Time-periodic weak solutions for the interaction of an incompressible fluid with a linear Koiter type shell under dynamic pressure boundary conditions

TL;DR

This paper proves the existence of time-periodic weak solutions for a fluid-structure interaction problem involving a Navier-Stokes fluid and a linear Koiter shell with dynamic pressure boundary conditions, using novel approximation methods.

Contribution

It introduces new approximation techniques and a-priori estimates to establish the existence of weak time-periodic solutions in a complex fluid-structure interaction model.

Findings

Existence of at least one weak time-periodic solution.

Applicable to fluid-structure systems with dynamic pressure boundaries.

New mathematical methods for analyzing periodic solutions.

Abstract

In many occurrences of fluid-structure interaction time-periodic motions are observed. We consider the interaction between a fluid driven by the three dimensional Navier-Stokes equation and a two dimensional linearized elastic Koiter shell situated at the boundary. The fluid-domain is a part of the solution and as such changing in time periodically. On a steady part of the boundary we allow for the physically relevant case of dynamic pressure boundary values, prominent to model inflow/outflow. We provide the existence of at least one weak time-periodic solution for given periodic external forces that are not too large. For that we introduce new approximation techniques and a-priori estimates.