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SUMMARY:Uniqueness and Lagrangianity for Solutions with Lack of Integrabil
 ity of the Continuity Equation
DTSTART:20190315T151500
DTEND:20190315T160000
DTSTAMP:20260511T165353Z
UID:d78a738dfa7e12ab8a3b9801e0b700abd49818703d19d7a84cb7f7ad
CATEGORIES:Conferences - Seminars
DESCRIPTION:Prof. Gianluca CRIPPA (Univ. Basel)\nAbstract:\nWe discuss the
  question of the uniqueness (and of the property of being transported by a
  flow) for solutions to the continuity equation with Sobolev velocity fiel
 d when the integrability of the solution is below that requested by the Di
 Perna-Lions theory. Striking examples of nonuniqueness have been recently 
 constructed in certain regimes by Modena and Székelyhidi. In this talk we
  present a strategy to show uniqueness of $L^1$ solutions in the case of $
 W^{1\,p}$ velocity fields\, where $p$ is larger than the space dimension\,
  under the additional assumption that the so-called "forward-backward inte
 gral curves" associated to the vector field are trivial for almost every s
 tarting point. Our approach is based on a disintegration argument combined
  with a Lipschitz extension lemma in which the extension procedure simulta
 neously preserves the Lipschitz continuity for two non-equivalent distance
 s. The two distances under consideration are the Euclidean distance and\, 
 roughly speaking\, the geodesic distance along integral curves of the flow
  of the velocity field. The Lipschitz constant for the geodesic distance o
 f the extension can be estimated in terms of the Lipschitz constant for th
 e geodesic distance of the original function. This is a work in collaborat
 ion with Laura Caravenna (University of Padova). 
LOCATION:MA B1 11 https://plan.epfl.ch/?room=MAB111
STATUS:CONFIRMED
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