Well-posedness theorems in fluid-structure interaction: perfectly elastic shells

TL;DR

This paper proves local-in-time existence and uniqueness of solutions for a fluid-structure interaction system involving a 3D incompressible fluid and a 2D elastic shell, introducing new analytical techniques beyond standard methods.

Contribution

It develops a novel approach to establish well-posedness for fluid-structure systems with perfectly elastic shells, extending to global solutions in specific 2D cases.

Findings

Constructed a local-in-time unique strong solution for the coupled PDE system.

Extended the solution globally in time for 2D fluid and 1D elastic shell interaction.

Introduced a new estimate for the acceleration of the coupled system.

Abstract

In this work, we consider the interaction of a 3D incompressible fluid with a 2D flexible shell that occupies (a part of) the boundary of the fluid domain. We assume that the shell is perfectly elastic while the fluid is governed by the Navier--Stokes equations. Consequently, damping within the coupled system comes entirely from the parabolic fluid subsystem. Our main result is the construction of a local-in-time unique strong solution to the system of PDEs. Standard techniques from the literature do not apply here. They are restricted to visco-elastic structures, where the corresponding solid phase is parabolic. Our construction relies on a different method built upon a new estimate for the acceleration of the system. In the case of a 2D viscous incompressible fluid interacting with a 1D perfectly elastic shell we can extend the local solution globally in time (until a possible self-intersection of the shell).