On the motion of a fluid-filled rigid body with Navier Boundary conditions

TL;DR

This paper studies the motion of a fluid-filled rigid body with Navier boundary conditions, proving existence, regularity, and stability of solutions, and analyzing their long-term behavior and equilibrium stability.

Contribution

It establishes the existence and regularity of solutions, characterizes their long-time convergence to equilibrium, and analyzes stability based on moments of inertia.

Findings

Weak solutions exist and are regular under certain conditions.

Solutions converge exponentially to equilibrium in $L_q$-topology.

Largest moment of inertia equilibria are stable, others are unstable.

Abstract

We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution \`a la Leray-Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.