Nonlinear stability and asymptotic behavior of periodic traveling waves of multidimensional viscous conservation laws in dimensions one and two

Abstract. Under natural spectral stability assumptions motivated by previous investigations of the associ-ated 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.