Nonlinear asymptotic stability of planar viscous shocks for 3D compressible Navier-Stokes equations with periodic perturbations

Abstract

This paper studies a Cauchy problem for the three-dimensional compressible isentropic Navier-Stokes equations, in which the initial data is a planar viscous shock with a periodic perturbation. It is shown that if the shock strength is weak and the perturbation is small and fulfills a zero-mass type condition, then the Cauchy problem admits a unique classical solution globally in time, which approaches the background planar viscous shock with a constant shift in the $W^{1,\infty}(\mathbb{R}^3)$ space as $t\to +\infty$. Moreover, an exponential decay rate of the non-zero mode of the solution is obtained in the $L^\infty(\mathbb{R}^3)$ space. The result reveals the nonlinear time-asymptotic stability of planar viscous shocks under the perturbations that oscillate in all spatial variables. The main ingredients of the proof consist of the construction of a suitable ansatz, a decomposition idea, an anti-derivative technique and a framework of $L^2$-theory.Comment: 47 page