Zero-viscosity-capillarity limit to the planar rarefaction wave for the 2D compressible Navier–Stokes–Korteweg equations

Abstract

We shall consider the two-dimensional (2D) isentropic Navier–Stokes–Korteweg equations which are used to model compressible fluids with internal capillarity. Formally, the 2D isentropic Navier–Stokes–Korteweg equations converge, as the viscosity and the capillarity vanish, to the corresponding 2D inviscid Euler equations, and we do justify this for the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. More precisely, it is proved that there exists a family of smooth solutions for the 2D isentropic compressible Navier–Stokes–Korteweg equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficient and the capillarity away from the initial time. The key ingredients of our proof are the re-scaling technique and energy estimate, in which we also introduce the hyperbolic wave to recover the physical viscosities and capillarity of the inviscid rarefaction wave profile.