**Title:** Flows of nonsmooth vector fields

**Speaker:** Camillo De Lellis

**Speaker Info:** Institute of Advanced Study

**Brief Description:**

**Special Note**:

**Abstract:**

Consider a vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-LindelĂ¶f) 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.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 third of the three lectures.

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