Title: Flows of nonsmooth vector fields
Speaker: Camillo De Lellis
Speaker Info: Institute of Advanced Study
Consider a vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindelöf) 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 loses 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 80's shows that there is a unique such flow under very reasonable conditions and for much less regular vector fields. This theory has a lot of repercussions to several important partial differential equations where the idea of "following the trajectories of particles" plays a fundamental role.Date: Wednesday, April 12, 2023
In these three lectures I will review the state of the art of the subject, touching upon a variety of related topics, such as the most recent surprising outcomes of convex-integration techniques and the most interesting applications to evolutionary PDEs.
This is the first of the three lectures.