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We consider the continuity equation $\partial_t \mu_t +\mathop{\mathrm{div}}(b \mu_t) = 0$, where $\{\mu_t\}_{t \in \mathbb R}$ is ameasurable 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 vectorfield (and the equation is understood in the sense of distributions). If themeasure-valued solution $\mu_t$ is non-negative, then the following\emph{superposition principle} holds: $\mu_t$ can be decomposed into asuperposition of measures concentrated along the integral curves of $b$. Forsmooth $b$ this result follows from the method of characteristics, and in thegeneral case it was established by L. Ambrosio. A partial extension of thisresult for signed measure-valued solutions $\mu_t$ was obtained in \cite{AB},where the following problem was proposed: does the superposition principle holdfor signed measure-valued solutions in presence of unique flow ofhomeomorphisms solving the associated ordinary differential equation? We answerto this question in the negative, presenting two counterexamples in whichuniqueness of the flow of the vector field holds but one can constructnon-trivial signed measure-valued solutions to the continuity equation withzero initial data.

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