Alberti, Giovanni and Crippa, Gianluca and Mazzucato, Anna L.. (2019) Exponential selfsimilar mixing by incompressible flows. Journal of the American Mathematical Society, 32 (2). pp. 445490.
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Official URL: https://edoc.unibas.ch/68252/
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Abstract
We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergencefree velocity field, which satisfies a bound in the Sobolev space Ws,p, where s ≥ 0 and 1 ≤ p ≤ ∞. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm H−1, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under selfsimilar mixing. For the case s = 1 and 1 ≤ p ≤ ∞ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophyconstrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa) 

UniBasel Contributors:  Crippa, Gianluca 
Item Type:  Article, refereed 
Article Subtype:  Research Article 
Publisher:  American Mathematical Society (AMS) 
ISSN:  04981707 
eISSN:  10886834 
Note:  Publication type according to Uni Basel Research Database: Journal article 
Language:  English 
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edoc DOI:  
Last Modified:  12 Feb 2019 09:55 
Deposited On:  12 Feb 2019 09:55 
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