Existence of weak solutions for incompressible fluid-Koiter shell interactions with Navier slip boundary condition

TL;DR

This paper proves the existence of global weak solutions for a complex 3D fluid-structure interaction involving incompressible Navier-Stokes fluid and a nonlinear elastic Koiter shell with Navier slip boundary conditions, addressing key mathematical challenges.

Contribution

It introduces novel approximation and compactness methods for the fluid-shell system with slip boundary conditions, enabling analysis of nonlinear Koiter shells in 3D.

Findings

Existence of weak solutions up to shell self-intersection

Handling of slip boundary conditions in a 3D setting

Strong convergence results for shell displacement derivatives

Abstract

We study a three-dimensional fluid-structure interaction problem describing the motion of an incompressible, viscous fluid coupled with a deformable elastic shell of Koiter type that forms part of the fluid boundary. The fluid motion is governed by the incompressible Navier--Stokes equations posed on a time-dependent domain, while the shell evolution is described by a nonlinear elastic model. At the fluid-structure interface, we impose Navier slip boundary conditions, allowing for tangential slip penalized by friction. Our main result establishes the global-in-time existence of weak solutions up to the first possible self-intersection of the shell, for arbitrarily large initial data with finite energy. The analysis is carried out in a fully three-dimensional setting and addresses the major mathematical challenges arising from the moving domain, the geometric nonlinearity of the shell, and the reduced regularization induced by the slip boundary condition. The proof relies on a careful construction of suitable approximation schemes, novel compactness arguments adapted to the slip framework, and a new extension operator for divergence-free test functions compatible with the fluid-shell coupling. As a further contribution, we provide a direct approach to the strong convergence of second-order spatial derivatives of the shell displacement, which allows us to treat nonlinear Koiter shell models within the same framework.