On the weak uniqueness of "viscous incompressible fluid + rigid body" system with Navier slip-with-friction conditions in a 2D bounded domain

TL;DR

This paper proves that in a 2D bounded domain, the weak solutions of the viscous incompressible fluid coupled with a rigid body with Navier slip-with-friction conditions are unique, continuous in time, and satisfy the energy equality, extending known results from fluid-only cases.

Contribution

It establishes the weak uniqueness, continuity, and energy equality for the 2D fluid-rigid body system with Navier slip conditions, filling a gap in the understanding of such coupled systems.

Findings

Weak solutions are unique in 2D with Navier slip conditions.

Solutions are continuous in time with $L^2$ regularity.

Solutions satisfy the energy equality.

Abstract

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.