Convergence to the superposition of rarefaction waves and contact discontinuity for the 1-D compressible Navier-Stokes-Korteweg system

This paper is concerned with the vanishing capillarity-viscosity limit for the one-dimensional compressible Navier-Stokes-Korteweg system to the Riemann solution of the Euler system that consists of the supposition of two rarefaction waves and a contact discontinuity. It is shown that there exists a family of smooth solutions to the compressible Navier-Stokes-Korteweg system which converge to the Riemann solution away from the initial time t=0 and the contact discontinuity located at x=0, as the coefficients of capillarity, viscosity and heat conductivity tend to zero. Moreover, a uniform convergence rate in terms of the above physical parameters is also obtained. Here, the strengths of both the rarefaction waves and the contact discontinuity are not required to be small.