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 $\mathcal{H}^s (\mathbb{R}^2)$, $s > - \frac 15$, which are randomly perturbed using Wiener randomization. This result shows "robustness" of nonlinear FSI problems/models, and provides confidence that even for the "rough data" (data in $\mathcal{H}^s$, $s > -\frac 1 5$) 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 $\mathcal{H}^s$ topology.