Uniqueness of signed measures solving the continuity equation for Osgood vector fields

TL;DR

This paper extends the uniqueness results for measure solutions of the continuity equation to signed measures under a two-sided Osgood condition, broadening the class of velocity fields for which uniqueness holds.

Contribution

It provides a partial extension of uniqueness results from nonnegative to signed measure solutions under a quantitative Osgood condition on the velocity field.

Findings

Uniqueness of signed measure solutions under Osgood condition

Extension of log-Lipschitz results to broader Osgood class

Quantitative two-sided Osgood condition ensures uniqueness

Abstract

Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this result to signed measure-valued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained for log-Lipschitz vector fields by Bahouri and Chemin.