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SUMMARY:Camillo De Lellis (IAS Princeton)
DTSTART:20210429T130000Z
DTEND:20210429T140000Z
DTSTAMP:20260423T035925Z
UID:SNPDEA/25
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SNPDEA/25/">
 Flows of vector fields: classical and modern</a>\nby Camillo De Lellis (IA
 S Princeton) as part of "Partial Differential Equations and Applications" 
 Webinar\n\n\nAbstract\nConsider a (possibly time-dependent) vector field $
 v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Pica
 rd-Lindelöf) Theorem states that\, if the vector field $v$ is Lipschitz i
 n space\, for every initial datum $x$ there is a unique trajectory $\\gamm
 a$ starting at $x$ at time $0$ and solving the ODE $\\dot{\\gamma} (t) = v
  (t\, \\gamma (t))$. The theorem looses its validity as soon as $v$ is sli
 ghtly less regular. However\, if we bundle all trajectories into a global 
 map allowing $x$ to vary\, a celebrated theory put forward by DiPerna and 
 Lions in the 80es show that there is a unique such flow under very reasona
 ble conditions and for much less regular vector fields. A long-standing op
 en question is whether this theory is the byproduct of a stronger classica
 l result which ensures the uniqueness of trajectories for almost every ini
 tial datum. I will give a complete answer to the latter question and draw 
 connections with partial differential equations\, harmonic analysis\, prob
 ability theory and Gromov's $h$-principle.\n
LOCATION:https://researchseminars.org/talk/SNPDEA/25/
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