Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations
type: Item
instance of: scholarly article
title: Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations
zbMATH Open document ID: 1235.35212
author: Shixiang Ma
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-01-04T00:00:00Z
review text: The one-dimensional model to the ideal polytropic gas is considered. The solutions to the Euler equations \[ \begin{cases} \frac{\partial v}{\partial t}-\frac{\partial u}{\partial x}=0,\quad \frac{\partial u}{\partial t}+\frac{\partial p}{\partial x}=0,\\ \frac{\partial }{\partial t}\left(e+\frac{u^2}{2}\right)+\frac{\partial }{\partial x}(pu)=0,\quad x\in \mathbb{R},\;t>0 \end{cases} \] and to the compressible Navier-Stokes equations \[ \begin{cases} \frac{\partial v}{\partial t}-\frac{\partial u}{\partial x}=0,\quad \frac{\partial u}{\partial t}+\frac{\partial p}{\partial x}=\mu\frac{\partial }{\partial x}\left(\frac{1}{v}\frac{\partial u}{\partial x}\right),\\ \frac{\partial }{\partial t}\left(e+\frac{u^2}{2}\right)+\frac{\partial }{\partial x}(pu)= k\frac{\partial }{\partial x}\left(\frac{1}{v}\frac{\partial\theta}{\partial x}\right)+ \mu\frac{\partial }{\partial x}\left(\frac{u}{v}\frac{\partial u}{\partial x}\right) ,\quad x\in \mathbb{R},\;t>0 \end{cases} \] are compared. Here \(v\) is the specific volume, \(u\) is the velocity, \(\theta\) is the temperature, \(p\) is the pressure, \(e\) is the internal energy. Let \((V,U,\Theta)\) be a contact discontinuity (at \(x=0\)) solution to the Euler system. Then there exists a sequence of smooth solutions to the Navier-Stokes system which converges to \((V,U,\Theta)\) at a rate of \(k^{3/4}\) as \(k\) tends to zero provided that the coefficient \(\mu\) tends to zero with order not lower than \(k.\)
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35Q31
Mathematics Subject Classification ID: 76N99
zbMATH DE Number: 5994361
zbMATH Keywords: compressible Navier-Stokes equations
zbMATH Keywords: compressible Euler system
zbMATH Keywords: zero dissipation limit
zbMATH Keywords: contact discontinuity
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.10.010
OpenAlex ID: W2011700411
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DOI: 10.1016/J.JMAA.2011.10.010
description: scientific article; zbMATH DE number 5994361