Consider the ODE $$ \begin{cases} \partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\ \Phi(0,x) = x, & x \in \mathbb R \end{cases} $$ where $f$ is function which is a non-increasing in the second variable (without other assumptions on regularity).

Then $\Phi$ exists and is Lipschitz with respect to space (Flow of ODE with monotone source).

How can one compute the a.e. space derivative of this Lipschitz flow $\Phi(t, \cdot)$?

**Remark.** Note that, if $f$ was Lipschitz, we would get that the space derivative of the flow $\partial_x \Phi$ satisfies

$$\partial_t \partial_x \Phi = \partial_x f(t,\Phi(t,x))\partial_x \Phi.$$

To reiterate, the question of this post is the following:

In general, how can we compute $\partial_x \Phi(t,\cdot)$ if we only assume that $f$ is function which is a non-increasing in the second variable (without other assumptions on regularity of $f$)?