On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids

TL;DR

This paper proves the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids in three dimensions, allowing vacuum regions and using energy and entropy bounds without damping.

Contribution

It establishes compactness of finite energy weak solutions for large initial data, including vacuum regions, without additional damping terms, advancing the mathematical understanding of the system.

Findings

Proved compactness of weak solutions in 3D for large data.

Allowed vacuum regions in the weak solution framework.

Used energy and BD entropy bounds for analysis.

Abstract

In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. In contrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy.