A uniqueness result for the decomposition of vector fields in Rd

Given a vector field ρ (1,b) ∈ L¹ₗoc(R⁺× Rᵈ,Rᵈ⁺¹) ρ ( 1 , b ) ∈ L loc 1 ( R + × R d , R d + 1 ) such that \,div\,t,x (ρ (1,b)) div t , x ( ρ ( 1 , b ) ) is a measure, we consider the problem of uniqueness of the representation η η of ρ (1,b) Lᵈ⁺¹ ρ ( 1 , b ) L d + 1 as a superposition of characteristics γ : (t⁻_γ ,t⁺_γ ) → Rᵈ γ : ( t γ - , t γ + ) → R d , γ ̇ (t)= b(t,γ (t)) γ ˙ ( t ) = b ( t , γ ( t ) ) . We give conditions in terms of a local structure of the representation η η on suitable sets in order to prove that there is a partition of Rᵈ⁺¹ R d + 1 into disjoint trajectories ₐ ℘ a , a∈ A a ∈ A , such that the PDE aligned \,div\,t,x ( u ρ (1,b) ) ∈ M(Rᵈ⁺¹), u ∈ L^∞ (R⁺× Rᵈ), aligned div t , x ( u ρ ( 1 , b ) ) ∈ M ( R d + 1 ) , u ∈ L ∞ ( R + × R d ) , can be disintegrated into a family of ODEs along ₐ ℘ a with measure r.h.s. The decomposition ₐ ℘ a is essentially unique. We finally show that b∈ L¹ₜ(\,BV\,ₓ)ₗoc b ∈ L t 1 ( BV x ) loc satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible \,BV\, BV vector fields.