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arXiv · 1001.0048

Nonlinear stability of periodic traveling wave solutions of viscous conservation laws in dimensions one and two

Mathew A. Johnson·Kevin Zumbrun
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Abstract

Extending results of Oh and Zumbrun in dimensions $d\ge 3$, we establish nonlinear stability and asymptotic behavior of spatially-periodic traveling-wave solutions of viscous systems of conservation laws in critical dimensions $d=1,2$, under a natural set of spectral stability assumptions introduced by Schneider in the setting of reaction diffusion equations. The key new steps in the analysis beyond that in dimensions $d\ge 3$ are a refined Green function estimate separating off translation as the slowest decaying linear mode and a novel scheme for detecting cancellation at the level of the nonlinear iteration in the Duhamel representation of a modulated periodic wave.

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Mathew A. Johnson, Kevin Zumbrun. 2010-01-07. Nonlinear stability of periodic traveling wave solutions of viscous conservation laws in dimensions one and two. https://arxiv.org/abs/1001.0048

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