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Under natural spectral stability assumptions motivated by previousinvestigations of the associated spectral stability problem, we determine sharp$L^p$ estimates on the linearized solution operator about a multidimensionalplanar periodic wave of a system of conservation laws with viscosity, yieldinglinearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d\ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and$L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can ingeneral be rather complicated, involving both convective (i.e., wave-like) anddiffusive effects.

Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions