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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.

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

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