*Published Paper*

**Inserted:** 8 jul 2014

**Last Updated:** 23 may 2017

**Journal:** Comptes Rendus MathÃ©matique

**Volume:** 352

**Number:** 11

**Pages:** 901-906

**Year:** 2014

**Doi:** 10.1016/j.crma.2014.08.021

**Abstract:**

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.

**Keywords:**
continuity equation, mixing, negative Sobolev norms, incompressible flows, self-similarity

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