Nonlinear Stability of Periodic Traveling-Wave Solutions of Viscous Conservation Laws in Dimensions One and Two | Zendy

Mathew A. Johnson | Zendy; Kevin Zumbrun | Zendy

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Nonlinear Stability of Periodic Traveling-Wave Solutions of Viscous Conservation Laws in Dimensions One and Two

Author(s) -

Mathew A. Johnson,

Kevin Zumbrun

Publication year - 2011

Publication title -

siam journal on applied dynamical systems

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.218

H-Index - 61

ISSN - 1536-0040

DOI - 10.1137/100781808

Subject(s) - nonlinear system , conservation law , stability (learning theory) , mathematics , mathematical analysis , traveling wave , representation (politics) , diffusion , function (biology) , physics , law , computer science , quantum mechanics , machine learning , evolutionary biology , politics , biology , political science , thermodynamics

Extending results of Oh and Zumbrun in dimensions $d\ge 3$, we establishnonlinear stability and asymptotic behavior of spatially-periodictraveling-wave solutions of viscous systems of conservation laws in criticaldimensions $d=1,2$, under a natural set of spectral stability assumptionsintroduced by Schneider in the setting of reaction diffusion equations. The keynew steps in the analysis beyond that in dimensions $d\ge 3$ are a refinedGreen function estimate separating off translation as the slowest decayinglinear mode and a novel scheme for detecting cancellation at the level of thenonlinear iteration in the Duhamel representation of a modulated periodic wave.

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