Time-asymptotic stability of planar Navier-Stokes shocks with spatial oscillations

This paper shows that for the three-dimensional compressible isentropic Navier-Stokes equations, the planar viscous shocks are time-asymptotically stable to suitably small initial perturbations with zero masses. In particular, the perturbations consist of not only $ H^3 $-perturbations, but also periodic ones that oscillate at spatial infinity. In the former case, the final shock locations can be predicted in terms of the initial conditions, while in the latter the locations are subject to the dynamics of the oscillations. The stability analysis is based on the $L^2$-energy method. The key point is that the bad effect due to the compression of the shock waves can be removed by a combination of an anti-derivative technique and the use of Poincar\'{e} inequality in the normal and transversal directions, respectively.