In this talk, we will introduce the recent results on the asymptotic stabilities of shocks and rarefaction waves under space-periodic perturbations for scalar 1-d convex conservation laws in both inviscid and viscous cases. Under arbitrary bounded periodic perturbation, both two kinds of waves are proved to be stable. However, there is a difference between the inviscid shock and the viscous shock profile, where the latter one will have a constant shift, but the former one will not. This shift is essentially due to the viscosity and the periodic perturbation, and it also depends on the structure of the flux. Besides, the shift will disappear as the viscosity vanishes. This work is a joint work with Prof. Zhouping Xin (CUHK) and Dr. Yuan Yuan (SCNU).

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