Caravenna, Laura and Crippa, Gianluca. (2021) A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation. Preprints Fachbereich Mathematik, 2021 (10).

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Abstract
We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multivalued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPernaLions theory of continuity equations in the case of vector fields in the Sobolev space $W^{1,p}$, where $p$ is larger than the space dimension, under the assumption that the socalled "forwardbackward integral curves" associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable.
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:  20 Jan 2023 08:08 
Deposited On:  08 Apr 2021 19:46 
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