  We consider transport of a passive scalar advected by an irregular divergence
free vector field. Given any non-constant initial data $\bar \rho \in
H^1_\text{loc}({\mathbb R}^d)$, $d\geq 2$, we construct a divergence free
advecting velocity field $v$ (depending on $\bar \rho$) for which the unique
weak solution to the transport equation does not belong to
$H^1_\text{loc}({\mathbb R}^d)$ for any positive positive time. The velocity
field $v$ is smooth, except at one point, controlled uniformly in time, and
belongs to almost every Sobolev space $W^{s,p}$ that does not embed into the
Lipschitz class. The velocity field $v$ is constructed by pulling back and
rescaling an initial data dependent sequence of sine/cosine shear flows on the
torus. This loss of regularity result complements that in [Ann. PDE, 5(1):Paper
No. 9, 19, 2019].
