  We consider the Cauchy problem for the continuity equation in space dimension
${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded
in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly
supported initial datum such that the unique solution to the continuity
equation with this initial datum and advecting field does not belong to any
Sobolev space of positive fractional order at any positive time. We also
construct velocity fields in $W^{r,p}$, with $r>1$, and solutions of the
continuity equation with these velocities that exhibit some loss of regularity,
as long as the Sobolev space $W^{r,p}$ does not embed in the space of Lipschitz
functions. Our constructions are based on examples of optimal mixers from the
companion paper "Exponential self-similar mixing by incompressible flows" (J.
Amer. Math. Soc. 32 (2019), no. 2), and have been announced in "Exponential
self-similar mixing and loss of regularity for continuity equations" (C. R.
Math. Acad. Sci. Paris, 352 (2014), no. 11).
