  We discuss in this short note the local-in-time strong well-posedness of the
compressible Navier-Stokes system for non-Newtonian fluids on the three
dimensional torus. We show that the result established recently by Kalousek,
M\'{a}cha, and Ne\v{c}asova in \doi{10.1007/s00208-021-02301-8} can be extended
to the case where vanishing density is allowed initially. Our proof builds on
the framework developed by Cho, Choe, and Kim in
\doi{10.1016/j.matpur.2003.11.004} for compressible Navier-Stokes equations in
the case of Newtonian fluids. To adapt their method, special attention is given
to the elliptic regularity of a challenging nonlinear elliptic system. We show
particular results in this direction, however, the main result of this paper is
proven in the general case when elliptic $W^{2,p}$-regularity is imposed as an
assumption. Also, we give a finite time blow-up criterion.
