Uniform stability of equilibria in the inviscid limit for the Navier-Stokes-Korteweg system

Abstract

This paper considers a stability result for the three-dimensional Navier-Stokes-Korteweg system uniformly in the inviscid limit. We obtain a unique global smooth solution close to the constant equilibria $(1,0)$, independent of the viscosity parameter ε, assuming that the potential part of the initial velocity is small independently of the viscosity parameter ε while the incompressible part of the initial velocity is small compared to ε. The proof is based on the parabolic energy estimates and dispersive properties involving the method of space time resonances.