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

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 $\rho $ and velocity field $u$ satisfy ${\parallel \nabla \rho \parallel}_{{L}^{\infty}(0,T;{W}^{1,q})}+{\parallel u\parallel}_{{L}^{s}(0,T;{L}_{\omega}^{r})}<\infty $ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r\le 1$, $3<r\le \infty ,$ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here ${L}_{\omega}^{r}$ denotes the weak ${L}^{r}$ space.