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In this paper, we study the nonlinear stability of the composite waveconsisting of planar rarefaction and planar contact waves for viscousconservation laws with degenerate flux under multi-dimensional periodicperturbations. To the level of our knowledge, it is the first stability resultof the composite wave for conservation laws in several dimensions. Moreover,the perturbations studied in the present paper are periodic, which keepconstantly oscillating at infinity. Suitable ansatz is constructed to overcomethe difficulty caused by this kind of perturbation and delicate estimates aredone on zero and non-zero modes of perturbations. We obtain satisfactory decayrates for zero modes and exponential decay rates for non-zero modes.

Nonlinear stability of the composite wave of planar rarefaction waves  and planar contact waves for viscous conservation laws with non-convex flux  under multi-dimensional periodic perturbations