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Journal Articles Nonlinear Analysis: Theory, Methods and Applications Year : 2009

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

Abstract

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 $\varepsilon ^{-1}\frac{\beta}{\alpha}$.
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Dates and versions

hal-01090385 , version 1 (03-12-2014)

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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⟩
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