Existence of Strong Solutions for a Perfect Elastic Beam Interacting with Navier–Stokes Equations

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 critical estimate in the spirit of Ladyzhenskaya applied directly to the in-time differentiated system.