A uniqueness result for the decomposition of vector fields in Rd

Given a vector field ρ(1,b)∈Lloc1(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:(tγ-,tγ+)→Rd, γ˙ (t) = b(t, γ(t)). We give conditions in terms of a local structure of the representation 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∈Lt1(BVx)loc satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible BV vector fields.