I'm reading the proof of Theorem 5.1 in this paper. I have a few questions about flux.
They say that for a vector field $\xi \in C^\infty_0(\mathbb{R}^n)$, the corresponding flux is $\{\Phi_\tau\}_{\tau \in \mathbb{R}}$ satisfying $$ \partial_\tau \Phi_\tau = \xi \circ \Phi_\tau \;\;\;\;\; \text{ for all }\tau \in \mathbb{R}.$$
Then, they say that $\Phi_\tau$ defines a pushforward mapping on measures, i.e. for a given measure $\rho^k(y)$, its pushforward under $\Phi_\tau$ is the measure $\rho_\tau (y)$ with $$ \int_{\mathbb{R}^n} \rho_\tau(y) \zeta(y)dy = \int_{\mathbb{R}^n} \rho^k (y) \zeta(\Phi_\tau(y))dy \forall \zeta \in C^0_0(\mathbb{R}^n).$$
I don't understand the first equation (definition of the flux). I thought that flux was a surface integral. Here, since the vector field has bounded support, I would guess that the corresponding flux would be the flux out of the boundary. I don't know what $\tau$ represents here, either. Furthermore, how can $\Phi_\tau$ be a mapping on measures?
To expand user480840's comment-answer (please correct/add more if there is anything wrong):
First note that $\Phi_\tau: \mathbb{R}^n\rightarrow \mathbb{R}^n$.
The equation $\partial_\tau \Phi_\tau = \xi \circ \Phi_\tau$ tells us that $\Phi_\tau$ is an evolving vector field where $\Phi_\tau$ is the vector field at time $\tau$. If the initial condition is $\Phi_\tau = $ Id (as it was in the original paper), then $\Phi_\tau(x)$ gives us the result of "flowing" a vector $x \in \mathbb{R}^n$ along the vector field $\xi$ for the amount of time $\tau$.
This made more sense to me in the simple notation in the case $n=1$: $$ x'(t) = \xi(x(t)) \;\;\;\; x(0) = x_0$$
This can be considered a pushforward measure because the evolution $x(t)$ is a transformation on the state space: for any initial measure $\mu$, we can define the pushforward measure $\nu$ under the transformation $x(t)$. This satisfies, for example, $\nu(x(t)) = \mu(x_0).$