# Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions

## TL;DR

This paper proves the existence and higher regularity of weak solutions for a 3D fluid interacting with a non-linear shell, extending the theory to complex shell models and introducing new analytical tools.

## Contribution

It extends weak solution existence theory to non-linear Koiter shell models and develops new tools for analyzing fluid-structure interactions.

## Key findings

- Higher regularity of shell displacement beyond energy estimates
- A generalized Aubin-Lions compactness result for fluid-structure interactions
- Existence of weak solutions independent of approximation methods

## Abstract

We study the unsteady incompressible Navier-Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter type. This leads to a coupled system of non-linear PDEs where the moving part of the boundary is an unknown of the problem. The known existence theory for weak solutions is extended to non-linear Koiter shell models. We introduce a-priori estimates that reveal higher regularity of the shell displacement beyond energy estimates. These are essential for non-linear Koiter shell models, since such shell models are non-convex (w.r.t.\ terms of highest order). The estimates are obtained by introducing new analytical tools that allow to exploit dissipative effects of the fluid for the (non-dissipative) solid. The regularity result depends on the geometric constitution alone and is independent of the approximation procedure; hence it holds for arbitrary weak solutions. The developed tools are further used to introduce a generalized Aubin-Lions type compactness result suitable for fluid-structure interactions.