A stochastically perturbed fluid-structure interaction problem modeled by a stochastic viscous wave equation

TL;DR

This paper establishes the well-posedness of a stochastic viscous wave equation modeling fluid-structure interaction, demonstrating existence, uniqueness, and regularity of solutions in one and two spatial dimensions, highlighting the regularizing effect of fluid viscosity.

Contribution

It provides the first well-posedness results for a stochastically perturbed fluid-structure interaction model with spacetime white noise, showing regularization effects in higher dimensions.

Findings

Existence and uniqueness of stochastic mild solutions in 1D and 2D.

Solutions are Hölder continuous with specific exponents.

Viscosity regularizes noise, enabling solutions in higher dimensions.

Abstract

We study well-posedness for fluid-structure interaction driven by stochastic forcing. This is of particular interest in real-life applications where forcing and/or data have a strong stochastic component. The prototype model studied here is a stochastic viscous wave equation, which arises in modeling the interaction between Stokes flow and an elastic membrane. To account for stochastic perturbations, the viscous wave equation is perturbed by spacetime white noise scaled by a nonlinear Lipschitz function, which depends on the solution. We prove the existence of a unique function-valued stochastic mild solution to the corresponding Cauchy problem in spatial dimensions one and two. Additionally, we show that up to a modification, the stochastic mild solution is $\alpha$-H\"{o}lder continuous for almost every realization of the solution's sample path, where $\alpha\in[0,1)$ for spatial dimension $n=1$, and $\alpha \in [0,1/2)$ for spatial dimension $n=2$. This result contrasts the known results for the heat and wave equations perturbed by spacetime white noise, including the damped wave equation perturbed by spacetime white noise, for which a function-valued mild solution exists only in spatial dimension one and not higher. Our results show that dissipation due to fluid viscosity, which is in the form of the Dirichlet-to-Neumann operator applied to the time derivative of the membrane displacement, sufficiently regularizes the roughness of white noise in the stochastic viscous wave equation to allow the stochastic mild solution to exist even in dimension two, which is the physical dimension of the problem. To the best of our knowledge, this is the first result on well-posedness for a stochastically perturbed fluid-structure interaction problem.