A regularity result for a solid–fluid system associated to the compressible Navier–Stokes equations

Abstract

In this paper we deal with a fluid-structure interaction problem for a compressible fluid and a rigid structure immersed in a regular bounded domain in dimension 3. The fluid is modelled by the compressible Navier-Stokes system in the barotropic regime with no-slip boundary conditions and the motion of the structure is described by the usual law of balance of linear and angular moment.The main result of the paper states that, for small initial data, we have the existence and uniqueness of global smooth solutions as long as no collisions occur. This result is proved in two steps; first, we prove the existence and uniqueness of local solution and then we establish some a priori estimates independently of time. RésuméDans cet article, nous considérons un problème d'interaction fluide-structure entre un fluide compressible et une structure rigide évoluant à l'intérieur d'un domaine borné et régulier en dimension 3. Le fluide est décrit par le système de Navier-Stokes compressible barotrope avec des conditions de non-glissement sur le bord et le mouvement de la structure est régi par les lois de conservation des moments linéaire et angulaire.Nous montrons, pour des données initiales petites, l'existence et l'unicité de solutions globales régulières tant qu'il n'y a pas de chocs. Ce résultat est obtenu en deux temps ; tout d'abord, nous prouvons l'existence et l'unicité de solutions locales puis nous démontrons des estimations a priori indépendamment du temps.

Mathematical analysis of the motion of a rigid body in a compressible Navier–Stokes–Fourier fluid

“…Since the domain of the fluid equation is one of the unknowns, we first rewrite the system in a fixed spatial domain. This can be achieved either by a “geometric” change of variables (see ), by using Lagrangian coordinates (see ) or by combining these two change of coordinates (see ). In the present work, we found more convenient to use Lagrangian variables.…”

“…Concerning three‐dimensional models, global existence of weak solutions for compressible fluid and rigid body interaction problems was studied by Desjardins and Esteban and Feireisl . Boulakia and Guerrero proved global existence and uniqueness of strong solutions for small initial data within the Hilbert space framework. Hieber and Murata proved local in time existence and uniqueness in a ‐ setting.…”

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

“…Observe that (ρ, 0, 0) is a stationary solution of system (1), (4) and 7- (8). Finally, we introduce the initial conditions ρ(0, •) = ρ 0 in Ω F (0), u(0, •) = u 0 in Ω F (0) (9) and ξ(0, •) = ξ 0 in Ω S (0), ξ t (0, •) = ξ 1 in Ω S (0) (10) which satisfy ρ 0 ∈ H 3 (Ω F (0)), ρ 0 ρ min > 0 in Ω F (0), u 0 ∈ H 4 (Ω F (0)), ξ 0 ∈ H 3 (Ω S (0)), ξ 1 ∈ H 2 (Ω S (0)). (11) together with the following compatibility conditions:…”

“…Concerning compressible fluids, the global existence of weak solutions for the interaction with a rigid structure is obtained in [8] (with γ 2) and in [11] (with γ > N/2). Moreover, in [4], the existence of global regular solutions is proved for small initial data. At last, for the interaction between a compressible fluid and an elastic structure, [2] proves the global existence of a weak solution in 3D for γ > 3/2.…”

Existence of weak solutions for a Bingham fluid-rigid body system

“…We can quote for instance [11,13,19,24,26,35,36,52,55,56,57], etc. Some works deal with different fluids [30,39,48] (incompressible perfect fluid), [5,6,14,18] (viscous compressible fluid), [21] (viscous multipolar fluid), [20,27] (incompressible non-Newtonian fluid). Let us also mention some results for the Navier-Stokes system but with other types of boundary conditions: [1,9,29]).…”