Large-Time Behavior of a Rigid Body of Arbitrary Shape in a Viscous Fluid Under the Action of Prescribed Forces and Torques

TL;DR

This paper proves that a rigid body of arbitrary shape moving in a viscous fluid under vanishing forces and torques has a unique global solution that tends to rest as time goes to infinity, extending previous results beyond spherical bodies.

Contribution

It generalizes known results on the long-time behavior of rigid bodies in fluids from spherical shapes to arbitrary shapes under small, regular data.

Findings

Existence of a global strong solution under small data conditions.

Solution converges to zero as time approaches infinity.

Extension of previous spherical-body results to arbitrary shapes.

Abstract

Let $\mathcal B$ be a sufficiently smooth rigid body (compact set of $\mathbb R^3$) of arbitrary shape moving in an unbounded Navier-Stokes liquid under the action of prescribed external force, $\textup{F}$, and torque, $\textup{M}$. We show that if the data are suitably regular and small, and $\textup{F}$ and $\textup{M}$ vanish for large times in the $L^2$-sense, there exists at least one global strong solution to the corresponding initial-boundary value problem. Moreover, this solution converges to zero as time approaches infinity. This type of results was known, so far, only when $\mathcal B$ is a ball.