On Vector Fields Describing the 2d Motion of a Rigid Body in a Viscous Fluid and Applications

Abstract

We present some properties of functions in suitable Sobolev spaces which arise naturally in the study of the motion of a rigid body in compressible and incompressible fluid. We relax the regularity assumption of the rigid body by allowing its boundary to be Lipschitz. In the case of a smooth rigid body we obtain a new estimate on the angular velocity. Our results extend and complement related results by V. Starovoitov and moreover we show that they are optimal. As an application we present an example where the rigid body collides with the boundary with non zero speed. Finally, we present a new non collision result concerning a smooth rotating body approaching the boundary, without assuming any special geometry on either the body or the container.