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

This study is concerned with the large time behaviour of the three-dimensional isentropic compressible Navier-Stokes-Korteweg equations, which are used to model viscous and compressible fluids with internal capillarity. Based on the fact that the rarefaction wave is nonlinearly stable to the one-dimensional isentropic compressible Navier-Stokes-Korteweg equations, the planar rarefaction wave to the three-dimensional isentropic compressible Navier-Stokes-Korteweg equations is first constructed. Then it is shown that the planar rarefaction wave is asymptotically stable in the case 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. The result indicate that the planar rarefaction wave of the inviscid Euler system is stable for the three-dimensional isentropic compressible fluids with physical viscosities and internal capillarity.