Nonunique characteristic curves of Sobolev vector fields

Abstract: 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 corollaryof 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.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

Lisbon webinar in analysis and differential equations

Series comments: The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field. All seminars will take place in the Zoom platform. The links for the planned seminars can be found at: wade.ulisboa.pt/ In order to get the password to access the seminars, please subscribe the announcements in the registration menu wade.ulisboa.pt/registration, or contact one of the organizers.

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