Under natural spectral stability assumptions motivated by previous
investigations of the associated spectral stability problem, we determine sharp
Lp estimates on the linearized solution operator about a multidimensional
planar periodic wave of a system of conservation laws with viscosity, yielding
linearized L1∩Lp→Lp stability for all p≥2 and dimensions d≥1 and nonlinear L1∩Hs→Lp∩Hs stability and
L2-asymptotic behavior for p≥2 and d≥3. The behavior can in
general be rather complicated, involving both convective (i.e., wave-like) and
diffusive effects