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

Abstract

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 initial perturbation is periodic on the torus ${\mathbb{T}}^3$ with zero average. To solve this Cauchy problem in which the initial data is periodic with respect to only $x_2$ and $x_3$ but not to $x_1,$ we construct a suitable ansatz carrying the oscillations of the solution in the $x_1$- direction, but remaining to be periodic in the transverse $x_2$- and $x_3$- directions. In such a way, the difference between the ansatz and the solution can be integrable on the region $\mathbb{R}\times {\mathbb{T}}^2,$ which allows us to utilize the energy method with the aid of a Gagliardo-Nirenberg type inequality on $\mathbb{R}\times {\mathbb{T}}^2$ to prove the result.