Loss of regularity for the continuity equation with non-Lipschitz velocity field

Alberti, Giovanni and Crippa, Gianluca and Mazzucato, Anna L.. (2019) Loss of regularity for the continuity equation with non-Lipschitz velocity field. Annals of PDE, 5 (1). p. 9.

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We consider the Cauchy problem for the continuity equation in space dimension d≥2. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces W1,p, for 1≤p<∞, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in Wr,p, with r>1, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space Wr,p does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper Exponential self-similar mixing by incompressible flows (Alberti et al. in J Am Math Soc 32(2):445–490, 2019), and have been announced in Exponential self-similar mixing and loss of regularity for continuity equations (Alberti et al. in Comptes Rendus Math Acad Sci Paris 352(11):901–906, 2014).
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
Note:Publication type according to Uni Basel Research Database: Journal article
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Last Modified:15 Jan 2020 13:53
Deposited On:13 Jan 2020 11:47

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