Non-uniqueness of signed measure-valued solutions to the continuity equation in presence of a unique flow

Abstract

We consider the continuity equation $\partial_t \mu_t + \mathop{\mathrm{div}}(b \mu_t) = 0$, where $\{\mu_t\}_{t \in \mathbb R}$ is a measurable family of (possibily signed) Borel measures on $\mathbb R^d$ and $b \colon \mathbb R \times \mathbb R^d \to \mathbb R^d$ is a bounded Borel vector field (and the equation is understood in the sense of distributions). If the measure-valued solution $\mu_t$ is non-negative, then the following \emph{superposition principle} holds: $\mu_t$ can be decomposed into a superposition of measures concentrated along the integral curves of $b$. For smooth $b$ this result follows from the method of characteristics, and in the general case it was established by L. Ambrosio. A partial extension of this result for signed measure-valued solutions $\mu_t$ was obtained in \cite{AB}, where the following problem was proposed: does the superposition principle hold for signed measure-valued solutions in presence of unique flow of homeomorphisms solving the associated ordinary differential equation? We answer to this question in the negative, presenting two counterexamples in which uniqueness of the flow of the vector field holds but one can construct non-trivial signed measure-valued solutions to the continuity equation with zero initial data.