Energy Equality and Uniqueness of Weak Solutions of a “Viscous Incompressible Fluid + 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 Gérard-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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L^{2} $$\end{document} 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.