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).

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Abstract
We consider transport of a passive scalar advected by an irregular divergence free vector field. Given any nonconstant 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 
Language:  English 
edoc DOI:  
Last Modified:  03 Nov 2021 07:44 
Deposited On:  03 Nov 2021 07:44 
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