Stability of the planar rarefaction wave to three-dimensional full compressible Navier-Stokes-Korteweg equations

Abstract

In this paper, we are concerned with the large time behavior of the three-dimensional full compressible Navier-Stokes-Korteweg equations, which is used to model compressible viscous and heat-conductive fluids with internal capillarity, i.e., the liquid-vapor phase mixtures endowed with a variable internal capillarity. First, we construct the planar rarefaction wave to the three-dimensional full compressible Navier-Stokes-Korteweg equations, which can be derived by the fact that the rarefaction wave is nonlinearly stable to the one-dimensional full compressible Navier-Stokes-Korteweg equations. Then it is shown that the planar rarefaction wave is asymptotically stable provided that the initial data are a suitably small perturbation of the planar rarefaction wave and the strength of the rarefaction wave is small. The proof is based on the delicate energy method.