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 ${\mathcal{H}}^s({\mathbb{R}}^2)$, $s>-\frac{1}{5}$, 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 ${\mathcal{H}}^s$, $s>-\frac{1}{5}$) 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.

Keywords:

• Viscous nonlinear wave equation
• wiener randomization
• probabilistic well-posedness

2020 Mathematics Subject Classifications:

• 35L05
• 35L71
• 35R60
• 60H15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Lemma 2.2 in the arXiv version.

2 Namely, given by a smooth Fourier multiplier.

3 Here, we did not show the continuity in time of $\overrightarrow{v}$ in ${\mathcal{H}}^1({\mathbb{R}}^2)$, but this can be done by a standard argument, which we omit.

Additional information

Funding

This work was partially supported by the National Science Foundation [grant numbers DMS-1853340, and DMS-2011319] (Čanić and Kuan), and by the European Research Council [grant number 864138] ‘SingStochDispDyn’ (Oh).