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Zero-viscosity–capillarity limit toward rarefaction wave with vacuum for the Navier–Stokes–Korteweg equations of compressible fluids

We shall consider the one-dimensional Navier–Stokes–Korteweg (NSK) equations that are used to model compressible fluids with internal capillarity. Formally, the NSK equations converge, as the viscosity and capillarity vanish, to the corresponding Euler equations, and we do justify this for the case that the Euler equations have a rarefaction wave with one-side vacuum state. To prove this result, we first construct a sequence of approximate solutions to the NSK equations and then show that the sequence converges, as both viscosity and capillarity tend simultaneously to zero, to this rarefaction wave. The uniform convergence rates are also obtained. The key ingredients of our proof are the re-scaling technique and energy estimates.