  We investigate the existence of weak solutions to a certain system of partial
differential equations, modelling the behaviour of a compressible non-Newtonian
fluid for small Reynolds number. We construct the weak solutions despite the
lack of the $L^\infty$ estimate on the divergence of the velocity field. The
result was obtained by combining the regularity theory for singular operators
with a certain logarithmic integral inequality for $BMO$ functions, which
allowed us to adjust the method from (Feireisl et al., 2015) to more relaxed
conditions on the velocity.
