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SUMMARY:Camillo De Lellis (IAS\, Princeton)
DTSTART:20200415T200000Z
DTEND:20200415T210000Z
DTSTAMP:20260423T024803Z
UID:RJWAPDE/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/RJWAPDE/1/">
 Flows of vector fields: classical and modern</a>\nby Camillo De Lellis (IA
 S\, Princeton) as part of Rio de Janeiro webinar on analysis and partial d
 ifferential equations\n\n\nAbstract\nConsider a (possibly time-dependent) 
 vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (a
 lso named Picard-Lindel\\"of) Theorem states that\, if the vector field $v
 $ is Lipschitz in space\, for every initial datum $x$ there is a unique tr
 ajectory $\\gamma$ starting at $x$ at time $0$ and solving the ODE $\\dot{
 \\gamma} (t) = v (t\, \\gamma (t))$. The theorem looses its validity as so
 on as $v$ is slightly less regular. However\, if we bundle all trajectorie
 s 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 un
 der very reasonable conditions and for much less regular vector fields. A 
 long-standing open question is whether this theory is the byproduct of a s
 tronger classical result which ensures the uniqueness of trajectories for 
 {\\em almost every} initial datum. I will give a complete answer to the la
 tter question and draw connections with partial differential equations\, h
 armonic analysis\, probability theory and Gromov's $h$-principle.\n
LOCATION:https://researchseminars.org/talk/RJWAPDE/1/
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