Conference on Calculus of Variations in Lille - 4th edition - June 24-28 2024

BV curves of measures and the continuity equation

Jun 25, 2024, 4:00 PM

1h

Polytech Lille, Chappe auditorium, Cité Scientifique (Laboratoire Paul Painlevé)

Speaker

Riccarda Rossi (Università degli Studi di Brescia)

Description

Representation results for Lipschitz (or even absolutely continuous) curves $\mu :[0,T]\to {\mathcal{P}}_p({\mathbb{R}}^d)$, $p>1$, with values in
the Wasserstein space $({\mathcal{P}}_p({\mathbb{R}}^d),W_p)$ of Borel probability measures in ${\mathbb{R}}^d$ with finite $p$-moment provide a crucial tool to study evolutionary PDEs and geometric problems in a measure-theoretic setting.

They are strictly related to corresponding representation results for measure-valued solutions to the continuity equation, as a superposition of absolutely continuous curves solving a suitable differential equation.

In this talk we discuss the validity and the appropriate formulation of the above results in the case $p=1$, for the space of probability measures with finite moment ${\mathcal{P}}_1({\mathbb{R}}^d)$ endowed with the metric $W_1.$ We will thus provide a suitable version of the superposition principle for curves of measures in ${\mathcal{P}}_1({\mathbb{R}}^d)$ that are only of bounded variation with respect to the time variable.

Joint work with Stefano Almi (Napoli) and Giuseppe Savaré (Milano).