Well-Posedness of Solutions to Stochastic Fluid–Structure Interaction

Abstract

In this paper we introduce a constructive approach to study well-posedness of solutions to stochastic fluid–structure interaction with stochastic noise. We focus on a benchmark problem in stochastic fluid–structure interaction, and prove the existence of a unique weak solution in the probabilistically strong sense. The benchmark problem consists of the 2D time-dependent Stokes equations describing the flow of an incompressible, viscous fluid interacting with a linearly elastic membrane modeled by the 1D linear wave equation. The membrane is stochastically forced by the time-dependent white noise. The fluid and the structure are linearly coupled. The constructive existence proof is based on a time-discretization via an operator splitting approach. This introduces a sequence of approximate solutions, which are random variables. We show the existence of a subsequence of approximate solutions which converges, almost surely, to a weak solution in the probabilistically strong sense. The proof is based on uniform energy estimates in terms of the expectation of the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence of probability measures associated with the approximate solutions. Probabilistic techniques based on the Skorohod representation theorem and the Gyöngy–Krylov lemma are then employed to obtain almost sure convergence of a subsequence of the random approximate solutions to a weak solution in the probabilistically strong sense. The result shows that the deterministic benchmark FSI model is robust to stochastic noise, even in the presence of rough white noise in time. To the best of our knowledge, this is the first well-posedness result for stochastic fluid–structure interaction.

Appendix: An Alternative Approach to the Existence Proof

In this appendix, we make some additional comments about an alternative approach to passing to the limit in the random approximate solutions constructed in Sect. 7. We would like to thank the anonymous reviewer, whose read the manuscript carefully and provided helpful suggestions which inspired the observations stated in this appendix. In particular, in the exposition presented in the current manuscript, we used compactness arguments based on showing tightness of the laws of the random approximate solutions, and stochastic PDE techniques involving the Skorohod representation theorem and the Gyöngy–Krylov lemma to pass to the limit. As emphasized throughout the manuscript, the rationale for using these compactness arguments, even in the case of a fully linear stochastic system of PDEs, was to develop a robust framework that is applicable to a wide variety of stochastic systems of PDEs. In particular, the compactness argument framework presented in this manuscript extends to the case of linearly coupled FSI with nonlinear dependence of the intensity of the random noise [36], and nonlinearly coupled stochastic FSI in which the fluid equations are posed on a time-dependent (random) moving fluid domain which is determined by the displacement of a stochastically forced elastic membrane [57].

However, we note that in the case of a genuinely fully linear stochastic system of PDEs as in the case of the current manuscript, having uniform boundedness of the random approximate solutions, even just in expectation, is sufficient to pass to the limit in the semidiscrete weak formulation, in order to obtain existence of a probabilistically strong solution directly on the original probability space, hence bypassing the need to use the Skorohod representation theorem and eliminating the need to transfer the problem to a different probability space. We illustrate this procedure below, starting from the uniform boundedness of the approximate solutions generated by the splitting scheme, stated in Proposition 7.2 and Proposition 7.3, where these approximate solutions satisfy the semidiscrete formulation (25). Though this procedure using weak convergence of the random approximate solutions in function spaces involving both the probability space and the spacetime function spaces works well for the case of this fully linear stochastic system, we emphasize that this approach does not generalize to more complex stochastic systems involving nonlinearities, which require compactness arguments of the type presented in this manuscript.

We start with the uniform boundedness result stated in Proposition 7.2 and Proposition 7.3, and we conclude that there exist limiting random variables $\eta$, v, and $\varvec{u}$ such that

• $\eta _{N}$ converges weakly star to $\eta$ in $L^{2}(\Omega ; L^{\infty }(0, T; H_{0}^{1}(\Gamma )))$.

• ${\overline{\eta }}_{N}$ converges weakly star to $\eta$ in $L^{2}(\Omega ; W^{1, \infty }(0, T; L^{2}(\Gamma )))$.

• $v_{N}$ converges weakly star to v in $L^{2}(\Omega ; L^{\infty }(0, T; L^{2}(\Gamma )))$.

• $v_{N}^{\Delta t}$ converge weakly to v in $L^{2}(\Omega ; L^{2}(0, T; H^{1/2}(\Gamma )))$.

• $v_{N}^{*}$ converges weakly star to v in $L^{2}(\Omega ; L^{\infty }(0, T; L^{2}(\Gamma )))$.

• $\varvec{u}_{N}$ converges weakly star to $\varvec{u}$ in $L^{2}(\Omega ; L^{\infty }(0, T; L^{2}(\Omega _{f})))$.

• $\varvec{u}_{N}^{\Delta t}$ converges weakly to $\varvec{u}$ in $L^{2}(\Omega ; L^{2}(0, T; H^{1}(\Omega _{f})))$.

In addition, by combining the uniform numerical dissipation estimates stated in Proposition 6.7 with the convergences stated above, we conclude that

• $\overline{\varvec{u}}_{N}$ converges to $\varvec{u}$ weakly in $L^{2}(\Omega ; L^{2}(0, T; L^{2}(\Omega _{f})))$.

• ${\overline{v}}_{N}$ converges to v weakly in $L^{2}(\Omega ; L^{2}(0, T; L^{2}(\Gamma )))$.

We recall that the (random) approximate solutions satisfy the semidiscrete formulation (25) almost surely for each (deterministic) test function $(\varvec{q}, \psi ) \in {\mathcal {Q}}(0, T)$:

$$\begin{aligned}{} & {} \int _{\Omega _{f}} \frac{\varvec{u}^{n + 1}_{N} - \varvec{u}^{n}_{N}}{\Delta t} \cdot \varvec{q} d\varvec{x} + 2\mu \int _{\Omega _{f}} \varvec{D}(\varvec{u}^{n + 1}_{N}): \varvec{D}(\varvec{q}) d\varvec{x} + \int _{\Gamma } \frac{v^{n + 1}_{N} - v^{n}_{N}}{\Delta t} \psi dz + \int _{\Gamma } \nabla \eta ^{n + 1}_{N} \cdot \nabla \psi dz \\{} & {} \quad = \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t} \psi dz + P^{n}_{N, in}\int _{0}^{R} (q_{z})|_{z = 0} dr - P^{n}_{N, out} \int _{0}^{R} (q_{z})|_{z = L} dr. \end{aligned}$$

We want to show that the limiting functions $(\eta , v, \varvec{u})$ satisfy the continuous weak formulation almost surely for each $(\varvec{q}, \psi ) \in {\mathcal {Q}}(0, T)$:

$$\begin{aligned}{} & {} - \int _{0}^{T} \int _{\Omega _{f}} \varvec{u} \cdot \partial _{t}\varvec{q} d\varvec{x} dt + 2\mu \int _{0}^{T} \int _{\Omega _{f}} \varvec{D}(\varvec{u}): \varvec{D}(\varvec{q}) d\varvec{x} dt - \int _{0}^{T} \int _{\Gamma } v\partial _{t}\psi dz dt + \int _{0}^{T} \int _{\Gamma } \nabla \eta \cdot \nabla \psi dz dt \\{} & {} \quad = \int _{0}^{T} P_{in}(t) \left( \int _{\Gamma _{in}} q_{z} dr\right) dt - \int _{0}^{T} P_{out}(t) \left( \int _{\Gamma _{out}} q_{z} dr\right) dt + \int _{\Omega _{f}} \varvec{u_{0}} \cdot \varvec{q}(0) d\varvec{x} \\{} & {} \qquad + \int _{\Gamma } v_{0}\psi (0) dz + \int _{0}^{T}\left( \int _{\Gamma } \psi dz\right) dW(t). \end{aligned}$$

We emphasize that the convergence of the approximate solutions to the limiting functions $(\eta , v, \varvec{u})$ is only convergence weakly and weakly star in function spaces involving the probability space itself since the uniform boundedness of the approximate solutions is only in expectation.

We consider a fixed but arbitrary deterministic test function $(\varvec{q}, \psi ) \in {\mathcal {Q}}(0, T)$ where the test space ${\mathcal {Q}}(0, T)$ is defined in (13), and we associate to this test function the following random variable (defined with the limiting weak formulation in consideration):

$$\begin{aligned}{} & {} X_{(\varvec{q}, \psi )}:= - \int _{0}^{T} \int _{\Omega _{f}} \varvec{u} \cdot \partial _{t}\varvec{q} d\varvec{x} dt + 2\mu \int _{0}^{T} \int _{\Omega _{f}} \varvec{D}(\varvec{u}): \varvec{D}(\varvec{q}) d\varvec{x} dt - \int _{0}^{T} \int _{\Gamma } v\partial _{t}\psi dz dt + \int _{0}^{T} \int _{\Gamma } \nabla \eta \cdot \nabla \psi dz dt \nonumber \\{} & {} \quad - \int _{0}^{T} P_{in}(t) \left( \int _{\Gamma _{in}} q_{z} dr\right) dt + \int _{0}^{T} P_{out}(t) \left( \int _{\Gamma _{out}} q_{z} dr\right) dt - \int _{\Omega _{f}} \varvec{u_{0}} \cdot \varvec{q}(0) d\varvec{x} \nonumber \\{} & {} \quad - \int _{\Gamma } v_{0}\psi (0) dz - \int _{0}^{T}\left( \int _{\Gamma } \psi dz\right) dW(t). \end{aligned}$$

(83)

Because of the function spaces that the limiting solution $(\eta , v, \varvec{u})$ belong to and the regularity of the test functions in the test space ${\mathcal {Q}}(0, T)$ which is defined in (13), we conclude that $X_{(\varvec{q}, \psi )} \in L^{2}(\Omega )$ is a square integrable real-valued random variable that is hence finite almost surely.

Because the semidiscrete weak formulation holds pathwise as a result of how the splitting scheme constructs the approximate solutions pathwise outcome by outcome in the probability space, we more generally have that the approximate solutions satisfy the following generalized semidiscrete formulation almost surely

$$\begin{aligned}{} & {} \int _{\Omega _{f}} \frac{\varvec{u}^{n + 1}_{N} - \varvec{u}^{n}_{N}}{\Delta t} \cdot \left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} + 2\mu \int _{\Omega _{f}} \varvec{D}(\varvec{u}^{n + 1}_{N}): \varvec{D}\left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} \nonumber \\{} & {} \quad + \int _{\Gamma } \frac{v^{n + 1}_{N} - v^{n}_{N}}{\Delta t} \left( X_{(\varvec{q}, \psi )} \psi \right) dz + \int _{\Gamma } \nabla \eta ^{n + 1}_{N} \cdot \nabla \left( X_{(\varvec{q}, \psi )} \psi \right) dz \nonumber \\{} & {} \quad - \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t} \left( X_{(\varvec{q}, \psi )} \psi \right) dz \nonumber \\{} & {} \quad - P^{n}_{N, in}\int _{0}^{R} \left( X_{(\varvec{q}, \psi )} q_{z}\right) |_{z = 0} dr + P^{n}_{N, out} \int _{0}^{R} \left( X_{(\varvec{q}, \psi )} q_{z}\right) |_{z = L} dr = 0, \end{aligned}$$

(84)

so that the test function $\left( X_{(\varvec{q}, \psi )} q, X_{(\varvec{q}, \psi )} \psi \right)$ in the semidiscrete weak formulation is now a random test function. Because all of the weak and weak star convergences that we have involve the probability space and hence involve convergence of quantities in expectation, we integrate from $n\Delta t$ to $(n + 1)\Delta t$ in time, sum from $n = 0$ to $n = N - 1$, and take the expectation of both sides of (84), and then pass to the limit as $N \rightarrow \infty$. We hence want to pass to the limit as $N \rightarrow \infty$ in the left hand side of the expression:

$$\begin{aligned}{} & {} \mathbb {E} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \Bigg [\int _{\Omega _{f}} \frac{\varvec{u}^{n + 1}_{N} - \varvec{u}^{n}_{N}}{\Delta t} \cdot \left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} + 2\mu \int _{\Omega _{f}} \varvec{D}(\varvec{u}^{n + 1}_{N}): \varvec{D}\left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} \nonumber \\{} & {} \quad + \int _{\Gamma } \frac{v^{n + 1}_{N} - v^{n}_{N}}{\Delta t} \left( X_{(\varvec{q}, \psi )} \psi \right) dz + \int _{\Gamma } \nabla \eta ^{n + 1}_{N} \cdot \nabla \left( X_{(\varvec{q}, \psi )} \psi \right) dz - \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t} \left( X_{(\varvec{q}, \psi )} \psi \right) dz \nonumber \\{} & {} \quad - P^{n}_{N, in}\int _{0}^{R} \left( X_{(\varvec{q}, \psi )} q_{z}\right) |_{z = 0} dr + P^{n}_{N, out} \int _{0}^{R} \left( X_{(\varvec{q}, \psi )} q_{z}\right) |_{z = L} dr\Bigg ] dt = 0, \end{aligned}$$

(85)

To handle the first term, we use an integration by parts in time as in (56) and the fact that $\varvec{q}$ is compactly supported in [0, T) to obtain:

$$\begin{aligned}{} & {} \mathbb {E} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Omega _{f}} \frac{\varvec{u}^{n + 1}_{N} - \varvec{u}^{n}_{N}}{\Delta t} \cdot \left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} dt = \mathbb {E} \int _{0}^{T} \int _{\Omega _{f}} \partial _{t}\overline{\varvec{u}}_{N} \cdot \left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} dt \nonumber \\{} & {} \quad = -\mathbb {E} \int _{0}^{T} \int _{\Omega _{f}} \overline{\varvec{u}}_{N} \cdot \left( X_{(\varvec{q}, \psi )} \partial _{t} \varvec{q}\right) d\varvec{x} dt - \mathbb {E} \int _{\Omega _{f}} \varvec{u}_{0} \cdot \left( X_{(\varvec{q}, \psi )} \varvec{q}(0)\right) \nonumber \\{} & {} \quad \rightarrow -\mathbb {E} \int _{0}^{T} \int _{\Omega _{f}} \varvec{u} \cdot \left( X_{(\varvec{q}, \psi )} \partial _{t}\varvec{q}\right) d\varvec{x} dt - \mathbb {E} \int _{\Omega _{f}} \varvec{u}_{0} \cdot \left( X_{(\varvec{q}, \psi )} \varvec{q}(0)\right) , \end{aligned}$$

(86)

by the convergence of $\overline{\varvec{u}}_{N}$ to $\varvec{u}$ weakly in $L^{2}(\Omega ; L^{2}(0, T; L^{2}(\Omega _{f})))$. Similarly, we have by the weak convergence of $v_{N}$ to v in $L^{2}(\Omega ; L^{2}(0, T; L^{2}(\Gamma )))$ that

$$\begin{aligned} \mathbb {E} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \frac{v^{n + 1}_{N} - v^{n}_{N}}{\Delta t} \left( X_{(\varvec{q}, \psi )} \psi \right) dz dt \rightarrow -\mathbb {E} \int _{0}^{T} \int _{\Gamma } v\left( X_{(\varvec{q}, \psi )} \partial _{t}\psi \right) dz dt - \mathbb {E} \int _{\Omega _{f}} v_{0} \left( X_{(\varvec{q}, \psi )} \psi (0)\right) .\nonumber \\ \end{aligned}$$

(87)

By the weak convergence of $\varvec{u}^{\Delta t}_{N}$ to $\varvec{u}$ in $L^{2}(\Omega ; L^{2}(0, T; H^{1}(\Omega _{f})))$, we obtain that

$$\begin{aligned} \mathbb {E} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} 2\mu \int _{\Omega _{f}} \varvec{D}(\varvec{u}^{n + 1}_{N}) : \varvec{D}\left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} dt&= \mathbb {E} \left( 2\mu \int _{0}^{T} \int _{\Omega _{f}} \varvec{D}\left( \varvec{u}^{\Delta t}_{N}\right) : \varvec{D}\left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} dt\right) \nonumber \\&\rightarrow \mathbb {E}\left( 2\mu \int _{0}^{T} \int _{\Omega _{f}} \varvec{D}\left( \varvec{u}\right) : \varvec{D}\left( X_{(\varvec{q}, \psi )} \varvec{q}\right) d\varvec{x} dt\right) . \end{aligned}$$

(88)

Similarly, by the weak convergence of $\eta ^{\Delta t}_{N}$ to $\eta$ in $L^{2}(\Omega ; L^{2}(0, T; H_{0}^{1}(\Gamma )))$ which follows from the uniform numerical dissipation estimates in Proposition 6.7, where $\eta ^{\Delta t}_{N}$ is defined in (30), we obtain that

$$\begin{aligned} \mathbb {E}\sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \nabla \eta ^{n + 1}_{N} \cdot \nabla \left( X_{(\varvec{q}, \psi )} \psi \right) dz dt \rightarrow \mathbb {E}\left( \int _{0}^{T} \int _{\Gamma } \nabla \eta \cdot \nabla \left( X_{(\varvec{q}, \psi )} \psi \right) dz dt\right) . \end{aligned}$$

(89)

For the stochastic integral, we claim that

$$\begin{aligned}{} & {} \mathbb {E} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t} \left( X_{(\varvec{q}, \psi )} \psi \right) dz dt \nonumber \\{} & {} \quad = \mathbb {E} \left( X_{(\varvec{q}, \psi )} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t} \psi dz dt\right) \rightarrow \mathbb {E} \left( X_{(\varvec{q}, \psi )} \int _{0}^{T} \int _{\Gamma } \psi (t, z) dz dW(t)\right) .\nonumber \\ \end{aligned}$$

(90)

In order to show this, we first observe that

$$\begin{aligned}{} & {} \mathbb {E} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t} \psi dz dt \nonumber \\{} & {} \quad = \mathbb {E} \left( \int _{0}^{T} \sum _{n = 0}^{N - 1} \left( \frac{1}{\Delta t}\int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \psi (s, z) dz ds \right) 1_{[n\Delta t, (n + 1)\Delta t)}(t) dW(t)\right) , \end{aligned}$$

(91)

and note that we have the following deterministic convergence:

$$\begin{aligned} \int _{0}^{T} \left( \int _{\Gamma } \psi (t, z) dz - \frac{1}{\Delta t} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \psi (s, z) dz ds 1_{[n\Delta t, (n + 1)\Delta t)}(t)\right) ^{2} dt \rightarrow 0, \qquad \text { as } N \rightarrow \infty ,\nonumber \\ \end{aligned}$$

(92)

by the fact that the function $\displaystyle \int _{\Gamma } \psi (t, z) dz$ is a uniformly continuous real-valued (deterministic) function on [0, T]. Hence, by Itô’s isometry and (91),

$$\begin{aligned} \mathbb {E}\sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t} \psi dz dt \rightarrow \mathbb {E} \left( \int _{0}^{T} \int _{\Gamma } \psi (t, z) dz dW(t)\right) . \end{aligned}$$

(93)

Then, because $X_{(\varvec{q}, \psi )} \in L^{2}(\Omega )$ and in addition,

$$\begin{aligned} \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \int _{\Gamma } \frac{W((n + 1)\Delta t) - W(n\Delta t)}{\Delta t}\psi dz dt, \quad \int _{0}^{T} \int _{\Gamma } \psi (t, z) dz dW(t) \in L^{2}(\Omega ), \end{aligned}$$

we conclude the desired convergence (90) by using (93) and the Cauchy-Schwarz inequality with $L^{2}(\Omega )$.

Finally, for the pressure term, we use the deterministic convergence (75) to conclude that

$$\begin{aligned}{} & {} \mathbb {E}\sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} \left( P^{n}_{N, in/out} \int _{0}^{R} \left( X_{(\varvec{q}, \psi )} q_{z}\right) |_{z = 0} dr\right) \nonumber \\{} & {} \quad = \left( \sum _{n = 0}^{N - 1} \int _{n\Delta t}^{(n + 1)\Delta t} P^{n}_{N, in/out} \int _{0}^{R} (q_{z})|_{z = 0} dr dt\right) \mathbb {E}\left( X_{(\varvec{q}, \psi )}\right) \nonumber \\{} & {} \quad \rightarrow \left( \int _{0}^{T} P_{in/out}(t) \int _{0}^{R} (q_{z})|_{z = 0} dr dt\right) \mathbb {E}\left( X_{(\varvec{q}, \psi )}\right) . \end{aligned}$$

(94)

Combining the convergences (86), (87), (88), (89), (90), and (94) and applying these convergences to take the limit in (85) as $N \rightarrow \infty$, we obtain that

$$\begin{aligned} \mathbb {E}\left( \left| X_{(\varvec{q}, \psi )}\right| ^{2}\right) = 0, \end{aligned}$$

since we can take the real-valued random variable $X_{(\varvec{q}, \psi )}$ out of any integrals involving space or time as a multiplicative constant. Hence, $X_{(\varvec{q}, \psi )} = 0$ almost surely for every fixed but arbitrary deterministic test function $(\varvec{q}, \psi ) \in {\mathcal {Q}}(0, T)$, which shows that the limiting functions $(\eta , v, \varvec{u})$ satisfy the continuous in time weak formulation. This is a result of the fact that $X_{(\varvec{q}, \psi )} = 0$ almost surely, and the definition of the real-valued random variable $X_{(\varvec{q}, \psi )}$ in (83).