Existence of weak martingale solutions to a stochastic fluid-structure interaction problem with a compressible viscous fluid

Abstract

We study the existence of weak martingale solutions to a stochastic moving boundary problem arising from the interaction between an isentropic compressible fluid and a viscoelastic structure. In the model, we consider a three-dimensional compressible isentropic fluid with adiabatic constant $\gamma > 3/2$ interacting dynamically with an elastic structure on the boundary of the fluid domain described by a plate equation, under the additional influence of stochastic perturbations which randomly force both the compressible fluid and elastic structure equations in time. The problem is nonlinearly coupled in the sense that the a priori unknown (and random) displacement of the elastic structure from its reference configuration determines the a priori unknown (and random) time-dependent fluid domain on which the compressible isentropic Navier-Stokes equations are posed. We use a splitting method, consisting of a fluid and structure subproblem, to construct random approximate solutions to an approximate Galerkin form of the problem with artificial viscosity and artificial pressure. We introduce stopped processes of structure displacements which handle the issues associated with potential fluid domain degeneracy. In this splitting scheme, we handle mathematical difficulties associated with the a priori unknown and time-dependent fluid domain by using an extension of the fluid equations to a fixed maximal domain and we handle difficulties associated with imposing the no-slip condition in the stochastic setting by using a novel penalty term defined on an external tubular neighborhood of the moving fluid-structure interface. To the best of our knowledge, this is the first well-posedness result for stochastic fluid-structure interaction with compressible fluids.