Archive for Rational Mechanics and Analysis

, Volume 133, Issue 3, pp 249-298

First online:

Stability of rarefaction waves in viscous media

  • Anders SzepessyAffiliated withRoyal Institute of TechnologyIndiana University
  • , Kevin ZumbrunAffiliated withRoyal Institute of TechnologyIndiana University

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We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional 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 Hopf-Cole 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 constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass \(\mathcal{O}\) (log (t)). These “diffusion waves” have amplitude \(\mathcal{O}\)(t -1/2logt) in linear degenerate transversal fields and \(\mathcal{O}\)(t -1/2logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.