Uniqueness and Non-Uniqueness of Signed Measure-Valued Solutions to the Continuity Equation - Bonicatto, Paolo (Discontinuity, Nonlinearity, and Complexity, 2026)
We consider the continuity equation $\partial_t μ_t + \text{div}(b μ_t) = 0$, where $\{μ_t\}_{t \in \mathbb R}$ is a measurable family of (possibily signed) Borel measures on $\mathbb R^d$ and $b \colon \mathbb R × \mathbb R^d → \mathbb R^d$ is a bounded Borel vector field (and the equation is understood in the sense of distributions). We discuss some uniqueness and non-uniqueness results for this equation: in particular, we report on some 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. This is based on a joint work with N.A. Gusev [1].