# Stability of Permanent Rotations and Long-Time Behavior of Inertial Motions of a Rigid Body with an Interior Liquid-Filled Cavity

## TL;DR

This paper analyzes the stability and long-term behavior of a rigid body with an internal liquid cavity, proving conditions for exponential stability of permanent rotations and confirming Zhukovsky's conjecture about decay to such rotations.

## Contribution

It provides rigorous conditions for stability and decay of inertial motions in a liquid-filled rigid body, confirming Zhukovsky's conjecture with a comprehensive mathematical analysis.

## Key findings

- Sufficient conditions for exponential stability of permanent rotations.
- Conditions for instability of certain rotations.
- All weak solutions decay exponentially to a permanent rotation after some time.

## Abstract

A rigid body, with an interior cavity entirely filled with a Navier-Stokes liquid, moves in absence of external torques relative to the center of mass of the coupled system body-liquid (inertial motions). The only steady-state motions allowed are then those where the system, as a whole rigid body, rotates uniformly around one of the central axes of inertia (permanent rotations). Objective of this article is two-fold. On the one hand, we provide sufficient conditions for the asymptotic, exponential stability of permanent rotations, as well as for their instability. On the other hand, we study the asymptotic behavior of the generic motion in the class of weak solutions and show that there exists a time $t_0$ after that all such solutions must decay exponentially fast to a permanent rotation. This result provides a {\em full} and rigorous explanation of Zhukovsky's conjecture, and explains, likewise, other interesting phenomena that are observed in both lab and numerical experiments.