Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

TL;DR

This paper proves that the smoothness of a rigid body's motion in an incompressible perfect fluid is limited only by the smoothness of the boundaries, showing analyticity for analytic boundaries and smooth dependence on initial data.

Contribution

It establishes the relationship between boundary smoothness and the regularity of the body's motion, including analyticity results and dependence on initial conditions.

Findings

Motion is analytic for analytic boundaries.

The body's motion depends smoothly on initial data.

Smoothness is limited only by boundary regularity.

Abstract

We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the Holder space C^{1,r}. In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic (till the classical solution exists and till the solid does not hit the boundary). Moreover in this case this motion depends smoothly on the initial data.