Navier-Stokes-Fourier fluids interacting with elastic shells. (English) Zbl 1525.76081
This article explores the behavior of a compressible heat-conducting fluid within a dynamic domain. The fluid occupies a torus-like region enclosed by an elastic boundary, whose movement is influenced by the fluid’s actions. The governing system arises as a coupling between the Navier-Stokes-Fourier equation and an equation describing the optimal response of the elastic shell to the forces acting upon it.
The main issue addressed in the article is the fact that the domain, where the Navier-Stokes-Fourier system is considered, is changing in time according to the unknown function. The introductory sections of the article provide a clear overview of the tools and techniques required to analyze this setup, including various compactness and embedding results.
The main theorem then states the existence of a weak solution up to the time where, possibly, the domain approaches a self-intersection or the elastic energy of the shell degenerates. The proof itself is based on two-level approximation: the authors add a viscous term into the continuity equation and they regularize the pressure by an additional term. This approximation is solved by means of the Galerkin approach.
The main issue addressed in the article is the fact that the domain, where the Navier-Stokes-Fourier system is considered, is changing in time according to the unknown function. The introductory sections of the article provide a clear overview of the tools and techniques required to analyze this setup, including various compactness and embedding results.
The main theorem then states the existence of a weak solution up to the time where, possibly, the domain approaches a self-intersection or the elastic energy of the shell degenerates. The proof itself is based on two-level approximation: the authors add a viscous term into the continuity equation and they regularize the pressure by an additional term. This approximation is solved by means of the Galerkin approach.
Reviewer: Václav Mácha (Praha)
MSC:
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 76N06 | Compressible Navier-Stokes equations |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74K25 | Shells |
| 35Q30 | Navier-Stokes equations |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
fluid-structure interaction; compressible heat-conducting fluid; weak solution existence; two-level approximation; Galerkin methodReferences:
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