Partial differential equations
Exponential self-similar mixing and loss of regularity for continuity equations
Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 901-906.

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.

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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2014.08.021

Giovanni Alberti 1; Gianluca Crippa 2; Anna L. Mazzucato 3

1 Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy
2 Departement Mathematik und Informatik, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
3 Department of Mathematics, Penn State University, McAllister Building, University Park, PA 16802, USA
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Giovanni Alberti; Gianluca Crippa; Anna L. Mazzucato. Exponential self-similar mixing and loss of regularity for continuity equations. Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 901-906. doi : 10.1016/j.crma.2014.08.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.08.021/

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