We generalize the work of Oh & Zumbrun and Serre on spectral stability of spatially periodic traveling waves of systems of viscous conservation laws from the one-dimensional to the multi-dimensional setting. Specifically, we extend to multi-dimensions the connection observed by Serre between the linearized dispersion relation near zero frequency of the linearized equations about the wave and the homogenized system obtained by slow modulation (WKB) approximation. This may be regarded as partial justification of the WKB expansion; an immediate consequence is that hyperbolicity of the multi-dimensional homogenized system is a necessary condition for stability of the waves. As pointed out by Oh & Zumbrun in one dimension, the description of the low-frequency dispersion relation is also a first step in the determination of time-asymptotic behavior.
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Myunghyun Oh, Kevin Zumbrun, Low-Frequency Stability Analysis of Periodic Traveling-Wave Solutions of Viscous Conservation Laws in Several Dimensions. Z. Anal. Anwend. 25 (2006), no. 1, pp. 1–21DOI 10.4171/ZAA/1275