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 _t μ_t + div(b μ_t) = 0, where {μ_t}_t R is a measurable family of (possibily signed) Borel measures on R^d and b R × R^d → 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].