| Abstract: |
| Given a vector field $\rho (1,{\mathbf b}) \in
L^1_{\mathrm{loc}}({\mathbb R}^+\times {\mathbb R}^{d},{\mathbb
R}^{d+1})$ 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 $\gamma : (t^-_\gamma,t^+_\gamma) \to {\mathbb R}^d$,
$\dot \gamma (t)= \b(t,\gamma(t))$. 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
$$
{\mathrm{div}}_{t,x} \big( u \rho (1,{\mathbf b}) \big) \in \mathcal
M({\mathbb R}^{d+1}), \qquad u \in L^\infty({\mathbb R}^+\times {\mathbb
R}^{d}),
$$
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 ${\mathbf b} \in
L^1_t({\mathrm{BV}}_x)_{\mathrm{loc}}$ satisfies this local structural
assumption and this yields, in particular, the renormalization property
for nearly incompressible ${\mathrm{BV}}$ vector fields. |
|