Given a vector field ρ(1, b) ∈ L_(loc)~1(R~+ × R~d ,R~(d+1)) such that div_(t,x)(ρ(1, b)) is a measure, we consider the problem of uniqueness of the representation η of ρ(1, b)L~(d+1) as a superposition of characteristics γ : (t_γ~- , t_γ~+ ) → R~d , 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 R~(d+1) into disjoint trajectories ?_a, a ∈ A, such that the PDE div_(t,x) (uρ(1, b)) ∈M(R~(d+1)), u ∈ L~∞(R~+ × R~d ), 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 ∈ L_t~1 (BV_x )_(loc) satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible BV vector fields.
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