Non-trivial examples of regular Lagrangian flow in BV case What is a concrete example  of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow explicitly. An example I provided in a related question seems to have much stronger solutions.

Another related question is this: Relationship between three different definitions of solutions for ODE with irregular coefficient.
 A: Here is what I believe is a relevant example: consider the vector field $X$ in $\mathbb R^3$,
$$
X=a_1(x_2, x_3)\frac{\partial}{\partial x_1}+a_2(x_1, x_3)\frac{\partial}{\partial x_2}+a_3(x_1,x_2)\frac{\partial}{\partial x_3}, \quad\text{so that div $X=0$.}
$$
Assume that $a_j\in L^\infty$, $\frac{\partial a_j}{\partial x_1}, \frac{\partial a_j}{\partial x_2}\in L^1$, and 
$\frac{\partial a_1}{\partial x_3}, \frac{\partial a_2}{\partial x_3}$ are Radon measures. Then bounded measurable solutions of the Cauchy problem for the PDE, $Xu=F$, $u=g$ given on $∑$ a transversal hypersurface to $X$ are locally uniquely determined by $F, g$.
In fact, it is even possible to say that the "generic" example in 3D is 
$$
X=a_1(x_1,x_2, x_3)\frac{\partial}{\partial x_1}+a_2(x_1,x_2, x_3)\frac{\partial}{\partial x_2}+a_3(x_1,x_2,x_3)\frac{\partial}{\partial x_3},
\text{with $a_j,\ $div $X\in L^\infty$}, 
$$
and
$$
\frac{\partial a_j}{\partial x_1}, \frac{\partial a_j}{\partial x_2}\in L^1, \quad 
\frac{\partial a_1}{\partial x_3} \text{ Radon measure},
\frac{\partial a_2}{\partial x_3}, \frac{\partial a_3}{\partial x_3}\in L^1. 
$$
