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

Understand

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.