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Nonlinear Stability of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Dimensions One and Two
Johnson, Mathew A.
Johnson, Mathew A.
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Abstract
Extending results of Oh and Zumbrun in dimensions $d\geq3$, 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 $d\geq3$ 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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This is the published version, also available here: http://dx.doi.org/10.1137/100781808.
Date
2011-01-01
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Society for Industrial and Applied Mathematics
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Keywords
periodic traveling waves, Bloch decomposition, modulated waves
Citation
Johnson, Mathew A. & Zumbrun, Kevin. "Nonlinear Stability of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Dimensions One and Two." SIAM J. Appl. Dyn. Syst., 10(1), 189–211. (23 pages). http://dx.doi.org/10.1137/100781808.