Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations

TL;DR

This paper proves the existence of unique strong solutions for a coupled system of a deforming elastic beam and Navier-Stokes fluid flow, handling large initial data and geometric changes.

Contribution

It establishes the existence of strong solutions for a hyperbolic-Navier-Stokes coupled system with large initial data and geometric degeneracy, introducing novel a-priori estimates.

Findings

Existence of unique strong solutions up to geometric degeneracy.

A-priori estimates on the time-derivative of the coupled solution.

Application of Ladyzhenskaya-type estimates to the differentiated system.

Abstract

A perfectly elastic beam is situated on top of a two dimensional fluid canister. The beam is deforming in accordance to an interaction with a Navier-Stokes fluid. Hence a hyperbolic equation is coupled to the Navier-Stokes equation. The coupling is partially of geometric nature, as the geometry of the fluid domain is changing in accordance to the motion of the beam. Here the existence of a unique strong solution for large initial data and all times up to geometric degeneracy is shown. For that an a-priori estimate on the time-derivative of the coupled solution is introduced. For the Navier-Stokes part it is a borderline estimate in the spirit of Ladyzhenskaya applied directly to the in-time differentiated system.