Events for 03/04/2024 from all calendars

The Foias Lectures

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Professor Camillo De Lellis, Institute of Advanced Studies

Title: Flows of nonsmooth vector fields

Abstract: Consider a vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field 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 started by DiPerna and Lions in the 1980es shows that there is a unique such flow under very reasonable conditions and for much less regular vector fields. This has a lot of repercussions to several important partial differential equations where the idea of ``following the trajectories of particles'' plays a fundamental role. In this lecture I will review the fundamental ideas of the original theory and an alternative approach due to Gianluca Crippa and myself.