# Weak-strong uniqueness for the compressible fluid-rigid body interaction

## TL;DR

This paper establishes the uniqueness of weak solutions for a coupled system modeling the interaction between a compressible viscous fluid and a moving rigid body, under certain conditions, using a relative energy method.

## Contribution

It introduces a novel approach to prove weak-strong uniqueness for fluid-rigid body interaction systems by defining a suitable relative energy and employing a coordinate transformation.

## Key findings

- Weak solutions coincide with strong solutions under certain conditions.
- The transformation used in the proof must be an identity, ensuring uniqueness.
- The method extends the understanding of fluid-structure interaction models.

## Abstract

In this work we study the coupled system of partial and ordinary differential equations describing the interaction between a compressible isentropic viscous fluid and a rigid body moving freely inside the fluid. In particular the position and velocity of the rigid body in the fluid are unknown and the motion of the rigid body is driven by the normal stress forces of the fluid acting on the boundary of the body. We prove that the strong solution, which is known to exist under certain smallness assumptions, is unique in the class of weak solutions to the problem. The proof relies on a correct definition of the relative energy, to use this tool we then have to introduce a change of coordinates to transform the strong solution to the domain of the weak solution in order to use it as a test function in the relative energy inequality. Estimating all arising terms we prove that the weak solution has to coincide with the transformed strong solution and finally that the transformation has to be in fact an identity.