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Comptes Rendus Mathématique
Volume 352, n° 11
pages 901-906 (novembre 2014)
Doi : 10.1016/j.crma.2014.08.021
Received : 8 July 2014 ;  accepted : 28 August 2014
Exponential self-similar mixing and loss of regularity for continuity equations
Mélange auto-similaire exponentiel et perte de régularité pour l'équation de continuité
 

Giovanni Alberti a , Gianluca Crippa b , Anna L. Mazzucato c
a Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy 
b Departement Mathematik und Informatik, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland 
c Department of Mathematics, Penn State University, McAllister Building, University Park, PA 16802, USA 

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.

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Résumé

Nous étudions le comportement de mélange de solutions de l'équation de continuité associée à un champ de vitesse à divergence nulle. Dans cette note, nous décrivons deux exemples explicites de décroissance exponentielle de l'échelle de mélange de la solution. Dans le cas des champs de vitesse Sobolev, nous montrons donc l'optimalité des estimations par dessous connues. Nous décrivons aussi comment utiliser de tels exemples pour construire des solutions de l'équation de continuité à champs de vitesse Sobolev mais non lipschitziens : ces solutions perdent immédiatement toute régularité Sobolev fractionnaire.

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