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Speaker: Camillo De Lellis (Institute for 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 γ starting at x at time 0 and solving the ODE γ'(t) = v(t, γ(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 80es 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, an alternative approach due to Gianluca Crippa and myself, and review a series of natural questions. Some of these questions will be answered in the second part.
This mini-course is followed by a lecture on Transport equations and anomalous dissipation.