  It has been known, since the pioneering works by Serre, Hoff,
Va\u{i}gant-Kazhikhov, Lions and Feireisl, among others, the regularizing
properties of the effective viscous flux and its characterization as the
function whose gradient is the gradient part in the Hodge decomposition of the
Newtonian force of the fluid, when the shear viscosity of the fluid is
constant. In this article, we explore further the connection between the Hodge
decomposition of the Newtonian force and the regularizing properties of its
gradient part, by addressing the problem of the global existence of weak
solutions for compressible Navier-Stokes equations with both viscosities
depending on a spatial mollification of the density.
