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

Abstract

We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier–Stokes–Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an Lp‐Lq setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the scriptR‐sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.

Analysis of a Simplified Model of Rigid Structure Floating in a Viscous Fluid

“…On the other hand, using an interpolation inequality one has following estimate (see, for instance estimate, (6.13) of [10])…”

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

“…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].…”

“…With this combined change of variables, we reformulate the problem in the reference domain F. In most of the existing literature, a geometric change of variables via the displacement of the fluid-structure interface is used to rewrite the problem in a fixed domain ( [29,22,5,30]). However, in the context of compressible fluid-structure systems, it is more convenient to use a Lagrangian (see for instance [24]) or a combination of geometric and Lagrangian change of coordinates ( [25]). In fact, such transformations allow us to use basic contraction mapping theorem.…”

“…This yields that the system (3.25) admits a unique solution v ∈ W 1,2 p,q ((0, T ); F) 3 . In order to show that the estimate (3.36) holds with a constant independent of T, we can proceed as [24,Proposition 2.2].…”

“…For more details about the proof of the above estimates, we refer to [24,Lemma 3.19]. From (4.12) and ( To details on the proof of (4.87) can be found in [24,Proposition 3.20].…”