Inviscid incompressible limit for capillary fluids with density dependent viscosity

The asymptotic limit of the Navier-Stokes-Korteweg system for barotropic capillary fluids with density dependent viscosities in the low Mach number and vanishing viscosity regime is established in $\mathbb{R}^d$, with $d = 2, 3$. In the relative energy framework, we prove the convergence of weak solutions of the Navier-Stokes-Korteweg system to the strong solution of the incompressible Euler system. The convergence is obtained through the use of suitable dispersive estimates for an acoustic system altered by the presence of the Korteweg tensor.