1
$\begingroup$

Is there someone who knows the following paper

"Local well-posedness for the Maxwell-Schrodinger system" by M.Nakamura and T.Wada.

I'm trying to study it but I've some doubts. In particular I'm not really understanding where they use the assumption $s\geq \frac{5}{3}$.

Improve this question
$\endgroup$
7
  • $\begingroup$ They use the assumption $s\geq 5/3$ in the estimate (6.10). They apply the fractional Leibniz rule to the product in combination with the Sobolev embedding of $H^{s-1,2}$ to $L^q$ where they use $1-1/\tilde{r} = 1/2 +1/q$ and choose $\tilde{r} = 3/(4-2s)$. The Sobolev embedding then only works for $s\geq 5/3$. $\endgroup$
    – gsa
    Commented Jan 27, 2015 at 16:25
  • $\begingroup$ @gsa Thank you. I stopped before that lemma and I was trying to understand it with previous computations. $\endgroup$
    – Sal
    Commented Jan 27, 2015 at 16:39
  • $\begingroup$ @gsa May I ask you to clarify another doubt I have? Why do you think they use estimates for Klein-Gordon equations and not estimates for the wave equation? $\endgroup$
    – Sal
    Commented Jan 28, 2015 at 8:21
  • $\begingroup$ Maybe it's possible to work with the wave equation as well. But the standard Strichartz estimates for the wave equation are stated in the homogeneous Sobolev spaces $\dot{H}^{s,p}$, which are a bit delicate. My guess is that they prefer to work with the inhomogeneous spaces $H^{s,p}$ and thus use the Strichartz estimates for Klein-Gordon. $\endgroup$
    – gsa
    Commented Jan 28, 2015 at 17:31
  • $\begingroup$ @gsa But the wave equation has also a conservation of energy which can be used. I have to think about it, because it's not very clear to me. $\endgroup$
    – Sal
    Commented Jan 28, 2015 at 18:47

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .