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.

The paper is dedicated to the global well-posedness of the barotropic compressible Navier-Stokes-Poisson system in the whole space ${ℝ}^N$ with N ≥ 3. The global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces. The initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations. The proof relies on a uniform estimate for a mixed hyperbolic/parabolic linear system with a convection term.