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A uniqueness result for the decomposition of vector fields in Rd

TitleA uniqueness result for the decomposition of vector fields in Rd
Publication TypePreprint
2017
AuthorsBianchini, S, Bonicatto, P
Document NumberSISSA;15/2017/MATE
InstitutionSISSA

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
\begin{equation*}
\dive_{t,x} \big( u \rho (1,\b) \big) \in \mathcal M(\R^{d+1}), \qquad u \in
L^\infty(\R^+\times \R^{d}),
\end{equation*}
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.

http://preprints.sissa.it/handle/1963/35274

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