@article {2017,
	title = {A uniqueness result for the decomposition of vector fields in Rd},
	number = {SISSA;15/2017/MATE},
	year = {2017},
	institution = {SISSA},
	abstract = {<p>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.</p>},
	url = {http://preprints.sissa.it/handle/1963/35274},
	author = {Stefano Bianchini and Paolo Bonicatto}
}
@article {2014,
	title = {A uniqueness result for the continuity equation in two dimensions: dedicated to constantine dafermos on the occasion of his 70th birthday},
	number = {Journal of the European Mathematical Society;Volume 16; issue 2; pp. 201-234;},
	year = {2014},
	publisher = {European Mathematical Society; Springer Verlag},
	abstract = {We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation ∂tu +div(bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain nonautonomous vector fields b with bounded divergence.},
	doi = {10.4171/JEMS/431},
	url = {http://urania.sissa.it/xmlui/handle/1963/34692},
	author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa}
}
@article {2011,
	title = {A uniqueness result for the continuity equation in two dimensions},
	number = {SISSA;52/2011/M},
	year = {2011},
	institution = {SISSA},
	url = {http://hdl.handle.net/1963/4663},
	author = {Giovanni Alberti and Stefano Bianchini and Gianluca Crippa}
}
