2
$\begingroup$

Suppose 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 does one prove that $$\tilde \Phi(x,y,t) = \left(\Phi(x,t), \frac{\Phi(x + r y,t) - \Phi(x,t)}{r} \right)$$ 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?

Also, in an answer to Prove that the flow of a divergence-free vector field is measure preserving, it was 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 measures $\{\mu_t\}_{t\in \mathbb R}$ satisfies Liouville equation $$ \begin{cases} \partial_t \mu_t + \operatorname{div\,} (f \mu_t) = 0 \\ \mu_0 = \mu \end{cases} $$ in the sense of distributions.

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

$\endgroup$
2
$\begingroup$

Question 1: Denoting $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ the components of $\tilde \Phi(x,y,t)$ , we have $U+rV=\Phi(x + r y,t) $, and $$\displaystyle\frac{f(U+r V,t) - f(U,t)}{r} = \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} .$$ So

$$\partial_t\tilde \Phi(x,y,t)= $$$$= \left(\partial_t \Phi(x,t) , \frac{\partial_t\Phi(x + r y,t) - \partial_t\Phi(x,t) }{r} \right)$$ $$= \left(f(\Phi(x,t),t), \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} \right)$$ $$= \left(f(U,t), \frac{f(U+r V,t) - f(U,t)}{r} \right)$$$$=\tilde{f}_r(U,V,t) $$ $$=\tilde{f}_r(\tilde \Phi(x,y,t),t).$$

Question 2: as to $\tilde\mu_t$, the same result of the quoted link apply in particular to the flow $\tilde\Phi$, therefore you still have the Liouville equation $\partial_t {\tilde\mu}_t + \operatorname{div\,} ({\tilde f} {\tilde\mu_t) }= 0$

$\endgroup$
  • 1
    $\begingroup$ Thank you. What do you mean by "conjugation" in this context? $\endgroup$ – Riku Apr 15 at 15:27
  • 1
    $\begingroup$ You're welcome. Is the answer clear to you? As to question 2, the evolution of $\tilde\mu_t$ is a particular case of the result for $\mu_t$. (As to the conjugation: it was not important, and didn't make it any clearer, I removed). $\endgroup$ – Pietro Majer Apr 15 at 16:12
  • $\begingroup$ Thank you. The answer to question 1 is clear. About question 2, I'm not quite sure about what the divergence in $\partial_t {\tilde\mu}_t + \operatorname{div\,} ({\tilde f} {\tilde\mu_t) }= 0$ really is. Could you write it out in terms of $f$ instead of $\tilde f$, please? $\endgroup$ – Riku Apr 15 at 17:46
  • $\begingroup$ Also, can this be made rigorous even when $f$ is a Sobolev or BV function? $\endgroup$ – Riku Apr 15 at 20:33
  • $\begingroup$ @PietroMajer I deleted my answer to the Borsuk problem. I wrote it without thinking much (rushing to a lecture). As a result your comment has been deleted so you should post it as a comment to the question or as an answer. $\endgroup$ – Piotr Hajlasz Apr 16 at 19:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.