# Difference quotients of solutions of ODE and PDE in Sobolev setting

In the post Difference quotient for solutions of ODE and Liouville equation, it was showed that if $$\Phi$$ is the solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$ then $$\tilde \Phi(x,y,t) = \left(\Phi(x,t), \frac{\Phi(x + r y,t) - \Phi(x,t)}{r} \right)$$ is the flow of the ODE with $$\tilde{f}_r(x,y,t) = \left(f(x,t), \frac{f(x+r y,t) - f(x,t)}{r} \right)$$ as a vector field and $$\tilde\mu_t = (\tilde\Phi_t)_{\sharp} \mu$$ solves the PDE $$\begin{cases} \partial_t \tilde\mu_t + \operatorname{div\,} (\tilde f \tilde\mu_t) = 0 \\ \tilde\mu_0 = \mu \end{cases}$$ in the sense of distributions.

Questions.

1. Is the same true if we assume $$f$$ Sobolev or BV and $$\Phi$$ regular Lagrangian flow of the ODE?

2. How can $$\operatorname{div}(\tilde f \tilde \mu_t)$$ be written out explicitly (and rigorously) in terms of $$f$$ instead of $$\tilde f$$?