On the stability of rarefaction wave solutions for viscous \(p\)-system with boundary effect
type: Item
instance of: scholarly article
title: On the stability of rarefaction wave solutions for viscous \(p\)-system with boundary effect
zbMATH Open document ID: 1138.35369
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2004-09-22T00:00:00Z
review text: This paper concerns the large time behavior of solutions of the isentropic compressible Navier-Stokes equations with a \(\gamma\)-law (\(\gamma\geq 1\)) in one space dimension, when the spatial domain is a half-line \(\{\,x>0\,\}\), and with boundary data corresponding to supersonic inflow (that is, the velocity of the fluid in the direction \(x\) is positive, and greater than the sound speed). From the hyperbolic point of view, supersonic inflow means that the two characteristics enter the domain: therefore, the hyperbolic Initial Boundary Value Problem belongs to a rather simple class in which both the density and the velocity should be prescribed at \(x=0\), as is done in the viscous case. This suggests that no boundary layer should be present in the viscous solutions. And this is indeed what is found in the paper under review: viscous solutions with fixed endstates (at \(x=0\) and \(x=+\infty\)) are shown to be asymptotic (when time goes to infinity) to inviscid smooth solutions (rarefaction waves). The proof relies on a reformulation of equations in Lagrangian coordinates and on elementary \(L^2\) \textit{a priori} estimates. The subsonic inflow problem (in which we do expect a boundary layer, because of an outgoing characteristic in the inviscid problem) was studied in an earlier work by \textit{A. Matsumura} and \textit{K. Nishihara} [Commun. Math. Phys. 222, No. 3, 449--474 (2001; Zbl 1018.76038)].
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76N10
zbMATH DE Number: 2101821
zbMATH Keywords: large time behavior
zbMATH Keywords: inflow problem
author: Xiao-ding Shi
MaRDI profile type: Publication
full work available at URL: s10255-003-0109-z
OpenAlex ID: W2006766350
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Wikidata QID: Q126265334
DOI: 10.1007/S10255-003-0109-Z
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description: scientific article; zbMATH DE number 2101821