The motion of a rigid body in a viscous fluid: new results for strong solutions, uniqueness and integrability properties

TL;DR

This paper establishes new regularity and uniqueness results for strong solutions of the rigid body in viscous fluid interaction, under specific integrability conditions on initial data and solutions.

Contribution

It proves that the time derivative of solutions belongs to a certain function space and establishes uniqueness under assumptions verified by initial data conditions.

Findings

Time derivative of solutions in L^2 space

Uniqueness of solutions under specific assumptions

Existence of solutions satisfying the assumptions

Abstract

In this note, we show two results in the setting of Galdi-Silvestre strong solutions for the rigid body-viscous fluid interaction. The former, under an additional integrability assumption on the gradient of the initial data, proves that the time derivative of the solution belongs to $L^2(0,T;L^2({\Omega}))$. The latter, thanks to a further assumption only on one solution, proves that the uniqueness holds in the quoted setting. However, our extra assumption for the uniqueness is certainly verified under the integrability assumption on the gradient of the initial data. Hence, the set of solutions enjoying the uniqueness is not empty.