Martingale solutions in stochastic fluid-structure interaction

We consider a viscous incompressible fluid interacting with a linearly elastic shell of Koiter type which is located at some part of the boundary. Recently models with stochastic perturbation in the shell equation have been proposed in the literature but only analysed in simplified cases. We investigate the full model with transport noise, where (a part of) the boundary of the fluid domain is randomly moving in time. We prove the existence of a weak martingale solution to the underlying system.


Introduction
The mathematical analysis of systems of partial differential equations arising from fluid-structure interaction has seen a vast progress in the last two decades.This is motivated by a variety of applications, for instance in biomechanics [1], hydro-dynamics [6], aero-elasticity [11] and hemo-mechanics [15].

Deterministic models
We are interested in the case where a viscous incompressible fluid interacts with an elastic structure located at a part of the boundary of the fluid's domain O ⊂ R 3 denoted by Γ.The structure reacts to the forces imposed by the fluid at the boundary.Assuming that this deformation only acts in the normal direction and denoting by n the outer unit normal at the reference domain, O is deformed to the domain O η(t) defined through its boundary ∂O η(t) := {y + η(t, y)n(y) : y ∈ Γ}. (1.1) Here η : (t, y) : I × Γ → η(t, y) ∈ R describes the deformation of the structure and I := (0, T ), for some T > 0 denotes a time interval.For technical simplification we will suppose that Γ is the whole boundary and identify it with the two-dimensional torus (the precise geometric set-up is presented in Section 2.2).As a prototype let us consider the following problem where the equation for the shell can be seen as a linearised version of Koiter's model (neglecting lower order terms for simplicity and setting all positive physical constants to 1).In the unknowns u : (t, x) : I × O η → u(t, x) ∈ R 3 , π : (t, x) : I × O η → π(t, x) ∈ R, accounting for the fluid's velocity field and pressure, respectively (defined on a moving space-time cylinder ), it reads as (for simplicity we neglect volume forces in the fluid equations) The system is complemented by the kinetic boundary condition at the fluid-structure interface as well as initial conditions for (1.3)-(1.4)and periodic boundary conditions for (1.4).Here T(u, π) = (∇ x u + ∇ x u ⊤ ) − πI 3×3 is the stress tensor of the fluid.The vectors n and n η denote the normal vectors on O and O η , respectively.The function ϕ η gives the coordinate transform from Γ → ∂O η .The existence of a weak solution to (1.2)-(1.5)has been shown in [26] (see also [27] for the case of a cylindrical shell model).It satisfies the energy balance for a.a.t ∈ I, from which one can easily deduce the function spaces in which the weak solution lives.The easier case of an elastic plate (where the reference geometry is flat) has been studied before in [17].The main advancement in [26] is a new compactness method which eventually allows to establish compactness of the velocity field.On account of the deformed space-time cylinder on which the problem is posed it is impossible to apply the standard Aubin-Lions compactness lemma in order to pass to the limit in the convective term of approximate solutions.Interestingly, this issue is ultimately linked to the divergence-free constraint (1.2).Without it, the compactness can simply be localised thus completely removing the difficulties posed by the moving boundary, see [5] where the compressible Navier-Stokes equations are studied.In [28] (where even the fully nonlinear Koiter model is considered) the compactness argument from [26] has been replaced by an abstract compactness criterion which is more in the spirit of the classical Aubin-Lions result and thus allows for wider applications.Let us finally remark that all the results just mentioned hold under the assumption that there is no self-intersection of the structure (which can always be avoided if η L ∞ y is not too large).

Stochastic models
It was recently suggested in [24] to consider a stochastic perturbation in (1.4) to account for random effects in real-life problems and uncertainty in the data.A first step towards a well-posedness theory for such stochastic fluid-structure interaction models is done in [25], where the 2D time-dependent Stokes equations are linearly coupled to a structure described by a stochastic 1D wave equation.Although this is only a simplified model (and the boundary is not moving in time) the analysis is already quite advanced.As already indicated above, the geometry breaks down if η causes a selfintersection of the domain.In the simplified case, where the reference domain is a box and the deformation only occurs in the vertical direction, this happens exactly when the value of −η coincides with the height of the box.If η has a Gaussian distribution as in [25], this can always happen (though maybe only with a low probability) no matter how short the time horizon is.This issue may be circumvented by studying the local-in-time well-posedness of the problem which is done in [32].The authors of [32] study the interaction of an elastic plate (the reference geometry is flat) with the 2D Navier-Stokes equation.The existence time is a random variable about which the only available information is P-a.s.positivity.
In this paper we aim for the natural next step by considering the full model (1.2)-(1.5)globally in time, where (1.4) is subject to some Gaussian noise. 1 We take a different perspective to [24,25,32] and do not consider stochasticity entering as an external force but as an intrinsic property of the system.Thus we consider transport noise in the shell equation (see (1.9) below).It has the very appealing feature of being energy conservative.If the initial data is deterministic (or simply bounded in probability) we have a pathwise control over the energy and the restrictions on the time interval are the same as in previous deterministic papers such as [17,26,28].Transport noise has a clear physical meaning in fluid mechanical transport processes, see [8,21,22] as well as [7,9,14].Depending on the particular structure, it can be conservative with respect to several important quantities such as energy, enstrophy and circulation.Note that this is excluded in the case of an Itô noise.Also, it has been observed that transport noise has regularising effects on certain ill-posed PDEs, see [12,13].Nevertheless, the role of transport noise for elastic materials must be further explored.Understanding the role of noise in the shell equation (1.9) is motivated by [24,25,32] and its understanding is only at the beginning stages.
Our goal is to construct on random space-time cylinders Ω×I ×O η and Ω×I ×Γ, a global weak solution triple (note that the pressure does not enter the weak formulation) representing the fluid's velocity, the fluid's pressure and the structure displacement of the coupled fluid-structure system given by div x u = 0, (1.7) with Here, Ω is a sample space of a filtered probability space (Ω, F, (F t ) t≥0 , P) with associated expectation E(•).Equation (1.9) contains a Stratonovich differential of a real-valued Brownian motion (B t ) and κ κ κ is a given solenoidal (incompressible) vector field (i.e.div y κ κ κ = 0)2 in R 2 .The initial conditions for (1.7)-(1.9)are With regards to boundary conditions, we supplement the shell equation (1.9) with periodic boundary conditions and impose at the fluid-structure interface.Note that (1.7)-(1.12) is a free-boundary problem where the boundary of the fluid domain is moving randomly in time.
In the 3D case regularity and uniqueness of solutions to (1.7)-(1.12) is certainly out of reach (at least globally in time) so that one can only hope for the existence of weak martingale solutions.Here weak refers to the analytical concept of distributional derivatives, whereas martingale solutions refers to solutions which are weak in the probabilistic sense (they do not exist on a given stochastic basis; the latter becomes an integral part of the solution).Such a concept is very common in stochastic evolutionary problems (even on the level of ordinary stochastic differential equations), whenever uniqueness of the underlying system is unavailable.

The weak formulation
A first rather philosophical question is to come up with an analytically weak formulation for the problem.In fluid-structure interaction problems the space of test-functions typically depends on the structure displacement η (the test-function for the fluid subproblem and the structure sub-problem must match at the interface as in (1.5) and (1.12)).On the other hand, in stochastic PDEs it is common to work with spatial test-functions.This is also our preferred point of view as an η-dependence of the testfunctions in our case means that they depend on time and are also random.The idea now is to start with a pair of test-functions (φ, φ) on the reference domain (that is φ : Γ → R and φ : O → R 3 ) with the correct boundary condition and transform φ to the moving domain.An obvious choice, therefore, is the Hanzawa transform Ψ η : O → O η which we formally introduce in Section 2.2.Unfortunately, it has the disadvantage of destroying the divergence-free constraint on the test-functions.At the level of weak solutions this cannot be remedied through the recovery of the pressure function as the latter only exists as a distribution on the solenoidal test-functions.Thus, we use instead the Piola transform which preserves the solenoidability of a functions.Using now (ι η φ, J η(t) φ) with ι η = (det∇ x Ψ η ) −1 we obtain the following weak formulation for all test-functions (φ, φ) (which clearly depend only on space), see Section 2.4 for the derivation.One easily shows that a dense set of pairs of test-functions (φ, φ) with the correct boundary condition leads to a dense set of pairs of test-functions (ι η φ, J η(t) φ) on the moving domain with the right boundary condition, see [26, page 237].Thus our weak formulation is consistent with the strong formulation.However, the Piola transform behaves like ∇ x Ψ η and inherits the regularity of ∇ y η so that we require more regularity on η.Since the embedding W 2,2 (Γ) ֒→ W 1,∞ (Γ) fails in two dimensions, ∇ x Ψ η is not bounded.Thus the information from (1.6) is not sufficient to give a meaning to all the terms in (1.14).Hence we need additional regularity.A crucial point in our approach is, therefore, to establish an estimate for the L 2 (I; W 2+s,2 (Γ))norm of η for all s ∈ (0, 1/2).One can easily check by using Hölder's inequality and Sobolev's embedding that this information, together with (1.6), is sufficient to define all integrals appearing in (1.14).Different to (1.6) this estimate is not independent of the transport noise and hence only holds in expectation.As it turns out, the regularity of the terms arising from the transport noise have just enough regularity to close the estimate.Details can be found in Section 5.1.Concluding this discussion, the additional fractional differentiability of the shell displacement must be included in the definition of a solution, see Definition 2.

Plan
This work straddles different fields of mathematics including fluid mechanics, partial differential equations, differential geometry and stochastic analysis.In order to make this work as self-contained as possible, we collect in Section 2 useful results in the different fields of mathematics that are essential in establishing our result.We begin by giving a rigorous interpretation of the Stratonovich integral in (1.9) after which we introduce the geometric setup for the fluid-structure system (1.7)-(1.9).We also present the functional analytic framework (function spaces on moving domains) and present some key results necessary for our analysis (extension operators).Finally, we make precise, the notion of a solution that we are interested in (Definition 2) and state our main result (Theorem 5).The proof of our main result can be summarized into three main steps.In Section 3, we consider an extension of the fluid-structure system that incorporate 'artificial' regularising terms in the shell equation (1.9).We then construct a solution to the linearised version of this extended system using a Galerkin approximation and stochastic compactness tools.We then move to Section 4 where we use a fixed-point argument to remove the linearisation performed in the previous section and obtain a solution to the fully nonlinear system (with the extra regularising terms in the shell equation).To complete the proof of the main result, we pass to the limit in the regularisation parameter in Section 5 to finally obtain a solution for (1.7)-(1.12).In each stage we apply a refined stochastic compactness method which is based on Jakubowski's extension of the Skorokhod representation theorem [23].In our case it is crucial to re-interpret the compactness lemma from [28] in the context of tightness of probability measures, see Sections 3.2 and 4.1.Remark 1 (The 2D case).One might wonder whether it is possible to show the global-in-time existence of strong pathwise solutions to (1.7)-(1.12)(solutions existing on a given stochastic basis).In the 2D case the existence of global-in-time strong solutions to (1.2)-(1.5)has been shown in [2] with additional dissipation (see also for [18] for previous results for elastic plates) and in [31] for the case of plates (but without dissipation).In all cases the approach is heavily based on taking temporal derivatives of u and ∂ t η, which is not possible for (1.7)-(1.12).Hence its is unclear if one should even expect such a result here.
2 Mathematical framework and the main result

Stratonovich integrals
Let (Ω, F, (F t ) t≥0 , P) be a stochastic basis with a complete, right-continuous filtration and let (B t ) represent a real-valued Brownian motion relative to (F t ).We consider a smooth solenoidal vector field κ κ κ for φ ∈ W 1,2 (Γ).Here •, • t denotes the cross variation.We compute now the cross variations by means of (1.9).If ξ = ∂ t η, where η solves (1.9), we have for all φ ∈ W 2,2 (Γ) and t ∈ I, Only the last term on the right contributes to the cross variation when inserted in (2.1).Plugging the previous considerations together we obtain to be understood in W −1,2 (Γ) or W −2,2 (Γ), respectively, depending on the regularity of ∂ t η.

Geometric setup
The spatial domain O is assumed to be an open bounded subset of R 3 with smooth boundary ∂O and an outer unit normal n.We assume that ∂O can be parametrised by an injective mapping ϕ ∈ C k (Γ; R 3 ) for some sufficiently large k ∈ N, where Γ is the two-dimensional torus.We suppose for all points y = (y 1 , y 2 ) ∈ Γ that the pair of vectors ∂ i ϕ(y), i = 1, 2, is linearly independent.For a point x in the neighbourhood of ∂O, we define the functions y and s by where we used the projection p(x) = ϕ(y(x)).We define L > 0 to be the largest number such that s, y and p are well-defined on S L , where Due to the smoothness of ∂O for L small enough we have |s(x)| = min y∈Γ |x − ϕ(y)| for all x ∈ S L .This implies that S L = {sn(y) + y : (s, y) ∈ (−L, L) × ω}.For a given function η : I × Γ → R we parametrise the deformed boundary by With an abuse of notation we define the deformed space-time cylinder as The corresponding function spaces for variable domains are defined as follows.Definition 1. (Function spaces) For I = (0, T ), T > 0, and η ∈ C(I × ω) with η L ∞ (I×Γ) < L we define for 1 ≤ p, r ≤ ∞ To establish a relationship between the fixed domain and the time-dependent domain, we introduce the Hanzawa transform Ψ η : O → O η defined by for any η : ω → (−L, L).Here φ ∈ C ∞ (R) is such that φ ≡ 0 in a neighborhood of −L and φ ≡ 1 in a neighborhood of 0. The other variables p, s and n are as defined earlier in this section.A straightforward verification shows that the inverse of In order to obtain a weak formulation for the fluid-structure system, we also introduce the Piola transform of a vector field v : O → R 3 with respect to a mapping ζ : Γ → R. The Piola transform is invertible with inverse It preserves vanishing boundary values as well as the divergence-free property of a function.In order to compensate for the additional factor (det∇ x Ψ ζ ) −1 in the trace of J ζ v we define the mapping Thus a pair of test-functions (φ, φ) with the correct boundary condition leads to a pair of test-functions (ι η φ, J η(t) φ) on the moving domain with the right boundary condition.Also, a dense set of test-functions on the reference domain leads to a dense set of test-functions on the moving domain, see [26, page 237].
We finish this section by recalling the following Aubin-Lions type lemma which is shown in [28, Theorem 5.1.& Remark 5.2.] and slightly reformulated for our purposes.Theorem 1.Let X, Z be two Banach spaces, such that X ′ ⊂ Z ′ .Assume that f n : I → X and g n : I → X ′ , such that g n ∈ L ∞ (I; Z ′ ) uniformly.Moreover assume the following: (a) The boundedness: The approximability-condition is satisfied: For every κ ∈ (0, 1] there exists a f n,κ ∈ L s (I; X) ∩ L 1 (I; Z), such that for every ǫ ∈ (0, 1) there exists some κ ǫ ∈ (0, 1) (depending only on ǫ) such that and for every κ ∈ (0, 1] there is some C(κ) such that (c) The equi-continuity of g n : We require that there exists some α ∈ (0, 1], functions A n with A n ∈ L 1 (I) uniformly, such that for every κ > 0 that there exist some C(κ) > 0 and some (d) The compactness assumption is satisfied: X ′ ֒→֒→ Z ′ .More precisely, every uniformly bounded sequence in X ′ has a strongly converging subsequence in Z ′ .
Then there is a subsequence, such that

Solenoidal extension
In this section, we present a linear solenoidal extension operator that maps boundary elements of a spatial domain into the interior.For this end, we first consider the corrector map where λ η ≥ 0 for (t, x) ∈ I × A κ is an appropriately chosen weight function, cf.[28, equ.(3. 3)], and for all q ∈ [1, ∞].The corrector K η above preconditions the boundary data to be compatible with the interior solenoidality.The following is proved in [28,Prop. 3.3] and it provides a solenoidal extension.For that we introduce the solenoidal space In particular, we have the estimates ) The following result is a consequence of Proposition 2. Corollary 3. Let the assumptions of Proposition 2 be satisfied and in addition, let a, r ∈ [2, ∞], p, q ∈ (1, ∞) and s ∈ [0, 1], and assume that η ∈ L r (I; W 2,a (Γ)) ∩ W 1,r (I; L a (Γ)).Let ξ ∈ W s,p (Γ) and let ξ δ be a smooth approximation of ξ in Γ.Then ) satisfies all the conclusions in Proposition 2. In particular, and holds uniformly in t ∈ I.
For the final statement of this subsection, borrowed from [28, Lemma 3.5], we first introduce the following fractional difference quotient in space in the direction e i given by ∆ s h f (y) = h −s (f (y + e i h) − f (y)) for some h > 0. Now, we define where s ∈ (0, 1 2 ) and the result is as follows: Lemma 4. Let the assumptions of Proposition 2 be satisfied and in addition, let and when Here, the constants only depends on α, L and η C 0,θ (Γ) .

Weak martingale solutions
We are interested in a solution to (1.7)-(1.9)that is weak in the probabilistic sense and also weak in the deterministic sense.From the probabilistic point of view, this means that the stochastic basis is also an unknown of the system and from the deterministic angle, we want a distributional solution of the system integrated against a We are now deriving the weak formulation of the coupled system assuming we have a sufficiently regular solution at hand.Since the momentum equation (1.8) is merely a random PDE rather than a SPDE, and advected by the large-scale incompressible vector field, we can directly apply Reynolds transport theorem [20] to obtain for (ι η φ, J η(t) φ) (recalling the definitions from Section 2.2) We can now use the momentum equation (1.8) and the divergence-free condition on φ (which transfers to J η(t) φ) to obtain with the latter satisfying To obtain a distributional formulation for the shell equation (1.9), we first transform it into the Itô equation cf. the discussion in Section 2.1.If we now use Itô's formula, we obtain where due to the periodicity of the boundary of Γ, ˆΓ ι η φ∆ 2 y η dy dt = ˆΓ ∆ y ι η φ∆ y η dy dt.
If we now use the identity (v (2.9) Note that div x (J η(t) φ) = 0 and thus, no pressure term appears in the weak formulation.The term containing ∂ t J η is still not well-defined and needs to be rewritten.First of all, we have By using Gauß theorem, we obtain and similarly where Ψ i −η is the i-th component of Ψ −η and n j η that of n η .The last term of ∂ t (J η φ) does not require such an integration by parts.Combining Hölder's inequality with Sobolev's embedding and using that Ψ η has the same regularity as η, one easily checks that for a weak solution with regularity as below, all terms are well-defined.

s. and and for all
(e) equation (2.9) holds a.s.for all (φ, φ) The energy inequality holds in the sense that The following is our main result.

The linearised problem
In the first instant, we wish to construct a weak solution to a system with a regularized geometry and a regularized convection term.Here, by a regularized geometry, we mean a regularization of a solution to a given shell equation and not the solution to our anticipated shell equation (1.9).Thus, we aim at solving the system div x u = 0, (3.1) in I × O ηǫ where L ′ is the operator given ´Γ L ′ (η)φ dy = ´Γ ∇3 y η : ∇ 3 y φ dy for all φ ∈ W 3,2 (Γ), and ǫ > 0 is a fixed regularisation parameter.With some slight abuse of notation we denote by f ǫ the regularisation of a function on the fluid domain (which is previously extended by zero to the whole space) as well as the regularization of a function defined on I × Γ.The regularisation is taken with respect to space and time, where the temporal regularization is taken backwards in extending functions to (−∞, T ) by their values at time 0. A martingale solution to (3.1)-(3.3)can be defined analogously to Definition 2. We aim to show the following result (the proof of Theorem 6 can be found in the next section).Theorem 6.Let (η 0 , η 1 , u 0 , κ κ κ) be a dataset such that (2.10) holds and we have addi- tionally η 0 ∈ W 3,2 (Γ).Then there is a weak martingale solution of (3.1)-(3.3)with data (η 0 , η 1 , u 0 , κ κ κ).The interval of existence is of the form I = (0, t), where t < T only if lim s→t η(s) L ∞ (Γ) = L a.s. in Ω 0 for some Ω 0 ⊂ Ω with P(Ω 0 ) > 0.
In order to solve (3.1)-(3.3)we linearize the problem by replacing the regularized velocity in the convective term with a regularization of a given velocity field v ∈ R 3 .We also replace the regularized geometry with a regularized geometry with respect to a given structure displacement ζ with an initial state ζ(0, •) = η 0 .The corresponding regularization of the pair (ζ, v) is denoted by (ζ ǫ , v ǫ ).The solution we seek will be constructed as the limit N → ∞ of the solution (η N , u N ) to a finite dimensional Galerkin approximation system incorporating these regularizing terms.Since this is a linear system we aim to construct a probabilistically strong solution defined on a stochastic basis (Ω, F, (F t ) t≥0 , P) and driven by a given Brownian motion (B t ) relative to (F t ).Suppose that (ζ, v) (and thus its regularization (ζ ǫ , v ǫ )) are a given pair of (F t )progressively measurable 3 random variables with values in C(I ×Γ)×L 2 (I; L 2 (O∪S α )) belonging to L p (Ω) for some sufficiently large p.where we suppose that ǫ is small enough such that ζ ǫ L ∞ (I×Γ) < α < L a.s.We now look for an (F t )-progressively measurable process (η, u) with values in the space Theorem 7. Let (η 0 , η 1 , u 0 , κ κ κ) be a dataset such that (2.10) holds and we have addi- tionally η 0 ∈ W 3,2 (Γ).Let (Ω, F, (F t ) t≥0 , P) be a stochastic basis with a complete, right-continuous filtration and let (B t ) be an (F t )-Brownian motion.Then there is a unique probabilistically strong solution of (3.5) with data (η 0 , η 1 , u 0 , κ κ κ).The interval of existence is of the form I = (0, t), where t < T only if lim s→t η(s) L ∞ (Γ) = L a.s. in Ω 0 for some Ω 0 ⊂ Ω with P(Ω 0 ) > 0.
It will turn out that the solution satisfies the energy equality The aim of the following subsection is to construct a Galerkin approximation of (3.1)-(3.3)on a given stochastic basis (Ω, F, (F) t≥0 , P), while its limit passage (and thus the proof of Theorem 6) can be found in the next section.

The linearised Galerkin problem
Let us now explain in which function spaces we seek the finite dimensional objects (η N , n N ).Let (Y i ) i∈N be a basis of W 2,2 (Γ) and let (X i ) i∈N be a basis of W 1,2 0,divx (O).Clearly, there exists divergence-free vectors fields Y i that are solving Stokes systems in the reference domain O with boundary data (Y i n) • ϕ −1 .We then set and set being (F t )-progressively measurable.We search for ( )ι ζǫ w j dy dB t (3.8)   for 1 ≤ j ≤ N with an initial condition α N i (0) which is such that Note that the derivation of the weak formulation (3.8) is slightly different to the derivation of (2.9) due to the differences in their respective advective terms.The treatment of the former advection term goes as follows.By using the trivial identity we rewrite the last term as follows where we have used u N • ϕ ζǫ = n∂ t η N and J ζǫ(t) w j • ϕ ζǫ = ι ζǫ w j n in the last step.This explains the presence of the Jacobian determinant in (3.8).
Moving on, we note that equation (3.8) is equivalent to To simplify notations, we drop the summation signs by employing Einstein's summation convention.Then for we can rewrite the above as the following system of SDEs a.s.for a.a.t ∈ I. From the above, we then obtain sup sup (3.17) In addition, for any s ∈ (0, 1 2 ), it follows from u N • ϕ ζǫ = n∂ t η N , (3.17) and the trace theorem that holds.

Tightness of ∂ t η N
The effort of this subsection is to prove tightness of the law of ∂ t η N on L 2 in order to pass to the limit in the stochastic integral.We define the projection P N and the extension F ζǫ N (for a given ζ : ω → (−L, L)) . We have by definition, The eigenvalue equation for the basis vectors implies additionally that By interpolation we obtain an estimate on W s,2 (Γ) for any s ∈ [0, 3], that is Finally, for ζ ǫ ∈ W 3,2 (Γ) with ζ ǫ L ∞ (Γ) < α for α ∈ (0, L) we have for all b ∈ W 3,2 (Γ).Following [26], we write We consider the space X := L 2 (Γ)×W −s,2 (O∪S α ) with s ∈ (0, 1/4).In order to apply Theorem 1 yielding tightness of the corresponding laws we need to equip L 2 (I; X ′ ×X) with an unconventional topology which we denote by τ ♯ and define the convergence → ♯ as follows.We say that and it holds Since this topology is finer than the weak topology on L 2 (I; X ′ × X) it is clear that (L 2 (I; X ′ × X), τ ♯ ) is a quasi-Polish space such that Jakubowski's version of the Skorokhod representation theorem applies.We obtain the following result concerning tightness.Lemma 8.The laws of Proof.According to Theorem 1 we first need boundedness in L 2 (I; X ′ ×X).By (3.13)-(3.18)and the properties of the projection and extension operators above this follows immediately (even uniformly in probability).Note that the extension by zero is a bounded operator on W s,2 for s < 1 4 .For (b) we observe that we may assume that a regularizer b → (b) κ exists such that for any s The estimate is well known for a, s ∈ N 0 , while the general case follows by interpolation and duality.Next we introduce the mollification operator on ∂ t η N by considering for κ > 0 and We find by the continuity of the mollification operator from (3.26), the continuity of the projection operator from (3.21) and the estimate for the extension operator (3.22) that for a.e.t ∈ I and s < s 0 < 1/2 which can be made arbitrarily small in L 2 by choosing κ appropriately, cf.(3.18).Similarly, we have Clearly, if f N ⇀ f in L 2 (I; X) then we can deduce a converging subsequence such that f N,κ ⇀ f κ (for some f κ ) in L 2 (I; X) for any κ > 0, which implies (b).
For (c) we have to control g N (t) − g N (s), f N,κ (t) , where g N (t) := (∂ t η N (t), I O ζǫ(t) u N (t)) and hence we decompose We begin estimating (II) to find that using (3.23).By (3.26) and (3.21) the last term can be estimated by which is bounded by (3.14).Due to this as well as (3.14) and (3.16) we further have that In teh last step we used standard properties of the mollification recalling that ǫ is a fixed parameter on this level.It follows by Markov's inequality for K > 0 which can be amde arbitrarily small for large K provided ζ ∈ L 3 (Ω; C(I × Γ)).The term (I) is estimated using the test function f N,κ obtaining ) except for the last one.Here we have by Markov's inequality and Itô-isometry (using also div y κ κ κ = 0) using (3.14).This can be made arbitrarily small for K large.Now we set Z := W s0,2 (Γ)×W s0,2 (O∪S α ), where s 0 ∈ (s, 1  4 ).Noticing that property (d) from Theorem 1 follows by the usual compactness in (negative) Sobolev spaces, we conclude tightness of the law of (( The tightness for ((0, I O ζǫ u N ), (0, u N − F ζǫ N ∂ t η N )) follows along the same line, the only difference being the regularisation of While F ζǫ N ∂ t η N can be replaced by F ζǫ(s) N (P N ((∂ t η N (t)) κ )) as above, we need to regularise u N accordingly to preserve the homogeneous boundary conditions of f N .Recalling the definition of w i from (3.7) we define X κ i ∈ W 1,2 0,divx (O) as a spatial regularisation of X i and Y κ i as the solution to the Stokes problem with boundary datum Y κ i n (which inherits the regularity of Y κ i ).Then we set Hence f N,κ has zero boundary conditions and thus only the fluid part is seen in the expression g N (t) − g N (s), f N,κ (t) (where g N := (∂ t η N , I O ζǫ u N ) as above).In particular, the noise is not seen and we conclude obtaining again the claimed tightness.

Stochastic compactness
With the bounds from (3.13)- (3.18) in hand, we wish to obtain compactness.For this, we define the path space where From (3.13)-(3.18)(together with Alaoglu-Bourbaki Theorem) we obtain the following.Lemma 9.For fixed ǫ > 0, the joint law ; N ∈ N is tight on χ.Now we use Jakubowski's version of the Skorokhod representation theorem, see [23], to infer the following result (we refer to [29,Theorem A.1] for a statement which combines Prokhorov's and Skorokhod's theorem for quasi-Polish spaces) which one obtains after taking a non-relabelled subsequence.
Proposition 10.There exists a complete probability space ( Ω, F, P) with χ-valued random variables for N ∈ N and such that (a) For all n ∈ N the law of ΘN on χ is given by as well as (recalling the definition of τ ♯ from (3.25))
Now we introduce the filtration on the new probability space, which ensures the correct measurabilities of the new random variables.Let ( Ft ) t≥0 and ( FN t ) t≥0 be the P-augmented canonical filtration on the variables Θ and ΘN , respectively, that is 6 for t ∈ I and similarly for FN t .By [3, Theorem 2.9.1] the weak equation continuous to hold on the new probability space.Combining (5.13) and (5.14) This is sufficient to pass to the limit in the weak formulation of the equations (note that all terms except for the stochastic integral can be treated by using (3.30)).
Since we have a linear problem at hand, we may now take appropriate subsequence and pass to the limit in (3.8).Thus we obtain a martingale solution to (3.5).In order to obtain a probabilistically strong solution we prove in the following subsections pathwise uniqueness as well as convergence in probability of the original sequence.

Pathwise uniqueness
We are now going to prove that any martingale solution satisfies the energy equality (3.6).Pathwise uniqueness is then a direct consequence of the linearity of the problem.Proposition 11.Suppose that (η, u) is a weak martingale solution to (3.5).Then it holds P-a.s.
6 Some of the variables are not continuous in time, for those one can deifne σt as the history of a random distribution, cf.[3,Chapter 2.8] Proof.We rewrite (3.5) as an equation for (∂ t η, v), where η := ´t 0 ι ζǫ ∂ t η ds and where Note that H is a Hilbert space (as closed subset of the Hilbert space W 2,2 (Γ) × W 1,2 divx (O)) and L ∈ H ′ .Hence we get from Itô's formula in Hilbert spaces (see [10,Theorem 4.17]) applied to the mapping Noticing various cancellations such as The claim follows on noticing that u = J ζε(t) ū and ∆ y (ι ζε ∂ t η) = ∂ t ∆ y η.

Convergence in probability
In order to complete the proof of Theorem 6, we make use of [3, Chapter 2, Theorem 2.10.3] which is a generalization of the Gyöngy-Krylov characterization of convergence in probability introduced in [19] to quasi-Polish spaces.It applies to situations where pathwise uniqueness and existence of a martingale solution are valid and allows to establish existence of a probabilistically strong solution.We consider two sequences (N n ), (N m ) ⊂ N diverging to infinity.Let , for N ∈ N and set ϑ n := ϑ Nn , ϑ m := ϑ Nm .We also set u n := u Nn , η n := η Nn and u m := u Nm , η m := η Nm .We consider the collection of joint laws of (ϑ n , ϑ m , v, ζ, B t ) on the extended path space Similarly to Lemma 9 we obtain tightness of Let us take any subsequence (ϑ n k , ϑ m k , v, ζ, B t ).By the Jakubowski-Skorokhod representation theorem we infer for a non-relabelled subsequence the existence of a probability space ( Ω, F, P) with a sequence of random variables ( θn k , θm k , v k , ζ k , B k t ) converging almost surely in χ ext to a random variable ( θ, θ, v, ζ, B t ).Moreover, on χ ext for all k ∈ N. Observe that, in particular, As in Section 3.3 we can show that the limit objects are martingale solutions to (3.5) defined on the same stochastic basis ( Ω, F, ( Ft ), P), where ( Ft ) t≥0 is the P-augmented canonical filtration of ( θ, θ, v, ζ, B t ).We employ the pathwise uniqueness result from Proposition 11.Therefore, the solutions (η, û) and (η, ǔ) coincide P-a.s. and we have Now, we have all in hand to apply the Gyöngy-Krylov theorem from [3, Chapter 2, Theorem 2.10.3].It implies that the original sequence ϑ N defined on the initial probability space (Ω, F, P) converges in probability in the topology of χ u × χ ∇u × χ η × χ 2 f,g to the random variable .
Therefore, we finally deduce that (η, u) is a probabilistically strong solution to (3.5).This completes the proof of Theorem 7.

The nonlinear regularised problem
The aim of this section is to obtain a solution of the regularised problem thus completing the proof of Theorem 6.This is done via a fixed point argument for which the main point is proving compactness of the mapping (ζ, v) → (η, u).Given a bounded sequence (ζ n , v n ) this is achieved in three steps: • We first need to establish tightness of the probability laws, where the main difficulty arises for the velocity field.• We apply the stochastic compactness method based on Jakubowski's extension of the Skorokhod representation theorem.This yields a.s.convergence on a new probability space.• We apply a Gyöngy-Krylov type argument to obtain convergence in probability of the original sequence on the original probability space.This requires the pathwise uniqueness from Proposition 11.
Suppose there is a sequence of processes (ζ n , v n ) which are (F t )-progressively measurable and bounded in for some sufficiently large p.Now apply Theorem 7 yielding a sequence (η n , u n ) of solutions to (3.5).By the energy equality from Proposition 11 we obtain sup sup 1, (4.4) In addition, for any s ∈ (0,

Tightness of the velocity sequence
The effort of this subsection is to prove tightness of the law of u n .Similarly to Lemma 8 we obtain the following result.Lemma 12.The laws of Proof.As in Lemma 8 we must verify the assumptions from Theorem 1.First of all, boundedness in L 2 (I; X ′ × X) follows from By which can be made arbitrarily small in L 2 by choosing κ appropriately, cf.(4.6).Similarly, we have Clearly, if f N ⇀ f in L 2 (I; X) then we can deduce a converging subsequence such that f N,κ ⇀ f κ (for some f κ ) in L 2 (I; X) for any κ > 0, which implies (b).For (c) we have to control g n (t) − g n (s), f n,κ (t) , where g n := (∂ t η n , I O ζ n ǫ u n ) and hence decompose We begin estimating (II) to find that using Corollary 3. By (4.2) and (4.4) we thus get As in the proof of Lemma 8 we conclude for K > 0 that for n ∈ N and such that (a) for all n ∈ N the law of Θn on χ is given by This yields due the uniform-in-probability estimates (for arbitrary p < ∞) which gives the desired compactness of the fixed-point map.
5 The limit ǫ → 0 For fixed ǫ > 0, Theorem 6 yields the existence of a probabilistically strong solution (η ǫ , u ǫ ) to the regularized fluid-structure system defined on a given stochastic basis.

Higher regularity
As already explained in the introduction, the regularity arising from (5.1)-(5.6) is not sufficient to control the terms involving the Piola transform in the weak equation.Thus we aim at improving the spatial regularity of η implementing ideas from [28].
Finally, for some h > 0, we let ∆ s h f (y) = h −s (f (y + e i h) − f (y)) for i = 1, 2 represent the fractional difference quotient in space in the direction e i .Now, for where s ∈ (0, 1 2 ), we consider the following as test function in the weak formulation of the regularized fluid-structure system (this can be justified by Itô's formula similarly to the proof of Proposition 11).By making the fourth order term the subject, we obtain and since the corrector K η ǫ is spatially independent, Similarly, the two remaining ǫ-terms are zero.We now wish to take the p-th moment of the time integral of (5.7) where p ≥ 1.To begin with, we have by Proposition 2, where the right-hand side is uniformly bounded by (5.1) and (5.4).Moreover, which is again uniformly bounded.Since we assume that η (and consequently also ∂ t η) has mean value zero and K η ǫ maps to spatially independent functions we have ´Γ ∂ t η ǫ ∂ t K η ǫ (∆ s −h ∆ s h η ǫ ) dy = 0. Thus it holds which is uniformly bounded in L p (Ω; L 1 (I)) by (5.6).Recalling the definition of ϕ η from (2.3) we further have by (5.1), (5.2) and 2D Sobolev embeddings p which is again bounded by (5.6).By using Lemma 4 (with p = p ′ = 2, θ = s and ã = 6), and (5.1)-(5.5)(together with the embedding W 2,2 (Γ) ֒→ C 0,θ (Γ) for all θ < 1) we have that (for δ > 0 arbitrary) where we have used the continuous embedding W 2,2 (Γ) ֒→ W s+1,3 (Γ), s ∈ (0, 1  2 ).The first term can be absorbed for δ small enough, while the second done is uniformly bounded, cf.(5.1) and (5.5).Next, it follows from Proposition 2 that for all p ≥ 1.