ), Hölder's inequality and (6.4) The term J 6 is estimated similarly. For J 7 we obtain which is uniformly bounded due to (6.8), (6.9) and (6.10). Similarly, J 8 is bounded by taking into account (6.19). The terms of J 9 , J 10 can be estimated using by the bounds on the operator ∇ −1 B and Hölder's and Young's inequalities. The same is used to estimate The latter bound is a consequence of the fact that ε ∈ L ∞ (J ; L β (B))), u ε ∈ L 2 (J ; L 6 (B))) uniformly in ε. Finally, J 12 can be estimated using (6.19) and (6.20). Plugging all of this together we obtain (6.15) uniformly in ε.
The standard method as used in the proof of Lemma 6.15 does not apply up to the boundary. Also, the usage of the Bogovskiȋ-operator-common in literature as well-does not help (recall that our boundary depends on time and is not Lipschitzcontinuous). In the following Lemma we show equi-integrability at the boundary related to the method from [29]: Lemma 6.4. Let κ > 0 be arbitrary. There is a measurable set A κ I × η such that we have for all ε ε 0 Proof. We construct a test-function which has a positive and arbitrarily large divergence. For this let ϕ ∈ C ∞ 0 (S L ; [0, 1]), such that χ S L 2 ϕ χ 0 ∪S L and |∇ϕ| c L . Since we know that |η ε | L 2 , we find that ϕ(x) ≡ 1 in S L 2 ∩ η ε . We extend ϕ by zero to R 3 and define where K > 0 will be chosen later. It is well defined, since ϕ = 0 only in S L , where the mapping x → (q(x), s(x)) is well defined, see Section 2.2. Observe that we take coordinates with respect to the reference geometry and with respect to the reference outer normal ν on ∂ . On account of ∇s(x) = ν(q(x)) we have Observe that ξ 1 and ξ 4 are uniformly bounded by some constant c ξ . Moreover, for every p ∈ (1, ∞), q > p, it holds that uniformly in ε, cf. (6.5). Estimating ξ 2 in a similar way we gain for all p < ∞ uniformly in ε. Finally, we use the fact that ∇q i are all living in the tangentplane of ∂ and are therefore orthogonal to ν(q(x)). Hence, we have ξ 3 j j = 0. This implies that for every K > 0 there is a κ such that we have Due to (6.6) we have for all r < 2, is a fashion similar to (6.22). Now, using ϕ ε as a test-function (note that ϕ ε = 0 on ∂ η ε ), we obtain by smooth approximation that for some fixed C > 0 and λ ∈ (0, 1), where C, λ are independent of ε. Here, we used the uniform integrability bounds of all other terms of the momentum equation (5.3) and (6.22). Taking (6.20) and β > 3 into account we see that the remaining integral in (6.25) is uniformly p-integrable for some exponent p > 1 in terms of (6.24) cf. (6.4). This means we have uniformly for some λ ∈ (0, 1). Now, we set with C given in (6.26). Note that such a K always exists if κ is small enough. As a consequence of (6.23) and (6.25) we gain The claim follows.
We connect Lemmas 6.3 and 6.4 to get the following corollary: at least for a subsequence. Additionally, for κ > 0 arbitrary, there is a measurable set A κ I × η such that p ∈ L 1 (A κ ) and Combining Corollary 6.5 with the convergences (6.5)-(6.14) we can pass to the limit in (5.1)-(5.2) and obtain the following. There is (η, u, , p) ∈ Y I × X I η × W I η × L 1 (I × η ) that satisfies (in the sense of Lemma 2.4) u(·, · + ην) = ∂ t ην η in I × ∂ , the continuity equation for all ψ ∈ C ∞ (I × R 3 ) and the coupled weak momentum equation with tr η ϕ = bν. It remains to show that p = a γ + δ β . This will be achieved in the following two subsections.

The Effective Viscous Flux
We fix ε 0 > 0 and consider in the following just ε ∈ (0, ε 0 ). Next, we define It is the aim of this subsection to show that for ψ ∈ C ∞ 0 (I × ε 0 ) we have as ε → 0. Testing the momentum equation with ψ∇ −1 (ψ ε ) implies ∇u ε : ∇ψ ⊗ ∇ −1 ψ ε dx dσ We rewrite as well as We define the operator R by R i j := ∂ j −1 ∂ i and obtain Similarly, we obtain by testing the limit equation (6.29) by ψ∇ −1 (ψ ) In a manner similar to (6.31) we obtain where u · ∇ψ ψ − u ⊗ ∇ψ : ∇ 2 −1 (ψ ) dx dσ, Hence we now have that We will now show that the right hand side converges to 0 as ε → 0. Observe that after this preparation everything is localized and the known approach can be enforced to our problem. Nevertheless, to keep the result self contained we repeat the main steps of the argument here. First, by the assumption β > 3 and the continuity of ∇ −1 we find that , C √ ε using, additionally, the fact that √ ε∇ ε , ∇u ε are uniformly bounded in L 2 , cf. (6.8) and (6.10). Similarly, we find |E 1 | C √ ε as well. Hence, we have E 1 , E 2 → 0.
All other couples converge to 0 (by the known weak and strong convergences we have) except for the last couple on the right hand side of (6.33). The crucial point is to estimate the commutator term. We will prove that ψ ε R[ψ ε u ε ] − ψ ε u ε R[ψ ε ] converges strongly in L 2 (W −1,2 ). Then the crucial term converges, since ψu converges weakly to 0 in L 2 (W 1,2 ). For the identification of the limit we make use of the div-curl lemma. From (6.9) and (6.13) we obtain that a.e. in I.
Hence we can apply [21,Lemma 3.4] (to the sequences ψ ε and ψ ε u j ε ) to conclude that a.e. in t, where 1 r As a consequence we have a.e. in t using the compact support of the involved functions. Moreover, it is possible to show that for some p > 2, dt C, using (6.13) and (6.15) together with β 4. This gives the desired convergence as ε → 0. Thus, we conclude that (6.34) and, accordingly, as ε → 0.

Renormalized Solutions
The aim of this subsection is to prove Lemma 6.2. Similarly to Lemma 3.1 b) the proof is based on mollification and Lions' commutator estimate. Due to (6.7), (6.9) and (6.13) it is easy to pass to the limit in (5.2). Hence, we obtain . We extend by zero to I × R 3 and u by means of the extension operator constructed in Lemma 2.5 where 1 < p < 2 (but may be chosen close to 2). Hence, we find that . Now, analogous to the proof in Theorem 3.1 we mollify the equation in space using a standard convolution with parameter κ > 0 in space. The following holds: where r κ = div( κ E η u) − div( E η u) κ . Due to β > 2 we can infer from the commutator lemma (see e.g. [36,Lemma 2.3]) that for a.e. t as well as a.e. in I . Now, we multiply (6.36) by θ ( κ ) where θ , satisfies θ(0) = 0 and obtain Due to the properties of the mollification and θ ∈ C 1 the terms θ( κ ) and θ ( κ ) converge to the correct limits (at least after taking a subsequence). Hence, multiplying (6.38) by ψ ∈ C ∞ (I × R 3 ) and integrating over I × R 3 implies (6.39)

Strong Convergence of the Density
In order to deal with the local nature of (6.35) we use ideas from [18]. First of all, by the monotonicity of the mapping z → az γ + δz β , we find for arbitrary non-negative ψ ∈ C ∞ 0 ( ε 0 ) that using (6.35). As ψ is arbitrary we conclude div u div u a.e. in I × η , (6.40) where div u ε ε η div u in L 1 ( ; L 1 ( η ε )); recall (6.8) and (6.9). Now, we compute both sides of (6.40) by means of the corresponding continuity equations. Due to Lemma 5.2 with θ(z) = z ln z and for any t ∈ I . This gives the claimed convergence ε → in L 1 (I × R 3 ) by convexity of z → z ln z. Consequently, we havep = a γ + δ β and the proof of Theorem 6.1 is complete.
provided that η 0 , η 1 , 0 , q 0 , f and g are regular enough to give sense to the righthand side, that 0 0 a.e. and (1.10) is satisfied.
where r := 3γ 2γ −3 . We proceed, using Sobolev's inequality (note that u δ = 0 on ), by We need to choose r such that r γ which is equivalent to 2 3 γ − 1. Now, the various a-priori bounds yield |J | c uniformly in δ.
In a fashion similar to Lemma 6.4 we can exclude concentrations of the pressure at the moving boundary. However, we have to assume γ > 12 7 for this. Lemma 7.4. Let γ > 12 7 (γ > 1 in two dimensions). Let κ > 0 be arbitrary. There is a measurable set A κ I × η such that we have for, all δ δ 0 , Proof. We follow the approach of Proposition 6.4 replacing ε by δ, so we test with The critical term is again Following the proof of Proposition 6.4 this can be estimated provided γ > 3. We want to improve on this. In order to do so we write By Lemma 2.4 and since ∇u δ is uniformly bounded in L 2 (recall (7.7)) we find that u δ • δ | ∂ ∈ L 2 (I ; L q (∂ )) ∀q < 4 (7.14) uniformly in δ (q < ∞ in two dimensions). In a manner similar to (6.24) we obtain for all r < q < 4 (all r < q < ∞ in two dimensions) uniformly in δ. Now, the proof can be finished as in Proposition 6.4. We take into account (which follows from the uniform a-priori bounds) and γ > 12 7 (which yields 6γ γ +6 > 4 3 ). We see that the integral in (7.13) is uniformly bounded by K 1−λ for some λ ∈ (0, 1) using Hölder's inequality and (7.15) (choosing r and q appropriately). Lemmas 7.3 and 7.4 imply equi-integrability of the sequence γ δ χ η δ . This yields the existence of a function p such that (for a subsequence) T k ( δ ) δ − T k ( δ ) div u δ (here it might be necessary to pass to a subsequence).
To be more precise, the following holds: Thus, letting δ → 0 in (7.24) yields ∂ t T 1,k + div T 1,k u + T 2,k = 0 (7.27) in the sense of distributions on I × R 3 . Note that we used This, in turn, is a consequence of the convergences ). (7.28) We remark that the former one follows from the compactness of the embedding C w (I ; L p (O)) → L 2 (I ; W −1,2 (O)) for O R 3 and (7.25) (with p > 6 5 ). Note that u δ is extended to R 3 by means of Lemma 2.5.
Next, we take Q with Q Q I × η n δ and a cut off function ψ ∈ C ∞ 0 (Q) with 0 ψ 1 and ψ ≡ 1 in Q = J × B. Now, we test (6.1) with ψ∇ −1 (ψ T k ( δ )) and (7.20) with ψ∇ −1 (ψ T 1,k ). Using similar arguments as in Section 6.2 we find that We have to remove ψ in order to conclude. For some given κ > 0 we choose a measurable set in accordance to Lemma 7.4 and Corollary 7.5 for δ 0 small enough (using the fact that η δ → η uniformly, cf. (7.6)). Without loss of generality we can assume that ∂ A κ is regular. Hence we can cover A κ with parabolic cubes They can be chosen in a way that we find a partition of unity (ψ i ) with respect to the family Q i such that ψ i ∈ C ∞ 0 (Q i ) and In particular, (7.29) holds with ψ = ψ i . We gain Using (7.7) and (7.12) the first integral on the right-hand side is bounded in terms of κ. Using (7.29) and (7.19) we find that is bounded bin terms of κ. As κ is arbitrary we finally conclude that (7.23) holds.

Renormalized Solutions
The aim of this section is to prove Lemma 7.2. In order to do so it suffices to use the continuity equation and (7.23) again on the whole space.
First, we observe that δ is renormalized solution to the continuity equation by Lemma 6.2, i.e. we have ∂ t θ( δ ) + div θ( δ )u δ + θ ( δ ) δ − θ( δ ) div u δ = 0 (7.30) in the sense of distributions on I × R 3 . Note that (7.30) holds in particular for θ(z) = z, which implies that the continuity equation can be regarded as a PDE on the whole-space. We are interested in the particular choice θ = T k , where the cut-off functions T k are given by (7.22). We have to show that, similar to (7.30), equation (7.27) actually holds globally. Thus, choosing θ = T k in (7.30) letting δ → 0 yields for all ψ ∈ C ∞ (I × R 3 ). This means that we have ∂ t T 1,k + div T 1,k u + T 2,k = 0 (7.32) in the sense of distributions on I × R 3 . Note that we extended by zero to R 3 . The next step is to show lim sup δ→0 I ×R 3 |T k ( δ ) − T k ( )| γ +1 dx dt C, (7.33) where C does not depend on k. The proof of (7.33) follows exactly the arguments from the classical setting with fixed boundary (see [21,Lemma 4.4] and [17]) using (7.23) and the uniform bounds on u. We explain the details for the convenience of the reader. First, note that we have By convexity of z → z γ + δz β we conclude that |T k ( δ ) − T k ( )| γ +1 dx dt. Now we combine (7.34) end (7.35) with (7.23) to conclude (7.33). By a standard smoothing procedure we can consider "renormalized solutions" for T 1,k and deduce from (7.32) that ∂ t θ(T 1,k ) + div θ(T 1,k )u + θ (T 1,k )T 1,k − θ(T 1,k ) div u + θ (T 1,k )T 2,k = 0 (7.36) in the sense of distributions I × R 3 . Here, we use that θ (z) = 0 for z M θ . We want to pass to the limit k → ∞. On account of (7.8), we have, for all p ∈ (1, γ ), so we have as k → ∞. Therefore, we are left to show that θ (T 1,k )T 2,k → 0 in L 1 (I × R 3 ) with k → ∞. (7.38) Recall that θ has to satisfy θ (z) = 0 for all z M for some M = M θ . We define Q k,M := (t, x) ∈ I × R 3 ; T 1,k M and gain by weak lower semicontinuety that

Strong Convergence of the Density
We introduce the functions L k by L k (z) = z ln z, 0 z < k z ln k + z z k T k (s)/s 2 ds, z k.
We can choose θ = L k in (7.42) such that ∂ t L k ( ) + div L k ( )u + T k ( ) div u = 0 (7.43) in the sense of distributions on I × R 3 . We also have that ∂ t L k ( δ ) + div L k ( δ )u δ + T k ( δ ) div u δ = 0 in the sense of distributions, cf. Lemma 6.2. Using the testfunction ψ ≡ 1 in both equations implies T k ( δ ) div u δ dx dσ 0 (7.44) and The difference of both equations reads as We have the following convergences for all p ∈ (1, γ ): L k ( δ ) → L 1,k in C w (I ; L p (R 3 )), δ → 0, δ ln( δ ) → L 2,k in C w (I ; L p (R 3 )), δ → 0, which is a consequence of the fundamental theorem on Young measures (see, for instance, [40,Thm. 4.2.1,Cor. 4.2.19]) and the convergence of δ in C w (I ; L β (R 3 )). The latter one follows from the a-priori information on from (7.8) in combination with the control of the distributional time derivative of δ coming from the continuity equation (considered on the whole-space), so we gain (using also the fact, that δ (0) = 0 = (0)) ln dx dt.
Convexity of z → z ln z yields strong convergence of δ . Hence, due to (7.20), the proof of Theorem 7.1 is shown, for the time interval [0, T * ], with T * depending on the data only (such that η(t) ∞ < L 2 in (0, T * )). In the next section we will show how the interval of existence can be prolongated by a change of coordinates.

Maximal Interval of Existence
The interval of existence in Theorem 7.1 is restricted by the quantities of the given data, as well as the geometry of ∂ . By our assumption on the initial geometry we find that η(T * ) has no self intersections. We define η * = (η(T * )) κ , where κ is a convolution operator in space. We define˜ = η * ∈ C 4 . If κ is conveniently small, then also˜ has no self intersection either. In particular, there exists somẽ L > 0 such that onSL Second, we wish to get a fixpoint by applying Theorem 4.2. The only modification is that the fixpoint mapping has to be adjusted slightly. Indeed, the fixed point has to be found in the set D := (ζ , v) ∈ C([T * , T * * ] × ∂˜ ) × L 2 ([T * , T * * ] × R 3 )) : Here K has to be adjusted to T * * in accordance with the proof of Theorem 4.2.
Finally, we set F : D → P(D), with (ζ, v) and satisfies the energy bounds , whereη is defined via the solution η byη = η(q −ν(q)η * (q))−η * (q) as introduced above. The rest of the argument of Theorem 4.2 does not change, since the L ∞ bounds of η, ζ (which are critical for the fixed point argument) do not change by coordinate transformations. Once the fixed point is established, we may pass to the limit with κ, ε and δ as before. Observe, that in Section 6.3 one has to use the extension operator from Lemma 2.5 with respect to the coordinate transformation η as it satisfies η ∞ <L 2 (here we use the fact that η = η by our construction). We remark that the solution η and R κ η are defined via the same reference coordinates ∂ . This means it truly extends the solution and we can extend the interval of existence. Finally, the above procedure can be iterated until a self intersection is approached. This finishes the proof of Theorem 7.1.