  We consider the mixing behaviour of the solutions of the continuity equation
associated with a divergence-free velocity field. In this announcement we
sketch two explicit examples of exponential decay of the mixing scale of the
solution, in case of Sobolev velocity fields, thus showing the optimality of
known lower bounds. We also describe how to use such examples to construct
solutions to the continuity equation with Sobolev but non-Lipschitz velocity
field exhibiting instantaneous loss of any fractional Sobolev regularity.
