Bressan’s Conjecture on compactness of flow for BV vector fields

When studying a systems of conservation laws in several space dimensions, A. Bressan conjectured that the flows $X^n(t)$ generated by a smooth vector fields $\mathbf b^n(t,x)$,
\[
\frac{d}{dt} X^n(t,y) = \mathbf b^n(t,X(t,y)),
\]
are compact in $L^1(I\!\!R^d)$ for all $t \in [0,T]$ if $\mathbf b^n \in L^\infty \cap \mathrm{BV}((0,t) \times I\!\!R^d)$ and they are nearly incompressible, i.e.
\[
\frac{1}{C} \leq \det(\nabla_y X(t,y)) \leq C
\]
for some constant $C$. This conjecture is implied by the uniqueness of the solution to the linear transport equation
\[
\partial_t \rho + \mathrm{div}_x(\rho \mathbf b) = 0, \quad \rho \in L^\infty((0,T) \times I\!\!R^d),
\]
and it is the natural extension of a series of results concerning vector fields $\mathbf b(t,x)$ with Sobolev regularity.

We will give a general framework to approach the uniqueness problem for the linear transport equation and to prove Bressan's conjecture.