On the interaction between a rigid-body and a viscous-fluid: existence of a weak solution and a suitable Th\'eor\`eme de Structure

TL;DR

This paper establishes the existence of weak solutions for the coupled system of a rigid body and viscous fluid, introducing new analytical techniques and providing a partial regularity result that applies for large times.

Contribution

It presents a novel proof of weak solution existence and a new approach to Leray's Thérème de Structure for fluid-structure interaction systems.

Findings

Existence of weak solutions for the coupled system.

Partial regularity of solutions for large times.

A new analytical method differing from classical Navier-Stokes techniques.

Abstract

In this paper, we prove the existence and a partial regularity of a weak solution to the system governing the interaction between a rigid body and a viscous incompressible Newtonian fluid. The evolution of the system body-fluid is studied in a frame attached to the body. The choice of this special frame becomes critical from an analytical point of view due to the presence of the term $\omega\times x\cdot\nabla u$ in the balance of momentum equation for the fluid. As a consequence, we are forced to look for a technique that is different from the ones usually employed both for the existence and for the partial regularity of a weak solution to the Navier-Stokes problem. Hence, we prove the existence of a weak solution in an original way and give a new proof of the celebrated Th\'eor\`eme de Structure due to Leray. However, the regularity obtained for our weak solution is only for large times, hence our result is weaker compared to the one obtained by Leray.