Feedback stabilization of a 3D fluid flow by shape deformations of an obstacle

Abstract

We consider a fluid flow in a time dependent domain $\Omega_f(t)=\Omega \setminus \Omega_s(t)\subset {\mathbb R}^3$, surrounding a deformable obstacle $\Omega_s(t)$. We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in $\Omega_f(t)$, $t>0$. We prove that, for any arbitrary exponential decay rate $\omega>0$, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary $\partial \Omega_s(t)$ can be chosen so that the fluid flow is stabilized to rest, and the obstacle to its initial shape and its initial location, with the exponential decay rate $\omega>0$.