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# CANCELED – Camillo De Lellis – CANCELED

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## CANCELED – Camillo De Lellis – CANCELED

Mini-course: Flows of nonsmooth vector fields

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.

Lecture: Transport equations and anomalous dissipation

After mentioning the fundamental theorems of DiPerna-Lions and Ambrosio on flows of Sobolev vector fields we will explore a number of sharpness questions related to them (a more detailed and elementary overiew of the theorems and of the sharpness questions will already have been given in the previous lecture, but anyway this seminar will be arranged so that it can be followed independently from the latter). Many of these questions have been at least partially answered in the last few years and I will first survey what is the overall understanding which we have gained on them. I will then focus on a particular one and highlight its link with a phenomenon which is poorly understood at the rigorous mathematical level: the occurrence of the so-called anomalous dissipation in incompressible fluid dynamics.

- Séminaire Analyse non linéaire et EDP

#### Détails :

Orateur / Oratrice : Camillo De Lellis
Date : 30 avril 2024
Horaire : 9h30 - 12h30
Lieu : Jussieu — salle 15-16-309