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

Maity, Debayan; Takahashi, Takéo

doi:10.48550/arxiv.2006.00488

Cited by 1 publication

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“…Results concerning the long-time existence of weak solutions about structure interactions with heat conducting are missing so far -even in the incompressible case. The existence of a unique local-in-time strong solution to compressible Navier-Stokes-Fourier system coupled with a damped linear plate equation has been established very recently in [34].…”

Section: Introductionmentioning

confidence: 99%

Navier-Stokes-Fourier fluids interacting with elastic shells

Breit¹,

Schwarzacher²

2021

Preprint

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We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a nonlinear flexible shell. The fluid is described by the full Navier-Stokes-Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality. Contents 1. Introduction 1.1. State of art 1.2. The model 1.3. Mathematical strategy 1.4. Overview of the paper 2. Preliminaries 2.1. Structural and constitutive assumptions 2.2. Function spaces on variable domains 2.3. Extensions on variable domains 2.4. Convergence in variable domains. 2.5. Weak solutions and main theorem 3. Equations for density and temperature in variable domains 3.1. The continuity equation 3.2. The internal energy equation 4. Construction of an approximate solution 4.1. Galerkin approximation 4.2. Total energy balance 4.3. Compactness of ∂ t η N 4.4. Compactness of the shell energy 5. Construction of a solution. 5.1. The limit system for ε → 0 5.2. Compactness of the shell energy 5.3. Proof of Theorem 2.16. References

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Section: Introductionmentioning

confidence: 99%