Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 74, pp. 1-18.

Title: Zero-viscosity-capillarity limit for the contact discontinuity for the 1-D full compressible  Navier-Stokes-Korteweg equations

Authors: Jiaxue Chen (Nantong Univ., Nantong, China)
         Yeping Li (Nantong Univ., Nantong, China)
         Rong Yin (Nantong Univ., Nantong, China)

Abstract:
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 $\epsilon^{1/4}$ as the the viscosity $\mu=\epsilon$,
heat-conductivity coefficient $\alpha=\nu\epsilon$ and the
capillarity $\kappa=\lambda\epsilon^2$ and $\epsilon$ tends to
zero. The proof is completed using the energy method and the scaling
technique.


Submitted May 19, 2025. Published July 16, 2025.
Math Subject Classifications: 76W05, 35B40.
Key Words: Full compressible Navier-Stokes-Korteweg equation; compressible Euler system; 
  zero-viscosity-capillarity limit; contact discontinuity.