# (Webinar) Nonunique characteristic curves of Sobolev vector fields

## Maria Colombo (EPFL Lausanne)

created by caroccia on 29 May 2020

2 jun 2020 -- 16:00   [open in google calendar]

Lisbon WADE — Webinar in Analysis and Differential Equations

To be held online at:

Given a vector field in $\mathbb{R}^d$, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth; this, in turn, translates in existence and uniqueness results for the transport equation. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. In this talk we give an overview of the topic and we provide a negative answer to this question. To show this result we exploit the connection with the transport equation, based on Ambrosio’s superposition principle, and a new ill-posedness result for positive solutions of the continuity equation.