# Compressible fluids interacting with a linear-elastic shell

## TL;DR

This paper proves the existence of weak solutions for a coupled system of compressible Navier--Stokes equations and a linear-elastic shell, extending incompressible results to the compressible case with specific adiabatic exponent conditions.

## Contribution

It establishes the existence of weak solutions for compressible fluid-shell interaction systems with a linearized elastic shell boundary, under certain conditions on the adiabatic exponent.

## Key findings

- Existence of weak solutions for $\gamma > 12/7$ in 3D
- Solution exists until boundary self-intersection occurs
- Extends incompressible shell interaction results to compressible fluids

## Abstract

We study the Navier--Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter's elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies $\gamma>\frac{12}{7}$ ($\gamma>1$ in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in [D. Lengeler, M. \Ruzicka, Weak Solutions for an Incompressible Newtonian Fluid Interacting with a Koiter Type Shell. Arch. Ration. Mech. Anal. 211 (2014), no. 1, 205--255] on incompressible Navier--Stokes equations.