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

Abstract

In this article, we consider a fluid-structure interaction system where the fluid is viscous and compressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier-Stokes system, whereas the structure displacement is described by a wave equation. We show that the corresponding coupled system admits a unique, strong solution for an initial fluid density and an initial fluid velocity in H 3 and for an initial deformation and an initial deformation velocity in H 4 and H 3 respectively. The reference configuration for the fluid domain is a rectangular cuboid with the elastic structure being the top face. We use a modified Lagrangian change of variables to transform the moving fluid domain into the rectangular cuboid and then analyze the corresponding linear system coupling a transport equation (for the density), a heat-type equation, and a wave equation. The corresponding results for this linear system and estimations of the coefficients coming from the change of variables allow us to perform a fixed point argument and to prove the existence and uniqueness of strong solutions for the nonlinear system, locally in time.

On the interaction problem between a compressible viscous fluid and a nonlinear thermoelastic plate

“…Mitra considered a 2D model where a viscoelastic beam interacts with a viscous compressible fluid and obtained a regular solution in [32]. We also mention very recent results for the interaction problem between the full Navier-Stokes-Fourier system and a viscoelastic plate in 3D ( [31]), and the interaction problem between a compressible viscous fluid and a wave equation in 3D ( [30]).…”

Compressible fluids interacting with plates -- regularity and weak-strong uniqueness

“…(3) In [28], the fluid conducts heat and it is governed by the Navier-Stokes-Fourier system, but the same result holds for the barotropic case used in this paper, as it was pointed out in [28,Remark 1.3(5)]. (4) In [27], the structure is actually governed by a wave equation. Nevertheless, the same result can be obtained for the plate equation, both for α = 0 and α > 0, as these cases have more regularity.…”

“…Here, we refer to known results concerning strong and regular solutions to the problem ( 1 stands for Besov spaces (see [1]). ( 2) In [27,29], the plate domain Γ is periodic, i.e. Γ = (R/L 1 Z) × (R/L 2 Z) and Γ = (R/L 1 Z) , respectively, for some L 1 , L 2 > 0, and ρ, u, w are periodic in horizontal coordinates as well.…”

“…Later, Mitra proved the existence of a unique local regular solution in 2D, where the beam is viscoelastic in [29]. Maity and Takahashi obtained a strong solution in L p − L q framework for the interaction of heat-conducting fluid with a viscoelastic plate, and later Maity, Roy and Takahashi obtained a unique local regular solution in 3D [27], where the structure is modeled by a wave equation and doesn't have any viscosity. Trifunović and Wang obtained the existence of a weak solution in 3D in [38], where the structure is modeled by a nonlinear thermoelastic (or just elastic) plate, by constructing a time-continuous splitting approximation scheme that decouples the fluid and the structure.…”

Navier-Stokes-Fourier fluids interacting with elastic shells

“…Eventually, a similar result has been shown by a time-stepping method [42], where the interaction of a compressible fluid with a thermoelastic plate is studied (compare also with with the numeric results from [40]). Results on the short-time existence of strong solutions for compressible fluid models coupled with one-dimensional linear elastic structures can be found in [33,35]. In [2] the author studies an elastic structure (with a regularised elasticity law) which is immersed into a compressible fluid and proves the existence of weak solutions to the underlying system.…”