Time-periodic weak solutions for an incompressible Newtonian fluid interacting with an elastic plate

TL;DR

This paper proves the existence of time-periodic weak solutions for a fluid-structure interaction problem involving an incompressible fluid and an elastic plate, using a novel fixed-point approach and new a-priori estimates.

Contribution

It introduces a two fixed-point methodology and develops new a-priori estimates for coupled fluid-structure systems with time-periodic geometry.

Findings

Existence of time-periodic weak solutions for given geometry.

Existence of coupled solutions for all data excluding self-intersection.

A-priori estimates exploit fluid dissipation for solid deformation.

Abstract

Under the action of a time-periodic external forces we prove the existence of at least one time-periodic weak solution for the interaction between a three-dimensional incompressible fluid, governed by the Navier- Stokes equation and a two dimensional elastic plate. The challenge is that the Eulerian domain for the fluid changes in time and is a part of the solution. We introduce a two fixed-point methodology: First we construct a time-periodic solutions for a given variable time-periodic geometry. Then in a second step a (set-valued) fixed point is performed w.r.t.\ the geometry of the domain. The existence relies on newly developed a-priori estimates applicable for both coupled and uncoupled variable geometries. Due to the expected weak regularity of the solutions such Eulerian estimates are unavoidable. Note in particular, that only the fluid is assumed to be dissipative. But the here produced a-priori estimates show that its possible to exploit the dissipative effects of the fluid also for the solid deformation. The existence of periodic solutions for a given geometry is valid for arbitrary large data. The existence of periodic coupled solutions to the fluid-structure interaction is valid for all data that excludes a self-intersection a-priori.