Stability of rarefaction waves in viscous media |
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Authors: | Anders Szepessy Kevin Zumbrun |
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Institution: | (1) Royal Institute of Technology, Stockholm;(2) Indiana University, Bloomington, Indiana |
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Abstract: | 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)=
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
(log (t)). These diffusion waves have amplitude
(t
-1/2logt) in linear degenerate transversal fields and
(t
-1/2logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof. |
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Keywords: | |
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