Given a vector field ρ(1, b) ∈ L1loc(R+ ×Rd, Rd+1) such that divt,x(ρ(1, b)) is a measure, we consider the problem of uniqueness of the representation η of ρ(1, b)Ld+1 as a superposition of characteristics γ : (t−γ , t+γ ) → Rd, γ ̇ = b(t, γ(t)). We give conditions in terms of a local structure of the repre- sentation η on suitable sets in order to prove that there is a partition of Rd+1 into disjoint trajectories ℘a, a ∈ A, such that the PDE

divt,x (uρ(1, b)) ∈ M(Rd+1), u ∈ L∞(R+ × Rd),

can be disintegrated into a family of ODEs along ℘a with measure r.h.s.. The decomposition ℘a is essentially unique. We finally show that b ∈ L1t (BVx)loc satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible BV vector fields.

This is a joint work with Paolo Bonicatto.

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