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 \ (^1/4\) as the the viscosity \ (=\), heat-conductivity coefficient \ (=\) and the capillarity \ (=²\) and \ (\) 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