Exponential self-similar mixing and loss of regularity for continuity equations

We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity eld. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity elds, 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 eld exhibiting instantaneous loss of any fractional Sobolev regularity.