Stabilization of an incompressible fluid-elastic structure system using a vacuum bubble

TL;DR

This paper demonstrates that a vacuum bubble within an incompressible fluid-elastic structure system stabilizes the system, enabling global existence and exponential decay of solutions through a priori estimates.

Contribution

It introduces a novel stabilization mechanism via a vacuum bubble in a fluid-structure interaction model, providing new insights into controlling such complex systems.

Findings

Vacuum bubble stabilizes the fluid-structure system.

Ensures control of the pressure average.

Leads to global existence and exponential decay of solutions.

Abstract

We prove a priori estimates for the system of partial differential equations modeling the interaction between an elastic body and an incompressible fluid in a 3D curved domain. The fluid is governed by the incompressible Navier-Stokes equations and contains a bubble whose interior is a vacuum. The elastic body is described by a damped wave equation, and interaction with the fluid takes place along a free interface whose initial domain is curved. We show that the presence of the vacuum bubble stabilizes the system in the sense that it provides control of the average of the pressure function, and hence allows global existence and exponential decay of smooth solutions for small data.