# A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

## TL;DR

This paper establishes a blow-up criterion for strong solutions to the 3D density-dependent Navier-Stokes-Korteweg equations, showing conditions under which solutions exist globally even with vacuum.

## Contribution

It introduces a Serrin's type blow-up criterion specifically for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum.

## Key findings

- Strong solutions exist globally under certain Serrin's type conditions.
- The criterion applies even in the presence of vacuum.
- Provides a new blow-up criterion for complex fluid models.

## Abstract

This paper proves a Serrin's type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density and velocity field satisfy some Serrin's type condition, then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally.