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

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, $\text{F}$, and torque, $\text{M}$. We show that if the data are suitably regular and small, and $\text{F}$ and $\text{M}$ 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.

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