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

Luigi Ambrosio (1) , Patrick Bernard (2)

1 SNS - Scuola Normale Superiore di Pisa
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Domaines: Systèmes dynamiques [math.DS]; Equations aux dérivées partielles [math.AP]

Luigi Ambrosio, Patrick Bernard. Uniqueness of signed measures solving the continuity equation for Osgood vector fields. Rendiconti Lincei. Matematica e Applicazioni, 2008, 19 (3), pp.237--245. ⟨10.4171/RLM/522⟩. ⟨hal-00294648⟩