Probabilistic global well-posedness for a viscous nonlinear wave equation modeling fluid–structure interaction

Abstract

We prove probabilistic well-posedness for a 2D viscous nonlinear wave equation modeling fluid–structure interaction between a 3D incompressible, viscous Stokes flow and nonlinear elastodynamics of a 2D stretched membrane. The focus is on (rough) data, often arising in real-life problems, for which it is known that the deterministic problem is ill-posed. We show that random perturbations of such data give rise almost surely to the existence of a unique solution. More specifically, we prove almost sure global well-posedness for a viscous nonlinear wave equation with the subcritical initial data in the Sobolev space , , which are randomly perturbed using Wiener randomization. This result shows ‘robustness’ of nonlinear fluid–structure interaction problems/models, and provides confidence that even for the ‘rough data’ (data in , ) random perturbations of such data (due to, e.g. randomness in real-life data, numerical discretization, etc.) will almost surely provide a unique solution which depends continuously on the data in the topology.