**abstract:**
Given a vector field $\rho (1,\mathbf b) \in L^{1}_{{\mathrm} loc}(\mathbb R^{+\times
}
\mathbb R^{{d},\R}^{{d+1})$} such that $\mathrm{div}_{{t,x}} (\rho (1,\mathbf b))$ 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 $\gamma :
(t^{}-_{\gamma,t}^{+}_{\gamma)} \to \mathbb R^{d$,} $\dot \gamma (t)= \mathbf b(t,\gamma(t))$.
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
\begin{equation**}
\mathrm{div} _{{t,x}} \big( u \rho (1,\mathbf b) \big) \in \mathcal M(\mathbb R^{{d+1}),} \qquad u
\in L^{\infty}(\mathrm R^{+\times} \mathrm R^{{d}),
}
\end{equation**}
can be disintegrated into a family of ODEs along $\wp

Mon 5 Feb, 9:05 - 9:40, Aula Dini

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