  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.
