Well-Posedness of Solutions to Stochastic Fluid–Structure Interaction
Well-Posedness of Solutions to Stochastic Fluid–Structure Interaction
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DOI：
10.1007/s00021-023-00839-y
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发表时间：
2022-03
影响因子：
1.3
通讯作者：
Jeffrey Kuan;S. Čanić
中科院分区：
文献类型：
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作者：
Jeffrey Kuan;S. Čanić
在本文中，我们介绍了一种构造性方法来研究随机流固耦合与随机噪声的解的适定性。我们关注随机流固耦合中的一个基准问题，并证明了强概率意义上唯一弱解的存在。基准问题由二维瞬态斯托克斯方程组成，描述不可压缩的粘性流体与由一维线性波动方程建模的线性弹性膜相互作用的流动。膜受到随时间变化的白噪声的随机作用。流体和结构是线性耦合的。构造性的存在证明基于通过算子分裂方法的时间离散化。这引入了一系列近似解，它们是随机变量。我们证明了近似解子序列的存在，该子序列几乎肯定会收敛到强概率意义上的弱解。该证明基于根据能量范数的期望进行的统一能量估计，这是弱紧性论证的支柱，产生与近似解相关的概率度量的弱收敛子序列。然后采用基于 Skorohod 表示定理和 Gyöngy-Krylov 引理的概率技术来获得随机近似解的子序列几乎肯定收敛于概率强意义上的弱解。结果表明，确定性基准 FSI 模型对随机噪声具有鲁棒性，即使及时存在粗糙白噪声也是如此。据我们所知，这是随机流固耦合的第一个适定结果。
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 theexpectationof the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence ofprobability measuresassociated 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.