BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Maria Colombo (EPFL) DTSTART:20200602T150000Z DTEND:20200602T160000Z DTSTAMP:20260404T121926Z UID:LisbonWADE/7 DESCRIPTION:Title: Nonunique characteristic curves of Sobolev vector fields\nby Maria Colombo (EPFL) as part of Lisbon webinar in analysis and differential equa tions\n\n\nAbstract\nGiven a vector field in $\\mathbb{R}^d$\, the classic al Cauchy-Lipschitz theorem shows existence and uniqueness of its flow pro vided the vector field is sufficiently smooth\; this\, in turn\, translate s in existence and uniqueness results for the transport equation. In 1989\ , Di Perna and Lions proved that Sobolev regularity for vector fields\, wi th bounded divergence and a growth assumption\, is sufficient to establish existence\, uniqueness and stability of a generalized notion of flow\, co nsisting of a suitable selection among the trajectories of the associated ODE. A long-standing open question is whether the uniqueness of the regula r Lagrangian flow is a corollaryof the uniqueness of the trajectory of the ODE for a.e. initial datum. In this talk we give an overview of the topic and we provide a negative answer to this question. To show this result we exploit the connection with the transport equation\, based on Ambrosio’ s superposition principle\, and a new ill-posedness result for positive so lutions of the continuity equation.\n LOCATION:https://researchseminars.org/talk/LisbonWADE/7/ END:VEVENT END:VCALENDAR