A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body

TL;DR

This paper extends Kato's theorem on the inviscid limit of Navier-Stokes equations to include a moving rigid body, showing that under certain conditions, both fluid and body velocities converge to their Euler counterparts.

Contribution

It establishes a Kato-type criterion ensuring convergence of fluid and body velocities in the inviscid limit with a moving rigid body, linking fixed boundary results to dynamic cases.

Findings

Kato-type condition implies convergence of fluid velocity

Kato-type condition implies convergence of body velocity

Results suggest fixed boundary analysis informs moving body scenarios

Abstract

The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Tosio Kato says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear if the additional difficulties linked to the body's dynamic make this issue more difficult or not. In this paper we consider the motion of a rigid body in an incompressible fluid occupying the complementary set in the space and we prove that a Kato type condition implies the convergence of the fluid velocity and of the body velocity as well, what seems to indicate that an answer in the case of a fixed boundary could also bring an answer to the case where there is a moving body in the fluid.