On the regularity of weak solutions to the fluid-rigid body interaction problem

TL;DR

This paper proves that weak solutions to 3D fluid-rigid body interaction problems become smooth under certain regularity conditions, extending classical Navier-Stokes regularity results to coupled fluid-solid systems.

Contribution

It generalizes classical Navier-Stokes regularity criteria to fluid-rigid body interactions, establishing smoothness of weak solutions under Prodi-Serrin conditions and bounded rigid body acceleration.

Findings

Prodi-Serrin conditions imply $W^{2,p}$ regularity for fluid velocity.

Solutions are $C^{inity}$ if rigid body acceleration is bounded.

Weak solutions become smooth under specified regularity assumptions.

Abstract

We study a 3D fluid-rigid body interaction problem. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations describing conservation of linear and angular momentum. Our aim is to prove that any weak solution satisfying certain regularity conditions is smooth. This is a generalization of the classical result for the $3D$ incompressible Navier-Stokes equations, which says that a weak solution that additionally satisfy Prodi - Serrin $L^r-L^s$ condition is smooth. We show that in the case of fluid - rigid body the Prodi - Serrin conditions imply $W^{2,p}$ and $W^{1,p}$ regularity for the fluid velocity and fluid pressure, respectively. Moreover, we show that solutions are $C^{\infty}$ if additionally we assume that the rigid body acceleration is bounded almost anywhere in time variable.