Calculus of Variations and Geometric Measure Theory

G. Alberti - G. Crippa - A. L. Mazzucato

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

created by crippa on 08 Jul 2014
modified by alberti on 23 May 2017


Published Paper

Inserted: 8 jul 2014
Last Updated: 23 may 2017

Journal: Comptes Rendus Mathématique
Volume: 352
Number: 11
Pages: 901-906
Year: 2014
Doi: 10.1016/j.crma.2014.08.021


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