Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$

Abstract

We consider the compressible barotropic Navier-Stokes-Korteweg system with friction in this paper.The global solutions and optimal convergence rates are obtained by pure energy method provided the initial perturbation around a constant state is small enough. In particular, the decay ratesof the higher-order spatial derivatives of the solution areobtained. Our proof is based on a family of scaled energy estimatesand interpolations among them without linear decay analysis.