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

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.