Growth of Sobolev norms and loss of regularity in transport equations

Crippa, Gianluca and Elgindi, Tarek and Iyer, Gautam and Mazzucato, Anna L.. (2021) Growth of Sobolev norms and loss of regularity in transport equations. Preprints Fachbereich Mathematik, 2021 (18).


Official URL: https://edoc.unibas.ch/84983/

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We consider transport of a passive scalar advected by an irregular divergence free vector field. Given any non-constant initial data $\bar \rho \in H^1_\text{loc}({\mathbb R}^d)$, $d\geq 2$, we construct a divergence free advecting velocity field $v$ (depending on $\bar \rho$) for which the unique weak solution to the transport equation does not belong to $H^1_\text{loc}({\mathbb R}^d)$ for any positive positive time. The velocity field $v$ is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space $W^{s,p}$ that does not embed into the Lipschitz class. The velocity field $v$ is constructed by pulling back and rescaling an initial data dependent sequence of sine/cosine shear flows on the torus. This loss of regularity result complements that in {\em Ann. PDE}, 5(1):Paper No. 9, 19, 2019.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Crippa, Gianluca
Item Type:Preprint
Publisher:Universität Basel
edoc DOI:
Last Modified:03 Nov 2021 07:44
Deposited On:03 Nov 2021 07:44

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