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

We consider a fluid flow in a time dependent domain Ω_f(t)=Ω\Ω_s(t)̅⊂ℝ³ \begin{equation*}\Omega_f(t)= \Omega\setminus \overline{\Omega_s(t)}\subset {\mathbb R}^3\end{equation*} Ω f (t) = Ω \ Ω s (t) ̅ ⊂ R 3 , surrounding a deformable obstacle Ω_s(t). We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in Ω_f(t), t > 0. We prove that, for any arbitrary exponential decay rate ω > 0, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary ∂Ω_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 ω > 0. [ABSTRACT FROM AUTHOR]