# 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.

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 $$\frac{\partial a_j}{\partial x_1}, \frac{\partial a_j}{\partial x_2}\in L^1, \quad \frac{\partial a_1}{\partial x_3} \text{ Radon measure}, \frac{\partial a_2}{\partial x_3}, \frac{\partial a_3}{\partial x_3}\in L^1.$$