TY - JOUR

T1 - Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field

AU - Alberti, Giovanni

AU - Crippa, Gianluca

AU - Mazzucato, Anna L.

N1 - Funding Information:
The authors thank the anonymous referees for a careful reading of the manuscript and insightful comments. This work was started during a visit of the first and third authors at the University of Basel. Their stay has been partially supported by the Swiss National Science Foundation Grants 140232 and 156112. The visits of the second author to Pisa have been supported by the University of Pisa PRA project “Metodi variazionali per problemi geometrici [Variational Methods for Geometric Problems]”. The second author was partially supported by the ERC Starting Grant 676675 FLIRT and the third author by the US National Science Foundation Grants DMS-1312727 and DMS-1615457.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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).

AB - 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).

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U2 - 10.1007/s40818-019-0066-3

DO - 10.1007/s40818-019-0066-3

M3 - Article

AN - SCOPUS:85076240777

VL - 5

JO - Annals of PDE

JF - Annals of PDE

SN - 2524-5317

IS - 1

M1 - 9

ER -