A uniqueness result for the decomposition of vector fields in (\R^{\MakeLowercase{d}}

Abstract: Given a vector field \(\rho (1,\b) \in L^1_\loc(\R^+\times

\R^{d},\R^{d+1})\) such that \(\dive_{t,x} (\rho (1,\b))\) is a measure, we

consider the problem of uniqueness of the representation \(\eta\) of \(\rho

(1,\b) \mathcal L^{d+1}\) as a superposition of characteristics \(\gamma :

(t^-_\gamma,t^+_\gamma) \to \R^d\), \(\dot \gamma (t)= \b(t,\gamma(t))\).

We give conditions in terms of a local structure of the representation

\(\eta\) on suitable sets in order to prove that there is a partition of

\(\R^{d+1}\) into disjoint trajectories \(\wp_\a\), \(\a \in \A\), such that

the PDE

\(\dive_{t,x} \big( u \rho (1,\b) \big) \in \mathcal M(\R^{d+1}), \qquad u

\in L^\infty(\R^+\times \R^{d}),

can be disintegrated into a family of ODEs along\($\wp_\a\) with measure

r.h.s.. The decomposition \(\wp_\a\) is essentially unique. We finally

show that \(\b \in L^1_t(\BV_x)_\loc\) satisfies this local structural

assumption and this yields, in particular, the renormalization property

for nearly incompressible \(\BV\) vector fields.