In https://arxiv.org/abs/math/0307289 (eq. (8)), for a (smooth) solution of the equation $$u_t - uu_x + Hu_{xx} = 0$$ (where $H$ denotes the Hilbert transform) the following estimate is stated (without computations): $$\frac{d}{dt}\int_{\mathbb R} (u_x^2 -\frac{3}{4} u^2 H u_x - \frac{1}{8} u^4 )dx = 0$$ I've tried to reproduce it, but I get extra terms that don't cancel out. How does one work out this estimate?
Bonus question: is it true/clear that $(u_x^2 -\frac{3}{4} u^2 H u_x - \frac{1}{8} u^4 )$ is nonnegative and/or that the estimate above is enough to bound $\|u\|_{L^4}$ in terms of the $L^4$ and $H^1$ norm of the initial data?
