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

Youcef Mammeri (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

We consider the periodic Benjamin-Ono equation, regularized in the same manner the Benjamin-Bona-Mahony equation is found from the Korteweg-de Vries one, namely the equation $u_t + u_x + \alpha u u_x + \beta H(u_{xt})=0,$ where $H$ is the Hilbert transform, $\alpha$ the quotient between the characteristic waves amplitude and the depth of the waves and $\beta$ the quotient between this depth and the wavelength. We show that the solution, starting from an initial datum of size $\varepsilon$, remains smaller than $\varepsilon$ for a time scale of order $\left(\varepsilon^{-1}\frac{\beta}{\alpha}\right)^2$, whereas the local well-posedness gives only a time of order $\varepsilon ^{-1}\frac{\beta}{\alpha}$.

Mots clés: BO-BBM equation; KP equations; local and global well-posedness; long time bounds

Domaines: Mathématiques [math]

Youcef Mammeri. Long time bounds for the periodic Benjamin–Ono–BBM equation. Nonlinear Analysis: Theory, Methods and Applications, 2009, 71 (10), pp.5010 - 5021. ⟨10.1016/j.na.2009.03.078⟩. ⟨hal-01090385⟩