Long time bounds for the periodic Benjamin–Ono–BBM equation

We consider the periodic Benjamin–Ono equation, regularized in the same way as the Benjamin–Bona–Mahony equation is found from the Korteweg–de Vries one [T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272 (1220) (1972) 47–78], namely the equation ut+ux+αuux+βH(uxt)=0ut+ux+αuux+βH(uxt)=0, where HH is the Hilbert transform, αα the quotient between the characteristic wave amplitude and the depth of the waves and ββ the quotient between this depth and the wavelength. We show that the solution, starting from an initial datum of size εε, remains smaller than εε for a time scale of order (ε−1β/α)2(ε−1β/α)2, whereas the local well-posedness gives only a time of order ε−1β/αε−1β/α.