On the long time behaviour of a system of several rigid bodies immersed in a viscous fluid

TL;DR

This paper proves the existence of global dissipative solutions for multiple rigid bodies in a viscous fluid and shows the system naturally tends to equilibrium over time, regardless of collisions or initial energy.

Contribution

It introduces a new concept of dissipative weak solutions for rigid bodies in fluid and establishes their global existence and long-term behavior.

Findings

Global-in-time dissipative solutions exist for connected rigid bodies.

System tends to static equilibrium as time approaches infinity.

Results hold regardless of collisions and initial energy levels.

Abstract

We consider several rigid bodies immersed in a viscous Newtonian fluid contained in a bounded domain in $R^3$. We introduce a new concept of dissipative weak solution of the problem based on a combination of the approach proposed by Judakov with a suitable form of energy inequality. We show that global--in--time dissipative solutions always exist as long as the rigid bodies are connected compact sets. In addition, in the absence of external driving forces, the system always tends to a static equilibrium as time goes to infinity. The results hold independently of possible collisions of rigid bodies and for any finite energy initial data.