A bound from below for the temperature in compressible Navier-Stokes equations
type: Item
instance of: scholarly article
title: A bound from below for the temperature in compressible Navier-Stokes equations
zbMATH Open document ID: 1173.35100
author: Antoine Mellet
author: Alexis F. Vasseur
published in: Monatshefte für Mathematik
publication date: 2009-06-30T00:00:00Z
review text: The paper considers the following full system of compressible Navier-Stokes equations for heat conducting fluid, filling a bounded three-dimensional domain \(\Omega\) in a time interval \([0,T]\): \[ \partial_t\rho + \text{div}(\rho u)=0, \] \[ \partial_t(\rho\theta) + \text{div}(\rho u \theta) - \text{div}(\kappa \nabla\theta)+R\rho\theta \operatorname{div}u=2\mu|Du|^2 + \lambda |\operatorname{div}u|^2, \] \[ \partial_t(\rho u) + \text{div}(\rho u u) - \text{div}(2\mu Du + \lambda \operatorname{div}u Id)+\nabla P(\rho\theta)=0, \] representing respectively the continuity, the temperature and the momentum equation, where \(\rho\geq 0\), \(\theta\geq 0\) and \(u(t,x)\in\mathbb R^3\) denotes respectively the density, temperature and velocity field of the fluid, and the pressure law is \[ P(\rho\theta)=p_e(\rho) + R\rho\theta, \] with \(\mu,\lambda\) the Lamé coefficients (depending on \(\rho\) and \(\theta\)). This PDE system is ended with the non-slip condition for the velocity and homogeneous Neumann condition for the temperature. Finally, initial conditions are denoted by \(\rho_0\), \(\theta_0\) and \(u_0\) respectively. Assuming the existence of the so-called admisible solutions, which satisfying the following energy (or entropy) inequality for such equations: \[ \frac{d}{dt}\int_\Omega \rho \varphi(\theta) - \int_\Omega \mu\varphi'(\theta)|Du|^2 - \int_\Omega \lambda \varphi'(\theta)|\operatorname{div}u|^2 + \int_\Omega \kappa\varphi ''(\theta)|\nabla\theta|^2 \leq -R \int_\Omega \rho\theta \varphi'(\theta)\operatorname{div}u \] for some appropiate functions \(\varphi\), under reasonable assumptions on the coefficients \[ \mu>0,\quad 2\mu+3\lambda\geq \nu(\theta)>0 \quad \forall \theta, \] \[ \nu(\theta)\geq C\theta, \quad \kappa(\theta)\geq \kappa_{\min}>0, \quad \text{for small }\theta, \] and on the initial data \[ \int_\Omega \rho_0 [\ln (1/\rho_0)]_+ < \infty \] (which imply in particular that the initial temperature may not vanish on an open set), the pape…
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76N10
zbMATH DE Number: 5571615
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: compressible flows
zbMATH Keywords: heat conducting fluids
reviewed by: Francisco Guillén-González
MaRDI profile type: Publication
full work available at URL: s00605-008-0021-y
OpenAlex ID: W2147362162
cites work: LPBounds of solutions of reaction-diffusion equations
cites work: Q5702675
cites work: On the motion of a viscous, compressible, and heat conducting fluid
cites work: \(L^p\) estimates for quantities advected by a compressible flow
cites work: A new proof of partial regularity of solutions to Navier-Stokes equations
DOI: 10.1007/S00605-008-0021-Y
description: scientific article; zbMATH DE number 5571615