Consider the Cauchy problem for the Benjamin-Ono equation $$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$ where $\mathcal H$ is the Hilbert transform. How can I compute the following? $$\frac{d}{d t} \left(\frac{1}{2} \int_{\mathbb R} u^2\right)$$ $$\frac{d}{d t }\left( \frac{1}{2} \int_{\mathbb R} |(-\Delta)^{1/2} u|^2 + \frac{1}{6} u^3\right)$$ I have seen in many references that these quantities are estimated, but always the final result, without the intermediate computations (and I'm confused about how to handle the fractional derivative and the Hilbert transform).
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$\begingroup$ Are you sure that you don't want, for the second integral, the expression for the energy of the B-O equation at $\beta=0$? Also, is there a reason you wrote the integrand with an imaginary fractional Laplacian, rather than with $u_{x}^{2}$? $\endgroup$– BuzzCommented Dec 29, 2021 at 4:25
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$\begingroup$ @Buzz I'm not sure what you mean $\endgroup$– RikuCommented Dec 31, 2021 at 11:56
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$\begingroup$ Your first integral is the total momentum $P$ carried by the B-O field. The second integral, with two terms in the integrand, is almost the corresponding energy $E$ carried by the field, but the relative sizes of the two terms in the integrand are wrong for that See, e.g., my paper arXiv:1902.04643 $\endgroup$– BuzzCommented Dec 31, 2021 at 17:13
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$\begingroup$ @Buzz I see. Could you please write up in an answer what the right constants are and, most importantly, how to estimate the momentum and energy evolution? $\endgroup$– RikuCommented Jan 1, 2022 at 17:38
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