Stability of large-amplitude viscous shock under periodic perturbation for 1-d isentropic Navier-Stokes equations

The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier-Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N-S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearly stable even though the perturbation oscillates at the far fields. The key point is to construct a suitable ansatz $ (\tilde{v},\tilde{u}) $, which carries the same oscillations of the solution $ (v,u) $ at the far fields, so that the difference $ (v-\tilde{v},u-\tilde{u}) $ belongs to the $ H^2(R) $ space for all $ t\geq 0. $