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