$$L^{p}$$ Theory for the Interaction Between the Incompressible Navier–Stokes System and a Damped Plate

Abstract

We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary can deform. We assume that the corresponding displacement follows a damped beam equation. Our main results are the existence and uniqueness of strong solutions for the corresponding fluid-structure interaction system in an L p -L q setting for small times or for small data. An important ingredient of the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped plate equation. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.

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

“…(1) Note that, in Theorem 1.1 we do not need initial displacement of the plate η 0 1 to be zero. This is a difference with respect to previous works, for instance [33] or our previous work [30] (with an incompressible fluid). Here we manage to handle this case by modifying our change of variables (see Section 3.1).…”

“…In the context of fluid-solid interaction problems, there are only few articles available in the literature that studies well-posedness in an L p − L q framework. Let us mention [20,32] (viscous incompressible fluid and rigid bodies), [25,31,24] (viscous compressible fluid and rigid bodies) and [30,13] (viscous incompressible fluid interacting with viscoelastic structure located at the boundary of the fluid domain). In fact, this article is a compressible counterpart of our previous work [30].…”

“…Let us mention [20,32] (viscous incompressible fluid and rigid bodies), [25,31,24] (viscous compressible fluid and rigid bodies) and [30,13] (viscous incompressible fluid interacting with viscoelastic structure located at the boundary of the fluid domain). In fact, this article is a compressible counterpart of our previous work [30].…”

Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation*

“…The work [22] is devoted to the global in time existence and uniqueness of solutions for the case of a damped beam equation and investigates, in particular, the possible contacts between the structure and the bottom of the domain. We also mention [17,32] where the authors obtain the existence of strong solutions within an 'L p − L q ' framework instead of a 'Hilbert' space framework.…”

Controllability of a Stokes system with a diffusive boundary condition

“…The vector fields n corresponds to the unit exterior normal to Ω ℓ(t) . This system has been studied by many authors: [15] (existence of weak solutions), [7], [38], [23] and [41] (existence of strong solutions), [51] (stabilization of strong solutions), [4] (stabilization of weak solutions around a stationary state). There are also some works in the case δ = 0, that is without damping on the beam equation: the existence of weak solutions is proved in [22] and in [45] (see also [60]).…”