Exponential self-similar mixing and loss of regularity for continuity equations |
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Authors: | Giovanni Alberti Gianluca Crippa Anna L Mazzucato |
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Institution: | 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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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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