Uniqueness and Non-Uniqueness of Signed Measure-Valued Solutions to the Continuity Equation

We consider the continuity equation ∂tμt + div(bμt) = 0, where {μt}t∈ℝ is a measurable family of (possibily signed) Borel measures on ℝd and b: ℝ X ℝd 萇 ℝd 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].