Existence of time-periodic strong solutions to a fluid–structure system

Jean-Jérôme Casanova

doi:10.3934/dcds.2019136

Abstract

We study a nonlinear coupled fluid–structure system modelling the blood flow through arteries. The fluid is described by the incompressible Navier–Stokes equations in a 2D rectangular domain where the upper part depends on a structure satisfying a damped Euler–Bernoulli beam equation. The system is driven by time-periodic source terms on the inflow and outflow boundaries. We prove the existence of time-periodic strong solutions for this problem under smallness assumptions for the source terms.

Highlights

In this paper we are interested in the existence of time-periodic solutions for a fluid–structure system involving the incompressible Navier–Stokes equations coupled with a damped Euler–Bernoulli beam equation located on a part of the fluid domain boundary
When the system is driven by periodic source terms, related for example to the periodic heartbeat, we expect a periodic response of the system
We prove the existence of time-periodic solutions for the fluid–structure system subject to small periodic impulses on the inflow and outflow boundaries

Summary

Introduction

In this paper we are interested in the existence of time-periodic solutions for a fluid–structure system involving the incompressible Navier–Stokes equations coupled with a damped Euler–Bernoulli beam equation located on a part of the fluid domain boundary. We prove that (A, D(A)) is the infinitesimal generator of an analytic semigroup and that its resolvent is compact At this stage we use the abstract results developed in the appendix to ensure the existence of a time-periodic solution for the linear system. Let us conclude this introduction with a brief history on the existence of time-periodic solutions for the Navier–Stokes equations This question was initially considered in 1960s in [13, 29, 30, 32]. Cρ([0, T ]; X) := {v|[0,T ] | v ∈ Cρ(R; X) is T -periodic}, Hρ(0, T ; X) := {v|[0,T ] | v ∈ Hlρoc(R; X) is T -periodic}

Linear system

Nonlinear system