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

Giovanni Alberti, Gianluca Crippa, Anna L. Mazzucato

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21 Scopus citations

Abstract

We consider the mixing behavior of the solutions to the continuity equation associated with a divergence-free velocity field. In this Note, 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.

Original languageEnglish (US)
Pages (from-to)901-906
Number of pages6
JournalComptes Rendus Mathematique
Volume352
Issue number11
DOIs
StatePublished - 2014

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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