A maximal regularity approach to the study of motion of a rigid body with a fluid-filled cavity

TL;DR

This paper analyzes the motion of a rigid body with a fluid-filled cavity, establishing stability of equilibria, convergence of solutions, and the role of maximal regularity in understanding the system's dynamics.

Contribution

It introduces a maximal regularity framework to study the stability and convergence of solutions for a rigid body with an interior viscous fluid.

Findings

Equilibria with the largest moment of inertia are stable.

All solutions converge exponentially to an equilibrium.

The critical spaces for the evolution equation are identified.

Abstract

We consider the inertial motion of a rigid body with an interior cavity that is completely filled with a viscous incompressible fluid. The equilibria of the system are characterized and their stability properties are analyzed. It is shown that equilibria associated with the largest moment of inertia are normally stable, while all other equilibria are normally hyperbolic. We show that every Leray-Hopf weak solution converges to an equilibrium at an exponential rate. In addition, we determine the critical spaces for the governing evolution equation, and we demonstrate how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity of solutions and their convergence to equilibria.