Compressible Fluids Interacting with a Linear-Elastic Shell

Abstract

We study the Navier-Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter's elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies γ >

A linearized viscous, compressible flow-plate interaction with non-dissipative coupling

“…In this paper we consider a possibly viscous compressible fluid flow in 3-D, interacting with a 2-D elastic structure. The recent works [9,28] deal with the analogous system to (3.1)-(??) in the fully nonlinear (isentropic case), but focus only on existence of weak solutions, whereas [12,13] deal with viscous incompressible fluids.…”

“…The surveys [18,22] provide a nice overview of the modeling, wellposedness, and long-time behavior results for the family of dynamics described above. In any analysis, allowing compressibility yields additional variables, and, as a result, well-posedness is not obtained straightforwardly [9,28]. The essential difficulty lies in showing the range condition of a generator, since one has to address this additional density/pressure component.…”

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

“…Using the above estimate and the definition of X (see (3.18)) it follows that ∇X − I 3 L ∞ (0,∞;W 1,q (F )) 9 ∇X 0 − I 3 W 2,q (F ) 9 + C ∇v L p β (0,∞;W 1,q (F )) 9 CR. (4.79)…”

“…Concerning compressible fluids interacting with plate/beam equations through boundary of the fluid domain, there are only few results available in the literature. Global existence of weak solutions until the structure touches the boundary of the fluid domain were proved in [19,9]. Local in time existence of strong solutions in the corresponding 2D/1D case was recently obtained in [33].…”

Eight(y) mathematical questions on fluids and structures

“…A uniqueness result has been obtained by Glass and Sueur [128] (both when the fluid is governed by the Euler equations or the Navier-Stokes equations). It is also worth mentioning that fluid-structure interaction problems have been con-sidered for compressible fluids in [41][42][43]46] and stabilisation or control issues have been tackled e.g. in [11,12,44,280].…”