Zero-viscosity-capillarity limit for the contact discontinuity for the 1-D full compressible Navier-Stokes-Korteweg 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 \(ε1/4\) 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. For more information see https://ejde.math.txstate.edu/Volumes/2025/74/abstr.html