Existence of martingale solutions to a nonlinearly coupled stochastic fluid-structure interaction problem

We study a nonlinear stochastic fluid-structure interaction problem with a multiplicative, white-in-time noise. The problem consists of the Navier-Stokes equations describing the flow of an incompressible, viscous fluid in a 2D cylinder interacting with an elastic lateral wall whose elastodynamics is described by a membrane equation. The flow is driven by the inlet and outlet data and by the stochastic forcing. The noise is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The fluid and the structure are <b>nonlinearly coupled</b> via the kinematic and dynamic conditions assumed at the moving interface, which is a random variable not known <i>a priori</i>. The geometric nonlinearity due to the nonlinear coupling requires the development of new techniques to capture martingale solutions for this class of stochastic FSI problems. Our analysis reveals a first-of-its-kind temporal regularity result for the solutions. This is the first result in the field of SPDEs that addresses the existence of solutions on moving domains involving incompressible fluids, where <b>the displacement of the boundary and the fluid domain are random variables that are not known <i>a priori</i> and are parts of the solution itself</b>.