Zero-viscosity-capillarity limit for the contact discontinuity for the 1-D full compressible Navier-Stokes-Korteweg equations - Jiaxue Chen; Yeping Li; Rong Yin (Electronic Journal of Differential Equations)
In this article, we study the zero-viscosity-capillarity limit problem for the one-dimensional full compressible Navier-Stokes-Korteweg equations. This equation models compressible viscous fluids with internal capillarity and heat conductivity. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the one-dimensional full compressible Navier-Stokes-Korteweg system which converge to the inviscid solution away from the contact discontinuity. It converges a rate of $ε^{1/4}$ as the the viscosity $μ=ε$, heat-conductivity coefficient $α=νε$ and the capillarity $κ=λε^2$ and $ε$ tends to zero. The proof is completed using the energy method and the scaling technique.