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Considering the space-periodic perturbations, we prove the time-asymptoticstability of the composite wave of a viscous contact wave and two rarefactionwaves for the Cauchy problem of 1-D compressible Navier-Stokes equations inthis paper. This kind of perturbations keep oscillating at the far field andare not integrable. The key is to construct a suitable ansatz carrying the sameoscillation %eliminating the oscillation of the solution as in\cite{HuangXuYuan2020,HuangYuan2021}, but due to the degeneration of contactdiscontinuity, the construction is more subtle. We find a way to use the sameweight function for different variables and wave patterns, which still ensurethe errors be controllable. Thus, this construction can be applied to contactdiscontinuity and composite waves. Finally, by the energy method, we prove thatthe Cauchy problem admits a unique global-in-time solution and the compositewave is still stable under the space-periodic perturbations.

Asymptotic stability of the combination of a viscous contact wave with  two rarefaction waves for 1-D Navier-Stokes equations under periodic  perturbations