Energy Equality and Uniqueness of Weak Solutions of a "Viscous Incompressible Fluid plus Rigid Body" System with Navier Slip-with-Friction Conditions in a 2D Bounded Domain

The existence of weak solutions to the viscous incompressible fluid + rigid body system with Navier slip-with-friction conditions in a 3D bounded domain has been recently proved by Gerard-Varet and Hillairet (Commun Pure Appl Math 67(12):2022-2076, 2014). In 2D for a fluid alone (without any rigid body) it is well-known since Leray that weak solutions are unique, continuous in time with L2 regularity in space and satisfy the energy equality. In this paper we prove that these properties also hold for the 2D viscous incompressible fluid + rigid body system with Navier slip-with-friction conditions.