The APDE seminar on Monday, 4/12, will be given by Jeffrey Kuan online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (gmoschidis@berkeley.edu) or Federico Pasqualotto (fpasqualotto@berkeley.edu).

Title: A stochastic fluid-structure interaction model given by a stochastic viscous wave equation

Abstract: We consider a stochastic fluid-structure interaction (FSI) model, given by a stochastic viscous wave equation perturbed by spacetime white noise. The wave equation part of the model describes the elastodynamics of a thin structure, such as an elastic membrane, while the viscous part, which is in the form of the Dirichlet-to-Neumann operator, describes the impact of a viscous, incompressible fluid in a two-way coupled fluid-structure interaction problem. The stochastic perturbation describes random deviations observed in real-life data. We prove that this stochastic viscous wave equation has a mild solution in dimension one, and also in dimension two, which is the physical dimension of the FSI problem (thin 2D membrane). This behavior contrasts that of the stochastic heat and the stochastic wave equations, which do not have function valued mild solutions in dimensions two and higher. This means that in the two dimensional model, unlike the heat and wave equations, dissipation due to fluid viscosity in the viscous wave equation, keeps the stochastically perturbed solution “in control”. We also consider Hölder continuity path properties of solutions and show that the solution is Hölder continuous up to Hölder exponent 1/2 in both space and time, after stochastic modification. This is joint work with Suncica Canic.