Inviscid incompressible limit for capillary fluids with density dependent viscosity

TL;DR

This paper proves that solutions of the Navier-Stokes-Korteweg system for capillary fluids with density-dependent viscosity converge to incompressible Euler solutions in low Mach and viscosity limits, using dispersive estimates.

Contribution

It establishes the inviscid incompressible limit for capillary fluids with density-dependent viscosities, extending previous results to include Korteweg tensor effects.

Findings

Weak solutions converge to strong incompressible Euler solutions

Dispersive estimates are key to establishing convergence

Results hold in both 2D and 3D settings

Abstract

The asymptotic limit of the 2D and 3D Navier-Stokes-Korteweg system for barotropic capillary fluids with density dependent viscosities in the low Mach number and vanishing viscosity regime is established. 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.