Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

International audienceWe consider a three--dimensional viscous incompressible fluid governed by the Navier--Stokes equations, interacting with an elastic plate located on one part of the fluid boundary. We do not neglect the deformation of the fluid domain which consequently depends on the displacement of the structure. The purpose of this work is to study the solutions of this unsteady fluid--structure interaction problem, as the coefficient modeling the viscoelasticity (resp. the rotatory inertia) of the plate tends to zero. As a consequence, we obtain the existence of at least one weak solution for the limit problem (Navier--Stokes equation coupled with a plate in flexion) as long as the structure does not touch the bottom of the fluid cavity

A note on the Trace Theorem for domains which are locally subgraph of a Holder continuous function

The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a H\" older continuous function. More precisely, let $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ and let $\Omega_{\eta}$ be a domain which is locally subgraph of a function $\eta$. We prove that mapping $\gamma_{\eta}:u\mapsto u({\bf x},\eta({\bf x}))$ can be extended by continuity to a linear, continuous mapping from $H^1(\Omega_{\eta})$ to $H^s(\omega)$, $s<\alpha/2$. This study is motivated by analysis of fluid-structure interaction problems.Comment: Networks and Heterogeneous Medi

LpL^p-theory for a fluid-structure interaction model

We consider a fluid-structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in $L^p$-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.Comment: 18 page

Regular solutions of a problem coupling a compressible fluid and an elastic structure

We are interested by the three-dimensional coupling between a compressible viscous fluid and an elastic structure immersed inside the fluid. They are contained in a fixed bounded set. The fluid motion is modelled by the compressible Navier-Stokes equations and the structure motion is described by the linearized elasticity equation. We establish the local in time existence and the uniqueness of regular solutions for this model. We emphasize that the equations do not contain extra regularizing term. The result is proved by first introducing a problem linearized and by proving that it admits a unique regular solution. The regularity is obtained thanks to successive estimates on the unknowns and their derivatives in time and thanks to elliptic estimates. At last, a fixed point theorem allows to prove the existence and uniqueness of regular solution of the nonlinear problem

Exponential decay properties of a mathematical model for a certain fluid-structure interaction

In this work, we derive a result of exponential stability for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation. In the case that the PDE plate component is rotational inertia-free, one will have that solutions of this fluid-structure PDE system exhibit an exponential rate of decay. By way of proving this decay, an estimate is obtained for the resolvent of the associated semigroup generator, an estimate which is uniform for frequency domain values along the imaginary axis. Subsequently, we proceed to discuss relevant point control and boundary control scenarios for this fluid-structure PDE model, with an ultimate view to optimal control studies on both finite and infinite horizon. (Because of said exponential stability result, optimal control of the PDE on time interval $(0,\infty)$ becomes a reasonable problem for contemplation.)Comment: 15 pages, 1 figure; submitte

Lp theory for the interaction between the incompressible Navier-Stokes system and a damped beam

We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary is moving as a damped beam under the action of the fluid. We prove the existence and uniqueness of global strong solutions for the corresponding fluid-structure interaction system in an Lp-Lq setting. The main point in the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped beam. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator

A global attractor for a fluid--plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main peculiarity of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. We also show that the corresponding linearized system generates exponentially stable $C_0$-semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system.Comment: 18 page