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

Abstract

We consider a fluid flow in a time dependent domain Ωf(t)=Ω\Ωs(t)̅⊂ℝ3, surrounding a deformable obstacle Ωs(t). We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in Ωf(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 ∂Ωs(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.