*Accepted Paper*

**Inserted:** 12 oct 2017

**Last Updated:** 23 dec 2019

**Journal:** Inventiones mathematicae

**Year:** 2019

**Doi:** https://doi.org/10.1007/s00222-019-00928-8

**Abstract:**

Given a vector field $\rho (1,\mathbf b) \in L^1_{\rm{loc}}(\mathbb R^+\times \mathbb R^{d},\mathbb R^{d+1})$ such that $\text{div}_{t,x} (\rho (1,\mathbf b))$ is a measure, we consider the problem of uniqueness of the representation $\eta$ of $\rho (1,\mathbf b) \mathcal L^{d+1}$ as a superposition of characteristics $\gamma : (t^-_\gamma,t^+_\gamma) \to \mathbb R^d$, with $\dot \gamma (t)= \mathbf b(t,\gamma(t))$. 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 $\mathbb R^{d+1}$ into disjoint trajectories $\wp_\mathfrak a$, $\mathfrak a \in \mathfrak A$, such that the PDE \[ \text{div}_{t,x} \big( u \rho (1,\mathbf b) \big) \in \mathcal M(\mathbb R^{d+1}), \qquad u \in L^\infty(\mathbb R^+\times \mathbb R^{d}), \] can be disintegrated into a family of ODEs along $\wp_\mathfrak a$ with measure r.h.s.. The decomposition $\wp_{\mathfrak a}$ is essentially unique. We finally show that $\mathbf b \in L^1_t(BV_x)_{\rm{loc}}$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible $BV$ vector fields.

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