Forward untangling and applications to the uniqueness problem for the continuity equation

We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure ρ(1, b), where ρ ∈ M+(Rd+1) and b: Rd+1 → Rd is a ρ-integrable vector field with divt,x(ρ(1, b)) = µ ∈ M(R × Rd): forward untangling formalizes the notion of forward uniqueness in the language of Lagrangian representations. We identify local conditions for a Lagrangian representation to be forward untangled, and we show how to derive global forward untangling from such local assumptions. We then show how to reduce the PDE divt,x(ρ(1, b)) = µ on a partition of R+ × Rd obtained concatenating the curves seen by the Lagrangian representation. As an application, we recover known well posedeness results for the flow of monotone vector fields and for the associated continuity equation.