On page 3 of [this preprint](https://arxiv.org/pdf/1611.05928.pdf), after recalling the definition of flow generated by a vector field, the authors remark that "a necessary condition for a flow $\varphi\_t(\cdot)$ generated by $a(t, \cdot)$ to be measure preserving is: $\mathrm{div}\, a = 0$ in a suitable sense".


Assuming $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$, is the "suitable sense" the sense of distributions? If yes, how can one prove that $\mathrm{div}\, a = 0$ in the sense of distributions implies $\phi\_t$ measure preserving?



Prove that the flow of a divergence-free vector field is measure preserving
Let $\mu\_t = (\varphi\_t)\_{\sharp} \mu$ denote the image of the measure $\mu$ by the flow of $a$ (where $\mu$ can be Lebesgue measure).
It is well-known that the family of measures $\{\mu\_t\}\_{t\in \mathbb R}$ satisfies Liouville equation (aka continuity equation)
$$
\partial\_t \mu\_t + \operatorname{div\,} (a \mu\_t) = 0 \tag{1}
$$
in the sense of distributions. Indeed, for any smooth compactly supported function $f=f(t,x)$
$$
\iint (\partial\_t f(t,y) + a(t,y) \nabla f(t,y)) \,d\mu\_t(y) \,dt =
\iint \bigl((\partial\_t f)(t,\varphi\_t(x)) + a(t, \varphi\_t(x)) (\nabla f)(t, \varphi\_t(x))\bigr) \, d\mu(x) \,dt =
\iint \partial\_t \bigl( f(t, \varphi\_t(x)) \bigr) \,dt \,d\mu(x) = \int 0 \,d\mu = 0,
$$
where we have used the chain rule and the fact that $\partial\_t\varphi\_t(x) = a(t, \varphi\_t(x))$ for a.e. $t$.


*Claim 1.* If $\varphi\_t$ preserves the measure $\mu$ then $\operatorname{div} (a\mu) = 0$.


Indeed, if the flow of $\varphi\_t$ preserves the measure $\mu$, i.e. $\mu\_t = \mu$ for all $t$, then by (1)
$$
\operatorname{div} (a \mu) = 0. \tag{2}
$$
In particular, if $\mu$ is the Lebesgue measure then $\operatorname{div} a = 0$.


*Claim 2.* Suppose that $\mu\_t$ is the *unique* solution of (2) with the initial condition $\mu\_t|\_{t=0} = \mu$ (e.g. in the class of non-negative measure-valued solutions, absolutely continuous with respect to Lebesgue measure). Then $\operatorname{div} (a\mu) = 0$ implies that the flow $\varphi\_t$ preserves the measure $\mu$.


Indeed, uniqueness implies $\mu\_t = \mu$ for a.e. $t$, that is $\mu$ is preserved by the flow of $a$.


(I learned this trick in some paper by E.O. Stepanov, but don't remember precisely which one.)


