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Nonlinear Analysis

Flows of vector fields: classical and modern

Camillo De Lellis, Institute for Advanced Study

Location:  Zoom
Date & time: Wednesday, 03 February 2021 at 9:30AM - 10:30AM

Abstract: Consider a (possibly time-dependent) vector field \(v\) on the Euclidean space. The classical Cauchy-Lipschitz (also 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 trajectory \(gamma\) 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 slightly 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 reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for {em almost every} initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory and Gromov's \(h\)-principle. 

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