I am studying the use of the commutator for finding the estimate of energy. During my looking through many papers I found that [this paper](https://www.aimsciences.org/article/doi/10.3934/dcds.2019145) contains a possible typo. [Here is the archive version](https://arxiv.org/abs/1809.02027) which has the same prospective typo!


Another question about the commutator. When I espand the follawing commutator $[J^s,u]\partial\_x u$ the last two terms are cancel eachohter! Are my calculations right?


\begin{align}
[J^s,u]\partial\_x u &= J^s(u\, \partial\_x u)- u J^s(\partial\_x u)\\
&=J^s(u) \partial\_x u + u J^s(\partial\_x u) - u J^s(\partial\_x u)\\
&=J^s(u) \partial\_x u 
\end{align}


Is the above right?



A question about the commutator $[J^s,u]\partial_x u$
The computation is not right in general. $J^s$ does not satisfy the Leibniz rule. The idea is that at least one of the derivatives hits the function $w$ in the paper, So the authors used the commutator to get rid of it.


