Motion of Rigid Bodies of Arbitrary Shape in a Viscous Incompressible Fluid: Wellposedness and Large Time Behaviour

We investigate the long-time behaviour of a coupled PDE–ODE system that describes the motion of a rigid body of arbitrary shape moving in a viscous incompressible fluid. We assume that the system formed by the rigid body and the fluid fills the entire space R3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{3}.$$\end{document} We extend in this way our previous results which were limited to the case when the rigid body was a ball. More precisely, we show that, under appropriate assumptions (in particular smallness ones) on the initial velocity field, the position of the rigid body converges to some final configuration as time goes to infinity. Finally, we show that our methodology can be applied in the case of several rigid bodies of arbitrary shapes moving in a viscous incompressible fluid.