We consider the Cauchy problem of third order Benjamin-Ono type equations on the torus. Nonlinear terms may yield derivative losses, which prevents us from using the classical energy method. In order to overcome that difficulty, we add a correction term into the energy. We also use the Bona-Smith type argument to show the continuous dependence.
"Local well-posedness for third order Benjamin-Ono type equations on the torus." Adv. Differential Equations 24 (9/10) 555 - 580, September/October 2019.