Assume that $ 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} $$

How can we prove that

$$ tilde Phi (x, y, t) = left ( Phi (x, t), frac { Phi (x + ry, t) – Phi (x, t)} {r} right) $$

is the flow of ODE with

$$ tilde {f} _r (x, y, t) = left (f (x, t), frac {f (x + ry, t) – f (x, t)} {r} right ) $$

as a vector field?

In addition, in a response to Demonstrate that the flow of a divergent vector field preserves the measurements, it has been proved that if $ mu_t = ( Phi ( cdot, t)) _ { sharp} mu $ denote the image of the measure $ mu $ by the flow of $ f $, then the family of measurements $ { mu_t } _ {t in mathbb R} $ satisfies Liouville's equation

$$

begin {cases}

partial_t mu_t + operatorname {div ,} (f mu_t) = 0 \

mu_0 = mu

end {cases}

$$

in the sense of distributions.

What does the PDE do? $ tilde mu_t = ( tilde Phi_t) _ { sharp} mu $ solve?