  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 G\'{e}rard-Varet and Hillairet in \cite{exi:GeH}.
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 $ L^{2} $ 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.
