Archive for Rational Mechanics and Analysis
, Volume 133, Issue 3, pp 249298
First online:
Stability of rarefaction waves in viscous media
 Anders SzepessyAffiliated withRoyal Institute of TechnologyIndiana University
 , Kevin ZumbrunAffiliated withRoyal Institute of TechnologyIndiana University
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We study the timeasymptotic behavior of weak rarefaction waves of systems of conservation laws describing onedimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, “Burgers” rarefaction wave, for initial perturbations w _{0} with small mass and localized as w _{0}(x)=\(\mathcal{O}(x^{  1} )\)
The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the HopfCole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error.
This pointwise method has been used successfully in studying stability of shock and constantstate solutions. New features in the rarefaction case are timevarying coefficients in the linearized equations and error waves of unbounded mass \(\mathcal{O}\) (log (t)). These “diffusion waves” have amplitude \(\mathcal{O}\)(t ^{1/2}logt) in linear degenerate transversal fields and \(\mathcal{O}\)(t ^{1/2}logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.
 Title
 Stability of rarefaction waves in viscous media
 Journal

Archive for Rational Mechanics and Analysis
Volume 133, Issue 3 , pp 249298
 Cover Date
 199609
 DOI
 10.1007/BF00380894
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Anders Szepessy ^{(1)} ^{(2)}
 Kevin Zumbrun ^{(1)} ^{(2)}
 Author Affiliations

 1. Royal Institute of Technology, Stockholm
 2. Indiana University, Bloomington, Indiana