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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 $\tilde f$?

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