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.

本研究涉及三维等熵可压缩 Navier-Stokes-Korteweg 方程的大时间行为，该方程用于模拟具有内部毛细作用的粘性和可压缩流体。基于稀疏波对一维等熵可压缩Navier-Stokes-Korteweg方程的非线性稳定这一事实，首先构造了对三维等熵可压缩Navier-Stokes-Korteweg方程的平面稀疏波。然后表明，在初始数据为平面稀疏波的适当小扰动且稀疏波的强度较小的情况下，平面稀疏波是渐近稳定的。证明是基于微妙的能量方法。