Small Moving Rigid Body into a Viscous Incompressible Fluid

Abstract

We consider a single disk moving under the influence of a two dimensional viscous fluid and we study the asymptotic as the size of the solid tends to zero. If the density of the solid is independent of \({\varepsilon}\), the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimal Lp−Lq decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, we will deduce the convergence to the solution of the Navier–Stokes equations in \({\mathbb{R}^{2}}\).