Existence of Global Strong Solutions to a Beam–Fluid Interaction System

Abstract

Abstract. We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.

Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

“…In this study, the mid-surface of the structure is not flat anymore and existence of weak solutions is obtained. More recently, existence of a unique global-in-time solution for a 2D/1D coupling with a damped beam has been proven in [15]. This result includes that there is no contact between the structure and the bottom boundary and the additional viscosity of the beam is a key ingredient of the proof.…”

“…• (C α,γ ) the case for which α > 0, γ > 0 and β = δ = 0 ; this last one models again a beam in flexion equation but with additional viscosity (already considered in [19,15]):…”

Gevrey Regularity for a System Coupling the Navier--Stokes System with a Beam Equation

“…Several works analyze such a model: [6] (existence of weak solutions), [4], [10] and [8] (existence of strong solutions), [12] (stabilization of strong solutions), [2] (stabilization of weak solutions around a stationary state). In all these works, the damping term −δ∂tssη is crucial.…”

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

“…For (PBC), the existence of global strong solutions without smallness assumptions on the initial data is proved in [11]. For a wide range of beam equations, depending on the positivity of the coefficients β, γ, α, the existence of local-in-time strong solutions without smallness assumptions is proved in [11,12].…”

“…The term 'strong solutions' is related to the spacial regularity of the solution, which is typically, for the fluid, H 2 . In the semigroup terminology of evolution equations, the solutions considered in [2,5,11,12,18,31] correspond to strict solutions in L 2 (see Definition 14 in the appendix). Motivated by the stabilization of (1.1) in a neighbourhood of a periodic solution, we prove the existence of a time-periodic strict solution in C 0 for (1.1) with Hölder regularity in time.…”