We generalize work of Oh & Zumbrun and Serre on spectral stability ofspatially periodic traveling waves of systems of viscous conservation laws fromthe one-dimensional to the multi-dimensional setting. Specifically, we extendto multi-dimensions the connection observed by Serre between the linearizeddispersion relation near zero frequency of the linearized equations about thewave and the homogenized system obtained by slow modulation (WKB)approximation. This may be regarded as partial justification of the WKBexpansion; an immediate consequence is that hyperbolicity of themulti-dimensional homogenized system is a necessary condition for stability ofthe wave. As pointed out by Oh & Zumbrun in one dimension, description of thelow-frequency dispersion relation is also a first step in the determination oftime-asymptotic behavior.
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