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

Abstract

We consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result [3] where we supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.

Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier–Stokes–Fourier fluid and a damped plate equation

“…We refer to, for instance [23] and references therein for a concise description of recent progress regarding incompressible flows interacting with deformable structure (beam or plate) located on a part of the fluid domain boundary. Moreover, in some recent articles ( [22,5,6]) existence and uniqueness of strong solutions (either local in time or for small initial data) were proved without the additional damping term (i.e., without the term −∆ s ∂ t η) in the beam/plate equation.…”

“…This is achieved thanks to a combination of a geometric change of variables (defined through the initial displacement of the structure) and a Lagrangian change of coordinates. With this combined change of variables, we reformulate the problem in the reference domain F. In most of the existing literature, a geometric change of variables via the displacement of the fluid-structure interface is used to rewrite the problem in a fixed domain ( [29,22,5,30]). However, in the context of compressible fluid-structure systems, it is more convenient to use a Lagrangian (see for instance [24]) or a combination of geometric and Lagrangian change of coordinates ( [25]).…”

An $$L^2$$ approach to viscous flow in the half space with free elastic surface

“…For local-in-time strong solutions in this latter setting we refer to Beiraõ da Veiga [3]. Moreover, strong solutions are found by Badra and Takahashi [2] using a non-analytic semigroup of Gevrey class. Finally, in [10] the present authors construct weak solutions to the fluid-structure interaction problem of a viscous fluid coupled with a damped elastic plate under the nonlinear Coulomb boundary friction condition.…”

On an existence theory for a fluid-beam problem encompassing possible contacts

“…When no viscosity is added and in case the dynamics of the structure displacement is governed by a membrane equation, existence and uniqueness of a local strong solution is obtained in [8]. The beam case with no additional viscosity is investigated in [2], where existence of strong solution locally in time (or for small data) is proved but with a gap between the regularity of the initial conditions and the propagated regularity of the structure displacement. Existence of weak solutions is obtained in [4] for 3D-2D coupling where the structure behaviour is described by a viscous plate equation and in [6,14] in the non-viscous case.…”