Zero‐viscosity‐capillarity limit towards rarefaction wave for the full Navier–Stokes–Korteweg system of compressible fluids

Abstract

We shall consider the one‐dimensional full Navier–Stokes–Korteweg (NSK) system which are used to model full compressible fluids with internal capillarity. Formally, the full NSK system converges, as the viscosity, heat‐conductivity, and capillary coefficient vanish, to the corresponding full Euler equation, and we do justify this for the case that the full Euler equation has a rarefaction wave. To prove this result, we first decompose the solution as a small perturbation of the smooth rarefaction wave, then make use of the rescaling technique and an elementary energy method on two time scales. Moreover, the uniform convergence rates are also obtained.