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We study the asymptotic stability of a planar rarefaction wave (in the $ x_1$- direction) for the 3-d isentropic Navier-Stokes equations, where the initialperturbation is periodic on the torus $ \mathbb{T}^3 $ with zero average. Tosolve this Cauchy problem in which the initial data is periodic with respect toonly $ x_2 $ and $ x_3 $ but not to $ x_1, $ we construct a suitable ansatzcarrying the oscillations of the solution in the $ x_1 $- direction, butremaining to be periodic in the transverse $ x_2 $- and $ x_3 $- directions. Insuch a way, the difference between the ansatz and the solution can beintegrable on the region $ \mathbb{R}\times\mathbb{T}^2, $ which allows us toutilize the energy method with the aid of a Gagliardo-Nirenberg type inequalityon $ \mathbb{R}\times\mathbb{T}^2 $ to prove the result.

Asymptotic stability of planar rarefaction waves for 3-d isentropic Navier-Stokes equations under periodic perturbations