Asymptotic stability of the rarefaction wave to one-dimensional compressible Navier-Stokes-Korteweg equations under space-periodic perturbations

Abstract

In this article, we are concerned with the asymptotic stability of the rarefaction wave for the one-dimensional isentropic compressible Navier-Stokes-Korteweg equations under initial perturbations, which tend to spaceperiodic functions at infinities. It is shown that the solution of the onedimensional isentropic compressible Navier-Stokes-Korteweg equations tends to background rarefaction wave as time tends to infinity, provided that the initial perturbation and strength of the rarefaction wave are suitably small. The proof is based on the delicate constructions of the proper ansatzes, which capture the interactions between the background waves and the periodic perturbations, and the energy method in Eulerian coordinates involving the effect of the Korteweg stress term.