The Motion of a Rigid Body in a Viscous Fluid: Results for Strong Solutions, Uniqueness and Integrability Properties

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 datum, proves that the time derivative of the solution belongs to $$L^2(0,T;L^2(\Omega ))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . 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 datum. Hence, the set of solutions enjoying the uniqueness is not empty.